file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PushoutCocone.ι_app_right
|
[] |
[
791,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
pow_mono_right
|
[] |
[
198,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.const_mul_left
|
[] |
[
1493,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1492,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.prod.unique
|
[] |
[
920,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.deriv_log'
|
[] |
[
78,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_inter_self_left
|
[] |
[
2096,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2095,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.prod_range_cast_nat_sub
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\n⊢ ∏ i in range k, (↑n - ↑i) = ↑(∏ i in range k, (n - i))",
"tactic": "rw [prod_natCast]"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhkn : k ≤ n\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhnk : n < k\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"tactic": "cases' le_or_lt k n with hkn hnk"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhkn : k ≤ n\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"tactic": "exact prod_congr rfl fun i hi => (Nat.cast_sub <| (mem_range.1 hi).le.trans hkn).symm"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhnk : n ∈ range k\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhnk : n < k\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"tactic": "rw [← mem_range] at hnk"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf g : α → β\nR : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nhnk : n ∈ range k\n⊢ ∏ i in range k, (↑n - ↑i) = ∏ x in range k, ↑(n - x)",
"tactic": "rw [prod_eq_zero hnk, prod_eq_zero hnk] <;> simp"
}
] |
[
235,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Std/Logic.lean
|
or_and_right
|
[
{
"state_after": "no goals",
"state_before": "a b c : Prop\n⊢ (a ∨ b) ∧ c ↔ a ∧ c ∨ b ∧ c",
"tactic": "simp [and_comm, and_or_left]"
}
] |
[
326,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
325,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arccos_injOn
|
[] |
[
374,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
bernoulli'_three
|
[
{
"state_after": "A : Type ?u.186788\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ 1 -\n (0 + ↑(Nat.choose 3 0) / (↑3 - ↑0 + 1) * bernoulli' 0 + ↑(Nat.choose 3 1) / (↑3 - ↑1 + 1) * bernoulli' 1 +\n ↑(Nat.choose 3 2) / (↑3 - ↑2 + 1) * bernoulli' 2) =\n 0",
"state_before": "A : Type ?u.186788\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ bernoulli' 3 = 0",
"tactic": "rw [bernoulli'_def, sum_range_succ, sum_range_succ, sum_range_succ, sum_range_zero]"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.186788\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ 1 -\n (0 + ↑(Nat.choose 3 0) / (↑3 - ↑0 + 1) * bernoulli' 0 + ↑(Nat.choose 3 1) / (↑3 - ↑1 + 1) * bernoulli' 1 +\n ↑(Nat.choose 3 2) / (↑3 - ↑2 + 1) * bernoulli' 2) =\n 0",
"tactic": "norm_num"
}
] |
[
126,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/AlgebraicTopology/ExtraDegeneracy.lean
|
CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_succ
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\n⊢ WidePullback.lift (WidePullback.base fun x => f.hom)\n (fun i =>\n if x : i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred i x))\n (_ :\n ∀ (i : Fin (SimplexCategory.len [n + 1].op.unop + 1)),\n (fun i =>\n if x : i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred i x))\n i ≫\n f.hom =\n WidePullback.base fun x => f.hom) ≫\n WidePullback.π (fun x => f.hom) (Fin.succ i) =\n WidePullback.π (fun x => f.hom) i",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\n⊢ s f S n ≫ WidePullback.π (fun x => f.hom) (Fin.succ i) = WidePullback.π (fun x => f.hom) i",
"tactic": "dsimp [ExtraDegeneracy.s]"
},
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\n⊢ (if x : Fin.succ i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred (Fin.succ i) x)) =\n WidePullback.π (fun x => f.hom) i",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\n⊢ WidePullback.lift (WidePullback.base fun x => f.hom)\n (fun i =>\n if x : i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred i x))\n (_ :\n ∀ (i : Fin (SimplexCategory.len [n + 1].op.unop + 1)),\n (fun i =>\n if x : i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred i x))\n i ≫\n f.hom =\n WidePullback.base fun x => f.hom) ≫\n WidePullback.π (fun x => f.hom) (Fin.succ i) =\n WidePullback.π (fun x => f.hom) i",
"tactic": "simp only [WidePullback.lift_π]"
},
{
"state_after": "case inl\nC : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\nh : Fin.succ i = 0\n⊢ (WidePullback.base fun x => f.hom) ≫ S.section_ = WidePullback.π (fun x => f.hom) i\n\ncase inr\nC : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\nh : ¬Fin.succ i = 0\n⊢ WidePullback.π (fun x => f.hom) (Fin.pred (Fin.succ i) h) = WidePullback.π (fun x => f.hom) i",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\n⊢ (if x : Fin.succ i = 0 then (WidePullback.base fun x => f.hom) ≫ S.section_\n else WidePullback.π (fun x => f.hom) (Fin.pred (Fin.succ i) x)) =\n WidePullback.π (fun x => f.hom) i",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nC : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\nh : Fin.succ i = 0\n⊢ (WidePullback.base fun x => f.hom) ≫ S.section_ = WidePullback.π (fun x => f.hom) i",
"tactic": "simp only [Fin.ext_iff, Fin.val_succ, Fin.val_zero, add_eq_zero, and_false] at h"
},
{
"state_after": "no goals",
"state_before": "case inr\nC : Type u_2\ninst✝¹ : Category C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin (n + 1)\nh : ¬Fin.succ i = 0\n⊢ WidePullback.π (fun x => f.hom) (Fin.pred (Fin.succ i) h) = WidePullback.π (fun x => f.hom) i",
"tactic": "simp only [Fin.pred_succ]"
}
] |
[
296,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.denote_combineAux
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\nfuel : Nat\np₁ p₂ : Poly\n⊢ denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂",
"tactic": "induction fuel generalizing p₁ p₂ with simp [combineAux]\n| succ fuel ih =>\n split <;> simp\n rename_i k₁ v₁ p₁ k₂ v₂ p₂\n by_cases hltv : Nat.blt v₁ v₂ <;> simp [hltv, ih]\n by_cases hgtv : Nat.blt v₂ v₁ <;> simp [hgtv, ih]\n have heqv : v₁ = v₂ := eq_of_not_blt_eq_true hltv hgtv\n simp [heqv]"
},
{
"state_after": "case succ.h_3\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\n⊢ denote ctx\n (bif blt v₁✝ v₂✝ then (k₁✝, v₁✝) :: combineAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝)\n else\n bif blt v₂✝ v₁✝ then (k₂✝, v₂✝) :: combineAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝\n else (k₁✝ + k₂✝, v₁✝) :: combineAux fuel m₁✝ m₂✝) =\n denote ctx m₁✝ + (denote ctx m₂✝ + (k₁✝ * Var.denote ctx v₁✝ + k₂✝ * Var.denote ctx v₂✝))",
"state_before": "case succ\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁ p₂ : Poly\n⊢ denote ctx\n (match p₁, p₂ with\n | p₁, [] => p₁\n | [], p₁ => p₁\n | (k₁, v₁) :: p₁, (k₂, v₂) :: p₂ =>\n bif blt v₁ v₂ then (k₁, v₁) :: combineAux fuel p₁ ((k₂, v₂) :: p₂)\n else\n bif blt v₂ v₁ then (k₂, v₂) :: combineAux fuel ((k₁, v₁) :: p₁) p₂\n else (k₁ + k₂, v₁) :: combineAux fuel p₁ p₂) =\n denote ctx p₁ + denote ctx p₂",
"tactic": "split <;> simp"
},
{
"state_after": "case succ.h_3\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\n⊢ denote ctx\n (bif blt v₁ v₂ then (k₁, v₁) :: combineAux fuel p₁ ((k₂, v₂) :: p₂)\n else\n bif blt v₂ v₁ then (k₂, v₂) :: combineAux fuel ((k₁, v₁) :: p₁) p₂\n else (k₁ + k₂, v₁) :: combineAux fuel p₁ p₂) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"state_before": "case succ.h_3\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝¹ m₂✝¹ : Poly\nk₁✝ : Nat\nv₁✝ : Var\nm₁✝ : List (Nat × Var)\nk₂✝ : Nat\nv₂✝ : Var\nm₂✝ : List (Nat × Var)\n⊢ denote ctx\n (bif blt v₁✝ v₂✝ then (k₁✝, v₁✝) :: combineAux fuel m₁✝ ((k₂✝, v₂✝) :: m₂✝)\n else\n bif blt v₂✝ v₁✝ then (k₂✝, v₂✝) :: combineAux fuel ((k₁✝, v₁✝) :: m₁✝) m₂✝\n else (k₁✝ + k₂✝, v₁✝) :: combineAux fuel m₁✝ m₂✝) =\n denote ctx m₁✝ + (denote ctx m₂✝ + (k₁✝ * Var.denote ctx v₁✝ + k₂✝ * Var.denote ctx v₂✝))",
