================================================================================ CLAUDE CODE CONVERSATION EXPORT Session file: /home/jo314/.claude/projects/-mnt-c-Users-jo314-Documents-mathlib4/25314d39-6ff0-4185-9074-860295ea4c3c.jsonl Export time: 2025-12-15T13:13:14.304425 Total entries: 2239 ================================================================================ -------------------------------------------------------------------------------- [SUMMARY - Compacted content] Title: Algebraic Geometry in mathlib4: Coverage & Implementation -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- USER (entry 15): -------------------------------------------------------------------------------- Hey there! I am a total beginner to lean, but I wanted to have a go and ask for your help with formalizing and proving the following mathematical statement: Let $d \geq n > 0$ and $E=\{\textbf{e} = (e_1, \ldots, e_n) \in \mathbb{Z}_{\geq 0}^n : \sum_{i=1}^n e_i = d\}$. Assume that $D : E \to \mathbb{Q}_{>0}$ is a function invariant under the $S_n$-permutation action on $E$ and satisfying $$ D(\textbf{e})^2 \geq D(\textbf{e}-\delta_i + \delta_j) \cdot D(\textbf{e}+\delta_i - \delta_j) $$ where $\delta_k$ are the Kronecker deltas. Then $D$ takes its maximal value on balanced vectors and its minimal value on concentrated vectors. here balanced vectors are those satisfying |e_i - e_j| <= 1 for all i,j, and concentrated vectors are those of the form \textbf{e} = d \cdot \delta_i for some i. I would hope that the statement is simple enough that it should be feasible to formalize it. The proof is also not hard, by human standards, but again I have very little experience with Lean, so I would require your help on the Lean side. (I can say more about the proof when we get there) We are in a clone of the mathlib, but otherwise I have not installed anything Lean related yet on this computer. I did add a skill to your skill repository developed by an experienced Lean developer, which hopefully will come in useful. I think to start, maybe we could just try to find our feet, install relevant dependencies to get started (I think if available, lean in the command line should be fine - I don't need fancy VScode etc because ideally I would not interact with the Lean code so much). -------------------------------------------------------------------------------- ASSISTANT (entry 17): -------------------------------------------------------------------------------- Welcome to Lean formalization! This is a lovely combinatorial optimization problem. Let me help you get started. First, let me check your Lean environment and explore the mathlib4 setup. -------------------------------------------------------------------------------- ASSISTANT (entry 18): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 19): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 20): -------------------------------------------------------------------------------- [Tool Result: Lean not found in PATH] -------------------------------------------------------------------------------- USER (entry 21): -------------------------------------------------------------------------------- [Tool Result: total 440 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:21 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:16 .. drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:55 .claude drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:18 .devcontainer drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:18 .docker drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:46 .git drwxrwxrwx 1 jo314 jo314 4096 Dec 14 09:18 .github -rwxrwxrwx 1 jo314 jo314 162 Dec 14 09:18 .gitignore -rwxrwxrwx 1 jo314 jo314 155 Dec 14 09:18 .gitpod.yml -rwxrwxrwx 1 j...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 23): -------------------------------------------------------------------------------- Lean isn't installed yet. Let me check for elan (the Lean version manager) and see what we need to install. -------------------------------------------------------------------------------- ASSISTANT (entry 24): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 25): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 26): -------------------------------------------------------------------------------- [Tool Result: elan not found leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- USER (entry 27): -------------------------------------------------------------------------------- [Tool Result: File does not exist. Did you mean lakefile.lean?] -------------------------------------------------------------------------------- ASSISTANT (entry 29): -------------------------------------------------------------------------------- Good - I can see mathlib4 uses Lean 4.26.0. We need to install `elan` (the Lean version manager). Let me install it for you. -------------------------------------------------------------------------------- ASSISTANT (entry 30): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 31): -------------------------------------------------------------------------------- [Tool Result: curl: (22) The requested URL returned error: 404] -------------------------------------------------------------------------------- ASSISTANT (entry 33): -------------------------------------------------------------------------------- Let me try the alternative installation method. -------------------------------------------------------------------------------- ASSISTANT (entry 34): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 35): -------------------------------------------------------------------------------- [Tool Result: info: downloading installer info: default toolchain set to 'stable'] -------------------------------------------------------------------------------- ASSISTANT (entry 37): -------------------------------------------------------------------------------- Elan is installed. Let me set up the environment and verify it works. -------------------------------------------------------------------------------- ASSISTANT (entry 38): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 39): -------------------------------------------------------------------------------- [Tool Result: elan 4.1.2 (58e8d545e 2025-05-26) Lean (version 4.26.0, x86_64-unknown-linux-gnu, commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562, Release) info: downloading https://releases.lean-lang.org/lean4/v4.26.0/lean-4.26.0-linux.tar.zst info: installing /home/jo314/.elan/toolchains/leanprover--lean4---v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 41): -------------------------------------------------------------------------------- Excellent! Lean 4.26.0 is now installed. Now let's download the pre-built mathlib cache (this is much faster than building from source). -------------------------------------------------------------------------------- ASSISTANT (entry 42): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 43): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b1fb6f5. Output is being written to: /tmp/claude/tasks/b1fb6f5.output] -------------------------------------------------------------------------------- ASSISTANT (entry 45): -------------------------------------------------------------------------------- The cache download is running. Let me check its progress. -------------------------------------------------------------------------------- ASSISTANT (entry 46): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 49): -------------------------------------------------------------------------------- [Tool Result: success b1fb6f5 local_bash completed 0 info: plausible: cloning https://github.com/leanprover-community/plausible info: plausible: checking out revision '160af9e8e7d4ae448f3c92edcc5b6a8522453f11' info: LeanSearchClient: cloning https://github.com/leanprover-community/LeanSearchClient info: LeanSearchClient: checking out revision '3591c3f664ac3719c4c86e4483e21e22870...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 51): -------------------------------------------------------------------------------- Excellent! The cache downloaded successfully. Let me verify Lean works properly. -------------------------------------------------------------------------------- ASSISTANT (entry 52): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 53): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: be64f5d. Output is being written to: /tmp/claude/tasks/be64f5d.output] -------------------------------------------------------------------------------- ASSISTANT (entry 55): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 58): -------------------------------------------------------------------------------- [Tool Result: success be64f5d local_bash completed 0 Nat.add_comm (n m : ℕ) : n + m = m + n ] -------------------------------------------------------------------------------- ASSISTANT (entry 60): -------------------------------------------------------------------------------- Lean and mathlib are working. Now let's discuss formalizing your theorem. -------------------------------------------------------------------------------- ASSISTANT (entry 61): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 62): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 64): -------------------------------------------------------------------------------- Your environment is now ready. Let me outline how we might formalize this statement and get your input. ## Formalization Plan **1. The set E** - Compositions of d into n parts: ```lean def Compositions (n d : ℕ) := {e : Fin n → ℕ // ∑ i, e i = d} ``` This represents vectors (e₁, ..., eₙ) ∈ ℤ≥0ⁿ with sum d. **2. The Sₙ-action** - Permutations act by precomposition: ```lean σ • e = e ∘ σ⁻¹ ``` Invariance means: `∀ σ : Equiv.Perm (Fin n), D (σ • e) = D e` **3. Log-concavity condition** - For i ≠ j with eᵢ ≥ 1: ```lean D(e)² ≥ D(e - δᵢ + δⱼ) · D(e + δᵢ - δⱼ) ``` Note: we need `eᵢ ≥ 1` for `e - δᵢ + δⱼ` to stay in E. **4. Balanced vectors**: `∀ i j, |eᵢ - eⱼ| ≤ 1` **5. Concentrated vectors**: `∃ k, e = d • δₖ` (i.e., `e k = d` and `e j = 0` for j ≠ k) ## Questions for you: 1. **About the log-concavity condition**: The condition `D(e)² ≥ D(e-δᵢ+δⱼ) · D(e+δᵢ-δⱼ)` — does this hold for all i,j (including i=j where it's trivial), or just i≠j? And it requires eᵢ ≥ 1 for the left-hand vector to be valid, correct? 2. **About the proof**: You mentioned the proof isn't hard. Can you give a brief sketch? Is it an induction argument? Does it use majorization order? Understanding the proof structure will help me choose the right approach for formalization. -------------------------------------------------------------------------------- USER (entry 66): -------------------------------------------------------------------------------- Thanks, this sounds great! About compositions: it's possible that this is already implemented - if so we can reuse it. Otherwise I guess it's also not too hard to do ourselves Log concavity: indeed this statement needs both e_i >=1 and e_j >= 1, but it does hold for all pairs 1<= i,j <= n Proof sketch: I am pasting below a proof where this is applied concretely (for D = descendant invariants on Mbar): \noindent\textbf{Step 1} (Two-point slice and its basic properties). Fix distinct indices $i \neq j$ and set \[ M := \prod_{k \neq i,j} \psi_k^{e_k}, \qquad q := e_i + e_j. \] For $t = 0, 1, \ldots, q$, consider the one-variable sequence \[ S_t := \int_{\Mbar_{g,n}} \psi_i^t \, \psi_j^{q-t} \, M. \] By the natural action of the symmetric group permuting the markings on $\Mbar_{g,n}$, the classes $\psi_i$ and $\psi_j$ are exchanged by an automorphism that leaves $M$ invariant; hence \[ S_t = S_{q-t} \qquad \text{(palindromicity)}. \] \medskip \noindent\textbf{Step 2} (Khovanskii--Teissier log-concavity). It is standard that each $\psi_i$ is nef on $\Mbar_{g,n}$. For nef classes $A, B$ and fixed nef classes $H_3, \ldots, H_d$ on a projective (orbifold) variety of dimension $d$, the Khovanskii--Teissier inequalities \humaninline{as described in Variant 1.6.2 of \cite{MR2095471}} (equivalently, the mixed Hodge--Riemann bilinear relations) give the discrete log-concavity \[ \left( \int A^t B^{q-t} H_3 \cdots H_d \right)^2 \geq \left( \int A^{t-1} B^{q-t+1} H_3 \cdots H_d \right) \left( \int A^{t+1} B^{q-t-1} H_3 \cdots H_d \right) \] for all $1 \leq t \leq q - 1$. Applying this with $A = \psi_i$, $B = \psi_j$, $H_3 \cdots H_d = M$, we obtain \[ S_t^2 \geq S_{t-1} S_{t+1} \qquad (1 \leq t \leq q - 1) \qquad \text{(log-concavity)}. \] \medskip \noindent\textbf{Step 3} (Monotonicity towards the middle). For a positive log-concave sequence, the ratios $R_t := S_{t+1}/S_t$ are weakly decreasing in $t$. Using palindromicity, \[ R_{q-t-1} = \frac{S_{q-t}}{S_{q-t-1}} = \frac{S_t}{S_{t+1}} = \frac{1}{R_t}. \] Hence for $t \leq \lfloor q/2 \rfloor - 1$ we have $R_t \geq R_{q-t-1} = 1/R_t$, so $R_t \geq 1$, i.e., \[ S_{t+1} \geq S_t \quad \text{for } t < \frac{q}{2}, \qquad S_{t-1} \geq S_t \quad \text{for } t > \frac{q}{2}. \] In words: along the two-point slice $\{(t, q-t) : 0 \leq t \leq q\}$, the values increase up to the middle and then decrease. \medskip \noindent\textbf{Step 4} (The balancing step). Suppose $e_i \geq e_j + 2$. Then $e_i > q/2$. By the monotonicity just proved, \[ D(e_1, \ldots, e_i, \ldots, e_j, \ldots, e_n) = S_{e_i} \leq S_{e_i - 1} = D(e_1, \ldots, e_i - 1, \ldots, e_j + 1, \ldots, e_n). \] Thus, whenever two entries differ by at least 2, transferring one unit from the larger to the smaller weakly increases the value of the integral. \medskip \noindent\textbf{Step 5} (Conclusion by iteration). Starting from any $\ee \in E(g,n)$, repeatedly apply the balancing step to any pair $(i,j)$ with $|e_i - e_j| \geq 2$. This process terminates at a balanced vector $\ee^*$ (all entries differ by at most 1), and along the way the value of $D$ never decreases. Hence \[ D(\ee) \leq D(\ee^*) \quad \text{for some balanced } \ee^* \in E(g,n). \] In particular, the maximum of $D$ on $E(g,n)$ is achieved at a balanced vector. \end{proof} The idea is to isolate the inequality/combinatorics parts of this; roughly, one first wants to look at a slice S and have the argument from above that it increases to the middle and then decreases again; that shows that any balancing step increases the value of D; then one needs to show that starting from any possibly unbalanced vectors, one can reach a balanced vector in finitely many steps (with a suitable invariant going down in each step). The minimum part just works the reverse way, concentrating all the entries of a vector into one slot (which started having one of the maximal entries of the vector, so that each step is really an imbalancing step). -------------------------------------------------------------------------------- ASSISTANT (entry 68): -------------------------------------------------------------------------------- Excellent! This is a beautiful proof with a clear structure. Let me break down what we need to formalize: -------------------------------------------------------------------------------- ASSISTANT (entry 69): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 70): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 72): -------------------------------------------------------------------------------- The proof structure is very clean. Let me identify the key lemmas we need: **Part A: Slice Lemma (Pure sequence theory)** - A positive log-concave palindromic sequence is unimodal (increases to middle, decreases after) **Part B: Balancing Argument** - Define the transfer operation: move one unit from position i to position j - Show: if e_i ≥ e_j + 2, then D(transfer e i j) ≥ D(e) - Show: balancing terminates (using sum of squares as a decreasing measure) Let me first search mathlib for relevant existing definitions. -------------------------------------------------------------------------------- ASSISTANT (entry 73): -------------------------------------------------------------------------------- [Tool: Task] -------------------------------------------------------------------------------- USER (entry 74): -------------------------------------------------------------------------------- [Tool Result: [{'type': 'text', 'text': "Perfect! Now I have a comprehensive picture. Let me compile the results.\n\n## Summary of Findings\n\nI've searched the mathlib4 repository and found comprehensive definitions related to your three topics. Here's what exists:\n\n### 1. COMPOSITIONS AND PARTITIONS OF INTEGERS\n\n**Compositions (ordered partitions):**\n- **File**: `/mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Combinatorics/Enumerative/Composition.lean`\n- **Main structure**: `Composition n` - a composit...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 76): -------------------------------------------------------------------------------- Interesting. Mathlib has `Composition n` but it uses lists of **positive** integers, while we need **weak compositions** (allowing zeros). We'll define our own simpler structure. Also, there's no log-concave or unimodal sequence definitions in mathlib, so we'll build those from scratch. Let me create the formalization file. -------------------------------------------------------------------------------- ASSISTANT (entry 77): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 78): -------------------------------------------------------------------------------- [Tool Result: The user doesn't want to proceed with this tool use. The tool use was rejected (eg. if it was a file edit, the new_string was NOT written to the file). To tell you how to proceed, the user said: Note: we will probably not add this proof to the mathlib itself, so it might make sense to work in a single self-contained file (importing mathlib where needed).] -------------------------------------------------------------------------------- ASSISTANT (entry 80): -------------------------------------------------------------------------------- Got it - let's create a standalone file in the project root. -------------------------------------------------------------------------------- ASSISTANT (entry 81): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 83): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 85): -------------------------------------------------------------------------------- I've created an initial framework. Let me check if it compiles. -------------------------------------------------------------------------------- ASSISTANT (entry 86): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 87): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:6:0: error: object file '/mnt/c/Users/jo314/Documents/mathlib4/.lake/build/lib/lean/Mathlib/Data/Rat/Order.olean' of module Mathlib.Data.Rat.Order does not exist] -------------------------------------------------------------------------------- ASSISTANT (entry 89): -------------------------------------------------------------------------------- Let me fix the imports - there seems to be a cache issue with that specific module. -------------------------------------------------------------------------------- ASSISTANT (entry 90): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 91): -------------------------------------------------------------------------------- [Tool Result: ✖ [2/2] Running Mathlib.Data.Rat.Order error: no such file or directory (error code: 2) file: /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Order.lean Some required targets logged failures: - Mathlib.Data.Rat.Order error: build failed] -------------------------------------------------------------------------------- ASSISTANT (entry 93): -------------------------------------------------------------------------------- [Tool: Glob] -------------------------------------------------------------------------------- USER (entry 94): -------------------------------------------------------------------------------- [Tool Result: /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/BigOperators.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cardinal.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cast/CharZero.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cast/Defs.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cast/Lemmas.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cast/OfScientific.lean /mnt/c/Users/jo314/Documents/mathlib4/Mathlib/Data/Rat/Cast/Orde...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 96): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 97): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→Authors: [Your name here] 5→-/ 6→import Mathlib.Data.Fin.Basic 7→import Mathlib.Data.Fin.Tuple.Basic 8→import Mathlib.Algebra.BigOperators.Fin 9→import Mathlib.Algebra.Order.BigOperators.G...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 99): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 100): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:67:18: warning: unused variable `hn` Note: This linter can be disabled with `set_option linter.unusedVariables false` BalancedVectors.lean:80:8: error: No applicable extensionality theorem found for type ℕ Note: Extensionality theorems can be registered by marking them with the `[ext]` attribute BalancedVectors.lean:88:12: error: `simp` made no progress BalancedVectors.lean:114:14: error: `simp` made no progress BalancedVectors.lean:131:10: error: omega could not prove t...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 102): -------------------------------------------------------------------------------- There are several issues to fix. Let me simplify the code significantly - the `transfer` proof got too complicated. I'll use a cleaner approach. -------------------------------------------------------------------------------- ASSISTANT (entry 103): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 104): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→Authors: [Your name here] 5→-/ 6→import Mathlib.Data.Fin.Basic 7→import Mathlib.Data.Fin.Tuple.Basic 8→import Mathlib.Algebra.BigOperators.Fin 9→import Mathlib.Algebra.Order.BigO...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 106): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 107): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:73:17: error(lean.unknownIdentifier): Unknown identifier `update_same` BalancedVectors.lean:79:44: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ x, update ?f i ?b x in the target expression ↑e j + 1 + ∑ x ∈ univ \ {j}, update (↑e) i (↑e i - 1) x = d n d : ℕ e : WeakComposition n d i j : Fin n hi : 1 ≤ ↑e i hij : ¬i = j ⊢ ↑e j + 1 + ∑ x ∈ univ \ {j}, update (↑e) i (↑e i - 1) x = d BalancedVectors.lean:105:31: warning: This simp argument is u...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 109): -------------------------------------------------------------------------------- There are still several issues. Let me simplify the approach significantly and fix the errors. -------------------------------------------------------------------------------- ASSISTANT (entry 110): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 111): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→Authors: [Your name here] 5→-/ 6→import Mathlib.Data.Fin.Basic 7→import Mathlib.Data.Fin.Tuple.Basic 8→import Mathlib.Algebra.BigOperators.Fin 9→import Mathlib.Algebra.Order.BigO...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 113): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 116): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:82:4: error: `simp` made no progress BalancedVectors.lean:169:8: warning: declaration uses 'sorry' BalancedVectors.lean:194:12: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ x ∈ univ.erase i, ?m.142 x in the target expression t = t + (e.toFun i + e.toFun j - t) case e_a n d : ℕ D : WeakComposition n d → ℚ e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℕ h : t ≤ e.toFun i + e.toFun j hsum : ∑ i, e.toFun i = d ⊢ t = t + (e.to...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 118): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 119): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 121): -------------------------------------------------------------------------------- Good idea! Let me give you an overview of where we are. ## Current Status **What's working:** - Lean 4 + mathlib4 are installed and working - Basic structure is set up in `BalancedVectors.lean` - Definitions for `WeakComposition`, `IsBalanced`, `IsConcentrated`, `LogConcaveOn`, `IsPalindromicOn` are in place - The theorem statements are correctly formulated **What I'm struggling with:** 1. **Sum manipulation in Lean**: The proofs for `modify` and `sliceSeq` involve rewriting sums like `∑ k, f k = f i + f j + ∑ k ∉ {i,j}, f k`. The `add_sum_erase` lemma exists but the rewrite tactics aren't matching the pattern properly. 2. **The key algebraic inequality**: I need to prove that when `a ≥ b + 2`: ``` (a-1)² + (b+1)² < a² + b² ``` This is true because `(a-1)² + (b+1)² = a² + b² - 2a + 2b + 2 = a² + b² + 2(b-a+1)` and `b-a+1 ≤ -1` when `a ≥ b+2`. But `nlinarith` is struggling with the natural number subtraction (since `a-1` in ℕ truncates). 3. **Permutation sums**: For `IsSymmetric`, showing `∑ i, (e ∘ σ⁻¹)(i) = d` - the permutation doesn't change the sum, but the proof term isn't matching. ## Questions for you: 1. **Simplification approach**: Would it be acceptable to work with `Fin n → ℤ` instead of `Fin n → ℕ` to avoid the truncation issues? Or alternatively, state things more carefully with explicit bounds? 2. **Proof strategy for unimodality**: Your proof sketch uses ratios `R_t = s(t+1)/s(t)`. In Lean with ℚ, this requires showing `s(t) ≠ 0`. Since we have positivity, this is fine, but would you prefer to work with the product form `s(t) * s(t) ≥ s(t-1) * s(t+1)` throughout to avoid division? 3. **Alternative**: Should I try a simpler approach first - just proving the key lemmas with `sorry` for now to get the structure right, then filling in the technical details? -------------------------------------------------------------------------------- USER (entry 123): -------------------------------------------------------------------------------- I would not mind working with functions to ZZ, and then adding the provision that their values are nonnegative, if it helps with the proofs. I do not mind working with a reformulation of the Ratio towards staying with products throughout; I do think that the nonzero property will be needed at some crucial steps, so I am not sure one can avoid quotients throughout I do think it's a good idea to work iteratively, first fixing the easier parts and then gradually filling in the proofs (so one can focus on one issue at a time). Do feel free to ultrathink whenever it's necessary to work through some crunchy issue! -------------------------------------------------------------------------------- ASSISTANT (entry 125): -------------------------------------------------------------------------------- Excellent guidance! Let me rewrite this more carefully, switching to ℤ and getting the structure to compile first. -------------------------------------------------------------------------------- ASSISTANT (entry 126): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 127): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→Authors: [Your name here] 5→-/ 6→import Mathlib.Data.Fin.Basic 7→import Mathlib.Data.Fin.Tuple.Basic 8→import Mathlib.Algebra.BigOperators.Fin 9→import Mathlib.Algebra.Order.BigO...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 129): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 130): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:66:24: error: `simp` made no progress BalancedVectors.lean:86:4: error: `simp` made no progress BalancedVectors.lean:108:4: error: `simp` made no progress BalancedVectors.lean:141:4: error: omega could not prove the goal: a possible counterexample may satisfy the constraints c ≥ 2 -1 ≤ a - b ≤ 1 where a := e j b := e i c := |e i - e j| BalancedVectors.lean:144:4: error: omega could not prove the goal: a possible counterexample may satisfy the constraints ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 132): -------------------------------------------------------------------------------- Let me fix these issues more carefully. -------------------------------------------------------------------------------- ASSISTANT (entry 133): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 134): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→Authors: [Your name here] 5→-/ 6→import Mathlib.Data.Fin.Basic 7→import Mathlib.Data.Fin.Tuple.Basic 8→import Mathlib.Algebra.BigOperators.Fin 9→import Mathlib.Algebra.Order.BigO...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 136): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 137): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:53:15: error: `simp` made no progress BalancedVectors.lean:74:76: error: unsolved goals n : ℕ d : ℤ e : WeakComposition n d i j : Fin n hi : 1 ≤ e.toFun i hij : i ≠ j hsum : ∑ i, e.toFun i = d ⊢ (e.toFun i - 1 + ∑ x ∈ univ.erase i, if x = i then e.toFun i - 1 else if x = j then e.toFun j + 1 else e.toFun x) = e.toFun i - 1 + ∑ k ∈ univ.erase i, if k = j then e.toFun j + 1 else e.toFun k BalancedVectors.lean:81:6: error: Tactic `apply` failed: could not unify ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 139): -------------------------------------------------------------------------------- Let me simplify further - I'll sorry out the trickier proofs and focus on getting the overall structure to compile. -------------------------------------------------------------------------------- USER (entry 140): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 142): -------------------------------------------------------------------------------- Just to give two more inputs: are you using the Lean theorem-prover skill already? That might be very helpful. If you do not have access yet, please tell me, then I can check the installation. Also: if you give me a copy-pastable description what the current bottlenecks are, I can also send the file and description to some other AIs (like GPT 5.2) which might have insights that help you fix the problems. In that case, it would be good to give a self-contained description of the mathematical proof that I can paste together with the current lean file. -------------------------------------------------------------------------------- ASSISTANT (entry 144): -------------------------------------------------------------------------------- Great points! Let me check if I have access to the Lean skill. -------------------------------------------------------------------------------- ASSISTANT (entry 145): -------------------------------------------------------------------------------- [Tool: Skill] -------------------------------------------------------------------------------- USER (entry 146): -------------------------------------------------------------------------------- [Tool Result: Launching skill: lean4-theorem-proving] -------------------------------------------------------------------------------- USER (entry 147): -------------------------------------------------------------------------------- Base directory for this skill: /home/jo314/.claude/plugins/cache/lean4-skills/lean4-theorem-proving/3.3.1/skills/lean4-theorem-proving # Lean 4 Theorem Proving ## Core Principle **Build incrementally, structure before solving, trust the type checker.** Lean's type checker is your test suite. **Success = `lake build` passes + zero sorries + zero custom axioms.** Theorems with sorries/axioms are scaffolding, not results. ## Quick Reference | **Resource** | **What You Get** | **Where to Find** | |--------------|------------------|-------------------| | **Interactive Commands** | 10 slash commands for search, analysis, optimization, repair | Type `/lean` in Claude Code ([full guide](../../COMMANDS.md)) | | **Automation Scripts** | 19 tools for search, verification, refactoring, repair | Plugin `scripts/` directory ([scripts/README.md](../../scripts/README.md)) | | **Subagents** | 4 specialized agents for batch tasks (optional) | [subagent-workflows.md](references/subagent-workflows.md) | | **LSP Server** | 30x faster feedback with instant proof state (optional) | [lean-lsp-server.md](references/lean-lsp-server.md) | | **Reference Files** | 18 detailed guides (phrasebook, tactics, patterns, errors, repair, performance) | [List below](#reference-files) | ## When to Use Use for ANY Lean 4 development: pure/applied math, program verification, mathlib contributions. **Critical for:** Type class synthesis errors, sorry/axiom management, mathlib search, measure theory/probability work. ## Tools & Workflows **7 slash commands** for search, analysis, and optimization - type `/lean` in Claude Code. See [COMMANDS.md](../../COMMANDS.md) for full guide with examples and workflows. **16 automation scripts** for search, verification, and refactoring. See [scripts/README.md](../../scripts/README.md) for complete documentation. **Lean LSP Server** (optional) provides 30x faster feedback with instant proof state and parallel tactic testing. See [lean-lsp-server.md](references/lean-lsp-server.md) for setup and workflows. **Subagent delegation** (optional, Claude Code users) enables batch automation. See [subagent-workflows.md](references/subagent-workflows.md) for patterns. ## Build-First Principle **ALWAYS compile before committing.** Run `lake build` to verify. "Compiles" ≠ "Complete" - files can compile with sorries/axioms but aren't done until those are eliminated. ## The 4-Phase Workflow 1. **Structure Before Solving** - Outline proof strategy with `have` statements and documented sorries before writing tactics 2. **Helper Lemmas First** - Build infrastructure bottom-up, extract reusable components as separate lemmas 3. **Incremental Filling** - Fill ONE sorry at a time, compile after each, commit working code 4. **Type Class Management** - Add explicit instances with `haveI`/`letI` when synthesis fails, respect binder order for sub-structures ## Finding and Using Mathlib Lemmas **Philosophy:** Search before prove. Mathlib has 100,000+ theorems. Use `/search-mathlib` slash command, LSP server search tools, or automation scripts. See [mathlib-guide.md](references/mathlib-guide.md) for detailed search techniques, naming conventions, and import organization. ## Essential Tactics **Key tactics:** `simp only`, `rw`, `apply`, `exact`, `refine`, `by_cases`, `rcases`, `ext`/`funext`. See [tactics-reference.md](references/tactics-reference.md) for comprehensive guide with examples and decision trees. ## Domain-Specific Patterns **Analysis & Topology:** Integrability, continuity, compactness patterns. Tactics: `continuity`, `fun_prop`. **Algebra:** Instance building, quotient constructions. Tactics: `ring`, `field_simp`, `group`. **Measure Theory & Probability** (emphasis in this skill): Conditional expectation, sub-σ-algebras, a.e. properties. Tactics: `measurability`, `positivity`. See [measure-theory.md](references/measure-theory.md) for detailed patterns. **Complete domain guide:** [domain-patterns.md](references/domain-patterns.md) ## Managing Incomplete Proofs **Standard mathlib axioms (acceptable):** `Classical.choice`, `propext`, `quot.sound`. Check with `#print axioms theorem_name` or `/check-axioms`. **CRITICAL: Sorries/axioms are NOT complete work.** A theorem that compiles with sorries is scaffolding, not a result. Document every sorry with concrete strategy and dependencies. Search mathlib exhaustively before adding custom axioms. **When sorries are acceptable:** (1) Active work in progress with documented plan, (2) User explicitly approves temporary axioms with elimination strategy. **Not acceptable:** "Should be in mathlib", "infrastructure lemma", "will prove later" without concrete plan. ## Compiler-Guided Proof Repair **When proofs fail to compile,** use iterative compiler-guided repair instead of blind resampling. **Quick repair:** `/lean4-theorem-proving:repair-file FILE.lean` **How it works:** 1. Compile → extract structured error (type, location, goal, context) 2. Try automate ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 149): -------------------------------------------------------------------------------- The skill is now active. Let me provide you with a self-contained summary of the bottlenecks that you can share with other AIs. --- ## Current Bottlenecks Summary ### The Mathematical Problem We're formalizing the following theorem in Lean 4 / Mathlib: **Theorem:** Let E = {e ∈ ℤ≥0^n : ∑ eᵢ = d} be weak compositions. Suppose D : E → ℚ₊ is: 1. **Sₙ-invariant**: D(σ·e) = D(e) for all permutations σ 2. **Log-concave**: D(e)² ≥ D(e - δᵢ + δⱼ) · D(e + δᵢ - δⱼ) when eᵢ, eⱼ ≥ 1 Then D is **maximized on balanced vectors** (|eᵢ - eⱼ| ≤ 1 for all i,j) and **minimized on concentrated vectors** (e = d·δₖ for some k). **Proof sketch:** 1. Fix indices i,j and consider the "slice" s(t) = D(e with t at position i, q-t at position j) where q = eᵢ + eⱼ 2. The slice is log-concave (from condition 2) and palindromic (from Sₙ-invariance) 3. A positive log-concave palindromic sequence is unimodal (increases to middle, decreases after) 4. If eᵢ ≥ eⱼ + 2, then eᵢ > q/2, so transferring one unit from i to j moves toward the middle and increases D 5. The sum of squares ∑ eᵢ² decreases with each balancing step, ensuring termination --- ### Current Lean 4 File ```lean /- Copyright (c) 2024. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -/ import Mathlib.Data.Fin.Basic import Mathlib.Data.Fin.Tuple.Basic import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.GroupTheory.Perm.Basic import Mathlib.Data.Rat.Defs import Mathlib.Algebra.Order.Field.Basic import Mathlib.Order.Monotone.Basic import Mathlib.Tactic open Finset BigOperators Function variable {n : ℕ} structure WeakComposition (n : ℕ) (d : ℤ) where toFun : Fin n → ℤ sum_eq : ∑ i, toFun i = d nonneg : ∀ i, 0 ≤ toFun i namespace WeakComposition variable {d : ℤ} instance : CoeFun (WeakComposition n d) (fun _ => Fin n → ℤ) := ⟨WeakComposition.toFun⟩ -- Need to define: modify (transfer one unit from i to j) def modify (e : WeakComposition n d) (i j : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) : WeakComposition n d where toFun := fun k => if k = i then e i - 1 else if k = j then e j + 1 else e k sum_eq := sorry -- BOTTLENECK 1 nonneg := sorry end WeakComposition ``` --- ### Specific Technical Bottlenecks **Bottleneck 1: Sum manipulation with conditional updates** Goal: Prove that when we modify a vector by subtracting 1 at position i and adding 1 at position j, the sum is preserved. ```lean -- Given: ∑ k, e k = d -- Prove: ∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k) = d ``` The natural approach is: ```lean rw [← add_sum_erase _ _ (mem_univ i)] ``` But Lean produces goals like: ``` ⊢ (e i - 1 + ∑ x ∈ univ.erase i, if x = i then e i - 1 else if x = j then e j + 1 else e x) = ... ``` The problem: `add_sum_erase` extracts the term at `i`, but the remaining sum still has `if x = i` clauses that should simplify but don't automatically. **Bottleneck 2: Permutation sum invariance** For `IsSymmetric`, we need: ```lean -- Given: ∑ i, e i = d -- Prove: ∑ i, (e ∘ σ.symm) i = d ``` The lemma `Equiv.sum_comp` should give this, but the types don't match up correctly. **Bottleneck 3: Disjunction from negated conjunction** After `push_neg` on `¬(e i ≤ e j + 1 ∧ e j ≤ e i + 1)`, Lean produces an implication `e i ≤ e j + 1 → e i + 1 < e j` rather than a disjunction. Need the right way to case split. --- ### What We Need 1. The correct idiom for proving sum-preservation under conditional point updates in Mathlib 2. How to apply `Equiv.sum_comp` when the function is a composition 3. The right tactic sequence for case-splitting negated conjunctions --- Now let me use the skill's repair tools to try fixing these issues. -------------------------------------------------------------------------------- ASSISTANT (entry 150): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 151): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:53:15: error: `simp` made no progress BalancedVectors.lean:74:76: error: unsolved goals n : ℕ d : ℤ e : WeakComposition n d i j : Fin n hi : 1 ≤ e.toFun i hij : i ≠ j hsum : ∑ i, e.toFun i = d ⊢ (e.toFun i - 1 + ∑ x ∈ univ.erase i, if x = i then e.toFun i - 1 else if x = j then e.toFun j + 1 else e.toFun x) = e.toFun i - 1 + ∑ k ∈ univ.erase i, if k = j then e.toFun j + 1 else e.toFun k BalancedVectors.lean:81:6: error: Tactic `apply` failed: could not unify the conclusi...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 153): -------------------------------------------------------------------------------- Now I can see the issues clearly. Let me fix them systematically. The key problems are: 1. After `add_sum_erase`, the remaining sum still has `if x = i` terms that need simplifying 2. `push_neg` on `¬(A ∧ B)` gives `A → ¬B`, not `¬A ∨ ¬B` 3. `Equiv.sum_comp` needs correct application -------------------------------------------------------------------------------- ASSISTANT (entry 154): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 155): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→impo...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 157): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 158): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:64:7: error: unexpected token 'in'; expected ',' BalancedVectors.lean:76:13: error: unexpected token 'in'; expected ',' BalancedVectors.lean:152:8: warning: declaration uses 'sorry' BalancedVectors.lean:164:38: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ i, e.toFun i in the target expression ∑ x, e.toFun ((Equiv.symm σ) x) = d n : ℕ d : ℤ D : WeakComposition n d → ℚ σ : Equiv.Perm (Fin n) e : WeakComposition n d ⊢ ∑ x, e.toFun...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 160): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 161): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 57→@[simp] 58→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 59→ simp only [concentrated, h, ite_false] 60→ 61→/-- Helper lemma: sum over erase with simplified if -/ 62→lemma sum_erase_ite_eq_sum {α : Type*} [AddCommMonoid α] (s : Finset (Fin n)) (i : Fin n) 63→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 163): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 164): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 70→ toFun := fun k => if k = i then e i - 1 else if k = j then e j + 1 else e k 71→ sum_eq := by 72→ have hsum := e.sum_eq 73→ -- Key: split the sum using sum_ite_eq' which handles the "if k = i" nicely 74→ calc ∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k) 75→ = (if i ∈ univ then ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 166): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 167): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 156→ sorry 157→ 158→/-! ### The Main Theorem Setup -/ 159→ 160→/-- A function on weak compositions that is symmetric under the Sₙ action. -/ 161→def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 162→ ∀ (σ : Equiv.Perm (Fin n)) (e : WeakComposition n d), 163→ D ⟨e ∘ σ.symm, 164→ by simp only...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 169): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 170): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 193→ (hij : i ≠ j) (hi : 1 ≤ e i) (hdiff : e j + 2 ≤ e i) : 194→ imbalance (e.modify i j hi hij).toFun < imbalance e.toFun := by 195→ sorry 196→ 197→/-- If a vector is not balanced, there exist i, j with e_i ≥ e_j + 2. -/ 198→lemma exists_imbalanced_pair (e : Fin n → ℤ) (h : ¬ IsBalanced e) : 199→ ∃ i j, i ≠ j ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 172): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 173): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 214→ (t : ℤ) (ht : 0 ≤ t) (htq : t ≤ e i + e j) : WeakComposition n d where 215→ toFun := fun k => if k = i then t else if k = j then e i + e j - t else e k 216→ sum_eq := by 217→ have hsum := e.sum_eq 218→ calc ∑ k, (if k = i then t else if k = j then e i + e j - t else e k) 219→ = (if i ∈ univ then t els...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 175): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 176): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:79:12: error: Tactic `apply` failed: could not unify the conclusion of `sum_congr rfl` Finset.sum ?m.340 ?m.337 = Finset.sum ?m.340 ?m.338 with the goal (∑ k, if k = i then e.toFun i - 1 else if k = j then e.toFun j + 1 else e.toFun k) = (∑ x, if x = i then e.toFun i - 1 else 0) + ∑ k ∈ univ.erase i, if k = j then e.toFun j + 1 else e.toFun k Note: The full type of `sum_congr rfl` is (∀ x ∈ ?m.340, ?m.337 x = ?m.338 x) → Finset.sum ?m.340 ?m.337 = Fins...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 178): -------------------------------------------------------------------------------- The sum manipulations are quite tricky. Let me simplify by using `sorry` for the technical sum proofs so we can focus on the main mathematical structure. -------------------------------------------------------------------------------- ASSISTANT (entry 179): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 180): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→impo...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 182): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 185): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 BalancedVectors.lean:76:4: error: `simp` made no progress BalancedVectors.lean:91:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ x ∈ univ.erase i, ?m.362 x in the target expression e i = e i + e j case e_a n : ℕ d : ℤ e : Fin n → ℤ i j : Fin n hij : i ≠ j hsum : ∑ k, e k = d key : (∑ k, if k = i then e i - 1 else if k = j then e j + 1 else e k) = e i - 1 + (e j + 1) + ∑ k ∈ (univ.erase i).erase j, e k ⊢ e i = e i + e j BalancedVectors.lean:...[truncated]] -------------------------------------------------------------------------------- USER (entry 187): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me go through the conversation chronologically to capture all important details: 1. **Initial Setup**: User is a beginner to Lean and wants to formalize a mathematical theorem about balanced/concentrated vectors achieving max/min of a symmetric log-concave function on weak compositions. 2. **Environment Setup**: - Installed elan (Lean version manager) - Downloaded mathlib4 cache (7728 files) - Verified Lean 4.26.0 works with mathlib 3. **Mathematical Problem**: - E = {e ∈ ℤ≥0^n : ∑ eᵢ = d} (weak compositions) - D: E → ℚ₊ is Sₙ-invariant and log-concave - Prove: D maximized on balanced vectors, minimized on concentrated vectors 4. **User provided proof sketch**: 5-step proof involving slice analysis, log-concavity, palindromicity, unimodality, and balancing steps. 5. **User feedback on approach**: - Suggested working with ℤ instead of ℕ to avoid truncation issues - Agreed to use product form instead of ratios - Encouraged iterative approach with sorries - Mentioned they added a Lean theorem-proving skill 6. **Multiple iterations of the Lean file** with various compilation errors: - `simp` not making progress - Sum manipulation issues with `add_sum_erase` - `push_neg` giving implications instead of disjunctions - `Equiv.sum_comp` type mismatches - `rcases`/`cases` on wrong types 7. **Key user input near end**: They provided ChatGPT 5.2's suggestions for fixing the three main bottlenecks with specific code patterns. Current file state has several remaining errors related to sum manipulation and the imbalance_decreases proof. Summary: 1. Primary Request and Intent: The user, a Lean beginner with mathematical expertise, wants to formalize and prove a theorem about symmetric log-concave functions on weak compositions. Specifically: for D: E → ℚ₊ that is Sₙ-invariant and satisfies D(e)² ≥ D(e-δᵢ+δⱼ)·D(e+δᵢ-δⱼ), prove that D is maximized on balanced vectors (|eᵢ-eⱼ| ≤ 1) and minimized on concentrated vectors (e = d·δₖ). They want to work iteratively, getting structure compiling with sorries first before filling in proofs. 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Weak compositions: vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - Log-concave sequences: s(t)² ≥ s(t-1)·s(t+1) - Palindromic sequences: s(t) = s(q-t) - Unimodality: increasing to middle, decreasing after - Sₙ-invariance (permutation symmetry) - Slice analysis: fixing two indices i,j and varying allocation between them - Imbalance measure: ∑eᵢ² (decreases with balancing steps) - Sum manipulation with Finset.sum_ite_eq', add_sum_erase - Equiv.sum_comp for permutation sum invariance 3. Files and Code Sections: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - Main formalization file, repeatedly edited - Contains WeakComposition structure, modify operation, IsBalanced, IsConcentrated definitions - Defines LogConcaveOn, IsPalindromicOn, IsPositiveOn for sequences - Key theorems: unimodal_of_logconcave_palindromic, balanced_maximizes, concentrated_minimizes - Current state (most recent version with remaining errors): ```lean /- Copyright (c) 2024. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -/ import Mathlib.Data.Fin.Basic import Mathlib.Data.Fin.Tuple.Basic import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.GroupTheory.Perm.Basic import Mathlib.Data.Rat.Defs import Mathlib.Algebra.Order.Field.Basic import Mathlib.Order.Monotone.Basic import Mathlib.Tactic open Finset BigOperators Function variable {n : ℕ} structure WeakComposition (n : ℕ) (d : ℤ) where toFun : Fin n → ℤ sum_eq : ∑ i, toFun i = d nonneg : ∀ i, 0 ≤ toFun i namespace WeakComposition variable {d : ℤ} instance : CoeFun (WeakComposition n d) (fun _ => Fin n → ℤ) := ⟨WeakComposition.toFun⟩ -- Key definitions: concentrated, modify, sliceComposition -- Key lemmas: sum_modify_eq, sum_slice_eq, imbalance_decreases, exists_imbalanced_pair -- Main theorems with sorry: unimodal_of_logconcave_palindromic, balanced_maximizes, concentrated_minimizes ``` 4. Errors and fixes: - **ℕ subtraction truncation**: Switched from `Fin n → ℕ` to `Fin n → ℤ` per user suggestion - **`simp` made no progress on sum_ite_eq'**: The pattern `simp only [sum_ite_eq', mem_univ, ite_true]` works for extracting single elements - **Sum manipulation with add_sum_erase**: Used conv tactic with sum_add_distrib pattern, but still having issues with `congr 1` creating wrong goals - **push_neg giving implication**: `push_neg` on `¬(A ∧ B)` gives `A → ¬B`, not `¬A ∨ ¬B`. Fix: use `simp only [not_forall, not_and_or, not_le]` or `classical` + `not_and_or` - **Equiv.sum_comp type mismatch**: Need `have : ∑ x, e.toFun (σ.symm x) = ∑ ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 189): -------------------------------------------------------------------------------- Let me read the current state of the file to see what needs to be fixed. -------------------------------------------------------------------------------- ASSISTANT (entry 190): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 191): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 193): -------------------------------------------------------------------------------- Let me check if the file compiles to see what errors remain. -------------------------------------------------------------------------------- ASSISTANT (entry 194): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 195): -------------------------------------------------------------------------------- [Tool Result: /bin/bash: line 1: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 197): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 198): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:76:4: error: `simp` made no progress BalancedVectors.lean:91:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ x ∈ univ.erase i, ?m.362 x in the target expression e i = e i + e j case e_a n : ℕ d : ℤ e : Fin n → ℤ i j : Fin n hij : i ≠ j hsum : ∑ k, e k = d key : (∑ k, if k = i then e i - 1 else if k = j then e j + 1 else e k) = e i - 1 + (e j + 1) + ∑ k ∈ (univ.erase i).erase j, e k ⊢ e i = e i + e j BalancedVectors.lean:148:8: warni...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 200): -------------------------------------------------------------------------------- I see the issues. The main problems are: 1. `sum_ite_eq` vs `sum_ite_eq'` - they differ in argument order 2. The `congr 1` tactic creates wrong goals 3. `split_ifs` creates contradictory cases Let me rewrite the `sum_modify_eq` proof with a cleaner approach using the ChatGPT 5.2 suggestions. -------------------------------------------------------------------------------- ASSISTANT (entry 202): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 204): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 206): -------------------------------------------------------------------------------- Let me check if that fix works and continue fixing the remaining issues. -------------------------------------------------------------------------------- ASSISTANT (entry 207): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 208): -------------------------------------------------------------------------------- [Tool Result: Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncomm...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 210): -------------------------------------------------------------------------------- The `split_ifs` creates cases where `k = i` and `k = j` simultaneously, which is impossible since `i ≠ j`. Let me fix this with a cleaner approach. -------------------------------------------------------------------------------- ASSISTANT (entry 211): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 212): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 214): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 215): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:67:77: error: unsolved goals case neg n : ℕ d : ℤ e : Fin n → ℤ i j k : Fin n hij : ¬i = j hsum : ∑ k, e k = d hi : k = i ⊢ e i - 1 = e i + -1 BalancedVectors.lean:72:8: error: No goals to be solved BalancedVectors.lean:130:8: warning: declaration uses 'sorry' Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abe...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 217): -------------------------------------------------------------------------------- The arithmetic goal `e i - 1 = e i + -1` needs `ring`. Let me combine `simp_all` with `ring`. -------------------------------------------------------------------------------- ASSISTANT (entry 218): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 219): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 221): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 222): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:71:39: error: Application type mismatch: The argument Eq.symm hj has type j = k but is expected to have type i = ?m.250 in the application Eq.trans hi (Eq.symm hj) Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. Balanced...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 224): -------------------------------------------------------------------------------- I see - the hypotheses are `k = i` and `k = j`, so I need to substitute using those. Let me fix this. -------------------------------------------------------------------------------- ASSISTANT (entry 225): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 226): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 228): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 229): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:75:16: error(lean.unknownIdentifier): Unknown identifier `hj` BalancedVectors.lean:80:8: error: No goals to be solved BalancedVectors.lean:138:8: warning: declaration uses 'sorry' Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 231): -------------------------------------------------------------------------------- The issue is that `split_ifs` doesn't always name hypotheses consistently. Let me use pattern matching instead. -------------------------------------------------------------------------------- ASSISTANT (entry 232): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 233): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 235): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 238): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:67:77: error: unsolved goals case neg n : ℕ d : ℤ e : Fin n → ℤ i j : Fin n hij : i ≠ j hsum : ∑ k, e k = d k : Fin n a✝ : k ∈ univ hki : k = i hkj : ¬k = j ⊢ e i - 1 = e i + -1 + if i = j then 1 else 0 case pos n : ℕ d : ℤ e : Fin n → ℤ i j : Fin n hij : i ≠ j hsum : ∑ k, e k = d k : Fin n a✝ : k ∈ univ hki : ¬k = i hkj : k = j ⊢ (if j = i then e i - 1 else e j + 1) = (e j + if j = i then -1 else 0) + 1 BalancedVectors.lean:73:8: error: No goals to be solved BalancedVector...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 240): -------------------------------------------------------------------------------- I see - the `simp` doesn't use the `hij` hypothesis. Let me try a cleaner approach. -------------------------------------------------------------------------------- ASSISTANT (entry 241): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 242): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[simp] 57→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 58→ simp only [concentrated, h, ite_false] 59→ 60→/-- Auxiliary: the sum of a function that modifies two positions. 61→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 62→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 244): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 245): -------------------------------------------------------------------------------- [Tool Result: Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. BalancedVectors.lean:71:6: error: unsolved goals case pos n : ℕ d : ℤ e : Fin n → ℤ i j : Fin n hij : i ≠ j hsum : ∑ k, e k = d k : Fin n hki : ¬k = i hkj : k = j ⊢ (if j = i then -1 + e i else 1 + e j...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 247): -------------------------------------------------------------------------------- Let me fix these issues - I need to use `hki` and `hkj` to derive that `j ≠ i`, and fix the sum rewrite. -------------------------------------------------------------------------------- USER (entry 248): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 250): -------------------------------------------------------------------------------- Hey, before continuing, can you give me an update? Thanks! -------------------------------------------------------------------------------- ASSISTANT (entry 252): -------------------------------------------------------------------------------- ## Status Update Here's where we stand with the Lean formalization: ### ✅ What's Working 1. **Basic structure**: `WeakComposition`, `IsBalanced`, `IsConcentrated` definitions compile 2. **`concentrated` constructor**: Works correctly with `sum_ite_eq'` 3. **`modify` definition**: Structure is correct, depends on `sum_modify_eq` 4. **`sq_transfer_decreases`**: Algebraic lemma proving `(a-1)² + (b+1)² < a² + b²` when `a ≥ b+2` ✓ 5. **`exists_imbalanced_pair`**: Works using `not_and_or` pattern ✓ 6. **`sliceSeq_positive`**: Compiles correctly ✓ 7. **Main theorem statements**: All correctly stated with `sorry` ### 🔧 Current Compilation Errors (3 main issues) 1. **`sum_modify_eq` (line 63-78)**: Sum preservation lemma - Problem: After `simp_rw`, the `sum_add_distrib` rewrite fails because the sum structure doesn't match the expected pattern - Also: In the `k = j` case, we have `if j = i then ...` that needs `hij.symm` to simplify 2. **`imbalance_decreases` (line ~184)**: - Problem: `split_ifs` creates contradictory cases (k = i AND k = j simultaneously) that `ring` can't handle - Need to explicitly handle the impossible case with `subst` + contradiction 3. **`sum_slice_eq` (line ~252)**: Similar pattern to `sum_modify_eq` - Same sum manipulation issues ### 📋 Outstanding Tasks | Task | Status | |------|--------| | Fix `sum_modify_eq` proof | In progress | | Fix `imbalance_decreases` proof | Pending | | Fix `sum_slice_eq` proof | Pending | | Prove `sliceSeq_palindromic` | `sorry` | | Prove `sliceSeq_logconcave` | `sorry` | | Prove `unimodal_of_logconcave_palindromic` | `sorry` | | Prove `balancing_increases_D` | `sorry` | | Prove `balanced_maximizes` | `sorry` | | Prove `concentrated_minimizes` | `sorry` | ### 🎯 The Core Technical Challenge The main blocker is **sum manipulation in Lean 4 / Mathlib4**. We need to prove things like: ``` ∑ k, f(k) = f(i) + f(j) + ∑ k ∈ (univ.erase i).erase j, f(k) ``` The standard approach using `sum_add_distrib` and `sum_ite_eq'` is finicky about: - Associativity of the sum expression - Handling contradictory cases from `split_ifs` when `i ≠ j` - Getting the right form for `simp` to recognize Would you like me to get GPT 5.2's input on these specific issues? The key questions would be: 1. Best pattern for proving `∑ k, (if k = i then a else if k = j then b else f k) = a + b + ∑ k ∉ {i,j}, f k` 2. How to handle `split_ifs` creating impossible cases when we have `i ≠ j` as a hypothesis -------------------------------------------------------------------------------- USER (entry 254): -------------------------------------------------------------------------------- Here is the GPT 5.2 input: No need to ask anyone else — the cleanest way to make *all three* of your “sum manipulation” pain points disappear is to factor out one reusable lemma: > **Extract `i`, then extract `j`, both via `Finset.sum_ite_eq'`.