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1
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommMagma G inst_2 : Add H inst_3 : AddConstMapClass F G H a b f : F x : G ⊢ (f : (a : G) → H) (a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b
(f : (a : G) → H) (x + a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b
rw
[]
rw [add_comm, map_add_const]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommMagma G\ninst_2 : Add H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\n⊢ (f : (a : G) → H) (a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b\n```\n...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommMagma G inst_2 : Add H inst_3 : AddConstMapClass F G H a b f : F x : G rw : (f : (a : G) → H) (x + a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b ⊢ (f : (a : G) → H) (a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b
HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b
rw₁
[]
rw [add_comm, map_add_const]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommMagma G\ninst_2 : Add H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nrw : (f : (a : G) → H) (x + a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) b\n⊢ ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommMonoid G inst_2 : AddMonoid H inst_3 : AddConstMapClass F G H a b f : F n : ℕ x : G ⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
(f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
rw
[]
rw [add_comm, map_add_nsmul]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommMonoid G\ninst_2 : AddMonoid H\ninst_3 : AddConstMapClass F G H a b\nf : F\nn : ℕ\nx : G\n⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f : (a : G) → ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommMonoid G inst_2 : AddMonoid H inst_3 : AddConstMapClass F G H a b f : F n : ℕ x : G rw : (f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b) ⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f :...
HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
rw₁
[]
rw [add_comm, map_add_nsmul]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommMonoid G\ninst_2 : AddMonoid H\ninst_3 : AddConstMapClass F G H a b\nf : F\nn : ℕ\nx : G\nrw : (f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G)...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F x : G n : ℕ | HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
((f : (a : G) → H) (x - n • a) : H)
rw
[]
rw [← sub_add_cancel x (n • a), map_add_nsmul, add_sub_cancel_right]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\n| HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)\n```\n\nLibrary theo...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F x : G n : ℕ | HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
((f : (a : G) → H) (x - n • a) : H)
rw
[]
rw [← sub_add_cancel x (n • a), map_add_nsmul, add_sub_cancel_right]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\n| HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)\n```\n\nLibrary theo...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F x : G n : ℕ | HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
((f : (a : G) → H) (x - n • a) : H)
rw
[]
rw [← sub_add_cancel x (n • a), map_add_nsmul, add_sub_cancel_right]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\n| HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)\n```\n\nLibrary theo...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F x : G n : ℕ ⊢ (f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
(f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
conv_rhs
[]
conv_rhs => rw [← sub_add_cancel x (n • a), map_add_nsmul, add_sub_cancel_right]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\n⊢ (f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f : (a : G) → H) x :...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F x : G n : ℕ conv_rhs : (f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b) ⊢ (f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f :...
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x - n • a) : H) ((f : (a : G) → H) (x - n • a) : H)
conv_rhs₁
[]
conv_rhs => rw [← sub_add_cancel x (n • a), map_add_nsmul, add_sub_cancel_right]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\nconv_rhs : (f : (a : G) → H) (x - n • a) = HSub.hSub (α := H) ((f : (a : G)...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F x : G n : ℤ ⊢ (f : (a : G) → H) (x + (↑n : G)) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x + (↑n : G)) : H) ((f : (a : G) → H) (x + HSMul.hSMul (β := G) n 1) : H)
rw
[]
rw [← map_add_zsmul f x n, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nx : G\nn : ℤ\n⊢ (f : (a : G) → H) (x + (↑n : G)) = HAdd.hAdd (α := H) ((f : (a : G) → H) ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F x : G n : ℤ rw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x + (↑n : G)) : H) ((f : (a : G) → H) (x + HSMul.hSMul (β := G) n 1) : H) ⊢ (f : (a : G) → H) (x + (↑n : G...
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x + (↑n : G)) : H) ((f : (a : G) → H) (x + (↑n : G)) : H)
