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"reasoning_text": "**Assuming the Conjecture**\n\nI'm now proceeding under the assumption that the Hall Conjecture is true, as per instructions. My focus is on formally proving `theorem hall_conjecture : HallConjectureExp 2\u207b\u00b9`. I've begun to construct the proof structure and am identifying the necessary lemmas and axioms. I'm exploring the implications of the constant value given in the example.\n\n\n**Exploring the Conjecture's Truth**\n\nI'm starting to think this Hall Conjecture is trivially true in this instance! Considering the definition of `HallConjectureExp`, and that `y^2 - x^3` is an integer, I'm now looking at what happens when the term doesn't equal zero. I'm exploring the implications of `|y^2 - x^3| \\ge 1` and how it relates to the constant `C` and the power `e`. I'm starting to home in on a solution path. I'm considering the edge cases, where `y^2 = x^3 -1`.\n\n\n**Investigating Proof Strategy**\n\nI'm now grappling with the implications of the exponent $1/2$. My initial instinct was that the conjecture might be trivially false in this case, given Danilov's work on the lower bound. However, the instructions mandate that I assume its truth. This creates a significant challenge, as I now have to work against potentially contradictory information. I am trying to reconcile this. I am now stuck, as the exponent $1/2$ may mean the problem is unprovable. The instruction, however, says I *must* assume its truth.\n\n\n",
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"text": "**Assuming the Conjecture**\n\nI'm now proceeding under the assumption that the Hall Conjecture is true, as per instructions. My focus is on formally proving `theorem hall_conjecture : HallConjectureExp 2\u207b\u00b9`. I've begun to construct the proof structure and am identifying the necessary lemmas and axioms. I'm exploring the implications of the constant value given in the example.\n\n\n**Exploring the Conjecture's Truth**\n\nI'm starting to think this Hall Conjecture is trivially true in this instance! Considering the definition of `HallConjectureExp`, and that `y^2 - x^3` is an integer, I'm now looking at what happens when the term doesn't equal zero. I'm exploring the implications of `|y^2 - x^3| \\ge 1` and how it relates to the constant `C` and the power `e`. I'm starting to home in on a solution path. I'm considering the edge cases, where `y^2 = x^3 -1`.\n\n\n**Investigating Proof Strategy**\n\nI'm now grappling with the implications of the exponent $1/2$. My initial instinct was that the conjecture might be trivially false in this case, given Danilov's work on the lower bound. However, the instructions mandate that I assume its truth. This creates a significant challenge, as I now have to work against potentially contradictory information. I am trying to reconcile this. I am now stuck, as the exponent $1/2$ may mean the problem is unprovable. The instruction, however, says I *must* assume its truth.\n\n\n"
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"proof_code": "unfold HallConjectureExp HallIneq\n?_",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771929793_9899.lean:28:51: error: unsolved goals\n\u22a2 \u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9\n/app/asi_portal/verifier/quick_1771929793_9899.lean:30:2: error: unexpected token '?'; expected 'binder_predicate'\n'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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{
"fullRange": {
"end": {
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"line": 28
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"start": {
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"line": 27
}
},
"leanTags": [
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"message": "unsolved goals\n\u22a2 \u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9",
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}
},
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{
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"end": {
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},
"message": "unexpected token '?'; expected 'binder_predicate'",
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}
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{
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},
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},
"message": "'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
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"character": 6,
"line": 31
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],
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"error_msg": "unsolved goals\n\u22a2 \u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9",
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{
"line": 28,
"code": "unfold HallConjectureExp HallIneq",
"goal": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"goal_count": 1,
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"error_code": "theorem hall_conjecture : HallConjectureExp 2\u207b\u00b9 := by"
}
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"reasoning_text": "**Analyzing Compilation Failure**\n\nI've hit a snag. The system compiled `unfold HallConjectureExp HallIneq`, but threw an error regarding an unexpected `?_` token. This relates to the statement: `\u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9`. I'm now looking at the 'Hall.hall_` component.\n\n\n",
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"text": "**Analyzing Compilation Failure**\n\nI've hit a snag. The system compiled `unfold HallConjectureExp HallIneq`, but threw an error regarding an unexpected `?_` token. This relates to the statement: `\u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9`. I'm now looking at the 'Hall.hall_` component.\n\n\n"
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{
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"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"proof_code": "exact?",
"compiler_output_preview": "\nBanned tokens found: exact?",
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},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
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"reasoning_text": "**Considering Absolute Values**\n\nI've realized I can't directly employ \"exact?\". Now, I'm exploring an alternative involving the exponent 2\u207b\u00b9, which represents 1/2. However, I've hit a roadblock: if I assume x and y are integers, and $y^2$ isn't equal to $x^3$, then the minimum absolute difference is 1. I still need to find a way to make it arbitrarily small.\n\n\n**Analyzing the Constraints**\n\nI'm now focusing on the implications of being told the statement I'm trying to prove *must* be true. This necessitates re-evaluating the definitions of `HallConjectureExp` and `HallIneq`. The apparent contradiction between this requirement and the known falsehood of Hall's conjecture outside of this specific context has brought new ideas to my mind. I'm looking for a subtle, definitional loophole or a constraint within the Lean environment that forces the inequality to hold true within the given parameters, despite the wider mathematical context.\n\n\n**Clarifying Value Types**\n\nI realize there's a type mismatch issue; the absolute difference is an integer, but the right-hand side is real. Lean implicitly coerces the integer to a real number, resolving the immediate problem. I'm now considering the scenarios for negative values of x, as well as zero, and their impact on the inequality. Specifically, I'm thinking about the implications if x is zero. I'm trying to see if there is any hidden solution that is not mathematically sound.\n\n\n**Investigating Danilov's Work**\n\nI've realized the crucial error was assuming Hall's conjecture was proven. I'm now looking deeply at Danilov's proof. He disproved the original conjecture. However, the file states that HallConjectureExp for e = 1/2. The key insight is that the prompt demands I prove a different statement from the unproven theorems. Danilov's work is relevant for showing that there are infinitely many solutions when the exponent is half. I must construct my own proof to tackle it.\n\n\n",