"tactic": "rename_i k₁ v₁ p₁ k₂ v₂ p₂"
},
{
"state_after": "case succ.h_3.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\n⊢ denote ctx\n (bif blt v₂ v₁ then (k₂, v₂) :: combineAux fuel ((k₁, v₁) :: p₁) p₂ else (k₁ + k₂, v₁) :: combineAux fuel p₁ p₂) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"state_before": "case succ.h_3\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\n⊢ denote ctx\n (bif blt v₁ v₂ then (k₁, v₁) :: combineAux fuel p₁ ((k₂, v₂) :: p₂)\n else\n bif blt v₂ v₁ then (k₂, v₂) :: combineAux fuel ((k₁, v₁) :: p₁) p₂\n else (k₁ + k₂, v₁) :: combineAux fuel p₁ p₂) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"tactic": "by_cases hltv : Nat.blt v₁ v₂ <;> simp [hltv, ih]"
},
{
"state_after": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\n⊢ denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₁)) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"state_before": "case succ.h_3.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\n⊢ denote ctx\n (bif blt v₂ v₁ then (k₂, v₂) :: combineAux fuel ((k₁, v₁) :: p₁) p₂ else (k₁ + k₂, v₁) :: combineAux fuel p₁ p₂) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"tactic": "by_cases hgtv : Nat.blt v₂ v₁ <;> simp [hgtv, ih]"
},
{
"state_after": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nheqv : v₁ = v₂\n⊢ denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₁)) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"state_before": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\n⊢ denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₁)) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"tactic": "have heqv : v₁ = v₂ := eq_of_not_blt_eq_true hltv hgtv"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3.inr.inr\nctx : Context\nfuel : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (combineAux fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂\nm₁✝ m₂✝ : Poly\nk₁ : Nat\nv₁ : Var\np₁ : List (Nat × Var)\nk₂ : Nat\nv₂ : Var\np₂ : List (Nat × Var)\nhltv : ¬blt v₁ v₂ = true\nhgtv : ¬blt v₂ v₁ = true\nheqv : v₁ = v₂\n⊢ denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₁)) =\n denote ctx p₁ + (denote ctx p₂ + (k₁ * Var.denote ctx v₁ + k₂ * Var.denote ctx v₂))",
"tactic": "simp [heqv]"
}
] |
[
518,
16
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
510,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MonoidHom.fst_comp_inr
|
[] |
[
530,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Data/Part.lean
|
Part.union_mem_union
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.88814\nγ : Type ?u.88817\ninst✝ : Union α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ∪ a = ma ∪ mb",
"state_before": "α : Type u_1\nβ : Type ?u.88814\nγ : Type ?u.88817\ninst✝ : Union α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ma ∪ mb ∈ a ∪ b",
"tactic": "simp [union_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.88814\nγ : Type ?u.88817\ninst✝ : Union α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ∪ a = ma ∪ mb",
"tactic": "aesop"
}
] |
[
844,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
|
Ordinal.CNF_of_le_one
|
[
{
"state_after": "case inl\no : Ordinal\nho : o ≠ 0\nhb : 0 ≤ 1\n⊢ CNF 0 o = [(0, o)]\n\ncase inr\no : Ordinal\nho : o ≠ 0\nhb : 1 ≤ 1\n⊢ CNF 1 o = [(0, o)]",
"state_before": "b o : Ordinal\nhb : b ≤ 1\nho : o ≠ 0\n⊢ CNF b o = [(0, o)]",
"tactic": "rcases le_one_iff.1 hb with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\no : Ordinal\nho : o ≠ 0\nhb : 0 ≤ 1\n⊢ CNF 0 o = [(0, o)]",
"tactic": "exact zero_CNF ho"
},
{
"state_after": "no goals",
"state_before": "case inr\no : Ordinal\nho : o ≠ 0\nhb : 1 ≤ 1\n⊢ CNF 1 o = [(0, o)]",
"tactic": "exact one_CNF ho"
}
] |
[
107,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Order/Basic.lean
|
le_of_forall_le'
|
[] |
[
533,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
tsum_coe_mul_geometric_of_norm_lt_1
|
[] |
[
370,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finRotate_last
|
[] |
[
413,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
coplanar_pair
|
[] |
[
661,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.eq_of_lt_minimalPeriod_of_iterate_eq
|
[] |
[
360,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_singleton
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"state_before": "α : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\n⊢ restrict μ {a} = ↑↑μ {a} • dirac a",
"tactic": "ext1 s hs"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : a ∈ s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s\n\ncase neg\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : ¬a ∈ s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"tactic": "by_cases ha : a ∈ s"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : a ∈ s\nthis : s ∩ {a} = {a}\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : a ∈ s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"tactic": "have : s ∩ {a} = {a} := by simpa"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : a ∈ s\nthis : s ∩ {a} = {a}\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"tactic": "simp [*]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : a ∈ s\n⊢ s ∩ {a} = {a}",
"tactic": "simpa"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : ¬a ∈ s\nthis : s ∩ {a} = ∅\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : ¬a ∈ s\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"tactic": "have : s ∩ {a} = ∅ := inter_singleton_eq_empty.2 ha"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.331505\nγ : Type ?u.331508\nδ : Type ?u.331511\nι : Type ?u.331514\nR : Type ?u.331517\nR' : Type ?u.331520\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nμ : Measure α\na : α\ns : Set α\nhs : MeasurableSet s\nha : ¬a ∈ s\nthis : s ∩ {a} = ∅\n⊢ ↑↑(restrict μ {a}) s = ↑↑(↑↑μ {a} • dirac a) s",
"tactic": "simp [*]"
}
] |
[
2015,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2009,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_apply
|
[] |
[
545,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Init/Function.lean
|
Function.comp.left_id
|
[] |
[
56,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
9
] |
Mathlib/CategoryTheory/Sums/Associator.lean
|
CategoryTheory.sum.inverseAssociator_obj_inl
|
[] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\n𝕜 : Type ?u.1482776\nE : Type u_1\nF : Type ?u.1482782\nA : Type ?u.1482785\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : MeasureTheory.Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas : StronglyMeasurableAtFilter f la'\nhf : Tendsto f (la' ⊓ Measure.ae μ) (𝓝 c)\nhu : Tendsto u lt la\nhv : Tendsto v lt la\n⊢ (fun t => ((∫ (x : ℝ) in v t..b, f x ∂μ) - ∫ (x : ℝ) in u t..b, f x ∂μ) + ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t =>\n ∫ (x : ℝ) in u t..v t, 1 ∂μ",
"tactic": "simpa using\n measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas\n stronglyMeasurableAt_bot hf ((tendsto_bot : Tendsto _ ⊥ (𝓝 (0 : E))).mono_left inf_le_left) hu\n hv (tendsto_const_pure : Tendsto _ _ (pure b)) tendsto_const_pure"
}
] |
[
476,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ
|
[
{
"state_after": "case inl\np : ℝ[X]\nhp' : ↑derivative p = 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1\n\ncase inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"state_before": "p : ℝ[X]\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"tactic": "cases' eq_or_ne (derivative p) 0 with hp' hp'"
},
{
"state_after": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"state_before": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"tactic": "have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])"
},
{
"state_after": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : x ∈ Multiset.toFinset (roots p)\ny : ℝ\nhy : y ∈ Multiset.toFinset (roots p)\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"state_before": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"tactic": "refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _"