rw₁
[]
rw [← map_add_zsmul f x n, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nx : G\nn : ℤ\nrw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x + (↑n : G)) : H) ((f : (a ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F x : G n : ℤ ⊢ (f : (a : G) → H) (x - (↑n : G)) = HSub.hSub (α := H) ((f : (a : G) → H) x : H) (n • b)
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x - (↑n : G)) : H) ((f : (a : G) → H) (x - HSMul.hSMul (β := G) n 1) : H)
rw
[]
rw [← map_sub_zsmul, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nx : G\nn : ℤ\n⊢ (f : (a : G) → H) (x - (↑n : G)) = HSub.hSub (α := H) ((f : (a : G) → H) ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F x : G n : ℤ rw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x - (↑n : G)) : H) ((f : (a : G) → H) (x - HSMul.hSMul (β := G) n 1) : H) ⊢ (f : (a : G) → H) (x - (↑n : G...
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x - (↑n : G)) : H) ((f : (a : G) → H) (x - (↑n : G)) : H)
rw₁
[]
rw [← map_sub_zsmul, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nx : G\nn : ℤ\nrw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) (x - (↑n : G)) : H) ((f : (a ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F n : ℤ x : G ⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
(f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
rw
[]
rw [add_comm, map_add_zsmul]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nn : ℤ\nx : G\n⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f : (a : G) → H)...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H a b f : F n : ℤ x : G rw : (f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b) ⊢ (f : (a : G) → H) (n • a + x) = HAdd.hAdd (α := H) ((f : (...
HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
rw₁
[]
rw [add_comm, map_add_zsmul]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H a b\nf : F\nn : ℤ\nx : G\nrw : (f : (a : G) → H) (x + n • a) = HAdd.hAdd (α := H) ((f : (a : G) →...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddCommGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F n : ℤ x : G ⊢ (f : (a : G) → H) ((↑n : G) + x) = HAdd.hAdd (α := H) ((f : (a : G) → H) x : H) (n • b)
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) ((↑n : G) + x) : H) ((f : (a : G) → H) (HSMul.hSMul (β := G) n 1 + x) : H)
rw
[]
rw [← map_zsmul_add, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddCommGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nn : ℤ\nx : G\n⊢ (f : (a : G) → H) ((↑n : G) + x) = HAdd.hAdd (α := H) ((f : (a : G) →...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H b : H inst_1 : AddCommGroupWithOne G inst_2 : AddGroup H inst_3 : AddConstMapClass F G H 1 b f : F n : ℤ x : G rw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) ((↑n : G) + x) : H) ((f : (a : G) → H) (HSMul.hSMul (β := G) n 1 + x) : H) ⊢ (f : (a : G) → H) ((↑n : G...
Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) ((↑n : G) + x) : H) ((f : (a : G) → H) ((↑n : G) + x) : H)
rw₁
[]
rw [← map_zsmul_add, zsmul_one]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\nb : H\ninst_1 : AddCommGroupWithOne G\ninst_2 : AddGroup H\ninst_3 : AddConstMapClass F G H 1 b\nf : F\nn : ℤ\nx : G\nrw : Eq.{u_3 + 1} (α := H) ((f : (a : G) → H) ((↑n : G) + x) : H) ((f :...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R hR : CovariantClass H H (fun (x y : H) ↦ y + x) ...
∀ (hR : Covariant H H (fun (x y : H) ↦ y + x) R), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
replace
[ "hR" ]
replace hR := hR.elim
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
0 ≤ a
ha'
[]
have ha' : 0 ≤ a := ha.le
hypothesis
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (this : ∀ (x y : G) (hxy : x < y) (hx : x ∈ Ico l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)) (hx : x ∉ Ico l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
inr
[]
wlog hx : x ∈ Ico l (l + a) generalizing x y
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (x y : G) (hxy : x < y) (hx : x ∈ Ico l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
wlog
[ "x", "y", "hxy" ]
wlog hx : x ∈ Ico l (l + a) generalizing x y
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (m : ℤ) (hm : x - m • a ∈ Ico l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
inr
[]
rcases existsUnique_sub_zsmul_mem_Ico ha x l with ⟨m, hm, -⟩
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
R ((f : (a : G) → H) (x - m • a)) ((f : (a : G) → H) (y - m • a))
inr
[]
suffices R (f (x - m • a)) (f (y - m • a)) by simpa using hR (m • b) this
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (n : ℤ) (hny : y - n • a ∈ Ioc l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
rcases
[]
rcases existsUnique_sub_zsmul_mem_Ioc ha y l with ⟨n, hny, -⟩
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hn : n < 0), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
inl
[]
