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/Data/Polynomial/FieldDivision.lean
|
Polynomial.prime_of_degree_eq_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp1 : degree p = 1\nhp0 : p = 0\n⊢ ¬degree 0 = 1",
"tactic": "simp"
}
] |
[
480,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.forall_sups_iff
|
[] |
[
109,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.inj_on_iUnion_of_directed
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx : x ∈ ⋃ (i : ι), s i\ny : α\nhy : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\n⊢ x = y",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\n⊢ InjOn f (⋃ (i : ι), s i)",
"tactic": "intro x hx y hy hxy"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\n⊢ x = y",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx : x ∈ ⋃ (i : ι), s i\ny : α\nhy : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\n⊢ x = y",
"tactic": "rcases mem_iUnion.1 hx with ⟨i, hx⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy✝ : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\nj : ι\nhy : y ∈ s j\n⊢ x = y",
"state_before": "case intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\n⊢ x = y",
"tactic": "rcases mem_iUnion.1 hy with ⟨j, hy⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy✝ : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\nj : ι\nhy : y ∈ s j\nk : ι\nhi : s i ⊆ s k\nhj : s j ⊆ s k\n⊢ x = y",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy✝ : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\nj : ι\nhy : y ∈ s j\n⊢ x = y",
"tactic": "rcases hs i j with ⟨k, hi, hj⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.191837\nι : Sort u_2\nι' : Sort ?u.191843\nι₂ : Sort ?u.191846\nκ : ι → Sort ?u.191851\nκ₁ : ι → Sort ?u.191856\nκ₂ : ι → Sort ?u.191861\nκ' : ι' → Sort ?u.191866\ns : ι → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ (i : ι), s i\ny : α\nhy✝ : y ∈ ⋃ (i : ι), s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\nj : ι\nhy : y ∈ s j\nk : ι\nhi : s i ⊆ s k\nhj : s j ⊆ s k\n⊢ x = y",
"tactic": "exact hf k (hi hx) (hj hy) hxy"
}
] |
[
1575,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1569,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.comp
|
[] |
[
531,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
8
] |
Mathlib/Order/BoundedOrder.lean
|
sup_eq_bot_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.26825\nδ : Type ?u.26828\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\na b : α\n⊢ a ≤ ⊥ ∧ b ≤ ⊥ ↔ a = ⊥ ∧ b = ⊥",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.26825\nδ : Type ?u.26828\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\na b : α\n⊢ a ⊔ b = ⊥ ↔ a = ⊥ ∧ b = ⊥",
"tactic": "rw [eq_bot_iff, sup_le_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.26825\nδ : Type ?u.26828\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\na b : α\n⊢ a ≤ ⊥ ∧ b ≤ ⊥ ↔ a = ⊥ ∧ b = ⊥",
"tactic": "simp"
}
] |
[
481,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.infinite_of_forall_exists_lt
|
[] |
[
1417,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1416,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.map_pow
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.980529\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.980535\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.980547\nQ : Type ?u.980550\nS : Type ?u.980553\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.981680\nQ' : Type ?u.981683\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] M\ng : N →ₗ[R] N\n⊢ map f g ^ Nat.zero = map (f ^ Nat.zero) (g ^ Nat.zero)\n\ncase succ\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.980529\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.980535\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.980547\nQ : Type ?u.980550\nS : Type ?u.980553\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.981680\nQ' : Type ?u.981683\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nn : ℕ\nih : map f g ^ n = map (f ^ n) (g ^ n)\n⊢ map f g ^ Nat.succ n = map (f ^ Nat.succ n) (g ^ Nat.succ n)",
"state_before": "R : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.980529\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.980535\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.980547\nQ : Type ?u.980550\nS : Type ?u.980553\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.981680\nQ' : Type ?u.981683\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nn : ℕ\n⊢ map f g ^ n = map (f ^ n) (g ^ n)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.980529\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.980535\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.980547\nQ : Type ?u.980550\nS : Type ?u.980553\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.981680\nQ' : Type ?u.981683\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] M\ng : N →ₗ[R] N\n⊢ map f g ^ Nat.zero = map (f ^ Nat.zero) (g ^ Nat.zero)",