** > Don’t go through `sum_add_distrib`, and don’t expand into “three independent ifs” (that’s what creates the impossible `k=i` and `k=j` branch). Below is a drop-in patch that fixes **(1) `sum_modify_eq`**, **(3) `sum_slice_eq` (your `sliceComposition.sum_eq`)**, and also gives a robust pattern for **(2) `imbalance_decreases`** without `split_ifs` contradictions. --- ## A reusable lemma: sum of a “two-point update” Put this somewhere near the top (e.g. inside `namespace WeakComposition` or globally): ```lean open Finset BigOperators lemma sum_ite_ite_eq_add_add_sum_erase_erase {n : ℕ} (f : Fin n → ℤ) (i j : Fin n) (hij : i ≠ j) (a b : ℤ) : (∑ k, (if k = i then a else if k = j then b else f k)) = a + b + ∑ k in (univ.erase i).erase j, f k := by classical have hjmem : j ∈ (univ.erase i : Finset (Fin n)) := mem_erase.mpr ⟨hij.symm, mem_univ j⟩ calc (∑ k, (if k = i then a else if k = j then b else f k)) = (if i ∈ (univ : Finset (Fin n)) then a else 0) + ∑ k in univ.erase i, (if k = j then b else f k) := by simpa [Finset.sum_ite_eq'] using (Finset.sum_ite_eq' (s := (univ : Finset (Fin n))) (a := i) (b := a) (f := fun k => if k = j then b else f k)) _ = a + ∑ k in univ.erase i, (if k = j then b else f k) := by simp [mem_univ] _ = a + ((if j ∈ (univ.erase i : Finset (Fin n)) then b else 0) + ∑ k in (univ.erase i).erase j, f k) := by congr 1 simpa [Finset.sum_ite_eq'] using (Finset.sum_ite_eq' (s := (univ.erase i : Finset (Fin n))) (a := j) (b := b) (f := f)) _ = a + (b + ∑ k in (univ.erase i).erase j, f k) := by simp [hjmem] _ = a + b + ∑ k in (univ.erase i).erase j, f k := by simp [add_assoc] ``` This is the exact “best pattern” for [ \sum_k (\text{if }k=i\text{ then }a\text{ else if }k=j\text{ then }b\text{ else }f,k) = a+b+\sum_{k\notin{i,j}} f,k. ] --- ## Fix 1) `sum_modify_eq` Replace your current `sum_modify_eq` proof with: ```lean lemma sum_modify_eq {n : ℕ} {d : ℤ} {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) : ∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k) = d := by classical have hmod : (∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k)) = (e i - 1) + (e j + 1) + ∑ k in (univ.erase i).erase j, e k := by simpa using (sum_ite_ite_eq_add_add_sum_erase_erase (f := e) (i := i) (j := j) hij (a := e i - 1) (b := e j + 1)) have hi_sum : e i + ∑ k in univ.erase i, e k = d := by simpa [hsum] using (add_sum_erase (s := (univ : Finset (Fin n))) (f := e) (mem_univ i)) have hj_sum : e j + ∑ k in (univ.erase i).erase j, e k = ∑ k in univ.erase i, e k := by have hjmem : j ∈ (univ.erase i : Finset (Fin n)) := mem_erase.mpr ⟨hij.symm, mem_univ j⟩ simpa using (add_sum_erase (s := (univ.erase i : Finset (Fin n))) (f := e) hjmem) have hdecomp : e i + e j + ∑ k in (univ.erase i).erase j, e k = d := by linarith [hi_sum, hj_sum] calc (∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k)) = (e i - 1) + (e j + 1) + ∑ k in (univ.erase i).erase j, e k := hmod _ = e i + e j + ∑ k in (univ.erase i).erase j, e k := by ring _ = d := hdecomp ``` No `sum_add_distrib`, no `simp_rw` gymnastics, and the “`if j = i`” simplification never even appears. --- ## Fix 3) `sum_slice_eq` (your `sliceComposition.sum_eq`) Inside `sliceComposition`, replace the entire `sum_eq := by ...` block with: ```lean sum_eq := by classical have hslice : (∑ k, (if k = i then t else if k = j then e i + e j - t else e k)) = t + (e i + e j - t) + ∑ k in (univ.erase i).erase j, e k := by simpa using (sum_ite_ite_eq_add_add_sum_erase_erase (f := e) (i := i) (j := j) hij (a := t) (b := e i + e j - t)) have hi_sum : e i + ∑ k in univ.erase i, e k = d := by simpa [e.sum_eq] using (add_sum_erase (s := (univ : Finset (Fin n))) (f := e) (mem_univ i)) have hj_sum : e j + ∑ k in (univ.erase i).erase j, e k = ∑ k in univ.erase i, e k := by have hjmem : j ∈ (univ.erase i : Finset (Fin n)) := mem_erase.mpr ⟨hij.symm, mem_univ j⟩ simpa using (add_sum_erase (s := (univ.erase i : Finset (Fin n))) (f := e) hjmem) have hdecomp : e i + e j + ∑ k in (univ.erase i).erase j, e k = d := by linarith [hi_sum, hj_sum] calc (∑ k, (if k = i then t else if k = j then e i + e j - t else e k)) = t + (e i + e j - t) + ∑ k in (univ.erase i).erase j, e k := hslice _ = e i + e j + ∑ k in (univ.erase i).erase j, e k := by ring _ = d := h ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 256): -------------------------------------------------------------------------------- Excellent! This is very helpful - the key insight is to factor out the "two-point update" pattern into a reusable lemma. Let me apply these fixes. -------------------------------------------------------------------------------- ASSISTANT (entry 257): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 258): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 260): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 261): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 263): -------------------------------------------------------------------------------- Now let me apply the GPT 5.2 fixes. First, I'll add the reusable lemma near the top of the file. -------------------------------------------------------------------------------- ASSISTANT (entry 264): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 265): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 19→symmetric and satisfy a log-concavity condition, the maximum is achieved on balanced 20→vectors and the minimum on concentrated vectors. 21→-/ 22→ 23→open Finset BigOperators Function 24→ 25→variable {n : ℕ} 26→ 27→/-! ### Reusable sum lemma for two-point updates -/ 28→ 29→/-- Key lemma: extract...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 267): -------------------------------------------------------------------------------- Now let me update `sum_modify_eq` with the new pattern. -------------------------------------------------------------------------------- ASSISTANT (entry 268): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 269): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 86→@[simp] 87→lemma concentrated_ne (hd : 0 ≤ d) (k j : Fin n) (h : j ≠ k) : (concentrated hd k) j = 0 := by 88→ simp only [concentrated, h, ite_false] 89→ 90→/-- Auxiliary: the sum of a function that modifies two positions. 91→ This captures: if we change e at i to (e i - 1) and at j to (e j + 1), 92→ the ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 271): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 272): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 274): -------------------------------------------------------------------------------- Now let me update `imbalance_decreases`. -------------------------------------------------------------------------------- ASSISTANT (entry 275): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 276): -------------------------------------------------------------------------------- [Tool Result: 180→ D ⟨e ∘ σ.symm, 181→ by simp only [comp_apply] 182→ have : ∑ x, e.toFun (σ.symm x) = ∑ x, e.toFun x := Equiv.sum_comp σ.symm e.toFun 183→ rw [this]; exact e.sum_eq, 184→ fun i => by simp only [comp_apply]; exact e.nonneg _⟩ = D e 185→ 186→/-- The log-concavity condition for D. -/ 187→def SatisfiesLogConcavity {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 188→ ∀ (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) 189→ (hi ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 278): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 279): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 204→ have expand : (a - 1)^2 + (b + 1)^2 = a^2 + b^2 + 2*(b - a + 1) := by ring 205→ have neg : b - a + 1 ≤ -1 := by omega 206→ omega 207→ 208→/-- If e_i ≥ e_j + 2, then modifying from i to j decreases imbalance. -/ 209→lemma imbalance_decreases {d : ℤ} (e : WeakComposition n d) (i j : Fin n) 210→ (hij : i ≠ j) (hi...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 281): -------------------------------------------------------------------------------- Now let me update `sum_slice_eq`. -------------------------------------------------------------------------------- ASSISTANT (entry 282): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 283): -------------------------------------------------------------------------------- [Tool Result: 255→ · exact ⟨i, j, fun heq => by simp only [heq] at h1; omega, by omega⟩ 256→ · exact ⟨j, i, fun heq => by simp only [heq] at h2; omega, by omega⟩ 257→ 258→/-! ### Slice Analysis -/ 259→ 260→/-- Auxiliary: sum is preserved when we put t at position i and (q-t) at position j. -/ 261→lemma sum_slice_eq {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) 262→ (t : ℤ) : 263→ ∑ k, (if k = i then t else if k = j then e i + e j - t else e k) = d := by ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 285): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 286): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 256→ · exact ⟨j, i, fun heq => by simp only [heq] at h2; omega, by omega⟩ 257→ 258→/-! ### Slice Analysis -/ 259→ 260→/-- Auxiliary: sum is preserved when we put t at position i and (q-t) at position j. -/ 261→lemma sum_slice_eq {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) 262→ (t : ℤ) : 263→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 288): -------------------------------------------------------------------------------- Now let me check if everything compiles. -------------------------------------------------------------------------------- ASSISTANT (entry 289): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 290): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:43:25: error: Invalid argument name `f` for function `sum_ite_eq'` Hint: Perhaps you meant one of the following parameter names: • `ι`: f̵ι̲ • `M`: f̵M̲ BalancedVectors.lean:43:22: error: Application type mismatch: The argument a has type ℤ but is expected to have type Fin n → ?m.216 in the application sum_ite_eq' univ i a BalancedVectors.lean:51:25: error: Invalid argument name `f` for function `sum_ite_eq'` Hint: Perhaps you meant one of the following paramet...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 292): -------------------------------------------------------------------------------- There are some issues with the reusable lemma and the argument ordering. Let me fix them. -------------------------------------------------------------------------------- ASSISTANT (entry 293): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 294): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Expected exactly one file name Lean (version 4.26.0, x86_64-unknown-linux-gnu, commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562, Release) Miscellaneous -h, --help display this message --features display features compiler provides (eg. LLVM support) -v, --version display version information -V, --short-version display short version number -g, --githash display the git commit hash number used to build this binary --run ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 296): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 297): -------------------------------------------------------------------------------- [Tool Result: Exit code 2 /bin/bash: eval: line 3: unexpected EOF while looking for matching `''] -------------------------------------------------------------------------------- ASSISTANT (entry 299): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 301): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /tmp/check.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 303): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 304): -------------------------------------------------------------------------------- [Tool Result: @Finset.sum_ite_eq' : ∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DecidableEq ι] (s : Finset ι) (a : ι) (b : ι → M), (∑ x ∈ s, if x = a then b x else 0) = if a ∈ s then b a else 0] -------------------------------------------------------------------------------- ASSISTANT (entry 306): -------------------------------------------------------------------------------- I see - `sum_ite_eq'` takes a function `b : ι → M`, not a value. Let me check `sum_ite_eq` (without the prime). -------------------------------------------------------------------------------- ASSISTANT (entry 307): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 308): -------------------------------------------------------------------------------- [Tool Result: The file /tmp/check.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→import Mathlib.Algebra.BigOperators.Fin 2→#check @Finset.sum_ite_eq 3→] -------------------------------------------------------------------------------- ASSISTANT (entry 310): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 311): -------------------------------------------------------------------------------- [Tool Result: @Finset.sum_ite_eq : ∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DecidableEq ι] (s : Finset ι) (a : ι) (b : ι → M), (∑ x ∈ s, if a = x then b x else 0) = if a ∈ s then b a else 0] -------------------------------------------------------------------------------- ASSISTANT (entry 313): -------------------------------------------------------------------------------- Both take a function. For a constant value, I need to use `fun _ => a`. Let me rewrite the reusable lemma correctly. -------------------------------------------------------------------------------- ASSISTANT (entry 314): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 315): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 23→open Finset BigOperators Function 24→ 25→variable {n : ℕ} 26→ 27→/-! ### Reusable sum lemma for two-point updates -/ 28→ 29→/-- Key lemma: extract two positions from a sum with nested if-then-else. -/ 30→lemma sum_ite_ite_eq_add_add_sum_erase_erase 31→ (f : Fin n → ℤ) (i j : Fin n) (hij : i ≠ j) (a b...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 317): -------------------------------------------------------------------------------- Now let me fix the `imbalance_decreases` proof - the issue is with `simpa` and the order of `add_lt_add_right`. -------------------------------------------------------------------------------- ASSISTANT (entry 318): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 319): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 236→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase 237→ (fun k => (e k)^2) i j hij ((e i - 1)^2) ((e j + 1)^2) 238→ _ = (e i - 1)^2 + (e j + 1)^2 + rest := rfl 239→ 240→ have hsum_orig : (∑ k, (e k)^2) = (e i)^2 + (e j)^2 + rest := by 241→ have h := sum_ite_ite_eq_add_add_sum_erase_eras...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 321): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 322): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:41:15: warning: This simp argument is unused: mem_univ Hint: Omit it from the simp argument list. simp only [̵m̵e̵m̵_̵u̵n̵i̵v̵,̵ ̵i̵t̵e̵_̵t̵r̵u̵e̵]̵[̲i̲t̲e̲_̲t̲r̲u̲e̲]̲ Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` BalancedVectors.lean:51:15: warning: This simp argument is unused: mem_erase Hint: Omit it from the simp argument list. simp only [m̵e̵m̵_̵e̵r̵a̵s̵e̵,̵ ̵ne_eq, mem_univ, and_true, hjmem, ite_true] Note: This linter can...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 324): -------------------------------------------------------------------------------- I see the issue - the cases aren't being handled correctly. Let me fix the `hsum_orig` proof. -------------------------------------------------------------------------------- ASSISTANT (entry 325): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 326): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 236→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase 237→ (fun k => (e k)^2) i j hij ((e i - 1)^2) ((e j + 1)^2) 238→ _ = (e i - 1)^2 + (e j + 1)^2 + rest := rfl 239→ 240→ have hsum_orig : (∑ k, (e k)^2) = (e i)^2 + (e j)^2 + rest := by 241→ have h := sum_ite_ite_eq_add_add_sum_erase_eras...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 328): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 329): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:41:15: warning: This simp argument is unused: BalancedVectors.lean:51:15: warning: This simp argument is unused: BalancedVectors.lean:51:26: warning: This simp argument is unused: BalancedVectors.lean:51:33: warning: This simp argument is unused: BalancedVectors.lean:51:43: warning: This simp argument is unused: BalancedVectors.lean:51:53: warning: This simp argument is unused: BalancedVectors.lean:106:4: warning: try 'simp' instead of 'simpa' BalancedVectors.lean:174:8: war...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 331): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 332): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:248:6: error: unsolved goals case pos n : ℕ d : ℤ e : WeakComposition n d i j : Fin n hij : i ≠ j hi : 1 ≤ e.toFun i hdiff : e.toFun j + 2 ≤ e.toFun i rest : ℤ := ∑ k ∈ (univ.erase i).erase j, e.toFun k ^ 2 hrest : rest = ∑ k ∈ (univ.erase i).erase j, e.toFun k ^ 2 hpow_mod : ∀ (k : Fin n), (e.modify i j hi hij).toFun k ^ 2 = if k = i then (e.toFun i - 1) ^ 2 else if k = j then (e.toFun j + 1) ^ 2 else e.toFun k ^ 2 hsum_mod : ∑ k, (e.modify i j hi hij).toFun k ^...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 334): -------------------------------------------------------------------------------- Let me be more explicit with the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 335): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 336): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 236→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase 237→ (fun k => (e k)^2) i j hij ((e i - 1)^2) ((e j + 1)^2) 238→ _ = (e i - 1)^2 + (e j + 1)^2 + rest := rfl 239→ 240→ have hsum_orig : (∑ k, (e k)^2) = (e i)^2 + (e j)^2 + rest := by 241→ have h := sum_ite_ite_eq_add_add_sum_erase_eras...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 338): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 339): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 341): -------------------------------------------------------------------------------- Excellent! No errors. Let me verify the full compilation. -------------------------------------------------------------------------------- ASSISTANT (entry 342): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 343): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:41:15: warning: This simp argument is unused: BalancedVectors.lean:51:15: warning: This simp argument is unused: BalancedVectors.lean:51:26: warning: This simp argument is unused: BalancedVectors.lean:51:33: warning: This simp argument is unused: BalancedVectors.lean:51:43: warning: This simp argument is unused: BalancedVectors.lean:51:53: warning: This simp argument is unused: BalancedVectors.lean:106:4: warning: try 'simp' instead of 'simpa' BalancedVectors.lean:174:8: war...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 345): -------------------------------------------------------------------------------- The file now compiles successfully. The only warnings are: - Unused simp arguments (style issues) - `sorry` placeholders for proofs that still need to be filled in Let me update the todo list and clean up the minor style issues. -------------------------------------------------------------------------------- ASSISTANT (entry 346): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 347): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 349): -------------------------------------------------------------------------------- Let me clean up the unused simp arguments. -------------------------------------------------------------------------------- ASSISTANT (entry 350): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 351): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 33→ = a + b + ∑ k ∈ (univ.erase i).erase j, f k := by 34→ classical 35→ have hjmem : j ∈ (univ.erase i : Finset (Fin n)) := 36→ mem_erase.mpr ⟨hij.symm, mem_univ j⟩ 37→ -- First, split the sum: extract the i-th term 38→ have step1 : ∑ k, (if k = i then a else if k = j then b else f k) = 39→ a + ∑ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 353): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 354): -------------------------------------------------------------------------------- [Tool Result: 100→ classical 101→ have hmod : (∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k)) 102→ = (e i - 1) + (e j + 1) + ∑ k ∈ (univ.erase i).erase j, e k := by 103→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase e i j hij (e i - 1) (e j + 1) 104→ 105→ have hi_sum : e i + ∑ k ∈ univ.erase i, e k = d := by 106→ simpa [hsum] using add_sum_erase (s := (univ : Finset (Fin n))) (f := e) (mem_univ i) 107→ 108→ have hj_sum : e j + ∑ k ∈ (univ.erase...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 356): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 357): -------------------------------------------------------------------------------- [Tool Result: Found 2 matches of the string to replace, but replace_all is false. To replace all occurrences, set replace_all to true. To replace only one occurrence, please provide more context to uniquely identify the instance. String: have hi_sum : e i + ∑ k ∈ univ.erase i, e k = d := by simpa [hsum] using add_sum_erase (s := (univ : Finset (Fin n))) (f := e) (mem_univ i)] -------------------------------------------------------------------------------- ASSISTANT (entry 359): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 360): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 97→ the total sum is preserved. -/ 98→lemma sum_modify_eq {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) : 99→ ∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k) = d := by 100→ classical 101→ have hmod : (∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k)) 102→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 362): -------------------------------------------------------------------------------- Let me fix the other `simpa` warnings and unused arguments. -------------------------------------------------------------------------------- ASSISTANT (entry 363): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 364): -------------------------------------------------------------------------------- [Tool Result: 220→ 221→ have hpow_mod : ∀ k, (e.modify i j hi hij).toFun k ^ 2 = 222→ (if k = i then (e i - 1)^2 else if k = j then (e j + 1)^2 else (e k)^2) := by 223→ intro k 224→ by_cases hki : k = i 225→ · subst hki; simp [WeakComposition.modify] 226→ · by_cases hkj : k = j 227→ · subst hkj; simp [WeakComposition.modify, hki, hij.symm] 228→ · simp [WeakComposition.modify, hki, hkj] 229→ 230→ have hsum_mod : (∑ k, (e.modify i j hi hij).toFun k ^...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 366): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 367): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 217→ classical 218→ unfold imbalance 219→ set rest : ℤ := ∑ k ∈ (univ.erase i).erase j, (e k)^2 with hrest 220→ 221→ have hpow_mod : ∀ k, (e.modify i j hi hij).toFun k ^ 2 = 222→ (if k = i then (e i - 1)^2 else if k = j then (e j + 1)^2 else (e k)^2) := by 223→ intro k 224→ by_cases hki : k = i 225→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 369): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 370): -------------------------------------------------------------------------------- [Tool Result: 270→ 271→/-- Auxiliary: sum is preserved when we put t at position i and (q-t) at position j. -/ 272→lemma sum_slice_eq {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) 273→ (t : ℤ) : 274→ ∑ k, (if k = i then t else if k = j then e i + e j - t else e k) = d := by 275→ classical 276→ have hslice : (∑ k, (if k = i then t else if k = j then e i + e j - t else e k)) 277→ = t + (e i + e j - t) + ∑ k ∈ (univ.erase i).erase j, e k := by 278→ s...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 372): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 373): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 276→ have hslice : (∑ k, (if k = i then t else if k = j then e i + e j - t else e k)) 277→ = t + (e i + e j - t) + ∑ k ∈ (univ.erase i).erase j, e k := by 278→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase e i j hij t (e i + e j - t) 279→ 280→ have hi_sum : e i + ∑ k ∈ univ.erase i, e k = d := by 281→ simp...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 375): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 376): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:106:16: warning: This simp argument is unused: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:281:16: warning: This simp argument is unused: BalancedVectors.lean:316:6: warning: declaration uses 'sorry' BalancedVectors.lean:322:6: warning: declaration uses 'sorry' BalancedVectors.lean:338:6: warning: declaration uses 'sorry' BalancedVectors.lean:345:8: warning: declaration uses 'sorry' BalancedVectors.lean:351:8: warning: declaration uses ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 378): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 379): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:106:16: warning: This simp argument is unused: add_sum_erase (s := (univ : Finset (Fin n))) (f := e) (mem_univ i) Hint: Omit it from the simp argument list.] -------------------------------------------------------------------------------- ASSISTANT (entry 381): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 382): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 101→ have hmod : (∑ k, (if k = i then e i - 1 else if k = j then e j + 1 else e k)) 102→ = (e i - 1) + (e j + 1) + ∑ k ∈ (univ.erase i).erase j, e k := by 103→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase e i j hij (e i - 1) (e j + 1) 104→ 105→ have hi_sum : e i + ∑ k ∈ univ.erase i, e k = d := by 106→ rw...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 384): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 385): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 276→ have hslice : (∑ k, (if k = i then t else if k = j then e i + e j - t else e k)) 277→ = t + (e i + e j - t) + ∑ k ∈ (univ.erase i).erase j, e k := by 278→ simpa using sum_ite_ite_eq_add_add_sum_erase_erase e i j hij t (e i + e j - t) 279→ 280→ have hi_sum : e i + ∑ k ∈ univ.erase i, e k = d := by 281→ rw [...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 387): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 388): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:316:6: warning: declaration uses 'sorry' BalancedVectors.lean:322:6: warning: declaration uses 'sorry' BalancedVectors.lean:338:6: warning: declaration uses 'sorry' BalancedVectors.lean:345:8: warning: declaration uses 'sorry' BalancedVectors.lean:351:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 390): -------------------------------------------------------------------------------- All style warnings are fixed. The file now compiles cleanly - the only remaining warnings are the expected `sorry` placeholders for proofs we haven't written yet. -------------------------------------------------------------------------------- ASSISTANT (entry 391): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 392): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- USER (entry 394): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 396): -------------------------------------------------------------------------------- Thanks a lot, this was great work! I think it might make sense to tackle the easier lemmas first for filling in the sorry statements. Here are some suggestions about this from GPT 5.2: I went after two of the “low-hanging fruit” sorries: the slice-sequence lemmas. They’re fairly direct rewrites + `simp`/`ext` once you pick the right permutation / identify the neighboring slices. I’ve updated your file in-place here: [BalancedVectors.lean](sandbox:/mnt/data/BalancedVectors.lean) ## 1) `sliceSeq_palindromic` Idea: for `q = e i + e j`, the slice at `t` has `(i↦t, j↦q-t)`. Swapping `i` and `j` turns that into `(i↦q-t, j↦t)`, i.e. the slice at `q-t`. Then use `IsSymmetric D` with `σ = Equiv.Perm.swap i j`. Key points in the proof: * `simp [sliceSeq]` on-range to reduce both sides to `D (sliceComposition ...)`. * Apply symmetry `hsym σ (sliceComposition ...)`. * Use `WeakComposition.ext` to identify the “swapped” composition with `sliceComposition ... (q-t)`. (You’ll see this fully implemented in the file.) ## 2) `sliceSeq_logconcave` Idea: on the valid range `1 ≤ t ≤ q-1`, the central slice `e0 := sliceComposition ... t` satisfies: * `(e0 i) = t ≥ 1` and `(e0 j) = q - t ≥ 1` * `e0.modify i j` is exactly the neighboring slice at `t-1` * `e0.modify j i` is exactly the neighboring slice at `t+1` Then `SatisfiesLogConcavity D` applied to `e0` *is literally* the log-concavity inequality for `sliceSeq` at `t`, once you rewrite the two `modify` terms into those neighboring slices. Key points: * `omega` to produce the various bound facts (`0 ≤ t`, `0 ≤ t±1`, etc.) * `ext k; by_cases k=i/j` to prove the equalities * `simp [sliceSeq, …]` to rewrite back to `sliceSeq` terms Again, this is now filled in in the file. --- # Comments / suggestions for the remaining sorries ## A) `balancing_increases_D` This should become a short wrapper around your unimodality lemma + the slice lemmas you already have: Skeleton: ```lean lemma balancing_increases_D ... := by classical -- let q := e i + e j -- let s := sliceSeq D e i j hij have hpal := sliceSeq_palindromic (D:=D) hsym e i j hij have hlcs := sliceSeq_logconcave (D:=D) hlc e i j hij have hposs := sliceSeq_positive (D:=D) hpos e i j hij -- apply unimodality to s on [0,q] have hunim := unimodal_of_logconcave_palindromic (s:=sliceSeq D e i j hij) (q:=e i + e j) (by omega) hposs hlcs hpal -- use the “right-half decreases” part at t = e i -- hdiff : e j + 2 ≤ e i gives 2*(e i) > (e i + e j) have hright : 2 * (e i) > (e i + e j) := by omega have hle : sliceSeq D e i j hij (e i) ≤ sliceSeq D e i j hij (e i - 1) := by -- this is hunim.2 with appropriate bounds -- watch the index: hunim.2 is `s t ≤ s (t - 1)` exact hunim.2 (t := e i) hright (by omega) -- rewrite sliceSeq at (e i) to D e, and at (e i - 1) to D (e.modify i j ...) -- using `sliceComposition` ext-lemmas: -- sliceComposition ... (e i) ... = e -- sliceComposition ... (e i - 1) ... = e.modify i j hi hij -- then finish by simp/linarith ``` The only “plumbing” you’ll still need is the two identification lemmas: * `sliceComposition e i j hij (e i) ... = e` * `sliceComposition e i j hij (e i - 1) ... = e.modify i j hi hij` Both are easy `ext k; by_cases k=i/j` + `simp [sliceComposition, WeakComposition.modify]`. ## B) `balanced_maximizes` You already set this up perfectly with: * `exists_imbalanced_pair` to find `i,j` with `e j + 2 ≤ e i` * `imbalance_decreases` to ensure a measure strictly decreases * `balancing_increases_D` to ensure `D` weakly increases What’s left is picking a *well-founded* measure. Since `imbalance` is currently `ℤ`, I strongly recommend switching to a `Nat` measure (or wrapping via `Int.toNat`) so you can use `WellFounded` induction cleanly. For example: ```lean def imbalanceNat (e : Fin n → ℤ) : ℕ := Int.toNat (imbalance e) ``` Then prove `imbalance_decreases` implies `imbalanceNat` decreases, and run `Nat` well-founded induction to repeatedly apply the balancing step until `IsBalanced`. ## C) `concentrated_minimizes` A practical route that tends to be easiest in Lean: * Fix a target coordinate `k` (e.g. an argmax of `e k`). * Repeatedly move one unit from any other coordinate `j` with `e j > 0` into `k`. * Use unimodality on the 2-slice `(k,j)` to show this *decreases* `D` (it’s the same unimodality inequality, just used in the direction `s (t+1) ≤ s t` on the right half). * Terminate by induction on the finite “mass outside `k`”, e.g. `m := d - e k` (converted to `Nat`). You’ll likely want a small helper lemma of the form: ```lean lemma move_to_max_decreases_D (hkmax : ∀ l, e l ≤ e k) (hjpos : 1 ≤ e j) (hjk : j ≠ k) : D (e.modify j k hjpos hjk) ≤ D e := ... ``` proved via slice/unimodality in the same style as `balancing_increases_D`. --- If you want, paste the *current* statement you’re aiming for in `unimodal_of_logconcave_palindromic` (or any local lemmas you already have about log-concavity/unimodality), ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 398): -------------------------------------------------------------------------------- Let me read the current file and then implement the suggested proofs for `sliceSeq_palindromic` and `sliceSeq_logconcave`. -------------------------------------------------------------------------------- ASSISTANT (entry 399): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 400): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 402): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 403): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 405): -------------------------------------------------------------------------------- Let me implement `sliceSeq_palindromic` first. The key is using the swap permutation. -------------------------------------------------------------------------------- ASSISTANT (entry 406): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 407): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 311→ if h : 0 ≤ t ∧ t ≤ e i + e j then 312→ D (sliceComposition e i j hij t h.1 h.2) 313→ else 0 314→ 315→/-- The slice sequence is palindromic when D is symmetric. -/ 316→lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) 317→ (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 409): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 410): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:349:8: error: `simp` made no progress BalancedVectors.lean:354:10: error: `simp` made no progress BalancedVectors.lean:357:12: error: `simp` made no progress BalancedVectors.lean:360:2: error: Type mismatch] -------------------------------------------------------------------------------- ASSISTANT (entry 412): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 413): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:349:8: error: `simp` made no progress BalancedVectors.lean:354:10: error: `simp` made no progress BalancedVectors.lean:357:12: error: `simp` made no progress BalancedVectors.lean:360:2: error: Type mismatch hswap has type D (sliceComposition e i j hij t ht htq) = D (sliceComposition e i j hij (q - t) ht' htq') but is expected to have type (if h : True ∧ t ≤ e.toFun i + e.toFun j then D (sliceComposition e i j hij t ⋯ ⋯) else 0) =] -------------------------------------------------------------------------------- ASSISTANT (entry 415): -------------------------------------------------------------------------------- Let me fix the proof - I need to handle the `dite` more explicitly. -------------------------------------------------------------------------------- ASSISTANT (entry 416): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 417): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 311→ if h : 0 ≤ t ∧ t ≤ e i + e j then 312→ D (sliceComposition e i j hij t h.1 h.2) 313→ else 0 314→ 315→/-- The slice sequence is palindromic when D is symmetric. -/ 316→lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) 317→ (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 419): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 422): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:348:6: error: `simp` made no progress BalancedVectors.lean:351:6: error: unsolved goals] -------------------------------------------------------------------------------- ASSISTANT (entry 424): -------------------------------------------------------------------------------- Good idea about saving output. Let me do that and fix the remaining issues. -------------------------------------------------------------------------------- ASSISTANT (entry 425): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 426): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:347:17: error(lean.unknownIdentifier): Unknown identifier `i` BalancedVectors.lean:348:6: error: `simp` made no progress BalancedVectors.lean:351:6: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d i : Fin n t : ℤ ht : 0 ≤ t k : Fin n hki : ¬k = i hij : i ≠ k q : ℤ := e.toFun i + e.toFun k htq : t ≤ q hq : q = e.toFun i + e.toFun k ht' : 0 ≤ q - t ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 428): -------------------------------------------------------------------------------- I see the issue - there's a variable name conflict. Let me use a different variable name for the `ext` proof. -------------------------------------------------------------------------------- ASSISTANT (entry 429): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 430): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 311→ if h : 0 ≤ t ∧ t ≤ e i + e j then 312→ D (sliceComposition e i j hij t h.1 h.2) 313→ else 0 314→ 315→/-- The slice sequence is palindromic when D is symmetric. -/ 316→lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) 317→ (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 432): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 433): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:347:37: error: unsolved goals BalancedVectors.lean:345:4: error: unsolved goals BalancedVectors.lean:353:39: error: unsolved goals BalancedVectors.lean:351:6: error: unsolved goals BalancedVectors.lean:357:10: error: `simp` made no progress] -------------------------------------------------------------------------------- ASSISTANT (entry 435): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 436): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:347:37: error: unsolved goals n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d j : Fin n t : ℤ ht : 0 ≤ t m : Fin n hij : m ≠ j q : ℤ := e.toFun m + e.toFun j htq : t ≤ q hq : q = e.toFun m + e.toFun j ht' : 0 ≤ q - t htq' : q - t ≤ q] -------------------------------------------------------------------------------- ASSISTANT (entry 438): -------------------------------------------------------------------------------- The variable scoping is confusing Lean. Let me try a different approach using `congr` and explicit arguments. -------------------------------------------------------------------------------- ASSISTANT (entry 439): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 440): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 311→ if h : 0 ≤ t ∧ t ≤ e i + e j then 312→ D (sliceComposition e i j hij t h.1 h.2) 313→ else 0 314→ 315→/-- The slice sequence is palindromic when D is symmetric. -/ 316→lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) 317→ (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 442): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 443): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 445): -------------------------------------------------------------------------------- `sliceSeq_palindromic` compiles. Let me check what warnings remain. -------------------------------------------------------------------------------- ASSISTANT (entry 446): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 447): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:366:6: warning: declaration uses 'sorry' BalancedVectors.lean:382:6: warning: declaration uses 'sorry' BalancedVectors.lean:389:8: warning: declaration uses 'sorry' BalancedVectors.lean:395:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 449): -------------------------------------------------------------------------------- Now let me implement `sliceSeq_logconcave`. -------------------------------------------------------------------------------- ASSISTANT (entry 450): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 451): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 453): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 454): -------------------------------------------------------------------------------- [Tool Result: 360→ have hswap_other : σ.symm k = k := by 361→ rw [hσ, Equiv.symm_swap, Equiv.swap_apply_of_ne_of_ne hki hkj] 362→ rw [hswap_other] 363→ simp only [hki, hkj, ite_false] 364→ 365→/-- The slice sequence is log-concave when D satisfies log-concavity. -/ 366→lemma sliceSeq_logconcave {d : ℤ} (D : WeakComposition n d → ℚ) 367→ (hlc : SatisfiesLogConcavity D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : 368→ LogConcaveOn (sliceSeq D e ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 456): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 457): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 361→ rw [hσ, Equiv.symm_swap, Equiv.swap_apply_of_ne_of_ne hki hkj] 362→ rw [hswap_other] 363→ simp only [hki, hkj, ite_false] 364→ 365→/-- The slice sequence is log-concave when D satisfies log-concavity. -/ 366→lemma sliceSeq_logconcave {d : ℤ} (D : WeakComposition n d → ℚ) 367→ (hlc : SatisfiesLogC...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 459): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 460): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:381:30: error: `simp` made no progress BalancedVectors.lean:382:34: error: `simp` made no progress BalancedVectors.lean:398:6: error: unsolved goals BalancedVectors.lean:403:36: error: No goals to be solved BalancedVectors.lean:405:6: error: unsolved goals BalancedVectors.lean:406:6: error: unsolved goals BalancedVectors.lean:415:2: error: No goals to be solved] -------------------------------------------------------------------------------- ASSISTANT (entry 462): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 463): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:381:30: error: `simp` made no progress BalancedVectors.lean:382:34: error: `simp` made no progress BalancedVectors.lean:398:6: error: unsolved goals case neg n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j] -------------------------------------------------------------------------------- ASSISTANT (entry 465): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 466): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 376→ have ht_lo : 0 ≤ t := by omega 377→ have ht_hi : t ≤ q := by omega 378→ -- The central slice e0 at position t 379→ set e0 := sliceComposition e i j hij t ht_lo ht_hi with he0 380→ -- e0 has i-value = t and j-value = q - t 381→ have he0_i : e0 i = t := by simp only [he0, sliceComposition, ite_true] 382→ have h...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 468): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 469): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:382:31: error: unsolved goals BalancedVectors.lean:398:6: error: unsolved goals BalancedVectors.lean:403:36: error: No goals to be solved BalancedVectors.lean:405:6: error: unsolved goals BalancedVectors.lean:406:6: error: unsolved goals BalancedVectors.lean:415:2: error: No goals to be solved] -------------------------------------------------------------------------------- ASSISTANT (entry 471): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 472): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:382:31: error: unsolved goals n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 1 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q - 1 hq : q = e.toFun i + e.toFun j htm1_lo : 0 ≤ t - 1 htm1_hi : t - 1 ≤ q htp1_lo : 0 ≤ t + 1] -------------------------------------------------------------------------------- ASSISTANT (entry 474): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 475): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:382:31: error: unsolved goals n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 1 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q - 1 hq : q = e.toFun i + e.toFun j htm1_lo : 0 ≤ t - 1 htm1_hi : t - 1 ≤ q htp1_lo : 0 ≤ t + 1 htp1_hi : t + 1 ≤ q ht_lo : 0 ≤ t ht_hi : t ≤ q e0 : WeakComposition n d := sliceComposition e i j hij t ht_lo ht_hi he0 : e0 = sliceComposition e i j hij t ht_lo ht_hi he0_i :...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 477): -------------------------------------------------------------------------------- I see the issues. Let me fix them - the simp doesn't know about `q` definition. -------------------------------------------------------------------------------- ASSISTANT (entry 478): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 479): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 376→ have ht_lo : 0 ≤ t := by omega 377→ have ht_hi : t ≤ q := by omega 378→ -- The central slice e0 at position t 379→ set e0 := sliceComposition e i j hij t ht_lo ht_hi with he0 380→ -- e0 has i-value = t and j-value = q - t 381→ have he0_i : e0 i = t := by simp only [he0, sliceComposition, ite_true] 382→ have h...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 481): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 482): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:400:6: error: unsolved goals BalancedVectors.lean:405:62: error: No goals to be solved BalancedVectors.lean:407:6: error: unsolved goals BalancedVectors.lean:408:6: error: unsolved goals BalancedVectors.lean:416:2: error: No goals to be solved] -------------------------------------------------------------------------------- ASSISTANT (entry 484): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 485): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:400:6: error: unsolved goals case neg n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 1 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q - 1 hq : q = e.toFun i + e.toFun j htm1_lo : 0 ≤ t - 1 htm1_hi : t - 1 ≤ q htp1_lo : 0 ≤ t + 1 htp1_hi : t + 1 ≤ q ht_lo : 0 ≤ t ht_hi : t ≤ q e0 : WeakComposition n d := sliceComposition e i j hij t ht_lo ht_hi] -------------------------------------------------------------------------------- ASSISTANT (entry 487): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 488): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:400:6: error: unsolved goals case neg n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 1 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q - 1 hq : q = e.toFun i + e.toFun j htm1_lo : 0 ≤ t - 1 htm1_hi : t - 1 ≤ q htp1_lo : 0 ≤ t + 1 htp1_hi : t + 1 ≤ q ht_lo : 0 ≤ t ht_hi : t ≤ q e0 : WeakComposition n d := sliceComposition e i j hij t ht_lo ht_hi he0 : e0 = sliceComposition e i j hij t ht_lo ht_hi...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 490): -------------------------------------------------------------------------------- I see - the `else` branch needs to show that the compositions preserve values at other positions. Let me fix this. -------------------------------------------------------------------------------- ASSISTANT (entry 491): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 492): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 389→ have hlc_app := hlc e0 i j hij hi_ge1 hj_ge1 390→ -- e0.modify i j gives i ↦ t-1, j ↦ (q-t)+1 = q-(t-1) 391→ -- e0.modify j i gives i ↦ t+1, j ↦ (q-t)-1 = q-(t+1) 392→ -- Show these equal the neighboring slice compositions 393→ have hmod_ij : e0.modify i j hi_ge1 hij = sliceComposition e i j hij (t - 1) htm1_lo htm1_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 494): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 495): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:407:62: error: No goals to be solved BalancedVectors.lean:409:6: error: unsolved goals BalancedVectors.lean:420:2: error: No goals to be solved] -------------------------------------------------------------------------------- ASSISTANT (entry 497): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 498): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:407:62: error: No goals to be solved BalancedVectors.lean:409:6: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i : Fin n t : ℤ ht : 1 ≤ t htm1_lo : 0 ≤ t - 1 htp1_lo : 0 ≤ t + 1 ht_lo : 0 ≤ t k : Fin n hki : ¬k = i hij : i ≠ k q : ℤ := e.toFun i + e.toFun k htq : t ≤ q - 1 hq : q = e.toFun i + e.toFun k htm1_hi : t - 1 ≤ q htp1_hi : t + 1 ≤ q ht_hi : t ≤ q e0 : WeakComposition n d := sliceComposit...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 500): -------------------------------------------------------------------------------- I see the variable scoping issue again - `k` from `ext` is shadowing `j`. Let me use `m` as the variable name. -------------------------------------------------------------------------------- ASSISTANT (entry 501): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 502): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 389→ have hlc_app := hlc e0 i j hij hi_ge1 hj_ge1 390→ -- e0.modify i j gives i ↦ t-1, j ↦ (q-t)+1 = q-(t-1) 391→ -- e0.modify j i gives i ↦ t+1, j ↦ (q-t)-1 = q-(t+1) 392→ -- Show these equal the neighboring slice compositions 393→ have hmod_ij : e0.modify i j hi_ge1 hij = sliceComposition e i j hij (t - 1) htm1_lo htm1_...[truncated]] -------------------------------------------------------------------------------- USER (entry 504): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me go through the conversation chronologically to capture all important details: 1. **Initial Context**: This is a continuation of a previous conversation about formalizing a mathematical theorem in Lean 4 / Mathlib4 about balanced/concentrated vectors achieving max/min of symmetric log-concave functions on weak compositions. 2. **Starting Point**: The file `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` had compilation errors related to sum manipulation proofs. 3. **First Major Fix**: Applied GPT 5.2's suggestions to create a reusable lemma `sum_ite_ite_eq_add_add_sum_erase_erase` for handling two-point updates in sums. 4. **Three Core Fixes Applied**: - Fixed `sum_modify_eq` using the new reusable lemma - Fixed `imbalance_decreases` using the new pattern - Fixed `sum_slice_eq` using the new pattern 5. **Style Cleanups**: Fixed unused simp arguments, replaced `simpa` with `simp` where appropriate. 6. **GPT 5.2 Second Input**: User provided suggestions for proving the "low-hanging fruit" - `sliceSeq_palindromic` and `sliceSeq_logconcave`. 7. **`sliceSeq_palindromic` Implementation**: - Used swap permutation `Equiv.swap i j` - Had variable scoping issues - `k` from `ext k` was shadowing variables - Fixed by using `m` as the variable name in `ext` - Successfully compiled 8. **`sliceSeq_logconcave` Implementation** (Current Work): - Key insight: the central slice at t, when modified, gives neighboring slices at t-1 and t+1 - Applying log-concavity condition to the central slice gives the inequality - Had similar variable scoping issues with `ext k` shadowing `j` - Changed to `ext m` to avoid conflicts - Still has compilation errors to fix 9. **User Messages**: - Asked for updates on status multiple times - Suggested saving Lean output to temp file for easier debugging - Provided GPT 5.2 inputs with detailed code patterns 10. **Current Errors**: Lines 407, 409, 420 still have errors related to: - "No goals to be solved" (likely `ring` applied when goal already closed) - Unsolved goals in the `m = j` case Summary: 1. Primary Request and Intent: The user (a Lean beginner with mathematical expertise) wants to formalize a theorem about symmetric log-concave functions on weak compositions in Lean 4 / Mathlib4. The theorem states: for D: E → ℚ₊ that is Sₙ-invariant and satisfies D(e)² ≥ D(e-δᵢ+δⱼ)·D(e+δᵢ-δⱼ), prove D is maximized on balanced vectors (|eᵢ-eⱼ| ≤ 1) and minimized on concentrated vectors (e = d·δₖ). The user wants to work iteratively, getting proofs compiling before filling in details. User requested following GPT 5.2's suggestions for implementing the "low-hanging fruit" proofs. 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Weak compositions: vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - Log-concave sequences: s(t)² ≥ s(t-1)·s(t+1) - Palindromic sequences: s(t) = s(q-t) - Sₙ-invariance (permutation symmetry) - Slice analysis: fixing two indices i,j and varying allocation - `Finset.sum_ite_eq'`, `add_sum_erase` for sum manipulation - `Equiv.swap` permutation for palindromicity proofs - `WeakComposition.ext` for proving composition equality - Variable scoping issues with `ext` tactic shadowing outer variables 3. Files and Code Sections: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - Main formalization file containing all definitions and proofs **Key Completed Sections:** - Reusable sum lemma (lines 30-60): ```lean lemma sum_ite_ite_eq_add_add_sum_erase_erase (f : Fin n → ℤ) (i j : Fin n) (hij : i ≠ j) (a b : ℤ) : (∑ k, (if k = i then a else if k = j then b else f k)) = a + b + ∑ k ∈ (univ.erase i).erase j, f k := by classical -- Uses add_sum_erase twice to extract i and j terms ``` - `sliceSeq_palindromic` (lines 315-363) - COMPLETED: ```lean lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : IsPalindromicOn (sliceSeq D e i j hij) (e i + e j) := by -- Uses Equiv.swap i j permutation -- Shows slice at t equals slice at q-t via symmetry ``` - `sliceSeq_logconcave` (lines 365-420) - IN PROGRESS WITH ERRORS: ```lean lemma sliceSeq_logconcave {d : ℤ} (D : WeakComposition n d → ℚ) (hlc : SatisfiesLogConcavity D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : LogConcaveOn (sliceSeq D e i j hij) (e i + e j) := by intro t ht htq set q := e i + e j with hq -- ... bounds setup ... set e0 := sliceComposition e i j hij t ht_lo ht_hi with he0 have he0_i : e0 i = t := by simp only [he0, sliceComposition, ite_true] have he0_j : e0 j = q - t := by simp only [he0, sliceCompositio ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 506): -------------------------------------------------------------------------------- Let me check the current state of the file and fix the remaining errors in `sliceSeq_logconcave`. -------------------------------------------------------------------------------- ASSISTANT (entry 507): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 508): -------------------------------------------------------------------------------- [Tool Result: 340→ fun k => by simp only [comp_apply]; exact (sliceComposition e i j hij (q - t) ht' htq').nonneg _⟩ : 341→ WeakComposition n d) = sliceComposition e i j hij t ht htq by 342→ rw [← h, hsym_app] 343→ -- Prove the compositions are equal pointwise 344→ refine WeakComposition.ext (fun k => ?_) 345→ simp only [sliceComposition, comp_apply] 346→ by_cases hki : k = i 347→ · -- k = i: σ.symm i = j, so we get the j-value from the (q-t) slice 348→ subst hk...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 510): -------------------------------------------------------------------------------- Let me run the lean compilation to see the current errors. -------------------------------------------------------------------------------- ASSISTANT (entry 511): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 512): -------------------------------------------------------------------------------- [Tool Result: /bin/bash: line 1: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 514): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 515): -------------------------------------------------------------------------------- [Tool Result: /bin/bash: line 1: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 517): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 518): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:349:26: error(lean.unknownIdentifier): Unknown identifier `i` BalancedVectors.lean:356:28: error(lean.unknownIdentifier): Unknown identifier `j` BalancedVectors.lean:407:62: error: No goals to be solved BalancedVectors.lean:409:6: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hlc : SatisfiesLogConcavity D e : WeakComposition n d i : Fin n t : ℤ ht : 1 ≤ t htm1_lo : 0 ≤ t - 1 htp1_lo : 0 ≤ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 520): -------------------------------------------------------------------------------- I see several issues. Let me read the full `sliceSeq_palindromic` section to understand the `subst` behavior. -------------------------------------------------------------------------------- ASSISTANT (entry 521): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 522): -------------------------------------------------------------------------------- [Tool Result: 315→/-- The slice sequence is palindromic when D is symmetric. -/ 316→lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) 317→ (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : 318→ IsPalindromicOn (sliceSeq D e i j hij) (e i + e j) := by 319→ intro t ht htq 320→ set q := e i + e j with hq 321→ -- Both t and q-t are in range 322→ have ht' : 0 ≤ q - t := by omega 323→ have htq' : q - t ≤ q := by omega 324→ -- Unfo...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 524): -------------------------------------------------------------------------------- I see the issues now. In `sliceSeq_palindromic`, after `subst hki` where `hki : k = i`, the variable `i` becomes unavailable (Lean 4's subst can eliminate either variable based on dependencies). Let me fix both proofs. -------------------------------------------------------------------------------- ASSISTANT (entry 525): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 526): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 340→ fun k => by simp only [comp_apply]; exact (sliceComposition e i j hij (q - t) ht' htq').nonneg _⟩ : 341→ WeakComposition n d) = sliceComposition e i j hij t ht htq by 342→ rw [← h, hsym_app] 343→ -- Prove the compositions are equal pointwise 344→ refine WeakComposition.ext (fun k => ?_) 345→ simp only [...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 528): -------------------------------------------------------------------------------- Now let me fix the `sliceSeq_logconcave` proof. Let me read the current state. -------------------------------------------------------------------------------- ASSISTANT (entry 529): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 530): -------------------------------------------------------------------------------- [Tool Result: 395→ · subst hmi; simp only [ite_true, he0_i] 396→ · by_cases hmj : m = j 397→ · subst hmj; simp only [hmi, ite_false, ite_true, he0_j]; ring 398→ · simp only [hmi, hmj, ite_false] 399→ -- e0 m = e m when m ≠ i and m ≠ j 400→ simp only [he0, sliceComposition, hmi, hmj, ite_false] 401→ have hmod_ji : e0.modify j i hj_ge1 hij.symm = sliceComposition e i j hij (t + 1) htp1_lo htp1_hi := by 402→ refine WeakComposition.ext (fun m => ?_) 403...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 532): -------------------------------------------------------------------------------- I see the issue - the same `subst` problem exists here. Let me rewrite both `hmod_ij` and `hmod_ji` to avoid using `subst`. -------------------------------------------------------------------------------- ASSISTANT (entry 533): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 534): -------------------------------------------------------------------------------- [Tool Result: 388→ -- e0.modify i j gives i ↦ t-1, j ↦ (q-t)+1 = q-(t-1) 389→ -- e0.modify j i gives i ↦ t+1, j ↦ (q-t)-1 = q-(t+1) 390→ -- Show these equal the neighboring slice compositions 391→ have hmod_ij : e0.modify i j hi_ge1 hij = sliceComposition e i j hij (t - 1) htm1_lo htm1_hi := by 392→ refine WeakComposition.ext (fun m => ?_) 393→ simp only [WeakComposition.modify, sliceComposition] 394→ by_cases hmi : m = i 395→ · subst hmi; simp only [ite_true, he0_i] ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 536): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 537): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 387→ have hlc_app := hlc e0 i j hij hi_ge1 hj_ge1 388→ -- e0.modify i j gives i ↦ t-1, j ↦ (q-t)+1 = q-(t-1) 389→ -- e0.modify j i gives i ↦ t+1, j ↦ (q-t)-1 = q-(t+1) 390→ -- Show these equal the neighboring slice compositions 391→ have hmod_ij : e0.modify i j hi_ge1 hij = sliceComposition e i j hij (t - 1) htm1_lo htm1_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 539): -------------------------------------------------------------------------------- Now let me also simplify the final `convert` block since it was producing errors. -------------------------------------------------------------------------------- ASSISTANT (entry 540): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 541): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 410→ · -- m = j: LHS is e0 j - 1 = (q-t) - 1, RHS is q - (t+1) 411→ simp only [hmi, hmj, ite_false, ite_true, he0_j]; ring 412→ · -- m ≠ i and m ≠ j: e0 m = e m 413→ simp only [hmi, hmj, ite_false, he0, sliceComposition] 414→ -- Rewrite the log-concavity inequality 415→ rw [hmod_ij, hmod_ji] at hlc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 543): -------------------------------------------------------------------------------- Now let me run the compilation to check the status. -------------------------------------------------------------------------------- ASSISTANT (entry 544): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 545): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:353:4: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 0 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q hq : q = e.toFun i + e.toFun j ht' : 0 ≤ q - t htq' : q - t ≤ q σ : Equiv.Perm (Fin n) := Equiv.swap i j hσ : σ = Equiv.swap i j hsym_app : D { toFun := (sliceComposition e i j hij (q - t) ht' htq').toFun ∘ ⇑...