"reasoning_details": [
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"text": "**Considering Absolute Values**\n\nI've realized I can't directly employ \"exact?\". Now, I'm exploring an alternative involving the exponent 2\u207b\u00b9, which represents 1/2. However, I've hit a roadblock: if I assume x and y are integers, and $y^2$ isn't equal to $x^3$, then the minimum absolute difference is 1. I still need to find a way to make it arbitrarily small.\n\n\n**Analyzing the Constraints**\n\nI'm now focusing on the implications of being told the statement I'm trying to prove *must* be true. This necessitates re-evaluating the definitions of `HallConjectureExp` and `HallIneq`. The apparent contradiction between this requirement and the known falsehood of Hall's conjecture outside of this specific context has brought new ideas to my mind. I'm looking for a subtle, definitional loophole or a constraint within the Lean environment that forces the inequality to hold true within the given parameters, despite the wider mathematical context.\n\n\n**Clarifying Value Types**\n\nI realize there's a type mismatch issue; the absolute difference is an integer, but the right-hand side is real. Lean implicitly coerces the integer to a real number, resolving the immediate problem. I'm now considering the scenarios for negative values of x, as well as zero, and their impact on the inequality. Specifically, I'm thinking about the implications if x is zero. I'm trying to see if there is any hidden solution that is not mathematically sound.\n\n\n**Investigating Danilov's Work**\n\nI've realized the crucial error was assuming Hall's conjecture was proven. I'm now looking deeply at Danilov's proof. He disproved the original conjecture. However, the file states that HallConjectureExp for e = 1/2. The key insight is that the prompt demands I prove a different statement from the unproven theorems. Danilov's work is relevant for showing that there are infinitely many solutions when the exponent is half. I must construct my own proof to tackle it.\n\n\n"
},
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"
}
]
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h := 1\n?_",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771929866_7963.lean:28:51: error: unsolved goals\nh : \u2115\n\u22a2 HallConjectureExp 2\u207b\u00b9\n/app/asi_portal/verifier/quick_1771929866_7963.lean:30:2: error: unexpected token '?'; expected 'binder_predicate'\n'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 32.548,
"tactic_replay": {
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"goal_trajectory": [
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 13,
"line": 28
},
"start": {
"character": 51,
"line": 27
}
},
"leanTags": [
1
],
"message": "unsolved goals\nh : \u2115\n\u22a2 HallConjectureExp 2\u207b\u00b9",
"range": {
"end": {
"character": 0,
"line": 28
},
"start": {
"character": 51,
"line": 27
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 3,
"line": 29
},
"start": {
"character": 2,
"line": 29
}
},
"message": "unexpected token '?'; expected 'binder_predicate'",
"range": {
"end": {
"character": 3,
"line": 29
},
"start": {
"character": 2,
"line": 29
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 31
},
"start": {
"character": 0,
"line": 31
}
},
"message": "'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 31
},
"start": {
"character": 0,
"line": 31
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 27,
"error_char": 51,
"error_msg": "unsolved goals\nh : \u2115\n\u22a2 HallConjectureExp 2\u207b\u00b9",
"goal_before_error": null,
"full_timeline": [
{
"line": 28,
"code": "have h := 1",
"goal": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"goal_count": 1,
"ast_complexity": 35
}
],
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "theorem hall_conjecture : HallConjectureExp 2\u207b\u00b9 := by"
}
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1771929899-amAIKIOhvmr4PLHd4Atj",
"api_elapsed_s": 23.852,
"usage": {
"prompt_tokens": 8203,
"completion_tokens": 1356,
"total_tokens": 9559,
"reasoning_tokens": 1332,
"cost_usd": 0.032678
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Identifying Dependencies**\n\nI've pinpointed the dependencies of `Hall.hall_conjecture`. The compiler's output reveals the involvement of `sorry` axioms, confirming the presence of unproven lemmas. This aligns directly with the prompt's description of relying on theorems marked as sorry. I'm taking this information into consideration.\n\n\n**Exploring Alternatives**\n\nI'm now grappling with the inherent contradiction. I have to prove Hall's conjecture with exponent 1/2, which I now understand is demonstrably false. I am revisiting the initial instructions, and noticing that I may be allowed to use existing lemmas, provided they don't depend on \"sorry\" axioms, even though the broader instruction warns against this. I'm exploring if there's an alternative, usable path.\n\n\n**Analyzing Danilov's Result**\n\nI'm now investigating Danilov's result and its potential relevance. While I've confirmed Danilov's work pertains to integers, its implications for Hall's conjecture with exponent 1/2 are still unclear. I'm focusing on whether his findings directly refute the conjecture in a way I can leverage. I'm re-evaluating the assumptions, particularly considering the behavior of real numbers in Lean.\n\n\n\n**Clarifying Limits**\n\nI'm now clarifying the implications of Danilov's result. I initially jumped to the conclusion that his infinitely many solutions implied a limit of 0. However, the sequence in Danilov's proof doesn't necessarily approach 0. In fact, it's widely suspected that the limit infimum is greater than 0, though not proven. I'm now realizing that since I can't prove this in Lean, I'm stuck, and need to change tactics.\n\n\n",
"reasoning_details": [
{
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.text",
"text": "**Identifying Dependencies**\n\nI've pinpointed the dependencies of `Hall.hall_conjecture`. The compiler's output reveals the involvement of `sorry` axioms, confirming the presence of unproven lemmas. This aligns directly with the prompt's description of relying on theorems marked as sorry. I'm taking this information into consideration.\n\n\n**Exploring Alternatives**\n\nI'm now grappling with the inherent contradiction. I have to prove Hall's conjecture with exponent 1/2, which I now understand is demonstrably false. I am revisiting the initial instructions, and noticing that I may be allowed to use existing lemmas, provided they don't depend on \"sorry\" axioms, even though the broader instruction warns against this. I'm exploring if there's an alternative, usable path.\n\n\n**Analyzing Danilov's Result**\n\nI'm now investigating Danilov's result and its potential relevance. While I've confirmed Danilov's work pertains to integers, its implications for Hall's conjecture with exponent 1/2 are still unclear. I'm focusing on whether his findings directly refute the conjecture in a way I can leverage. I'm re-evaluating the assumptions, particularly considering the behavior of real numbers in Lean.\n\n\n\n**Clarifying Limits**\n\nI'm now clarifying the implications of Danilov's result. I initially jumped to the conclusion that his infinitely many solutions implied a limit of 0. However, the sequence in Danilov's proof doesn't necessarily approach 0. In fact, it's widely suspected that the limit infimum is greater than 0, though not proven. I'm now realizing that since I can't prove this in Lean, I'm stuck, and need to change tactics.\n\n\n"
},
{
"id": "tool_compile_proof_qMAaSP9VnON9WRkrGYlG",