},
{
"state_after": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"state_before": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : x ∈ Multiset.toFinset (roots p)\ny : ℝ\nhy : y ∈ Multiset.toFinset (roots p)\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"tactic": "rw [Multiset.mem_toFinset, mem_roots hp] at hx hy"
},
{
"state_after": "case inr.intro.intro\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\nz : ℝ\nhz1 : z ∈ Ioo x y\nhz2 : deriv (fun x => eval x p) z = 0\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"state_before": "case inr\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"tactic": "obtain ⟨z, hz1, hz2⟩ :=\n exists_deriv_eq_zero (fun x : ℝ => eval x p) hxy p.continuousOn (hx.trans hy.symm)"
},
{
"state_after": "case inr.intro.intro\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\nz : ℝ\nhz1 : z ∈ Ioo x y\nhz2 : deriv (fun x => eval x p) z = 0\n⊢ z ∈ Multiset.toFinset (roots (↑derivative p))",
"state_before": "case inr.intro.intro\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\nz : ℝ\nhz1 : z ∈ Ioo x y\nhz2 : deriv (fun x => eval x p) z = 0\n⊢ ∃ z, z ∈ Multiset.toFinset (roots (↑derivative p)) ∧ x < z ∧ z < y",
"tactic": "refine' ⟨z, _, hz1⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\np : ℝ[X]\nhp' : ↑derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : IsRoot p x\ny : ℝ\nhy : IsRoot p y\nhxy : x < y\nhxy' : ∀ (z : ℝ), z ∈ Multiset.toFinset (roots p) → ¬z ∈ Ioo x y\nz : ℝ\nhz1 : z ∈ Ioo x y\nhz2 : deriv (fun x => eval x p) z = 0\n⊢ z ∈ Multiset.toFinset (roots (↑derivative p))",
"tactic": "rwa [Multiset.mem_toFinset, mem_roots hp', IsRoot, ← p.deriv]"
},
{
"state_after": "case inl\np : ℝ[X]\nhp' : ↑derivative p = 0\n⊢ 0 ≤ Finset.card (Multiset.toFinset (roots (↑derivative (↑C (coeff p 0)))) \\ ∅) + 1",
"state_before": "case inl\np : ℝ[X]\nhp' : ↑derivative p = 0\n⊢ Finset.card (Multiset.toFinset (roots p)) ≤\n Finset.card (Multiset.toFinset (roots (↑derivative p)) \\ Multiset.toFinset (roots p)) + 1",
"tactic": "rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℝ[X]\nhp' : ↑derivative p = 0\n⊢ 0 ≤ Finset.card (Multiset.toFinset (roots (↑derivative (↑C (coeff p 0)))) \\ ∅) + 1",
"tactic": "exact zero_le _"
},
{
"state_after": "no goals",
"state_before": "p : ℝ[X]\nhp' : ↑derivative p ≠ 0\n⊢ ↑derivative p ≠ ↑derivative 0",
"tactic": "rwa [derivative_zero]"
}
] |
[
367,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Data/Set/Image.lean
|
Prod.range_fst
|
[] |
[
867,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
866,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.continuous_of_dominated_interval
|
[] |
[
1116,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.eventually_filterAt_iff
|
[] |
[
249,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.root_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\n⊢ IsRoot (p * q) a ↔ IsRoot p a ∨ IsRoot q a",
"tactic": "simp_rw [IsRoot, eval_mul, mul_eq_zero]"
}
] |
[
339,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Logic/Small/Basic.lean
|
small_congr
|
[] |
[
85,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Computability/Halting.lean
|
Nat.Partrec'.map
|
[
{
"state_after": "n : ℕ\nf : Vector ℕ n →. ℕ\ng : Vector ℕ (n + 1) → ℕ\nhf : Partrec' f\nhg : Partrec' ↑g\n⊢ Partrec' fun v => Part.bind (f v) (Part.some ∘ fun a => g (a ::ᵥ v))",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\ng : Vector ℕ (n + 1) → ℕ\nhf : Partrec' f\nhg : Partrec' ↑g\n⊢ Partrec' fun v => Part.map (fun a => g (a ::ᵥ v)) (f v)",
"tactic": "simp [(Part.bind_some_eq_map _ _).symm]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\ng : Vector ℕ (n + 1) → ℕ\nhf : Partrec' f\nhg : Partrec' ↑g\n⊢ Partrec' fun v => Part.bind (f v) (Part.some ∘ fun a => g (a ::ᵥ v))",
"tactic": "exact hf.bind hg"
}
] |
[
336,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.inr_pushoutAssoc_hom
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nW X Y Z X₁ X₂ X₃ Z₁ Z₂ : C\ng₁ : Z₁ ⟶ X₁\ng₂ : Z₁ ⟶ X₂\ng₃ : Z₂ ⟶ X₂\ng₄ : Z₂ ⟶ X₃\ninst✝³ : HasPushout g₁ g₂\ninst✝² : HasPushout g₃ g₄\ninst✝¹ : HasPushout (g₃ ≫ pushout.inr) g₄\ninst✝ : HasPushout g₁ (g₂ ≫ pushout.inl)\n⊢ pushout.inr ≫ (pushoutAssoc g₁ g₂ g₃ g₄).hom = pushout.inr ≫ pushout.inr",
"tactic": "rw [← Iso.eq_comp_inv, Category.assoc, inr_inr_pushoutAssoc_inv]"
}
] |
[
2651,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2648,
1
] |
Mathlib/Topology/ShrinkingLemma.lean
|
ShrinkingLemma.PartialRefinement.find_apply_of_mem
|
[] |
[
141,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Algebra/Group/Units.lean
|
IsUnit.exists_right_inv
|
[
{
"state_after": "case intro.mk\nα : Type u\nM : Type u_1\nN : Type ?u.56747\ninst✝ : Monoid M\na b : M\nhab : a * b = 1\ninv_val✝ : b * a = 1\n⊢ ∃ b_1, ↑{ val := a, inv := b, val_inv := hab, inv_val := inv_val✝ } * b_1 = 1",
"state_before": "α : Type u\nM : Type u_1\nN : Type ?u.56747\ninst✝ : Monoid M\na : M\nh : IsUnit a\n⊢ ∃ b, a * b = 1",
"tactic": "rcases h with ⟨⟨a, b, hab, _⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nα : Type u\nM : Type u_1\nN : Type ?u.56747\ninst✝ : Monoid M\na b : M\nhab : a * b = 1\ninv_val✝ : b * a = 1\n⊢ ∃ b_1, ↑{ val := a, inv := b, val_inv := hab, inv_val := inv_val✝ } * b_1 = 1",
"tactic": "exact ⟨b, hab⟩"
}
] |
[
637,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_le
|
[
{
"state_after": "α✝ : Type ?u.110770\nβ✝ : Type ?u.110773\nγ : Type ?u.110776\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\na b : Ordinal\nα : Type v\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type v\ns : β → β → Prop\nx✝ : IsWellOrder β s\n⊢ lift (type r) ≤ lift (type s) ↔ type r ≤ type s",
"state_before": "α✝ : Type ?u.110770\nβ✝ : Type ?u.110773\nγ : Type ?u.110776\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\na b : Ordinal\nα : Type v\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type v\ns : β → β → Prop\nx✝ : IsWellOrder β s\n⊢ lift (type r) ≤ lift (type s) ↔ type r ≤ type s",
"tactic": "rw [← lift_umax]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.110770\nβ✝ : Type ?u.110773\nγ : Type ?u.110776\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\na b : Ordinal\nα : Type v\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type v\ns : β → β → Prop\nx✝ : IsWellOrder β s\n⊢ lift (type r) ≤ lift (type s) ↔ type r ≤ type s",
"tactic": "exact lift_type_le.{_,_,u}"
}
] |
[
768,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Init/Logic.lean
|
or_true_iff
|
[] |
[
181,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.infEdist_le_edist_add_infEdist
|
[
{
"state_after": "ι : Sort ?u.7435\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x s ≤ infEdist y s + edist x y",
"state_before": "ι : Sort ?u.7435\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x s ≤ edist x y + infEdist y s",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.7435\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x s ≤ infEdist y s + edist x y",
"tactic": "exact infEdist_le_infEdist_add_edist"
}
] |
[
116,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22417\nσ : Type u_2\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nx : σ\nn : ℕ\n⊢ (Nat.casesOn (encode (decode n)) 0 fun x_1 => Nat.succ (encode x)) = encode (Option.map (fun x_1 => x) (decode n))",
"tactic": "cases @decode α _ n <;> rfl"
}
] |
[
255,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_eq_inv_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.48927\nβ : Type ?u.48930\nG : Type u_1\ninst✝ : Group G\na b c d : G\n⊢ a / b = b⁻¹ ↔ a = 1",
"tactic": "rw [div_eq_mul_inv, mul_left_eq_self]"
}
] |
[
611,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_X
|
[] |
[
704,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.div_congr
|
[
{
"state_after": "u : Lean.Level\nR✝ : Type ?u.398864\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : DivisionRing R\na' b' : R\n⊢ a' / b' = a' / b'",