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 547): -------------------------------------------------------------------------------- I see the issue now. The `if` expressions with conditions like `if j = i` or `if i = j` aren't being simplified because `simp` doesn't automatically use `hij : i ≠ j` for those. I need to use `if_neg hij` and `if_neg hij.symm` to handle these conditionals. -------------------------------------------------------------------------------- ASSISTANT (entry 548): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 549): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 348→ have hswap_k : σ.symm k = j := by rw [hki, hσ, Equiv.symm_swap, Equiv.swap_apply_left] 349→ rw [hswap_k] 350→ simp only [hki, hij.symm, ite_false, ite_true] 351→ ring 352→ · by_cases hkj : k = j 353→ · -- k = j: σ.symm k = i, so we get the i-value from the (q-t) slice 354→ have hswap_k : σ.symm ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 551): -------------------------------------------------------------------------------- Now let me fix the `sliceSeq_logconcave` proof's conditional simplifications. -------------------------------------------------------------------------------- ASSISTANT (entry 552): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 553): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 387→ have hlc_app := hlc e0 i j hij hi_ge1 hj_ge1 388→ -- e0.modify i j gives i ↦ t-1, j ↦ (q-t)+1 = q-(t-1) 389→ -- e0.modify j i gives i ↦ t+1, j ↦ (q-t)-1 = q-(t+1) 390→ -- Show these equal the neighboring slice compositions 391→ have hmod_ij : e0.modify i j hi_ge1 hij = sliceComposition e i j hij (t - 1) htm1_lo htm1_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 555): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 556): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:353:4: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 0 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q hq : q = e.toFun i + e.toFun j ht' : 0 ≤ q - t htq' : q - t ≤ q σ : Equiv.Perm (Fin n) := Equiv.swap i j hσ : σ = Equiv.swap i j hsym_app : D { toFun := (sliceComposition e i j hij (q - t) ht' htq').toFun ∘ ⇑...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 558): -------------------------------------------------------------------------------- Good progress! Just two issues left: 1. Line 353 needs `ring` to close (goal is `q - t = e i + e j - t` where `q = e i + e j`) 2. Line 407 has unnecessary `ring` -------------------------------------------------------------------------------- ASSISTANT (entry 559): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 560): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 348→ have hswap_k : σ.symm k = j := by rw [hki, hσ, Equiv.symm_swap, Equiv.swap_apply_left] 349→ rw [hswap_k] 350→ simp only [hki, hij.symm, ite_false, ite_true] 351→ ring 352→ · by_cases hkj : k = j 353→ · -- k = j: σ.symm k = i, so we get the i-value from the (q-t) slice 354→ have hswap_k : σ.symm ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 562): -------------------------------------------------------------------------------- Now let me remove the unnecessary `ring` from the m = i case in hmod_ji. -------------------------------------------------------------------------------- ASSISTANT (entry 563): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 564): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 401→ simp only [hmi, hmj, ite_false, he0, sliceComposition] 402→ have hmod_ji : e0.modify j i hj_ge1 hij.symm = sliceComposition e i j hij (t + 1) htp1_lo htp1_hi := by 403→ refine WeakComposition.ext (fun m => ?_) 404→ simp only [WeakComposition.modify, sliceComposition] 405→ by_cases hmi : m = i 406→ ·...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 566): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 567): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:432:6: warning: declaration uses 'sorry' BalancedVectors.lean:439:8: warning: declaration uses 'sorry' BalancedVectors.lean:445:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 569): -------------------------------------------------------------------------------- Excellent! Both `sliceSeq_palindromic` and `sliceSeq_logconcave` now compile successfully! The only warnings are for the `sorry` declarations in proofs that still need to be completed. Let me update the todo list and check what's at line 174. -------------------------------------------------------------------------------- ASSISTANT (entry 570): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 571): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 573): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 574): -------------------------------------------------------------------------------- [Tool Result: 168→def IsPositiveOn (s : ℤ → ℚ) (q : ℤ) : Prop := 169→ ∀ t, 0 ≤ t → t ≤ q → 0 < s t 170→ 171→/-! ### Key Lemma: Log-concave palindromic sequences are unimodal -/ 172→ 173→/-- The main unimodality lemma. -/ 174→theorem unimodal_of_logconcave_palindromic {s : ℤ → ℚ} {q : ℤ} (hq : 0 ≤ q) 175→ (hpos : IsPositiveOn s q) (hlc : LogConcaveOn s q) (hpal : IsPalindromicOn s q) : 176→ (∀ t, 0 ≤ t → 2 * t < q → s t ≤ s (t + 1)) ∧ 177→ (∀ t, 2 * t > q → t ≤ q → s t ≤...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 576): -------------------------------------------------------------------------------- Now let me work on `unimodal_of_logconcave_palindromic`. This is a key mathematical lemma that requires careful proof. Let me read the current state and implement the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 577): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 578): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 580): -------------------------------------------------------------------------------- The proof of `unimodal_of_logconcave_palindromic` requires showing that log-concave palindromic positive sequences are unimodal. The key insight is: 1. Log-concavity implies the ratio r(t) = s(t+1)/s(t) is non-increasing 2. Palindromicity implies r(t) · r(q-1-t) = 1 3. Together these show r(t) ≥ 1 before midpoint and r(t) ≤ 1 after midpoint -------------------------------------------------------------------------------- ASSISTANT (entry 581): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 582): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 169→ ∀ t, 0 ≤ t → t ≤ q → 0 < s t 170→ 171→/-! ### Key Lemma: Log-concave palindromic sequences are unimodal -/ 172→ 173→/-- The main unimodality lemma. -/ 174→theorem unimodal_of_logconcave_palindromic {s : ℤ → ℚ} {q : ℤ} (hq : 0 ≤ q) 175→ (hpos : IsPositiveOn s q) (hlc : LogConcaveOn s q) (hpal : IsPalindromicOn s ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 584): -------------------------------------------------------------------------------- The `unimodal_of_logconcave_palindromic` proof requires careful handling. Let me verify the file still compiles with the current skeleton and then continue with the remaining proofs. The unimodality lemma is mathematically involved - we can fill it in later. -------------------------------------------------------------------------------- ASSISTANT (entry 585): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 586): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:484:6: warning: declaration uses 'sorry' BalancedVectors.lean:491:8: warning: declaration uses 'sorry' BalancedVectors.lean:497:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 588): -------------------------------------------------------------------------------- Good - file compiles. Let me check the remaining theorems that need proofs. -------------------------------------------------------------------------------- ASSISTANT (entry 589): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 590): -------------------------------------------------------------------------------- [Tool Result: 475→ (hpos : IsStrictlyPositive D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : 476→ IsPositiveOn (sliceSeq D e i j hij) (e i + e j) := by 477→ intro t ht htq 478→ simp only [sliceSeq, ht, htq, and_self, ↓reduceDIte] 479→ exact hpos _ 480→ 481→/-! ### Main Theorems -/ 482→ 483→/-- The balancing step weakly increases D when entries differ by ≥ 2. -/ 484→lemma balancing_increases_D {d : ℤ} (D : WeakComposition n d → ℚ) 485→ (hpos : IsStrictly...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 592): -------------------------------------------------------------------------------- Now let me implement `balancing_increases_D` using the slice unimodality. -------------------------------------------------------------------------------- ASSISTANT (entry 593): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 594): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 479→ exact hpos _ 480→ 481→/-! ### Main Theorems -/ 482→ 483→/-- The balancing step weakly increases D when entries differ by ≥ 2. -/ 484→lemma balancing_increases_D {d : ℤ} (D : WeakComposition n d → ℚ) 485→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) 486→ (e : WeakComp...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 596): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 599): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. BalancedVectors.lean:515:6: error: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hpos : IsStrictlyPositive D hsym : IsSym...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 601): -------------------------------------------------------------------------------- I see the issues. The `sliceSeq` unfolding needs to be handled differently, and the `if j = i` conditions need `if_neg hij.symm`. Let me fix the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 602): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 603): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 479→ exact hpos _ 480→ 481→/-! ### Main Theorems -/ 482→ 483→/-- The balancing step weakly increases D when entries differ by ≥ 2. -/ 484→lemma balancing_increases_D {d : ℤ} (D : WeakComposition n d → ℚ) 485→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) 486→ (e : WeakComp...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 605): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 606): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:537:2: error: Type mismatch hdecr has type (if h : True ∧ e.toFun i ≤ e.toFun i + e.toFun j then D e else 0) ≤ if h : True ∧ e.toFun i - 1 ≤ e.toFun i + e.toFun j then D (e.modify i j hi hij) else 0 but is expected to have type D e ≤ D (e.modify i j hi hij) BalancedVectors.lean:535:31: warning: This simp argument is unused: hei_hi Hint: Omit it from the simp argument list. simp only [sliceSeq, hei_l...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 608): -------------------------------------------------------------------------------- I see - the `dif_pos` needs explicit proofs. Let me fix the final step. -------------------------------------------------------------------------------- ASSISTANT (entry 609): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 610): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 524→ · simp only [hki, ite_true] 525→ · by_cases hkj : k = j 526→ · simp only [hkj, ite_true, if_neg hij.symm]; ring 527→ · simp only [hki, hkj, ite_false] 528→ -- Use the decreasing part of unimodality 529→ -- Condition: 2 * (e i) > q and e i ≤ q 530→ have h2ei_gt_q : 2 * (e i) > q := by simp only [hq...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 612): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 613): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:535:31: error: Invalid `⟨...⟩` notation: The expected type of this term could not be determined BalancedVectors.lean:535:57: error: Invalid `⟨...⟩` notation: The expected type of this term could not be determined BalancedVectors.lean:538:6: error: Tactic `rewrite` failed: motive is not type correct: fun _a => D _a ≤ D (_a.modify i j hi hij) Error: Application type mismatch: The argument hi has type 1 ≤ e.toF...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 615): -------------------------------------------------------------------------------- The `dif_pos` and rewrite don't work well here. Let me use a cleaner approach with `simp` and explicit lemmas. -------------------------------------------------------------------------------- ASSISTANT (entry 616): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 617): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 524→ · simp only [hki, ite_true] 525→ · by_cases hkj : k = j 526→ · simp only [hkj, ite_true, if_neg hij.symm]; ring 527→ · simp only [hki, hkj, ite_false] 528→ -- Use the decreasing part of unimodality 529→ -- Condition: 2 * (e i) > q and e i ≤ q 530→ have h2ei_gt_q : 2 * (e i) > q := by simp only [hq...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 619): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 620): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:535:102: error: unsolved goals n : ℕ d : ℤ D : WeakComposition n d → ℚ hpos : IsStrictlyPositive D hsym : IsSymmetric D hlc : SatisfiesLogConcavity D e : WeakComposition n d i j : Fin n hij : i ≠ j hi : 1 ≤ e.toFun i hdiff : e.toFun j + 2 ≤ e.toFun i q : ℤ := e.toFun i + e.toFun j hq_def : q = e.toFun i + e.toFun j hpal : IsPalindromicOn (sliceSeq D e i j hij) (e.toFun i + e.toFun j) hlc' : LogConcaveOn (sliceSeq ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 622): -------------------------------------------------------------------------------- I see - `simp` is simplifying `0 ≤ e.toFun i` to `True` but not closing the `dif`. Let me use explicit `dif_pos` handling. -------------------------------------------------------------------------------- ASSISTANT (entry 623): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 624): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 524→ · simp only [hki, ite_true] 525→ · by_cases hkj : k = j 526→ · simp only [hkj, ite_true, if_neg hij.symm]; ring 527→ · simp only [hki, hkj, ite_false] 528→ -- Use the decreasing part of unimodality 529→ -- Condition: 2 * (e i) > q and e i ≤ q 530→ have h2ei_gt_q : 2 * (e i) > q := by simp only [hq...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 626): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 627): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:546:8: warning: declaration uses 'sorry' BalancedVectors.lean:552:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 629): -------------------------------------------------------------------------------- Excellent! `balancing_increases_D` now compiles. Let me update the todo list and continue with `balanced_maximizes`. -------------------------------------------------------------------------------- ASSISTANT (entry 630): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 631): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 633): -------------------------------------------------------------------------------- [Tool: Grep] -------------------------------------------------------------------------------- USER (entry 634): -------------------------------------------------------------------------------- [Tool Result: 255:/-- The "imbalance" of a vector: sum of squares. -/ 256:def imbalance (e : Fin n → ℤ) : ℤ := ∑ i, (e i) ^ 2 265:/-- If e_i ≥ e_j + 2, then modifying from i to j decreases imbalance. -/ 266:lemma imbalance_decreases {d : ℤ} (e : WeakComposition n d) (i j : Fin n) 268: imbalance (e.modify i j hi hij).toFun < imbalance e.toFun := by 270: unfold imbalance 312:lemma exists_imbalanced_pair (e : Fin n → ℤ) (h : ¬ IsBalanced e) :] -------------------------------------------------------------------------------- ASSISTANT (entry 636): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 637): -------------------------------------------------------------------------------- [Tool Result: 255→/-- The "imbalance" of a vector: sum of squares. -/ 256→def imbalance (e : Fin n → ℤ) : ℤ := ∑ i, (e i) ^ 2 257→ 258→/-- Key algebraic fact: (a-1)² + (b+1)² < a² + b² when a ≥ b + 2. -/ 259→lemma sq_transfer_decreases {a b : ℤ} (h : a ≥ b + 2) : 260→ (a - 1)^2 + (b + 1)^2 < a^2 + b^2 := by 261→ have expand : (a - 1)^2 + (b + 1)^2 = a^2 + b^2 + 2*(b - a + 1) := by ring 262→ have neg : b - a + 1 ≤ -1 := by omega 263→ omega 264→ 265→/-- If e_i ≥ e_j + 2, t...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 639): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 640): -------------------------------------------------------------------------------- [Tool Result: 543→ exact hdecr 544→ 545→/-- Any vector can be transformed to a balanced vector while increasing D. -/ 546→theorem balanced_maximizes {d : ℤ} (D : WeakComposition n d → ℚ) 547→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 548→ ∀ e : WeakComposition n d, ∃ e' : WeakComposition n d, IsBalanced e'.toFun ∧ D e ≤ D e' := by 549→ sorry 550→ 551→/-- Any vector can be transformed to a concentrated vector while decreasing D. -/...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 642): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 643): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 541→ rw [dif_pos ⟨hem1_lo, hem1_hi⟩] 542→ rw [hslice_ei, hslice_em1, ← he_eq_slice, ← hmod_eq_slice] at hdecr 543→ exact hdecr 544→ 545→/-- Imbalance is non-negative for weak compositions. -/ 546→lemma imbalance_nonneg {d : ℤ} (e : WeakComposition n d) : 0 ≤ imbalance e.toFun := by 547→ unfold imbalance 548→ ap...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 645): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 646): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:562:2: error: Invalid alternative name `ind`: Expected `h` BalancedVectors.lean:561:2: error: Alternative `h` has not been provided BalancedVectors.lean:585:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 648): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 649): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 557→ intro e 558→ -- Convert imbalance to ℕ for well-founded recursion 559→ have h_nonneg : 0 ≤ imbalance e.toFun := imbalance_nonneg e 560→ generalize hm : (imbalance e.toFun).toNat = m 561→ induction m using Nat.strong_induction_on generalizing e with 562→ | _ m ih => 563→ by_cases hbal : IsBalanced e.toFun ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 651): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 652): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:580:14: error: Function expected at Int.toNat_lt_toNat ?m.161 but this term has type Int.toNat ?m.159 < Int.toNat ?m.160 ↔ ?m.159 < ?m.160 Note: Expected a function because this term is being applied to the argument h_nonneg BalancedVectors.lean:580:33: error: Application type mismatch: The argument himb_decr has type imbalance e1.toFun < imbalance e.toFun but is expected to have type 0 < ?m.160 in th...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 654): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 655): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 574→ have himb_decr : imbalance e1.toFun < imbalance e.toFun := 575→ imbalance_decreases e i j hij hi hdiff 576→ -- Apply induction hypothesis 577→ have h1_nonneg : 0 ≤ imbalance e1.toFun := imbalance_nonneg e1 578→ have hm1 : (imbalance e1.toFun).toNat < m := by 579→ rw [← hm] 580→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 657): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 658): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:580:31: error: Application type mismatch: The argument h_nonneg has type 0 ≤ imbalance e.toFun but is expected to have type 0 < imbalance e.toFun in the application Int.toNat_lt_toNat h_nonneg BalancedVectors.lean:586:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 660): -------------------------------------------------------------------------------- I need `0 < imbalance e.toFun` for the lemma. Since e is not balanced, imbalance is actually positive. -------------------------------------------------------------------------------- ASSISTANT (entry 661): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 662): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 572→ have hD_incr : D e ≤ D e1 := balancing_increases_D D hpos hsym hlc e i j hij hi hdiff 573→ -- Imbalance decreases 574→ have himb_decr : imbalance e1.toFun < imbalance e.toFun := 575→ imbalance_decreases e i j hij hi hdiff 576→ -- Apply induction hypothesis 577→ have h1_nonneg : 0 ≤ imbal...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 664): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 665): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:580:33: error: omega could not prove the goal: a possible counterexample may satisfy the constraints h ≥ 0 g ≥ 0 g - h ≤ -1 f ≥ 0 f - g ≥ 1 f - h ≤ -1 c ≥ 0 b ≤ -1 a ≥ 1 a - c ≥ 1 where a := ↑m b := e.toFun j c := imbalance e1.toFun f := ↑↑j g := ↑↑i h := ↑n BalancedVectors.lean:582:63: error: unsolved goals n : ℕ d : ℤ D : WeakComposition n d → ℚ hpos : IsStrictlyPositive D hsym : IsSymm...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 667): -------------------------------------------------------------------------------- I need to use the non-negativity of weak composition entries. Let me fix the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 668): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 669): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 574→ have himb_decr : imbalance e1.toFun < imbalance e.toFun := 575→ imbalance_decreases e i j hij hi hdiff 576→ -- Apply induction hypothesis 577→ have h1_nonneg : 0 ≤ imbalance e1.toFun := imbalance_nonneg e1 578→ -- Need to show imbalance e > 0 (since e is not balanced) 579→ have h_pos : 0...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 671): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 672): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:584:10: error: Tactic `apply` failed: could not unify the conclusion of `@single_le_sum` ?f ?a ≤ ∑ x ∈ ?s, ?f x with the goal e.toFun i ^ 2 ≤ ∑ k, e.toFun k ^ 2 Note: The full type of `@single_le_sum` is ∀ {ι : Type ?u.20484} {N : Type ?u.20483} [inst : AddCommMonoid N] [inst_1 : PartialOrder N] {f : ι → N} {s : Finset ι} [AddLeftMono N], (∀ i ∈ s, 0 ≤ f i) → ∀ {a : ι}, a ∈ s → f a ≤ ∑ x ∈ s, f x n : ℕ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 674): -------------------------------------------------------------------------------- I need to explicitly use `Finset.univ`. Let me fix this. -------------------------------------------------------------------------------- ASSISTANT (entry 675): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 676): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 574→ have himb_decr : imbalance e1.toFun < imbalance e.toFun := 575→ imbalance_decreases e i j hij hi hdiff 576→ -- Apply induction hypothesis 577→ have h1_nonneg : 0 ≤ imbalance e1.toFun := imbalance_nonneg e1 578→ -- Need to show imbalance e > 0 (since e is not balanced) 579→ have h_pos : 0...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 678): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 679): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:585:37: error: Application type mismatch: The argument mem_univ i has type i ∈ univ of sort `Prop` but is expected to have type Fin n of sort `Type` in the application @single_le_sum (Fin n) ℤ Int.instAddCommMonoid instLatticeInt.toSemilatticeInf.toPartialOrder (fun k => e.toFun k ^ 2) univ ?m.243 (fun k x => sq_nonneg (e.toFun k)) (mem_univ i) BalancedVectors.lean:586:28: error(lean.unknownIdentifier)...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 681): -------------------------------------------------------------------------------- Let me simplify this proof using a cleaner approach. -------------------------------------------------------------------------------- ASSISTANT (entry 682): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 683): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 574→ have himb_decr : imbalance e1.toFun < imbalance e.toFun := 575→ imbalance_decreases e i j hij hi hdiff 576→ -- Apply induction hypothesis 577→ have h1_nonneg : 0 ≤ imbalance e1.toFun := imbalance_nonneg e1 578→ -- Need to show imbalance e > 0 (since e is not balanced) 579→ have h_pos : 0...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 685): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 688): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:596:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 690): -------------------------------------------------------------------------------- Excellent! `balanced_maximizes` now compiles! Let me update the todo list, create a git commit to save progress, and then continue with `concentrated_minimizes`. -------------------------------------------------------------------------------- ASSISTANT (entry 691): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 692): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 694): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 695): -------------------------------------------------------------------------------- [Tool Result: On branch master Your branch is up to date with 'origin/master'. Untracked files: (use "git add ..." to include in what will be committed) BalancedVectors.lean nothing added to commit but untracked files present (use "git add" to track)] -------------------------------------------------------------------------------- ASSISTANT (entry 697): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 698): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors 775c29d25e] wip: Balanced vectors formalization progress 1 file changed, 600 insertions(+) create mode 100644 BalancedVectors.lean Switched to a new branch 'balanced_vectors'] -------------------------------------------------------------------------------- ASSISTANT (entry 700): -------------------------------------------------------------------------------- Good, progress saved on branch `balanced_vectors`. Now let me implement `concentrated_minimizes`. Let me first check the `IsConcentrated` definition. -------------------------------------------------------------------------------- ASSISTANT (entry 701): -------------------------------------------------------------------------------- [Tool: Grep] -------------------------------------------------------------------------------- USER (entry 702): -------------------------------------------------------------------------------- [Tool Result: 152- 153-/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 154:def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 155- ∃ k, ∀ i, e i = if i = k then d else 0 156- 157-/-! ### Log-Concave and Palindromic Sequences -/ 158- 159-/-- A sequence `s : ℤ → ℚ` is log-concave on `[0, q]`. -/ -- 597- (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 598- ∀ e : WeakComposition n d, ∃ e' : WeakComposition n d, 599: IsConcentrated d e'.toFun...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 704): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 705): -------------------------------------------------------------------------------- [Tool Result: 593→ exact ⟨e', hbal', le_trans hD_incr hD'⟩ 594→ 595→/-- Any vector can be transformed to a concentrated vector while decreasing D. -/ 596→theorem concentrated_minimizes {d : ℤ} (hn : 0 < n) (hd : 0 ≤ d) (D : WeakComposition n d → ℚ) 597→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 598→ ∀ e : WeakComposition n d, ∃ e' : WeakComposition n d, 599→ IsConcentrated d e'.toFun ∧ D e' ≤ D e := by 600→ sorry 601→ e_j, moving mass from j to i (concentrating) decreases D. -/ 596→lemma concentrating_decreases_D {d : ℤ} (D : WeakComposition n d → ℚ) ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 711): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 712): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:677:24: error: Application type mismatch: The argument Ne.symm hik has type k ≠ i but is expected to have type i ≠ k in the application hcontra i k (Ne.symm hik) BalancedVectors.lean:686:24: error(lean.unknownIdentifier): Unknown identifier `k` BalancedVectors.lean:686:52: error(lean.unknownIdentifier): Unknown identifier `k` BalancedVectors.lean:687:52: error(lean.unknownIdentifier): Unknown identifier `k...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 714): -------------------------------------------------------------------------------- Let me fix these errors step by step. -------------------------------------------------------------------------------- ASSISTANT (entry 715): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 716): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 656→ -- Not concentrated means at least two indices have positive values 657→ by_contra hcontra 658→ push_neg at hcontra 659→ -- All positive entries are at most one 660→ have hone : ∃ k, ∀ i, i ≠ k → e i = 0 := by 661→ by_cases hall : ∀ i, e i = 0 662→ · -- All zero contradicts d > 0 663→ have hsum_eq...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 718): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 719): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 699→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 700→ ∀ e : WeakComposition n d, ∃ e' : WeakComposition n d, 701→ IsConcentrated d e'.toFun ∧ D e' ≤ D e := by 702→ intro e 703→ -- Handle d = 0 case: only composition is all zeros 704→ by_cases hd0 : d = 0 705→ · us...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 721): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 722): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 725→ · -- e is not concentrated: find two positive entries and concentrate 726→ obtain ⟨i, j, hij, hi_pos, hj_pos⟩ := exists_two_positive hd_pos e hconc 727→ -- Move mass from the smaller to the larger 728→ by_cases hdiff : e j < e i 729→ · -- e.j < e.i: modify j → i 730→ have hj_ge1 : 1 ≤ e j ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 724): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 725): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 739→ -- For now, we use a sorry 740→ sorry 741→ obtain ⟨e', hconc', hD'⟩ := ih (nonzeroCount e1.toFun) hm1 e1 rfl 742→ exact ⟨e', hconc', le_trans hD' hD_decr⟩ 743→ · -- e.j ≥ e.i: need to handle the symmetric case 744→ push_neg at hdiff 745→ by_cases heq : e j = e i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 727): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 728): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:677:12: error: Function expected at hcontra i k hik hpos_i but this term has type e.toFun k ≤ 0 Note: Expected a function because this term is being applied to the argument hpos_k BalancedVectors.lean:686:31: error(lean.unknownIdentifier): Unknown identifier `k` BalancedVectors.lean:686:59: error(lean.unknownIdentifier): Unknown identifier `k` BalancedVectors.lean:687:52: error(lean.unknownIdentifier): Unkn...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 730): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 731): -------------------------------------------------------------------------------- [Tool Result: 655→ -- If concentrated, all mass is on one index 656→ -- Not concentrated means at least two indices have positive values 657→ by_contra hcontra 658→ push_neg at hcontra 659→ -- All positive entries are at most one 660→ have hone : ∃ k, ∀ i, i ≠ k → e i = 0 := by 661→ by_cases hall : ∀ i, e i = 0 662→ · -- All zero contradicts d > 0 663→ have hsum_eq : d = ∑ i, e i := e.sum_eq.symm 664→ simp only [hall, Finset.sum_const_zero] at hsum_eq 665...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 733): -------------------------------------------------------------------------------- The issue is with variable shadowing and how `hcontra` is used. Let me fix the `exists_two_positive` lemma. -------------------------------------------------------------------------------- ASSISTANT (entry 734): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 735): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 649→/-- Count of non-zero entries. -/ 650→def nonzeroCount (e : Fin n → ℤ) : ℕ := (Finset.univ.filter (fun i => e i ≠ 0)).card 651→ 652→/-- If e is not concentrated and d > 0, there exist distinct i, j with e.i > 0 and e.j > 0. -/ 653→lemma exists_two_positive {d : ℤ} (hd : 0 < d) (e : WeakComposition n d) 654→ (h : ¬ Is...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 737): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 738): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:685:31: error(lean.unknownIdentifier): Unknown identifier `k₀` BalancedVectors.lean:685:60: error(lean.unknownIdentifier): Unknown identifier `k₀` BalancedVectors.lean:686:52: error(lean.unknownIdentifier): Unknown identifier `k₀` BalancedVectors.lean:688:41: error(lean.unknownIdentifier): Unknown identifier `k₀` BalancedVectors.lean:697:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 740): -------------------------------------------------------------------------------- The `subst` tactic is causing issues with variable scoping. Let me avoid using `subst` and restructure the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 741): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 742): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 673→ have hpos_k₀ : 0 < e k₀ := by have := e.nonneg k₀; omega 674→ have hpos_i : 0 < e i := by have := e.nonneg i; omega 675→ have hcontr_app := hcontra i k₀ hik hpos_i 676→ omega 677→ obtain ⟨k₀, hk₀⟩ := hone 678→ apply h 679→ use k₀ 680→ intro i 681→ by_cases hik : i = k₀ 682→ · -- i ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 744): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 745): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:174:8: warning: declaration uses 'sorry' BalancedVectors.lean:697:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 747): -------------------------------------------------------------------------------- The file now compiles. Let me commit this progress and give you a status update. -------------------------------------------------------------------------------- ASSISTANT (entry 748): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 749): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors 6db876d018] wip: Add concentrated_minimizes and helper lemmas 1 file changed, 169 insertions(+), 1 deletion(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 751): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 752): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 754): -------------------------------------------------------------------------------- **Status Update** I've made significant progress on the formalization. Here's the current state: **Completed proofs (no sorry):** 1. `sliceSeq_palindromic` - using swap permutation 2. `sliceSeq_logconcave` - using modify identification 3. `balancing_increases_D` - using slice unimodality 4. `balanced_maximizes` - using well-founded induction on imbalance 5. `concentrating_decreases_D` - lemma showing concentration decreases D 6. `exists_two_positive` - helper for concentration 7. `imbalance_nonneg` - helper for balanced_maximizes **Proofs with sorry (structure complete):** 1. **`unimodal_of_logconcave_palindromic`** (line 174) - The core mathematical lemma stating that log-concave palindromic sequences are unimodal. This is the mathematical heart of the proof and requires showing: - The ratio r(t) = s(t+1)/s(t) is non-increasing (from log-concavity) - r(t) · r(q-1-t) = 1 (from palindromicity) - Together these give unimodality 2. **`concentrated_minimizes`** (line 697) - Has several internal `sorry`s: - `nonzeroCount` bounds after modify operations - D decrease in the equal case (e.j = e.i) **Git progress saved on branch `balanced_vectors`:** - Commit 1: Core structure and `balanced_maximizes` - Commit 2: `concentrated_minimizes` and helpers The main remaining mathematical work is proving `unimodal_of_logconcave_palindromic`. Would you like me to continue working on that, or would GPT 5.2 input be helpful for the proof strategy? -------------------------------------------------------------------------------- USER (entry 758): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me chronologically analyze the conversation: 1. **Initial Context**: This is a continuation from a previous session about formalizing a theorem in Lean 4 / Mathlib4 about balanced/concentrated vectors achieving max/min of symmetric log-concave functions on weak compositions. 2. **Starting Point**: The file `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` had compilation errors related to variable scoping issues with `subst` and `ext` tactics. 3. **First Major Fix - sliceSeq_palindromic**: - Errors at lines 349, 356: "Unknown identifier `i`" and "Unknown identifier `j`" - Problem: After `subst hki` where `hki : k = i`, the variable was being eliminated - Fix: Changed from using `subst` to working with variables directly using `rw [hki, ...]` 4. **Second Major Fix - sliceSeq_logconcave**: - Multiple errors with `subst`, `if_neg`, and variable scoping - Problem: `simp` wasn't simplifying `if j = i` conditions - Fix: Added `if_neg hij.symm` and `if_neg hij` to handle the conditionals - Also removed unnecessary `ring` calls where `simp` already closed goals 5. **Third Major Fix - balancing_increases_D**: - Issues with `sliceSeq` unfolding and `dif_pos` handling - Problem: The `sliceSeq` definition uses `dif` which wasn't simplifying correctly - Fix: Used explicit `unfold sliceSeq; rw [dif_pos ⟨..., ...⟩]` pattern 6. **Fourth Major Fix - balanced_maximizes**: - Needed well-founded induction on imbalance - Problem: `Nat.strong_induction_on` had different alternative name than expected - Fix: Changed `| ind m ih =>` to `| _ m ih =>` - Also needed to prove `0 < imbalance e.toFun` (not just `0 ≤`) - Used a calculation showing imbalance ≥ 4 > 0 when e is not balanced 7. **Fifth Major Work - concentrated_minimizes**: - Created helper lemmas: `concentrating_decreases_D`, `nonzeroCount`, `exists_two_positive` - Multiple variable scoping issues with `k`, `k₀` - Fixed by renaming variables and avoiding `subst` - Still has `sorry` placeholders for nonzeroCount bounds 8. **User Requests**: - Initial: Continue working on the formalization - Mid-session: Asked for status updates - Suggested saving Lean output to temp files - Asked to create git commits on branch `balanced_vectors` 9. **Git Commits Made**: - Commit 1: Core structure with `balanced_maximizes` - Commit 2: `concentrated_minimizes` and helpers 10. **Current State**: - File compiles with only `sorry` warnings - Two main remaining `sorry` blocks: - `unimodal_of_logconcave_palindromic` (core mathematical lemma) - `concentrated_minimizes` (internal sorries for nonzeroCount bounds) Summary: 1. Primary Request and Intent: The user (a Lean beginner with mathematical expertise) wants to formalize a theorem in Lean 4 / Mathlib4 about symmetric log-concave functions on weak compositions. The theorem states: for D: E → ℚ₊ that is Sₙ-invariant and satisfies D(e)² ≥ D(e-δᵢ+δⱼ)·D(e+δᵢ-δⱼ), prove D is maximized on balanced vectors (|eᵢ-eⱼ| ≤ 1) and minimized on concentrated vectors (e = d·δₖ). The user wants iterative development, getting proofs compiling before filling in all details. User also requested creating git commits to save progress. 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Weak compositions: vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - Log-concave sequences: s(t)² ≥ s(t-1)·s(t+1) - Palindromic sequences: s(t) = s(q-t) - Sₙ-invariance (permutation symmetry) - Slice analysis: fixing two indices i,j and varying allocation - `Equiv.swap` permutation for palindromicity proofs - `WeakComposition.ext` for proving composition equality - `subst` tactic variable elimination issues in Lean 4 - `if_neg` for conditional simplification - `dif_pos` for dependent if-then-else - Well-founded induction using `Nat.strong_induction_on` - Imbalance measure: sum of squares ∑(eᵢ)² 3. Files and Code Sections: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - Main formalization file (~770 lines) **Key Completed Sections:** - `sliceSeq_palindromic` (lines 315-362) - Uses swap permutation: ```lean lemma sliceSeq_palindromic {d : ℤ} (D : WeakComposition n d → ℚ) (hsym : IsSymmetric D) (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) : IsPalindromicOn (sliceSeq D e i j hij) (e i + e j) := by -- Key pattern: avoid subst, use rw [hki, ...] instead have hswap_k : σ.symm k = j := by rw [hki, hσ, Equiv.symm_swap, Equiv.swap_apply_left] ``` - `sliceSeq_logconcave` (lines 363-420) - Uses modify identification: ```lean -- Key pattern: use if_neg for conditional simplification simp only [hmj, ite_true, if_neg hij.symm, he0_j]; ring ``` - `balancing_increases_D` (lines 483-543) - Uses slice unimo ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- USER (entry 763): -------------------------------------------------------------------------------- Welcome back! The last status update you gave me is: ● Status Update I've made significant progress on the formalization. Here's the current state: Completed proofs (no sorry): 1. sliceSeq_palindromic - using swap permutation 2. sliceSeq_logconcave - using modify identification 3. balancing_increases_D - using slice unimodality 4. balanced_maximizes - using well-founded induction on imbalance 5. concentrating_decreases_D - lemma showing concentration decreases D 6. exists_two_positive - helper for concentration 7. imbalance_nonneg - helper for balanced_maximizes Proofs with sorry (structure complete): 1. unimodal_of_logconcave_palindromic (line 174) - The core mathematical lemma stating that log-concave palindromic sequences are unimodal. This is the mathematical heart of the proof and requires showing: - The ratio r(t) = s(t+1)/s(t) is non-increasing (from log-concavity) - r(t) · r(q-1-t) = 1 (from palindromicity) - Together these give unimodality 2. concentrated_minimizes (line 697) - Has several internal sorrys: - nonzeroCount bounds after modify operations - D decrease in the equal case (e.j = e.i) Git progress saved on branch balanced_vectors: - Commit 1: Core structure and balanced_maximizes - Commit 2: concentrated_minimizes and helpers The main remaining mathematical work is proving unimodal_of_logconcave_palindromic. Would you like me to continue working on that, or would GPT 5.2 input be helpful for the proof strategy? Here are some proposals by GPT-5.2 how to continue: Here’s a **drop-in replacement** for your `unimodal_of_logconcave_palindromic` proof that follows the “ratio is antitone + palindromic reciprocal” strategy. I wrote it in a way that should fit your current definitions (`LogConcaveOn`, `IsPositiveOn`, `IsPalindromicOn`) and tactics (`omega`, `ring`, `simp`). Paste this in place of the current theorem body (starting at `theorem unimodal_of_logconcave_palindromic ... := by`). ```lean /-- The main unimodality lemma. -/ theorem unimodal_of_logconcave_palindromic {s : ℤ → ℚ} {q : ℤ} (hq : 0 ≤ q) (hpos : IsPositiveOn s q) (hlc : LogConcaveOn s q) (hpal : IsPalindromicOn s q) : (∀ t, 0 ≤ t → 2 * t < q → s t ≤ s (t + 1)) ∧ (∀ t, 2 * t > q → t ≤ q → s t ≤ s (t - 1)) := by classical -- Ratio r(t) = s(t+1)/s(t) let r : ℤ → ℚ := fun t => s (t + 1) / s t have hs_pos : ∀ t, 0 ≤ t → t ≤ q → 0 < s t := by intro t ht0 htq exact hpos t ht0 htq have hr_pos : ∀ t, 0 ≤ t → t + 1 ≤ q → 0 < r t := by intro t ht0 ht1q have hst : 0 < s t := hs_pos t ht0 (by omega) have hst1 : 0 < s (t + 1) := hs_pos (t + 1) (by omega) (by omega) simp [r, div_pos hst1 hst] -- From log-concavity: r(t+1) ≤ r(t) on the valid range. have hr_step : ∀ t, 0 ≤ t → t + 2 ≤ q → r (t + 1) ≤ r t := by intro t ht0 ht2q -- apply log-concavity at (t+1) have ht1 : 1 ≤ t + 1 := by omega have ht1q : t + 1 ≤ q - 1 := by omega have hlc' := hlc (t + 1) ht1 ht1q -- positivity for denominators have hst : 0 < s t := hs_pos t ht0 (by omega) have hst1 : 0 < s (t + 1) := hs_pos (t + 1) (by omega) (by omega) -- convert hlc' into ratio inequality via `div_le_div_iff` have hmul : s (t + 2) * s t ≤ s (t + 1) * s (t + 1) := by -- hlc' is: s t * s(t+2) ≤ s(t+1)^2 (because it was written with ≥) have hlc'' : s t * s (t + 2) ≤ s (t + 1) ^ 2 := by simpa using hlc' calc s (t + 2) * s t = s t * s (t + 2) := by simp [mul_comm] _ ≤ s (t + 1) ^ 2 := hlc'' _ = s (t + 1) * s (t + 1) := by simp [pow_two] have hdiv : s (t + 2) / s (t + 1) ≤ s (t + 1) / s t := by -- a/b ≤ c/d <-> a*d ≤ c*b (with b,d>0) refine (div_le_div_iff hst1 hst).2 ?_ -- need: s(t+2)*s(t) ≤ s(t+1)*s(t+1) simpa [mul_comm, mul_left_comm, mul_assoc] using hmul simpa [r] using hdiv -- Antitone version of the ratio along integer steps (uses Int.le_induction). have hr_antitone : ∀ {a b : ℤ}, 0 ≤ a → b + 1 ≤ q → a ≤ b → r b ≤ r a := by intro a b ha0 hbq hab -- P(k): r k ≤ r a let P : ℤ → Prop := fun k => r k ≤ r a have hbase : P a := by simp [P] have hstep : ∀ k, a ≤ k → k < b → P k → P (k + 1) := by intro k hak hkb hkP have hk0 : 0 ≤ k := le_trans ha0 hak have hk2q : k + 2 ≤ q := by have : k + 2 ≤ b + 1 := by omega exact le_trans this hbq have hrat : r (k + 1) ≤ r k := by -- hr_step is stated as r(t+1) ≤ r(t); here t=k simpa [add_assoc] using (hr_step k hk0 hk2q) -- r(k+1) ≤ r(k) ≤ r(a) exact le_trans hrat hkP -- NOTE: if the arguments to Int.le_induction don’t match on your Mathlib version, -- swap the order to: `exact Int.le_induction hab hbase hstep`. have : P b := by exact Int.le_induction (m := a) (n := b) hbase hstep hab simpa [P] using this -- Palindromicity gives the reciprocal relation: r(q-1-t) = (r t)⁻¹. have hr_pal : ∀ t, 0 ≤ t → t + 1 ≤ q → r (q - 1 - t) = (r t)⁻¹ := by intro t ht0 ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 765): -------------------------------------------------------------------------------- Thank you for the kind words and for the GPT-5.2 suggestions! The ratio-based proof strategy for `unimodal_of_logconcave_palindromic` looks solid. Let me read the current state of the file and implement these proofs. -------------------------------------------------------------------------------- ASSISTANT (entry 766): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 767): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 769): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 770): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 772): -------------------------------------------------------------------------------- Let me check what's available in Mathlib4 for integer induction and the ratio inequality lemmas. -------------------------------------------------------------------------------- ASSISTANT (entry 773): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 774): -------------------------------------------------------------------------------- [Tool Result: /bin/bash: line 29: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 776): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 777): -------------------------------------------------------------------------------- [Tool Result: /bin/bash: line 29: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 779): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 780): -------------------------------------------------------------------------------- [Tool Result: Exit code 127 /bin/bash: line 29: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 782): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 783): -------------------------------------------------------------------------------- [Tool Result: Exit code 127 /bin/bash: line 29: lake: command not found] -------------------------------------------------------------------------------- ASSISTANT (entry 785): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 788): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b8267f5. Output is being written to: /tmp/claude/tasks/b8267f5.output] -------------------------------------------------------------------------------- ASSISTANT (entry 790): -------------------------------------------------------------------------------- Good idea! Let me create a CLAUDE.md with best practices for this project. -------------------------------------------------------------------------------- ASSISTANT (entry 791): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 793): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/mathlib4/CLAUDE.md] -------------------------------------------------------------------------------- ASSISTANT (entry 795): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 798): -------------------------------------------------------------------------------- [Tool Result: success b8267f5 local_bash failed 1 @Int.le_induction : ∀ {m : ℤ} {motive : (n : ℤ) → m ≤ n → Prop}, motive m ⋯ → (∀ (n : ℤ) (hmn : m ≤ n), motive n hmn → motive (n + 1) ⋯) → ∀ (n : ℤ) (hmn : m ≤ n), motive n hmn /tmp/check_lemmas.lean:6:8: error(lean.unknownIdentifier): Unknown identifier `one_le_sq_iff` /tmp/check_lemmas.lean:8:8: error(lean.unknownIdentifier):...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 800): -------------------------------------------------------------------------------- Let me search for the correct lemmas in Mathlib4. -------------------------------------------------------------------------------- ASSISTANT (entry 801): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 802): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: bd2afee. Output is being written to: /tmp/claude/tasks/bd2afee.output] -------------------------------------------------------------------------------- ASSISTANT (entry 804): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 807): -------------------------------------------------------------------------------- [Tool Result: success bd2afee local_bash failed 1 /tmp/check_lemmas2.lean:4:8: error(lean.unknownIdentifier): Unknown identifier `sq_le_one'` /tmp/check_lemmas2.lean:5:8: error(lean.unknownIdentifier): Unknown identifier `one_le_sq` @sq_nonneg : ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [PosMulMono R] [AddLeftMono R] (a : R), 0 ≤ a ^ 2 /tm...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 809): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 812): -------------------------------------------------------------------------------- [Tool Result: Try this: [apply] exact div_le_div_iff₀ hb hd /tmp/check_lemmas3.lean:8:2: error: `exact?` could not close the goal. Try `apply?` to see partial suggestions.] -------------------------------------------------------------------------------- ASSISTANT (entry 814): -------------------------------------------------------------------------------- Let me try a more direct approach. I found `div_le_div_iff₀` for the division lemma. Let me write a complete proof. -------------------------------------------------------------------------------- ASSISTANT (entry 815): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 816): -------------------------------------------------------------------------------- [Tool Result: /tmp/test_unimodal.lean:67:9: error: invalid 'calc' step, left-hand side is s (t + 2) * s t : Rat but is expected to be s (t + 1 + 1) * s t : Rat /tmp/test_unimodal.lean:78:4: error: Invalid target: hab has type a ≤ b of sort `Prop` but is expected to have type ℤ of sort `Type` /tmp/test_unimodal.lean:99:4: error: No goals to be solved /tmp/test_unimodal.lean:122:8: error(lean.unknownIdentifier): Unknown identifier `one_le_div_iff₀`] -------------------------------------------------------------------------------- ASSISTANT (entry 818): -------------------------------------------------------------------------------- Let me fix those errors. -------------------------------------------------------------------------------- ASSISTANT (entry 819): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 820): -------------------------------------------------------------------------------- [Tool Result: /tmp/test_unimodal.lean:80:6: error: Invalid target: han has type a ≤ n of sort `Prop` but is expected to have type ℤ of sort `Type` /tmp/test_unimodal.lean:103:25: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern s (q - (t + 1)) in the target expression s t / s (q - 1 - t) = (s (t + 1) / s t)⁻¹ s : ℤ → ℚ q : ℤ hq : 0 ≤ q hpos : IsPositiveOn s q hlc : LogConcaveOn s q hpal : IsPalindromicOn s q r : ℤ → ℚ := fun t => s (t + 1) / s t hs_pos : ∀ (t : ℤ), 0 ≤ t → ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 822): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 823): -------------------------------------------------------------------------------- [Tool Result: /tmp/test_unimodal.lean:78:4: error: failed to create binder due to failure when reverting variable dependencies] -------------------------------------------------------------------------------- USER (entry 824): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 826): -------------------------------------------------------------------------------- Note: it's ok if you want to create temporary files in the current folder to debug any lemmas you might need, then it might be easier to fix errors by modifying the lines instead of writing the whole code from scratch every time. But as you prefer! -------------------------------------------------------------------------------- ASSISTANT (entry 828): -------------------------------------------------------------------------------- Great idea! Let me create a temporary file in the mathlib4 folder to debug this more easily. -------------------------------------------------------------------------------- ASSISTANT (entry 829): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 831): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/mathlib4/TempUnimodal.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 833): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 834): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 836): -------------------------------------------------------------------------------- No errors. Let me verify the output file is empty (meaning successful compilation). -------------------------------------------------------------------------------- ASSISTANT (entry 837): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 838): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 840): -------------------------------------------------------------------------------- The proof compiles. Now let me update the main file with this working proof. -------------------------------------------------------------------------------- ASSISTANT (entry 841): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 842): -------------------------------------------------------------------------------- [Tool Result: 160→def LogConcaveOn (s : ℤ → ℚ) (q : ℤ) : Prop := 161→ ∀ t, 1 ≤ t → t ≤ q - 1 → s t ^ 2 ≥ s (t - 1) * s (t + 1) 162→ 163→/-- A sequence is palindromic on `[0, q]`. -/ 164→def IsPalindromicOn (s : ℤ → ℚ) (q : ℤ) : Prop := 165→ ∀ t, 0 ≤ t → t ≤ q → s t = s (q - t) 166→ 167→/-- A sequence is positive on `[0, q]`. -/ 168→def IsPositiveOn (s : ℤ → ℚ) (q : ℤ) : Prop := 169→ ∀ t, 0 ≤ t → t ≤ q → 0 < s t 170→ 171→/-! ### Key Lemma: Log-concave palindromic sequence...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 844): -------------------------------------------------------------------------------- Now I need to replace the proof in the main file. Let me add the helper lemmas and replace the theorem. -------------------------------------------------------------------------------- ASSISTANT (entry 845): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 846): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 167→/-- A sequence is positive on `[0, q]`. -/ 168→def IsPositiveOn (s : ℤ → ℚ) (q : ℤ) : Prop := 169→ ∀ t, 0 ≤ t → t ≤ q → 0 < s t 170→ 171→/-! ### Key Lemma: Log-concave palindromic sequences are unimodal -/ 172→ 173→-- Helper: 1 ≤ x² and x > 0 implies 1 ≤ x 174→private lemma one_le_of_one_le_sq_pos {x : ℚ} (hx : 0...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 848): -------------------------------------------------------------------------------- Now let me compile the main file to check it works. -------------------------------------------------------------------------------- ASSISTANT (entry 849): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 850): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:758:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 852): -------------------------------------------------------------------------------- Excellent! Only one sorry left in `concentrated_minimizes`. Let me update my todo list and check what's remaining. -------------------------------------------------------------------------------- ASSISTANT (entry 853): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 854): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 856): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 857): -------------------------------------------------------------------------------- [Tool Result: 740→ use k₀ 741→ intro i 742→ by_cases hik : i = k₀ 743→ · -- i = k₀: show e i = d 744→ simp only [hik, ite_true] 745→ have hsum : d = ∑ m, e m := e.sum_eq.symm 746→ have hsplit : ∑ m, e m = e k₀ + ∑ m ∈ Finset.univ.erase k₀, e m := by 747→ rw [Finset.add_sum_erase _ _ (Finset.mem_univ k₀)] 748→ rw [hsplit] at hsum 749→ have hrest : ∑ m ∈ Finset.univ.erase k₀, e m = 0 := by 750→ apply Finset.sum_eq_zero 751→ intro m hm 752→...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 862): -------------------------------------------------------------------------------- I see the issues. The main challenges are: 1. `nonzeroCount` doesn't always decrease (only when e.j = 1) 2. The equal case (e.j = e.i) needs special handling since `concentrating_decreases_D` requires strict inequality Let me follow GPT-5.2's suggestion and restructure using `maxEntry` as the measure, and also handle the equal case properly. First let me create a temp file to work on this. -------------------------------------------------------------------------------- USER (entry 865): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 867): -------------------------------------------------------------------------------- Amazing! Feel free to also create another git commit. Here is some further input from GPT 5.2: Below is a **Lean sketch** for the “use `maxEntry` as the termination measure” approach (so you can delete the `nonzeroCount`-based sorries). It’s written to match your file’s conventions (`Fin n → ℤ`, `WeakComposition`, `modify`, `IsConcentrated`). You can paste this as a new section (e.g. right before `concentrated_minimizes`) and then refactor the proof accordingly. --- ## 1) Define `maxEntry` and basic lemmas ```lean /-! ### Concentration measure via `maxEntry` -/ open Finset BigOperators variable {n : ℕ} noncomputable def maxEntry (e : Fin n → ℤ) (hn : 0 < n) : ℤ := ( (Finset.univ : Finset (Fin n)).image e ).max' (by classical -- nonempty because `n>0`, so `⟨0,hn⟩ : Fin n` exists refine ⟨e ⟨0, hn⟩, ?_⟩ refine Finset.mem_image.2 ?_ refine ⟨⟨0, hn⟩, by simp, rfl⟩) lemma exists_eq_maxEntry (e : Fin n → ℤ) (hn : 0 < n) : ∃ i : Fin n, e i = maxEntry e hn := by classical -- `maxEntry e hn` is in the image, so comes from some index unfold maxEntry set s : Finset ℤ := (Finset.univ : Finset (Fin n)).image e have hs : s.Nonempty := by refine ⟨e ⟨0, hn⟩, ?_⟩ refine Finset.mem_image.2 ?_ refine ⟨⟨0, hn⟩, by simp, rfl⟩ have hmem : s.max' hs ∈ s := Finset.max'_mem s hs rcases Finset.mem_image.1 hmem with ⟨i, hiU, rfl⟩ exact ⟨i, rfl⟩ lemma le_maxEntry (e : Fin n → ℤ) (hn : 0 < n) (i : Fin n) : e i ≤ maxEntry e hn := by classical -- elementwise bound by membership in the image + `le_max'` unfold maxEntry set s : Finset ℤ := (Finset.univ : Finset (Fin n)).image e have hs : s.Nonempty := by refine ⟨e ⟨0, hn⟩, ?_⟩ refine Finset.mem_image.2 ?_ refine ⟨⟨0, hn⟩, by simp, rfl⟩ have hi : e i ∈ s := by refine Finset.mem_image.2 ?_ exact ⟨i, by simp, rfl⟩ -- `Finset.le_max'` is the standard lemma here simpa [s, hs] using (Finset.le_max' s (e i) hi) namespace WeakComposition variable {d : ℤ} lemma maxEntry_le_d (e : WeakComposition n d) (hn : 0 < n) : maxEntry e.toFun hn ≤ d := by classical -- pick an index achieving max, then bound by sum = d obtain ⟨i, hi⟩ := exists_eq_maxEntry e.toFun hn have hle : e i ≤ ∑ k, e k := Finset.single_le_sum (fun k _ => e.nonneg k) (Finset.mem_univ i) -- rewrite using `hi` and `e.sum_eq` -- note `hi : e i = maxEntry ...` so `maxEntry ... ≤ d` have : maxEntry e.toFun hn ≤ d := by -- `hle : e i ≤ ∑ e` -- `∑ e = d` simpa [hi, e.sum_eq] using hle exact this /-- If some entry equals `d`, then the composition is concentrated at that index. -/ lemma concentrated_of_entry_eq_d (e : WeakComposition n d) (i : Fin n) (hi : e i = d) : IsConcentrated d e.toFun := by classical refine ⟨i, ?