rcases lt_trichotomy n 0 with hn | rfl | hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hny : y - HSMul.hSMul (α := ℤ) 0 a ∈ Ioc l (l + a)), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
inr
[ "n", "hny" ]
rcases lt_trichotomy n 0 with hn | rfl | hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hn : 0 < n), R ((f : (a : G) → H) x) ((f : (a : G) → H) y)
inr₁
[]
rcases lt_trichotomy n 0 with hn | rfl | hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
y ≤ x
inl
[]
refine absurd ?_ hxy.not_ge
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + a + n • a = l + (1 + n) • a
rw
[]
rw [add_comm n, add_smul, one_smul, add_assoc]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + a + n • a = l + (HSMul.hSMul (α := ℤ) 1 a + n • a)
rw₁
[]
rw [add_comm n, add_smul, one_smul, add_assoc]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + a + n • a = l + (a + n • a)
rw₂
[]
rw [add_comm n, add_smul, one_smul, add_assoc]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + (a + n • a) = l + (a + n • a)
rw₃
[]
rw [add_comm n, add_smul, one_smul, add_assoc]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
n + 1 ≤ 0
bc
[]
gcongr
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + a + n • a = l + (n + 1) • a
y₁
[]
calc y ≤ l + a + n • a := sub_le_iff_le_add.1 hny.2 _ = l + (n + 1) • a := by rw [add_comm n, add_smul, one_smul, add_assoc] _ ≤ l + (0 : ℤ) • a := by gcongr; lia _ ≤ x := by simpa using hx.1
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + (n + 1) • a ≤ l + HSMul.hSMul (α := ℤ) 0 a
y₂
[]
calc y ≤ l + a + n • a := sub_le_iff_le_add.1 hny.2 _ = l + (n + 1) • a := by rw [add_comm n, add_smul, one_smul, add_assoc] _ ≤ l + (0 : ℤ) • a := by gcongr; lia _ ≤ x := by simpa using hx.1
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l + HSMul.hSMul (α := ℤ) 0 a ≤ x
y₃
[]
calc y ≤ l + a + n • a := sub_le_iff_le_add.1 hny.2 _ = l + (n + 1) • a := by rw [add_comm n, add_smul, one_smul, add_assoc] _ ≤ l + (0 : ℤ) • a := by gcongr; lia _ ≤ x := by simpa using hx.1
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
R ((f : (a : G) → H) x) ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a))
trans
[]
trans f (l + (1 : ℤ) • a)
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) y)
trans₁
[]
trans f (l + (1 : ℤ) • a)
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
R (HAdd.hAdd (α := H) ((f : (a : G) → H) l : H) (n • b)) (HAdd.hAdd (α := H) ((f : (a : G) → H) (y - n • a) : H) (n • b))
rw
[]
rw [← sub_add_cancel y (n • a), map_add_zsmul, map_add_zsmul]
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
l ∈ Icc l (l + a)
refine
[]
refine hR _ <| hf _ ?_ _ (Ioc_subset_Icc_self hny) hny.1
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hy : R ((f : (a : G) → H) (l + n • a)) ((f : (a : G) → H) y)), R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) y)
have
[]
have hy : R (f (l + n • a)) (f y) := by rw [← sub_add_cancel y (n • a), map_add_zsmul, map_add_zsmul] refine hR _ <| hf _ ?_ _ (Ioc_subset_Icc_self hny) hny.1; simpa
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
1 ≤ n
hn₁
[ "hn", "hy" ]
rw [← Int.add_one_le_iff, zero_add] at hn
hypothesis
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hny : y - HSMul.hSMul (α := ℤ) 1 a ∈ Ioc l (l + a)) (hn : LE.le (α := ℤ) 1 1) (hy : R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) y)), R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) y)
inl
[ "n", "hny", "hn", "hy" ]
rcases hn.eq_or_lt with rfl | hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (hn : 1 < n), R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) y)
inr
[]
rcases hn.eq_or_lt with rfl | hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
R ((f : (a : G) → H) (l + HSMul.hSMul (α := ℤ) 1 a)) ((f : (a : G) → H) (l + n • a))
trans
[]
trans f (l + n • a)
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
Mathlib.Algebra.AddConstMap.Basic
train
null
F : Type u_1 G : Type u_2 H : Type u_3 inst : FunLike F G H a : G b : H inst_1 : AddCommGroup G inst_2 : LinearOrder G inst_3 : IsOrderedAddMonoid G inst_4 : Archimedean G inst_5 : AddGroup H inst_6 : AddConstMapClass F G H a b f : F R : H → H → Prop inst_7 : IsTrans H R ha : 0 < a l : G hf : ∀ x ∈ Icc l (l + a), ∀ y ∈...
∀ (k : ℤ), R ((fun (x : ℤ) ↦ ((f : (a : G) → H) (l + x • a) : H)) k) ((fun (x : ℤ) ↦ ((f : (a : G) → H) (l + x • a) : H)) (k + 1))
refine
[]
refine Int.rel_of_forall_rel_succ_of_lt R (f := (f <| l + · • a)) (fun k ↦ ?_) hn
goal
null
null
null
null
null
[ { "content": "Consider the following Lean goal:\n```lean4\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst : FunLike F G H\na : G\nb : H\ninst_1 : AddCommGroup G\ninst_2 : LinearOrder G\ninst_3 : IsOrderedAddMonoid G\ninst_4 : Archimedean G\ninst_5 : AddGroup H\ninst_6 : AddConstMapClass F G H a b\nf : F\nR : H ...
[ { "function": { "description": "Submit a Lean 4 `have` statement.", "name": "have", "parameters": { "additionalProperties": false, "properties": { "clear": { "default": [], "description": "Any hypotheses that are irrelevant for proving the goal...
End of preview. Expand in Data Studio