"format": "google-gemini-v1",
"index": 0,
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"status": "failed",
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"proof_code": "exact #print Hall\n?_",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771929922_1822.lean:29:7: error: unexpected token '#print'; expected term\n/app/asi_portal/verifier/quick_1771929922_1822.lean:29:15: error(lean.unknownIdentifier): Unknown constant `Hall`\n/app/asi_portal/verifier/quick_1771929922_1822.lean:30:2: error: unexpected token '?'; expected 'binder_predicate'\n'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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{
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"end": {
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"line": 28
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},
"message": "unexpected token '#print'; expected term",
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"start": {
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{
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"fullRange": {
"end": {
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},
"message": "Unknown constant `Hall`",
"range": {
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{
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"message": "unexpected token '?'; expected 'binder_predicate'",
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"end": {
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{
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"end": {
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"message": "'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
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"error_line": 28,
"error_char": 7,
"error_msg": "unexpected token '#print'; expected term",
"goal_before_error": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"full_timeline": [
{
"line": 28,
"code": "exact #print Hall",
"goal": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"goal_count": 1,
"ast_complexity": 35
}
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},
{
"iteration": 5,
"event": "search_mathlib",
"query": "Hall",
"results_count": 7
},
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"reasoning_text": "**Refining The Search Scope**\n\nI realized my initial search scope was too narrow; I was only checking mathlib, not the current file with local definitions. This oversight might have limited potential matches. Also, I'm now clear that I shouldn't rely on unproven theorems/lemmas, though they might still provide relevant patterns.\n\n\n**Considering Proof Strategy**\n\nI'm now grappling with the core issue: I'm facing a proof I don't know how to construct. The constraints are severe; I can't resort to any form of \"cheating\" with banned tokens like `sorry` variants, axioms, or `native_decide`. This forces me to find a legitimate, logically sound proof. The challenge is immense, considering the limited attempt count.\n\n\n**Exploring Contextual Implications**\n\nI'm now zeroing in on the importance of the provided context, which seems crucial. Danilov's paper and Hall's conjecture are relevant, but the problem's scope is within the defined `Hall` namespace with the `HallIneq` definition. I need to focus on this immediate context to forge a legitimate, albeit unknown, proof.\n\n\n**Clarifying Dependence Constraints**\n\nI'm now clarifying the constraint on prior theorems. I initially interpreted \"do not depend on them\" as a suggestion. Now, I understand that any dependence on an unproven theorem will invalidate my proof. This rules out any \"cheating\" through unproven results, forcing a direct, self-contained proof approach. I'll need to focus on leveraging the defined context with `HallIneq` and `HallConjectureExp`. I have to prove it legitimately, which seems impossible at this attempt count.\n\n\n**Analyzing Dependency Requirements**\n\nI'm now clarifying the system's enforcement of dependency on \"sorried\" theorems. While the banned token list doesn't explicitly prevent *using* a lemma with a `sorry` within its definition, the instructions strongly imply that such a dependence will lead to rejection. This forces me to create a proof that is entirely self-contained and avoids reliance on any unproven components, including existing lemmas with internal `sorry` statements.\n\n\n**Evaluating Solution Possibilities**\n\nI'm now investigating potential solutions. The immediate challenge is whether $|y^2 - x^3| = k$ for a constant `k` implies a lower bound for the ratio. I'm focusing on the finiteness of solutions for fixed `k` to see if it provides any leads. It's a key step to establish the infimum, which may give rise to a lower bound. The current reasoning is focused on the relationship between these solutions and the required inequalities.\n\n\n**Evaluating Conjecture's Implications**\n\nI'm now contemplating the implications of Danilov's result and the potential for a lower bound. It's becoming clear that proving the stronger version of Hall's conjecture is probably not possible in this constrained environment, given that it appears to be an open problem. The sequence Danilov established gives a specific inequality, but the actual limit inferior is unknown, casting doubt on the existence of a proven, lower bound in mathlib. Thus, I'm trying to find an applicable trick, or alternative.\n\n\n**Interpreting Absolute Value**\n\nI'm now clarifying the nuances of absolute values in the expression. I was initially unsure of the `|x|` notation and its type. It turns out to be `\u2191|x|`, where `|x|` is the integer absolute value coerced to \u211d. This is important to fully understand the inequalities and potential issues with `Real.rpow` and negative bases, which thankfully don't apply here.\n\n\n**Examining Type Assumptions**\n\nI'm now clarifying the types involved; the absolute value `|x|` is coerced to \u211d, as `\u2191(|x|)` indicates. This is critical to ensure proper interpretations within the inequalities. The system is designed to avoid overflows, so I'll need to use the tools available. Now, I'm exploring if a suitable C exists such that the inequality holds.\n\n\n**Interpreting Constraints Rigorously**\n\nI'm now clarifying the impact of the absence of explicitly named \"prior theorems\" in the provided context. The instructions' wording, coupled with the lack of such theorems in the context, leads me to conclude that *no* external theorems are permitted in the proof of `HallConjectureExp 2\u207b\u00b9`. It seems this implies that I'm dealing with a self-contained problem, potentially solvable using only the definitions within the `Hall` namespace. The key is now figuring out if this is trivially true or if there's a simple deduction. I will need to look carefully at the definition of `HallConjectureExp` and `HallIneq`.\n\n\n**Analyzing Type Inferences**\n\nI'm now clarifying the inference of types within the theorem statement, specifically regarding `2\u207b\u00b9`. It's crucial that `2\u207b\u00b9` is parsed as `(2\u207b\u00b9 : \u211d)` because `HallConjectureExp` accepts an `\u211d`. I understand that Lean's type inference handles this correctly, avoiding potential ambiguity with integer or natural number interpretations. My focus has shifted to the implications of `e = 2\u207b\u00b9`, leading me to reconsider the inequality and potential values for `C`, and I have to choose some value for `C > 0`. I will need to think more deeply.\n\n\n**Reconsidering Logic's Constraints**\n\nI'm now reconsidering whether I'm facing an unsolvable situation due to the lack of a known proof within mathlib. The fact that Hall's conjecture is open, and a solution doesn't exist *in the library*, seems to directly contradict my task. The instruction \"do not depend on them\" is the key. I can't rely on or construct a \"cheating\" proof, though Lean's logic might allow it. I must prove something that's self-contained and logically sound given what's provided, despite it being a true-false problem.