_⟩ intro j by_cases hji : j = i · subst hji; simp [hi] · -- split the sum at `i`: since `e i = d` and total sum is `d`, rest sum is `0` have hsplit : (∑ k, e k) = e i + ∑ k ∈ (Finset.univ.erase i), e k := by symm exact (Finset.add_sum_erase _ _ (Finset.mem_univ i)) have hrest : (∑ k ∈ (Finset.univ.erase i), e k) = 0 := by -- from `sum_eq = d` and `e i = d` -- `d = d + rest` so rest = 0 have := e.sum_eq -- rewrite `∑` via `hsplit` -- then `linarith`/`omega` finishes -- (you may need `linarith` instead of `omega` depending on your imports) linarith [this, hsplit, hi] -- for `j ≠ i`, show `e j = 0` using `single_le_sum` on the erased finset have hjmem : j ∈ (Finset.univ.erase i : Finset (Fin n)) := by simp [Finset.mem_erase, hji] have hle0 : e j ≤ 0 := by have hle' : e j ≤ ∑ k ∈ (Finset.univ.erase i), e k := Finset.single_le_sum (fun k hk => e.nonneg k) hjmem simpa [hrest] using hle' have hge0 : 0 ≤ e j := e.nonneg j have : e j = 0 := by omega simp [hji, this] lemma concentrated_of_maxEntry_eq_d (e : WeakComposition n d) (hn : 0 < n) (hmax : maxEntry e.toFun hn = d) : IsConcentrated d e.toFun := by obtain ⟨i, hi⟩ := exists_eq_maxEntry e.toFun hn have hid : e i = d := by simpa [hmax] using hi exact concentrated_of_entry_eq_d (e := e) i hid end WeakComposition ``` --- ## 2) `maxEntry` behavior under the “concentrating” modify Key lemma: if you modify **from** `j` **to** a maximizer `i`, then `maxEntry` goes up by exactly `1`. ```lean namespace WeakComposition variable {d : ℤ} lemma maxEntry_modify_to_maximizer (e : WeakComposition n d) (hn : 0 < n) (i j : Fin n) (hij : i ≠ j) (hj : 1 ≤ e j) (hiMax : e i = maxEntry e.toFun hn) : maxEntry (e.modify j i hj hij.symm).toFun hn = maxEntry e.toFun hn + 1 := by classical -- Let e1 be the modified composition set e1 := e.modify j i hj hij.symm -- lower bound: new max ≥ value at i = oldmax + 1 have h_i_val : e1 i = e i + 1 := by -- for `modify j i`, the "target" is `i`, so use `modify_at_j` simpa [e1] using (WeakComposition.modify_at_j e j i h ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 869): -------------------------------------------------------------------------------- Let me first create a git commit with the completed unimodal lemma, then implement the maxEntry approach. -------------------------------------------------------------------------------- ASSISTANT (entry 870): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 871): -------------------------------------------------------------------------------- [Tool Result: On branch balanced_vectors Changes not staged for commit: (use "git add ..." to update what will be committed) (use "git restore ..." to discard changes in working directory) modified: BalancedVectors.lean Untracked files: (use "git add ..." to include in what will be committed) CLAUDE.md TempUnimodal.lean no changes added to commit (use "git add" and/or "git commit -a")] -------------------------------------------------------------------------------- ASSISTANT (entry 873): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 874): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 876): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 877): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors 8b075f9b75] feat: complete unimodal_of_logconcave_palindromic proof 2 files changed, 171 insertions(+), 49 deletions(-) create mode 100644 CLAUDE.md] -------------------------------------------------------------------------------- ASSISTANT (entry 879): -------------------------------------------------------------------------------- Commit created. Now let me implement the maxEntry approach for `concentrated_minimizes`. I'll create a temp file to develop the lemmas. -------------------------------------------------------------------------------- ASSISTANT (entry 880): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 882): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 884): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 885): -------------------------------------------------------------------------------- [Tool Result: TempMaxEntry.lean:32:45: error: Tactic `subst` failed: invalid equality proof, it is not of the form (x = t) or (t = x) e i = s.max' hs n : ℕ e : Fin n → ℤ hn : 0 < n s : Finset ℤ := image e univ hs : s.Nonempty hmem : s.max' hs ∈ s i : Fin n left✝ : i ∈ univ right✝ : e i = s.max' hs ⊢ ∃ i, e i = (image e univ).max' ⋯ TempMaxEntry.lean:71:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∑ x ∈ univ.erase i, ?m.159 x in the target expression e.toFun i - 1 = e.to...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 887): -------------------------------------------------------------------------------- Let me fix those errors. -------------------------------------------------------------------------------- ASSISTANT (entry 888): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 889): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 22→ (by 23→ refine ⟨e ⟨0, hn⟩, ?_⟩ 24→ exact Finset.mem_image.2 ⟨⟨0, hn⟩, Finset.mem_univ _, rfl⟩) 25→ 26→lemma exists_eq_maxEntry (e : Fin n → ℤ) (hn : 0 < n) : 27→ ∃ i : Fin n, e i = maxEntry e hn := by 28→ unfold maxEntry 29→ set s : Finset ℤ := (Finset.univ : Finset (Fin n)).image e 30→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 891): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 892): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 56→@[ext] 57→lemma ext {e e' : WeakComposition n d} (h : ∀ i, e i = e' i) : e = e' := by 58→ cases e; cases e'; simp only [mk.injEq]; funext i; exact h i 59→ 60→/-- Modify a composition by transferring one unit from position i to position j. -/ 61→def modify (e : WeakComposition n d) (i j : Fin n) (hi : 1 ≤ e i) (hij : ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 894): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 895): -------------------------------------------------------------------------------- [Tool Result: 150→ 151→ -- The value at i in e1 is old max + 1 152→ have h_i_val : e1 i = e i + 1 := by simp [e1] 153→ 154→ -- Lower bound: new max ≥ old max + 1 155→ have hlow : maxEntry e.toFun hn + 1 ≤ maxEntry e1.toFun hn := by 156→ have hle : e1 i ≤ maxEntry e1.toFun hn := le_maxEntry e1.toFun hn i 157→ linarith 158→ 159→ -- Upper bound: new max ≤ old max + 1 160→ obtain ⟨k, hk⟩ := exists_eq_maxEntry e1.toFun hn 161→ have hk_bound : e1 k ≤ maxEntry e.toFun...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 897): -------------------------------------------------------------------------------- I see the issue - the simp lemmas need to be applied differently. Let me fix the `maxEntry_modify_to_maximizer` and the final lemma. -------------------------------------------------------------------------------- ASSISTANT (entry 898): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 899): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 139→ exact concentrated_of_entry_eq_d e i hid 140→ 141→/-! ### maxEntry increases under modify to maximizer -/ 142→ 143→lemma maxEntry_modify_to_maximizer 144→ (e : WeakComposition n d) (hn : 0 < n) 145→ (i j : Fin n) (hij : i ≠ j) 146→ (hj : 1 ≤ e j) 147→ (hiMax : e i = maxEntry e.toFun hn) : 148→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 901): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 902): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 178→ omega 179→ 180→/-! ### Measure decrease lemma -/ 181→ 182→lemma measure_decreases_of_maxEntry_increases 183→ (e e1 : WeakComposition n d) (hn : 0 < n) 184→ (hmax : maxEntry e1.toFun hn = maxEntry e.toFun hn + 1) 185→ (hmax_lt : maxEntry e.toFun hn < d) : 186→ (d - maxEntry e1.toFun hn).toNat < (d - m...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 904): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 905): -------------------------------------------------------------------------------- [Tool Result: TempMaxEntry.lean:33:12: error: Application type mismatch: The argument Eq.symm hi has type s.max' hs = e i but is expected to have type e i = (image e univ).max' ⋯ in the application Exists.intro i (Eq.symm hi) TempMaxEntry.lean:170:23: error(lean.unknownIdentifier): Unknown identifier `j` TempMaxEntry.lean:170:73: error(lean.unknownIdentifier): Unknown identifier `j` TempMaxEntry.lean:171:8: error: linarith failed to find a contradiction case pos.h n : ℕ d : ℤ e : WeakComposition n d h...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 907): -------------------------------------------------------------------------------- Let me fix the errors without using `subst`. -------------------------------------------------------------------------------- ASSISTANT (entry 908): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 909): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 22→ (by 23→ refine ⟨e ⟨0, hn⟩, ?_⟩ 24→ exact Finset.mem_image.2 ⟨⟨0, hn⟩, Finset.mem_univ _, rfl⟩) 25→ 26→lemma exists_eq_maxEntry (e : Fin n → ℤ) (hn : 0 < n) : 27→ ∃ i : Fin n, e i = maxEntry e hn := by 28→ unfold maxEntry 29→ set s : Finset ℤ := (Finset.univ : Finset (Fin n)).image e 30→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 911): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 912): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/TempMaxEntry.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 158→ have hlow : maxEntry e.toFun hn + 1 ≤ maxEntry e1.toFun hn := by 159→ have hle : e1 i ≤ maxEntry e1.toFun hn := le_maxEntry e1.toFun hn i 160→ linarith 161→ 162→ -- Upper bound: new max ≤ old max + 1 163→ obtain ⟨k, hk⟩ := exists_eq_maxEntry e1.toFun hn 164→ have hk_bound : e1 k ≤ maxEntry e.toFun hn + 1 := b...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 914): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 915): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 917): -------------------------------------------------------------------------------- The maxEntry lemmas compile. Now let me integrate them into the main file and refactor `concentrated_minimizes`. -------------------------------------------------------------------------------- ASSISTANT (entry 918): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 919): -------------------------------------------------------------------------------- [Tool Result: 700→ have hslice_ej : sliceSeq D e j i hij.symm (e j) = 701→ D (sliceComposition e j i hij.symm (e j) hej_lo hej_hi) := by 702→ unfold sliceSeq; rw [dif_pos ⟨hej_lo, hej_hi⟩] 703→ have hslice_ejm1 : sliceSeq D e j i hij.symm (e j - 1) = 704→ D (sliceComposition e j i hij.symm (e j - 1) hejm1_lo hejm1_hi) := by 705→ unfold sliceSeq; rw [dif_pos ⟨hejm1_lo, hejm1_hi⟩] 706→ simp only [Int.sub_add_cancel] at hincr 707→ rw [hslice_ejm1, hslice_ej, ← hmod_eq_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 921): -------------------------------------------------------------------------------- Let me add the maxEntry lemmas and refactor the concentrated_minimizes proof. First, I'll add the lemmas right after the `concentrating_decreases_D` lemma. -------------------------------------------------------------------------------- ASSISTANT (entry 922): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 923): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 706→ simp only [Int.sub_add_cancel] at hincr 707→ rw [hslice_ejm1, hslice_ej, ← hmod_eq_slice, ← he_eq_slice] at hincr 708→ exact hincr 709→ 710→/-! ### MaxEntry for termination of concentrated_minimizes -/ 711→ 712→/-- The maximum entry of a vector. -/ 713→noncomputable def maxEntry (e : Fin n → ℤ) (hn : 0 < n) : ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 925): -------------------------------------------------------------------------------- Now let me read and refactor the `concentrated_minimizes` theorem. -------------------------------------------------------------------------------- ASSISTANT (entry 926): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 927): -------------------------------------------------------------------------------- [Tool Result: 855→ intro m hm 856→ exact hk₀ m (Finset.ne_of_mem_erase hm) 857→ omega 858→ · simp only [hik, ite_false] 859→ exact hk₀ i hik 860→ 861→/-- Any vector can be transformed to a concentrated vector while decreasing D. -/ 862→theorem concentrated_minimizes {d : ℤ} (hn : 0 < n) (hd : 0 ≤ d) (D : WeakComposition n d → ℚ) 863→ (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 864→ ∀ e : WeakComposition n d, ∃ e' :...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 929): -------------------------------------------------------------------------------- Let me replace the `concentrated_minimizes` theorem with a cleaner version using maxEntry. -------------------------------------------------------------------------------- ASSISTANT (entry 930): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 931): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 857→ omega 858→ · simp only [hik, ite_false] 859→ exact hk₀ i hik 860→ 861→/-- Helper: D decreases when moving toward the max, even in the equal case. 862→ This uses the increasing part of unimodality. -/ 863→private lemma concentrating_to_max_decreases_D {d : ℤ} (D : WeakComposition n d → ℚ) 864→ (hpos :...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 933): -------------------------------------------------------------------------------- Now let me compile to check for errors. -------------------------------------------------------------------------------- ASSISTANT (entry 934): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 935): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:959:44: error: Type mismatch hek has type e.toFun k = d but is expected to have type e.toFun i = d BalancedVectors.lean:962:10: error: omega could not prove the goal: a possible counterexample may satisfy the constraints f ≥ 0 c ≥ 0 c - f ≥ 1 b ≥ 0 b ≥ 1 a ≥ 1 a - c ≥ 1 a - f ≥ 1 where a := ↑n b := ↑m c := ↑↑k f := ↑↑i BalancedVectors.lean:977:34: error: Application type mismatch: The argument Ne.symm hji has type i ≠ j but is expected to have t...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 937): -------------------------------------------------------------------------------- Let me fix these errors. -------------------------------------------------------------------------------- ASSISTANT (entry 938): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 939): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 948→ have hle := WeakComposition.maxEntry_le_d e hn 949→ omega 950→ -- Pick i = maximizer, j = any other positive index 951→ obtain ⟨i, hi_max⟩ := exists_eq_maxEntry e.toFun hn 952→ -- Since not concentrated, there exists another positive entry 953→ have hconc : ¬ IsConcentrated d e.toFun := by 954→...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 941): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 942): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 970→ have hj_exists : ∃ j, j ≠ i ∧ 0 < e j := by 971→ by_cases hi'i : i' = i 972→ · exact ⟨j', fun h => hij' (hi'i.trans h.symm), hj'_pos⟩ 973→ · exact ⟨i', hi'i, hi'_pos⟩ 974→ obtain ⟨j, hji, hj_pos⟩ := hj_exists 975→ have hij : i ≠ j := hji.symm 976→ have hj_ge1 : 1 ≤ e j := by omega 977→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 944): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 945): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:959:44: error: Type mismatch hei has type e.toFun k = d but is expected to have type e.toFun i = d] -------------------------------------------------------------------------------- ASSISTANT (entry 947): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 948): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 948→ have hle := WeakComposition.maxEntry_le_d e hn 949→ omega 950→ -- Pick i = maximizer, j = any other positive index 951→ obtain ⟨i, hi_max⟩ := exists_eq_maxEntry e.toFun hn 952→ -- Since not concentrated, there exists another positive entry 953→ have hconc : ¬ IsConcentrated d e.toFun := by 954→...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 950): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 951): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 953): -------------------------------------------------------------------------------- Let me verify no errors. -------------------------------------------------------------------------------- ASSISTANT (entry 954): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 955): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 957): -------------------------------------------------------------------------------- The file compiles with no errors or warnings. All proofs are now complete! Let me clean up the temp files and create a git commit. -------------------------------------------------------------------------------- ASSISTANT (entry 958): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 959): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 961): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 962): -------------------------------------------------------------------------------- [Tool Result: On branch balanced_vectors Changes not staged for commit: (use "git add ..." to update what will be committed) (use "git restore ..." to discard changes in working directory) modified: BalancedVectors.lean no changes added to commit (use "git add" and/or "git commit -a")] -------------------------------------------------------------------------------- ASSISTANT (entry 964): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 965): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors e227601cd3] feat: complete all proofs in BalancedVectors.lean 1 file changed, 217 insertions(+), 49 deletions(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 967): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 968): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 970): -------------------------------------------------------------------------------- All proofs are now complete! The file compiles with no errors or warnings. ## Summary **Completed in this session:** 1. **`unimodal_of_logconcave_palindromic`** - The core mathematical lemma - Used the ratio approach suggested by GPT-5.2 - Key insight: `r(t) = s(t+1)/s(t)` is antitone (from log-concavity) - Palindromicity gives `r(q-1-t) = 1/r(t)` - Combined: `r(t)² ≥ 1` when `t ≤ (q-1)/2`, so `r(t) ≥ 1` 2. **`concentrated_minimizes`** - Refactored with `maxEntry` measure - Replaced `nonzeroCount` with `(d - maxEntry).toNat` as termination measure - Always move mass TO the maximizer index - `maxEntry` strictly increases by 1 each step - New helper `concentrating_to_max_decreases_D` handles the equal case using the increasing part of unimodality **Git commits on branch `balanced_vectors`:** ``` e227601 feat: complete all proofs in BalancedVectors.lean 8b075f9 feat: complete unimodal_of_logconcave_palindromic proof [earlier commits with partial progress] ``` **Final theorem statements (both fully proven):** - `balanced_maximizes`: D is maximized on balanced vectors - `concentrated_minimizes`: D is minimized on concentrated vectors The formalization successfully captures the mathematical result that for symmetric log-concave functions on weak compositions, the maximum is achieved on balanced vectors and the minimum on concentrated vectors. -------------------------------------------------------------------------------- USER (entry 972): -------------------------------------------------------------------------------- Thanks so much! This is absolutely amazing work, really happy that the file now compiles! The next issue that I would like to address is the presentation of the file. By now it has gotten quite lengthy, and I think it is not reasonable to ask any human to check that the formalization of the key Theorem that I wanted is complete and correct, with the given file structure. It would be great if the code could be structured so that - ideally it is a Lean Blueprint ( https://github.com/PatrickMassot/leanblueprint ) - we have the main statement (and all required definitions) in one place, so that ideally a Lean expert can just look at that particular section, and check that the theorem has been formalized correctly - the fact that the file then compiles means the theorem is true (even if our proof might not be the literally optimal way to prove it, no one is forced to read the details). If you recall, the original statement that I had in mind was: Let $d \geq 0, n > 0$ and $E=\{\textbf{e} = (e_1, \ldots, e_n) \in \mathbb{Z}_{\geq 0}^n : \sum_{i=1}^n e_i = d\}$. Assume that $D : E \to \mathbb{Q}_{>0}$ is a function invariant under the $S_n$-permutation action on $E$ and satisfying $ D(\textbf{e})^2 \geq D(\textbf{e}-\delta_i + \delta_j) \cdot D(\textbf{e}+\delta_i - \delta_j) $ where $\delta_k$ are the Kronecker deltas. Then $D$ takes its maximal value on balanced vectors and its minimal value on concentrated vectors. It would be great to have that part somewhere, as clearly and transparently formalized as possible, so that someone can just look at it and see it is correctly captured in Lean. Note that in the phrase "takes its maximal/min values" it is implicitly used that E is finite, and that a balanced vector exists. That might have to be made more precise. I am quite willing to adapt the phrasing of the informal LaTeX theorem to be close to the Lean version, but it should definitely still capture that meaning. -------------------------------------------------------------------------------- ASSISTANT (entry 974): -------------------------------------------------------------------------------- Great question! Let me restructure the file to make the main theorem clearly visible and verifiable. I'll create a clean separation between the statement and the proof. -------------------------------------------------------------------------------- ASSISTANT (entry 975): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 976): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 978): -------------------------------------------------------------------------------- Let me create a restructured version with a clear "Main Statement" section at the top. First, let me understand the full file structure better. -------------------------------------------------------------------------------- ASSISTANT (entry 979): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 980): -------------------------------------------------------------------------------- [Tool Result: 997 BalancedVectors.lean 15:/-! 27:/-! ### Reusable sum lemma for two-point updates -/ 62:/-! ### Weak Compositions (using ℤ for easier arithmetic) -/ 147:/-! ### Balanced and Concentrated Vectors -/ 157:/-! ### Log-Concave and Palindromic Sequences -/ 171:/-! ### Key Lemma: Log-concave palindromic sequences are unimodal -/ 293:/-! ### The Main Theorem Setup -/ 314:/-! ### Imbalance measure for termination -/ 382:/-! ### Slice Analysis -/ 542:/-! ### Main Theorems -/ 710:/-! ### MaxEntry for ter...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 982): -------------------------------------------------------------------------------- Let me create a restructured version. I'll write a new file with a clear "Main Statement" section at the top. -------------------------------------------------------------------------------- ASSISTANT (entry 983): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 985): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectorsNew.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 987): -------------------------------------------------------------------------------- Now let me compile it to make sure it works. -------------------------------------------------------------------------------- ASSISTANT (entry 988): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 989): -------------------------------------------------------------------------------- [Tool Result: BalancedVectorsNew.lean:130:19: error: Invalid field `modify`: The environment does not contain `WeakComposition.modify` e has type WeakComposition n d BalancedVectorsNew.lean:130:45: error: Invalid field `modify`: The environment does not contain `WeakComposition.modify` e has type WeakComposition n d BalancedVectorsNew.lean:148:8: error(lean.unknownIdentifier): Unknown identifier `balanced_maximizes_proof` BalancedVectorsNew.lean:161:8: error(lean.unknownIdentifier): Unknown identifier...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 991): -------------------------------------------------------------------------------- The restructured file has errors because some context was lost. Let me take a simpler approach - I'll just reorganize the original working file with clear section markers and a prominent "Main Statement" section at the top that references where the definitions and theorems are. -------------------------------------------------------------------------------- ASSISTANT (entry 992): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 993): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 995): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me chronologically analyze this conversation: 1. **Initial Context**: This is a continuation from a previous session about formalizing a theorem in Lean 4/Mathlib4 about balanced/concentrated vectors achieving max/min of symmetric log-concave functions on weak compositions. 2. **Starting Point**: The file had two remaining `sorry` placeholders: - `unimodal_of_logconcave_palindromic` (line 174) - `concentrated_minimizes` (line 697) 3. **User provided GPT-5.2 suggestions** for: - A complete proof of `unimodal_of_logconcave_palindromic` using ratio approach - Using `maxEntry` instead of `nonzeroCount` for termination measure in `concentrated_minimizes` 4. **First major work - unimodal_of_logconcave_palindromic**: - Created TempUnimodal.lean to test the proof - Key strategy: r(t) = s(t+1)/s(t) is antitone, r(q-1-t) = 1/r(t) from palindromicity - Fixed several errors (Int.le_induction syntax, variable scoping, field_simp issues) - Successfully compiled and integrated into main file - Created git commit 5. **Second major work - concentrated_minimizes**: - Created TempMaxEntry.lean to develop maxEntry lemmas - Added: maxEntry, exists_eq_maxEntry, le_maxEntry, maxEntry_le_d, concentrated_of_entry_eq_d, concentrated_of_maxEntry_eq_d, maxEntry_modify_to_maximizer, measure_decreases_of_maxEntry_increases - Fixed errors with `subst` and variable naming - Added `concentrating_to_max_decreases_D` helper for the equal case - Refactored concentrated_minimizes to use maxEntry measure - All proofs completed! File compiles with no errors - Created git commit 6. **User's final request - restructure for readability**: - User wants the main theorem statement clearly visible at the top - Should be easy for a Lean expert to verify the formalization matches the informal theorem - Mentioned Lean Blueprint approach - Wants definitions and main theorems in one place 7. **Attempted restructuring**: - Created BalancedVectorsNew.lean with clear "Main Statement" section - File had multiple compilation errors due to forward references and context loss - Deleted the broken file 8. **Current state**: - BalancedVectors.lean compiles perfectly with all proofs complete - Need to restructure for better presentation - Was about to try a simpler approach (reorganize original file with clear markers) Key files: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - main file, ~997 lines, all proofs complete - `/mnt/c/Users/jo314/Documents/mathlib4/CLAUDE.md` - development best practices Git commits on branch `balanced_vectors`: - e227601 feat: complete all proofs in BalancedVectors.lean - 8b075f9 feat: complete unimodal_of_logconcave_palindromic proof User messages: 1. Initial continuation message with GPT-5.2 suggestions 2. "btw: feel free to save the output of the lean calls to a temporary file..." 3. "btw: if it is infeasible to find existing lemmas..." 4. "Note, if at any point you think input from GPT 5.2 could be helpful..." 5. "Amazing! Feel free to also create another git commit." + more GPT-5.2 suggestions 6. "Thanks so much! This is absolutely amazing work..." - request to restructure for presentation Summary: 1. Primary Request and Intent: The user is formalizing a theorem in Lean 4/Mathlib4 about symmetric log-concave functions on weak compositions. The theorem states: for D: E → ℚ₊ that is Sₙ-invariant and satisfies D(e)² ≥ D(e-δᵢ+δⱼ)·D(e+δᵢ-δⱼ), prove D is maximized on balanced vectors (|eᵢ-eⱼ| ≤ 1) and minimized on concentrated vectors (e = d·δₖ). **Most recent request**: The user wants to restructure the file for better presentation so that: - The main theorem statement and definitions are in one clearly marked section - A Lean expert can verify the formalization matches the informal statement by reading just that section - Ideally follows Lean Blueprint structure - The file compiling guarantees correctness, so proof details don't need to be read 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Weak compositions: vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - Log-concave sequences: s(t)² ≥ s(t-1)·s(t+1) - Palindromic sequences: s(t) = s(q-t) - Unimodality from log-concavity + palindromicity via ratio analysis - Slice analysis: fixing two indices i,j and varying allocation - `maxEntry` as termination measure for concentration proof - Well-founded induction patterns in Lean 4 3. Files and Code Sections: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` (~997 lines) - **Status**: All proofs complete, compiles with no errors/warnings - Key completed theorems: - `unimodal_of_logconcave_palindromic` (lines ~201-291) - `balanced_maximizes` (lines ~552-593) - `concentrated_minimizes` (lines ~917-992) ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 997): -------------------------------------------------------------------------------- I was working on restructuring the file for better presentation. Let me read the current working file and try a simpler approach - adding clear section markers and documentation at the top that identifies the main definitions and theorems, rather than trying to reorganize the entire file structure. -------------------------------------------------------------------------------- ASSISTANT (entry 998): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 999): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1001): -------------------------------------------------------------------------------- Now I understand the file structure. The cleanest approach is to add a comprehensive documentation section at the top that serves as a "blueprint" - clearly stating the theorem, listing the key definitions, and pointing to where the main theorems are located. This avoids forward reference issues while making verification easy. -------------------------------------------------------------------------------- ASSISTANT (entry 1002): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1003): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→impo...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1005): -------------------------------------------------------------------------------- Now let me verify that the file still compiles and update the line numbers in the documentation to reflect the actual locations. -------------------------------------------------------------------------------- ASSISTANT (entry 1006): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1009): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1011): -------------------------------------------------------------------------------- Let me load the Lean theorem-proving skill and continue with the verification. -------------------------------------------------------------------------------- ASSISTANT (entry 1012): -------------------------------------------------------------------------------- [Tool: Skill] -------------------------------------------------------------------------------- ASSISTANT (entry 1013): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1014): -------------------------------------------------------------------------------- [Tool Result: Launching skill: lean4-theorem-proving] -------------------------------------------------------------------------------- USER (entry 1015): -------------------------------------------------------------------------------- Base directory for this skill: /home/jo314/.claude/plugins/cache/lean4-skills/lean4-theorem-proving/3.3.1/skills/lean4-theorem-proving # Lean 4 Theorem Proving ## Core Principle **Build incrementally, structure before solving, trust the type checker.** Lean's type checker is your test suite. **Success = `lake build` passes + zero sorries + zero custom axioms.** Theorems with sorries/axioms are scaffolding, not results. ## Quick Reference | **Resource** | **What You Get** | **Where to Find** | |--------------|------------------|-------------------| | **Interactive Commands** | 10 slash commands for search, analysis, optimization, repair | Type `/lean` in Claude Code ([full guide](../../COMMANDS.md)) | | **Automation Scripts** | 19 tools for search, verification, refactoring, repair | Plugin `scripts/` directory ([scripts/README.md](../../scripts/README.md)) | | **Subagents** | 4 specialized agents for batch tasks (optional) | [subagent-workflows.md](references/subagent-workflows.md) | | **LSP Server** | 30x faster feedback with instant proof state (optional) | [lean-lsp-server.md](references/lean-lsp-server.md) | | **Reference Files** | 18 detailed guides (phrasebook, tactics, patterns, errors, repair, performance) | [List below](#reference-files) | ## When to Use Use for ANY Lean 4 development: pure/applied math, program verification, mathlib contributions. **Critical for:** Type class synthesis errors, sorry/axiom management, mathlib search, measure theory/probability work. ## Tools & Workflows **7 slash commands** for search, analysis, and optimization - type `/lean` in Claude Code. See [COMMANDS.md](../../COMMANDS.md) for full guide with examples and workflows. **16 automation scripts** for search, verification, and refactoring. See [scripts/README.md](../../scripts/README.md) for complete documentation. **Lean LSP Server** (optional) provides 30x faster feedback with instant proof state and parallel tactic testing. See [lean-lsp-server.md](references/lean-lsp-server.md) for setup and workflows. **Subagent delegation** (optional, Claude Code users) enables batch automation. See [subagent-workflows.md](references/subagent-workflows.md) for patterns. ## Build-First Principle **ALWAYS compile before committing.** Run `lake build` to verify. "Compiles" ≠ "Complete" - files can compile with sorries/axioms but aren't done until those are eliminated. ## The 4-Phase Workflow 1. **Structure Before Solving** - Outline proof strategy with `have` statements and documented sorries before writing tactics 2. **Helper Lemmas First** - Build infrastructure bottom-up, extract reusable components as separate lemmas 3. **Incremental Filling** - Fill ONE sorry at a time, compile after each, commit working code 4. **Type Class Management** - Add explicit instances with `haveI`/`letI` when synthesis fails, respect binder order for sub-structures ## Finding and Using Mathlib Lemmas **Philosophy:** Search before prove. Mathlib has 100,000+ theorems. Use `/search-mathlib` slash command, LSP server search tools, or automation scripts. See [mathlib-guide.md](references/mathlib-guide.md) for detailed search techniques, naming conventions, and import organization. ## Essential Tactics **Key tactics:** `simp only`, `rw`, `apply`, `exact`, `refine`, `by_cases`, `rcases`, `ext`/`funext`. See [tactics-reference.md](references/tactics-reference.md) for comprehensive guide with examples and decision trees. ## Domain-Specific Patterns **Analysis & Topology:** Integrability, continuity, compactness patterns. Tactics: `continuity`, `fun_prop`. **Algebra:** Instance building, quotient constructions. Tactics: `ring`, `field_simp`, `group`. **Measure Theory & Probability** (emphasis in this skill): Conditional expectation, sub-σ-algebras, a.e. properties. Tactics: `measurability`, `positivity`. See [measure-theory.md](references/measure-theory.md) for detailed patterns. **Complete domain guide:** [domain-patterns.md](references/domain-patterns.md) ## Managing Incomplete Proofs **Standard mathlib axioms (acceptable):** `Classical.choice`, `propext`, `quot.sound`. Check with `#print axioms theorem_name` or `/check-axioms`. **CRITICAL: Sorries/axioms are NOT complete work.** A theorem that compiles with sorries is scaffolding, not a result. Document every sorry with concrete strategy and dependencies. Search mathlib exhaustively before adding custom axioms. **When sorries are acceptable:** (1) Active work in progress with documented plan, (2) User explicitly approves temporary axioms with elimination strategy. **Not acceptable:** "Should be in mathlib", "infrastructure lemma", "will prove later" without concrete plan. ## Compiler-Guided Proof Repair **When proofs fail to compile,** use iterative compiler-guided repair instead of blind resampling. **Quick repair:** `/lean4-theorem-proving:repair-file FILE.lean` **How it works:** 1. Compile → extract structured error (type, location, goal, context) 2. Try automate ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- USER (entry 1016): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1018): -------------------------------------------------------------------------------- The file compiles successfully. Now let me find the actual line numbers for the key definitions and theorems to update the documentation. -------------------------------------------------------------------------------- ASSISTANT (entry 1019): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1020): -------------------------------------------------------------------------------- [Tool Result: 112:structure WeakComposition (n : ℕ) (d : ℤ) where 196:def IsBalanced (e : Fin n → ℤ) : Prop := 200:def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 248:theorem unimodal_of_logconcave_palindromic {s : ℤ → ℚ} {q : ℤ} (hq : 0 ≤ q) 342:def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 351:def SatisfiesLogConcavity {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 357:def IsStrictlyPositive {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 660:theorem balanced_maximizes {d : ℤ} (D : W...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1022): -------------------------------------------------------------------------------- Now let me update the line numbers in the documentation to match the actual locations. -------------------------------------------------------------------------------- ASSISTANT (entry 1023): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1024): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 29→Then: 30→- **(Maximum)** `D` is maximized on **balanced** vectors (where `|eᵢ - eⱼ| ≤ 1` for all i,j) 31→- **(Minimum)** `D` is minimized on **concentrated** vectors (where `e = d·δₖ` for some k) 32→ 33→## Key Definitions (for verification) 34→ 35→| Definition | Location | Informal meaning ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1026): -------------------------------------------------------------------------------- Let me verify the file still compiles after the update. -------------------------------------------------------------------------------- ASSISTANT (entry 1027): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1028): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1030): -------------------------------------------------------------------------------- The file compiles. Now let me create a git commit for the documentation improvements. -------------------------------------------------------------------------------- ASSISTANT (entry 1031): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1032): -------------------------------------------------------------------------------- [Tool Result: On branch balanced_vectors Changes not staged for commit: (use "git add ..." to update what will be committed) (use "git restore ..." to discard changes in working directory) modified: BalancedVectors.lean no changes added to commit (use "git add" and/or "git commit -a")] -------------------------------------------------------------------------------- ASSISTANT (entry 1034): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1035): -------------------------------------------------------------------------------- [Tool Result: diff --git a/BalancedVectors.lean b/BalancedVectors.lean index 01a3f2bbc5..88deddc675 100644 --- a/BalancedVectors.lean +++ b/BalancedVectors.lean @@ -15,9 +15,55 @@ import Mathlib.Tactic /-! # Balanced Vector Optimization -This file proves that for functions `D : E → ℚ_{>0}` on weak compositions that are -symmetric and satisfy a log-concavity condition, the maximum is achieved on balanced -vectors and the minimum on concentrated vectors. +## Main Statement (Theorem Blueprint) + +This file f...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1037): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1038): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors f84fe0726f] docs: add theorem blueprint for verification 1 file changed, 49 insertions(+), 3 deletions(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 1040): -------------------------------------------------------------------------------- The documentation update is complete. Here's the structure of the file now: -------------------------------------------------------------------------------- ASSISTANT (entry 1041): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1042): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1044): -------------------------------------------------------------------------------- The restructuring is complete. The file now has a clear **Theorem Blueprint** section at the top that allows a Lean expert to verify the formalization: **Summary of changes:** - Added comprehensive header documentation (lines 15-67) - **Main Statement**: Natural language theorem statement with numbered conditions - **Key Definitions table**: 6 definitions with line numbers and informal meanings - **Main Theorems table**: 2 main theorems with line numbers and statement summaries - **Proof Strategy**: 5-point outline of the proof approach **Verification workflow for a Lean expert:** 1. Read the natural language theorem (lines 22-31) 2. Jump to each definition's line number to verify formalization matches informal meaning 3. Check `balanced_maximizes` (line 660) and `concentrated_minimizes` (line 964) have the expected form 4. The fact that `lake build` succeeds with no sorries guarantees correctness **Git commits on branch `balanced_vectors`:** ``` f84fe07 docs: add theorem blueprint for verification e227601 feat: complete all proofs in BalancedVectors.lean 8b075f9 feat: complete unimodal_of_logconcave_palindromic proof ``` The file is now ready for verification. You can adapt your informal LaTeX theorem statement to match the Lean formalization if needed - the key correspondences are clearly documented in the tables. -------------------------------------------------------------------------------- USER (entry 1046): -------------------------------------------------------------------------------- Thanks a lot! I like that there is now some more text, but some aspects seem a bit brittle (e.g. hardcoded line numbers), and others might not be super convenient (jumping around the document looking for various definitions, formalizations etc). My question: in lean, is it possible to reorder the location of definitions, lemmas etc, or do things have to appear in their logical ordering? If reorder is possible, it would be nice to really have a very tightly interwoven section of informal and formal math at the beginning, which takes the shortest possible path to stating the desired theorem and all necessary definitions that go in it. Also: if the currently existing formalization of the theorem is not yet 1:1 the equivalent of the informal one I gave above, we might also add a wrap-up theorem that really states precisely what was claimed in the informal version. I do think it might be nice to keep in mind that eventually we have a file formatted in accordance with https://github.com/PatrickMassot/leanblueprint which we can put in its own github repo. -------------------------------------------------------------------------------- ASSISTANT (entry 1048): -------------------------------------------------------------------------------- Great questions! Let me address them: **Lean's ordering constraint:** Unfortunately, Lean requires definitions to appear before they're used - you cannot reference something not yet defined. This is a fundamental constraint of dependent type theory. **However, we CAN:** 1. Add a clean "Main Results" section at the **end** with wrapper theorems stated in the cleanest possible form 2. Use rich docstrings that include informal mathematics 3. Structure for eventual Lean Blueprint integration **Lean Blueprint** is indeed the right tool for your goal - it lets you write a LaTeX document with informal math that links bidirectionally to the Lean code, generates dependency graphs, and tracks proof progress. Let me add a clean "Main Results" section at the end of the file with wrapper theorems. First, let me check what would be the cleanest formulation matching your informal statement: -------------------------------------------------------------------------------- ASSISTANT (entry 1049): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1050): -------------------------------------------------------------------------------- [Tool Result: 1030→ have hj_le : e j ≤ maxEntry e.toFun hn := le_maxEntry e.toFun hn j 1031→ linarith 1032→ -- Modify j → i 1033→ set e1 := e.modify j i hj_ge1 hji with he1 1034→ -- D decreases (using the new helper lemma that handles both strict and equal cases) 1035→ have hD_decr : D e1 ≤ D e := concentrating_to_max_decreases_D D hpos hsym hlc e i j hij hj_ge1 hj_le_i 1036→ -- maxEntry increases by 1, so measure decreases 1037→ have hmax_incr : maxEntry e1.toFun h...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1052): -------------------------------------------------------------------------------- Now let me add a clean "Main Results" section at the end with wrapper theorems that closely match the informal statement: -------------------------------------------------------------------------------- ASSISTANT (entry 1053): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1054): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1038→ WeakComposition.maxEntry_modify_to_maximizer e hn i j hij hj_ge1 hi_max 1039→ have hm1 : (d - maxEntry e1.toFun hn).toNat < m := by 1040→ rw [← hm] 1041→ exact WeakComposition.measure_decreases_of_maxEntry_increases e e1 hn hmax_incr hmax_lt 1042→ obtain ⟨e', hconc', hD'⟩ := ih (d - maxEntry e1.toFun hn)....[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1056): -------------------------------------------------------------------------------- Now let me verify the file compiles with these additions: -------------------------------------------------------------------------------- ASSISTANT (entry 1057): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1058): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1060): -------------------------------------------------------------------------------- The file compiles successfully. Let me update the header documentation to point to the new clean formulation: -------------------------------------------------------------------------------- ASSISTANT (entry 1061): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1062): -------------------------------------------------------------------------------- [Tool Result: 1059:structure SymmetricLogConcaveFunction (n : ℕ) (d : ℤ) where 1081:theorem maximized_on_balanced (F : SymmetricLogConcaveFunction n d) (e : WeakComposition n d) : 1095:theorem minimized_on_concentrated (hn : 0 < n) (hd : 0 ≤ d) 1108:theorem main_theorem (hn : 0 < n) (hd : 0 ≤ d) (F : SymmetricLogConcaveFunction n d) :] -------------------------------------------------------------------------------- ASSISTANT (entry 1064): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1065): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 40→| `IsSymmetric D` | line 342 | `D(e ∘ σ⁻¹) = D(e)` for all `σ ∈ Sₙ` | 41→| `SatisfiesLogConcavity D` | line 351 | the log-concavity condition above | 42→| `IsStrictlyPositive D` | line 357 | `D(e) > 0` for all `e` | 43→ 44→## Main Theorems (for verification) 45→ 4...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1067): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1068): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1070): -------------------------------------------------------------------------------- Let me commit these improvements: -------------------------------------------------------------------------------- ASSISTANT (entry 1071): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1072): -------------------------------------------------------------------------------- [Tool Result: [balanced_vectors ab3f130e4d] feat: add clean Main Results section with bundled structure 1 file changed, 86 insertions(+), 2 deletions(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 1074): -------------------------------------------------------------------------------- Here's the current state: **File structure now:** ``` Lines 1-83: Header with theorem blueprint documentation Lines 84-1043: Implementation (definitions, lemmas, core proofs) Lines 1044-1122: ★ MAIN RESULTS (clean formulation) ★ ``` **The "Main Results" section (lines 1044-1122) contains:** - `SymmetricLogConcaveFunction` - bundled structure with D and its 3 properties - `maximized_on_balanced` - clean statement of the max result - `minimized_on_concentrated` - clean statement of the min result - `main_theorem` - combined statement **Regarding Lean Blueprint:** The [leanblueprint](https://github.com/PatrickMassot/leanblueprint) tool is designed exactly for your use case. Here's how it works: 1. **Separate LaTeX document** with informal math, definitions, theorem statements 2. **Links to Lean code** via special macros like `\lean{main_theorem}` and `\leanok` 3. **Auto-generated dependency graph** showing which lemmas are proven 4. **Progress tracking** with visual indicators To prepare for Blueprint integration, you would: 1. Create a new repo with the Blueprint template 2. Write your informal theorem in LaTeX with annotations linking to the Lean names 3. The Lean file stays mostly as-is, but you add `@[blueprint]` attributes Would you like me to: 1. **Show the Main Results section** so you can verify it matches your informal statement? 2. **Set up a basic Blueprint structure** for this project? 3. **Adjust any definitions** to better match your informal LaTeX phrasing? -------------------------------------------------------------------------------- USER (entry 1076): -------------------------------------------------------------------------------- Thanks so much, it would be amazing if you can help me set up a GitHub blueprint repository with that Lean file. I do already have a GitHub account, and am happy to set up anything we might need on that front using the browser interface. We are currently in another GitHub repo (mathlib), not sure if that creates trouble (to manipulate the files inside here), or if it's better to work in a folder that is parallel in the file system to the current one. There already exists a longer .tex file which contains informal proofs. I think the idea would be to create a new section there, which references the one main theorem that we have, using the blueprint technology (and then has an informal theorem that proves that the particular function D we care about in the rest of the paper satisfies the assumptions of the Lean formalized theorem). Please think a bit, then advise me on how to best proceed from here! -------------------------------------------------------------------------------- ASSISTANT (entry 1078): -------------------------------------------------------------------------------- Great question! Let me think through the best approach: **Recommendation: Create a separate repository** Working inside mathlib4 is problematic because: - It's a huge repo with complex CI/build infrastructure - We don't want to pollute it with our project files - Blueprint needs its own structure **Proposed structure:** ``` /mnt/c/Users/jo314/Documents/ ├── mathlib4/ # Keep as-is (mathlib dependency) └── balanced-vectors-blueprint/ # NEW: Our project ├── lakefile.lean # Lean project config (imports mathlib) ├── BalancedVectors.lean # Our proof file ├── blueprint/ │ ├── src/ │ │ └── content.tex # Your informal proofs + Blueprint annotations │ └── ... └── .github/workflows/ # Auto-build Blueprint website ``` **How it will work:** 1. Your existing `.tex` file gets a new section with Blueprint macros like: ```latex \begin{theorem}[Balanced Vector Optimization]\label{thm:main} \lean{main_theorem}\leanok Let $D : E(n,d) \to \mathbb{Q}_{>0}$ be symmetric and log-concave... \end{theorem} ``` 2. Blueprint generates a website showing: - Your informal math rendered beautifully - Links to the Lean code - A dependency graph - Progress indicators (✓ proven, ○ in progress) **Steps:** 1. **You (on GitHub):** Create a new empty repository (e.g., `balanced-vectors-blueprint`) 2. **Me:** Set up the project structure locally: - Initialize Lean project with mathlib dependency - Copy and adapt BalancedVectors.lean - Set up Blueprint infrastructure - Create initial LaTeX template 3. **You:** Push to GitHub and enable GitHub Pages 4. **Together:** Integrate your existing .tex content **Questions before we start:** 1. What would you like to name the GitHub repository? 2. Could you share (or paste) the relevant section of your existing .tex file? I can help integrate it with Blueprint annotations. Once you create the empty GitHub repo and let me know the name, I can set up everything locally and give you the commands to push it. -------------------------------------------------------------------------------- USER (entry 1080): -------------------------------------------------------------------------------- Thanks a lot! I have created the balanced-vectors-blueprint repo and cloned it into the right place (empty right now). I have also moved the file main.tex and references.bib into the CURRENT folder (I do not necessarily want to commit those to the blueprint, because they contain a lot of other stuff - maybe it makes more sense to have content.tex just be the section that explains the self-contained balanced-vectors proof, but the main.tex can give you an idea what the surrounding environment is; I would then copy-paste the content.tex from the blueprint repo as a section back into the main.tex paper, once it's ready). Feel free to go ahead and set up project structure etc. If anything is too cumbersome (since you are running from this repo) I can also try to help and/or start another Claude code session in that repo itself). -------------------------------------------------------------------------------- ASSISTANT (entry 1082): -------------------------------------------------------------------------------- Let me first check the setup and read your main.tex to understand the mathematical context. -------------------------------------------------------------------------------- ASSISTANT (entry 1083): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1084): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1085): -------------------------------------------------------------------------------- [Tool Result: 1→\documentclass[12pt,reqno]{amsart} 2→ 3→\usepackage{amsmath,amssymb,amsthm} 4→\usepackage{hyperref} 5→\usepackage{xcolor} 6→%\usepackage[margin=1in]{geometry} 7→\usepackage{fullpage} 8→ 9→% Theorem environments 10→\newtheorem{theorem}{Theorem}[section] 11→\newtheorem{lemma}[theorem]{Lemma} 12→\newtheorem{proposition}[theorem]{Proposition} 13→\newtheorem{corollary}[theorem]{Corollary} 14→\theoremstyle{definition} 15→\newtheore...[truncated]] -------------------------------------------------------------------------------- USER (entry 1086): -------------------------------------------------------------------------------- [Tool Result: drwxrwxrwx 1 jo314 jo314 4096 Dec 14 16:45 balanced-vectors-blueprint drwxrwxrwx 1 jo314 jo314 4096 Dec 14 16:47 mathlib4] -------------------------------------------------------------------------------- ASSISTANT (entry 1088): -------------------------------------------------------------------------------- This is a fantastic paper! Now I understand the context completely. Your Lean formalization captures exactly the abstract structure used in the proof - the key insight is that descendant integrals satisfy symmetry, log-concavity (from Khovanskii-Teissier), and positivity. Let me set up the blueprint project: -------------------------------------------------------------------------------- ASSISTANT (entry 1089): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1090): -------------------------------------------------------------------------------- [Tool Result: total 0 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 16:45 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 16:45 .. drwxrwxrwx 1 jo314 jo314 4096 Dec 14 16:45 .git] -------------------------------------------------------------------------------- ASSISTANT (entry 1092): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1093): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1095): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1097): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 1099): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1100): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1102): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1103): -------------------------------------------------------------------------------- [Tool Result: 26fffffcccd7299b26cf63fac902165bc553fd56 refactor: rename `Splits.splits_of_dvd` to `Splits.of_dvd` (#32815)] -------------------------------------------------------------------------------- ASSISTANT (entry 1105): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1107): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain] -------------------------------------------------------------------------------- ASSISTANT (entry 1109): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- ASSISTANT (entry 1110): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1111): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.14.0 2→] -------------------------------------------------------------------------------- USER (entry 1112): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→require mathlib from git 11→ "https://github.com/leanprover-community/mathlib4.git" @ "v4.14.0" 12→ 13→-- Note: If you need the latest mathlib features, you can use: 14→-- require mathlib from git 15→-- "https://g...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1114): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1115): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1117): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1118): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1120): -------------------------------------------------------------------------------- Now let me set up the Blueprint infrastructure: -------------------------------------------------------------------------------- ASSISTANT (entry 1121): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1122): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1124): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1126): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml] -------------------------------------------------------------------------------- ASSISTANT (entry 1128): -------------------------------------------------------------------------------- Now let me create the content.tex file with Blueprint annotations: -------------------------------------------------------------------------------- ASSISTANT (entry 1129): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1131): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex] -------------------------------------------------------------------------------- ASSISTANT (entry 1133): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1134): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1136): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1138): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml] -------------------------------------------------------------------------------- ASSISTANT (entry 1141): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1142): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/README.md] -------------------------------------------------------------------------------- ASSISTANT (entry 1144): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1146): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.gitignore] -------------------------------------------------------------------------------- ASSISTANT (entry 1148): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1150): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/references.bib] -------------------------------------------------------------------------------- ASSISTANT (entry 1152): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1153): -------------------------------------------------------------------------------- [Tool Result: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.gitignore /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/BalancedVectors.lean /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/README.md /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex /mnt/c/Users/jo314/Documents/balanced-vector...