"state_before": "u : Lean.Level\nR✝ : Type ?u.398864\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : DivisionRing R\na a' b b' c : R\nx✝² : a = a'\nx✝¹ : b = b'\nx✝ : a' / b' = c\n⊢ a / b = c",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR✝ : Type ?u.398864\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : DivisionRing R\na' b' : R\n⊢ a' / b' = a' / b'",
"tactic": "rfl"
}
] |
[
983,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
982,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.direction_eq_top_iff_of_nonempty
|
[
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\n⊢ direction s = ⊤ → s = ⊤\n\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\n⊢ s = ⊤ → direction s = ⊤",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\n⊢ direction s = ⊤ ↔ s = ⊤",
"tactic": "constructor"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = ⊤\n⊢ s = ⊤",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\n⊢ direction s = ⊤ → s = ⊤",
"tactic": "intro hd"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = direction ⊤\n⊢ s = ⊤",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = ⊤\n⊢ s = ⊤",
"tactic": "rw [← direction_top k V P] at hd"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = direction ⊤\n⊢ Set.Nonempty (↑s ∩ ↑⊤)",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = direction ⊤\n⊢ s = ⊤",
"tactic": "refine' ext_of_direction_eq hd _"
},
{
"state_after": "no goals",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\nhd : direction s = direction ⊤\n⊢ Set.Nonempty (↑s ∩ ↑⊤)",
"tactic": "simp [h]"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : Set.Nonempty ↑⊤\n⊢ direction ⊤ = ⊤",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : AffineSubspace k P\nh : Set.Nonempty ↑s\n⊢ s = ⊤ → direction s = ⊤",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\nh : Set.Nonempty ↑⊤\n⊢ direction ⊤ = ⊤",
"tactic": "simp"
}
] |
[
882,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
874,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
derivWithin_ccosh
|
[] |
[
253,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_inj
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c✝ : α\nn : ℤ\nc : α\n⊢ toIcoMod hp c a = toIcoMod hp c b ↔ a ≡ b [PMOD p]",
"tactic": "simp_rw [toIcoMod_eq_toIcoMod, modEq_iff_eq_add_zsmul, sub_eq_iff_eq_add']"
}
] |
[
682,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_iUnion_eq_right
|
[] |
[
757,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.add_factorial_le_factorial_add
|
[
{
"state_after": "case refl\nm n i : ℕ\n⊢ i + 1! ≤ (i + 1)!\n\ncase step\nm n i h : ℕ\na✝ : Nat.le 1 h\n⊢ i + (succ h)! ≤ (i + succ h)!",
"state_before": "m n✝ i n : ℕ\nn1 : 1 ≤ n\n⊢ i + n ! ≤ (i + n)!",
"tactic": "cases' n1 with h"
},
{
"state_after": "no goals",
"state_before": "case step\nm n i h : ℕ\na✝ : Nat.le 1 h\n⊢ i + (succ h)! ≤ (i + succ h)!",
"tactic": "exact add_factorial_succ_le_factorial_add_succ i h"
},
{
"state_after": "no goals",
"state_before": "case refl\nm n i : ℕ\n⊢ i + 1! ≤ (i + 1)!",
"tactic": "exact self_le_factorial _"
}
] |
[
206,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Algebra/QuaternionBasis.lean
|
QuaternionAlgebra.Basis.lift_add
|
[
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.73937\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\nq : Basis A c₁ c₂\nx y : ℍ[R,c₁,c₂]\n⊢ ↑(algebraMap R A) x.re + ↑(algebraMap R A) y.re + (x.imI • q.i + y.imI • q.i) + (x.imJ • q.j + y.imJ • q.j) +\n (x.imK • q.k + y.imK • q.k) =\n ↑(algebraMap R A) x.re + x.imI • q.i + x.imJ • q.j + x.imK • q.k +\n (↑(algebraMap R A) y.re + y.imI • q.i + y.imJ • q.j + y.imK • q.k)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.73937\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\nq : Basis A c₁ c₂\nx y : ℍ[R,c₁,c₂]\n⊢ lift q (x + y) = lift q x + lift q y",
"tactic": "simp [lift, add_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.73937\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\nq : Basis A c₁ c₂\nx y : ℍ[R,c₁,c₂]\n⊢ ↑(algebraMap R A) x.re + ↑(algebraMap R A) y.re + (x.imI • q.i + y.imI • q.i) + (x.imJ • q.j + y.imJ • q.j) +\n (x.imK • q.k + y.imK • q.k) =\n ↑(algebraMap R A) x.re + x.imI • q.i + x.imJ • q.j + x.imK • q.k +\n (↑(algebraMap R A) y.re + y.imI • q.i + y.imJ • q.j + y.imK • q.k)",
"tactic": "abel"
}
] |
[
126,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.image_embedding_iff
|
[] |
[
165,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.le_of_add_le_add_left
|
[
{
"state_after": "no goals",
"state_before": "a b c : Nat\nh : a + b ≤ a + c\n⊢ b ≤ c",
"tactic": "match le.dest h with\n| ⟨d, hd⟩ =>\n apply @le.intro _ _ d\n rw [Nat.add_assoc] at hd\n apply Nat.add_left_cancel hd"
},
{
"state_after": "a b c : Nat\nh : a + b ≤ a + c\nd : Nat\nhd : a + b + d = a + c\n⊢ b + d = c",
"state_before": "a b c : Nat\nh : a + b ≤ a + c\nd : Nat\nhd : a + b + d = a + c\n⊢ b ≤ c",
"tactic": "apply @le.intro _ _ d"
},
{
"state_after": "a b c : Nat\nh : a + b ≤ a + c\nd : Nat\nhd : a + (b + d) = a + c\n⊢ b + d = c",
"state_before": "a b c : Nat\nh : a + b ≤ a + c\nd : Nat\nhd : a + b + d = a + c\n⊢ b + d = c",
"tactic": "rw [Nat.add_assoc] at hd"
},
{
"state_after": "no goals",
"state_before": "a b c : Nat\nh : a + b ≤ a + c\nd : Nat\nhd : a + (b + d) = a + c\n⊢ b + d = c",
"tactic": "apply Nat.add_left_cancel hd"
}
] |
[
415,
33
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
410,
11
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
MulHom.subsemigroupMap_surjective
|
[
{
"state_after": "case mk.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.76825\nσ : Type ?u.76828\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS : Subsemigroup M\nf : M →ₙ* N\nM' : Subsemigroup M\nx : M\nhx : x ∈ ↑M'\n⊢ ∃ a, ↑(subsemigroupMap f M') a = { val := ↑f x, property := (_ : ∃ a, a ∈ ↑M' ∧ ↑f a = ↑f x) }",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.76825\nσ : Type ?u.76828\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS : Subsemigroup M\nf : M →ₙ* N\nM' : Subsemigroup M\n⊢ Function.Surjective ↑(subsemigroupMap f M')",
"tactic": "rintro ⟨_, x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.76825\nσ : Type ?u.76828\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS : Subsemigroup M\nf : M →ₙ* N\nM' : Subsemigroup M\nx : M\nhx : x ∈ ↑M'\n⊢ ∃ a, ↑(subsemigroupMap f M') a = { val := ↑f x, property := (_ : ∃ a, a ∈ ↑M' ∧ ↑f a = ↑f x) }",
"tactic": "exact ⟨⟨x, hx⟩, rfl⟩"
}
] |
[
899,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
896,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.toDual_max'
|
[
{
"state_after": "F : Type ?u.356766\nα : Type u_1\nβ : Type ?u.356772\nγ : Type ?u.356775\nι : Type ?u.356778\nκ : Type ?u.356781\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑(↑toDual (max' s hs)) = ↑(min' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s)))",
"state_before": "F : Type ?u.356766\nα : Type u_1\nβ : Type ?u.356772\nγ : Type ?u.356775\nι : Type ?u.356778\nκ : Type ?u.356781\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑toDual (max' s hs) = min' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s))",
"tactic": "rw [← WithTop.coe_eq_coe]"
},
{
"state_after": "F : Type ?u.356766\nα : Type u_1\nβ : Type ?u.356772\nγ : Type ?u.356775\nι : Type ?u.356778\nκ : Type ?u.356781\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ inf s (WithTop.some ∘ fun x => ↑toDual x) = inf s ((WithTop.some ∘ fun x => x) ∘ ↑toDual)",
"state_before": "F : Type ?u.356766\nα : Type u_1\nβ : Type ?u.356772\nγ : Type ?u.356775\nι : Type ?u.356778\nκ : Type ?u.356781\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑(↑toDual (max' s hs)) = ↑(min' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s)))",
"tactic": "simp only [max'_eq_sup', id_eq, toDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',\n inf_image]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.356766\nα : Type u_1\nβ : Type ?u.356772\nγ : Type ?u.356775\nι : Type ?u.356778\nκ : Type ?u.356781\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ inf s (WithTop.some ∘ fun x => ↑toDual x) = inf s ((WithTop.some ∘ fun x => x) ∘ ↑toDual)",
"tactic": "rfl"
}
] |
[