"tactic": "simp only [Nat.zero_eq, pow_zero, map_one]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.980529\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.980535\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.980547\nQ : Type ?u.980550\nS : Type ?u.980553\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.981680\nQ' : Type ?u.981683\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nn : ℕ\nih : map f g ^ n = map (f ^ n) (g ^ n)\n⊢ map f g ^ Nat.succ n = map (f ^ Nat.succ n) (g ^ Nat.succ n)",
"tactic": "simp only [pow_succ', ih, map_mul]"
}
] |
[
788,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
11
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.card_image₂_iff
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.9016\nβ : Type u_3\nβ' : Type ?u.9022\nγ : Type u_1\nγ' : Type ?u.9028\nδ : Type ?u.9031\nδ' : Type ?u.9034\nε : Type ?u.9037\nε' : Type ?u.9040\nζ : Type ?u.9043\nζ' : Type ?u.9046\nν : Type ?u.9049\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ card (image₂ f s t) = card (s ×ˢ t) ↔ InjOn (fun x => f x.fst x.snd) ↑(s ×ˢ t)",
"state_before": "α : Type u_2\nα' : Type ?u.9016\nβ : Type u_3\nβ' : Type ?u.9022\nγ : Type u_1\nγ' : Type ?u.9028\nδ : Type ?u.9031\nδ' : Type ?u.9034\nε : Type ?u.9037\nε' : Type ?u.9040\nζ : Type ?u.9043\nζ' : Type ?u.9046\nν : Type ?u.9049\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ card (image₂ f s t) = card s * card t ↔ InjOn (fun x => f x.fst x.snd) (↑s ×ˢ ↑t)",
"tactic": "rw [← card_product, ← coe_product]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.9016\nβ : Type u_3\nβ' : Type ?u.9022\nγ : Type u_1\nγ' : Type ?u.9028\nδ : Type ?u.9031\nδ' : Type ?u.9034\nε : Type ?u.9037\nε' : Type ?u.9040\nζ : Type ?u.9043\nζ' : Type ?u.9046\nν : Type ?u.9049\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ card (image₂ f s t) = card (s ×ˢ t) ↔ InjOn (fun x => f x.fst x.snd) ↑(s ×ˢ t)",
"tactic": "exact card_image_iff"
}
] |
[
65,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Set/BoolIndicator.lean
|
Set.not_mem_iff_boolIndicator
|
[
{
"state_after": "α : Type u_1\ns : Set α\nx : α\n⊢ ¬x ∈ s ↔ (if x ∈ s then true else false) = false",
"state_before": "α : Type u_1\ns : Set α\nx : α\n⊢ ¬x ∈ s ↔ boolIndicator s x = false",
"tactic": "unfold boolIndicator"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Set α\nx : α\n⊢ ¬x ∈ s ↔ (if x ∈ s then true else false) = false",
"tactic": "split_ifs with h <;> simp [h]"
}
] |
[
37,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_symmDiff_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.68209\nα : Type u_1\nβ : Type ?u.68215\nπ : ι → Type ?u.68220\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ (c \\ (a ⊔ b) ⊔ c ⊓ a ⊓ b) = a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ c \\ (a ⊔ b) ⊔ a ⊓ b ⊓ c",
"tactic": "ac_rfl"
}
] |
[
465,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Topology/UniformSpace/Compact.lean
|
Continuous.uniformContinuous_of_tendsto_cocompact
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\n⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\n⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α",
"tactic": "obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\n⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\n⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α",
"tactic": "obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx <| mem_nhds_left _ ht)"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\n⊢ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r} ⊆ {x | (f x.fst, f x.snd) ∈ r}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\n⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α",
"tactic": "apply\n mem_of_superset\n (symmetrize_mem_uniformity <|\n (hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.continuousAt) <|\n symmetrize_mem_uniformity hr)"
},
{
"state_after": "case intro.intro.intro.intro.intro.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\n⊢ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r} ⊆ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "rintro ⟨b₁, b₂⟩ h"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : b₁ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"state_before": "case intro.intro.intro.intro.intro.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "by_cases h₁ : b₁ ∈ s"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : b₂ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : ¬b₂ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "by_cases h₂ : b₂ ∈ s"
},
{
"state_after": "case neg.a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : ¬b₂ ∈ s\n⊢ (f (b₁, b₂).fst, f (b₁, b₂).snd) ∈ t ○ t",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : ¬b₂ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "apply htr"
},
{
"state_after": "no goals",
"state_before": "case neg.a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : ¬b₂ ∈ s\n⊢ (f (b₁, b₂).fst, f (b₁, b₂).snd) ∈ t ○ t",
"tactic": "exact ⟨x, htsymm.mk_mem_comm.1 (hst h₁), hst h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : b₁ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "exact (h.1 h₁).1"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44329\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SymmetricRel t\nhtr : t ○ t ⊆ r\ns : Set α\nhs : IsCompact s\nhst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}\nb₁ b₂ : α\nh : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}\nh₁ : ¬b₁ ∈ s\nh₂ : b₂ ∈ s\n⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}",