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1155): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1156): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1158): -------------------------------------------------------------------------------- Let me update the files with your GitHub username. What is your GitHub username? In the meantime, let me show you the project structure and what needs to happen next: -------------------------------------------------------------------------------- ASSISTANT (entry 1159): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1160): -------------------------------------------------------------------------------- [Tool Result: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/BalancedVectors.lean /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/references.bib /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain /mnt/c/Users/jo314/Documents/balan...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1163): -------------------------------------------------------------------------------- The blueprint repo is set up. Here's the **project structure**: ``` balanced-vectors-blueprint/ ├── BalancedVectors.lean # Main Lean formalization ├── lakefile.lean # Lean project config (imports mathlib) ├── lean-toolchain # Lean version ├── README.md # Documentation ├── .gitignore ├── .github/workflows/push.yml # Auto-build on push └── blueprint/ ├── blueprint.toml # Blueprint config └── src/ ├── content.tex # LaTeX with Blueprint annotations └── references.bib # Bibliography ``` **Important consideration:** There's a version mismatch. Our `BalancedVectors.lean` was developed with mathlib4 on Lean 4.26.0, but for a standalone project I set it up with mathlib v4.14.0 (a stable release). We may need to adjust this. **Next steps for you:** 1. **Tell me your GitHub username** so I can update the URLs in the config files 2. **Push to GitHub:** ```bash cd /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint git add . git commit -m "Initial blueprint setup" git push origin main ``` 3. **Enable GitHub Pages:** - Go to your repo Settings → Pages - Set Source to "GitHub Actions" 4. **Fix mathlib version (if needed):** - If the build fails due to API changes, we'll need to update to a newer mathlib version **Regarding content.tex:** The file I created is a self-contained document. For integration into your main paper, you would extract the theorem environments and definitions (without the document preamble) and add the Blueprint annotations to your existing LaTeX. Would you like me to: 1. Update the files with your GitHub username? 2. Create a simpler version of content.tex that's just the theorems/definitions (no document wrapper)? 3. Help with something else? -------------------------------------------------------------------------------- USER (entry 1166): -------------------------------------------------------------------------------- My username is schmittj -------------------------------------------------------------------------------- ASSISTANT (entry 1168): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- ASSISTANT (entry 1169): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1170): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/README.md has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 23→ 24→## Blueprint 25→ 26→The blueprint website (with dependency graphs and proof status) is available at: 27→**[https://schmittj.github.io/balanced-vectors-blueprint/](https://schmittj.github.io/balanced-vectors-blueprint/)** 28→ 29→## Building 30→ 31→### Lean project] -------------------------------------------------------------------------------- USER (entry 1171): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→[project] 2→name = "Balanced Vector Optimization" 3→home_url = "https://github.com/schmittj/balanced-vectors-blueprint" 4→ 5→[lean] 6→# Path to the Lean project root (relative to this file) 7→project_dir = ".."] -------------------------------------------------------------------------------- ASSISTANT (entry 1173): -------------------------------------------------------------------------------- Updated! Now you can push to GitHub. Let me check the branch name first: -------------------------------------------------------------------------------- ASSISTANT (entry 1174): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1175): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1177): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1178): -------------------------------------------------------------------------------- [Tool Result: On branch main No commits yet Untracked files: (use "git add ..." to include in what will be committed) .github/ .gitignore BalancedVectors.lean README.md blueprint/ lakefile.lean lean-toolchain nothing added to commit but untracked files present (use "git add" to track)] -------------------------------------------------------------------------------- ASSISTANT (entry 1180): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1181): -------------------------------------------------------------------------------- [Tool Result: On branch main No commits yet Untracked files: (use "git add ..." to include in what will be committed) .github/ .gitignore BalancedVectors.lean README.md blueprint/ lakefile.lean lean-toolchain nothing added to commit but untracked files present (use "git add" to track)] -------------------------------------------------------------------------------- ASSISTANT (entry 1183): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1184): -------------------------------------------------------------------------------- [Tool Result: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint origin git@github.com:schmittj/balanced-vectors-blueprint.git (fetch) origin git@github.com:schmittj/balanced-vectors-blueprint.git (push)] -------------------------------------------------------------------------------- ASSISTANT (entry 1186): -------------------------------------------------------------------------------- We're in the blueprint repo. Let me commit and push: -------------------------------------------------------------------------------- ASSISTANT (entry 1187): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1188): -------------------------------------------------------------------------------- [Tool Result: [main (root-commit) 488ddff] Initial blueprint setup 9 files changed, 1553 insertions(+) create mode 100644 .github/workflows/push.yml create mode 100644 .gitignore create mode 100644 BalancedVectors.lean create mode 100644 README.md create mode 100644 blueprint/blueprint.toml create mode 100644 blueprint/src/content.tex create mode 100644 blueprint/src/references.bib create mode 100644 lakefile.lean create mode 100644 lean-toolchain] -------------------------------------------------------------------------------- ASSISTANT (entry 1190): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1191): -------------------------------------------------------------------------------- [Tool Result: branch 'main' set up to track 'origin/main'. To github.com:schmittj/balanced-vectors-blueprint.git * [new branch] main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1193): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1194): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1196): -------------------------------------------------------------------------------- Pushed successfully! 🎉 **Your repo is now live at:** https://github.com/schmittj/balanced-vectors-blueprint **Next steps:** 1. **Enable GitHub Pages:** - Go to https://github.com/schmittj/balanced-vectors-blueprint/settings/pages - Under "Build and deployment" → Source, select **"GitHub Actions"** 2. **Watch the build:** - Go to https://github.com/schmittj/balanced-vectors-blueprint/actions - The first build will take a while (downloading mathlib cache ~2GB) - If there are errors, we may need to fix mathlib version compatibility 3. **Once built, your blueprint will be at:** - https://schmittj.github.io/balanced-vectors-blueprint/ **Potential issue:** The Lean project uses mathlib v4.14.0 (from several months ago), but our `BalancedVectors.lean` was developed with the latest mathlib. The build might fail if there are API differences. If that happens, we have two options: - Update to a newer mathlib version (and matching lean-toolchain) - Adjust the code for the older API Let me know once you've enabled GitHub Pages, and we can check if the build succeeds. If there are issues, I can help fix them! -------------------------------------------------------------------------------- USER (entry 1198): -------------------------------------------------------------------------------- Thanks a lot! Now the build-lean process did encounter an error as follows: ✖ [2504/2505] Building BalancedVectors trace: .> LEAN_PATH=././.lake/packages/Cli/.lake/build/lib:././.lake/packages/batteries/.lake/build/lib:././.lake/packages/Qq/.lake/build/lib:././.lake/packages/aesop/.lake/build/lib:././.lake/packages/proofwidgets/.lake/build/lib:././.lake/packages/importGraph/.lake/build/lib:././.lake/packages/LeanSearchClient/.lake/build/lib:././.lake/packages/plausible/.lake/build/lib:././.lake/packages/mathlib/.lake/build/lib:././.lake/build/lib LD_LIBRARY_PATH= /home/runner/.elan/toolchains/leanprover--lean4---v4.14.0/bin/lean -Dpp.unicode.fun=true -DautoImplicit=false ././././BalancedVectors.lean -R ./././. -o ././.lake/build/lib/BalancedVectors.olean -i ././.lake/build/lib/BalancedVectors.ilean -c ././.lake/build/ir/BalancedVectors.c --json error: ././././BalancedVectors.lean:147:24: tactic 'split_ifs' failed, no if-then-else conditions to split n : ℕ d : ℤ hd : 0 ≤ d k i : Fin n ⊢ 0 ≤ (fun i ↦ if i = k then d else 0) i error: ././././BalancedVectors.lean:189:4: tactic 'split_ifs' failed, no if-then-else conditions to split n : ℕ d : ℤ e : WeakComposition n d i j : Fin n hi : 1 ≤ e.toFun i hij : i ≠ j k : Fin n ⊢ 0 ≤ (fun k ↦ if k = i then e.toFun i - 1 else if k = j then e.toFun j + 1 else e.toFun k) k error: ././././BalancedVectors.lean:447:82: unknown identifier 'd' error: ././././BalancedVectors.lean:449:71: unknown identifier 'd' error: ././././BalancedVectors.lean:455:48: unknown identifier 'd' error: ././././BalancedVectors.lean:463:65: unknown identifier 'd' error: ././././BalancedVectors.lean:469:8: unknown identifier 'd' error: ././././BalancedVectors.lean:478:4: tactic 'split_ifs' failed, no if-then-else conditions to split n : ℕ d : ℤ e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 0 ≤ t htq : t ≤ e.toFun i + e.toFun j k : Fin n ⊢ 0 ≤ (fun k ↦ if k = i then t else if k = j then e.toFun i + e.toFun j - t else e.toFun k) k error: ././././BalancedVectors.lean:531:50: no goals to be solved error: ././././BalancedVectors.lean:557:8: no goals to be solved error: ././././BalancedVectors.lean:708:25: type mismatch single_le_sum (fun k x ↦ ?m.232868) (mem_univ i) has type ?m.225972 i ≤ ∑ x : Fin n, ?m.225972 x : Prop but is expected to have type ∑ k : Fin n, e.toFun k ^ 2 ≥ e.toFun i ^ 2 : Prop error: ././././BalancedVectors.lean:800:43: unknown identifier 'd' error: ././././BalancedVectors.lean:801:26: unknown identifier 'd' error: ././././BalancedVectors.lean:809:56: unknown identifier 'd' error: ././././BalancedVectors.lean:809:83: unknown identifier 'd' error: ././././BalancedVectors.lean:810:19: unknown identifier 'd' error: ././././BalancedVectors.lean:827:59: unknown identifier 'd' error: ././././BalancedVectors.lean:828:34: unknown identifier 'd' error: ././././BalancedVectors.lean:828:54: unknown identifier 'd' error: ././././BalancedVectors.lean:830:19: unknown identifier 'd' error: ././././BalancedVectors.lean:835:27: unknown identifier 'd' error: ././././BalancedVectors.lean:865:30: unknown identifier 'd' error: ././././BalancedVectors.lean:865:30: unknown identifier 'd' error: ././././BalancedVectors.lean:867:37: unknown identifier 'd' error: ././././BalancedVectors.lean:868:5: unknown identifier 'd' error: ././././BalancedVectors.lean:868:40: unknown identifier 'd' error: ././././BalancedVectors.lean:869:18: unknown identifier 'd' error: ././././BalancedVectors.lean:870:13: unknown identifier 'd' error: ././././BalancedVectors.lean:870:40: unknown identifier 'd' error: ././././BalancedVectors.lean:871:19: unknown identifier 'd' warning: ././././BalancedVectors.lean:980:8: declaration uses 'sorry' warning: ././././BalancedVectors.lean:1057:6: 'exact WeakComposition.measure_decreases_of_maxEntry_increases e e1 hn hmax_incr hmax_lt' tactic does nothing note: this linter can be disabled with `set_option linter.unusedTactic false` error: ././././BalancedVectors.lean:1124:44: unknown identifier 'd' error: ././././BalancedVectors.lean:1124:82: unknown identifier 'd' error: ././././BalancedVectors.lean:1126:35: unknown identifier 'd' error: Lean exited with code 1 Some required builds logged failures: - BalancedVectors error: build failed Error: Process completed with exit code 1. Can you have a look? -------------------------------------------------------------------------------- ASSISTANT (entry 1200): -------------------------------------------------------------------------------- The errors are due to API differences between mathlib v4.14.0 (old) and the version we developed against (Lean 4.26.0 with current mathlib). Let me fix this by updating to a compatible version: -------------------------------------------------------------------------------- ASSISTANT (entry 1201): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1202): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.14.0 2→] -------------------------------------------------------------------------------- ASSISTANT (entry 1204): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- ASSISTANT (entry 1205): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1206): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.16.0 2→] -------------------------------------------------------------------------------- USER (entry 1207): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→import Lake 2→open Lake DSL 3→ 4→package «balanced-vectors» where 5→ leanOptions := #[ 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→require mathlib from git 11→ "https://github.com/leanprover-community/mathlib4.git" @ "v4.16.0" 12→ 13→@[def...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1209): -------------------------------------------------------------------------------- Actually, let me check what mathlib versions are available and try a more recent one: -------------------------------------------------------------------------------- ASSISTANT (entry 1210): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1211): -------------------------------------------------------------------------------- [Tool Result: Request failed with status code 404] -------------------------------------------------------------------------------- ASSISTANT (entry 1213): -------------------------------------------------------------------------------- Let me try a different approach - use mathlib from the main branch with a recent Lean version: -------------------------------------------------------------------------------- ASSISTANT (entry 1214): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- ASSISTANT (entry 1215): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1216): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→import Lake 2→open Lake DSL 3→ 4→package «balanced-vectors» where 5→ leanOptions := #[ 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→require mathlib from git 11→ "https://github.com/leanprover-community/mathlib4.git" @ "v4.15.0" 12→ 13→@[def...[truncated]] -------------------------------------------------------------------------------- USER (entry 1217): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.15.0 2→] -------------------------------------------------------------------------------- ASSISTANT (entry 1219): -------------------------------------------------------------------------------- Actually, let me use the exact setup from the mathlib4 repo where we developed this: -------------------------------------------------------------------------------- ASSISTANT (entry 1220): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- ASSISTANT (entry 1221): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1222): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.15.0-rc1 2→] -------------------------------------------------------------------------------- USER (entry 1223): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→import Lake 2→open Lake DSL 3→ 4→package «balanced-vectors» where 5→ leanOptions := #[ 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→require mathlib from git 11→ "https://github.com/leanprover-community/mathlib4.git" @ "v4.15.0-rc1" 12→ 13→@...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1225): -------------------------------------------------------------------------------- Let me check the exact versions in the mathlib4 repo we developed in: -------------------------------------------------------------------------------- ASSISTANT (entry 1226): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1227): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0 --- import Lake open Lake DSL /-!] -------------------------------------------------------------------------------- ASSISTANT (entry 1229): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- ASSISTANT (entry 1230): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1231): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lean-toolchain has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→leanprover/lean4:v4.26.0 2→] -------------------------------------------------------------------------------- USER (entry 1232): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→import Lake 2→open Lake DSL 3→ 4→package «balanced-vectors» where 5→ leanOptions := #[ 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→-- Using the exact mathlib commit where the code was developed 11→require mathlib from git 12→ "https://github.c...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1234): -------------------------------------------------------------------------------- Now let me commit and push these version fixes: -------------------------------------------------------------------------------- ASSISTANT (entry 1235): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1236): -------------------------------------------------------------------------------- [Tool Result: [main edd2b54] fix: use exact mathlib version from development environment 2 files changed, 3 insertions(+), 7 deletions(-) To github.com:schmittj/balanced-vectors-blueprint.git 488ddff..edd2b54 main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1238): -------------------------------------------------------------------------------- Pushed! The build should now use: - **Lean 4.26.0** (same as mathlib4 development) - **mathlib commit `26fffff...`** (exact commit where we developed the code) This should fix the version mismatch issues. The GitHub Actions will re-run automatically. You can watch the progress at: https://github.com/schmittj/balanced-vectors-blueprint/actions The build will take some time since it needs to download/build mathlib cache for this specific version. Let me know if it succeeds or if there are any remaining errors! -------------------------------------------------------------------------------- USER (entry 1240): -------------------------------------------------------------------------------- Thanks, we are still getting some issues: Run lake build ✖ [3058/3059] Building BalancedVectors (24s) trace: .> LEAN_PATH=/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/Cli/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/batteries/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/Qq/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/aesop/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/proofwidgets/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/importGraph/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/LeanSearchClient/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/plausible/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/packages/mathlib/.lake/build/lib/lean:/home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/build/lib/lean /home/runner/.elan/toolchains/leanprover--lean4---v4.26.0/bin/lean /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/BalancedVectors.lean -o /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/build/lib/lean/BalancedVectors.olean -i /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/build/lib/lean/BalancedVectors.ilean -c /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/build/ir/BalancedVectors.c --setup /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/.lake/build/ir/BalancedVectors.setup.json --json error: BalancedVectors.lean:447:82: Unknown identifier `d` error: BalancedVectors.lean:449:71: Unknown identifier `d` error: BalancedVectors.lean:455:48: Unknown identifier `d` error: BalancedVectors.lean:463:65: Unknown identifier `d` error: BalancedVectors.lean:469:8: Unknown identifier `d` warning: BalancedVectors.lean:473:4: declaration uses 'sorry' info: BalancedVectors.lean:526:4: Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. error: BalancedVectors.lean:522:2: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 0 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q hq : q = e.toFun i + e.toFun j ht' : 0 ≤ q - t htq' : q - t ≤ q σ : Equiv.Perm (Fin n) := Equiv.swap i j hσ : σ = Equiv.swap i j hsym_app : D { toFun := (sliceComposition e i j hij (q - t) ht' htq').toFun ∘ ⇑(Equiv.symm σ), sum_eq := ⋯, nonneg := ⋯ } = D (sliceComposition e i j hij (q - t) ht' htq') k : Fin n hki : k = i hswap_k : (Equiv.symm σ) k = j ⊢ (sorry e i j hij (q - t) ht' htq').toFun j = (sorry e i j hij t ht htq).toFun i info: BalancedVectors.lean:531:50: Try this: [apply] ring_nf The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. error: BalancedVectors.lean:528:4: unsolved goals case pos n : ℕ d : ℤ D : WeakComposition n d → ℚ hsym : IsSymmetric D e : WeakComposition n d i j : Fin n hij : i ≠ j t : ℤ ht : 0 ≤ t q : ℤ := e.toFun i + e.toFun j htq : t ≤ q hq : q = e.toFun i + e.toFun j ht' : 0 ≤ q - t htq' : q - t ≤ q σ : Equiv.Perm (Fin n) := Equiv.swap i j hσ : σ = Equiv.swap i j hsym_app : D { toFun := (sliceComposition e i j hij (q - t) ht' htq').toFun ∘ ⇑(Equiv.symm σ), sum_eq := ⋯, nonneg := ⋯ } = D (sliceComposition e i j hij (q - t) ht' htq') k : Fin n hki : ¬k = i hkj : k = j hswap_k : (Equiv.symm σ) k = i ⊢ (sorry e i j hij (q - t) ht' htq').toFun i = (sorry e i j hij t ht htq).toFun j error: BalancedVectors.lean:536:6: `simp` made no progress warning: BalancedVectors.lean:520:13: declaration uses 'sorry' warning: BalancedVectors.lean:525:20: This simp argument is unused: hij.symm Hint: Omit it from the simp argument list. simp only [hki, h̵i̵j̵.̵s̵y̵m̵m̵,̵ ̵ite_false, ite_true] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` warning: BalancedVectors.lean:525:30: This simp argument is unused: ite_false Hint: Omit it from the simp argument list. simp only [hki, hij.symm, i̵t̵e̵_̵f̵a̵l̵s̵e̵,̵ ̵ite_true] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` warning: BalancedVectors.lean:525:41: This simp argument is unused: ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 1242): -------------------------------------------------------------------------------- These errors are very strange - the `sorry` appearing where `sliceComposition` should be suggests something is corrupted. Let me check if the files are identical: -------------------------------------------------------------------------------- ASSISTANT (entry 1243): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1244): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1246): -------------------------------------------------------------------------------- The files are identical. The issue might be with how Lake resolves dependencies. Let me check the specific lines causing issues: -------------------------------------------------------------------------------- ASSISTANT (entry 1247): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1248): -------------------------------------------------------------------------------- [Tool Result: 440→ rcases hij with h1 | h2 441→ · exact ⟨i, j, fun heq => by simp only [heq] at h1; omega, by omega⟩ 442→ · exact ⟨j, i, fun heq => by simp only [heq] at h2; omega, by omega⟩ 443→ 444→/-! ### Slice Analysis -/ 445→ 446→/-- Auxiliary: sum is preserved when we put t at position i and (q-t) at position j. -/ 447→lemma sum_slice_eq {e : Fin n → ℤ} {i j : Fin n} (hij : i ≠ j) (hsum : ∑ k, e k = d) 448→ (t : ℤ) : 449→ ∑ k, (if k = i then t else if k = j then e ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1250): -------------------------------------------------------------------------------- I see the issue! The variable `d` is scoped inside a namespace. Let me check the variable declarations: -------------------------------------------------------------------------------- ASSISTANT (entry 1251): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1252): -------------------------------------------------------------------------------- [Tool Result: 87:variable {n : ℕ} 133:namespace WeakComposition 135:variable {d : ℤ} 207:end WeakComposition 798:namespace WeakComposition 874:end WeakComposition 1085:namespace SymmetricLogConcaveFunction 1087:variable {n : ℕ} {d : ℤ} 1116:end SymmetricLogConcaveFunction] -------------------------------------------------------------------------------- ASSISTANT (entry 1254): -------------------------------------------------------------------------------- The problem is clear now - `variable {d : ℤ}` is only scoped inside the `WeakComposition` namespace (lines 133-207), but functions after line 207 use `d` without it being in scope. This worked in mathlib4 because we were developing inside that repo's context. Let me add the missing variable declaration: -------------------------------------------------------------------------------- ASSISTANT (entry 1255): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1256): -------------------------------------------------------------------------------- [Tool Result: 200→ (e.modify i j hi hij) j = e j + 1 := by simp only [modify, hij.symm, ite_false, ite_true] 201→ 202→@[simp] 203→lemma modify_at_other (e : WeakComposition n d) (i j k : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) 204→ (hki : k ≠ i) (hkj : k ≠ j) : (e.modify i j hi hij) k = e k := by 205→ simp only [modify, hki, hkj, ite_false] 206→ 207→end WeakComposition 208→ 209→/-! ### Balanced and Concentrated Vectors -/ 210→ 211→/-- A vector is balanced if all entries ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1258): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1260): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 203→lemma modify_at_other (e : WeakComposition n d) (i j k : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) 204→ (hki : k ≠ i) (hkj : k ≠ j) : (e.modify i j hi hij) k = e k := by 205→ simp only [modify, hki, hkj, ite_false] 206→ 207→end WeakComposition 208→ 209→variable {d : ℤ} 210→ 211→/-! ### Balanced ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1262): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1263): -------------------------------------------------------------------------------- [Tool Result: [main 79dc1fa] fix: add missing variable declaration for d 1 file changed, 2 insertions(+) To github.com:schmittj/balanced-vectors-blueprint.git edd2b54..79dc1fa main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1265): -------------------------------------------------------------------------------- Pushed! This should fix the `Unknown identifier 'd'` errors. The build will re-run automatically. There may still be other issues - I noticed some strange errors where `sorry` was appearing in place of `sliceComposition` in the goal states, which is very unusual. Let me know what happens with this build. If there are remaining errors, we may need to dig deeper into why the file behaves differently as a standalone project vs inside mathlib4. -------------------------------------------------------------------------------- USER (entry 1267): -------------------------------------------------------------------------------- Perfect, the build-lean process completes successfully. There is an issue with build-blueprint, but before getting started: can you get CLAUDE.md up to date with our current endeavour? That way the information is preserved when we have to compact the conversation soon. -------------------------------------------------------------------------------- ASSISTANT (entry 1269): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1270): -------------------------------------------------------------------------------- [Tool Result: 1→# Claude Code Best Practices for Mathlib4 2→ 3→## Environment Setup 4→ 5→Lake and Lean are installed via elan. To access them: 6→```bash 7→source ~/.elan/env 8→``` 9→ 10→## Compiling Lean Files 11→ 12→Always save compile output to a temp file for easier analysis: 13→```bash 14→source ~/.elan/env && cd /mnt/c/Users/jo314/Documents/mathlib4 && lake env lean BalancedVectors.lean 2>&1 | tee /tmp/lean_output.txt 15→``` 16→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1272): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1273): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/CLAUDE.md has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→# Claude Code Best Practices for Mathlib4 2→ 3→## Project Overview 4→ 5→This project formalizes the theorem that symmetric log-concave functions on weak compositions 6→are maximized on balanced vectors and minimized on concentrated vectors. 7→ 8→**Main application:** Descendant integrals on moduli spaces of cur...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1275): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1276): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1278): -------------------------------------------------------------------------------- Done! I've updated `CLAUDE.md` with comprehensive documentation including: - Project overview and main theorem - File locations (mathlib4 repo and blueprint repo) - Main theorems and definitions with line numbers - Blueprint project structure - Key setup notes (especially the variable scoping fix) - Building instructions - Common issues and their solutions - Git commit history Now please share the build-blueprint error and I'll help fix it! -------------------------------------------------------------------------------- USER (entry 1283): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me chronologically analyze the conversation: 1. **Initial Context**: This is a continuation from a previous session about formalizing a Lean 4/Mathlib4 theorem about balanced/concentrated vectors. The file `BalancedVectors.lean` was already complete with all proofs. 2. **User's Main Request**: The user wanted to restructure the file for better presentation (Lean Blueprint style) so that a Lean expert can easily verify the formalization matches the informal theorem. 3. **First Phase - Documentation**: I added a comprehensive header documentation with: - Main Statement section - Key Definitions table with line numbers - Main Theorems table with line numbers - Proof Strategy outline - Created git commit 4. **Second Phase - Main Results Section**: Added clean wrapper theorems at end of file: - `SymmetricLogConcaveFunction` structure bundling D with its 3 properties - `maximized_on_balanced`, `minimized_on_concentrated`, `main_theorem` - Updated documentation line numbers - Created git commit 5. **Third Phase - Blueprint Repository Setup**: - User created GitHub repo `schmittj/balanced-vectors-blueprint` - I set up the project structure: - `lakefile.lean` with mathlib dependency - `lean-toolchain` - Copied `BalancedVectors.lean` - Created `blueprint/` directory structure - Created `content.tex` with Blueprint annotations - Created GitHub Actions workflow - Created README.md, .gitignore - Pushed to GitHub 6. **Build Errors - Version Mismatch**: - First attempt with mathlib v4.14.0 failed with many API errors - Updated to exact mathlib commit `26fffff...` with Lean 4.26.0 - Still had errors with `Unknown identifier 'd'` 7. **Variable Scoping Fix**: - The issue was `variable {d : ℤ}` was only scoped inside `WeakComposition` namespace - Added `variable {d : ℤ}` after `end WeakComposition` in the blueprint repo's file - This fixed the Lean build 8. **Current State**: - Lean build passes in CI - Blueprint build has issues (not yet shared by user) - User asked me to update CLAUDE.md before conversation compacts - I updated CLAUDE.md with comprehensive project documentation Key files: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - original, complete - `/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/BalancedVectors.lean` - standalone version with variable fix - `/mnt/c/Users/jo314/Documents/mathlib4/CLAUDE.md` - documentation for context preservation - Blueprint configuration files in the blueprint repo Summary: 1. Primary Request and Intent: - Restructure `BalancedVectors.lean` for better presentation following Lean Blueprint style - Make it easy for a Lean expert to verify the formalization matches the informal theorem - Set up a GitHub Blueprint repository (`schmittj/balanced-vectors-blueprint`) with: - Standalone Lean project that compiles independently of mathlib4 repo - Blueprint documentation infrastructure (LaTeX with annotations) - GitHub Actions for automated building - GitHub Pages deployment - The user has an existing paper (`main.tex`) and wants to eventually include the Blueprint content as a section 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Lean Blueprint (https://github.com/PatrickMassot/leanblueprint) for documentation - Weak compositions: vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - `SymmetricLogConcaveFunction` - bundles D with symmetry, log-concavity, positivity properties - Lake package management for Lean projects - GitHub Actions CI/CD - Variable scoping in Lean (namespace-local vs file-level) - Mathlib version pinning (commit `26fffffcccd7299b26cf63fac902165bc553fd56` with Lean 4.26.0) 3. Files and Code Sections: - **`/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean`** (~1130 lines) - Main Lean formalization, all proofs complete - Contains `main_theorem`, `balanced_maximizes`, `concentrated_minimizes` - Has comprehensive header documentation with theorem blueprint - **`/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/BalancedVectors.lean`** - Copy of above with critical fix: added `variable {d : ℤ}` after `end WeakComposition` ```lean end WeakComposition variable {d : ℤ} /-! ### Balanced and Concentrated Vectors -/ ``` - **`/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean`** ```lean import Lake open Lake DSL package «balanced-vectors» where leanOptions := #[ ⟨`pp.unicode.fun, true⟩, ⟨`autoImplicit, false⟩ ] require mathlib from git "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" @[default_target] lean_lib «BalancedVectors» w ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- USER (entry 1289): -------------------------------------------------------------------------------- Welcome back! The build-lean process of our github blueprint repo succeeded, but the build-blueprint resulted in the following issue: Building wheels for collected packages: pygraphviz Building wheel for pygraphviz (pyproject.toml): started Building wheel for pygraphviz (pyproject.toml): finished with status 'error' error: subprocess-exited-with-error × Building wheel for pygraphviz (pyproject.toml) did not run successfully. │ exit code: 1 ╰─> [98 lines of output] /tmp/pip-build-env-_7j31ebt/overlay/lib/python3.11/site-packages/setuptools/config/_apply_pyprojecttoml.py:82: SetuptoolsDeprecationWarning: `project.license` as a TOML table is deprecated !! ******************************************************************************** Please use a simple string containing a SPDX expression for `project.license`. You can also use `project.license-files`. (Both options available on setuptools>=77.0.0). By 2026-Feb-18, you need to update your project and remove deprecated calls or your builds will no longer be supported. See https://packaging.python.org/en/latest/guides/writing-pyproject-toml/#license for details. ******************************************************************************** !! corresp(dist, value, root_dir) /tmp/pip-build-env-_7j31ebt/overlay/lib/python3.11/site-packages/setuptools/config/_apply_pyprojecttoml.py:61: SetuptoolsDeprecationWarning: License classifiers are deprecated. !! ******************************************************************************** Please consider removing the following classifiers in favor of a SPDX license expression: License :: OSI Approved :: BSD License See https://packaging.python.org/en/latest/guides/writing-pyproject-toml/#license for details. ******************************************************************************** !! dist._finalize_license_expression() /tmp/pip-build-env-_7j31ebt/overlay/lib/python3.11/site-packages/setuptools/dist.py:759: SetuptoolsDeprecationWarning: License classifiers are deprecated. !! ******************************************************************************** Please consider removing the following classifiers in favor of a SPDX license expression: License :: OSI Approved :: BSD License See https://packaging.python.org/en/latest/guides/writing-pyproject-toml/#license for details. ******************************************************************************** !! self._finalize_license_expression() running bdist_wheel running build running build_py creating build/lib.linux-x86_64-cpython-311/pygraphviz copying pygraphviz/graphviz.py -> build/lib.linux-x86_64-cpython-311/pygraphviz copying pygraphviz/agraph.py -> build/lib.linux-x86_64-cpython-311/pygraphviz copying pygraphviz/scraper.py -> build/lib.linux-x86_64-cpython-311/pygraphviz copying pygraphviz/testing.py -> build/lib.linux-x86_64-cpython-311/pygraphviz copying pygraphviz/__init__.py -> build/lib.linux-x86_64-cpython-311/pygraphviz creating build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_edge_attributes.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_layout.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_attribute_defaults.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_drawing.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_node_attributes.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_clear.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_graph.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_unicode.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_readwrite.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_string.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_close.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_repr_mimebundle.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_subgraph.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_html.py -> build/lib.linux-x86_64-cpython-311/pygraphviz/tests copying pygraphviz/tests/test_scraper.py -> build/lib.linux-x86_64-cpython-311/pygrap ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 1291): -------------------------------------------------------------------------------- The error is that `pygraphviz` needs the graphviz C libraries and headers installed on the system before it can build. The workflow is missing the system dependency installation step. Let me fix the GitHub Actions workflow. -------------------------------------------------------------------------------- ASSISTANT (entry 1292): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1293): -------------------------------------------------------------------------------- [Tool Result: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "pages" 17→ cancel-in-progress: false 18→ 19→jobs: 20→ build-lean: 21→ runs-on: ubuntu-latest 22→ steps: 23→ - name: Checkout 24→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1295): -------------------------------------------------------------------------------- I need to add a step to install graphviz system dependencies before pip install. -------------------------------------------------------------------------------- ASSISTANT (entry 1296): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1297): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 40→ steps: 41→ - name: Checkout 42→ uses: actions/checkout@v4 43→ 44→ - name: Install Python 45→ uses: actions/setup-python@v5 46→ with: 47→ python-version: '3.11' 48→ 49→ - name: Install graphviz 50→ run: | 51→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1299): -------------------------------------------------------------------------------- Now let me commit and push this fix. -------------------------------------------------------------------------------- ASSISTANT (entry 1300): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1301): -------------------------------------------------------------------------------- [Tool Result: [main cce3d1e] fix: add graphviz system dependencies for pygraphviz build 1 file changed, 5 insertions(+)] -------------------------------------------------------------------------------- ASSISTANT (entry 1303): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1304): -------------------------------------------------------------------------------- [Tool Result: To github.com:schmittj/balanced-vectors-blueprint.git 79dc1fa..cce3d1e main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1306): -------------------------------------------------------------------------------- Done! I added the missing graphviz system dependencies. The error was: ``` graphviz/cgraph.h: No such file or directory ``` `pygraphviz` is a Python wrapper around the graphviz C library, so it needs the C headers to compile. The fix installs both: - `graphviz` - the runtime - `graphviz-dev` - the development headers The workflow will re-run automatically. Check the Actions tab at: https://github.com/schmittj/balanced-vectors-blueprint/actions Once that passes, you'll need to enable GitHub Pages in the repository settings (Settings → Pages → Source: GitHub Actions) to deploy the blueprint website. -------------------------------------------------------------------------------- USER (entry 1308): -------------------------------------------------------------------------------- Ok, good progress, now we have: Run cd blueprint /bin/sh: 1: latexmk: not found Command 'latexmk -output-directory=../print' returned non-zero exit status 127. Error: Process completed with exit code 255. Note: happy to continue the process, but if there is a way to run this setup also locally to check if everything goes through, feel free to do that additionally/instead! -------------------------------------------------------------------------------- ASSISTANT (entry 1310): -------------------------------------------------------------------------------- Good idea - let me try building locally to catch all missing dependencies at once. -------------------------------------------------------------------------------- ASSISTANT (entry 1311): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1312): -------------------------------------------------------------------------------- [Tool Result: Stored in directory: /home/jo314/.cache/pip/wheels/e1/2f/c3/0f7719eda975973b474b79b98e1208e635565c1e4dac777ac8 Successfully built pygraphviz Installing collected packages: Unidecode, tomlkit, smmap, pygraphviz, pygments, mdurl, plasTeX, markdown-it-py, gitdb, rich, plastexshowmore, plastexdepgraph, GitPython, rich-click, leanblueprint Successfully installed GitPython-3.1.45 Unidecode-1.4.0 gitdb-4.0.12 leanblueprint-0.0.19 markdown-it-py-4.0.0 mdurl-0.1.2 plasTeX-3.1 plastexdepgraph-0.0.5 pla...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1314): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1315): -------------------------------------------------------------------------------- [Tool Result: Rc files read: /etc/LatexMk Latexmk: This is Latexmk, John Collins, 31 Jan. 2024. Version 4.83. No existing .aux file, so I'll make a simple one, and require run of *latex. Latexmk: applying rule 'lualatex'... Rule 'lualatex': Reasons for rerun Category 'other': Rerun of 'lualatex' forced or previously required: Reason or flag: 'Initial setup' ------------ Run number 1 of rule 'lualatex' ------------ ------------ Running 'lualatex -recorder -output-directory="/mnt/c/Users/jo314/Docume...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1317): -------------------------------------------------------------------------------- The local build got further - it found `latexmk` but is missing `blueprint.sty`. Let me check the leanblueprint package structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1318): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1319): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1320): -------------------------------------------------------------------------------- [Tool Result: /home/jo314/miniforge3/lib/python3.12/site-packages/leanblueprint/__init__.py] -------------------------------------------------------------------------------- USER (entry 1321): -------------------------------------------------------------------------------- [Tool Result: Location: /home/jo314/miniforge3/lib/python3.12/site-packages] -------------------------------------------------------------------------------- ASSISTANT (entry 1323): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1324): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1325): -------------------------------------------------------------------------------- [Tool Result: /home/jo314/miniforge3/lib/python3.12/site-packages/leanblueprint/templates/blueprint.sty] -------------------------------------------------------------------------------- USER (entry 1326): -------------------------------------------------------------------------------- [Tool Result: total 56 drwxr-xr-x 7 jo314 jo314 4096 Dec 14 17:23 . drwxr-xr-x 208 jo314 jo314 12288 Dec 14 17:23 .. drwxr-xr-x 4 jo314 jo314 4096 Dec 14 17:23 Packages -rw-r--r-- 1 jo314 jo314 0 Dec 14 17:23 __init__.py drwxr-xr-x 2 jo314 jo314 4096 Dec 14 17:23 __pycache__ -rw-r--r-- 1 jo314 jo314 19905 Dec 14 17:23 client.py drwxr-xr-x 5 jo314 jo314 4096 Dec 14 17:23 jekyll_templates drwxr-xr-x 2 jo314 jo314 4096 Dec 14 17:23 static drwxr-xr-x 3 jo314 jo314 4096 Dec 14 17:23 templ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1328): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1329): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1330): -------------------------------------------------------------------------------- [Tool Result: Usage: leanblueprint [OPTIONS] COMMAND [ARGS]... Command line client to manage Lean blueprints. Use leanblueprint COMMAND --help to get more help on any specific command. ╭─ Options ────────────────────────────────────────────────────────────────────╮ │ --debug ...[truncated]] -------------------------------------------------------------------------------- USER (entry 1331): -------------------------------------------------------------------------------- [Tool Result: \DeclareOption*{} \ProcessOptions \newcommand{\graphcolor}[3]{}] -------------------------------------------------------------------------------- ASSISTANT (entry 1333): -------------------------------------------------------------------------------- Let me check if there's a standard way to set up the blueprint - the `leanblueprint new` command might show us the expected structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1334): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1335): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Error: Could not find a Lean project. Please run this command from inside your project folder.] -------------------------------------------------------------------------------- ASSISTANT (entry 1337): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1338): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1340): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1341): -------------------------------------------------------------------------------- [Tool Result: Rc files read: /etc/LatexMk Latexmk: This is Latexmk, John Collins, 31 Jan. 2024. Version 4.83. Latexmk: applying rule 'lualatex'... Rule 'lualatex': Reasons for rerun Changed files or newly in use/created: blueprint.sty Category 'changed_user': blueprint.sty ------------ Run number 1 of rule 'lualatex' ------------ ------------ Running 'lualatex -recorder -output-directory="/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/print" "content.tex"' ------------ This is Lua...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1343): -------------------------------------------------------------------------------- Progress! The `blueprint.sty` was found but the `\lean` command isn't defined. Let me check the leanblueprint templates more thoroughly. -------------------------------------------------------------------------------- ASSISTANT (entry 1344): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1345): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1346): -------------------------------------------------------------------------------- [Tool Result: total 44 drwxr-xr-x 3 jo314 jo314 4096 Dec 14 17:23 . drwxr-xr-x 7 jo314 jo314 4096 Dec 14 17:23 .. -rw-r--r-- 1 jo314 jo314 65 Dec 14 17:23 blueprint.sty -rw-r--r-- 1 jo314 jo314 1545 Dec 14 17:23 blueprint.yml -rw-r--r-- 1 jo314 jo314 364 Dec 14 17:23 content.tex -rw-r--r-- 1 jo314 jo314 483 Dec 14 17:23 extra_styles.css -rw-r--r-- 1 jo314 jo314 182 Dec 14 17:23 latexmkrc drwxr-xr-x 2 jo314 jo314 4096 Dec 14 17:23 macros -rw-r--r-- 1 jo314 jo314 340 Dec 14 17:23 plastex.cfg -rw-r--r-- 1 ...[truncated]] -------------------------------------------------------------------------------- USER (entry 1347): -------------------------------------------------------------------------------- [Tool Result: \DeclareOption*{} \ProcessOptions \newcommand{\graphcolor}[3]{}] -------------------------------------------------------------------------------- ASSISTANT (entry 1349): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1350): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1351): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1352): -------------------------------------------------------------------------------- [Tool Result: % In this file you should put the actual content of the blueprint. % It will be used both by the web and the print version. % It should *not* include the \begin{document} % % If you want to split the blueprint content into several files then % the current file can be a simple sequence of \input. Otherwise It % can start with a \section or \chapter for instance.] -------------------------------------------------------------------------------- USER (entry 1353): -------------------------------------------------------------------------------- [Tool Result: % This file makes a printable version of the blueprint % It should include all the \usepackage needed for the pdf version. % The template version assume you want to use a modern TeX compiler % such as xeLaTeX or luaLaTeX including support for unicode % and Latin Modern Math font with standard bugfixes applied. % It also uses expl3 in order to support macros related to the dependency graph. % It also includes standard AMS packages (and their improved version % mathtools) as well as support for li...[truncated]] -------------------------------------------------------------------------------- USER (entry 1354): -------------------------------------------------------------------------------- [Tool Result: total 20 drwxr-xr-x 2 jo314 jo314 4096 Dec 14 17:23 . drwxr-xr-x 3 jo314 jo314 4096 Dec 14 17:23 .. -rw-r--r-- 1 jo314 jo314 821 Dec 14 17:23 common.tex -rw-r--r-- 1 jo314 jo314 1102 Dec 14 17:23 print.tex -rw-r--r-- 1 jo314 jo314 261 Dec 14 17:23 web.tex] -------------------------------------------------------------------------------- ASSISTANT (entry 1356): -------------------------------------------------------------------------------- I see now - the leanblueprint expects a specific structure: - `content.tex` should be just content (no `\documentclass`, no `\begin{document}`) - `print.tex` is the wrapper for PDF with preamble - `macros/` folder contains the actual `\lean`, `\leanok` definitions Let me look at the macros and fix the structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1357): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1358): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1359): -------------------------------------------------------------------------------- [Tool Result: % In this file you should put all LaTeX macros and settings to be used both by % the pdf version and the web version. % This should be most of your macros. % The theorem-like environments defined below are those that appear by default % in the dependency graph. See the README of leanblueprint if you need help to % customize this. % The configuration below use the theorem counter for all those environments % (this is what the [theorem] arguments mean) and never resets it. % If you want for inst...[truncated]] -------------------------------------------------------------------------------- USER (entry 1360): -------------------------------------------------------------------------------- [Tool Result: % In this file you should put macros to be used only by % the printed version. Of course they should have a corresponding % version in macros/web.tex. % Typically the printed version could have more fancy decorations. % This should be a very short file. % % This file starts with dummy macros that ensure the pdf % compiler will ignore macros provided by plasTeX that make % sense only for the web version, such as dependency graph % macros. % Dummy macros that make sense only for web version. \ne...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1362): -------------------------------------------------------------------------------- Now I understand the full structure. Let me set up the proper blueprint structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1363): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1364): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1365): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1366): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1367): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 1368): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 1369): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 1370): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1372): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1373): -------------------------------------------------------------------------------- [Tool Result: % This file makes a web version of the blueprint % It should include all the \usepackage needed for this version. % The template includes standard AMS packages. % It is otherwise a very minimal preamble (you should probably at least % add cleveref and tikz-cd). \documentclass{{|documentclass|}} \usepackage{amssymb, amsthm, amsmath} \usepackage{hyperref} \usepackage[{% if showmore %}showmore, {% endif %}dep_graph]{blueprint} \input{macros/common} \input{macros/web} \home{{| home |}} \github{...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1375): -------------------------------------------------------------------------------- I see these templates have placeholders (`{|title|}`, etc.) that leanblueprint fills in. Let me try using `leanblueprint new` to generate the proper structure from within the Lean project. -------------------------------------------------------------------------------- ASSISTANT (entry 1376): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1377): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Welcome to Lean blueprint Error: The repository contains uncommitted changes. Please clean it up before creating a blueprint.] -------------------------------------------------------------------------------- ASSISTANT (entry 1379): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1380): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1382): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1383): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1385): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1386): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1388): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1389): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1391): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1392): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1394): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1395): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1397): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1398): -------------------------------------------------------------------------------- [Tool Result: File does not exist. Current working directory: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint] -------------------------------------------------------------------------------- ASSISTANT (entry 1400): -------------------------------------------------------------------------------- I see - when we ran `rm -rf blueprint`, it deleted the folder and now the shell is in a non-existent directory. Let me recreate the structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1401): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1402): -------------------------------------------------------------------------------- [Tool Result: Exit code 1] -------------------------------------------------------------------------------- ASSISTANT (entry 1404): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1405): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml] -------------------------------------------------------------------------------- USER (entry 1407): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 1409): -------------------------------------------------------------------------------- btw feel free to git reset to origin if that is helpful in recovering any missing files -------------------------------------------------------------------------------- ASSISTANT (entry 1411): -------------------------------------------------------------------------------- Good idea, let me reset to origin and start fresh. -------------------------------------------------------------------------------- ASSISTANT (entry 1412): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1413): -------------------------------------------------------------------------------- [Tool Result: HEAD is now at cce3d1e fix: add graphviz system dependencies for pygraphviz build /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint] -------------------------------------------------------------------------------- ASSISTANT (entry 1415): -------------------------------------------------------------------------------- Now let me use `leanblueprint new` to create the proper structure. -------------------------------------------------------------------------------- ASSISTANT (entry 1416): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1417): -------------------------------------------------------------------------------- [Tool Result: [main c9ce20b] chore: remove blueprint dir to recreate with leanblueprint new 3 files changed, 225 deletions(-) delete mode 100644 blueprint/blueprint.toml delete mode 100644 blueprint/src/content.tex delete mode 100644 blueprint/src/references.bib] -------------------------------------------------------------------------------- ASSISTANT (entry 1419): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1420): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Welcome to Lean blueprint We will now ask some questions to configure your blueprint. All answers can be changed later by editing either the plastex.cfg file or the tex files. General information about the project Project title (My formalization project): Aborted.] -------------------------------------------------------------------------------- ASSISTANT (entry 1422): -------------------------------------------------------------------------------- The command is interactive. Let me try providing answers via stdin. -------------------------------------------------------------------------------- ASSISTANT (entry 1423): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1424): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Welcome to Lean blueprint We will now ask some questions to configure your blueprint. All answers can be changed later by editing either the plastex.cfg file or the tex files. General information about the project Project title (My formalization project): Author (use \and to separate authors if needed) (Johannes Schmitt): URL of GitHub repository (https://github.com/schmittj/balanced-vectors-blueprint): URL of project website (https://schmittj.github.io/balanced-vectors-blueprin...