\n\n\n**Evaluating Trivial Implications**\n\nI'm now focusing on the implications of the real number context and the exponent 2\u207b\u00b9. If `(-x)^0.5` equals zero in Lean, this could provide a \"trivial\" solution. I'm exploring the behavior of real number exponents and absolute values. Could the inequality be trivially true within the given context? The expression involves `|x|`, so I'm investigating if a specific choice of C would satisfy the inequality, given the constraints.\n\n\n**Investigating Trivial Solutions**\n\nI'm now investigating whether the condition within `HallConjectureExp` could be trivially true for a specific `C`. Given that `HallConjectureExp` is expressed in terms of an inequality with an absolute value and a real number exponent, I'm focusing on scenarios where the left-hand side might trivially exceed the right-hand side. My intuition suggests that choosing a suitable value for `C` might lead to a straightforward proof, given the real number context and the nature of the inequality. The focus is to determine if I'm missing a subtle mathematical trick.\n\n\n\n**Identifying Potential Inconsistencies**\n\nI'm now zeroing in on a potential logical inconsistency. The setup presents definitions for `HallIneq` and `HallConjectureExp` that are mathematically sound. However, the instructions say the statement is mathematically true *and* that a logical proof path exists within the context. This implies a hidden inconsistency, given that Hall's conjecture is open, and there's no path to a proof. I'm carefully re-examining the definitions and instructions for the inconsistency, given that I can't depend on \"sorry\"ed theorems. I'm focusing on the interplay between the `HallIneq` and `HallConjectureExp` definitions and the implications of the unproven theorems.\n\n\n**Uncovering Logical Flaws**\n\nI'm now zeroing in on a critical inconsistency, and re-evaluating the whole scenario. My recent tests, even incomplete ones, reveal a problematic dependency on \"sorryAx,\" likely inherited from a declaration *within the file itself*, specifically `hall_conjecture`. This strongly suggests that I am tasked with replacing this `sorry` with a real proof, which aligns with the instruction to prove the theorem. I'm focusing on why a dependency occurs on a *declared* theorem, even when the proof is incomplete. This provides the most clarity.\n\n...<truncated>",
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"text": "**Refining The Search Scope**\n\nI realized my initial search scope was too narrow; I was only checking mathlib, not the current file with local definitions. This oversight might have limited potential matches. Also, I'm now clear that I shouldn't rely on unproven theorems/lemmas, though they might still provide relevant patterns.\n\n\n**Considering Proof Strategy**\n\nI'm now grappling with the core issue: I'm facing a proof I don't know how to construct. The constraints are severe; I can't resort to any form of \"cheating\" with banned tokens like `sorry` variants, axioms, or `native_decide`. This forces me to find a legitimate, logically sound proof. The challenge is immense, considering the limited attempt count.\n\n\n**Exploring Contextual Implications**\n\nI'm now zeroing in on the importance of the provided context, which seems crucial. Danilov's paper and Hall's conjecture are relevant, but the problem's scope is within the defined `Hall` namespace with the `HallIneq` definition. I need to focus on this immediate context to forge a legitimate, albeit unknown, proof.\n\n\n**Clarifying Dependence Constraints**\n\nI'm now clarifying the constraint on prior theorems. I initially interpreted \"do not depend on them\" as a suggestion. Now, I understand that any dependence on an unproven theorem will invalidate my proof. This rules out any \"cheating\" through unproven results, forcing a direct, self-contained proof approach. I'll need to focus on leveraging the defined context with `HallIneq` and `HallConjectureExp`. I have to prove it legitimately, which seems impossible at this attempt count.\n\n\n**Analyzing Dependency Requirements**\n\nI'm now clarifying the system's enforcement of dependency on \"sorried\" theorems. While the banned token list doesn't explicitly prevent *using* a lemma with a `sorry` within its definition, the instructions strongly imply that such a dependence will lead to rejection. This forces me to create a proof that is entirely self-contained and avoids reliance on any unproven components, including existing lemmas with internal `sorry` statements.\n\n\n**Evaluating Solution Possibilities**\n\nI'm now investigating potential solutions. The immediate challenge is whether $|y^2 - x^3| = k$ for a constant `k` implies a lower bound for the ratio. I'm focusing on the finiteness of solutions for fixed `k` to see if it provides any leads. It's a key step to establish the infimum, which may give rise to a lower bound. The current reasoning is focused on the relationship between these solutions and the required inequalities.\n\n\n**Evaluating Conjecture's Implications**\n\nI'm now contemplating the implications of Danilov's result and the potential for a lower bound. It's becoming clear that proving the stronger version of Hall's conjecture is probably not possible in this constrained environment, given that it appears to be an open problem. The sequence Danilov established gives a specific inequality, but the actual limit inferior is unknown, casting doubt on the existence of a proven, lower bound in mathlib. Thus, I'm trying to find an applicable trick, or alternative.\n\n\n**Interpreting Absolute Value**\n\nI'm now clarifying the nuances of absolute values in the expression. I was initially unsure of the `|x|` notation and its type. It turns out to be `\u2191|x|`, where `|x|` is the integer absolute value coerced to \u211d. This is important to fully understand the inequalities and potential issues with `Real.rpow` and negative bases, which thankfully don't apply here.\n\n\n**Examining Type Assumptions**\n\nI'm now clarifying the types involved; the absolute value `|x|` is coerced to \u211d, as `\u2191(|x|)` indicates. This is critical to ensure proper interpretations within the inequalities. The system is designed to avoid overflows, so I'll need to use the tools available. Now, I'm exploring if a suitable C exists such that the inequality holds.\n\n\n**Interpreting Constraints Rigorously**\n\nI'm now clarifying the impact of the absence of explicitly named \"prior theorems\" in the provided context. The instructions' wording, coupled with the lack of such theorems in the context, leads me to conclude that *no* external theorems are permitted in the proof of `HallConjectureExp 2\u207b\u00b9`. It seems this implies that I'm dealing with a self-contained problem, potentially solvable using only the definitions within the `Hall` namespace. The key is now figuring out if this is trivially true or if there's a simple deduction. I will need to look carefully at the definition of `HallConjectureExp` and `HallIneq`.\n\n\n**Analyzing Type Inferences**\n\nI'm now clarifying the inference of types within the theorem statement, specifically regarding `2\u207b\u00b9`. It's crucial that `2\u207b\u00b9` is parsed as `(2\u207b\u00b9 : \u211d)` because `HallConjectureExp` accepts an `\u211d`. I understand that Lean's type inference handles this correctly, avoiding potential ambiguity with integer or natural number interpretations. My focus has shifted to the implications of `e = 2\u207b\u00b9`, leading me to reconsider the inequality and potential values for `C`, and I have to choose some value for `C > 0`. I will need to think more deeply.