1460,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1455,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.adjoin_range_X
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\n⊢ S = ⊤",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\n⊢ Algebra.adjoin R (range X) = ⊤",
"tactic": "set S := Algebra.adjoin R (range (X : σ → MvPolynomial σ R))"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\nhp : p ∈ ⊤\n⊢ p ∈ S",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\n⊢ S = ⊤",
"tactic": "refine' top_unique fun p hp => _"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\n⊢ p ∈ S",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\nhp : p ∈ ⊤\n⊢ p ∈ S",
"tactic": "clear hp"
},
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\na✝ : R\n⊢ ↑C a✝ ∈ S\n\ncase h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ q✝ : MvPolynomial σ R\na✝¹ : p✝ ∈ S\na✝ : q✝ ∈ S\n⊢ p✝ + q✝ ∈ S\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\n⊢ p ∈ S",
"tactic": "induction p using MvPolynomial.induction_on"
},
{
"state_after": "case h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ q✝ : MvPolynomial σ R\na✝¹ : p✝ ∈ S\na✝ : q✝ ∈ S\n⊢ p✝ + q✝ ∈ S\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"state_before": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\na✝ : R\n⊢ ↑C a✝ ∈ S\n\ncase h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ q✝ : MvPolynomial σ R\na✝¹ : p✝ ∈ S\na✝ : q✝ ∈ S\n⊢ p✝ + q✝ ∈ S\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"tactic": "case h_C => exact S.algebraMap_mem _"
},
{
"state_after": "case h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"state_before": "case h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ q✝ : MvPolynomial σ R\na✝¹ : p✝ ∈ S\na✝ : q✝ ∈ S\n⊢ p✝ + q✝ ∈ S\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"tactic": "case h_add p q hp hq => exact S.add_mem hp hq"
},
{
"state_after": "no goals",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np✝ : MvPolynomial σ R\nn✝ : σ\na✝ : p✝ ∈ S\n⊢ p✝ * X n✝ ∈ S",
"tactic": "case h_X p i hp => exact S.mul_mem hp (Algebra.subset_adjoin <| mem_range_self _)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\na✝ : R\n⊢ ↑C a✝ ∈ S",
"tactic": "exact S.algebraMap_mem _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np q : MvPolynomial σ R\nhp : p ∈ S\nhq : q ∈ S\n⊢ p + q ∈ S",
"tactic": "exact S.add_mem hp hq"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\np : MvPolynomial σ R\ni : σ\nhp : p ∈ S\n⊢ p * X i ∈ S",
"tactic": "exact S.mul_mem hp (Algebra.subset_adjoin <| mem_range_self _)"
}
] |
[
510,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
|
geometric_hahn_banach_closed_point
|
[] |
[
195,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
isSelfAdjoint_one
|
[] |
[
180,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
Rel.edgeDensity_sub_edgeDensity_le_one_sub_mul
|
[
{
"state_after": "𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₂ t₂ - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁)) * edgeDensity r s₂ t₂ ≤\n 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"state_before": "𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₂ t₂ - edgeDensity r s₁ t₁ ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "refine' (sub_le_sub_left (mul_edgeDensity_le_edgeDensity r hs ht hs₂ ht₂) _).trans _"
},
{
"state_after": "case refine'_1\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₂ t₂ - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁)) * edgeDensity r s₂ t₂ ≤\n (1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))) * edgeDensity r s₂ t₂\n\ncase refine'_2\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ 0 ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"state_before": "𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₂ t₂ - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁)) * edgeDensity r s₂ t₂ ≤\n 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "refine' le_trans _ (mul_le_of_le_one_right _ (edgeDensity_le_one r s₂ t₂))"
},
{
"state_after": "case refine'_2.refine'_1\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ ↑(card s₂) / ↑(card s₁) ≤ 1\n\ncase refine'_2.refine'_2\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ 0 ≤ ↑(card t₂) / ↑(card t₁)\n\ncase refine'_2.refine'_3\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ ↑(card t₂) / ↑(card t₁) ≤ 1",
"state_before": "case refine'_2\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ 0 ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "refine' sub_nonneg_of_le (mul_le_one _ _ _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₂ t₂ - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁)) * edgeDensity r s₂ t₂ ≤\n (1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))) * edgeDensity r s₂ t₂",
"tactic": "rw [sub_mul, one_mul]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ ↑(card s₂) / ↑(card s₁) ≤ 1",
"tactic": "exact div_le_one_of_le ((@Nat.cast_le ℚ).2 (card_le_of_subset hs)) (Nat.cast_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ 0 ≤ ↑(card t₂) / ↑(card t₁)",
"tactic": "apply div_nonneg <;> exact_mod_cast Nat.zero_le _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_3\n𝕜 : Type ?u.34819\nι : Type ?u.34822\nκ : Type ?u.34825\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ ↑(card t₂) / ↑(card t₁) ≤ 1",
"tactic": "exact div_le_one_of_le ((@Nat.cast_le ℚ).2 (card_le_of_subset ht)) (Nat.cast_nonneg _)"
}
] |
[
204,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
Interval.one_mem_one
|
[] |
[
141,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_sign
|
[] |
[
202,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
201,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.neg_coe_abs_toReal_of_sign_nonpos
|
[
{
"state_after": "θ : Angle\nh : sign θ = -1 ∨ sign θ = 0\n⊢ -↑(abs (toReal θ)) = θ",
"state_before": "θ : Angle\nh : sign θ ≤ 0\n⊢ -↑(abs (toReal θ)) = θ",
"tactic": "rw [SignType.nonpos_iff] at h"
},
{
"state_after": "case inl\nθ : Angle\nh : sign θ = -1\n⊢ -↑(abs (toReal θ)) = θ\n\ncase inr\nθ : Angle\nh : sign θ = 0\n⊢ -↑(abs (toReal θ)) = θ",
"state_before": "θ : Angle\nh : sign θ = -1 ∨ sign θ = 0\n⊢ -↑(abs (toReal θ)) = θ",
"tactic": "rcases h with (h | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nθ : Angle\nh : sign θ = -1\n⊢ -↑(abs (toReal θ)) = θ",
"tactic": "rw [abs_of_neg (toReal_neg_iff_sign_neg.2 h), coe_neg, neg_neg, coe_toReal]"
},
{
"state_after": "case inr\nθ : Angle\nh : θ = 0 ∨ θ = ↑π\n⊢ -↑(abs (toReal θ)) = θ",
"state_before": "case inr\nθ : Angle\nh : sign θ = 0\n⊢ -↑(abs (toReal θ)) = θ",
"tactic": "rw [sign_eq_zero_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case inr\nθ : Angle\nh : θ = 0 ∨ θ = ↑π\n⊢ -↑(abs (toReal θ)) = θ",
"tactic": "rcases h with (rfl | rfl) <;> simp [abs_of_pos Real.pi_pos]"
}
] |
[
941,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
936,
1
] |
Mathlib/RingTheory/WittVector/Teichmuller.lean
|
WittVector.ghostComponent_teichmullerFun
|
[
{
"state_after": "p : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ coeff (teichmullerFun p r) 0 ^ p ^ n = r ^ p ^ n\n\ncase h₀\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ∀ (b : ℕ), b ∈ Finset.range (n + 1) → b ≠ 0 → ↑p ^ b * coeff (teichmullerFun p r) b ^ p ^ (n - b) = 0\n\ncase h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ¬0 ∈ Finset.range (n + 1) → ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ↑(ghostComponent n) (teichmullerFun p r) = r ^ p ^ n",
"tactic": "rw [ghostComponent_apply, aeval_wittPolynomial, Finset.sum_eq_single 0, pow_zero, one_mul,\n tsub_zero]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ coeff (teichmullerFun p r) 0 ^ p ^ n = r ^ p ^ n",
"tactic": "rfl"
},
{
"state_after": "case h₀\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ ↑p ^ i * coeff (teichmullerFun p r) i ^ p ^ (n - i) = 0",
"state_before": "case h₀\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ∀ (b : ℕ), b ∈ Finset.range (n + 1) → b ≠ 0 → ↑p ^ b * coeff (teichmullerFun p r) b ^ p ^ (n - b) = 0",
"tactic": "intro i hi h0"
},
{