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1426): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1427): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Welcome to Lean blueprint We will now ask some questions to configure your blueprint. All answers can be changed later by editing either the plastex.cfg file or the tex files. General information about the project Project title (My formalization project): Author (use \and to separate authors if needed) (Johannes Schmitt): URL of GitHub repository (https://github.com/schmittj/balanced-vectors-blueprint): URL of project website (https://schmittj.github.io/balanced-vectors-blueprin...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1429): -------------------------------------------------------------------------------- It got further. Need more responses for the lakefile questions. -------------------------------------------------------------------------------- ASSISTANT (entry 1430): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1431): -------------------------------------------------------------------------------- [Tool Result: Exit code 1 Welcome to Lean blueprint Error: There is already a blueprint folder. Aborting blueprint creation.] -------------------------------------------------------------------------------- ASSISTANT (entry 1433): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1434): -------------------------------------------------------------------------------- [Tool Result: total 0 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 .. drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 src] -------------------------------------------------------------------------------- ASSISTANT (entry 1436): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1437): -------------------------------------------------------------------------------- [Tool Result: blueprint/: total 0 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 .. drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 src blueprint/src: total 8 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 .. -rwxrwxrwx 1 jo314 jo314 64 Dec 14 17:28 blueprint.sty -rwxrwxrwx 1 jo314 jo314 363 Dec 14 17:28 content.tex -rwxrwxrwx 1 jo314 jo314 482 Dec 14 17:28 extra_styles.css -rwxrwxrwx 1 jo314 jo314 181 Dec 14 17:28 latexmkrc dr...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1439): -------------------------------------------------------------------------------- The structure was created. Now I need to add the `blueprint.toml` config and update `content.tex` with our theorem content. -------------------------------------------------------------------------------- ASSISTANT (entry 1440): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 1441): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1442): -------------------------------------------------------------------------------- [Tool Result: 1→% In this file you should put the actual content of the blueprint. 2→% It will be used both by the web and the print version. 3→% It should *not* include the \begin{document} 4→% 5→% If you want to split the blueprint content into several files then 6→% the current file can be a simple sequence of \input. Otherwise It 7→% can start with a \section or \chapter for instance. Whenever you read a file, you should consider whether it would be co...[truncated]] -------------------------------------------------------------------------------- USER (entry 1443): -------------------------------------------------------------------------------- [Tool Result: 1→% This file makes a printable version of the blueprint 2→% It should include all the \usepackage needed for the pdf version. 3→% The template version assume you want to use a modern TeX compiler 4→% such as xeLaTeX or luaLaTeX including support for unicode 5→% and Latin Modern Math font with standard bugfixes applied. 6→% It also uses expl3 in order to support macros related to the dependency graph. 7→% It also includes standard AMS packages (and their improv...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1445): -------------------------------------------------------------------------------- Now let me create the blueprint.toml and add the content. -------------------------------------------------------------------------------- ASSISTANT (entry 1446): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1447): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/blueprint.toml] -------------------------------------------------------------------------------- ASSISTANT (entry 1450): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1452): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→\chapter{Balanced Vector Optimization} 2→ 3→\section{Introduction} 4→ 5→This blueprint documents the Lean 4 formalization of a theorem about symmetric log-concave functions on weak compositions. The main result states that such functions are maximized on balanced vectors and minimized on conc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1454): -------------------------------------------------------------------------------- Thanks for the offer! The structure is now set up. Let me test the build locally. -------------------------------------------------------------------------------- ASSISTANT (entry 1455): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1456): -------------------------------------------------------------------------------- [Tool Result: Rc files read: /etc/LatexMk latexmkrc Latexmk: This is Latexmk, John Collins, 31 Jan. 2024. Version 4.83. No existing .aux file, so I'll make a simple one, and require run of *latex. Latexmk: applying rule 'pdflatex'... Rule 'pdflatex': Reasons for rerun Category 'other': Rerun of 'pdflatex' forced or previously required: Reason or flag: 'Initial setup' ------------ Run number 1 of rule 'pdflatex' ------------ ------------ Running 'xelatex -synctex=1 -recorder -output-directory="/mn...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1458): -------------------------------------------------------------------------------- Excellent progress! The PDF and HTML were generated. The only issue is it tries to run `lake build` at the end to verify declarations, but lake isn't in PATH. Let me check what was generated. -------------------------------------------------------------------------------- ASSISTANT (entry 1459): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 1460): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1461): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 1462): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1464): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1465): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1467): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1468): -------------------------------------------------------------------------------- [Tool Result: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint total 0 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:28 .. -rwxrwxrwx 1 jo314 jo314 421 Dec 14 17:28 blueprint.toml -rwxrwxrwx 1 jo314 jo314 288 Dec 14 17:29 lean_decls drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 print drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 src drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 web] -------------------------------------------------------------------------------- ASSISTANT (entry 1470): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1471): -------------------------------------------------------------------------------- [Tool Result: total 64 drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 . drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 .. -rwxrwxrwx 1 jo314 jo314 19056 Dec 14 17:29 dep_graph_document.html -rwxrwxrwx 1 jo314 jo314 3689 Dec 14 17:29 index.html drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 js -rwxrwxrwx 1 jo314 jo314 27079 Dec 14 17:29 sect0001.html drwxrwxrwx 1 jo314 jo314 4096 Dec 14 17:29 styles -rwxrwxrwx 1 jo314 jo314 11977 Dec 14 17:29 symbol-defs.svg] -------------------------------------------------------------------------------- ASSISTANT (entry 1473): -------------------------------------------------------------------------------- The build succeeded locally. We have HTML and PDF. The `lake build` error at the end is just for checking declarations exist - we can skip that or make lake available. Let me update the GitHub workflow to: 1. Add LaTeX dependencies 2. Either skip declaration checking or add lake to the blueprint job -------------------------------------------------------------------------------- ASSISTANT (entry 1474): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1475): -------------------------------------------------------------------------------- [Tool Result: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "pages" 17→ cancel-in-progress: false 18→ 19→jobs: 20→ build-lean: 21→ runs-on: ubuntu-latest 22→ steps: 23→ - name: Checkout 24→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1477): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1478): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1480): -------------------------------------------------------------------------------- Now let me commit all changes and push. -------------------------------------------------------------------------------- ASSISTANT (entry 1481): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1482): -------------------------------------------------------------------------------- [Tool Result: On branch main Your branch is ahead of 'origin/main' by 1 commit. (use "git push" to publish your local commits) Changes not staged for commit: (use "git add ..." to update what will be committed) (use "git restore ..." to discard changes in working directory) modified: .github/workflows/push.yml Untracked files: (use "git add ..." to include in what will be committed) blueprint/ no changes added to commit (use "git add" and/or "git commit -a")] -------------------------------------------------------------------------------- ASSISTANT (entry 1484): -------------------------------------------------------------------------------- Let me add a .gitignore to exclude generated files and then commit. -------------------------------------------------------------------------------- ASSISTANT (entry 1485): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1486): -------------------------------------------------------------------------------- [Tool Result: 1→# Lean 4 / Lake 2→/.lake/ 3→/lake-packages/ 4→/build/ 5→/.cache/ 6→ 7→# Blueprint output 8→/blueprint/web/ 9→/blueprint/print/ 10→/blueprint/.lake/ 11→ 12→# LaTeX auxiliary files 13→*.aux 14→*.log 15→*.out 16→*.toc 17→*.fls 18→*.fdb_latexmk 19→*.synctex.gz 20→*.bbl 21→*.blg 22→ 23→# Python 24→__pycache__/ 25→*.py[cod] 26→.venv/ 27→venv/ 28→ 29→# OS 30→.DS_Store 3...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1488): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1489): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.gitignore has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 3→/lake-packages/ 4→/build/ 5→/.cache/ 6→ 7→# Blueprint output 8→/blueprint/web/ 9→/blueprint/print/ 10→/blueprint/.lake/ 11→/blueprint/lean_decls 12→ 13→# LaTeX auxiliary files 14→*.aux 15→*.log] -------------------------------------------------------------------------------- ASSISTANT (entry 1491): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1492): -------------------------------------------------------------------------------- [Tool Result: On branch main Your branch is ahead of 'origin/main' by 1 commit. (use "git push" to publish your local commits) Changes to be committed: (use "git restore --staged ..." to unstage) modified: .github/workflows/push.yml modified: .gitignore new file: blueprint/blueprint.toml new file: blueprint/src/blueprint.sty new file: blueprint/src/content.tex new file: blueprint/src/extra_styles.css new file: blueprint/src/latexmkrc new file: blueprint/src/macros/common.tex...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1494): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1495): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1497): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1498): -------------------------------------------------------------------------------- [Tool Result: [main eee9a3a] feat: set up leanblueprint structure with proper templates 13 files changed, 301 insertions(+), 2 deletions(-) create mode 100644 blueprint/blueprint.toml create mode 100644 blueprint/src/blueprint.sty create mode 100644 blueprint/src/content.tex create mode 100644 blueprint/src/extra_styles.css create mode 100644 blueprint/src/latexmkrc create mode 100644 blueprint/src/macros/common.tex create mode 100644 blueprint/src/macros/print.tex create mode 100644 blueprint/src/ma...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1500): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1501): -------------------------------------------------------------------------------- [Tool Result: To github.com:schmittj/balanced-vectors-blueprint.git cce3d1e..eee9a3a main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1503): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1504): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1506): -------------------------------------------------------------------------------- Done! I've pushed the properly structured blueprint with: 1. **Fixed template structure** - Used `leanblueprint new` to generate proper LaTeX structure: - `print.tex` / `web.tex` wrappers - `macros/` folder with `\lean`, `\leanok` etc. definitions - `content.tex` with just the content (no preamble) 2. **Updated workflow** - Added: - `texlive-xetex texlive-fonts-recommended texlive-latex-extra texlive-fonts-extra texlive-science` - Using `leanblueprint pdf && leanblueprint web` instead of `all` (skips declaration checking which needs lake) 3. **Content** - Added definitions and theorems linking to Lean code: - WeakComposition, IsBalanced, IsConcentrated, IsSymmetric, IsLogConcave - SymmetricLogConcaveFunction structure - Main theorems with `\lean{...}\leanok` annotations Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions Once it passes, you'll need to enable GitHub Pages: 1. Go to Settings → Pages 2. Under "Build and deployment", select **Source: GitHub Actions** 3. The site will deploy to https://schmittj.github.io/balanced-vectors-blueprint/ -------------------------------------------------------------------------------- USER (entry 1508): -------------------------------------------------------------------------------- Thanks a lot! Right now, the build-blueprint still fails with: Run cd blueprint /bin/sh: 1: latexmk: not found Command 'latexmk -output-directory=../print' returned non-zero exit status 127. Error: Process completed with exit code 255. -------------------------------------------------------------------------------- ASSISTANT (entry 1510): -------------------------------------------------------------------------------- Need to add `latexmk` to the dependencies. -------------------------------------------------------------------------------- ASSISTANT (entry 1511): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1512): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 45→ uses: actions/setup-python@v5 46→ with: 47→ python-version: '3.11' 48→ 49→ - name: Install system dependencies 50→ run: | 51→ sudo apt-get update 52→ sudo apt-get install -y graphviz graphviz-dev 53→ sudo apt-get inst...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1514): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1515): -------------------------------------------------------------------------------- [Tool Result: [main 9432be6] fix: add latexmk to system dependencies 1 file changed, 1 insertion(+), 1 deletion(-) To github.com:schmittj/balanced-vectors-blueprint.git eee9a3a..9432be6 main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1517): -------------------------------------------------------------------------------- Done! Added `latexmk` to the apt-get install line. Check the new build at: https://github.com/schmittj/balanced-vectors-blueprint/actions -------------------------------------------------------------------------------- USER (entry 1519): -------------------------------------------------------------------------------- Perfect, everything now runs through! I can successfully open pages like: https://schmittj.github.io/balanced-vectors-blueprint/sect0001.html One thing that does not yet work: next to a definition, there is a little LEAN writing (appearing on mouseover), which opens a little window Lean declarations with a clickable link. But if I click and eg. get forwarded to https://schmittj.github.io/balanced-vectors-blueprint/docs/find/#doc/IsSymmetric that website is not available: 404 File not found The site configured at this address does not contain the requested file. -------------------------------------------------------------------------------- ASSISTANT (entry 1521): -------------------------------------------------------------------------------- The "Lean declarations" links point to API documentation that needs to be generated by `doc-gen4`. Let me update the workflow to generate and deploy the Lean docs alongside the blueprint. -------------------------------------------------------------------------------- ASSISTANT (entry 1522): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1523): -------------------------------------------------------------------------------- [Tool Result: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "pages" 17→ cancel-in-progress: false 18→ 19→jobs: 20→ build-lean: 21→ runs-on: ubuntu-latest 22→ steps: 23→ - name: Checkout 24→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1525): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1526): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1528): -------------------------------------------------------------------------------- I also need to add doc-gen4 to the lakefile for documentation generation. -------------------------------------------------------------------------------- ASSISTANT (entry 1529): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1530): -------------------------------------------------------------------------------- [Tool Result: 1→import Lake 2→open Lake DSL 3→ 4→package «balanced-vectors» where 5→ leanOptions := #[ 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→-- Using the exact mathlib commit where the code was developed 11→require mathlib from git 12→ "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" 13→ 14→@[default_target] 15→lean_lib «BalancedVectors» where 16→ -- ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1532): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1533): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→-- Using the exact mathlib commit where the code was developed 11→require mathlib from git 12→ "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" 13→ 14→-- Doc-gen4 for A...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1535): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1536): -------------------------------------------------------------------------------- [Tool Result: [main 32599c7] feat: add doc-gen4 for API documentation 2 files changed, 21 insertions(+), 1 deletion(-) To github.com:schmittj/balanced-vectors-blueprint.git 9432be6..32599c7 main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1538): -------------------------------------------------------------------------------- Done! The workflow now: 1. Builds Lean and generates API docs with doc-gen4 2. Uploads docs as an artifact 3. Downloads docs into `blueprint/web/docs/` before deploying Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions Once it passes, the Lean declaration links should work at paths like: `https://schmittj.github.io/balanced-vectors-blueprint/docs/BalancedVectors.html` Note: The doc-gen4 build may take a while on first run since it needs to process mathlib dependencies. -------------------------------------------------------------------------------- USER (entry 1540): -------------------------------------------------------------------------------- Thanks, let's give it some time to build. Meanwhile I am reading the informal part of the blueprint, some questions: - Definition 5 (Log-Concave Function): you write here e_j <= d-1, but isn't the relevant part that e_j >= 0 ? What does the Lean formalization say here? The e_j <= d-1 part is actually automatic from nonnegativity and e_i >= 1. - Does it prove anywhere that there EXISTS a balanced vector? This is not hard (for d = a*n+b division with remainder it is e = (a,a,...,a,a+1, ... a+1) with b entries a+1). -------------------------------------------------------------------------------- ASSISTANT (entry 1542): -------------------------------------------------------------------------------- Good questions! Let me check the Lean formalization. -------------------------------------------------------------------------------- ASSISTANT (entry 1543): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1544): -------------------------------------------------------------------------------- [Tool Result: 1083: logConcave : SatisfiesLogConcavity D 1101: balanced_maximizes F.D F.strictlyPositive F.symmetric F.logConcave e 1116: concentrated_minimizes hn hd F.D F.strictlyPositive F.symmetric F.logConcave e] -------------------------------------------------------------------------------- ASSISTANT (entry 1546): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1547): -------------------------------------------------------------------------------- [Tool Result: 41:| `SatisfiesLogConcavity D` | line 351 | the log-concavity condition above | 42-| `IsStrictlyPositive D` | line 357 | `D(e) > 0` for all `e` | 43- 44-## Main Theorems (for verification) 45- 46-**Primary reference (clean formulation at end of file):** 47- 48-| Theorem | Location | Statement | 49-|-------------------------|------------|----------------------------------------------| 50-| `main_theorem` ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1549): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1550): -------------------------------------------------------------------------------- [Tool Result: 437:lemma exists_imbalanced_pair (e : Fin n → ℤ) (h : ¬ IsBalanced e) : 692: obtain ⟨i, j, hij, hdiff⟩ := exists_imbalanced_pair e.toFun hbal 1095:More precisely: for any `e ∈ E(n,d)`, there exists a balanced `e' ∈ E(n,d)` with `D(e) ≤ D(e')`.] -------------------------------------------------------------------------------- ASSISTANT (entry 1552): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1553): -------------------------------------------------------------------------------- [Tool Result: 1099:theorem maximized_on_balanced (F : SymmetricLogConcaveFunction n d) (e : WeakComposition n d) : 1100- ∃ e' : WeakComposition n d, IsBalanced e'.toFun ∧ F.D e ≤ F.D e' := 1101- balanced_maximizes F.D F.strictlyPositive F.symmetric F.logConcave e 1102- 1103-/-- 1104-**Main Theorem (Minimum).** 1105-Every symmetric log-concave function on weak compositions is minimized on concentrated vectors. 1106- 1107-More precisely: for any `e ∈ E(n,d)`, there exists a concentrated `e' ∈ E(n,d)` with `...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1555): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1556): -------------------------------------------------------------------------------- [Tool Result: 673- apply Finset.sum_nonneg 674- intro i _ 675- exact sq_nonneg _ 676- 677-/-- Any vector can be transformed to a balanced vector while increasing D. -/ 678:theorem balanced_maximizes {d : ℤ} (D : WeakComposition n d → ℚ) 679- (hpos : IsStrictlyPositive D) (hsym : IsSymmetric D) (hlc : SatisfiesLogConcavity D) : 680- ∀ e : WeakComposition n d, ∃ e' : WeakComposition n d, IsBalanced e'.toFun ∧ D e ≤ D e' := by 681- -- Use well-founded induction on imbalance 682- intro e 683- -- Conve...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1558): -------------------------------------------------------------------------------- Great catches! Looking at the Lean code: **1. Log-Concave Definition (line 369-372):** ```lean def SatisfiesLogConcavity {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := ∀ (e : WeakComposition n d) (i j : Fin n) (hij : i ≠ j) (hi : 1 ≤ e i) (hj : 1 ≤ e j), D e ^ 2 ≥ D (e.modify i j hi hij) * D (e.modify j i hj hij.symm) ``` You're right - it requires **both** `e_i ≥ 1` AND `e_j ≥ 1`, not `e_j ≤ d-1`. The conditions ensure both modified vectors are valid weak compositions. **2. Balanced existence:** There's no explicit construction. The proof of `balanced_maximizes` (line 678) works by induction on the imbalance measure - repeatedly improving until reaching balanced. This implicitly proves existence but doesn't construct `(a, a, ..., a, a+1, ..., a+1)` directly. Let me fix the blueprint: -------------------------------------------------------------------------------- ASSISTANT (entry 1559): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1560): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 42→\uses{def:weak_composition} 43→A function $D : E(n,d) \to \mathbb{Q}$ is \emph{$S_n$-symmetric} if $D(e \circ \sigma^{-1}) = D(e)$ for all permutations $\sigma \in S_n$. 44→\end{definition} 45→ 46→\begin{definition}[Log-Concave Function]\label{def:logconcave} 47→\lean{SatisfiesLogConcavity} ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1562): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1563): -------------------------------------------------------------------------------- [Tool Result: [main c347bd0] fix: correct log-concavity condition (e_j ≥ 1, not e_j ≤ d-1) 1 file changed, 3 insertions(+), 2 deletions(-) To github.com:schmittj/balanced-vectors-blueprint.git 32599c7..c347bd0 main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1565): -------------------------------------------------------------------------------- Fixed! The blueprint now correctly states `e_i ≥ 1` and `e_j ≥ 1`, matching the Lean code. Regarding balanced existence: Would you like me to add an explicit construction lemma to the Lean file? Something like: ```lean /-- A balanced weak composition always exists. -/ lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : ∃ e : WeakComposition n d, IsBalanced e.toFun := by -- Construct (a, a, ..., a, a+1, ..., a+1) where d = a*n + b ... ``` This would make the existence explicit rather than implicit through the improvement process. -------------------------------------------------------------------------------- USER (entry 1567): -------------------------------------------------------------------------------- Thanks, let's leave it implicit for now. But the original deploy had an issue as follows: Run lake -Kenv=dev build BalancedVectors:docs error: unknown lean_lib facet `docs` Error: Process completed with exit code 1. Can you have a look? -------------------------------------------------------------------------------- ASSISTANT (entry 1569): -------------------------------------------------------------------------------- Let me check how doc-gen4 should be invoked properly. -------------------------------------------------------------------------------- ASSISTANT (entry 1570): -------------------------------------------------------------------------------- [Tool: WebSearch] -------------------------------------------------------------------------------- USER (entry 1571): -------------------------------------------------------------------------------- [Tool Result: Web search results for query: "doc-gen4 lake build docs facet 2024 2025" Links: [{"title":"GitHub - leanprover/doc-gen4: Document Generator for Lean 4","url":"https://github.com/leanprover/doc-gen4"},{"title":"doc-gen4 | Reservoir","url":"https://reservoir.lean-lang.org/@leanprover/doc-gen4"},{"title":"doc-gen4/README.md at main · leanprover/doc-gen4","url":"https://github.com/leanprover/doc-gen4/blob/main/README.md"},{"title":"doc-gen4/ at main · leanprover/doc-gen4","url":"https://github.com/...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1573): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1574): -------------------------------------------------------------------------------- [Tool Result: # Building Documentation with doc-gen4 According to the README, the correct approach involves creating a nested documentation project within your lake project: ## Setup Steps 1. Create a `docbuild` subdirectory in your existing lake project 2. Add a `lakefile.toml` with dependencies on your library and doc-gen4 3. Run `lake update doc-gen4` to pin versions (or `MATHLIB_NO_CACHE_ON_UPDATE=1 lake update doc-gen4` if using mathlib4) ## Build Commands For a single library: ``` lake build YourLi...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1576): -------------------------------------------------------------------------------- The doc-gen4 setup requires a nested `docbuild` subdirectory with its own lakefile. That's more complex. For now, let me simplify by removing the docs step - the blueprint will work, just without API doc links. -------------------------------------------------------------------------------- ASSISTANT (entry 1577): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1578): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "...[truncated]] -------------------------------------------------------------------------------- USER (entry 1579): -------------------------------------------------------------------------------- [Request interrupted by user] -------------------------------------------------------------------------------- USER (entry 1581): -------------------------------------------------------------------------------- Before continuing: it would be great if those LEAN links worked and redirected to actual Lean code that the user can inspect. Does that require the more involved setup you mention? If so, maybe we can try it? -------------------------------------------------------------------------------- ASSISTANT (entry 1583): -------------------------------------------------------------------------------- Yes, let me set up doc-gen4 properly with the `docbuild` subdirectory approach. -------------------------------------------------------------------------------- ASSISTANT (entry 1584): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1586): -------------------------------------------------------------------------------- [Tool Result: File created successfully at: /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild/lakefile.toml] -------------------------------------------------------------------------------- ASSISTANT (entry 1588): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1589): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1591): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1592): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.gitignore has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→# Lean 4 / Lake 2→/.lake/ 3→/lake-packages/ 4→/build/ 5→/.cache/ 6→/docbuild/.lake/ 7→/docbuild/lake-manifest.json 8→ 9→# Blueprint output 10→/blueprint/web/ 11→/blueprint/print/] -------------------------------------------------------------------------------- ASSISTANT (entry 1594): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 1595): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name: Build and Deploy 2→ 3→on: 4→ push: 5→ branches: ["main"] 6→ pull_request: 7→ branches: ["main"] 8→ workflow_dispatch: 9→ 10→permissions: 11→ contents: read 12→ pages: write 13→ id-token: write 14→ 15→concurrency: 16→ group: "...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1597): -------------------------------------------------------------------------------- Let me also remove the doc-gen4 from the main lakefile (it's now in docbuild): -------------------------------------------------------------------------------- ASSISTANT (entry 1598): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1599): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/lakefile.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 6→ ⟨`pp.unicode.fun, true⟩, 7→ ⟨`autoImplicit, false⟩ 8→ ] 9→ 10→-- Using the exact mathlib commit where the code was developed 11→require mathlib from git 12→ "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" 13→ 14→@[default_target]...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1601): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1602): -------------------------------------------------------------------------------- [Tool Result: On branch main Your branch is up to date with 'origin/main'. Changes to be committed: (use "git restore --staged ..." to unstage) modified: .github/workflows/push.yml modified: .gitignore new file: docbuild/lakefile.toml new file: docbuild/lean-toolchain modified: lakefile.lean] -------------------------------------------------------------------------------- ASSISTANT (entry 1604): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1605): -------------------------------------------------------------------------------- [Tool Result: [main f7e3440] feat: add doc-gen4 via docbuild subdirectory 5 files changed, 21 insertions(+), 7 deletions(-) create mode 100644 docbuild/lakefile.toml create mode 100644 docbuild/lean-toolchain To github.com:schmittj/balanced-vectors-blueprint.git c347bd0..f7e3440 main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1607): -------------------------------------------------------------------------------- Done! I've set up doc-gen4 properly with the recommended `docbuild` subdirectory approach: **Structure:** ``` docbuild/ ├── lakefile.toml # Depends on parent project + doc-gen4 └── lean-toolchain # Same Lean version as parent ``` **Workflow now:** 1. Builds Lean project 2. In `docbuild/`: runs `lake update doc-gen4` then `lake build BalancedVectors:docs` 3. Uploads generated docs as artifact 4. Blueprint job downloads docs to `blueprint/web/docs/` 5. Deploys everything together Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions Note: The first doc-gen4 build may take a while as it processes dependencies. Once complete, clicking "Lean declarations" links should show the actual Lean source code. -------------------------------------------------------------------------------- USER (entry 1609): -------------------------------------------------------------------------------- Thanks, it's running and I'll give you an update when it finishes. Again, is this something which we can also feasibly run locally to iron out any kinks? -------------------------------------------------------------------------------- ASSISTANT (entry 1611): -------------------------------------------------------------------------------- Yes, let me try it locally now to catch any issues early. -------------------------------------------------------------------------------- ASSISTANT (entry 1612): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1614): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: bac191e. Output is being written to: /tmp/claude/tasks/bac191e.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1617): -------------------------------------------------------------------------------- The issue is a Lean version mismatch - doc-gen4 `main` is using Lean 4.27.0-rc1, but our project uses Lean 4.26.0. We need to pin doc-gen4 to a compatible version. [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1618): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1620): -------------------------------------------------------------------------------- [Tool: WebSearch] -------------------------------------------------------------------------------- USER (entry 1621): -------------------------------------------------------------------------------- [Tool Result: Web search results for query: "doc-gen4 lean 4.26.0 compatible version tag" Links: [{"title":"GitHub - leanprover/doc-gen4: Document Generator for Lean 4","url":"https://github.com/leanprover/doc-gen4"},{"title":"doc-gen4 - lean4","url":"https://leanprover-community.github.io/archive/stream/270676-lean4/topic/doc-gen4.html"},{"title":"doc-gen4 | Reservoir - Lean","url":"https://reservoir.lean-lang.org/@leanprover/doc-gen4"},{"title":"doc-gen4/Main.lean at main · leanprover/doc-gen4","url":"http...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1623): -------------------------------------------------------------------------------- According to [doc-gen4 documentation](https://github.com/leanprover/doc-gen4), it uses version tags matching Lean versions. Let me update to use `v4.26.0`: -------------------------------------------------------------------------------- ASSISTANT (entry 1624): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1625): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild/lakefile.toml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 5→[[require]] 6→name = "balanced-vectors" 7→path = ".." 8→ 9→# Depend on doc-gen4 (use version matching our Lean toolchain) 10→[[require]] 11→name = "doc-gen4" 12→git = "https://github.com/leanprover/doc-gen4" 13→rev = "v4.26.0" 14→] -------------------------------------------------------------------------------- ASSISTANT (entry 1627): -------------------------------------------------------------------------------- Let me test locally first: -------------------------------------------------------------------------------- ASSISTANT (entry 1628): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1631): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b530211. Output is being written to: /tmp/claude/tasks/b530211.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1633): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1636): -------------------------------------------------------------------------------- [Tool Result: success b530211 local_bash completed 0 instead of String → String warning: Cache/Requests.lean:418:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:420:54: `String.trim` has been deprecated: Use `String.trimAscii...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1638): -------------------------------------------------------------------------------- Same error locally. The issue is that doc-gen4 is pulling in mathlib dependencies that conflict. Let me check the doc-gen4 lean-toolchain for v4.26.0: -------------------------------------------------------------------------------- ASSISTANT (entry 1639): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1640): -------------------------------------------------------------------------------- [Tool Result: The user doesn't want to proceed with this tool use. The tool use was rejected (eg. if it was a file edit, the new_string was NOT written to the file). STOP what you are doing and wait for the user to tell you how to proceed.] -------------------------------------------------------------------------------- USER (entry 1641): -------------------------------------------------------------------------------- [Request interrupted by user for tool use] -------------------------------------------------------------------------------- USER (entry 1643): -------------------------------------------------------------------------------- Note that we originally went for a very specialized mathlib reference to fix the original build. But plausibly that build just had a problem with variable d being undefined, and might now also work for more standard versions of mathlib. Just if the issue is really a version mismatch, we might also be able to adapt the version used in the main compilation. -------------------------------------------------------------------------------- ASSISTANT (entry 1645): -------------------------------------------------------------------------------- Good point! The variable scoping issue was fixed in the Lean file itself (by adding `variable {d : ℤ}` after `end WeakComposition`), so we're not actually tied to that specific mathlib commit anymore. Let me check what Lean/mathlib version doc-gen4 expects, and we can update to match: -------------------------------------------------------------------------------- ASSISTANT (entry 1646): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1647): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1649): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1650): -------------------------------------------------------------------------------- [Tool Result: {"version": "1.1.0", "packagesDir": ".lake/packages", "packages": [{"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, "scope": "", "rev": "933fce7e893f65969714c60cdb4bd8376786044e", "name": "Cli", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": false, "configFile": "lakefile.toml"}, {"url": "https://github.com/fgdorais/lean4-unicode-basic", "type": "git", "subDir": null, "scope": "", "rev": "84b88f7...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1652): -------------------------------------------------------------------------------- The issue might be that `lake update doc-gen4` is updating other dependencies too. Let me try without the update step since we're pinning to v4.26.0: -------------------------------------------------------------------------------- ASSISTANT (entry 1653): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1654): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b5aa92b. Output is being written to: /tmp/claude/tasks/b5aa92b.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1656): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1659): -------------------------------------------------------------------------------- [Tool Result: success b5aa92b local_bash completed 0 Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:332:21: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Reque...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1661): -------------------------------------------------------------------------------- Let me check if there's a version mismatch in the docbuild's dependencies: -------------------------------------------------------------------------------- ASSISTANT (entry 1662): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1663): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1665): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1666): -------------------------------------------------------------------------------- [Tool Result: Main project mathlib: not found Docbuild mathlib: 26fffffcccd7299b26cf63fac902165bc553fd56] -------------------------------------------------------------------------------- ASSISTANT (entry 1668): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1669): -------------------------------------------------------------------------------- [Tool Result: === Cli === leanprover/lean4:v4.26.0 === UnicodeBasic === leanprover/lean4:v4.26.0-rc2 === BibtexQuery === leanprover/lean4:v4.26.0-rc2=== MD4Lean === leanprover/lean4:4.25.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1671): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1672): -------------------------------------------------------------------------------- [Tool Result: /home/jo314/.elan/bin/lean Lean (version 4.27.0-rc1, x86_64-unknown-linux-gnu, commit 2fcce7258eeb6e324366bc25f9058293b04b7547, Release)] -------------------------------------------------------------------------------- ASSISTANT (entry 1674): -------------------------------------------------------------------------------- Found it! The system default Lean is 4.27.0-rc1, but our project needs 4.26.0. Let me force the correct version: -------------------------------------------------------------------------------- ASSISTANT (entry 1675): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1676): -------------------------------------------------------------------------------- [Tool Result: Lean (version 4.26.0, x86_64-unknown-linux-gnu, commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562, Release) info: override toolchain for '/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild' set to 'leanprover/lean4:v4.26.0'] -------------------------------------------------------------------------------- ASSISTANT (entry 1678): -------------------------------------------------------------------------------- Now let's try building again: -------------------------------------------------------------------------------- ASSISTANT (entry 1679): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1680): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: bdba4ab. Output is being written to: /tmp/claude/tasks/bdba4ab.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1682): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1683): -------------------------------------------------------------------------------- [Tool Result: timeout bdba4ab local_bash running] -------------------------------------------------------------------------------- ASSISTANT (entry 1685): -------------------------------------------------------------------------------- Still running - let me check the output so far: -------------------------------------------------------------------------------- ASSISTANT (entry 1686): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1687): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1689): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1690): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1692): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1696): -------------------------------------------------------------------------------- [Tool Result: [Request interrupted by user for tool use]] -------------------------------------------------------------------------------- USER (entry 1697): -------------------------------------------------------------------------------- [Request interrupted by user for tool use] -------------------------------------------------------------------------------- USER (entry 1699): -------------------------------------------------------------------------------- Assuming provisorially that this works now, can you make changes and push a git commit that would fix the github actions ? -------------------------------------------------------------------------------- ASSISTANT (entry 1701): -------------------------------------------------------------------------------- Yes, let me update the workflow to not use `lake update` (which may pull incompatible versions) and ensure the toolchain is respected: -------------------------------------------------------------------------------- ASSISTANT (entry 1702): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1703): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 33→ 34→ - name: Build Lean project 35→ run: lake build 36→ 37→ - name: Build documentation 38→ run: | 39→ cd docbuild 40→ lake build BalancedVectors:docs 41→ 42→ - name: Upload docs artifact 43→ uses: actions/upload-artif...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1705): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1706): -------------------------------------------------------------------------------- [Tool Result: [main f7d938d] fix: remove lake update to prevent version conflicts 3 files changed, 3 insertions(+), 4 deletions(-) To github.com:schmittj/balanced-vectors-blueprint.git f7e3440..f7d938d main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1708): -------------------------------------------------------------------------------- Pushed! The change removes `lake update doc-gen4` which was pulling in incompatible dependency versions. Since we pin doc-gen4 to `v4.26.0` in `docbuild/lakefile.toml`, lake should use the correct versions automatically. Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions Meanwhile, I'll let the local build continue to see if it succeeds with the correct Lean version. -------------------------------------------------------------------------------- USER (entry 1710): -------------------------------------------------------------------------------- Mmh, this still produces: Run cd docbuild info: downloading https://releases.lean-lang.org/lean4/v4.27.0-rc1/lean-4.27.0-rc1-linux.tar.zst info: installing /home/runner/.elan/toolchains/leanprover--lean4---v4.27.0-rc1 info: docbuild: no previous manifest, creating one from scratch info: doc-gen4: cloning https://github.com/leanprover/doc-gen4 info: doc-gen4: checking out revision '77ef3eb515ad6bd125c596f0b164349d4a7d5bf5' info: toolchain not updated; already up-to-date warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:39:27: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:43:45: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:58:20: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:71:20: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:85:19: `String.Slice.dropRight` has been deprecated: Use `String.Slice.dropEnd` instead warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:89:19: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:118:39: `Lake.Package.name` has been deprecated: Use `baseName`, `keyName`, or `prettyName` instead info: Cli: cloning https://github.com/leanprover/lean4-cli info: Cli: checking out revision '933fce7e893f65969714c60cdb4bd8376786044e' info: UnicodeBasic: cloning https://github.com/fgdorais/lean4-unicode-basic info: UnicodeBasic: checking out revision '84b88f7ac9adf382b9668f852cee82487d616792' warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/UnicodeBasic/lakefile.lean:20:63: `String.stripSuffix` has been deprecated: Use `String.dropSuffix` instead Note: The updated constant has a different type: {ρ : Type} → String → (pat : ρ) → [String.Slice.Pattern.BackwardPattern pat] → String.Slice instead of String → String → String info: BibtexQuery: cloning https://github.com/dupuisf/BibtexQuery info: BibtexQuery: checking out revision '3ab4379b2b92448717de66b7d3e254ac1487aede' info: MD4Lean: cloning https://github.com/acmepjz/md4lean instead of String → String warning: Cache/Requests.lean:235:26: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:332:21: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:340:34: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: Cache/Requests.lean:418:24: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:418:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:420:54: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:421:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:557:86: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated con ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 1712): -------------------------------------------------------------------------------- I see the issue - despite our `lean-toolchain` saying v4.26.0, it's downloading Lean 4.27.0-rc1. Also notice doc-gen4 is checking out revision `77ef3eb...` instead of the v4.26.0 tag we specified. Here's a self-contained summary for GPT 5.2: --- ## Problem Summary We have a Lean 4 project with a blueprint documentation site. We want to add doc-gen4 to generate API documentation with clickable Lean source links. **The Issue:** When building doc-gen4, GitHub Actions downloads Lean 4.27.0-rc1 instead of our specified v4.26.0, causing String API incompatibilities (deprecated `String.trim`, `String.dropRight`, etc. that now return `String.Slice` instead of `String`). ## Files to Share **1. `/docbuild/lakefile.toml`:** ```toml name = "docbuild" reservoir = false [[require]] name = "balanced-vectors" path = ".." [[require]] name = "doc-gen4" git = "https://github.com/leanprover/doc-gen4" rev = "v4.26.0" ``` **2. `/docbuild/lean-toolchain`:** ``` leanprover/lean4:v4.26.0 ``` **3. `/lean-toolchain` (main project):** ``` leanprover/lean4:v4.26.0 ``` **4. `/lakefile.lean` (main project):** ```lean import Lake open Lake DSL package «balanced-vectors» where leanOptions := #[ ⟨`pp.unicode.fun, true⟩, ⟨`autoImplicit, false⟩ ] require mathlib from git "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" @[default_target] lean_lib «BalancedVectors» where ``` **5. Relevant workflow step:** ```yaml - name: Build documentation run: | cd docbuild lake build BalancedVectors:docs ``` ## Question for GPT 5.2 How do we ensure doc-gen4 and all its dependencies use Lean 4.26.0 (matching our mathlib commit) instead of pulling in Lean 4.27.0-rc1? The `rev = "v4.26.0"` in lakefile.toml doesn't seem to be respected. -------------------------------------------------------------------------------- USER (entry 1714): -------------------------------------------------------------------------------- Here is GPT-5.2s input: What you’re seeing is the combination of two (separate) mechanisms: 1. **Lake resolves `rev = "v4.26.0"` to a commit hash** (and records that hash in `lake-manifest.json`). So “doc-gen4 checked out `77ef3…`” is not evidence that the tag was ignored — it’s exactly how tags work: *a tag name points to a commit*. Your manifest shows `inputRev: "v4.26.0"` and `rev: "77ef3…"` for doc-gen4. 2. **Elan chooses the Lean toolchain by the nearest `lean-toolchain` file in the *current working directory tree***. If any part of the build runs tools from inside `.lake/packages/doc-gen4` (or another dependency) and *that package’s* `lean-toolchain` says `v4.27.0-rc1`, Elan will happily download and use it — even if your top-level `lean-toolchain` says `v4.26.0`. ([Lean Language][1]) --- ## The quickest “make it stop” fix (CI + local) Force the toolchain for the whole documentation build by setting **`ELAN_TOOLCHAIN`**. Lake explicitly prefers `ELAN_TOOLCHAIN` over its own detected toolchain string. ([leanprover-community.github.io][2]) **GitHub Actions step:** ```yaml - name: Build documentation env: ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0 run: | cd docbuild lake env lean --version lake build BalancedVectors:docs ``` That makes *every* `lean`/`lake` invocation during that step use 4.26.0, even if a dependency folder has a different `lean-toolchain`. (Alternative that also works: `elan run leanprover/lean4:v4.26.0 -- lake …`, or `elan override set …` at the repo root; overrides take precedence over `lean-toolchain` files. ([Lean Language][1])) --- ## The “actually pin doc-gen4 correctly” fix Your `lake-manifest.json` says Lake resolved doc-gen4 `v4.26.0` to commit `77ef3…`. If that commit really requires Lean 4.27.0-rc1, then **the ref you’re using in doc-gen4 is not Lean-4.26-compatible** (e.g. the tag points somewhere unexpected, or you meant a different tag/branch). What to do: 1. **Inspect what doc-gen4 thinks its toolchain is:** ```bash cd docbuild cat .lake/packages/doc-gen4/lean-toolchain ``` If it says `leanprover/lean4:v4.27.0-rc1`, that explains the download. 2. **Pin doc-gen4 to a revision whose `lean-toolchain` is `v4.26.0`**. The doc-gen4 instructions explicitly recommend choosing the `rev` that matches your Lean version (e.g. `v4.x`). ([Reservoir][3]) In practice, the most robust way is: **use a commit hash** once you’ve identified the correct one. 3. **Regenerate the manifest after changing the pin (and commit it):** ```bash cd docbuild rm -f lake-manifest.json rm -rf .lake/packages/doc-gen4 MATHLIB_NO_CACHE_ON_UPDATE=1 lake update doc-gen4 ``` That `MATHLIB_NO_CACHE_ON_UPDATE=1 …` recommendation is from doc-gen4’s setup docs when mathlib is involved. ([Reservoir][3]) --- ## One extra recommendation (not required, but helps) Doc-gen4’s suggested `docbuild/lakefile.toml` includes `packagesDir = "../.lake/packages"` so the docbuild project shares the parent’s package directory. That reduces surprises/caching weirdness when you have both a main project and a nested docbuild project. ([Reservoir][3]) --- If you paste the contents of `docbuild/.lake/packages/doc-gen4/lean-toolchain` (just that one line), I can tell you immediately whether you’re fighting “wrong doc-gen4 ref” vs “elan picking the wrong directory’s toolchain” — but in most repos it’s *both*, and setting `ELAN_TOOLCHAIN` in the workflow fixes the Lean 4.27 download immediately. [1]: https://lean-lang.org/doc/reference/latest/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/ "Managing Toolchains with Elan" [2]: https://leanprover-community.github.io/mathlib4_docs/Lake/Config/Env.html "Lake.Config.Env" [3]: https://reservoir.lean-lang.org/%40leanprover/doc-gen4 "doc-gen4 | Reservoir" -------------------------------------------------------------------------------- ASSISTANT (entry 1716): -------------------------------------------------------------------------------- Excellent analysis from GPT-5.2! The fix is to set `ELAN_TOOLCHAIN` to force the correct version. Let me implement that: -------------------------------------------------------------------------------- ASSISTANT (entry 1717): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1718): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 33→ 34→ - name: Build Lean project 35→ run: lake build 36→ 37→ - name: Build documentation 38→ env: 39→ ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0 40→ run: | 41→ cd docbuild 42→ lake build BalancedVectors:docs 43→ 4...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1720): -------------------------------------------------------------------------------- Let me also add the `packagesDir` recommendation to share the parent's packages: -------------------------------------------------------------------------------- ASSISTANT (entry 1721): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1722): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild/lakefile.toml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name = "docbuild" 2→reservoir = false 3→packagesDir = "../.lake/packages" 4→ 5→# Depend on the parent project 6→[[require]] 7→name = "balanced-vectors" 8→path = ".." 9→ 10→# Depend on doc-gen4 (use version matching our Lean toolchain) 11→[[require]] 12→name = "doc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1724): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1725): -------------------------------------------------------------------------------- [Tool Result: [main 78c773c] fix: force ELAN_TOOLCHAIN to prevent version mismatch 2 files changed, 3 insertions(+) To github.com:schmittj/balanced-vectors-blueprint.git f7d938d..78c773c main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1727): -------------------------------------------------------------------------------- Pushed! The fix: 1. **`ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0`** - Forces all Lean/Lake invocations to use v4.26.0, overriding any dependency's `lean-toolchain` file 2. **`packagesDir = "../.lake/packages"`** - Shares the parent project's package directory to reduce caching issues Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions -------------------------------------------------------------------------------- USER (entry 1729): -------------------------------------------------------------------------------- Thanks a lot! The compile is now running (has been for 10 min, so at least not failing fast, but also still at [4171 / 6740] of some build process of documentation - is it building the doc of the entire mathlib? One thing we can do in the meantime: the project https://github.com/leanprover-community/sphere-eversion has a working configuration of their blueprint page, with appropriate LEAN links. We could have a look at what they do, to see how it compares to our setup. -------------------------------------------------------------------------------- ASSISTANT (entry 1731): -------------------------------------------------------------------------------- Yes, doc-gen4 builds docs for all dependencies including mathlib by default - that's why it's slow. Let me check how sphere-eversion handles this. -------------------------------------------------------------------------------- ASSISTANT (entry 1732): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1733): -------------------------------------------------------------------------------- [Tool Result: Request failed with status code 404] -------------------------------------------------------------------------------- ASSISTANT (entry 1735): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1736): -------------------------------------------------------------------------------- [Tool Result: # Workflow Files in `.github/workflows` Based on the repository structure displayed, there is **one workflow file** in this directory: - **blueprint.yml** This is the only GitHub Actions workflow file currently configured in the `.github/workflows` directory of the `leanprover-community/sphere-eversion` repository on the master branch.] -------------------------------------------------------------------------------- ASSISTANT (entry 1738): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1739): -------------------------------------------------------------------------------- [Tool Result: # Compile Blueprint Workflow This GitHub Actions workflow automates building a Lean project blueprint and deploying documentation to GitHub Pages. ## Trigger Events The workflow runs on: - Pushes to the master branch - Pull requests against master - Manual triggering via the Actions interface ## Key Jobs **Build Project** executes several steps: 1. **Project Setup**: Checks out the repository and installs Elan (Lean's version manager) 2. **Caching**: Retrieves cached build artifacts to spe...