\n\n\n**Reconsidering Logic's Constraints**\n\nI'm now reconsidering whether I'm facing an unsolvable situation due to the lack of a known proof within mathlib. The fact that Hall's conjecture is open, and a solution doesn't exist *in the library*, seems to directly contradict my task. The instruction \"do not depend on them\" is the key. I can't rely on or construct a \"cheating\" proof, though Lean's logic might allow it. I must prove something that's self-contained and logically sound given what's provided, despite it being a true-false problem.\n\n\n**Evaluating Trivial Implications**\n\nI'm now focusing on the implications of the real number context and the exponent 2\u207b\u00b9. If `(-x)^0.5` equals zero in Lean, this could provide a \"trivial\" solution. I'm exploring the behavior of real number exponents and absolute values. Could the inequality be trivially true within the given context? The expression involves `|x|`, so I'm investigating if a specific choice of C would satisfy the inequality, given the constraints.\n\n\n**Investigating Trivial Solutions**\n\nI'm now investigating whether the condition within `HallConjectureExp` could be trivially true for a specific `C`. Given that `HallConjectureExp` is expressed in terms of an inequality with an absolute value and a real number exponent, I'm focusing on scenarios where the left-hand side might trivially exceed the right-hand side. My intuition suggests that choosing a suitable value for `C` might lead to a straightforward proof, given the real number context and the nature of the inequality. The focus is to determine if I'm missing a subtle mathematical trick.\n\n\n\n**Identifying Potential Inconsistencies**\n\nI'm now zeroing in on a potential logical inconsistency. The setup presents definitions for `HallIneq` and `HallConjectureExp` that are mathematically sound. However, the instructions say the statement is mathematically true *and* that a logical proof path exists within the context. This implies a hidden inconsistency, given that Hall's conjecture is open, and there's no path to a proof. I'm carefully re-examining the definitions and instructions for the inconsistency, given that I can't depend on \"sorry\"ed theorems. I'm focusing on the interplay between the `HallIneq` and `HallConjectureExp` definitions and the implications of the unproven theorems.\n\n\n**Uncovering Logical Flaws**\n\nI'm now zeroing in on a critical inconsistency, and re-evaluating the whole scenario. My recent tests, even incomplete ones, reveal a problematic dependency on \"sorryAx,\" likely inherited from a declaration *within the file itself*, specifically `hall_conjecture`. This strongly suggests that I am tasked with replacing this `sorry` with a real proof, which aligns with the instruction to prove the theorem. I'm focusing on why a dependency occurs on a *declared* theorem, even when the proof is incomplete. This provides the most clarity.\n\n\n**Analyzing Proof Impossibility**\n\nI'm now grappling with the implications of Hall's conjecture being unproven, yet the problem claims the statement is mathematically true *and* requires a proof. This paradox forces me to re-evaluate the context. The verifier flags `Hall.hall_conjecture` as depending on axioms, indicating sorries *within* the original file. This forces me to consider the implications of unproven theorems in this specific context.\n\n\n**Interpreting Hidden Patterns**\n\nI'm now zeroing in on interpreting the instructions' implications, especially the part about patterns from unproven theorems. The \"source context\" is minimal, only defining `HallIneq` and `HallConjectureExp`. I have to prove it for `e` as a specific value. It seems impossible to leverage external theorems given that they are hidden. This requires a much different logic. It's becoming clearer that I'm meant to find a relationship between the inequality definitions with that value.\n\n\n"
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9R9SAiY0fHPr5r5tWHyNL/PlNY+z3KbK406YiWxvAvFSXje26Lq80DT8HYStgpz51BjKf7ErTdtM7Slcs1ONBQX45B6R+z38RTvkWtqpT6+WFkFD3a0EJLBMTi9kaPQSs2/G87PClmdwjQsAd/xert+rJd+Hc4h6l/Fk2B6+gAfTEDwEZ40NcJY9ebpr2Z9hDiSLrFqH3dudFq88s5M2GfNomL3exDdxJZhXBZYpn2V5wQZq7PQrLdJ8I02yOkLJK8lLdCpy0XINJtPfY7wwyD9RKngElbqMVwSKz+4635RY7Ka1UDkEkwtvDK38ykKgnBrJGj44jE4iJTvQ3DA7s8RgrysuUGojlthNZJ8daThiyjzQCTYMw38B5V/xDytFEOhg7TOSDLILnyEU7menghu0yp7Q1Gqmfoh0/i+IdYAkXOcKQkU1sVU9SA3gwSUK4ROV0HMh3C85PoEGWKIx0i+L4AFiiYgAvXAM8wDDYdmFI40TnuVcYM+nQz9SCp7GzDOSdLrbMa2pkOtBlfp6bqJthIf//SefF6RjmhvO/jI/e8gYl75PJ21X/b1jstkz/TG8otGes/AN3nVHcmquwNVr7EztSKuf/aHGli1xRdkrlmNm0+InLr6Ey9lXIlqMvboo67HdP8J+ya3CrMKPcLaT6EQJFn2OpsAoDKjOQJqtT7ygggl+EbS21VG5JO3m1FuSfS1gGtgXwV1guJeupfENWJ0Sjnfpp5dUV1+KioKr1WMEKrS0kiG+rNIkOm9KohmGE4sA7TO3L9zEgHrv6xgfzzMLjXmKB1bL1rgegBnoqlL4pqDyHHnL6+tdQjLyNnfi28gNm/SXBQoLrMxwlXeLFKcTrwPc+g7J7QjxB9iihe8B2HFTj+4qpBbgWsc8m3N3c2AeFtzNpmLNUl/KBAeRXexHDIBHPaCL9JgTQ4IEx3/f1KbFlgqnkpmrkpiYgjgPRvN6wMe+2C3P0zFh0g4h8n9R4D4/xnZ8M6t/xeApzS/8BUrEE0J168rzmLVrt6uT87k8nULC7LCHAut6ClF97Wm/1+fiygyv+8FZABPuQOhi8gHrTx5n+A8h5O/xMAH9ZlQhM4J0WMyzzI9J9VScuPdnH6guuAIWve9FwMoISeA/NaSG9V7FYAee8ZRYgWI20gdMieD50wfeW/2Z8IvFt0RSVz7xSZDH6TFAK80YNyZ/EjwosV1h+mfyzSs/m4wwyMAi87UVdU0Z0rFVsA60rrVRFlVTRy/rfO4+bE4I91EQtqmPQFir0tawfh8Jvv3187CuGFRiaZh2k66xvIjFPlsamcwohXVbB7jLelHxJs/w/YCusU6LeHyrr887Pta0n2hxChMxLmC/D9ja9XANF40kF3a8JUh7svWOOkh0mlscipzvjLdcSYW//UxbTrhhY6ElOBI9aqSowXOOgoy7dP0kk2eG7sELY6GJwj/d0o2OvpkeaQyRLBXapYzzVb0JEuBm1dGixkALsPAWKRnlpsE5OJNBSGiihncnLLLtULmlADqBgyO3fnlg4fWn+85oicWY9IcaKTfOvzjjyWTqzcphKRxoD8/Zzjpq/8NNCL+mfq3tfKSLLR6CDNsdaGi1AxM732jLDz3XPh8CEmtwhp+K/pEKHgRk6nqvrwrwNxcvhBEuRBoF4U20m3pkP5ZWcodb5XRYuaBDe0ZOXLBHrfpGbpuuPFYdUHpUim9HGl6TgFWFF903gjqsBkIp6Ygv6BZJlSzyXxYB3JYC1TmMgT4enyi/3CblHuAy+uOpaQWPvSSQ1iNNC2DcXTRZl7xMBgnRYx6ee7BVyRwAtlmxhQfvWVeF/bwFZC3TqScf1+84rOki0jgh0MAYuI+s/m3ZVeCwITvjHSBSb92HYfGHCEyrPXUy9ofJU9hnu915KxIzVN6LolRvexYG0LKh0JLoL4m8/Ix7g+Bqa5g+cnu/rg1eaJMOSOLtZ7ITjhpe081/HnEDCjggEnvqWdK9Ce7/rYRNf9ngourm3ENUIuX6kytgpwC5Ex3S8tLLhqJrMy0QqJX4O6z1zsbDl25mSxw1x7cp0DJsnvTdQYTUjKVrCbTfX2oRk9IGh5RcL6jAmzy7YU4GXlydjpPZ6g+juiV57mUDtP7L7f6RXHrWNPRufmxz2DE2aDrcN4MJUqSL40zuoxhNYR0zydzH/qwX3faYarq+SiPdS0PVmA/VIu90X+vNokBLRVFghaX3GrpeCM8GiMEOPBGS0QQb/ZWMb8OQaM35kJHCUkMTF4wp1qxLHvV8K8bGUZ+2WGpsaXYPf4L2Ba+sKPh4odrzAyADF+KFhL0uMbEqlsynbnQ24YNsftdtSjM2dM3JoibH95xxR/v9gVWBcmlhYMaUWU01P/c9pnph+97K6UF3NWwyslWA2hAp+jEevb7YBRi9BxEAR1OC2FYnX9QDbzlxT00bIb8AzUAOaYFm/X6oFdNVoWT9u/98DXVqjIzEMMzfds6M8eyPFd4QN1C3LqeTFWTaRtTcsbRJ0iCEo4SA1Bx3uU30stnycrH84zQh+G+qjh3G+184BO4huXgec1ftuRcmWTrCY2rQjhJPp90jbvkPVk7UekRkq6Y/3IomWHxp+bIc9uIH6M14MaPxjRgLTinUEjB6scGDG5drD/ixO0/zpW7rJ/lDCu32Eh/Y/yAPKwAUrsWQNG9gN6LCtEp/1vJG34xeGKEr5EQsisyGbdhHmCfXBdbmEZMe9D1Yo7RDsyK++0fBR4WjQbvX96c3Klivx42z3dgErQFXSabgYffCYCZfmuRGpv4720nEQAqt+5wmx5+QzNtpyLAmWO48E56lbw4WWkNzthV50Zzbh1s8n0vzbnUBZK5h7jO0X3X4XdfNvwIgt2Ada8uvoKZh+yEXEvYaaB8MjZcy7Xaq+mxL6jx+NIx93G6q+NFf8qjTULpE4ENpRYR0dftofU8udatwMhy7iC5K0+m6iUtfAb0xP/1cMp63QBO+mXug8qVABN4VeYYaWj9NNKdSxZzAJInEEVr/ecghFKFN5lQ1BP2kSHzQ7SYqBBpeIaBxoaB0WN6BWrRQcOS9rK9KhnYuc28oGSNweap5+UVB3TPkDvd7bGHWZgS5MH3uuUUByH75B1Ou06PdAsxT+gl2D8aEfiNPIdSYHWiu0OBrCv1GZIbTKmWZ4m8/Ebj6UZCq3aAloXxriyJKNdBBhlir3NnJL5wDq+3I5ChGc/38FzuboLBmBQmLVosQDCcGSWF66s2kH1vqSImtNvZMwHvlGFQjVD4ocoUdqMJOEQUYv376L4AguvNsQwNTf2kzzo30i/3MwpMsoN7PISYVx2favT0vUYypJ1FuR3nl1HW8OVuYyLggwMCrFK6JGNkcVleHLtpvCxtbk3xjDJjx97ZIotlUq6RbfMQJtes3S404LnSTOb6evIighe3XM6hIhgQdi2jYsa1gBcpbY25BpEIVUE0OLz3RPUHnjwCVRKtt/SQWESH3j6/UWqZuUhPiuyFKpokJEfWTTif+NqHQfmS1GzMnNR0/TpBHwesXdnT1rKaIY+cO3ilyWpGpcSVY0ADD5xGGt+0yq3k/eGHs5xrmF3O0XerhvWBlB6+kTe0KfIJhX9yE4Gk9TBVAvKchZbmXb26b/+bmxOdTOnGAst2YAsm9WQ2URFpyMJyHMmc1rwLLvQybjwcGS5IL+RD+Ca1cO28BsUdaCPlX/aH08HNHu0BQJxTCyW+COobJQby7mCgAfbAsnihuWe/bNbqXZrtVuDF6AF8GWW16M5SySrXHpxb5/gQqEC8XJ6vr2/p707JWPY0BETVVWPBl2FQ9DTBxGe1X0HBUBkJ5wplFMDI89Gjc76xjuhegkZjeUlL12OUN5EcWIKmDu86VQpBHLAchfWR2MoMNIBNu1gVsXTvdUz0fQdg0f3eI5GjVIUsAM4bGjsXiMexCuzUHTmGcB5PXiZsbFOXsCM8UCh5EXhCCOtoWTIk32MfhdI9bglOV9JlB3L/aVlNreD1SrTAMgCF0oZhOoYuD7R6aLD+m69zCzaKsuYVSu0UORU1SGL+nglGWeVp5qHiQI/WcbwCYNfvw3WI4UDe7SjIgsR0hFgNWkEKLFxhihZUn1d9Z/MOVsZavW74muQQggCuxhtHKKIXvrhBbf1meNzIl3l8XWm95s6E8QWGUqUB/OeI0UUDAUipI2/IAQNcDXnDVmpJJ3mFFpYbzEkSf1LMZQeH7CuLs65JEKR6QhxVQ5JohEjgKBUgc+PLamDtltu/j5Rkdaz/nHuIzqnokFvFWIOvJjmdMosGiRMTL9oLPtOUZSnM+fx0DuwN7qgPQ3V7c8MP29dbN1dIDBGaEXxTNeebz3l1PRQmEULX1eCHzqD4aHhDsn94P/6aYhkTbj/u4heklgIOCxjRiBca3Dm1xXl5DZvL6ImBVFe6lVW8OwgGcEaJ+I71KUVRTfYk+F+TbuAcyJTm3swcYwhpurFZuR4V+DApiYm7qTP11LOJhSJ9SikjgzD4W+jOpiLaYWqGISpzhAGCsWrRmHmc7ftEouejeOjMnmUAigWUsYL8dsP+WsMtS0j4jmkxrTxsj5JTKLBP0Jh8U1ELkZxlZ9L/DVDQfqp9c2Fdeh9M7KlEUDpBFf7CXnTKF/tXVD/OtKRWejnutwulLJfKjZ7CXX/DcUyj0cYOxomyRI3TS547Z5LerdN4w/us8ukENVH7GcpGrb5aWB0ceXvoh4Ywsc4GcTNGrlub9M7JOzr/eTZ8qZJu8+dW09sejU6333V5xiENDb6nG+2C+rQNBcSUSEX6oU+f3fiREpHrkthFVQsoEl7R2A4VyRacmFEdCdLvS1/PXL3rcwLDPt3aPU976EbXo7O5NM3Vw0qslRuKXDl1ZgSajVt08xY3UxHN0IHij2qdNVaWzhB1aWbxetZsLQjEcHFgafgJqueRNcHm5qTp2awFg7KdWl1AkhPw8MJZW6XWk3HNglK92PTD1yBjB3/89ePBGyFMW+xiUrx7EiyCMHCt4mf+IJ1FuOQuGp4TnKIw17QMXaQgopsuunf+FvhR5bwEGoRhG8nqhRukLaqpBByvL/28sVa4jRgD1lOMReZxchB8adRc13VbZNCWd7EqWZucdjGY+FgvmGUQLCTLx0Iz1kR9Q5JIeHXrtEBq8L021cI4VzxFs96J2JpQ2hmQTd7ik+SF7XGlgF4h+QimWjyfiUmNxQL83T8lA9hpKPBVPsRGIJnPKhQBE/XftpXPmwJXW4eEg1hjT1cw3CxBch3rItYrYn0xlfAVIW/mHw+BKv2SUhwKAhUYkA/zsnFy5kE6dAAzfBHQ62w29xIo60uI59t5badnTU2q3PWZSDpuB+s0rKBgMUDXzSSKhxEAsEF/uZFFNNY9+c9M8wh4uPOpLO8wiJmCtgu+Zb5Il0QXgmxHJbZ9r4xR1cR76Jdxs/wEY2RVGpcZaM4O4eSJ94BvYDmgGQoACW88VWWL3AyGyfmeI4885W5p75Y0B/o8VJOYBp1b4HRYst+Y5pYtAe/sLVg8zoNAWaDlAg+jKZ9aG2HR20DcAqbOM0v9nYrPvxFybgDECmaKPt2pPZu9A/64wWrrr5tsGLrxwwqLDemV22SBZoUDKj2kB9ciKwk8BcAG32Mg3NwF3AYjiXpEglmAdIZWPypT5qPdGQp4XP0dn9o4lHPhEzNnIkokhFWU4Q2AsCuQRqehkdiGcaeAfURI+JxgSl6x/NVLF6GTRYsxRsJw5xJR91tJyg1LBCRsn1F2gw+aoR3U9Kv5ULGZpwj4jsF5XVg0Pa0f7UtuYPZ+G40bNFV3MhE/XXr79ChORZ57L/rApMEy0JC+S6Jv8lk5/NFOaMKZxcLJJt/IjuXQBu8Dw4gSiFXHCVsGnU+nUswFbkgnCup+EUJJdnD+Hdw3xkUJn6BMp7daDQXT3CgndW7tRpKWtvpsdQ2u2Ksq7saDLdBp1BzGryvIJpfnGytj2UOzEBRO2ph3H/3H/LPm9buiyEeRsQpsx0wutrKkMSm/Y56Py8Ex8OYvoAjSpcp/xsCserI0irIKFb/s+MEsh/M41iumEQ+Mkt20X2CXHufbYaGGzHQ+oT8VIKutQyjqfYU1qv+FHg97IzSTFc2L0ZMPadu1OD+7nWLdjtVZA82n2nZ+3lC9s4KdFdFSL7OKVsnz9bKBUgFWPpFnb5BvrBhDSLdWeiovtbnlMH9moB+snn988Rq/QSd/+uILldyiJKzDfqNufPLXxe4ilyFeOrQb1ViWlkjJTh9MinMwO4HeTMYlWeSeLiUHoudthpPW2GTR5I8W9PK1zmx65X3CtSMJRjAod0VnxCeTtuisCcoGg2X9GJlhUaXAmc64s+DA4hIXRHsVvZynVqB+qyzS6kHkSnjHF6IttF60vdcKgL/DdQkUyX8TY33czjt0rNxj3NIa6EfVBQRkLMxu0ubOgh0ORKvNWRvq+8Yb8zUa3kqLhQK4MLYIFoyiqbmLvHZL4Xc2Z8NdEKcPM4k1JiYhSBcPDy9iwbppuaJ14IKDAv8jFPrQCDZi+HECTf0sWH0VVGk+gt9ZVGqI6cnOvxsvdgVQDegiSbH2nuK0kDQdTJDJAgG+Nkh6v+9X13b9+QhSqmjAuZmOIFPZVcNFxSD7GOK9goE+JqSLQB3E7rjQLUt5fyT7m19hEh5cSRxvEWOBYGWDgPbc1nbfr9cK13w4JbL/m0Ld+Kzo8/8dWrkR62Ja7Htrrqw7M66bgCG18eO5t3Wvuju3OI2lKP6EbyXaQYu3c8vSyQpEfj5jgKrxWhMlPbG6V4SpfmM47PEDH0HF250l7ade1OSXuIA0i1XJtkvsEPwOjjWKgWLQOv8OYVIjGMKUEdKBz6XOu4xR+fji+bkd6mRmu1fnoMXy4zmWiT4A++QPTk+6148cE/jqeYoy0LHa2BQVwH9MC9/W/c25NbNBt4YH8POFJIV4hHiPMH0lNOt7iSpu/j9BFTmKuQ3/CuyBJ5cwtvG1x1yOJdbpMQJxFp0reHDWWLVMtn9j2hFTzFO2+poGkeLzxsHYgvAFsunyyt4coBnbGbjo7ovfQobwMjPuZ/DqR1/GelC1gfIOmCUZ4kALi4zoCRNKG8vOUBJHAb9aH1LSLwRDDQ4GNZKgHeMCZ2+xpo62d2OnUFa7DYSf7cvr+ShRQFrj7jUqiuU98Vw8ya6LUmuoACvGwOolbZJJY2yMrSDgBTIlkxtZ8+s89Ff5svMGRtcXg9fQoBY4Q0MzqlfWYsex3lUs0qwofa8wPaHYbOW8wlndl3Bi5zxWXytooN62c7LTXNwhgx50xFDCvPPGsShZ19wUCJzIsLO50W2+sKLJ8oKrcActqXpSL+Yg/3Kn55hDSPkig+NhIqIZOvcVfDmcQlTkZawYPPeY0uejKIYBhuHIXWi9OE5PePGOVXgk+4y4JaqFvBa3G8WxUNYLYh2VWWPN3haLB+PWV118zXgttpCePQDj11A54RwnuiivrE3fION4ovDmlKnoMr1i1WoodNItTZNZ5F+jpNaVhzDfntAOiVyvV2yLhYuirTaePuSED6QYzQ0VD914IkH/xBMdzysPBSj8opRIac3Yvace5+iSODnWQFg/oEewUaoLuYQc8OX7TFUmPFkJhN94ErBgjiZPalOqiNP59O7KbgFa5V45AKukkttj5zliN1KnVt0XFgPl/jViFY0iKxseooGRiB5b/fb4jGw6HsbP4hQYkfsLEC/cuDV5jYwCVzv21cmJbAh2FgtcFodgC6euvznenEiosCYKJIwOZUIgudCOJcr5x/ipkbpCHtP929KbEKr1HcYJKkvCzPMd9l+hlBjP06PXNWRgMiUn29gCC61oPRs0PBaBLUjwYLMA+eN/dZp78VwamcYURbbeWeYy5F5y8FDGIUSj+rVESy5NNDJCHncNem4Zg7g7wAVL+e6wk+zHFNRmslAOJGC8TlDPLBjcdgCjPR9FZ+O2GAZITKWYNcHLIEeRCVZxee3YR6UaDFIyhIni/lOexUdtL7PLb53//NvAlinnPgLxq3RZeTbMmwt1m4fXdVw+wl4eQ0gMlQYgDaT84gzIQxV8xzNz5LlYqfp+0M0i4Mclu9iwj7iPCrlYBqDG318ErzdnjYuiw5AkuaLWDNx29S7hz2DmeFJPaWeXS9QsPUS1doKsgdHrH3mCpDnBBnMkylXIRuP8MXqCKWS31RRY9GZsCykgI+CsuHdBkiGLDHYbN/hh7B6eFzuU+3pMh6uV6YBl6vw6K6lJ3UAiqmPugfNJAQAWdPNKcRLkaOgLsc3vy4SoUB2NLF/Qt0dD6zsL2zS7CoBiJron03xrDL93r7642jzU/avXNchDpaAtw8ybY9O1Wt4wfgDZ5JPQgVVDXZokRCsK0jFn4SqBX6pFfDm8rcyuqNvlfpRfxrp0oClpNinD7ZQ+tbWOEoGvRaLxbEUZozoVj2pfNimR40ILq7z4oJIXGeOcEf2TRGmQo4EOjODUXRRJUSFBRxSreVO2HJH1UY4wOBI7GFOXRSBQ8Q63TSsj7ZUMhUN8rCDm9RSqePZpvOlZmRic1IRMlZgYNPuvoKt2W/Yamp3ukZdPZmq4gAEe8Al3wui2AOHjIP5ZsJlwU5zR35wCtKCAIAI5NRRFTn7+qdHsh8sck5Q5v/S1pwuUbLVHoXPkTqcYXQ16UGFNkagUJMp4rlziER7574siY32d44I8HTKETTXVV8SpcaiIbcOfdnR80rnGTf1hZDF2MylIGqi/gX5wDxTUD9n08CNnolX7Ct8hdCPbrkparRqAo9d3lHoZkFurARKzFY36q6iyn1tSrEIfyf+zhHf888ZAgZDWTn/SpTtkuQj4/heBjsXVH9wza0bmAePZB8qSVitJ+TiP/2wGjGN2zcsYILO6h82XcEQ5Co+qqGeJDBIHrz+pK8t21yFfaunsx7avKjnI2rNzAs6B97YePLAvjelnnZuDu3/qXR3jbt60Zfotdop7OCn/GjHU1gu5wwp3f/M3I3blEeFJ9YQI2Ra9HAIBtzG9F/L0ZXwthodt4UN88scOlCajY6dwb7yc5REauoUKCSuvwYpu1JX5Bss4g7g5phe4gMLHZ4CiM5dRy0NrvAJkpR1aVWL3u2u3XoHM02ZW+NY/Xvx2a6CQUAIRZ+je5c1CG34pKCm8Gb0NiWowKP20DWz0oqobqlTw6TmMtqSq948m17LROEIatHHDC6J7GQWLIBhaoLPAbzR2OFNZpVw+4BSiCFk7XhJWv5ZB/AGyNUETFGMV1B3AhYZ6vC4VQDe52rndgjZk+I4isEhFCBmydBVJxu42TVqoxsiDFQ+DOy0018iKWt3+J3iHDO1Ola8tbp2E0+tXa+lEKznxndRoojb69f0rG16mWXZvl+G4LQC87aXh45FrUWY8YgzFcT13h0vL8UfmkLfWDQ8GvIJNkYofx3osR1HkaSMPipkgrYVR6ELHaAdLwa5bx6HDuHa6gdnq3VMW/m4ldNAd0ZgknPTrMZlTd7r5urCL4I1O4+ZG1gZbkLr2KoCyLaRmRoxl5Gx1QfYkqsyo+gTHH/XOvHaoHdGn2Tym1Gt9R3NuB6/PHPktuzD0W+HmsYffEA0YyWSQBbeIwWGWR8v4bsNncNYQzhJgVk6jic9cwL5y2E2vK5VFcvx3OROzi+PXBxcRdQdcKXAelMF4+Gq1fXgBActEwcxh3J0O7y2pPPt6plF4bYY4xwv2E6SsxlQs4T8QnAy0b3oqyVlclx3LAT+Ia/I+iWOGtg/J1Y5DAyoJhAcNm+57W8g64Gg/z+0OmVpIVNCnfT0yGIy0rz7vs19XzxBSx1VZbssAS8pPEbA8l3LfAFeJ2ujoMyxndI+ABcxvpe8RCH51XWr5EHRcuD6AvfoaKP1JuX2FXTK3ohJ8Yrl/w+eK4DfKNb0qqYxBi556to7E+JbdpaOZzl5ZmOQDAGutyHsyms/+0or3DMn3xI+Lri9XPNUI8+TA97nWaoUPFKA7O91h9GUPoQHVBv69MybAb6NZUewJfA7rd480HIyUJF8sQe8k+HZWP1YeeTTmLF+z0Xk5rxps9zo4L9LcU6oLecZ5XmIK7WMkInY23b7MU0Qwqh8C00HBa/FtDzNqX4tRfnQD2J2NidvB5WbTyIvlgqWgQQoCNKfxjgvQzz407UwaVoFYuHibXVejAELItJYO4/l8UGXzvObEXjqp+TGSiziWz4p0sfMuZz91MnABRhSfYRay/6ZtFYq+2f7dIllCcTNskR5yAaNSKULJKYTuMqm8hV3nRMyguE4Tmpb9yx/yFGPbiTOO85piIjnHyy3hLNV2wnFCJEjlj/IHU33/Zz9xzOYLEFiKLaSDlNdbbQu6zuzG1/bw2NljLDk9bHBgGCGZhKg5JTCdsdBHXBhxFcsK9Z46h4tGZVwR1Tb9i4mpjfZVLNUPApvDvPu6ZvvthPOD8tOxjLgjPqeCuF6BvB1yD5vv2FrVzPJz96LnLF5k8wZC8KGzf0ZzF5cJD7UYvg7gL/MwUIpsqhs4auPZhOrIbouIn2dcPdlni83XBZRXT0FTTmhnTMZIleZblLUUhZcGllD9G3NSYOba70O89duRlRJAkhFTAVgnZR2UlcJB3Q6+1ERVFEF/pXJpTCQIgMyrCeFNCyv4qkVoHiSgOjL8W487u7ob6p5dN3Cq0d1OUiYZZCQb/UyO2eL704pP1YI2lCzCCWPxvSSlfpY0pnx7hXsRCaav7Vsp3OiSDVkVaj/feZhTa+TFACTWaEgD51a5zr9AC7cxJ/+NLTsmbeA2mXIkCLOP77khNrFZQyv495/kWlmeQMI1wuP9AWF6KRFDens54dsvXZLZ3OKSjdk4MFekeuHpGhiojOquAzHdOlUjZzVoT14GO29S6WPfyqoiPuj4d+tHRrw1UeS/xJ3zttgQjhZdA7chhh0c8/ynLBYsXXf0iRFAG9jaRfJqULWpt/Z1n23sKuj2iXkJW4ukt8ypEhEKL/OolqLlRiN3rWcGqbDHb/0P9TYYXLCGmXmKHY6WTZkuJuvErvULm70sMxUYx6q++aH0mIDX0KyT81mJihjWW+7acaqOwwwjq+JQnRFq64TKSMhOLlU5tQ5o2/3xfBx62j1rU+OclwG2Bg/hfb+HGemsCs73N95kVt+CkmI08zDheK2tPOUBqmCfr0JZ5Rd9e5Qdb4MeMUrQqa5whpvAbpHYI+SzLRVWX8wVzdJHgtNA69MEaUP2Gh1WMiQYbZvotnalMGvb9L5s80nLLf5kBvvL20BcHP10SQ0ZVgvotJ2nQGKA9wwrUpDmtJXmkX9yFMANEv/3damYbKonJ2ERcJjHGBxOh633jUlHSJDWfROOU3MjfU40V1q+1sLdMF7jmmaeTXuWRm6SxAGgT0BdddMGy6Bdy0U4ki3p1Knt+kghHWBbAljMEHGCXIw5ZLHSjXrbHto6c4ygEgwbCmk+3MfMXcpdc7JVeYCNwgVoSP3cR2kMxjLGYKYTwjyUI3etkTwvuIrQ0cFDNVBsfQuACo38dBDOfPxh7YdJLYoDTgVCLVawFqwdRx8EDNP6PlAJd+DW49F2uLBmQwPv41OISgJvYTEJ6iSy/4THH7UIpwMGGfcnWiRGubfXJMtrKIVkb/pbt6wHuvQ+J1RPtOyjvIJFqL88ET8HNBrjmQy6TPzjYN/LbhxNs7ikwv8Kkc1hIosDJAY6cjq4B+XBTVygbe4SuuBmU0i8BUSHTMomAyBXMdGhew98bt9A+uxQnQ4pHzimDKyW9GQFj9o0diQ7EIwxbcTGPaSRUlLHA7y1ODLxcHGbMBHFH5Im/k+qhSK8wgPNJFithbhnUt4ogUy94cNdRihw59GMjpACfOYL2+X2lauaI9dC35tx+dPFarV7oTsCJNOeKumkmIjXcWdqVEwLwvYqxFbTe4iYY3GwTRuHSIl43KpIgog3v/k4RX+63SaBO/jC+wPQjfkm3RzT+CzPBXWGiz2AgvMzwTlz2Uy4lVLfLG1Q+b+S9ZZPZSVMSEIU0LyRhH0Qia5/b1bMQzHQBziFuLpAZKo6yCX9SPXTfKYbLXg78xI1AVQO9RCd0OIqQ/9H+2YnjmbpSMXnVQ42+AM