"tactic": "exact (h.2 h₂).2"
}
] |
[
221,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.of_thinkN_terminates
|
[] |
[
426,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.degree_C_mul_T
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ Finset.max (↑C a * T n).support = ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ degree (↑C a * T n) = ↑n",
"tactic": "rw [degree]"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\nthis : (↑C a * T n).support = {n}\n⊢ Finset.max (↑C a * T n).support = ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ Finset.max (↑C a * T n).support = ↑n",
"tactic": "have : Finsupp.support (C a * T n) = {n} := by\n refine' support_eq_singleton.mpr _\n rw [← single_eq_C_mul_T]\n simp only [single_eq_same, a0, Ne.def, not_false_iff, eq_self_iff_true, and_self_iff]"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\nthis : (↑C a * T n).support = {n}\n⊢ Finset.max {n} = ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\nthis : (↑C a * T n).support = {n}\n⊢ Finset.max (↑C a * T n).support = ↑n",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\nthis : (↑C a * T n).support = {n}\n⊢ Finset.max {n} = ↑n",
"tactic": "exact Finset.max_singleton"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ ↑(↑C a * T n) n ≠ 0 ∧ ↑C a * T n = Finsupp.single n (↑(↑C a * T n) n)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ (↑C a * T n).support = {n}",
"tactic": "refine' support_eq_singleton.mpr _"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ ↑(Finsupp.single n a) n ≠ 0 ∧ Finsupp.single n a = Finsupp.single n (↑(Finsupp.single n a) n)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ ↑(↑C a * T n) n ≠ 0 ∧ ↑C a * T n = Finsupp.single n (↑(↑C a * T n) n)",
"tactic": "rw [← single_eq_C_mul_T]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\na : R\na0 : a ≠ 0\n⊢ ↑(Finsupp.single n a) n ≠ 0 ∧ Finsupp.single n a = Finsupp.single n (↑(Finsupp.single n a) n)",
"tactic": "simp only [single_eq_same, a0, Ne.def, not_false_iff, eq_self_iff_true, and_self_iff]"
}
] |
[
511,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
balancedCore_nonempty_iff
|
[] |
[
138,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSpace.map_inf
|
[] |
[
171,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Analysis/NormedSpace/IndicatorFunction.lean
|
norm_indicator_le_norm_self
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\ns t : Set α\nf : α → E\na : α\n⊢ indicator s (fun a => ‖f a‖) a ≤ ‖f a‖",
"state_before": "α : Type u_2\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\ns t : Set α\nf : α → E\na : α\n⊢ ‖indicator s f a‖ ≤ ‖f a‖",
"tactic": "rw [norm_indicator_eq_indicator_norm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\ns t : Set α\nf : α → E\na : α\n⊢ indicator s (fun a => ‖f a‖) a ≤ ‖f a‖",
"tactic": "apply indicator_norm_le_norm_self"
}
] |
[
49,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup'_congr
|
[
{
"state_after": "F : Type ?u.262044\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.262053\nι : Type ?u.262056\nκ : Type ?u.262059\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : Finset.Nonempty t\nh₂ : ∀ (x : β), x ∈ t → f x = g x\n⊢ sup' t H f = sup' t (_ : Finset.Nonempty t) g",
"state_before": "F : Type ?u.262044\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.262053\nι : Type ?u.262056\nκ : Type ?u.262059\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf✝ : β → α\nt : Finset β\nf g : β → α\nh₁ : s = t\nh₂ : ∀ (x : β), x ∈ s → f x = g x\n⊢ sup' s H f = sup' t (_ : Finset.Nonempty t) g",
"tactic": "subst s"
},
{
"state_after": "F : Type ?u.262044\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.262053\nι : Type ?u.262056\nκ : Type ?u.262059\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : Finset.Nonempty t\nh₂ : ∀ (x : β), x ∈ t → f x = g x\nc : α\n⊢ sup' t H f ≤ c ↔ sup' t (_ : Finset.Nonempty t) g ≤ c",
"state_before": "F : Type ?u.262044\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.262053\nι : Type ?u.262056\nκ : Type ?u.262059\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : Finset.Nonempty t\nh₂ : ∀ (x : β), x ∈ t → f x = g x\n⊢ sup' t H f = sup' t (_ : Finset.Nonempty t) g",
"tactic": "refine' eq_of_forall_ge_iff fun c => _"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.262044\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.262053\nι : Type ?u.262056\nκ : Type ?u.262059\ninst✝ : SemilatticeSup α\nf✝ : β → α\nt : Finset β\nf g : β → α\nH : Finset.Nonempty t\nh₂ : ∀ (x : β), x ∈ t → f x = g x\nc : α\n⊢ sup' t H f ≤ c ↔ sup' t (_ : Finset.Nonempty t) g ≤ c",
"tactic": "simp (config := { contextual := true }) only [sup'_le_iff, h₂]"
}
] |
[
880,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
876,
1
] |
Mathlib/Algebra/DirectSum/Ring.lean
|
DirectSum.mulHom_apply
|
[] |
[