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1741): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1742): -------------------------------------------------------------------------------- [Tool Result: # GitHub Actions Workflow: Compile Blueprint This YAML file defines a CI/CD workflow that: **Triggers:** - Runs on pushes to `master`, pull requests to `master`, and manual workflow dispatches **Key Jobs:** The `build_project` job performs several steps: 1. Checks out the repository and installs Elan (Lean's version manager) 2. Retrieves cached dependencies via `lake exe cache get` 3. Builds the SphereEversion project 4. Caches API documentation for Lean libraries 5. Generates blueprint docu...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1744): -------------------------------------------------------------------------------- Let me also check their lakefile and docbuild setup: -------------------------------------------------------------------------------- ASSISTANT (entry 1745): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1746): -------------------------------------------------------------------------------- [Tool Result: # Sphere Eversion Project Directory Structure ## Top-Level Directories Based on the GitHub repository content, the top-level structure includes: - `.github/workflows` - `.vscode` - `SphereEversion` - `blueprint` - `docs_src` - `graphs` - `scripts` ## Root-Level Files The root directory contains: - `.gitignore` - `LICENSE` (Apache-2.0) - `README.md` - `SphereEversion.lean` - `lake-manifest.json` - `lakefile.toml` - `lean-toolchain` - `noshake.json` - `tasks.py` ## Notable Findings **No `d...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1748): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1749): -------------------------------------------------------------------------------- [Tool Result: # Complete lakefile.toml Content The provided web page contains the full lakefile.toml file for the SphereEversion project. Here's the complete content: ```toml name = "SphereEversion" defaultTargets = ["SphereEversion"] [leanOptions] pp.unicode.fun = true autoImplicit = false relaxedAutoImplicit = false linter.mathlibStandardSet = true [[require]] name = "mathlib" scope = "leanprover-community" [[require]] name = "checkdecls" git = "https://github.com/PatrickMassot/checkdecls.git" [[lean_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1751): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1752): -------------------------------------------------------------------------------- [Tool Result: # Files in the Scripts Directory Based on the GitHub repository page, the `scripts` directory contains **2 files**: 1. **build_docs.sh** 2. **mk_all.sh** These are shell script files located in the `scripts` folder of the leanprover-community/sphere-eversion repository on the master branch.] -------------------------------------------------------------------------------- ASSISTANT (entry 1754): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1755): -------------------------------------------------------------------------------- [Tool Result: # Complete Script Content This bash script builds HTML documentation for the SphereEversion project. Here's what it does: **Setup Phase:** The script begins with strict error handling via `set -euo pipefail`, ensuring immediate failure on errors, unset variables, and pipeline issues. **Template Generation:** A function creates a lakefile.toml configuration with: - Project metadata (name: "docbuild", version "0.1.0") - Dependencies on SphereEversion and doc-gen4 (v4.25.1) - Custom packages dir...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1757): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1758): -------------------------------------------------------------------------------- [Tool Result: # LaTeX Document Analysis This is a **LaTeX document preamble** for a report on "The sphere eversion project" by Patrick Massot, Oliver Nash, and Floris van Doorn. ## Content Configuration **No dochome or URL configuration is present.** The document references: - External macro files: `macros_common` and `macros_print` (loaded via `\input{}`) - Content file: `content` (also loaded via `\input{}`) These are local file includes, not web-based resources. ## Key Settings The document specifie...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1760): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1761): -------------------------------------------------------------------------------- [Tool Result: # Sphere Eversion Project Configuration This LaTeX document defines a mathematical project with the following URL configurations: **Project URLs:** - **Home:** `https://leanprover-community.github.io/sphere-eversion/` - **GitHub:** `https://github.com/leanprover-community/sphere-eversion/` - **Documentation:** `https://leanprover-community.github.io/sphere-eversion/docs` **Project Details:** The document is titled "The sphere eversion project" and credits three authors: Patrick Massot, Oliver...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1763): -------------------------------------------------------------------------------- Found it! Their `web.tex` has `\dochome{...}` pointing to where the API docs are deployed. Let me check and update ours: -------------------------------------------------------------------------------- ASSISTANT (entry 1764): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1765): -------------------------------------------------------------------------------- [Tool Result: 1→% This file makes a web version of the blueprint 2→% It should include all the \usepackage needed for this version. 3→% The template includes standard AMS packages. 4→% It is otherwise a very minimal preamble (you should probably at least 5→% add cleveref and tikz-cd). 6→ 7→\documentclass{report} 8→ 9→\usepackage{amssymb, amsthm, amsmath} 10→\usepackage{hyperref} 11→\usepackage[showmore, dep_graph]{blueprint} 12→ 13→ 14→\input{ma...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1767): -------------------------------------------------------------------------------- Our `web.tex` already has the correct configuration: ```latex \dochome{https://schmittj.github.io/balanced-vectors-blueprint/docs} ``` **Key findings from sphere-eversion:** 1. **Similar approach** - They dynamically create a `docbuild` directory with a script that generates a lakefile.toml (similar to what we have) 2. **They pin doc-gen4 version** - They use `v4.25.1` explicitly, not `main` 3. **They use `MATHLIB_NO_CACHE_ON_UPDATE=1`** when running lake update 4. **Their `\dochome`** points to `/docs` subdirectory, same as ours 5. **Doc building is slow for them too** - They run doc-gen4 which builds docs for all dependencies The main difference is they generate the docbuild config dynamically via a script, but the principle is the same. Our setup should work once the current build completes. One potential optimization: if the full mathlib doc build is too slow (~30+ min), we could look into only building docs for our library, not all dependencies. But let's first see if the current build succeeds. How's the GitHub Action progressing? -------------------------------------------------------------------------------- USER (entry 1779): -------------------------------------------------------------------------------- bdba4ab /tmp/claude/tasks/bdba4ab.output killed Background command "Retry doc build with correct Lean version" was killed. Read the output file to retrieve the output. -------------------------------------------------------------------------------- ASSISTANT (entry 1782): -------------------------------------------------------------------------------- The issue is a Lean version mismatch - doc-gen4 `main` is using Lean 4.27.0-rc1, but our project uses Lean 4.26.0. We need to pin doc-gen4 to a compatible version. -------------------------------------------------------------------------------- ASSISTANT (entry 1783): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1784): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1786): -------------------------------------------------------------------------------- [Tool: WebSearch] -------------------------------------------------------------------------------- USER (entry 1787): -------------------------------------------------------------------------------- [Tool Result: Web search results for query: "doc-gen4 lean 4.26.0 compatible version tag" Links: [{"title":"GitHub - leanprover/doc-gen4: Document Generator for Lean 4","url":"https://github.com/leanprover/doc-gen4"},{"title":"doc-gen4 - lean4","url":"https://leanprover-community.github.io/archive/stream/270676-lean4/topic/doc-gen4.html"},{"title":"doc-gen4 | Reservoir - Lean","url":"https://reservoir.lean-lang.org/@leanprover/doc-gen4"},{"title":"doc-gen4/Main.lean at main · leanprover/doc-gen4","url":"http...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1789): -------------------------------------------------------------------------------- According to [doc-gen4 documentation](https://github.com/leanprover/doc-gen4), it uses version tags matching Lean versions. Let me update to use `v4.26.0`: -------------------------------------------------------------------------------- ASSISTANT (entry 1790): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1791): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild/lakefile.toml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 5→[[require]] 6→name = "balanced-vectors" 7→path = ".." 8→ 9→# Depend on doc-gen4 (use version matching our Lean toolchain) 10→[[require]] 11→name = "doc-gen4" 12→git = "https://github.com/leanprover/doc-gen4" 13→rev = "v4.26.0" 14→] -------------------------------------------------------------------------------- ASSISTANT (entry 1793): -------------------------------------------------------------------------------- Let me test locally first: -------------------------------------------------------------------------------- ASSISTANT (entry 1794): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1795): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b530211. Output is being written to: /tmp/claude/tasks/b530211.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1797): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1798): -------------------------------------------------------------------------------- [Tool Result: success b530211 local_bash completed 0 instead of String → String warning: Cache/Requests.lean:418:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:420:54: `String.trim` has been deprecated: Use `String.trimAscii...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1800): -------------------------------------------------------------------------------- Same error locally. The issue is that doc-gen4 is pulling in mathlib dependencies that conflict. Let me check the doc-gen4 lean-toolchain for v4.26.0: -------------------------------------------------------------------------------- ASSISTANT (entry 1801): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1802): -------------------------------------------------------------------------------- [Tool Result: The user doesn't want to proceed with this tool use. The tool use was rejected (eg. if it was a file edit, the new_string was NOT written to the file). STOP what you are doing and wait for the user to tell you how to proceed.] -------------------------------------------------------------------------------- USER (entry 1803): -------------------------------------------------------------------------------- [Request interrupted by user for tool use] -------------------------------------------------------------------------------- USER (entry 1804): -------------------------------------------------------------------------------- Note that we originally went for a very specialized mathlib reference to fix the original build. But plausibly that build just had a problem with variable d being undefined, and might now also work for more standard versions of mathlib. Just if the issue is really a version mismatch, we might also be able to adapt the version used in the main compilation. -------------------------------------------------------------------------------- ASSISTANT (entry 1806): -------------------------------------------------------------------------------- Good point! The variable scoping issue was fixed in the Lean file itself (by adding `variable {d : ℤ}` after `end WeakComposition`), so we're not actually tied to that specific mathlib commit anymore. Let me check what Lean/mathlib version doc-gen4 expects, and we can update to match: -------------------------------------------------------------------------------- ASSISTANT (entry 1807): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1808): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1810): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1811): -------------------------------------------------------------------------------- [Tool Result: {"version": "1.1.0", "packagesDir": ".lake/packages", "packages": [{"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, "scope": "", "rev": "933fce7e893f65969714c60cdb4bd8376786044e", "name": "Cli", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": false, "configFile": "lakefile.toml"}, {"url": "https://github.com/fgdorais/lean4-unicode-basic", "type": "git", "subDir": null, "scope": "", "rev": "84b88f7...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1813): -------------------------------------------------------------------------------- The issue might be that `lake update doc-gen4` is updating other dependencies too. Let me try without the update step since we're pinning to v4.26.0: -------------------------------------------------------------------------------- ASSISTANT (entry 1814): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1815): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: b5aa92b. Output is being written to: /tmp/claude/tasks/b5aa92b.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1817): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1818): -------------------------------------------------------------------------------- [Tool Result: success b5aa92b local_bash completed 0 Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:332:21: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Reque...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1820): -------------------------------------------------------------------------------- Let me check if there's a version mismatch in the docbuild's dependencies: -------------------------------------------------------------------------------- ASSISTANT (entry 1821): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1822): -------------------------------------------------------------------------------- [Tool Result: leanprover/lean4:v4.26.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1824): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1825): -------------------------------------------------------------------------------- [Tool Result: Main project mathlib: not found Docbuild mathlib: 26fffffcccd7299b26cf63fac902165bc553fd56] -------------------------------------------------------------------------------- ASSISTANT (entry 1827): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1828): -------------------------------------------------------------------------------- [Tool Result: === Cli === leanprover/lean4:v4.26.0 === UnicodeBasic === leanprover/lean4:v4.26.0-rc2 === BibtexQuery === leanprover/lean4:v4.26.0-rc2=== MD4Lean === leanprover/lean4:4.25.0] -------------------------------------------------------------------------------- ASSISTANT (entry 1830): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1831): -------------------------------------------------------------------------------- [Tool Result: /home/jo314/.elan/bin/lean Lean (version 4.27.0-rc1, x86_64-unknown-linux-gnu, commit 2fcce7258eeb6e324366bc25f9058293b04b7547, Release)] -------------------------------------------------------------------------------- ASSISTANT (entry 1833): -------------------------------------------------------------------------------- Found it! The system default Lean is 4.27.0-rc1, but our project needs 4.26.0. Let me force the correct version: -------------------------------------------------------------------------------- ASSISTANT (entry 1834): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1835): -------------------------------------------------------------------------------- [Tool Result: Lean (version 4.26.0, x86_64-unknown-linux-gnu, commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562, Release) info: override toolchain for '/mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild' set to 'leanprover/lean4:v4.26.0'] -------------------------------------------------------------------------------- ASSISTANT (entry 1837): -------------------------------------------------------------------------------- Now let's try building again: -------------------------------------------------------------------------------- ASSISTANT (entry 1838): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1839): -------------------------------------------------------------------------------- [Tool Result: Command running in background with ID: bdba4ab. Output is being written to: /tmp/claude/tasks/bdba4ab.output] -------------------------------------------------------------------------------- ASSISTANT (entry 1841): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1842): -------------------------------------------------------------------------------- [Tool Result: timeout bdba4ab local_bash running] -------------------------------------------------------------------------------- ASSISTANT (entry 1844): -------------------------------------------------------------------------------- Still running - let me check the output so far: -------------------------------------------------------------------------------- ASSISTANT (entry 1845): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1846): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1848): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1849): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 1851): -------------------------------------------------------------------------------- [Tool: TaskOutput] -------------------------------------------------------------------------------- USER (entry 1852): -------------------------------------------------------------------------------- [Tool Result: [Request interrupted by user for tool use]] -------------------------------------------------------------------------------- USER (entry 1853): -------------------------------------------------------------------------------- [Request interrupted by user for tool use] -------------------------------------------------------------------------------- USER (entry 1854): -------------------------------------------------------------------------------- Assuming provisorially that this works now, can you make changes and push a git commit that would fix the github actions ? -------------------------------------------------------------------------------- ASSISTANT (entry 1856): -------------------------------------------------------------------------------- Yes, let me update the workflow to not use `lake update` (which may pull incompatible versions) and ensure the toolchain is respected: -------------------------------------------------------------------------------- ASSISTANT (entry 1857): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1858): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 33→ 34→ - name: Build Lean project 35→ run: lake build 36→ 37→ - name: Build documentation 38→ run: | 39→ cd docbuild 40→ lake build BalancedVectors:docs 41→ 42→ - name: Upload docs artifact 43→ uses: actions/upload-artif...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1860): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1861): -------------------------------------------------------------------------------- [Tool Result: [main f7d938d] fix: remove lake update to prevent version conflicts 3 files changed, 3 insertions(+), 4 deletions(-) To github.com:schmittj/balanced-vectors-blueprint.git f7e3440..f7d938d main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1863): -------------------------------------------------------------------------------- Pushed! The change removes `lake update doc-gen4` which was pulling in incompatible dependency versions. Since we pin doc-gen4 to `v4.26.0` in `docbuild/lakefile.toml`, lake should use the correct versions automatically. Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions Meanwhile, I'll let the local build continue to see if it succeeds with the correct Lean version. -------------------------------------------------------------------------------- USER (entry 1864): -------------------------------------------------------------------------------- Mmh, this still produces: Run cd docbuild info: downloading https://releases.lean-lang.org/lean4/v4.27.0-rc1/lean-4.27.0-rc1-linux.tar.zst info: installing /home/runner/.elan/toolchains/leanprover--lean4---v4.27.0-rc1 info: docbuild: no previous manifest, creating one from scratch info: doc-gen4: cloning https://github.com/leanprover/doc-gen4 info: doc-gen4: checking out revision '77ef3eb515ad6bd125c596f0b164349d4a7d5bf5' info: toolchain not updated; already up-to-date warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:39:27: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:43:45: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:58:20: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:71:20: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:85:19: `String.Slice.dropRight` has been deprecated: Use `String.Slice.dropEnd` instead warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:89:19: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/doc-gen4/lakefile.lean:118:39: `Lake.Package.name` has been deprecated: Use `baseName`, `keyName`, or `prettyName` instead info: Cli: cloning https://github.com/leanprover/lean4-cli info: Cli: checking out revision '933fce7e893f65969714c60cdb4bd8376786044e' info: UnicodeBasic: cloning https://github.com/fgdorais/lean4-unicode-basic info: UnicodeBasic: checking out revision '84b88f7ac9adf382b9668f852cee82487d616792' warning: /home/runner/work/balanced-vectors-blueprint/balanced-vectors-blueprint/docbuild/.lake/packages/UnicodeBasic/lakefile.lean:20:63: `String.stripSuffix` has been deprecated: Use `String.dropSuffix` instead Note: The updated constant has a different type: {ρ : Type} → String → (pat : ρ) → [String.Slice.Pattern.BackwardPattern pat] → String.Slice instead of String → String → String info: BibtexQuery: cloning https://github.com/dupuisf/BibtexQuery info: BibtexQuery: checking out revision '3ab4379b2b92448717de66b7d3e254ac1487aede' info: MD4Lean: cloning https://github.com/acmepjz/md4lean instead of String → String warning: Cache/Requests.lean:235:26: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:332:21: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:340:34: `String.dropRight` has been deprecated: Use `String.dropEnd` instead Note: The updated constant has a different type: String → Nat → String.Slice instead of String → Nat → String warning: Cache/Requests.lean:418:24: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:418:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:420:54: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:421:51: `String.trim` has been deprecated: Use `String.trimAscii` instead Note: The updated constant has a different type: String → String.Slice instead of String → String warning: Cache/Requests.lean:557:86: `String.trimRight` has been deprecated: Use `String.trimAsciiEnd` instead Note: The updated con ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 1866): -------------------------------------------------------------------------------- I see the issue - despite our `lean-toolchain` saying v4.26.0, it's downloading Lean 4.27.0-rc1. Also notice doc-gen4 is checking out revision `77ef3eb...` instead of the v4.26.0 tag we specified. Here's a self-contained summary for GPT 5.2: --- ## Problem Summary We have a Lean 4 project with a blueprint documentation site. We want to add doc-gen4 to generate API documentation with clickable Lean source links. **The Issue:** When building doc-gen4, GitHub Actions downloads Lean 4.27.0-rc1 instead of our specified v4.26.0, causing String API incompatibilities (deprecated `String.trim`, `String.dropRight`, etc. that now return `String.Slice` instead of `String`). ## Files to Share **1. `/docbuild/lakefile.toml`:** ```toml name = "docbuild" reservoir = false [[require]] name = "balanced-vectors" path = ".." [[require]] name = "doc-gen4" git = "https://github.com/leanprover/doc-gen4" rev = "v4.26.0" ``` **2. `/docbuild/lean-toolchain`:** ``` leanprover/lean4:v4.26.0 ``` **3. `/lean-toolchain` (main project):** ``` leanprover/lean4:v4.26.0 ``` **4. `/lakefile.lean` (main project):** ```lean import Lake open Lake DSL package «balanced-vectors» where leanOptions := #[ ⟨`pp.unicode.fun, true⟩, ⟨`autoImplicit, false⟩ ] require mathlib from git "https://github.com/leanprover-community/mathlib4.git" @ "26fffffcccd7299b26cf63fac902165bc553fd56" @[default_target] lean_lib «BalancedVectors» where ``` **5. Relevant workflow step:** ```yaml - name: Build documentation run: | cd docbuild lake build BalancedVectors:docs ``` ## Question for GPT 5.2 How do we ensure doc-gen4 and all its dependencies use Lean 4.26.0 (matching our mathlib commit) instead of pulling in Lean 4.27.0-rc1? The `rev = "v4.26.0"` in lakefile.toml doesn't seem to be respected. -------------------------------------------------------------------------------- USER (entry 1867): -------------------------------------------------------------------------------- Here is GPT-5.2s input: What you’re seeing is the combination of two (separate) mechanisms: 1. **Lake resolves `rev = "v4.26.0"` to a commit hash** (and records that hash in `lake-manifest.json`). So “doc-gen4 checked out `77ef3…`” is not evidence that the tag was ignored — it’s exactly how tags work: *a tag name points to a commit*. Your manifest shows `inputRev: "v4.26.0"` and `rev: "77ef3…"` for doc-gen4. 2. **Elan chooses the Lean toolchain by the nearest `lean-toolchain` file in the *current working directory tree***. If any part of the build runs tools from inside `.lake/packages/doc-gen4` (or another dependency) and *that package’s* `lean-toolchain` says `v4.27.0-rc1`, Elan will happily download and use it — even if your top-level `lean-toolchain` says `v4.26.0`. ([Lean Language][1]) --- ## The quickest “make it stop” fix (CI + local) Force the toolchain for the whole documentation build by setting **`ELAN_TOOLCHAIN`**. Lake explicitly prefers `ELAN_TOOLCHAIN` over its own detected toolchain string. ([leanprover-community.github.io][2]) **GitHub Actions step:** ```yaml - name: Build documentation env: ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0 run: | cd docbuild lake env lean --version lake build BalancedVectors:docs ``` That makes *every* `lean`/`lake` invocation during that step use 4.26.0, even if a dependency folder has a different `lean-toolchain`. (Alternative that also works: `elan run leanprover/lean4:v4.26.0 -- lake …`, or `elan override set …` at the repo root; overrides take precedence over `lean-toolchain` files. ([Lean Language][1])) --- ## The “actually pin doc-gen4 correctly” fix Your `lake-manifest.json` says Lake resolved doc-gen4 `v4.26.0` to commit `77ef3…`. If that commit really requires Lean 4.27.0-rc1, then **the ref you’re using in doc-gen4 is not Lean-4.26-compatible** (e.g. the tag points somewhere unexpected, or you meant a different tag/branch). What to do: 1. **Inspect what doc-gen4 thinks its toolchain is:** ```bash cd docbuild cat .lake/packages/doc-gen4/lean-toolchain ``` If it says `leanprover/lean4:v4.27.0-rc1`, that explains the download. 2. **Pin doc-gen4 to a revision whose `lean-toolchain` is `v4.26.0`**. The doc-gen4 instructions explicitly recommend choosing the `rev` that matches your Lean version (e.g. `v4.x`). ([Reservoir][3]) In practice, the most robust way is: **use a commit hash** once you’ve identified the correct one. 3. **Regenerate the manifest after changing the pin (and commit it):** ```bash cd docbuild rm -f lake-manifest.json rm -rf .lake/packages/doc-gen4 MATHLIB_NO_CACHE_ON_UPDATE=1 lake update doc-gen4 ``` That `MATHLIB_NO_CACHE_ON_UPDATE=1 …` recommendation is from doc-gen4’s setup docs when mathlib is involved. ([Reservoir][3]) --- ## One extra recommendation (not required, but helps) Doc-gen4’s suggested `docbuild/lakefile.toml` includes `packagesDir = "../.lake/packages"` so the docbuild project shares the parent’s package directory. That reduces surprises/caching weirdness when you have both a main project and a nested docbuild project. ([Reservoir][3]) --- If you paste the contents of `docbuild/.lake/packages/doc-gen4/lean-toolchain` (just that one line), I can tell you immediately whether you’re fighting “wrong doc-gen4 ref” vs “elan picking the wrong directory’s toolchain” — but in most repos it’s *both*, and setting `ELAN_TOOLCHAIN` in the workflow fixes the Lean 4.27 download immediately. [1]: https://lean-lang.org/doc/reference/latest/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/ "Managing Toolchains with Elan" [2]: https://leanprover-community.github.io/mathlib4_docs/Lake/Config/Env.html "Lake.Config.Env" [3]: https://reservoir.lean-lang.org/%40leanprover/doc-gen4 "doc-gen4 | Reservoir" -------------------------------------------------------------------------------- ASSISTANT (entry 1869): -------------------------------------------------------------------------------- Excellent analysis from GPT-5.2! The fix is to set `ELAN_TOOLCHAIN` to force the correct version. Let me implement that: -------------------------------------------------------------------------------- ASSISTANT (entry 1870): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1871): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/.github/workflows/push.yml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 33→ 34→ - name: Build Lean project 35→ run: lake build 36→ 37→ - name: Build documentation 38→ env: 39→ ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0 40→ run: | 41→ cd docbuild 42→ lake build BalancedVectors:docs 43→ 4...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1873): -------------------------------------------------------------------------------- Let me also add the `packagesDir` recommendation to share the parent's packages: -------------------------------------------------------------------------------- ASSISTANT (entry 1874): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1875): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/docbuild/lakefile.toml has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→name = "docbuild" 2→reservoir = false 3→packagesDir = "../.lake/packages" 4→ 5→# Depend on the parent project 6→[[require]] 7→name = "balanced-vectors" 8→path = ".." 9→ 10→# Depend on doc-gen4 (use version matching our Lean toolchain) 11→[[require]] 12→name = "doc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1877): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1878): -------------------------------------------------------------------------------- [Tool Result: [main 78c773c] fix: force ELAN_TOOLCHAIN to prevent version mismatch 2 files changed, 3 insertions(+) To github.com:schmittj/balanced-vectors-blueprint.git f7d938d..78c773c main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 1880): -------------------------------------------------------------------------------- Pushed! The fix: 1. **`ELAN_TOOLCHAIN: leanprover/lean4:v4.26.0`** - Forces all Lean/Lake invocations to use v4.26.0, overriding any dependency's `lean-toolchain` file 2. **`packagesDir = "../.lake/packages"`** - Shares the parent project's package directory to reduce caching issues Check the build at: https://github.com/schmittj/balanced-vectors-blueprint/actions -------------------------------------------------------------------------------- USER (entry 1881): -------------------------------------------------------------------------------- Thanks a lot! The compile is now running (has been for 10 min, so at least not failing fast, but also still at [4171 / 6740] of some build process of documentation - is it building the doc of the entire mathlib? One thing we can do in the meantime: the project https://github.com/leanprover-community/sphere-eversion has a working configuration of their blueprint page, with appropriate LEAN links. We could have a look at what they do, to see how it compares to our setup. -------------------------------------------------------------------------------- ASSISTANT (entry 1883): -------------------------------------------------------------------------------- Yes, doc-gen4 builds docs for all dependencies including mathlib by default - that's why it's slow. Let me check how sphere-eversion handles this. -------------------------------------------------------------------------------- ASSISTANT (entry 1884): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1885): -------------------------------------------------------------------------------- [Tool Result: Request failed with status code 404] -------------------------------------------------------------------------------- ASSISTANT (entry 1887): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1888): -------------------------------------------------------------------------------- [Tool Result: # Workflow Files in `.github/workflows` Based on the repository structure displayed, there is **one workflow file** in this directory: - **blueprint.yml** This is the only GitHub Actions workflow file currently configured in the `.github/workflows` directory of the `leanprover-community/sphere-eversion` repository on the master branch.] -------------------------------------------------------------------------------- ASSISTANT (entry 1890): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1891): -------------------------------------------------------------------------------- [Tool Result: # Compile Blueprint Workflow This GitHub Actions workflow automates building a Lean project blueprint and deploying documentation to GitHub Pages. ## Trigger Events The workflow runs on: - Pushes to the master branch - Pull requests against master - Manual triggering via the Actions interface ## Key Jobs **Build Project** executes several steps: 1. **Project Setup**: Checks out the repository and installs Elan (Lean's version manager) 2. **Caching**: Retrieves cached build artifacts to spe...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1893): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1894): -------------------------------------------------------------------------------- [Tool Result: # GitHub Actions Workflow: Compile Blueprint This YAML file defines a CI/CD workflow that: **Triggers:** - Runs on pushes to `master`, pull requests to `master`, and manual workflow dispatches **Key Jobs:** The `build_project` job performs several steps: 1. Checks out the repository and installs Elan (Lean's version manager) 2. Retrieves cached dependencies via `lake exe cache get` 3. Builds the SphereEversion project 4. Caches API documentation for Lean libraries 5. Generates blueprint docu...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1896): -------------------------------------------------------------------------------- Let me also check their lakefile and docbuild setup: -------------------------------------------------------------------------------- ASSISTANT (entry 1897): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1898): -------------------------------------------------------------------------------- [Tool Result: # Sphere Eversion Project Directory Structure ## Top-Level Directories Based on the GitHub repository content, the top-level structure includes: - `.github/workflows` - `.vscode` - `SphereEversion` - `blueprint` - `docs_src` - `graphs` - `scripts` ## Root-Level Files The root directory contains: - `.gitignore` - `LICENSE` (Apache-2.0) - `README.md` - `SphereEversion.lean` - `lake-manifest.json` - `lakefile.toml` - `lean-toolchain` - `noshake.json` - `tasks.py` ## Notable Findings **No `d...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1900): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1901): -------------------------------------------------------------------------------- [Tool Result: # Complete lakefile.toml Content The provided web page contains the full lakefile.toml file for the SphereEversion project. Here's the complete content: ```toml name = "SphereEversion" defaultTargets = ["SphereEversion"] [leanOptions] pp.unicode.fun = true autoImplicit = false relaxedAutoImplicit = false linter.mathlibStandardSet = true [[require]] name = "mathlib" scope = "leanprover-community" [[require]] name = "checkdecls" git = "https://github.com/PatrickMassot/checkdecls.git" [[lean_...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1903): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1904): -------------------------------------------------------------------------------- [Tool Result: # Files in the Scripts Directory Based on the GitHub repository page, the `scripts` directory contains **2 files**: 1. **build_docs.sh** 2. **mk_all.sh** These are shell script files located in the `scripts` folder of the leanprover-community/sphere-eversion repository on the master branch.] -------------------------------------------------------------------------------- ASSISTANT (entry 1906): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1907): -------------------------------------------------------------------------------- [Tool Result: # Complete Script Content This bash script builds HTML documentation for the SphereEversion project. Here's what it does: **Setup Phase:** The script begins with strict error handling via `set -euo pipefail`, ensuring immediate failure on errors, unset variables, and pipeline issues. **Template Generation:** A function creates a lakefile.toml configuration with: - Project metadata (name: "docbuild", version "0.1.0") - Dependencies on SphereEversion and doc-gen4 (v4.25.1) - Custom packages dir...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1909): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1910): -------------------------------------------------------------------------------- [Tool Result: # LaTeX Document Analysis This is a **LaTeX document preamble** for a report on "The sphere eversion project" by Patrick Massot, Oliver Nash, and Floris van Doorn. ## Content Configuration **No dochome or URL configuration is present.** The document references: - External macro files: `macros_common` and `macros_print` (loaded via `\input{}`) - Content file: `content` (also loaded via `\input{}`) These are local file includes, not web-based resources. ## Key Settings The document specifie...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1912): -------------------------------------------------------------------------------- [Tool: WebFetch] -------------------------------------------------------------------------------- USER (entry 1913): -------------------------------------------------------------------------------- [Tool Result: # Sphere Eversion Project Configuration This LaTeX document defines a mathematical project with the following URL configurations: **Project URLs:** - **Home:** `https://leanprover-community.github.io/sphere-eversion/` - **GitHub:** `https://github.com/leanprover-community/sphere-eversion/` - **Documentation:** `https://leanprover-community.github.io/sphere-eversion/docs` **Project Details:** The document is titled "The sphere eversion project" and credits three authors: Patrick Massot, Oliver...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1915): -------------------------------------------------------------------------------- Found it! Their `web.tex` has `\dochome{...}` pointing to where the API docs are deployed. Let me check and update ours: -------------------------------------------------------------------------------- ASSISTANT (entry 1916): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1917): -------------------------------------------------------------------------------- [Tool Result: 1→% This file makes a web version of the blueprint 2→% It should include all the \usepackage needed for this version. 3→% The template includes standard AMS packages. 4→% It is otherwise a very minimal preamble (you should probably at least 5→% add cleveref and tikz-cd). 6→ 7→\documentclass{report} 8→ 9→\usepackage{amssymb, amsthm, amsmath} 10→\usepackage{hyperref} 11→\usepackage[showmore, dep_graph]{blueprint} 12→ 13→ 14→\input{ma...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1919): -------------------------------------------------------------------------------- Our `web.tex` already has the correct configuration: ```latex \dochome{https://schmittj.github.io/balanced-vectors-blueprint/docs} ``` **Key findings from sphere-eversion:** 1. **Similar approach** - They dynamically create a `docbuild` directory with a script that generates a lakefile.toml (similar to what we have) 2. **They pin doc-gen4 version** - They use `v4.25.1` explicitly, not `main` 3. **They use `MATHLIB_NO_CACHE_ON_UPDATE=1`** when running lake update 4. **Their `\dochome`** points to `/docs` subdirectory, same as ours 5. **Doc building is slow for them too** - They run doc-gen4 which builds docs for all dependencies The main difference is they generate the docbuild config dynamically via a script, but the principle is the same. Our setup should work once the current build completes. One potential optimization: if the full mathlib doc build is too slow (~30+ min), we could look into only building docs for our library, not all dependencies. But let's first see if the current build succeeds. How's the GitHub Action progressing? -------------------------------------------------------------------------------- USER (entry 1926): -------------------------------------------------------------------------------- bdba4ab /tmp/claude/tasks/bdba4ab.output killed Background command "Retry doc build with correct Lean version" was killed. Read the output file to retrieve the output. -------------------------------------------------------------------------------- ASSISTANT (entry 1927): -------------------------------------------------------------------------------- No response requested. -------------------------------------------------------------------------------- USER (entry 1928): -------------------------------------------------------------------------------- The GitHub Action finished successfully, so the first version of the Lean Blueprint is now up and running! :-) Meanwhile, there was a comment by a colleague that we might be able to address: There does seem to be an inconsistency between the statement in the paper and the latex statement in the blueprint. The blueprint statement asserts that for any $e \in E(n,d)$ and balanced $b \in E(n,d)$, then $D(e) \le D(b)$. The Lean statement is that for any $e \in E(n,d)$, there exists a balanced $b \in E(n,d)$, then $D(e) \le D(b)$. On the other hand, the statement in the paper says informally that $D$ is maximized on a balanced vector. To me, it seems that the natural formalization would be: there exists a balanced $b \in E(n,d)$ such that for all $a \in E(n,d)$, $D(b) \ge D(a)$. This is equivalent to your formalized statement because $E(n,d)$ is finite, but my suggestion seems to match the statement in the paper better. I do think it would be best to have this version in the Lean (which makes existence of a balanced vector explicit). This might or might not involve proving that balanced vectors are unique up to permutation. I propose we reformulate the lean file in the current mathlib folder and try it out until it works, then transfer to the blueprint folder. -------------------------------------------------------------------------------- ASSISTANT (entry 1930): -------------------------------------------------------------------------------- Great news that the blueprint is working! Let me look at the current theorem formulation and work on reformulating it to match the paper better. -------------------------------------------------------------------------------- ASSISTANT (entry 1931): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1932): -------------------------------------------------------------------------------- [Tool Result: 1097:theorem maximized_on_balanced (F : SymmetricLogConcaveFunction n d) (e : WeakComposition n d) : 1098- ∃ e' : WeakComposition n d, IsBalanced e'.toFun ∧ F.D e ≤ F.D e' := 1099- balanced_maximizes F.D F.strictlyPositive F.symmetric F.logConcave e 1100- 1101-/-- 1102-**Main Theorem (Minimum).** 1103-Every symmetric log-concave function on weak compositions is minimized on concentrated vectors. 1104- 1105-More precisely: for any `e ∈ E(n,d)`, there exists a concentrated `e' ∈ E(n,d)` with `...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1934): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 1935): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 1937): -------------------------------------------------------------------------------- Let me understand the structure better and check what lemmas we already have about balanced vectors: -------------------------------------------------------------------------------- ASSISTANT (entry 1938): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1939): -------------------------------------------------------------------------------- [Tool Result: 30:- **(Maximum)** `D` is maximized on **balanced** vectors (where `|eᵢ - eⱼ| ≤ 1` for all i,j) 38:| `IsBalanced e` | line 196 | `∀ i j, eᵢ ≤ eⱼ + 1 ∧ eⱼ ≤ eᵢ + 1` | 51:| `maximized_on_balanced` | line 1081 | `∀ e, ∃ e', IsBalanced e' ∧ D(e) ≤ D(e')` | 64:| `balanced_maximizes` | line 660 | Core proof for maximum result | 211:/-- A vector is balanced if all entries differ by at most 1. -/ 212:def IsBalanced (e : Fin n → ℤ) : Prop := 434:/-- If a vector is...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1941): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1942): -------------------------------------------------------------------------------- [Tool Result: 205→ simp only [modify, hki, hkj, ite_false] 206→ 207→end WeakComposition 208→ 209→/-! ### Balanced and Concentrated Vectors -/ 210→ 211→/-- A vector is balanced if all entries differ by at most 1. -/ 212→def IsBalanced (e : Fin n → ℤ) : Prop := 213→ ∀ i j, e i ≤ e j + 1 ∧ e j ≤ e i + 1 214→ 215→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 216→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 217→ ∃ k, ∀ i, e i = if i = k then d...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1944): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1945): -------------------------------------------------------------------------------- [Tool Result: 356- 357-/-- A function on weak compositions that is symmetric under the Sₙ action. -/ 358:def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 359- ∀ (σ : Equiv.Perm (Fin n)) (e : WeakComposition n d), 360- D ⟨e ∘ σ.symm, 361- by simp only [comp_apply] 362- have : ∑ x, e.toFun (σ.symm x) = ∑ x, e.toFun x := Equiv.sum_comp σ.symm e.toFun 363- rw [this]; exact e.sum_eq, 364- fun i => by simp only [comp_apply]; exact e.nonneg _⟩ = D e 365- 366-/-- The l...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1947): -------------------------------------------------------------------------------- Let me check where the Main Results section starts so I can add the new lemmas there: -------------------------------------------------------------------------------- ASSISTANT (entry 1948): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1949): -------------------------------------------------------------------------------- [Tool Result: 1060→ 1061→/-! ========================================================================== 1062→## Main Results (Clean Formulation) 1063→ 1064→This section provides the main theorems in their cleanest form, suitable for 1065→citation and verification against informal statements. 1066→ 1067→**Informal Theorem.** Let D : E(n,d) → ℚ₊ be a function on weak compositions 1068→satisfying Sₙ-symmetry, the log-concavity condition D(e)² ≥ D(e-δᵢ+δⱼ)·D(e+δᵢ-δⱼ), 1069→and strict positivit...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1951): -------------------------------------------------------------------------------- I'll add the new lemmas. Let me first find a good location to add them - after the `IsBalanced` definition: -------------------------------------------------------------------------------- ASSISTANT (entry 1952): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1953): -------------------------------------------------------------------------------- [Tool Result: 211→/-- A vector is balanced if all entries differ by at most 1. -/ 212→def IsBalanced (e : Fin n → ℤ) : Prop := 213→ ∀ i j, e i ≤ e j + 1 ∧ e j ≤ e i + 1 214→ 215→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 216→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 217→ ∃ k, ∀ i, e i = if i = k then d else 0 218→ 219→/-! ### Log-Concave and Palindromic Sequences -/ 220→ 221→/-- A sequence `s : ℤ → ℚ` is log-concave on `[0, q]`. -/ 222→...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1955): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1956): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 211→/-- A vector is balanced if all entries differ by at most 1. -/ 212→def IsBalanced (e : Fin n → ℤ) : Prop := 213→ ∀ i j, e i ≤ e j + 1 ∧ e j ≤ e i + 1 214→ 215→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 216→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 217→ ∃ k, ∀ i, e i = if i = k t...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1958): -------------------------------------------------------------------------------- Let me compile to see what errors we have: -------------------------------------------------------------------------------- ASSISTANT (entry 1959): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1960): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:230:59: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ?a • ?b in the target expression n • (d / ↑n) + #{x | ↑x < (d % ↑n).toNat} • 1 = d n✝ n : ℕ d : ℤ hn : 0 < n hd : 0 ≤ d hr : 0 ≤ d % ↑n hrn : d % ↑n < ↑n ⊢ n • (d / ↑n) + #{x | ↑x < (d % ↑n).toNat} • 1 = d BalancedVectors.lean:244:4: error: `simp` made no progress BalancedVectors.lean:226:74: warning: This simp argument is unused: smul_eq_mul Hint: Omit it from the simp argument list. ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1962): -------------------------------------------------------------------------------- There are several API issues. Let me simplify the approach - I'll use a more direct proof structure: -------------------------------------------------------------------------------- ASSISTANT (entry 1963): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1964): -------------------------------------------------------------------------------- [Tool Result: File has not been read yet. Read it first before writing to it.] -------------------------------------------------------------------------------- ASSISTANT (entry 1966): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1967): -------------------------------------------------------------------------------- [Tool Result: 215→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 216→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 217→ ∃ k, ∀ i, e i = if i = k then d else 0 218→ 219→/-! ### Existence and Uniqueness of Balanced Vectors -/ 220→ 221→/-- Construct a balanced vector: if d = q*n + r with 0 ≤ r < n, the vector has 222→ r entries equal to (q+1) and (n-r) entries equal to q. -/ 223→def balancedVector (n : ℕ) (d : ℤ) (hn : 0 < n) (hd : 0 ≤ d) : WeakCompositi...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1969): -------------------------------------------------------------------------------- Let me simplify the proofs - keep the structure but use sorry for the technical parts that have API issues: -------------------------------------------------------------------------------- ASSISTANT (entry 1970): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1971): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 215→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 216→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 217→ ∃ k, ∀ i, e i = if i = k then d else 0 218→ 219→/-! ### Existence and Uniqueness of Balanced Vectors -/ 220→ 221→/-- A balanced vector exists when n > 0 and d ≥ 0. 222→ Construction...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1973): -------------------------------------------------------------------------------- Now let me update the Main Results section with the reformulated theorem: -------------------------------------------------------------------------------- ASSISTANT (entry 1974): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1975): -------------------------------------------------------------------------------- [Tool Result: 940→ intro i hik 941→ by_contra hi 942→ have hpos_k₀ : 0 < e k₀ := by have := e.nonneg k₀; omega 943→ have hpos_i : 0 < e i := by have := e.nonneg i; omega 944→ have hcontr_app := hcontra i k₀ hik hpos_i 945→ omega 946→ obtain ⟨k₀, hk₀⟩ := hone 947→ apply h 948→ use k₀ 949→ intro i 950→ by_cases hik : i = k₀ 951→ · -- i = k₀: show e i = d 952→ simp only [hik, ite_true] 953→ have hsum : d = ∑ m, e m := e.sum_eq.sy...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1977): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1978): -------------------------------------------------------------------------------- [Tool Result: 950→ by_cases hik : i = k₀ 951→ · -- i = k₀: show e i = d 952→ simp only [hik, ite_true] 953→ have hsum : d = ∑ m, e m := e.sum_eq.symm 954→ have hsplit : ∑ m, e m = e k₀ + ∑ m ∈ Finset.univ.erase k₀, e m := by 955→ rw [Finset.add_sum_erase _ _ (Finset.mem_univ k₀)] 956→ rw [hsplit] at hsum 957→ have hrest : ∑ m ∈ Finset.univ.erase k₀, e m = 0 := by 958→ apply Finset.sum_eq_zero 959→ intro m hm 960→ exact hk₀ m (Finset.ne_of_m...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1980): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 1981): -------------------------------------------------------------------------------- [Tool Result: 1100→ obtain ⟨e', hconc', hD'⟩ := ih (d - maxEntry e1.toFun hn).toNat hm1 e1 rfl 1101→ exact ⟨e', hconc', le_trans hD' hD_decr⟩ 1102→ 1103→/-! ========================================================================== 1104→## Main Results (Clean Formulation) 1105→ 1106→This section provides the main theorems in their cleanest form, suitable for 1107→citation and verification against informal statements. 1108→ 1109→**Informal Theorem.** Let D : E(n,d) → ℚ₊ be a function ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1983): -------------------------------------------------------------------------------- Now let me add the reformulated theorem - the cleaner version that matches the paper: -------------------------------------------------------------------------------- ASSISTANT (entry 1984): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1985): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1156→ concentrated_minimizes hn hd F.D F.strictlyPositive F.symmetric F.logConcave e 1157→ 1158→end SymmetricLogConcaveFunction 1159→ 1160→/-- 1161→**Combined Main Theorem (Original formulation).** 1162→For any symmetric log-concave function D on weak compositions E(n,d): 1163→- The maximum of D is achieved on balanced vecto...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1987): -------------------------------------------------------------------------------- Now let me compile to check for errors: -------------------------------------------------------------------------------- ASSISTANT (entry 1988): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 1989): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:235:10: error: Tactic `split_ifs` failed: no if-then-else conditions to split n : ℕ d : ℤ hn : 0 < n hd : 0 ≤ d q : ℤ := d / ↑n r : ℕ := (d % ↑n).toNat i : Fin n hq : 0 ≤ q ⊢ 0 ≤ (fun i => q + if ↑i < r then 1 else 0) i BalancedVectors.lean:242:6: warning: declaration uses 'sorry' BalancedVectors.lean:251:58: error: Function expected at IsSymmetric but this term has type ?m.2 Note: Expected a function because this term is being applied to the argument D BalancedVecto...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 1991): -------------------------------------------------------------------------------- Let me fix the errors: -------------------------------------------------------------------------------- ASSISTANT (entry 1992): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 1993): -------------------------------------------------------------------------------- [Tool Result: File has not been read yet. Read it first before writing to it.] -------------------------------------------------------------------------------- USER (entry 1995): -------------------------------------------------------------------------------- This session is being continued from a previous conversation that ran out of context. The conversation is summarized below: Analysis: Let me chronologically analyze this conversation: 1. **Initial Context**: This is a continuation from a previous session about a Lean 4/Mathlib4 theorem formalization. The blueprint is now deployed at https://schmittj.github.io/balanced-vectors-blueprint/ 2. **Blueprint Build Success**: The GitHub Action for doc-gen4 completed successfully after fixing the ELAN_TOOLCHAIN issue (based on GPT-5.2's advice). 3. **Main Request - Theorem Reformulation**: A colleague pointed out an inconsistency between the paper statement and the Lean formalization: - Current Lean: `∀ e, ∃ b balanced, D(e) ≤ D(b)` (for any e, there exists balanced b with D(e) ≤ D(b)) - Paper says: "D is maximized on balanced vectors" - Proposed: `∃ b balanced, ∀ e, D(e) ≤ D(b)` (there exists balanced b such that for all e, D(e) ≤ D(b)) 4. **User's explicit request**: "I do think it would be best to have this version in the Lean (which makes existence of a balanced vector explicit). This might or might not involve proving that balanced vectors are unique up to permutation. I propose we reformulate the lean file in the current mathlib folder and try it out until it works, then transfer to the blueprint folder." 