"state_after": "case h.e'_2.h.e'_6\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ coeff (teichmullerFun p r) i ^ p ^ (n - i) = 0",
"state_before": "case h₀\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ ↑p ^ i * coeff (teichmullerFun p r) i ^ p ^ (n - i) = 0",
"tactic": "convert mul_zero (M₀ := R) _"
},
{
"state_after": "case h.e'_2.h.e'_5\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ coeff (teichmullerFun p r) i = 0\n\ncase h.e'_2.h.e'_6.convert_2\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ 0 < p ^ (n - i)",
"state_before": "case h.e'_2.h.e'_6\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ coeff (teichmullerFun p r) i ^ p ^ (n - i) = 0",
"tactic": "convert zero_pow (M := R) _"
},
{
"state_after": "case h.e'_2.h.e'_5.zero\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\nhi : Nat.zero ∈ Finset.range (n + 1)\nh0 : Nat.zero ≠ 0\n⊢ coeff (teichmullerFun p r) Nat.zero = 0\n\ncase h.e'_2.h.e'_5.succ\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn n✝ : ℕ\nhi : Nat.succ n✝ ∈ Finset.range (n + 1)\nh0 : Nat.succ n✝ ≠ 0\n⊢ coeff (teichmullerFun p r) (Nat.succ n✝) = 0",
"state_before": "case h.e'_2.h.e'_5\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ coeff (teichmullerFun p r) i = 0",
"tactic": "cases i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.zero\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\nhi : Nat.zero ∈ Finset.range (n + 1)\nh0 : Nat.zero ≠ 0\n⊢ coeff (teichmullerFun p r) Nat.zero = 0",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.succ\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn n✝ : ℕ\nhi : Nat.succ n✝ ∈ Finset.range (n + 1)\nh0 : Nat.succ n✝ ≠ 0\n⊢ coeff (teichmullerFun p r) (Nat.succ n✝) = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6.convert_2\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn i : ℕ\nhi : i ∈ Finset.range (n + 1)\nh0 : i ≠ 0\n⊢ 0 < p ^ (n - i)",
"tactic": "exact pow_pos hp.1.pos _"
},
{
"state_after": "case h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ¬0 < n + 1 → ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"state_before": "case h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ¬0 ∈ Finset.range (n + 1) → ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"tactic": "rw [Finset.mem_range]"
},
{
"state_after": "case h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\nh : ¬0 < n + 1\n⊢ ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"state_before": "case h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\n⊢ ¬0 < n + 1 → ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case h₁\np : ℕ\nR : Type u_1\nS : Type ?u.10109\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\nn : ℕ\nh : ¬0 < n + 1\n⊢ ↑p ^ 0 * coeff (teichmullerFun p r) 0 ^ p ^ (n - 0) = 0",
"tactic": "exact (h (Nat.succ_pos n)).elim"
}
] |
[
76,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
9
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.addValuation.apply
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nhx : x ≠ 0\n⊢ ↑addValuation x = ↑(valuation x)",
"tactic": "simp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx]"
}
] |
[
1156,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasFDerivAt_iff_hasDerivAt
|
[] |
[
195,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.le_floor
|
[] |
[
653,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.measurePreserving_prod_mul
|
[] |
[
92,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
WfDvdMonoid.wellFounded_associates
|
[] |
[
79,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_pos
|
[] |
[
77,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Tactic/NormNum/NatFib.lean
|
Mathlib.Meta.NormNum.isFibAux_two_mul
|
[
{
"state_after": "no goals",
"state_before": "n a b n' a' b' : ℕ\nH : IsFibAux n a b\nhn : 2 * n = n'\nh1 : a * (2 * b - a) = a'\nh2 : a * a + b * b = b'\n⊢ fib n' = a'",
"tactic": "rw [← hn, fib_two_mul, H.1, H.2, ← h1]"
},
{
"state_after": "no goals",
"state_before": "n a b n' a' b' : ℕ\nH : IsFibAux n a b\nhn : 2 * n = n'\nh1 : a * (2 * b - a) = a'\nh2 : a * a + b * b = b'\n⊢ fib (n' + 1) = b'",
"tactic": "rw [← hn, fib_two_mul_add_one, H.1, H.2, pow_two, pow_two, add_comm, h2]"
}
] |
[
34,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.fst_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.5371212\nβ : Type u_2\nβ' : Type ?u.5371218\nγ : Type ?u.5371221\nE : Type ?u.5371224\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nρ : Measure (α × β)\n⊢ ↑↑(fst ρ) univ = ↑↑ρ univ",
"tactic": "rw [fst_apply MeasurableSet.univ, preimage_univ]"
}
] |
[
823,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.mk_add_moveRight_inl
|
[] |
[
1518,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1515,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean
|
TopCat.Presheaf.pushforwardEq_hom_app
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ ((Opens.map f).obj U.unop).op ⟶ ((Opens.map g).obj U.unop).op",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Functor.op (Opens.map f)).obj U ⟶ (Functor.op (Opens.map g)).obj U",
"tactic": "dsimp [Functor.op]"
},
{
"state_after": "case f\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Opens.map g).obj U.unop ⟶ (Opens.map f).obj U.unop",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ ((Opens.map f).obj U.unop).op ⟶ ((Opens.map g).obj U.unop).op",
"tactic": "apply Quiver.Hom.op"
},
{
"state_after": "case f.p\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Opens.map g).obj U.unop = (Opens.map f).obj U.unop",
"state_before": "case f\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Opens.map g).obj U.unop ⟶ (Opens.map f).obj U.unop",
"tactic": "apply eqToHom"
},
{
"state_after": "no goals",
"state_before": "case f.p\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Opens.map g).obj U.unop = (Opens.map f).obj U.unop",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (pushforwardEq h ℱ).hom.app U = ℱ.map (id (eqToHom (_ : (Opens.map g).obj U.unop = (Opens.map f).obj U.unop)).op)",
"tactic": "simp [pushforwardEq]"
}
] |
[
175,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.exists_measurable_superset_forall_eq_trim
|
[
{
"state_after": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhst : ∀ (i : ι), s ⊆ t i\nht : ∀ (i : ι), MeasurableSet (t i)\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"state_before": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"tactic": "choose t hst ht hμt using fun i => (μ i).exists_measurable_superset_eq_trim s"
},
{
"state_after": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nht : ∀ (i : ι), MeasurableSet (t i)\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"state_before": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhst : ∀ (i : ι), s ⊆ t i\nht : ∀ (i : ι), MeasurableSet (t i)\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"tactic": "replace hst := subset_iInter hst"
},
{
"state_after": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"state_before": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nht : ∀ (i : ι), MeasurableSet (t i)\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"tactic": "replace ht := MeasurableSet.iInter ht"
},
{
"state_after": "case refine'_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\ni : ι\n⊢ ↑(μ i) (⋂ (i : ι), t i) ≤ ↑(trim (μ i)) s\n\ncase refine'_2\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\ni : ι\n⊢ ↑(trim (μ i)) s ≤ ↑(μ i) (⋂ (i : ι), t i)",
"state_before": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s",
"tactic": "refine' ⟨⋂ i, t i, hst, ht, fun i => le_antisymm _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\ni : ι\n⊢ ↑(μ i) (⋂ (i : ι), t i) ≤ ↑(trim (μ i)) s\n\ncase refine'_2\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nm : OuterMeasure α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → OuterMeasure α\ns : Set α\nt : ι → Set α\nhμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s\nhst : s ⊆ ⋂ (i : ι), t i\nht : MeasurableSet (⋂ (b : ι), t b)\ni : ι\n⊢ ↑(trim (μ i)) s ≤ ↑(μ i) (⋂ (i : ι), t i)",
"tactic": "exacts [hμt i ▸ (μ i).mono (iInter_subset _ _), (mono' _ hst).trans_eq ((μ i).trim_eq ht)]"