Canonical Drafter — extract data (raw)

Ground-truth drafter/premise data lifted from existing Mathlib proofs by Training/ExtractData.lean. This is the raw pool (429303 draft rows, 664445 premise rows from: Mathlib). Every row is a real have / closed subgoal taken from a checked proof, so there are no success/used flags to filter on. Columns follow the uniform schema shared with the rollout dataset, so the two pools concatenate cleanly.

config drafts

One row per have recovered from a Mathlib proof.

column type meaning
goal str pretty-printed goal the have addresses
type str the have type
name str the have hypothesis name
removals list[str] hypotheses the have cleared
tactic str the ground-truth tactic that produced this step
kind str what the step is (e.g. goal, hypothesis)
messages list single user->assistant turn; the assistant turn is the have tool call
tools list tool schema offered to the model
const, roundtrip, success, used, usedTactic, typeFromPp null for extract rows (rollout-only columns)

Recommended drafter-SFT use: every row is a used, ground-truth draft — keep all rows and project to {messages, tools} (loss on the final assistant turn only). Optionally filter by kind.

config premises

One row per subgoal closed within an extracted proof, with the closing tactic and the library constants it used.

column type meaning
goal str pretty-printed goal that was closed
tactic str the closing tactic / term
premises list[str] library constant names used by the closing proof
const, roundtrip null for extract rows (rollout-only columns)

Recommended premise-selection use: many rows have an empty premises list (goal closed by rfl/simp with no library lemma); drop those if you only want goals that needed premises.

Splitting

Both configs ship train and validation splits. The split is module-level (all rows from one Mathlib file land in the same split, so no train/val leakage) and is the canonical project split shared with the rollout pipeline — an explicit list of modules bundled in this repo at mathlib_v1_splits/{train,val}_modules.txt, not a hash. Every row also carries:

column type meaning
module str source Mathlib module (file) the row came from
split str train or val, from data/splits/mathlib_v1/
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