5. **My approach**: I added new lemmas to support the reformulation: - `balanced_exists`: A balanced vector exists when n > 0, d ≥ 0 - `balanced_permute`: Two balanced vectors are related by a permutation - `balanced_D_eq`: All balanced vectors have the same D value (via symmetry) - `exists_balanced_maximizer`: Paper-style theorem for maximum - `exists_concentrated_minimizer`: Paper-style theorem for minimum - `main_theorem_paper`: Combined paper-style theorem 6. **Compilation errors encountered**: - Various API issues with Finset, Int operations - `split_ifs` tactic failures - Type inference issues with `IsSymmetric` - I was in the process of fixing these when the summary was requested 7. **Files modified**: - `/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean` - the main Lean file being modified 8. **Current state**: The file has compilation errors that need to be fixed. The new theorem structure is in place but the proofs have `sorry` placeholders and some tactic errors. Summary: 1. Primary Request and Intent: The user received feedback from a colleague about an inconsistency between the paper's informal statement and the Lean formalization. The paper says "D is maximized on balanced vectors" which should be formalized as "there exists a balanced b such that for all e, D(e) ≤ D(b)" (existential statement followed by universal). The current Lean formalization has it as "for all e, there exists balanced b with D(e) ≤ D(b)" (universal followed by existential). The user wants to: - Add the paper-style formulation to the Lean file - Make existence of balanced vectors explicit - Possibly prove balanced vectors are unique up to permutation - Work in the mathlib4 folder first, then transfer to blueprint 2. Key Technical Concepts: - Lean 4 / Mathlib4 theorem proving - Weak compositions E(n,d): vectors e ∈ ℤⁿ with ∑eᵢ = d and eᵢ ≥ 0 - Balanced vectors: all entries differ by at most 1 - Concentrated vectors: all mass at single index (d·δₖ) - Symmetric log-concave functions - Permutation symmetry of D - Equivalence of existential/universal quantifier orderings (for finite sets) - Leanblueprint documentation system - doc-gen4 for API documentation 3. Files and Code Sections: - **`/mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean`** - Main file being modified - Added new section after `IsConcentrated` definition (line ~219): ```lean /-! ### Existence and Uniqueness of Balanced Vectors -/ /-- A balanced vector exists when n > 0 and d ≥ 0. -/ lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : ∃ e : WeakComposition n d, IsBalanced e.toFun := by sorry /-- Two balanced vectors are related by a permutation. -/ lemma balanced_permute {e₁ e₂ : WeakComposition n d} (h₁ : IsBalanced e₁.toFun) (h₂ : IsBalanced e₂.toFun) (hn : 0 < n) (hd : 0 ≤ d) : ∃ σ : Equiv.Perm (Fin n), ∀ i, e₁ i = e₂ (σ i) := by sorry /-- For a symmetric function D, all balanced vectors have the same D value. -/ lemma balanced_D_eq {d : ℤ} {D : WeakComposition n d → ℚ} (hsym : IsSymmetric D) {e₁ e₂ : WeakComposition n d} (h₁ : IsBalanced e₁.toFun) (h₂ : IsBalanced e₂.toFun) (hn : 0 < n) (hd : 0 ≤ d) : D e₁ = D e₂ := by obtain ⟨σ, hσ⟩ := balanced_permute h₁ h₂ hn hd have heq : e₁ = ⟨e₂ ∘ σ, by simp [e₂.sum_eq, Equiv.sum_comp], fun i => e₂.nonneg _⟩ := by ext i simp [hσ i] rw [heq] exact (hsym σ e₂).symm ``` - Added new theorems at end (after line ~1169): ```lean /-- **Main ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 1998): -------------------------------------------------------------------------------- I'll continue fixing the compilation errors in the reformulated theorem. Let me first read the current state of the file to understand what needs to be fixed. -------------------------------------------------------------------------------- ASSISTANT (entry 1999): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 2000): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2002): -------------------------------------------------------------------------------- [Tool Result: 220→ 221→/-- A balanced vector exists when n > 0 and d ≥ 0. 222→ Construction: if d = q*n + r with 0 ≤ r < n, the vector has 223→ r entries equal to (q+1) and (n-r) entries equal to q. -/ 224→lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : 225→ ∃ e : WeakComposition n d, IsBalanced e.toFun := by 226→ -- Construct the balanced vector explicitly 227→ let q := d / ↑n 228→ let r := (d % ↑n).toNat 229→ -- The vector: first r indices get q+1, rest get q ...[truncated]] -------------------------------------------------------------------------------- USER (entry 2003): -------------------------------------------------------------------------------- [Tool Result: 1170→ 1171→/-- 1172→**Main Theorem (Paper formulation - Maximum).** 1173→There exists a balanced vector that maximizes D over all of E(n,d). 1174→ 1175→This formulation matches the informal statement: "D is maximized on balanced vectors." 1176→-/ 1177→theorem exists_balanced_maximizer (hn : 0 < n) (hd : 0 ≤ d) 1178→ (F : SymmetricLogConcaveFunction n d) : 1179→ ∃ b : WeakComposition n d, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b := by 1180→ -- Get any balanced vector (...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2005): -------------------------------------------------------------------------------- Let me load the Lean theorem-proving skill and continue fixing the errors. -------------------------------------------------------------------------------- ASSISTANT (entry 2006): -------------------------------------------------------------------------------- [Tool: Skill] -------------------------------------------------------------------------------- USER (entry 2007): -------------------------------------------------------------------------------- [Tool Result: Launching skill: lean4-theorem-proving] -------------------------------------------------------------------------------- USER (entry 2008): -------------------------------------------------------------------------------- Base directory for this skill: /home/jo314/.claude/plugins/cache/lean4-skills/lean4-theorem-proving/3.3.1/skills/lean4-theorem-proving # Lean 4 Theorem Proving ## Core Principle **Build incrementally, structure before solving, trust the type checker.** Lean's type checker is your test suite. **Success = `lake build` passes + zero sorries + zero custom axioms.** Theorems with sorries/axioms are scaffolding, not results. ## Quick Reference | **Resource** | **What You Get** | **Where to Find** | |--------------|------------------|-------------------| | **Interactive Commands** | 10 slash commands for search, analysis, optimization, repair | Type `/lean` in Claude Code ([full guide](../../COMMANDS.md)) | | **Automation Scripts** | 19 tools for search, verification, refactoring, repair | Plugin `scripts/` directory ([scripts/README.md](../../scripts/README.md)) | | **Subagents** | 4 specialized agents for batch tasks (optional) | [subagent-workflows.md](references/subagent-workflows.md) | | **LSP Server** | 30x faster feedback with instant proof state (optional) | [lean-lsp-server.md](references/lean-lsp-server.md) | | **Reference Files** | 18 detailed guides (phrasebook, tactics, patterns, errors, repair, performance) | [List below](#reference-files) | ## When to Use Use for ANY Lean 4 development: pure/applied math, program verification, mathlib contributions. **Critical for:** Type class synthesis errors, sorry/axiom management, mathlib search, measure theory/probability work. ## Tools & Workflows **7 slash commands** for search, analysis, and optimization - type `/lean` in Claude Code. See [COMMANDS.md](../../COMMANDS.md) for full guide with examples and workflows. **16 automation scripts** for search, verification, and refactoring. See [scripts/README.md](../../scripts/README.md) for complete documentation. **Lean LSP Server** (optional) provides 30x faster feedback with instant proof state and parallel tactic testing. See [lean-lsp-server.md](references/lean-lsp-server.md) for setup and workflows. **Subagent delegation** (optional, Claude Code users) enables batch automation. See [subagent-workflows.md](references/subagent-workflows.md) for patterns. ## Build-First Principle **ALWAYS compile before committing.** Run `lake build` to verify. "Compiles" ≠ "Complete" - files can compile with sorries/axioms but aren't done until those are eliminated. ## The 4-Phase Workflow 1. **Structure Before Solving** - Outline proof strategy with `have` statements and documented sorries before writing tactics 2. **Helper Lemmas First** - Build infrastructure bottom-up, extract reusable components as separate lemmas 3. **Incremental Filling** - Fill ONE sorry at a time, compile after each, commit working code 4. **Type Class Management** - Add explicit instances with `haveI`/`letI` when synthesis fails, respect binder order for sub-structures ## Finding and Using Mathlib Lemmas **Philosophy:** Search before prove. Mathlib has 100,000+ theorems. Use `/search-mathlib` slash command, LSP server search tools, or automation scripts. See [mathlib-guide.md](references/mathlib-guide.md) for detailed search techniques, naming conventions, and import organization. ## Essential Tactics **Key tactics:** `simp only`, `rw`, `apply`, `exact`, `refine`, `by_cases`, `rcases`, `ext`/`funext`. See [tactics-reference.md](references/tactics-reference.md) for comprehensive guide with examples and decision trees. ## Domain-Specific Patterns **Analysis & Topology:** Integrability, continuity, compactness patterns. Tactics: `continuity`, `fun_prop`. **Algebra:** Instance building, quotient constructions. Tactics: `ring`, `field_simp`, `group`. **Measure Theory & Probability** (emphasis in this skill): Conditional expectation, sub-σ-algebras, a.e. properties. Tactics: `measurability`, `positivity`. See [measure-theory.md](references/measure-theory.md) for detailed patterns. **Complete domain guide:** [domain-patterns.md](references/domain-patterns.md) ## Managing Incomplete Proofs **Standard mathlib axioms (acceptable):** `Classical.choice`, `propext`, `quot.sound`. Check with `#print axioms theorem_name` or `/check-axioms`. **CRITICAL: Sorries/axioms are NOT complete work.** A theorem that compiles with sorries is scaffolding, not a result. Document every sorry with concrete strategy and dependencies. Search mathlib exhaustively before adding custom axioms. **When sorries are acceptable:** (1) Active work in progress with documented plan, (2) User explicitly approves temporary axioms with elimination strategy. **Not acceptable:** "Should be in mathlib", "infrastructure lemma", "will prove later" without concrete plan. ## Compiler-Guided Proof Repair **When proofs fail to compile,** use iterative compiler-guided repair instead of blind resampling. **Quick repair:** `/lean4-theorem-proving:repair-file FILE.lean` **How it works:** 1. Compile → extract structured error (type, location, goal, context) 2. Try automate ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 2010): -------------------------------------------------------------------------------- Now let me examine the `IsSymmetric` definition and fix the errors systematically. The user mentioned it's okay to use `sorry` for the helper lemmas to first verify the overall structure works. -------------------------------------------------------------------------------- ASSISTANT (entry 2011): -------------------------------------------------------------------------------- [Tool: Grep] -------------------------------------------------------------------------------- ASSISTANT (entry 2012): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2013): -------------------------------------------------------------------------------- [Tool Result: 1050→ -- Not concentrated: maxEntry < d 1051→ have hmax_lt : maxEntry e.toFun hn < d := by 1052→ have hle := WeakComposition.maxEntry_le_d e hn 1053→ omega 1054→ -- Pick i = maximizer, j = any other positive index 1055→ obtain ⟨i, hi_max⟩ := exists_eq_maxEntry e.toFun hn 1056→ -- Since not concentrated, there exists another positive entry 1057→ have hconc : ¬ IsConcentrated d e.toFun := by 1058→ intro hc 1059→ obtain ⟨k, hk⟩ := hc 1060→...[truncated]] -------------------------------------------------------------------------------- USER (entry 2014): -------------------------------------------------------------------------------- [Tool Result: 400:def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop :=] -------------------------------------------------------------------------------- ASSISTANT (entry 2016): -------------------------------------------------------------------------------- Let me look at the `IsSymmetric` definition and the structure more carefully: -------------------------------------------------------------------------------- ASSISTANT (entry 2017): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 2018): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2019): -------------------------------------------------------------------------------- [Tool Result: 395→ exact ⟨hincr, hdecr⟩ 396→ 397→/-! ### The Main Theorem Setup -/ 398→ 399→/-- A function on weak compositions that is symmetric under the Sₙ action. -/ 400→def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 401→ ∀ (σ : Equiv.Perm (Fin n)) (e : WeakComposition n d), 402→ D ⟨e ∘ σ.symm, 403→ by simp only [comp_apply] 404→ have : ∑ x, e.toFun (σ.symm x) = ∑ x, e.toFun x := Equiv.sum_comp σ.symm e.toFun 405→ rw [this]; ex...[truncated]] -------------------------------------------------------------------------------- USER (entry 2020): -------------------------------------------------------------------------------- [Tool Result: 1115→/-- A function `D` on weak compositions satisfying the three conditions: 1116→ Sₙ-symmetry, log-concavity, and strict positivity. -/ 1117→structure SymmetricLogConcaveFunction (n : ℕ) (d : ℤ) where 1118→ /-- The function D : E(n,d) → ℚ -/ 1119→ D : WeakComposition n d → ℚ 1120→ /-- D(e ∘ σ⁻¹) = D(e) for all permutations σ -/ 1121→ symmetric : IsSymmetric D 1122→ /-- D(e)² ≥ D(e - δᵢ + δⱼ) · D(e + δᵢ - δⱼ) when eᵢ, eⱼ ≥ 1 -/ 1123→ logConcave : SatisfiesLogConcavity...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2023): -------------------------------------------------------------------------------- Let me understand the variable scope around line 220 to fix the type inference issues: -------------------------------------------------------------------------------- ASSISTANT (entry 2024): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 2025): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2027): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- USER (entry 2028): -------------------------------------------------------------------------------- [Tool Result: 195→lemma modify_at_i (e : WeakComposition n d) (i j : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) : 196→ (e.modify i j hi hij) i = e i - 1 := by simp only [modify, ite_true] 197→ 198→@[simp] 199→lemma modify_at_j (e : WeakComposition n d) (i j : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) : 200→ (e.modify i j hi hij) j = e j + 1 := by simp only [modify, hij.symm, ite_false, ite_true] 201→ 202→@[simp] 203→lemma modify_at_other (e : WeakComposition n d) (i j k : Fin n) (hi : 1 ≤ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2030): -------------------------------------------------------------------------------- Good point! Let me focus on getting the structure right with `sorry` placeholders for the technical lemmas. The main issue is the variable scoping - we need to add variable declarations after `end WeakComposition`. Let me fix that and simplify the lemmas. -------------------------------------------------------------------------------- ASSISTANT (entry 2031): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2032): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 203→lemma modify_at_other (e : WeakComposition n d) (i j k : Fin n) (hi : 1 ≤ e i) (hij : i ≠ j) 204→ (hki : k ≠ i) (hkj : k ≠ j) : (e.modify i j hi hij) k = e k := by 205→ simp only [modify, hki, hkj, ite_false] 206→ 207→end WeakComposition 208→ 209→variable {n : ℕ} {d : ℤ} 210→ 211→/-! ### Balanced and Concen...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2034): -------------------------------------------------------------------------------- Now let me simplify `balanced_exists` to use `sorry` so we can verify the overall structure: -------------------------------------------------------------------------------- ASSISTANT (entry 2035): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2036): -------------------------------------------------------------------------------- [Tool Result: 221→/-! ### Existence and Uniqueness of Balanced Vectors -/ 222→ 223→/-- A balanced vector exists when n > 0 and d ≥ 0. 224→ Construction: if d = q*n + r with 0 ≤ r < n, the vector has 225→ r entries equal to (q+1) and (n-r) entries equal to q. -/ 226→lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : 227→ ∃ e : WeakComposition n d, IsBalanced e.toFun := by 228→ -- Construct the balanced vector explicitly 229→ let q := d / ↑n 230→ let r := (d % ↑n).toNat ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2038): -------------------------------------------------------------------------------- Let me simplify `balanced_exists` to use `sorry` for the whole proof to get the structure working first: -------------------------------------------------------------------------------- ASSISTANT (entry 2039): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2040): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 219→ ∃ k, ∀ i, e i = if i = k then d else 0 220→ 221→/-! ### Existence and Uniqueness of Balanced Vectors -/ 222→ 223→/-- A balanced vector exists when n > 0 and d ≥ 0. 224→ Construction: if d = q*n + r with 0 ≤ r < n, the vector has 225→ r entries equal to (q+1) and (n-r) entries equal to q. -/ 226→lemma balan...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2042): -------------------------------------------------------------------------------- Now let me simplify the `exists_concentrated_minimizer` proof: -------------------------------------------------------------------------------- ASSISTANT (entry 2043): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2044): -------------------------------------------------------------------------------- [Tool Result: 1180→ 1181→/-- 1182→**Main Theorem (Paper formulation - Minimum).** 1183→There exists a concentrated vector that minimizes D over all of E(n,d). 1184→ 1185→This formulation matches the informal statement: "D is minimized on concentrated vectors." 1186→-/ 1187→theorem exists_concentrated_minimizer (hn : 0 < n) (hd : 0 ≤ d) 1188→ (F : SymmetricLogConcaveFunction n d) : 1189→ ∃ c : WeakComposition n d, IsConcentrated d c.toFun ∧ ∀ e, F.D c ≤ F.D e := by 1190→ -- Get any...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2046): -------------------------------------------------------------------------------- Let me simplify `exists_concentrated_minimizer` to use `sorry` for the technical parts: -------------------------------------------------------------------------------- ASSISTANT (entry 2047): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2048): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1177→ -- By balanced_D_eq, D(b') = D(b) since both are balanced 1178→ have heq := balanced_D_eq F.symmetric hb'_bal hb_bal hn hd 1179→ linarith 1180→ 1181→/-- 1182→**Main Theorem (Paper formulation - Minimum).** 1183→There exists a concentrated vector that minimizes D over all of E(n,d). 1184→ 1185→This formulation match...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2050): -------------------------------------------------------------------------------- Now let me compile to check the remaining errors: -------------------------------------------------------------------------------- ASSISTANT (entry 2051): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2052): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:226:6: warning: declaration uses 'sorry' BalancedVectors.lean:233:6: warning: declaration uses 'sorry' BalancedVectors.lean:242:58: error: Function expected at IsSymmetric but this term has type ?m.1 Note: Expected a function because this term is being applied to the argument D BalancedVectors.lean:1178:7: error: failed to infer `have` declaration type BalancedVectors.lean:1170:74: error: unsolved goals case right n : ℕ d : ℤ hn : 0 < n hd : 0 ≤ d F : SymmetricLogConc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2054): -------------------------------------------------------------------------------- Let me look at the `balanced_D_eq` definition to fix the type error: -------------------------------------------------------------------------------- ASSISTANT (entry 2055): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2056): -------------------------------------------------------------------------------- [Tool Result: 238→ -- Technical permutation construction 239→ sorry 240→ 241→/-- For a symmetric function D, all balanced vectors have the same D value. -/ 242→lemma balanced_D_eq {D : WeakComposition n d → ℚ} (hsym : IsSymmetric D) 243→ {e₁ e₂ : WeakComposition n d} (h₁ : IsBalanced e₁.toFun) (h₂ : IsBalanced e₂.toFun) 244→ (hn : 0 < n) (hd : 0 ≤ d) : D e₁ = D e₂ := by 245→ obtain ⟨σ, hσ⟩ := balanced_permute h₁ h₂ hn hd 246→ have heq : e₁ = ⟨e₂ ∘ σ, by simp [e₂.sum_eq, Eq...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2058): -------------------------------------------------------------------------------- The issue is a naming conflict between the section variable `d` and the implicit parameter in `IsSymmetric`. Let me simplify `balanced_D_eq` to use `sorry`: -------------------------------------------------------------------------------- ASSISTANT (entry 2059): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2060): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 237→ -- (entries are either ⌊d/n⌋ or ⌈d/n⌉, with counts determined by d mod n) 238→ -- Technical permutation construction 239→ sorry 240→ 241→/-- For a symmetric function D, all balanced vectors have the same D value. 242→ Since balanced vectors have the same multiset of values, they're permutations of each other, ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2062): -------------------------------------------------------------------------------- Now I need to update the `exists_balanced_maximizer` proof to match the new `balanced_D_eq` signature: -------------------------------------------------------------------------------- ASSISTANT (entry 2063): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2064): -------------------------------------------------------------------------------- [Tool Result: 1160→ 1161→/-- 1162→**Main Theorem (Paper formulation - Maximum).** 1163→There exists a balanced vector that maximizes D over all of E(n,d). 1164→ 1165→This formulation matches the informal statement: "D is maximized on balanced vectors." 1166→-/ 1167→theorem exists_balanced_maximizer (hn : 0 < n) (hd : 0 ≤ d) 1168→ (F : SymmetricLogConcaveFunction n d) : 1169→ ∃ b : WeakComposition n d, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b := by 1170→ -- Get any balanced vector (...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2066): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2067): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1172→ use b, hb_bal 1173→ intro e 1174→ -- By maximized_on_balanced, there exists balanced b' with D(e) ≤ D(b') 1175→ obtain ⟨b', hb'_bal, hD⟩ := F.maximized_on_balanced e 1176→ -- By balanced_D_eq, D(b') = D(b) since both are balanced 1177→ have heq := balanced_D_eq F.D F.symmetric hb'_bal hb_bal hn hd 1178→ linarith ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2069): -------------------------------------------------------------------------------- Let me compile again to check: -------------------------------------------------------------------------------- ASSISTANT (entry 2070): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2071): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:226:6: warning: declaration uses 'sorry' BalancedVectors.lean:233:6: warning: declaration uses 'sorry' BalancedVectors.lean:244:6: warning: declaration uses 'sorry' BalancedVectors.lean:1186:8: warning: declaration uses 'sorry'] -------------------------------------------------------------------------------- ASSISTANT (entry 2073): -------------------------------------------------------------------------------- The file now compiles successfully with only `sorry` warnings. The structure is correct. Let me verify the final theorems: -------------------------------------------------------------------------------- ASSISTANT (entry 2074): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2075): -------------------------------------------------------------------------------- [Tool Result: 1155→-/ 1156→theorem main_theorem (hn : 0 < n) (hd : 0 ≤ d) (F : SymmetricLogConcaveFunction n d) : 1157→ (∀ e, ∃ e_bal, IsBalanced e_bal.toFun ∧ F.D e ≤ F.D e_bal) ∧ 1158→ (∀ e, ∃ e_conc, IsConcentrated d e_conc.toFun ∧ F.D e_conc ≤ F.D e) := 1159→ ⟨F.maximized_on_balanced, F.minimized_on_concentrated hn hd⟩ 1160→ 1161→/-- 1162→**Main Theorem (Paper formulation - Maximum).** 1163→There exists a balanced vector that maximizes D over all of E(n,d). 1164→ 1165→This for...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2077): -------------------------------------------------------------------------------- The structure now compiles. Let me summarize what we have and what needs to be filled in: -------------------------------------------------------------------------------- ASSISTANT (entry 2078): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 2079): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2080): -------------------------------------------------------------------------------- [Tool Result: 230: sorry 239: sorry 249: sorry 1192: sorry] -------------------------------------------------------------------------------- USER (entry 2081): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2083): -------------------------------------------------------------------------------- The reformulation is complete and compiles. Here's the summary: **New Paper-Style Theorems (lines 1167-1206):** 1. **`exists_balanced_maximizer`**: ```lean ∃ b, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b ``` "There exists a balanced b such that for all e, D(e) ≤ D(b)" 2. **`exists_concentrated_minimizer`**: ```lean ∃ c, IsConcentrated d c.toFun ∧ ∀ e, F.D c ≤ F.D e ``` "There exists a concentrated c such that for all e, D(c) ≤ D(e)" 3. **`main_theorem_paper`**: Combines both results **Helper lemmas with `sorry` (4 total) that need GPT 5.2 help:** | Line | Lemma | Purpose | |------|-------|---------| | 226 | `balanced_exists` | Construct balanced vector: r entries of ⌈d/n⌉, (n-r) entries of ⌊d/n⌋ | | 233 | `balanced_permute` | Two balanced vectors are permutations of each other | | 244 | `balanced_D_eq` | Symmetric functions have same value on all balanced vectors | | 1186 | `exists_concentrated_minimizer` | Construct concentrated vector and use symmetry | **The key insight:** The proof of `exists_balanced_maximizer` works because: 1. `balanced_exists` gives us some balanced vector `b` 2. `maximized_on_balanced` (already proven) gives `b'` balanced with `D(e) ≤ D(b')` 3. `balanced_D_eq` shows `D(b') = D(b)` since both are balanced Would you like me to attempt filling in any of these `sorry` proofs now, or would you prefer to get GPT 5.2's help with the technical details? -------------------------------------------------------------------------------- USER (entry 2085): -------------------------------------------------------------------------------- Below I paste some of the suggestions from GPT-5.2. I can’t see the “paper-style” theorem names you quoted in the copy of `BalancedVectors.lean` I have in front of me (the attached file still ends with `main_theorem` and doesn’t contain `exists_balanced_maximizer`, etc.). **But** the 4 helper lemmas you listed are very standard, and there are two good ways forward: 1. **keep your current plan** (canonical balanced vector + “all balanced are permutations” + symmetry ⇒ same `D`), or 2. **delete most of that work** by proving the “paper” global max/min using **finiteness + choice of a global maximizer/minimizer** (no `balanced_permute`, no `balanced_D_eq`, and also no “concentrated symmetry” lemma). Below are concrete proposals + code skeletons Claude can paste and then “massage” into your file. --- ## Option A (keep your current plan): make a canonical balanced vector and classify balanced vectors ### A1. Make a canonical `WeakComposition.balanced` once, then `balanced_exists` is trivial Instead of proving `balanced_exists` “from scratch”, define: ```lean namespace WeakComposition open Finset BigOperators noncomputable def balanced (n : ℕ) (d : ℤ) (hn : 0 < n) (hd : 0 ≤ d) : WeakComposition n d := by classical -- work in Nat since d ≥ 0 let dn : ℕ := Int.toNat d let q : ℕ := dn / n let r : ℕ := dn % n -- indicator: first r indices get +1 refine { toFun := fun i => (q : ℤ) + (if (i.1 : ℕ) < r then (1 : ℤ) else 0) sum_eq := ?_ nonneg := ?_ } · -- nonneg intro i; by_cases h : (i.1 : ℕ) < r <;> simp [h] · -- sum_eq -- goal: ∑ (q + ind) = d -- 1) split sums have hsum : (∑ i : Fin n, ((q : ℤ) + (if (i.1 : ℕ) < r then (1 : ℤ) else 0))) = (n : ℤ) * (q : ℤ) + (∑ i : Fin n, (if (i.1 : ℕ) < r then (1 : ℤ) else 0)) := by simp [Finset.sum_add_distrib, mul_comm, mul_left_comm, mul_assoc] -- 2) show indicator sum = r have hind : (∑ i : Fin n, (if (i.1 : ℕ) < r then (1 : ℤ) else 0)) = (r : ℤ) := by -- rewrite as range-sum -- then convert to a filtered sum and use card = r (since r < n from mod_lt) -- (the details are a short but slightly fiddly `simp`/`ext` exercise) sorry -- 3) use dn = q*n + r and d = dn (since hd) have hdiv : q * n + r = dn := by -- Nat.div_add_mod: dn / n * n + dn % n = dn simpa [q, r, Nat.mul_comm, Nat.add_comm, Nat.add_left_comm, Nat.add_assoc] using (Nat.div_add_mod dn n) -- finish -- you’ll need one of the standard lemmas rewriting `Int.toNat` back to `d` -- e.g. `Int.toNat_of_nonneg hd` or `Int.ofNat_of_nonneg hd` depending on simp set sorry end WeakComposition ``` Then your `balanced_exists` lemma is literally: ```lean lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : ∃ e : WeakComposition n d, IsBalanced e.toFun := by refine ⟨WeakComposition.balanced n d hn hd, ?_⟩ -- show IsBalanced of that explicit toFun: values are q or q+1 intro i j -- split on whether i simp [WeakComposition.balanced, hi, hj] · by_cases hj : (j.1 : ℕ) < (Int.toNat d % n) <;> simp [WeakComposition.balanced, hi, hj] ``` The only “real” work is that `hind` and the final `sum_eq` arithmetic need a bit of `Fin.sum_univ_eq_sum_range` + `Finset.sum_filter` + `Nat.div_add_mod`. --- ### A2. `balanced_permute`: reduce to matching the “high” indices and use `Equiv.sumCompl` Once you’ve proved the *classification*: > any balanced `e` has entries exactly `{q, q+1}` with a fixed count of `q+1` then permutation construction is clean. A good pattern is: * define predicates `p₁ i := e₁ i = q+1`, `p₂ i := e₂ i = q+1` * prove `Fintype.card {i // p₁ i} = Fintype.card {i // p₂ i}` * get `ePlus : {i // p₁ i} ≃ {i // p₂ i}` via `Fintype.equivOfCardEq` * similarly for complements * glue them into a permutation of `Fin n` using `Equiv.sumCompl` + `Equiv.sumCongr`. Sketch: ```lean lemma balanced_permute {e₁ e₂ : WeakComposition n d} (h₁ : IsBalanced e₁.toFun) (h₂ : IsBalanced e₂.toFun) (hn : 0 < n) (hd : 0 ≤ d) : ∃ σ : Equiv.Perm (Fin n), ∀ i, e₁ i = e₂ (σ i) := by classical -- define q,r from dn := d.toNat let dn : ℕ := Int.toNat d let q : ℤ := (dn / n : ℕ) -- NOTE: coe to ℤ happens automatically in simp if you write `(dn / n : ℕ)` as Nat and then cast. -- 1) prove value classification for each balanced vector: have hv1 : ∀ i, e₁ i = (q : ℤ) ∨ e₁ i = (q : ℤ) + 1 := by sorry have hv2 : ∀ i, e₂ i = (q : ℤ) ∨ e₂ i = (q : ℤ) + 1 := by sorry -- 2) predicate for “high” indices let p₁ : Fin n → Prop := fun i => e₁ i = (q : ℤ) + 1 let p₂ : Fin n → Prop := fun i => e₂ i = (q : ℤ) + 1 -- 3) show the high-counts are equal (use sum_eq + hv1/hv2) have hcard : Fintype.card {i : Fin n // p₁ i} = Fintype.card {i : Fin n // p₂ i} := by -- standard: sum = n*q + (#high) sorry ...[message truncated at 5000 chars] -------------------------------------------------------------------------------- ASSISTANT (entry 2087): -------------------------------------------------------------------------------- Excellent suggestions from GPT-5.2! Option B (finiteness + global max/min) is indeed much cleaner. Let me implement it: -------------------------------------------------------------------------------- ASSISTANT (entry 2088): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2089): -------------------------------------------------------------------------------- [Tool Result: 221→/-! ### Existence and Uniqueness of Balanced Vectors -/ 222→ 223→/-- A balanced vector exists when n > 0 and d ≥ 0. 224→ Construction: if d = q*n + r with 0 ≤ r < n, the vector has 225→ r entries equal to (q+1) and (n-r) entries equal to q. -/ 226→lemma balanced_exists (hn : 0 < n) (hd : 0 ≤ d) : 227→ ∃ e : WeakComposition n d, IsBalanced e.toFun := by 228→ -- Technical: explicit construction of balanced vector 229→ -- Can be filled in with GPT 5.2 help ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2091): -------------------------------------------------------------------------------- Let me implement Option B - the finiteness approach. I'll replace the helper lemmas section: -------------------------------------------------------------------------------- ASSISTANT (entry 2092): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2093): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 217→/-- A vector is concentrated if it equals `d • δₖ` for some `k`. -/ 218→def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 219→ ∃ k, ∀ i, e i = if i = k then d else 0 220→ 221→/-! ### Finiteness of Weak Compositions -/ 222→ 223→/-- Each entry of a weak composition is bounded by d. -/ 224→lemma WeakComposition.e...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2095): -------------------------------------------------------------------------------- Now let me update the paper-style theorems to use the finiteness approach: -------------------------------------------------------------------------------- ASSISTANT (entry 2096): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2097): -------------------------------------------------------------------------------- [Tool Result: 1160→end SymmetricLogConcaveFunction 1161→ 1162→/-- 1163→**Combined Main Theorem (Original formulation).** 1164→For any symmetric log-concave function D on weak compositions E(n,d): 1165→- The maximum of D is achieved on balanced vectors 1166→- The minimum of D is achieved on concentrated vectors 1167→-/ 1168→theorem main_theorem (hn : 0 < n) (hd : 0 ≤ d) (F : SymmetricLogConcaveFunction n d) : 1169→ (∀ e, ∃ e_bal, IsBalanced e_bal.toFun ∧ F.D e ≤ F.D e_bal) ∧ 1170→ (...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2099): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2100): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1169→ (∀ e, ∃ e_bal, IsBalanced e_bal.toFun ∧ F.D e ≤ F.D e_bal) ∧ 1170→ (∀ e, ∃ e_conc, IsConcentrated d e_conc.toFun ∧ F.D e_conc ≤ F.D e) := 1171→ ⟨F.maximized_on_balanced, F.minimized_on_concentrated hn hd⟩ 1172→ 1173→/-- 1174→**Main Theorem (Paper formulation - Maximum).** 1175→There exists a balanced vector that ma...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2102): -------------------------------------------------------------------------------- Now let me compile to check if the finiteness approach works: -------------------------------------------------------------------------------- ASSISTANT (entry 2103): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2104): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:231:70: warning: `Finset.not_mem_erase` has been deprecated: Use `Finset.notMem_erase` instead BalancedVectors.lean:230:75: error: unsolved goals n : ℕ d : ℤ e : WeakComposition n d hd : 0 ≤ d i : Fin n hsum : ∑ i, e.toFun i = d hi_nonneg : 0 ≤ e.toFun i hrest : 0 ≤ ∑ j ∈ univ.erase i, e.toFun j ⊢ e.toFun i + ∑ x ∈ univ.erase i, e.toFun x = e.toFun i + ∑ j ∈ (insert i (univ.erase i)).erase i, e.toFun j BalancedVectors.lean:237:33: warning: unused variable `hn` Note: This li...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2106): -------------------------------------------------------------------------------- Let me fix the errors. First, let me fix `entry_le_d`: -------------------------------------------------------------------------------- ASSISTANT (entry 2107): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2108): -------------------------------------------------------------------------------- [Tool Result: File has not been read yet. Read it first before writing to it.] -------------------------------------------------------------------------------- ASSISTANT (entry 2110): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2111): -------------------------------------------------------------------------------- [Tool Result: 221→/-! ### Finiteness of Weak Compositions -/ 222→ 223→/-- Each entry of a weak composition is bounded by d. -/ 224→lemma WeakComposition.entry_le_d (e : WeakComposition n d) (hd : 0 ≤ d) (i : Fin n) : 225→ e i ≤ d := by 226→ have hsum := e.sum_eq 227→ have hi_nonneg := e.nonneg i 228→ have hrest : 0 ≤ ∑ j ∈ Finset.univ.erase i, e j := 229→ Finset.sum_nonneg (fun j _ => e.nonneg j) 230→ have hsplit : ∑ j : Fin n, e j = e i + ∑ j ∈ Finset.univ.erase i, e j...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2113): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2114): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 219→ ∃ k, ∀ i, e i = if i = k then d else 0 220→ 221→/-! ### Finiteness of Weak Compositions -/ 222→ 223→/-- Each entry of a weak composition is bounded by d. -/ 224→lemma WeakComposition.entry_le_d (e : WeakComposition n d) (hd : 0 ≤ d) (i : Fin n) : 225→ e i ≤ d := by 226→ have hsum := e.sum_eq 227→ have hr...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2116): -------------------------------------------------------------------------------- Now let me fix the `nonempty` lemma: -------------------------------------------------------------------------------- ASSISTANT (entry 2117): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2118): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 249→ have h1 := e₁.nonneg i 250→ have h2 := e₂.nonneg i 251→ omega 252→ 253→/-- WeakComposition n d is nonempty when n > 0 and d ≥ 0. -/ 254→lemma WeakComposition.nonempty (hn : 0 < n) (hd : 0 ≤ d) : Nonempty (WeakComposition n d) := by 255→ classical 256→ -- The "all mass at index 0" vector works 257→ refine...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2120): -------------------------------------------------------------------------------- Let me compile again: -------------------------------------------------------------------------------- ASSISTANT (entry 2121): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2122): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:224:60: warning: unused variable `hd` Note: This linter can be disabled with `set_option linter.unusedVariables false` BalancedVectors.lean:235:33: warning: unused variable `hn` Note: This linter can be disabled with `set_option linter.unusedVariables false` BalancedVectors.lean:261:13: error: Tactic `split_ifs` failed: no if-then-else conditions to split case refine_2 n : ℕ d : ℤ hn : 0 < n hd : 0 ≤ d i : Fin n ⊢ 0 ≤ (fun i => if i = ⟨0, hn⟩ then d else 0) i] -------------------------------------------------------------------------------- ASSISTANT (entry 2124): -------------------------------------------------------------------------------- The `split_ifs` fails because the function needs to be applied first. Let me fix this: -------------------------------------------------------------------------------- ASSISTANT (entry 2125): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2126): -------------------------------------------------------------------------------- [Tool Result: File has not been read yet. Read it first before writing to it.] -------------------------------------------------------------------------------- ASSISTANT (entry 2128): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2129): -------------------------------------------------------------------------------- [Tool Result: 253→/-- WeakComposition n d is nonempty when n > 0 and d ≥ 0. -/ 254→lemma WeakComposition.nonempty (hn : 0 < n) (hd : 0 ≤ d) : Nonempty (WeakComposition n d) := by 255→ classical 256→ -- The "all mass at index 0" vector works 257→ refine ⟨⟨fun i => if i = ⟨0, hn⟩ then d else 0, ?_, ?_⟩⟩ 258→ · -- sum_eq: ∑ i, (if i = 0 then d else 0) = d 259→ simp only [Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte] 260→ · -- nonneg 261→ intro i; split_ifs <;> omega 26...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2131): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2132): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 256→ -- The "all mass at index 0" vector works 257→ refine ⟨⟨fun i => if i = ⟨0, hn⟩ then d else 0, ?_, ?_⟩⟩ 258→ · -- sum_eq: ∑ i, (if i = 0 then d else 0) = d 259→ simp only [Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte] 260→ · -- nonneg 261→ intro i 262→ show 0 ≤ if i = ⟨0, hn⟩ then d else 0 263→ ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2134): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2135): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean:224:60: warning: unused variable `hd` Note: This linter can be disabled with `set_option linter.unusedVariables false` BalancedVectors.lean:235:33: warning: unused variable `hn` Note: This linter can be disabled with `set_option linter.unusedVariables false`] -------------------------------------------------------------------------------- ASSISTANT (entry 2137): -------------------------------------------------------------------------------- Excellent! The file compiles successfully with no errors and no `sorry`! Let me verify: -------------------------------------------------------------------------------- ASSISTANT (entry 2138): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2139): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 2141): -------------------------------------------------------------------------------- No `sorry` remaining. Let me view the final paper-style theorems: -------------------------------------------------------------------------------- ASSISTANT (entry 2142): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 2143): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2144): -------------------------------------------------------------------------------- [Tool Result: 1173→ ⟨F.maximized_on_balanced, F.minimized_on_concentrated hn hd⟩ 1174→ 1175→/-- 1176→**Main Theorem (Paper formulation - Maximum).** 1177→There exists a balanced vector that maximizes D over all of E(n,d). 1178→ 1179→This formulation matches the informal statement: "D is maximized on balanced vectors." 1180→ 1181→*Proof strategy (Option B from GPT-5.2):* 1182→1. The domain is finite, so a global maximizer eMax exists 1183→2. By maximized_on_balanced, there's balanced b w...[truncated]] -------------------------------------------------------------------------------- USER (entry 2145): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2147): -------------------------------------------------------------------------------- The reformulation is complete using GPT-5.2's Option B (finiteness approach). Here's the summary: ## Results **The file compiles with no `sorry` and no errors!** ### New Paper-Style Theorems | Theorem | Statement | |---------|-----------| | `exists_balanced_maximizer` | `∃ b, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b` | | `exists_concentrated_minimizer` | `∃ c, IsConcentrated d c.toFun ∧ ∀ e, F.D c ≤ F.D e` | | `main_theorem_paper` | Both combined | ### Key Infrastructure Added (lines 221-263) | Lemma/Instance | Purpose | |----------------|---------| | `WeakComposition.entry_le_d` | Each entry ≤ d (using `Finset.single_le_sum`) | | `WeakComposition.finite` | Domain is finite via injection into `Fin n → Fin (d+1)` | | `WeakComposition.nonempty` | Domain is nonempty (concentrated at index 0) | ### Why Option B Works 1. **Finiteness** gives us `Finite.exists_max F.D` → global maximizer `eMax` 2. **Existing theorem** `maximized_on_balanced` gives balanced `b` with `D(eMax) ≤ D(b)` 3. **Global maximality** gives `D(b) ≤ D(eMax)` 4. **Hence** `D(b) = D(eMax)` and `b` is the balanced global maximizer This completely avoids the need for `balanced_permute`, `balanced_D_eq`, and the permutation combinatorics! ### Next Steps Would you like me to: 1. Transfer this to the blueprint repository? 2. Clean up the two warnings about unused variables (`hd` in `entry_le_d`, `hn` in `finite`)? 3. Update the blueprint LaTeX to document the new theorem names? -------------------------------------------------------------------------------- USER (entry 2149): -------------------------------------------------------------------------------- Thanks a lot! Maybe you can: - clean up the warnings - update the line numbers in the file to point to right places - update the LaTeX in the blueprint; in fact the Maximize / Minimize Theorems stated before the Main theorem can go to the Key Lemmas section (maybe call the section: Key Lemmas and Theorems), so that Main Results only states the Main Theorem; then also please reformulate the informal version of the theorem to very directly align with the new formalized version - then we can put everything in a commit of the blueprint repo and push -------------------------------------------------------------------------------- ASSISTANT (entry 2151): -------------------------------------------------------------------------------- I'll work through these tasks systematically. -------------------------------------------------------------------------------- ASSISTANT (entry 2152): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- ASSISTANT (entry 2153): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2154): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- USER (entry 2155): -------------------------------------------------------------------------------- [Tool Result: 221→/-! ### Finiteness of Weak Compositions -/ 222→ 223→/-- Each entry of a weak composition is bounded by d. -/ 224→lemma WeakComposition.entry_le_d (e : WeakComposition n d) (hd : 0 ≤ d) (i : Fin n) : 225→ e i ≤ d := by 226→ have hsum := e.sum_eq 227→ have hrest : 0 ≤ ∑ j ∈ Finset.univ.erase i, e j := 228→ Finset.sum_nonneg (fun j _ => e.nonneg j) 229→ -- Use Finset.single_le_sum: e i ≤ ∑ j, e j = d 230→ have h := Finset.single_le_sum (fun j _ => e.nonne...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2157): -------------------------------------------------------------------------------- Let me fix the unused variable warnings by removing the unnecessary parameters: -------------------------------------------------------------------------------- ASSISTANT (entry 2158): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2159): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 219→ ∃ k, ∀ i, e i = if i = k then d else 0 220→ 221→/-! ### Finiteness of Weak Compositions -/ 222→ 223→/-- Each entry of a weak composition is bounded by d. -/ 224→lemma WeakComposition.entry_le_d (e : WeakComposition n d) (i : Fin n) : e i ≤ d := by 225→ -- Use Finset.single_le_sum: e i ≤ ∑ j, e j = d 226→ have ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2161): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2162): -------------------------------------------------------------------------------- [Tool Result: 229→/-- The domain WeakComposition n d is finite when n > 0 and d ≥ 0. 230→ Each entry satisfies 0 ≤ e i ≤ d, so we can inject into Fin n → Fin (d.toNat + 1). -/ 231→instance WeakComposition.finite (hn : 0 < n) (hd : 0 ≤ d) : Finite (WeakComposition n d) := by 232→ classical 233→ -- Inject into (Fin n → Fin (d.toNat + 1)), which is finite 234→ let bound := d.toNat + 1 235→ let f : WeakComposition n d → (Fin n → Fin bound) := fun e i => 236→ ⟨(e i).toNat, by 2...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2164): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2165): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 225→ -- Use Finset.single_le_sum: e i ≤ ∑ j, e j = d 226→ have h := Finset.single_le_sum (fun j _ => e.nonneg j) (Finset.mem_univ i) 227→ linarith [e.sum_eq] 228→ 229→/-- The domain WeakComposition n d is finite when d ≥ 0. 230→ Each entry satisfies 0 ≤ e i ≤ d, so we can inject into Fin n → Fin (d.toNat + 1). -/ 2...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2167): -------------------------------------------------------------------------------- Now I need to update the call sites in the paper theorems: -------------------------------------------------------------------------------- ASSISTANT (entry 2168): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2169): -------------------------------------------------------------------------------- [Tool Result: 1183→theorem exists_balanced_maximizer (hn : 0 < n) (hd : 0 ≤ d) 1184→ (F : SymmetricLogConcaveFunction n d) : 1185→ ∃ b : WeakComposition n d, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b := by 1186→ classical 1187→ haveI : Finite (WeakComposition n d) := WeakComposition.finite hn hd 1188→ haveI : Nonempty (WeakComposition n d) := WeakComposition.nonempty hn hd 1189→ -- Get a global maximizer using finiteness 1190→ obtain ⟨eMax, heMax⟩ := Finite.exists_max F.D 1191→ -- ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2171): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2172): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1183→theorem exists_balanced_maximizer (hn : 0 < n) (hd : 0 ≤ d) 1184→ (F : SymmetricLogConcaveFunction n d) : 1185→ ∃ b : WeakComposition n d, IsBalanced b.toFun ∧ ∀ e, F.D e ≤ F.D b := by 1186→ classical 1187→ haveI : Finite (WeakComposition n d) := WeakComposition.finite hd 1188→ haveI : Nonempty (WeakComposition n d)...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2174): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2175): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1206→This formulation matches the informal statement: "D is minimized on concentrated vectors." 1207→ 1208→*Proof strategy:* Same as maximum but with global minimizer. 1209→-/ 1210→theorem exists_concentrated_minimizer (hn : 0 < n) (hd : 0 ≤ d) 1211→ (F : SymmetricLogConcaveFunction n d) : 1212→ ∃ c : WeakComposition n d,...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2177): -------------------------------------------------------------------------------- Let me compile to verify the warnings are fixed: -------------------------------------------------------------------------------- ASSISTANT (entry 2178): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2179): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- ASSISTANT (entry 2181): -------------------------------------------------------------------------------- No output means no errors or warnings. Let me now update the line numbers in the file header: -------------------------------------------------------------------------------- ASSISTANT (entry 2182): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- ASSISTANT (entry 2183): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2184): -------------------------------------------------------------------------------- [Tool Result: 1→/- 2→Copyright (c) 2024. All rights reserved. 3→Released under Apache 2.0 license as described in the file LICENSE. 4→-/ 5→import Mathlib.Data.Fin.Basic 6→import Mathlib.Data.Fin.Tuple.Basic 7→import Mathlib.Algebra.BigOperators.Fin 8→import Mathlib.Algebra.Order.BigOperators.Group.Finset 9→import Mathlib.GroupTheory.Perm.Basic 10→import Mathlib.Data.Rat.Defs 11→import Mathlib.Algebra.Order.Field.Basic 12→import Mathlib.Order.Monotone.Ba...[truncated]] -------------------------------------------------------------------------------- USER (entry 2185): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2187): -------------------------------------------------------------------------------- Let me find the actual line numbers for all the key definitions and theorems: -------------------------------------------------------------------------------- ASSISTANT (entry 2188): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2189): -------------------------------------------------------------------------------- [Tool Result: 128:structure WeakComposition (n : ℕ) (d : ℤ) where 214:def IsBalanced (e : Fin n → ℤ) : Prop := 218:def IsConcentrated (d : ℤ) (e : Fin n → ℤ) : Prop := 306:theorem unimodal_of_logconcave_palindromic {s : ℤ → ℚ} {q : ℤ} (hq : 0 ≤ q) 400:def IsSymmetric {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 409:def SatisfiesLogConcavity {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 415:def IsStrictlyPositive {d : ℤ} (D : WeakComposition n d → ℚ) : Prop := 718:theorem balanced_maximizes {d : ℤ} (D : W...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2191): -------------------------------------------------------------------------------- Now let me update the header with correct line numbers: -------------------------------------------------------------------------------- ASSISTANT (entry 2192): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2193): -------------------------------------------------------------------------------- [Tool Result: File has not been read yet. Read it first before writing to it.] -------------------------------------------------------------------------------- ASSISTANT (entry 2195): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2196): -------------------------------------------------------------------------------- [Tool Result: 33→## Key Definitions (for verification) 34→ 35→| Definition | Location | Informal meaning | 36→|-------------------------|----------|-----------------------------------------------| 37→| `WeakComposition n d` | line 112 | n-tuple `e : Fin n → ℤ` with `∑eᵢ = d`, `eᵢ ≥ 0` | 38→| `IsBalanced e` | line 196 | `∀ i j, eᵢ ≤ eⱼ + 1 ∧ eⱼ ≤ eᵢ + 1` | 39→| `IsConcentrated d e` | line 200 | `∃ k, eᵢ = if i = k the...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2198): -------------------------------------------------------------------------------- [Tool: Edit] -------------------------------------------------------------------------------- USER (entry 2199): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/mathlib4/BalancedVectors.lean has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 29→Then: 30→- **(Maximum)** `D` is maximized on **balanced** vectors (where `|eᵢ - eⱼ| ≤ 1` for all i,j) 31→- **(Minimum)** `D` is minimized on **concentrated** vectors (where `e = d·δₖ` for some k) 32→ 33→## Key Definitions (for verification) 34→ 35→| Definition | Location | Informal meaning ...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2201): -------------------------------------------------------------------------------- Let me verify it still compiles, then move on to the blueprint LaTeX: -------------------------------------------------------------------------------- ASSISTANT (entry 2202): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 2203): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2204): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 2205): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2207): -------------------------------------------------------------------------------- Now let me read and update the blueprint LaTeX: -------------------------------------------------------------------------------- ASSISTANT (entry 2208): -------------------------------------------------------------------------------- [Tool: Read] -------------------------------------------------------------------------------- USER (entry 2209): -------------------------------------------------------------------------------- [Tool Result: 1→\chapter{Balanced Vector Optimization} 2→ 3→\section{Introduction} 4→ 5→This blueprint documents the Lean 4 formalization of a theorem about symmetric log-concave functions on weak compositions. The main result states that such functions are maximized on balanced vectors and minimized on concentrated vectors. 6→ 7→The primary application is to descendant integrals on moduli spaces of curves: 8→\[ 9→\langle \tau_{e_1} \cdots \tau_{e_n} \rangle_g = \i...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2211): -------------------------------------------------------------------------------- [Tool: Write] -------------------------------------------------------------------------------- USER (entry 2212): -------------------------------------------------------------------------------- [Tool Result: The file /mnt/c/Users/jo314/Documents/balanced-vectors-blueprint/blueprint/src/content.tex has been updated. Here's the result of running `cat -n` on a snippet of the edited file: 1→\chapter{Balanced Vector Optimization} 2→ 3→\section{Introduction} 4→ 5→This blueprint documents the Lean 4 formalization of a theorem about symmetric log-concave functions on weak compositions. The main result states that such functions are maximized on balanced vectors and minimized on conc...[truncated]] -------------------------------------------------------------------------------- ASSISTANT (entry 2214): -------------------------------------------------------------------------------- Now let me copy the updated Lean file to the blueprint repo and commit: -------------------------------------------------------------------------------- ASSISTANT (entry 2215): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- ASSISTANT (entry 2216): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2217): -------------------------------------------------------------------------------- [Tool Result: ] -------------------------------------------------------------------------------- USER (entry 2218): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2220): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2221): -------------------------------------------------------------------------------- [Tool Result: On branch main Your branch is up to date with 'origin/main'. Changes not staged for commit: (use "git add ..." to update what will be committed) (use "git restore ..." to discard changes in working directory) modified: BalancedVectors.lean modified: blueprint/src/content.tex Untracked files: (use "git add ..." to include in what will be committed) balanced-vectors-claude-code-session.txt no changes added to commit (use "git add" and/or "git commit -a")] -------------------------------------------------------------------------------- ASSISTANT (entry 2223): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2224): -------------------------------------------------------------------------------- [Tool Result: BalancedVectors.lean | 152 +++++++++++++++++++++++++++++++++++++++++----- blueprint/src/content.tex | 54 ++++++++-------- 2 files changed, 163 insertions(+), 43 deletions(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 2226): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2227): -------------------------------------------------------------------------------- [Tool Result: [main 13de9ee] feat: add paper-style theorem formulation (∃ maximizer/minimizer) 2 files changed, 163 insertions(+), 43 deletions(-)] -------------------------------------------------------------------------------- ASSISTANT (entry 2229): -------------------------------------------------------------------------------- [Tool: Bash] -------------------------------------------------------------------------------- USER (entry 2230): -------------------------------------------------------------------------------- [Tool Result: To github.com:schmittj/balanced-vectors-blueprint.git 78c773c..13de9ee main -> main] -------------------------------------------------------------------------------- ASSISTANT (entry 2232): -------------------------------------------------------------------------------- [Tool: TodoWrite] -------------------------------------------------------------------------------- USER (entry 2233): -------------------------------------------------------------------------------- [Tool Result: Todos have been modified successfully. Ensure that you continue to use the todo list to track your progress. Please proceed with the current tasks if applicable] -------------------------------------------------------------------------------- ASSISTANT (entry 2235): -------------------------------------------------------------------------------- All done! Here's the summary: ## Completed Tasks ### 1. Cleaned up warnings - Removed unused `hd` parameter from `WeakComposition.entry_le_d` - Removed unused `hn` parameter from `WeakComposition.finite` ### 2. Updated line numbers in file header All line references now point to correct locations (e.g., `WeakComposition` at line 128, `main_theorem_paper` at line 1238). ### 3. Restructured Blueprint LaTeX - **Main Result** section now contains only `main_theorem_paper` - **Key Lemmas and Theorems** section contains: - `thm:maximized_on_balanced` (linked to `exists_balanced_maximizer`) - `thm:minimized_on_concentrated` (linked to `exists_concentrated_minimizer`) - `lem:unimodal`, `lem:balanced_maximizes`, `lem:concentrated_minimizes` - **Informal statement now matches formal version:** > - **Maximum:** There exists a balanced vector $b \in E(n,d)$ such that $D(e) \leq D(b)$ for all $e \in E(n,d)$. > - **Minimum:** There exists a concentrated vector $c \in E(n,d)$ such that $D(c) \leq D(e)$ for all $e \in E(n,d)$. ### 4. Committed and pushed ``` commit 13de9ee feat: add paper-style theorem formulation (∃ maximizer/minimizer) ``` The GitHub Actions will now rebuild the blueprint. You can check the progress at: https://github.com/schmittj/balanced-vectors-blueprint/actions Once complete, the updated blueprint will be at: https://schmittj.github.io/balanced-vectors-blueprint/ ================================================================================ END OF CONVERSATION EXPORT ================================================================================