207,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.eq_iInf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.61688\nγ : Type ?u.61691\nι : Sort u_2\nι' : Sort ?u.61697\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nh : HasBasis l (fun x => True) s\n⊢ l = ⨅ (i : ι), 𝓟 (s i)",
"tactic": "simpa only [iInf_true] using h.eq_biInf"
}
] |
[
758,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
gramSchmidtOrthonormalBasis_apply
|
[] |
[
353,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.le_comap_of_map_le
|
[] |
[
1427,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1426,
1
] |
Mathlib/Order/InitialSeg.lean
|
PrincipalSeg.codRestrict_apply
|
[] |
[
404,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_of_Iic
|
[
{
"state_after": "case hf\nα : Type u_2\nβ : Type ?u.1088754\nγ : Type ?u.1088757\nγ₂ : Type ?u.1088760\nδ : Type u_1\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)\n⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)",
"state_before": "α : Type u_2\nβ : Type ?u.1088754\nγ : Type ?u.1088757\nγ₂ : Type ?u.1088760\nδ : Type u_1\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)\n⊢ Measurable f",
"tactic": "apply measurable_of_Ioi"
},
{
"state_after": "case hf\nα : Type u_2\nβ : Type ?u.1088754\nγ : Type ?u.1088757\nγ₂ : Type ?u.1088760\nδ : Type u_1\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)\n⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)",
"state_before": "case hf\nα : Type u_2\nβ : Type ?u.1088754\nγ : Type ?u.1088757\nγ₂ : Type ?u.1088760\nδ : Type u_1\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)\n⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)",
"tactic": "simp_rw [← compl_Iic, preimage_compl, MeasurableSet.compl_iff]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_2\nβ : Type ?u.1088754\nγ : Type ?u.1088757\nγ₂ : Type ?u.1088760\nδ : Type u_1\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)\n⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)",
"tactic": "assumption"
}
] |
[
1066,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1063,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_dvd_iff_pow_eq_one
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : orderOf x ∣ n\n⊢ x ^ n = 1",
"tactic": "rw [pow_eq_mod_orderOf, Nat.mod_eq_zero_of_dvd h, _root_.pow_zero]"
}
] |
[
248,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.prod_le_prod_iff_le
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ p ≤ q",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\n⊢ prod p ≤ prod q → p ≤ q",
"tactic": "rintro ⟨c, eqc⟩"
},
{
"state_after": "case intro.refine'_1\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\n⊢ Irreducible x\n\ncase intro.refine'_2\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ prod q = prod (p + factors c)",
"state_before": "case intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ p ≤ q",
"tactic": "refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩"
},
{
"state_after": "case intro.refine'_1.inl\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\nh : x ∈ p\n⊢ Irreducible x\n\ncase intro.refine'_1.inr\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\nh : x ∈ factors c\n⊢ Irreducible x",
"state_before": "case intro.refine'_1\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\n⊢ Irreducible x",
"tactic": "obtain h | h := Multiset.mem_add.1 hx"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1.inl\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\nh : x ∈ p\n⊢ Irreducible x",
"tactic": "exact hp x h"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1.inr\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nx : Associates α\nhx : x ∈ p + factors c\nh : x ∈ factors c\n⊢ Irreducible x",
"tactic": "exact irreducible_of_factor _ h"
},
{
"state_after": "case intro.refine'_2\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ prod p * c = prod p * prod (factors c)",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ prod q = prod (p + factors c)",
"tactic": "rw [eqc, Multiset.prod_add]"
},
{
"state_after": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ c = prod (factors c)",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ prod p * c = prod p * prod (factors c)",
"tactic": "congr"
},
{
"state_after": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nhc : c = 0\n⊢ False",
"state_before": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\n⊢ c = prod (factors c)",
"tactic": "refine' associated_iff_eq.mp (factors_prod fun hc => _).symm"
},
{
"state_after": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nhc : c = 0\n⊢ 0 ∈ q",
"state_before": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nhc : c = 0\n⊢ False",
"tactic": "refine' not_irreducible_zero (hq _ _)"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2.e_a\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\np q : Multiset (Associates α)\nhp : ∀ (a : Associates α), a ∈ p → Irreducible a\nhq : ∀ (a : Associates α), a ∈ q → Irreducible a\nc : Associates α\neqc : prod q = prod p * c\nhc : c = 0\n⊢ 0 ∈ q",
"tactic": "rw [← prod_eq_zero_iff, eqc, hc, MulZeroClass.mul_zero]"