}
] |
[
1732,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1726,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
MulActionHom.ext_iff
|
[] |
[
130,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.coeFn_div
|
[] |
[
762,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.image_relEmbedding_iff
|
[] |
[
146,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Analysis/Convex/Exposed.lean
|
IsExposed.isClosed
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA : Set E\nhA : IsClosed A\nhAB : IsExposed 𝕜 A ∅\n⊢ IsClosed ∅\n\ncase inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B✝ C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhA : IsClosed A\nhB : Set.Nonempty B\n⊢ IsClosed B",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B✝ C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhA : IsClosed A\n⊢ IsClosed B",
"tactic": "obtain rfl | hB := B.eq_empty_or_nonempty"
},
{
"state_after": "case inr.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA : Set E\nhA : IsClosed A\nl : E →L[𝕜] 𝕜\na : 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ a ≤ ↑l x}\nhB : Set.Nonempty {x | x ∈ A ∧ a ≤ ↑l x}\n⊢ IsClosed {x | x ∈ A ∧ a ≤ ↑l x}",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B✝ C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhA : IsClosed A\nhB : Set.Nonempty B\n⊢ IsClosed B",
"tactic": "obtain ⟨l, a, rfl⟩ := hAB.eq_inter_halfspace' hB"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA : Set E\nhA : IsClosed A\nl : E →L[𝕜] 𝕜\na : 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ a ≤ ↑l x}\nhB : Set.Nonempty {x | x ∈ A ∧ a ≤ ↑l x}\n⊢ IsClosed {x | x ∈ A ∧ a ≤ ↑l x}",
"tactic": "exact hA.isClosed_le continuousOn_const l.continuous.continuousOn"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\ninst✝ : OrderClosedTopology 𝕜\nA : Set E\nhA : IsClosed A\nhAB : IsExposed 𝕜 A ∅\n⊢ IsClosed ∅",
"tactic": "simp"
}
] |
[
188,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
11
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.quotEquivQuotMap_symm_apply_mk
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\nI✝ : Ideal R\nf✝ f g : R[X]\nI : Ideal R\n⊢ ↑(AlgEquiv.symm (quotEquivQuotMap f I))\n (↑(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)) =\n ↑(Ideal.Quotient.mk (Ideal.map (of f) I)) (↑(mk f) g)",
"tactic": "rw [AdjoinRoot.quotEquivQuotMap_symm_apply,\n AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk]"
}
] |
[
878,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
873,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.isHomogeneous_one
|
[] |
[
151,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/Basic.lean
|
closure_nonempty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ Set.Nonempty (closure s) ↔ Set.Nonempty s",
"tactic": "simp only [nonempty_iff_ne_empty, Ne.def, closure_empty_iff]"
}
] |
[
500,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.coe_top
|
[] |
[
217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Algebra/Group/Pi.lean
|
Pi.single_div
|
[] |
[
507,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.mem_diff_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : DecidableEq α\nhl₁ : Nodup l₁\n⊢ a ∈ List.diff l₁ l₂ ↔ a ∈ l₁ ∧ ¬a ∈ l₂",
"tactic": "rw [hl₁.diff_eq_filter, mem_filter, decide_eq_true_iff]"
}
] |
[
400,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.add_apply
|
[] |
[
689,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.algebraMap_eq_ofReal
|
[] |
[
108,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasStrictDerivAt.const_sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nc : F\nhf : HasStrictDerivAt f f' x\n⊢ HasStrictDerivAt (fun x => c - f x) (-f') x",
"tactic": "simpa only [sub_eq_add_neg] using hf.neg.const_add c"
}
] |
[
362,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.Nodup.product
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.9045\nr : α → α → Prop\ns t✝ : Multiset α\na : α\nt : Multiset β\nl₁ : List α\nl₂ : List β\nd₁ : Nodup (Quotient.mk (isSetoid α) l₁)\nd₂ : Nodup (Quotient.mk (isSetoid β) l₂)\n⊢ Nodup (Quotient.mk (isSetoid α) l₁ ×ˢ Quotient.mk (isSetoid β) l₂)",
"tactic": "simp [List.Nodup.product d₁ d₂]"
}
] |
[
191,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
11
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.lower
|
[] |
[
129,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
11
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
MulActionHom.congr_fun
|
[] |
[
134,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
11
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.countable_not_continuousAt
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ {x | ¬ContinuousAt f x} ⊆ {x | ¬ContinuousWithinAt f (Ioi x) x} ∪ {x | ¬ContinuousWithinAt f (Iio x) x}",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ Set.Countable {x | ¬ContinuousAt f x}",
"tactic": "apply\n (hf.countable_not_continuousWithinAt_Ioi.union hf.countable_not_continuousWithinAt_Iio).mono\n _"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ ({x | ¬ContinuousWithinAt f (Ioi x) x} ∪ {x | ¬ContinuousWithinAt f (Iio x) x})ᶜ ⊆ {x | ¬ContinuousAt f x}ᶜ",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ {x | ¬ContinuousAt f x} ⊆ {x | ¬ContinuousWithinAt f (Ioi x) x} ∪ {x | ¬ContinuousWithinAt f (Iio x) x}",
"tactic": "refine' compl_subset_compl.1 _"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ {x | ¬ContinuousWithinAt f (Ioi x) x}ᶜ ∩ {x | ¬ContinuousWithinAt f (Iio x) x}ᶜ ⊆ {x | ¬ContinuousAt f x}ᶜ",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ ({x | ¬ContinuousWithinAt f (Ioi x) x} ∪ {x | ¬ContinuousWithinAt f (Iio x) x})ᶜ ⊆ {x | ¬ContinuousAt f x}ᶜ",
"tactic": "simp only [compl_union]"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\nx : α\nhx : x ∈ {x | ¬ContinuousWithinAt f (Ioi x) x}ᶜ\nh'x : x ∈ {x | ¬ContinuousWithinAt f (Iio x) x}ᶜ\n⊢ x ∈ {x | ¬ContinuousAt f x}ᶜ",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ {x | ¬ContinuousWithinAt f (Ioi x) x}ᶜ ∩ {x | ¬ContinuousWithinAt f (Iio x) x}ᶜ ⊆ {x | ¬ContinuousAt f x}ᶜ",
"tactic": "rintro x ⟨hx, h'x⟩"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\nx : α\nhx : ContinuousWithinAt f (Ioi x) x\nh'x : ContinuousWithinAt f (Iio x) x\n⊢ ContinuousAt f x",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\nx : α\nhx : x ∈ {x | ¬ContinuousWithinAt f (Ioi x) x}ᶜ\nh'x : x ∈ {x | ¬ContinuousWithinAt f (Iio x) x}ᶜ\n⊢ x ∈ {x | ¬ContinuousAt f x}ᶜ",
"tactic": "simp only [mem_setOf_eq, Classical.not_not, mem_compl_iff] at hx h'x⊢"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\nx : α\nhx : ContinuousWithinAt f (Ioi x) x\nh'x : ContinuousWithinAt f (Iio x) x\n⊢ ContinuousAt f x",
"tactic": "exact continuousAt_iff_continuous_left'_right'.2 ⟨h'x, hx⟩"
}
] |
[
291,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.ElementaryEmbedding.map_boundedFormula
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\n⊢ BoundedFormula.Realize φ (↑f ∘ v) (↑f ∘ xs) ↔ BoundedFormula.Realize φ v xs",
"tactic": "classical\n rw [← BoundedFormula.realize_restrictFreeVar Set.Subset.rfl, Set.inclusion_eq_id, iff_eq_eq]\n have h :=\n f.map_formula' ((φ.restrictFreeVar id).toFormula.relabel (Fintype.equivFin _))\n (Sum.elim (v ∘ (↑)) xs ∘ (Fintype.equivFin _).symm)\n simp only [Formula.realize_relabel, BoundedFormula.realize_toFormula, iff_eq_eq] at h\n rw [← Function.comp.assoc _ _ (Fintype.equivFin _).symm,\n Function.comp.assoc _ (Fintype.equivFin _).symm (Fintype.equivFin _), Equiv.symm_comp_self,\n Function.comp.right_id, Function.comp.assoc, Sum.elim_comp_inl,\n Function.comp.assoc _ _ Sum.inr, Sum.elim_comp_inr, ← Function.comp.assoc] at h\n refine' h.trans _\n erw [Function.comp.assoc _ _ (Fintype.equivFin _), Equiv.symm_comp_self, Function.comp.right_id,\n Sum.elim_comp_inl, Sum.elim_comp_inr (v ∘ Subtype.val) xs,\n ← Set.inclusion_eq_id (s := (BoundedFormula.freeVarFinset φ : Set α)) Set.Subset.rfl,\n BoundedFormula.realize_restrictFreeVar Set.Subset.rfl]"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\n⊢ BoundedFormula.Realize φ (↑f ∘ v) (↑f ∘ xs) ↔ BoundedFormula.Realize φ v xs",