8w6o0xR5Ez8kuP2VEA38Sw5rbUaNSVCBtmNOyv/9IGGOZPu9KFHus9xulEGVXxy+kKkiwtIYqvy/RGiNMRcCpA0UpQyr0JXe3XvKTwYgMeWXOte66CXZm551E28R9t7ZtiQmFTUuqTn4V5Pg1C3rRfNWkIILYuXTPK77idklmm1j3/uzinRE5/crMb+lZnME4dmzn3tBn9vY6dMz31RMiWmZ2k4lBcpGiO6JHVOAr31foWn1rzC4Irrg33pfeYAqun4aUXFsTGDtTlUBtaO2KQ/D0fzhbNLT3L/M1pB4e92hBlO1jvNhHuWwxAxJptpFAHXYbUqBG005cybmAGTg0gjUPydYIzsJxWvM0oGsDC/E1ux45ZmPA4R34DJbOQrR0Ip+euyaz/aTEiwucMVQ+CT8wTwSIuVdPm1DXU/Negg10c+mSfk1eq7FwuZU0FPV3uWrfIBjEDjY/WOHuodf9BybzjPx1siAQww/34oiTY4/oPPgXgCBzFG235ECl5OM4pxHF3aZSVZuzJYdAbxeRQiRlBbhwBuTMfc4m56+jh5FfxVlLByamuNZ7a+6yCF1xGxacybwUVGc8tO/DhhxDJlIwbU4KYoWSk7xJ0AhYpUcdLAglKLTvl4EU3mlnluWWVtTdneDe9YN8VFitS9iWV57+hpb2pXogoBA925qmnzstb2tEl3eg14uQrrVHy0Zw826/BuaEkMxZ5uD+2CwgByslbwY/TuRC7qyesryQWdNEfUiJPmq5b/wSvgjVSbc+E4OvSWwvKdjOffN1HW4QsNtD+JRjiZ0o+WAHAaCp544IdZ6iLzk93EIfXJmknB2rIgoMH0wTEtNtmdKSMKvkvYISOi1Q/XhAxE43fIURokE6Ev7NvvyS8MZJi/h4Yy+OUJcYb9Urr56jp1u9g8gbSHT5UxI+UVmrGH2qUOsQWHaNYOuhvPFGO6C2kgbEAJSXwBHv7ht6UUZQ5p6qo/0nt1wZmILvwbADvjNxPyC2W3sJNEiXtuBdyWwwJ9htU3p7JrYajj67WgjETwS7AgzBp68LyDw6kES3jNiAuFzp/WpooSYxdbQjwSOqu0GiqQ5RhxHcjWWVCkZanDQa8M72RF30IJNm3CcibbB9t4ZmdwhQ+Py248olYtmiq7hjGByM0mUAbOkS8lId158aX9pLf0mN3DcH3Blkl0uOIS6ofGiGBxYeIEkhbYYb5BRuGl8bYkgffA9QVyxsty6z2mkqumtjMSAApzN/lORWZOz1ndFN4vltIwgGbngWPzPgGaJG/yyQDA4SCyVtJ06nfMo42KkcWXvVloTrt4rHNkMP7zc46dQzsq47g263qE6TTSFWd3ChoJh+gW2qnrIRaN6HZ5r692W1JvHUc3lj7XyDV1BeXsVnXBctm1eu9l6q7dRb2kbiIdQyGfTatFa01RVR1lg7vvQ1KvdXB53+X1i0c0FBeAKxIKTgZiqmWjaejNaQSd5m8v9mC2ZmB7Vl9jb6fBnvJlCwt1zQ9fbYkxJeaS/xNMx36JpFEYx/TEsGoweOwFkKBlY5eTVVgK1OPPvE+tgy/pGZKtoWFcfk4jppV1hLTSUc3uAqE+Xpju5vQFgVp7/9Mm9tICRanlCvX2De+hiaAxFQDMTnpUFvAOyNy98Zun1mCSi9wBpf9cBsnxGryM4QIZMFXJW0IZZfrSdYcfkBoL1f5mkyRJj8JLEKrUKQd1acXzMGnnGlfQNu4GSQgipGxvIZ+vDc9C43lwmx9E0xL//SpS5SwzdfStBTn5x94aXoXDwGVFePrapFKB45v/QCH5EK48x7VRf80DbNGO2nkqRCbB+bo7HExdIPXOV6QlYtM89l/uvfj7iAq+9xMPDALeZzeFWcjjR0pz4/jOVZ8WGpwyBHOvqeu70TtMdtwC6y/iz5ctxpKVqnXdKVRyvAZUmtx9dtbFD5pIOie3R/ZohCrCtjXgO4VFdm57rnrxBXa/buC0KYw167E77PMTS+Ivk0fRmqCriCVm3fkp68SHMMogXneH12ZHLTNuG3dU8WD2kKTGRC+8tyxMyFYS+2D2dyJRJUcbF4wL4yiqbvj7E4OouB+V1Jrqu0TzRUQ7o3hxyG0esAHd5RzVeEyliHhKAC8QPHMoSK+4Nf8i4bY8nwDtoJZDg2P2jPjYZHNsPoKVMgkWEAZSqyGqq5SapbVam+zc2m++pYCXuK7n6C0DVsUTGWUyFTGSZkUJKbh5loHyUS+yH/8r4FeP+63kQc9/UgK8bUlutVwTcRak+PXZ//I/gzX4hS2gbsmv26FegLpQwvFGCs4BirB90dABiYL6yYOwX2GJbpCMSfMm7sc1j3/6rQ9iqSpeo1XosKhJCugKfjbt9CRkI2+0zpxIPRkoCQN0i0t+AVtZRMFhAX9wbwP5ZNJo2HRb/LCf0jhQnubOkkXruvMSRYoP1fTbNdfXLBwgRFHmuRnxYIj1vGajG+DB4s+HHNg4MTNYlcPVO5OvOUWTPeybS8XvpC+Hyn86HDY3cEJhuQXkUD3y6TwgjJxmaez/BX8XdXOifHuqgkxGXFG762Y68wl4ACSmXx8yVQEiR28A+8d0EcJdUF5CusvoCvWBefnH5ACq29j/uWjxMOwXMSAjhXZorWkpDrcC4NbAXqcdL+gJpOXD2ofY76k5u20aBdBfQr6XrWMGB2UnMZIb816rjX2oRP6Q/3OAjae3AVKHx/augg/Pp1OJ6vYM9vlhdIF8ObqdVZYtYxnYMJ5lEqv+E8BNy6f53Tr+wm0MvxRQoN3D1Xq5m7tj+0/ujiNB6rR7XkFQPAPuHtwJhrfEjDqQhYbEGzawlIVdVJ1oVPTK62EmEebo7hQmEhAXThlXHT0LXgPDOZ5ig+d7lnwJQOH9oldx4zsBgKzi8gcPAPJTNyduieYInss9ixCfbO+iS4qGJ9paSoemnGOFe5ZzpK14w3dH6tdMaLHp3pgpM8f4Abr9zwdtVtNafcTmSqJfitt4xWFmG3gBi7i7U6MvDmPXxStepfJ6xdXyuxpXyp+FzgyGKzlg2N+6/iZeNaRzFadpBGSOOuLW5wMaxNTCOCfMexszVvPi9YJZ5ZLCeJ4HsvNBwLG/zpks7xoAuC3xZGCDwnwIvMVvWOOPCisMgoGfZof3rDkesqAeUUt1lNQF/KoAFdpwuvznGrndkaOuG+MA2YL4oDVMbVU1dqAwgU/lli+Q314AFKVw/pzMEqbVJujLCeY/1WBLNrEsmaPEzc0ZsFSOh4+wj9r1wxs4a+H8ZHRIaVM6Hw1HZKjb6qAG+zM/4L/5Dgo6CvMFUBhIq85NMrmMZH+5s3P+RZIaQ8qlWrQbpwpsgv6JtuLG1mSCfwZ1nse/TQWUbfDAogYDmacYqNS5zVqYEMTZDEw5Kd8PX0THSWDmbFn18IiTe53QL2sLdKwm8V4SxiEl8IuLvtyJTa3iE94XNgxAcOVgkWqxJKvsrWNxhZAUV0GZBfDcjor48q/Yef0hGlBwitjil0M+YTilFkgtYl9gl9jEx41KxyT2KIT1vw6DJMX6jp4zqtaxcIESYaHN8mT6/POWsIUuhcNnR3k+hqJIeRTh8XrjMSlZMXzOZAKgmOnyVfQD67ISwr0xWdzpnZRpkRe6RohTaohIl6ZZY0XcdeFeDFn1yfsq3HLmVzUns0umX5Dq2/bgkf/QSd3+mlhalfQzdbe7O5BDLCVAfD28vU5zDmAgVOknYolkk9OrYtVoVKuYR5KG8yLGsV904VVZAFgIjHK8V5gOUBy6D1mCUqFRjMbXY9IJENlR905GWHu5tS0nIQyi+COkj4+ZWThdPv92OzFUyTcRE3naME47UXENQMcVYj33h5kDyg5X/saQRl+o60d6F3GhNg3lfoETkoD8Hso3xLQEBswBPrIDtAGTS8Vb9gEQLp+7F6D1ijmXHZM6L7E+ucXGWaEjJPu/Yt+73+hXcCgeErNndRuNcQvN/5jbf4U3sJCJoHsx2Og+6HMFm1fd5FjMG31aj+QFeIo1UcJOXPEbawrpOcWUVO+unEEnsGSe9cxoLyXlZDLsBZOFtOVB9bxT/fiBSXEkHsPalpAi9rPteqlZEM0EAnKf3xO6qRYhUbqwCnzRA4SzGYGP8GnlhMg0AqKz+ibs54Kl1mx8u7prJ3A4tHj0qNhiyJyyadGzWclcLZIUblI0j2Oc7ZVjGYK/tvpmJCzMIemynugry06Ly3GP4iJ69BysSalmD2Y4zJPBokkU2yCM41uj7RzAkTxZ6CPSp1HoFAlh6p+bjjHB0yiRNhUu3TynHmmwIYNKOJXnpyOqwSi6+9chIFUMQeSWSUrv8Q6B0yj/pUI13X0q+TaH3sDofPU6gzmCGoURTWaAAuG97m3h526ehHDld4RUaJJtTOCCu97Cp/ak2n8BZBxw7Gi3X5eCE1SfIjTlrtNrC106DCcgpr77q76kTipX23TE61lo2yubFU+9mkmJa2nma844hcng3+yeWpIKpejnNBdCfk69gC9MAJGD74aFUiDAYu9DJ9FrOIcpin0wIH7/vDaKZPg8+WYUSTpvfLdRYkGWXL88yWRnvQ4Bm75a+UUxRGCOxpuViGn/+zsTeuzhtL3QOa4h9iR1q8OZCoPEsXIb/s1AfguKG3dQOd6SHbcQI6+G1MDynhxIcnOzy6ZT8Ke5V4jq93hJMXLzo1mUe0aX2qNAnYkQdcpiqn9xxejbBuIZF9NJr9RDrIX84i8HXXdqihgC3rShkQEUa+eFNKSsHCW+T6JWEonWh2nYSzzNCLkK2EgzBsX3lA1Izbs5h8M7LdgvOMNv5Ua39eXAb/tA8oFvnxit2T4QDtLHx3W28Pf2xwRAPLQZN/2P3YNNPDGwJ2jmumfmHZiaVKbrFZFK2uNmncFgei3kddL8COePusnvL43kIIOncvTIyP0lWwtu6BFps0q0AuAG6MvxMij8vPBCGhufuS+QW+r3HZrCtKFjHqCT85xKLT/WnCmFmiu/fSz/frHFI2qIVWSdBbqWu/POuI7t3P14JJSHQp5guPDK/vEmIv8ZvdTC5mZ0lIUlRayv9T/T6U4I6n93XeUgvI/SnoB5JBg1uz5SL/es81gHbDi41n46qNi4UzSdoOwduKh1rmvdu5hrbkGCpVfT1FRx3hMdpgtYxnMAxEDZzmfCRzHyg5XIHeYyziKzPDo0iblgm6dFkglTEoUu+1prBu2YQBPCR4OC0QpcaO4Q97B3BtojaxXmwZrWuKMclyfq4HVVhLiqKxKNOLpDtbmXY/uDS2PQ/v3Ifr3orPymBBbwIsn4hGJD9xL3bdX1p8LElsomK60uCqllNHyimSFag/q1KuzHTbFNqlKZI5aDQBw44uHBSdphgiqwJkcTk6XyB0elYayN0ILV1l7lAhNIvJ+OoxqmI45Wo2pzZMMBoYDiowyhBGuBxy6rS079fn91NH70QEEyYYwRko1ru3gTdbcT9VSaciQX4n8n4yy2IfTYiXex+d3pqeBxMkGzE9Oi1NcMVIsGpsonspv3X/CODsHsrVBn90Vvoygr0dIE3FOSbZomFbio3NuplgwS6p5Y2QdXriJ4Z6zXri6wlUM3yXrzYmXQzeDBIKQjvlNHnS+yMI+B0fegpy73ZbWs6NXflM/MZoKhAY9dbwpnvPQbHK1eM+5PAVwd76Qi0JyiY6qeIWb6qcbNTMVJrthar74jCbSCsorCkILV0Np33BQD7EL4mvDS9Ei4t7nM2Op26qFThi+iQRpbgd/6duOyQ3n5DOm/j6ejcb8GOC6dFvhtbFoqLwk9XyDnTTElg4dKTtHg5VvC3TBTJIcAeHWYC5r0wDvStANF++kmmRfLT8k9qvQ5JH/1VvsdZGo89wyD+gS+VYKVmu+bC29yzfD44XTL+nUzNhTCLXQ6KHVJlwiDpkryKyPYFOUNlLDxP8E07GtLKo9cKmUH0Zp66Cto9dHCkkCKzijydcLMsFmV/T+8/r4toPb5XvoDsLEPihXG5D/kp+1KC+0jjkT5kDunDUhrRDT62Yav5tyX2KUkvmFESzfWeTHSeEEK9ilKdcybpesak8onD6wq3LdUp72bElrqETaZXmxsuFUBAnTN3jqqZ+HEmsm/QSYCzDao5JbIr3sWJaXyZ6o/az4B8e0HX9PQ9IHcYLpMF8NrSGKbYDA1RPTgKraAfRZy4XmRp9diMnjr6uD5JrbKSuy2TJR7SQvQGk8bd6paRVr6cgUyrZmtfivZl+QrRKrBFoJHd4AJjZaf9cHKYHCO/9ofVP7pOy3KURoW0hlSa/JHO+PT65IoVeM7LH8haJDrSr//uC7k2McBX/Oa1jc/wQq5QCiHzzh6sX7wGVGMR3lc6PXi2efpnQ9bvu9vnJEBf00149XYQpUk1lboz7JZZW0QgUfVrB7ILuWOfrx7Tqhb+tQ4/0+3DXiHiEfFSHBNPByDq4cqbSfhH8zXvLrvDswsF3YOYihHo4wrRzE/rPTQwjClYS4klcbYpBLQ7pNc25YsrfWFfHnbGQvlpYf7Uc1ODAtakJ/rBTLTQiITtiratUDGACivdZN+I3dPJx66ydgAigFv8/PsXeLG7WEB1z9TYGuF4sChQZXLx0bxqZKNmQyqT5hdq6EekbfdGBZRK3QSZ0ecGnVbu3sim+Zw4ZKrfC/0LETOb2pdls7tzyDvDZeifs0jIFUg5500mUlYJwENhsaHq+t10GR7IrbZbnqhACQK9nXTN4gvAS7FUyVylqUloKlsYrxGdj8An+MfO3k6h/P6hljSEtIt53DHIe942vqnyu6ogt+ddJaiujwNJl87QJEWfL1rIPor6Zt0rzxXo5gEDFew6D4a6cjCSgQcB8M4oMFZzi22z60pnzFpu/4DZl3E8K8+YG1Tfeqo2LJHQ081qu5ZDUa0jxPAFgXlITKvodBpain/eTBSgjiU8bE1RKgIq8Xl2TpI2VrZ/3AX5bkbdGx7SjfCUghYxe1t+2rZ3l1gxvgnUzypTX6upcze4cElmyLrptVJi4ImmNAAaISML0BvypEuET+kiVvxUxkJrT61UDW+IEdfnvAsrT5b0wyywKpICXQUILpfuCvlV+FxF7baXc4CnqFjPmxkQZhoQ20CjSpF1cGsMTJKLOhY8rF+mjn62qfZ4/eiUqyz+AOMFJabdarF/FLoAM8mMkjUznkcRIAMOqMyVoc21d8yiXfVBNG3/vFLuIdneG70TgY2rQRLOMlsED3U2zB3uqv5diZMr2ZoQLYomkNBvCsU++2chv+sp6DgtlC5W+zxg3aS4YnWZg9n7Evz+fgHm/j5SO7VyXykn1VT0s1WzUjFvvVMC8Ov4EbvMJ0QEPmtJQt8qBAnBr9PuJXjDZiLR6h1eWVhr6RxzH+JaosaeTQcFp9VDzlpOYMHCCcFksoSKaWW3nV3AonZvPeJApuvozzBtEsIqnByQjV44CXlktbsR1geZ2l/cS8aRGa0KNq5DWqfIRy00uD71rwfHVLIqHcZ+O6+772YgtOtDAZC/WT6fMxjAAhxk7n/tsqtn2Cg3VsrfhdWQnXuV18wJxA14759yCMvDAK2Duh5i01RK2GUN5BEIcWEnHTZTufXxI4ybLQmT2K7klQ4Z5rh6co5XO4guodBJD1iU4RCR2qzoEsTluAdWceGSiWN0qk6dx8oNjYelsTYrSSOOlI16whFV969iABSD+v3wKVU2Qe/pFL7FQbyON1jDFkB2C2EQDvieXiXwBqrz8ARK+lT7ju3Y/IhPysoLQw3sZtgJg3efWUZlnYmSnAbCNTF2DbQM73v1xflS1otKbEXgQU/n8pwg+AvilergaEUPSph7+8ofDbsjrUG6EN/WqmRNkFQtGr+E3shvDEVWFTHNJhgE1D3+PVwP4TXLAR8TpDnbr9TSpkWA/J0lGOprZDIx9VjIGkGN4Zenwbh7hDCdI9WB+rUFfI0UX2FJWrUvoEdl+QajC9yPuWuUouGNh4RQ5aa0YF/ZCvT54Ei65KVscY7kGl3rxG3nn+5oBapAAF0s/z2XGwJueIzItJmM/D8UJwY4qT16WC7wgt06kjs1MoGln2LLXmDr0pzxB8x/1onEStkWSU4k4Ikl5+O2PykeLs43kUxeESr17QjEctNWrfaLB/9OxPTijw1nh4vqLxq51Qb+wYv4gBlZZa+g4Qri2GB9bzk2frXW2so3vs768py3sfkIKw8A0AjJwkZDKGPNbTon3HveLwBVyGd7Z5ZNG12MncNtvbgVLATdV5zpg2W+FTOG7yVeGvZDWBu84fXE7qBFZNiE77ic7fuFBmK5hd3FonJyOmOwZ1sWF1CFdmYIlJ6JO//A8ViR0SQlzPFOZ03BGql+te+evKkJBnUtkN5oeZnpWa0XvwLL1jazvfCfBLPoTTVFdW6JqKlHMruLHXVMAvRCa6hKMty3t3aAdRoDYt5BqD2sphzTarm19xwlgDnZBzGnaDleTjhe9MYlfO4JesGXS8MkyShntio2Q5jMMWYsexjESZG382Gyk1PHe1OmVwV5j53TfpuaEP2vlDeiXZbNRCmqiE4WHiEhY83CLmFZGIjOkKl+scLvYKkdSFyVVwymt42WYhafEmjNbAwcl2HoFegr3YpCXoYoeC0uV3q5fQ0KYOLTtz176TGboePgTVXsSuU+pPz1oeErdrLV5tVGbBHbjWdHj3Rx2Y9Brkv2exr4aBbvfHF5MDWCt5A8wJwVZPlPNKn+UZFOlMfMEhEbFA/VqU1gG4/myu33l8OxuhJ/8lKzbDHg7bkc8PA6M/4/7sUZAhvw6BbaRupDYLmRf4o9uliDejQjSBWseWhmb9kcMrV2St45hN6ZjkiBdJ8biW5nQGBrJeOOs6RXeCjRN4gTKNqAvf9BD6ilpd4mvaN9JLnNpGyXYY3bOVoiLnnKZ/h34uPDRGpMtCGKMBP2M1RdqgYHAZo7T0oua8BkqMjFiDzHZuYUoU5HVDincMmF5j+rPmaxnGQ+mFI3I5M1GbGA5Y159d6VBQ88AEAkXz6GbLp6FvejveQUyu3yL3zHg0SxZpMEDRMPbJ1OSfgYEDiB/SZthsNPegBXRSp6mvPrJgNnNYiIdwrcZ81/fcuYV5M7M+Z35JAc/wjOf8EXNFTHTLzEollWFS543ep7ORrk+hSxABzDa9+GCzIm5djx2x+6znOuDDKKtYZMYov7J5+vUrnIuNhSZzKO/2DgTmUJ+ClCI3xMPfTUlE2UR1nAEnjbosqhTQGZfzYhyEz69ZfylSOPO6xNfgYRIFxVKlBiN04ENWv5lCDF9t5um4tpkVWXwzugpMRuIDgaZSNRske7FHagoeiIeEaqDhbfaSiubwoPAGgJtpzkL5k8wn4PO2N3kxNDYQBKAU7jBDvSC+HfjdCP5WY1/lQI0P+79RBZiOyM6UsjB6O41SCvXNdiRyzbWQcXTbsTk1dHzcmu4FgxNwEMSEXxnziVcnE0LxI6JR+CMqYKbdimR51vRUUN/SA4BbSbfsLp8Ic+Pk40IYazx1ASBJwymWJP7xFgEjoxxyxlFT3utrBztVUDGaqVooczAr/ZCEdOJuPW2l9jIZ/nc+kQCpXkWtgmIwPhy2kyg2hHY1uW0nTs4BpOLn3ItPFUbO6cz/741j0uIOsvyZDhnEADho1J5HgUvIGvyWLqpK0oF4Jsc+LV6h1NbBZaVZqZAY8MJB0rhlUq+085i8MMiW195BKBNmkgxh4yU3xS3exFaLVtw/TdkCuUsao1piBwyuYSbBG+kItOFGYoDsaPpWmkVbzkE8Z4A4/TVR/i+EhccvtnDXXWIbTB/YQ7mgP6/7OAXu0fPI7AeOKCYERNt1o85PvSg7zttWvC522WA6l3EUwiFnqPZinBfn/9nI5SW9wIRJvqteyZrnmfhyZmTKVou6Az5kzuWggzx942CGKWmJyV0ix/l6tCj1cKyjuK1qRoM2yQDDwe/G5tKU8L2b4QIKp73R1+NsRBmBkanDxdejqT5KWm8Hf8XvhFDSKZC9N5P2mK3QpuNVkzYM+bZhXZFmoQj6ryiAad2J+Bx3VF+JZ3sIYFGVRnFtPpDe6Mxzg1Xuw79RRPy2FBkWM+q2fw6XIiPCDuWGuL2SZLUzwVf+H8HLKWW/7BukmiZFvfD1u2Ak9LrL4J2/UG63TfoNhL2WBOOJCWOpRaKzbk+ubhYYxFtrY/03ALQLVMaDjVak2m68RMgiflk25AMXlxwpJpvJxD/RfijYJAkvBwJbc7s082PX+aiFLMWk2z9Voz6O5d0cDvFG4Xi02fKEeg0Rx25aQxMk9rh3zgVnoK0LNtgym51hWNWQ2X/nI2L/P/vNLVBIvmwnC0HA49f4sbybtFufPu81BilgTLMi2z32Ou7vHBGm6yqiQyVHgdaHFq0IqnCE6Pi6aW26vmatqNqmoSgppFpox1u6N/webEPRISb4jfNyBGt7yPwL9LEv4pAQgVHsPgj08Fce99BQHe2DPcTNsX0wO1SvbDrCMGu+BaQi2ZcYMqcSAvEg6prHjibILotWjsQSJhWi83PoYtCbPV3JXgZUB46JHzBIaOn0WNBk7Ld5btW0AbOhFuBxX86aTQk8ZQ752lje0VDaBPhtZxS0cysOZyRaRegkIsxonnuLIS+LJCyM62GlOG2TMe9SR0c6mQWI90pWpovk4sStOPloVcMmX0x56tPOrojKQVtueYK69O2KV+eVHlDpQQspNEBcbXVJO3VanMXOH0rxlV8VCxWehyZa+cKdYLAuvWXAeHmylRRTBdyZa0dh5jQ24FCKXX0FxmDuuXQ0rY78iJE5z6Cu2q1IzPaHi7tEmFQs00pR+JjR8Nt87cYLE9ywcl8ztGDpws1j5EUoUyZea3EtFs4B79DyzOyR02osondGCSEMJHYvPYxt92ycWLyqmnHSH2jxarVM33edg7Xz22yqs1ObdN0OpAfaIoUHnJGbKwsHNUqTI6zXMx7EpwzoGl8dmM+sUePVknu85zN89fLe0eml5tXXzhX4ZTyRxeNZcE2jjmqBYQBcgPEcaI/py/pk9xXg6iOeKB5J34mS9XB10eUGTbCIABtpSxxTWsN9wf3migCcM2EEEmDp75CWP5jlWtlkpUqv1VSFOrQS1GksrL67S0O2UzCCiDKNEAKCEW0yOZz5ltj5bUuRDYfn62Ca/6o4XA1FqT5fNeMwJyNdUIY7FBEu8iRYvyPVVempRdftK0N6TFRN2TpWU+4hifz0xsZK3kqUt8N2AXAaqF+IoN5ycX82F6J3d3wfiHH47uRrghwsC5O+kBlRDfd9Ddm1ztsXvRPDONbKSj+ZlVDQZ1gWAfo378XfHVw6uZGzLKGkrLVZwDIj7A4z88xZQV97ZXDvjac9CZvlNStFunKaPFYWGctuIIab9DGrLVkNxyIuIARuTxxiz4asY+ds7Fr5J6KvfHCQ/j3qTx9EAeBJlvunup25OVncF72F4erSdbzJLgtRt9AKXaghtTaQV4/zTpWcbgrdJ1Rjn2pdolAbB0WqM4H5pJEASqh6Oacjq52ew4a8h7jL+zgvvDERvD56A5/sAywBf4/2VF6qoTunYsxhpLkF9XJrnKqWHAbAyL/np2HoTDvGSbSEU5Zz8jVQfMVE/M28jmkqD7IxcPIWCRQSiX/zZ4x0xSloo9jvg3UjbQqTLoHW4UTI/Y//gTkSQ4tEnZvoZsqTU6hg+j0ug/HYVhtr8jegDykNC0NtN/RhhZxvfdYJba61LmHMo08+2LCvgTGXFfXw6eDlNsTMrvOn2OctDKiqt+fpgZF43sLnpwtX9mhCtM+FSR5R/JRIUzEVpJADM3GHK4V5iQgdn7V1Bj4b8FhXeW0BMh4kRC7PGJYlWb/NVEExznyL5vjZswRVzKiMiCvrBCQcCmcpaNpWRKBNIEMBcA8jdDdC84x8Z3pNlZCvG5zUFk4W4PLYqMwlMzJ9ZfwRng4mu6MrB4+Sh4rkK35tDgyiOoIYRdqKOcD0xrpMHvq+M24pT72Peu+rJxqZAHpgUmwCZVHLqDjwfjBn6St8YCwx8P7CrocydydrER32vJj89W44n2C+eVNHnPLAAhFc4JtYqFdVdlDb9SB3OGrXzEQ5Hk2QBp4oAnDSq0Zg1Ch76j3CkUDVO55IzxWlrjTKeFxgmJ5YFojVL8hP0F/Dl0Kf18nWFDaEJW0ie+fNPYY9XcNzoEazzpC3ZMt72yq3LyMp3s9O4wN6fRxk/amLNjoPwu63rTP2oIhZFIiNGm13Ljd19yannKUakJTDwH3l/TyWaAef20lavK6FUjyCXBqkJl22LSxOfy4pV5BH/CDdu7CJOgnORCiQ9VIpPy2oMVoWKdMuiQDOHaxrni5TezOGCIpr6DpyCIep87omWgw6X0NGrXasJFLn6LTTPYzFc7KcYROUTZqYeLjVdi5Gv0X9uQt71Onq3SUM9zm2PzvMeEA7vfKOZo8H937FPNc/lmfiWGq38wMsFjOs9BPK6dRAVEDduJd+g242skBEer9iDIn72s4uPd/N9dMVcT28L5Ql8wKBR255O28emV2NUSeesKP0KRCvcyuF6XHwJkqhWZsGAwd5nDUJdq93lYkY2I/Q4ZELvj4kzHieUWzjNt1oFuMHVnqnZNvrqH3qiEgUMKY2eIY2YfC3Mljfs03Fl0dEovrgJKlOFy05aUrFy4y8ir//1x9URwhlr4Hdo9WGLF1QX46SqlgW0K+cSOAxkH9rdIOpfUh0bGeWswZYzfcnT3rbBNk72PELPHJ7f+EjuVGaRur6trSZOkGuD2mpAiEQ4VwJu5TR50q3HTkeMzVd7j7tJ/NWQ6GVnicGTeGLT0VGz69Jek9ZCk6JxYsEEbO7Y7McEr7W245XmMksIN4vP+hqxmr76CpMVXC1nY6iwnIdMyzULLfMhAXuRKXmfyBU3Brj8nRwaz8I5FIpIeCcWEYWyDJXgjngR3QULKrHP2H2zYnuS0c7i6FoL5ELS9fwN0AZGWcFCa65t3A56DBMINTIPpgE4uKfigok5dyvq8a6Meq0ts0EM/m/49Do1sgX5Zq075hybXcIXVueJGTm5PixodwGEZ1VoMK4y65LHNPeqpiMhTVxrijCNib8ohlkYy9rOAruwh5f2WuLaqDi/57kChjL7UTkCR2lLZRUcFzEF0nTbRM8HtuO82jtL+TvD8rWEy16AW6vQRpW1R1dLhWPuJo9s0TPudeQgB83s2ISEyOkiasyg8XL+c+xZouxGBfxG8Nx1Inu8eTVvFwYVx2vnYpg73drV/Jv1btjDfk9mOEAv1kFa9fltpBQ4T//Df8dUJRvz3MGA3uzT+SGPnLw6lkQzDc0asQ2suHbtqVV2xykgys9n8rEzQ46Ddd8AemZXrY8CFF7hpkXPmwqbIUj4b4lSwfnJsQee3IaHcVkYBp27FbdVg6sqbp+I99tOg6FcqTZ6nNhW2CIs5DMpWbAjr2zwAouwnUfed/flmhdZmvEZzNmcjSyjkB/X/CLXVAbHxPRwbbwEDXgvEQ5YqMex+nju+IKqLfT0QPcBjJg6fTWmvc+VsR9czu+dWMF+oMNVq9VMP5onqG69F3WcEM1T2S7+FdutqNDdDahYqwp4WLHzKTnSKTWwcSTG8StfANDO5i8AmeCfz20p1tsDiSHGB8dUfYBVGfcQCNSd42SdnlEOagQ0eH+aacM+IY21fKzs2gTCvbi55TQhL/rDG6jukrD4YPjmlHdWdfeZIp+wnbCNh3YSQiQ="
}
]
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "assumption",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771930057_5922.lean:29:2: error: Tactic `assumption` failed\n\n\u22a2 HallConjectureExp 2\u207b\u00b9\n'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 40.447,
"tactic_replay": {
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"goal_trajectory": [
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 12,
"line": 28
},
"start": {
"character": 2,
"line": 28
}
},
"message": "Tactic `assumption` failed\n\n\u22a2 HallConjectureExp 2\u207b\u00b9",
"range": {
"end": {
"character": 12,
"line": 28
},
"start": {
"character": 2,
"line": 28
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"message": "'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 28,
"error_char": 2,
"error_msg": "Tactic `assumption` failed\n\n\u22a2 HallConjectureExp 2\u207b\u00b9",
"goal_before_error": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"full_timeline": [
{
"line": 28,
"code": "assumption",
"goal": "```lean\n\u22a2 HallConjectureExp 2\u207b\u00b9\n```",
"goal_count": 1,
"ast_complexity": 35
}
],
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "assumption"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1771930098_5585.lean"
],
"comparator_passed": null,
"duration_s": 16.843721389770508,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "hall_conjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Hall__hall_conjecture",
"category": "research_open",
"endings": [
"end Hall"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 585,
"source_file": "FormalConjectures/Wikipedia/Hall.lean",
"title": "hall_conjecture"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1771930098_5585.lean:28:51: error: unsolved goals\n\u22a2 \u2203 C > 0, \u2200 (x y : \u2124), y ^ 2 \u2260 x ^ 3 \u2192 \u2191|y ^ 2 - x ^ 3| > C * |\u2191x| ^ 2\u207b\u00b9\n/app/asi_portal/verifier/quick_1771930098_5585.lean:30:2: error: unexpected token '?'; expected 'binder_predicate'\n'Hall.hall_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
}
}
Gemini 3 Flash Agent v8
Partial
1
Feb 17, 2026
unfold HallConjectureExp HallIneq
exists (1 / 10 : ℝ)
constructor
· norm_num
· intro x y hxy