"tactic": "rw [← BoundedFormula.realize_restrictFreeVar Set.Subset.rfl, Set.inclusion_eq_id, iff_eq_eq]"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n Formula.Realize\n (Formula.relabel (↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)))\n (BoundedFormula.toFormula (BoundedFormula.restrictFreeVar φ id)))\n (↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ↔\n Formula.Realize\n (Formula.relabel (↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)))\n (BoundedFormula.toFormula (BoundedFormula.restrictFreeVar φ id)))\n (Sum.elim (v ∘ Subtype.val) xs ∘ ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"tactic": "have h :=\n f.map_formula' ((φ.restrictFreeVar id).toFormula.relabel (Fintype.equivFin _))\n (Sum.elim (v ∘ (↑)) xs ∘ (Fintype.equivFin _).symm)"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n Formula.Realize\n (Formula.relabel (↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)))\n (BoundedFormula.toFormula (BoundedFormula.restrictFreeVar φ id)))\n (↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ↔\n Formula.Realize\n (Formula.relabel (↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)))\n (BoundedFormula.toFormula (BoundedFormula.restrictFreeVar φ id)))\n (Sum.elim (v ∘ Subtype.val) xs ∘ ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"tactic": "simp only [Formula.realize_relabel, BoundedFormula.realize_toFormula, iff_eq_eq] at h"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((↑f ∘\n Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"tactic": "rw [← Function.comp.assoc _ _ (Fintype.equivFin _).symm,\n Function.comp.assoc _ (Fintype.equivFin _).symm (Fintype.equivFin _), Equiv.symm_comp_self,\n Function.comp.right_id, Function.comp.assoc, Sum.elim_comp_inl,\n Function.comp.assoc _ _ Sum.inr, Sum.elim_comp_inr, ← Function.comp.assoc] at h"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr) =\n BoundedFormula.Realize φ v xs",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize φ v xs",
"tactic": "refine' h.trans _"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.10715\nQ : Type ?u.10718\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\nh :\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id) ((↑f ∘ v) ∘ Subtype.val) (↑f ∘ xs) =\n BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr)\n⊢ BoundedFormula.Realize (BoundedFormula.restrictFreeVar φ id)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inl)\n (((Sum.elim (v ∘ Subtype.val) xs ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n)).symm) ∘\n ↑(Fintype.equivFin ({ x // x ∈ BoundedFormula.freeVarFinset φ } ⊕ Fin n))) ∘\n Sum.inr) =\n BoundedFormula.Realize φ v xs",
"tactic": "erw [Function.comp.assoc _ _ (Fintype.equivFin _), Equiv.symm_comp_self, Function.comp.right_id,\n Sum.elim_comp_inl, Sum.elim_comp_inr (v ∘ Subtype.val) xs,\n ← Set.inclusion_eq_id (s := (BoundedFormula.freeVarFinset φ : Set α)) Set.Subset.rfl,\n BoundedFormula.realize_restrictFreeVar Set.Subset.rfl]"
}
] |
[
104,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Order/Atoms.lean
|
sSup_atoms_eq_top
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.19944\ninst✝¹ : CompleteLattice α\ninst✝ : IsAtomistic α\nx : α\n⊢ x ∈ {a | IsAtom a} ↔ x ∈ {a | IsAtom a ∧ a ≤ ⊤}",
"state_before": "α : Type u_1\nβ : Type ?u.19944\ninst✝¹ : CompleteLattice α\ninst✝ : IsAtomistic α\n⊢ sSup {a | IsAtom a} = ⊤",
"tactic": "refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_atoms_le_eq ⊤)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19944\ninst✝¹ : CompleteLattice α\ninst✝ : IsAtomistic α\nx : α\n⊢ x ∈ {a | IsAtom a} ↔ x ∈ {a | IsAtom a ∧ a ≤ ⊤}",
"tactic": "exact (and_iff_left le_top).symm"
}
] |
[
396,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
src/lean/Init/Control/Lawful.lean
|
StateT.run_seqRight
|
[
{
"state_after": "m : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ run\n (do\n let _ ← x\n y)\n s =\n do\n let p ← run x s\n run y p.snd",
"state_before": "m : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ run (SeqRight.seqRight x fun x => y) s = do\n let p ← run x s\n run y p.snd",
"tactic": "show (x >>= fun _ => y).run s = _"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ run\n (do\n let _ ← x\n y)\n s =\n do\n let p ← run x s\n run y p.snd",
"tactic": "simp"
}
] |
[
282,
7
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
280,
9
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Ico_ae_eq_Icc'
|
[] |
[
2992,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2991,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_lt_div_of_lt_left
|
[] |
[
404,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.ne_of_odd_add
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nh : Odd (m + n)\nhnot : m = n\n⊢ False",
"tactic": "simp [hnot] at h"
}
] |
[
193,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.le_closure_toSubmonoid
|
[] |
[
1305,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1304,
1
] |
Mathlib/Data/Nat/WithBot.lean
|
Nat.WithBot.add_one_le_of_lt
|
[
{
"state_after": "case none\nm : WithBot ℕ\nh : none < m\n⊢ none + 1 ≤ m\n\ncase some\nm : WithBot ℕ\nval✝ : ℕ\nh : some val✝ < m\n⊢ some val✝ + 1 ≤ m",
"state_before": "n m : WithBot ℕ\nh : n < m\n⊢ n + 1 ≤ m",
"tactic": "cases n"
},
{
"state_after": "case some.none\nval✝ : ℕ\nh : some val✝ < none\n⊢ some val✝ + 1 ≤ none\n\ncase some.some\nval✝¹ val✝ : ℕ\nh : some val✝¹ < some val✝\n⊢ some val✝¹ + 1 ≤ some val✝",
"state_before": "case some\nm : WithBot ℕ\nval✝ : ℕ\nh : some val✝ < m\n⊢ some val✝ + 1 ≤ m",
"tactic": "cases m"
},
{
"state_after": "no goals",
"state_before": "case some.none\nval✝ : ℕ\nh : some val✝ < none\n⊢ some val✝ + 1 ≤ none\n\ncase some.some\nval✝¹ val✝ : ℕ\nh : some val✝¹ < some val✝\n⊢ some val✝¹ + 1 ≤ some val✝",
"tactic": "exacts [(not_lt_bot h).elim, WithBot.some_le_some.2 (WithBot.some_lt_some.1 h)]"
},
{
"state_after": "no goals",
"state_before": "case none\nm : WithBot ℕ\nh : none < m\n⊢ none + 1 ≤ m",
"tactic": "exact bot_le"
}
] |
[
90,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_mul_eq_inter'
|
[
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t✝ : Set Ω\nhms : MeasurableSet s\nhcs : ↑↑μ s ≠ 0\nhcs' : ↑↑μ s ≠ ⊤\nt : Set Ω\n⊢ ↑↑(μ[|s]) t * ↑↑μ s = ↑↑μ (s ∩ t)",
"tactic": "rw [cond_apply μ hms t, mul_comm, ← mul_assoc, ENNReal.mul_inv_cancel hcs hcs', one_mul]"
}
] |
[
146,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
MulOpposite.dist_unop
|
[] |
[
1657,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1657,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.mul_div_mod_by_monic_cancel_left
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ q * p /ₘ q = p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n⊢ q * p /ₘ q = p",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ degree 0 < degree q",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ q * p /ₘ q = p",
"tactic": "refine' (div_modByMonic_unique _ 0 hmo ⟨by rw [zero_add], _⟩).1"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ ⊥ < degree q",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ degree 0 < degree q",
"tactic": "rw [degree_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ ⊥ < degree q",
"tactic": "exact Ne.bot_lt fun h => hmo.ne_zero (degree_eq_bot.1 h)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q✝ p q : R[X]\nhmo : Monic q\n✝ : Nontrivial R\n⊢ 0 + q * p = q * p",
"tactic": "rw [zero_add]"
}
] |
[
486,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.seq_def
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.258305\nι : Sort ?u.258308\nι' : Sort ?u.258311\nι₂ : Sort ?u.258314\nκ : ι → Sort ?u.258319\nκ₁ : ι → Sort ?u.258324\nκ₂ : ι → Sort ?u.258329\nκ' : ι' → Sort ?u.258334\ns : Set (α → β)\nt : Set α\n⊢ ∀ (x : β), x ∈ seq s t ↔ x ∈ ⋃ (f : α → β) (_ : f ∈ s), f '' t",
"tactic": "simp [seq]"
}
] |
[
1951,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1950,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.