}
] |
[
1391,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1376,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.ascFactorial_eq_factorial_mul_choose
|
[
{
"state_after": "n k : ℕ\n⊢ ascFactorial n k = choose (n + k) k * k !",
"state_before": "n k : ℕ\n⊢ ascFactorial n k = k ! * choose (n + k) k",
"tactic": "rw [mul_comm]"
},
{
"state_after": "n k : ℕ\n⊢ ascFactorial n k * (n + k - k)! = choose (n + k) k * k ! * (n + k - k)!",
"state_before": "n k : ℕ\n⊢ ascFactorial n k = choose (n + k) k * k !",
"tactic": "apply mul_right_cancel₀ (factorial_ne_zero (n + k - k))"
},
{
"state_after": "n k : ℕ\n⊢ k ≤ n + k",
"state_before": "n k : ℕ\n⊢ ascFactorial n k * (n + k - k)! = choose (n + k) k * k ! * (n + k - k)!",
"tactic": "rw [choose_mul_factorial_mul_factorial, add_tsub_cancel_right, ← factorial_mul_ascFactorial,\n mul_comm]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ k ≤ n + k",
"tactic": "exact Nat.le_add_left k n"
}
] |
[
244,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Tactic/Ring/RingNF.lean
|
Mathlib.Tactic.RingNF.nat_rawCast_0
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ Nat.rawCast 0 = 0",
"tactic": "simp"
}
] |
[
114,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.not_isFiniteMeasure_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.604437\nγ : Type ?u.604440\nδ : Type ?u.604443\nι : Type ?u.604446\nR : Type ?u.604449\nR' : Type ?u.604452\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : ¬IsFiniteMeasure μ\n⊢ ↑↑μ univ = ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.604437\nγ : Type ?u.604440\nδ : Type ?u.604443\nι : Type ?u.604446\nR : Type ?u.604449\nR' : Type ?u.604452\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\n⊢ ¬IsFiniteMeasure μ ↔ ↑↑μ univ = ⊤",
"tactic": "refine' ⟨fun h => _, fun h => fun h' => h'.measure_univ_lt_top.ne h⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.604437\nγ : Type ?u.604440\nδ : Type ?u.604443\nι : Type ?u.604446\nR : Type ?u.604449\nR' : Type ?u.604452\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : ¬IsFiniteMeasure μ\nh' : ¬↑↑μ univ = ⊤\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.604437\nγ : Type ?u.604440\nδ : Type ?u.604443\nι : Type ?u.604446\nR : Type ?u.604449\nR' : Type ?u.604452\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : ¬IsFiniteMeasure μ\n⊢ ↑↑μ univ = ⊤",
"tactic": "by_contra h'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.604437\nγ : Type ?u.604440\nδ : Type ?u.604443\nι : Type ?u.604446\nR : Type ?u.604449\nR' : Type ?u.604452\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : ¬IsFiniteMeasure μ\nh' : ¬↑↑μ univ = ⊤\n⊢ False",
"tactic": "exact h ⟨lt_top_iff_ne_top.mpr h'⟩"
}
] |
[
3042,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3039,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
Submodule.span_algebraMap_image_of_tower
|
[] |
[
340,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_smul
|
[] |
[
248,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
eq_one_of_inv_eq'
|
[] |
[
1129,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1121,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean
|
Metric.Sigma.dist_ne
|
[] |
[
336,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translationNumber_eq_of_tendsto_aux
|
[] |
[
653,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.preimage_closure
|
[] |
[
1842,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1841,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisInsertion.leftInverse_l_u
|
[] |
[
522,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.mul_inv_cancel
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬LimZero f\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv (↑(f * inv f hf - 1) j) < ε",
"tactic": "simpa [(abv_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using ε0"
}
] |
[
646,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Order/CompleteLattice.lean
|
disjoint_sSup_left
|
[] |
[
1950,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1948,
1
] |
Mathlib/Data/Set/Pointwise/Finite.lean
|
Group.card_pow_eq_card_pow_card_univ
|
[
{
"state_after": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩"
},
{
"state_after": "case pos\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : S = ∅\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)\n\ncase neg\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "by_cases hS : S = ∅"
},
{
"state_after": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"state_before": "case neg\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS"
},
{
"state_after": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],\n (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by\n refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _\n rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc\n exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))"
},
{
"state_after": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=\n monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb"
},
{
"state_after": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\n⊢ ∀ (b : G), b ∈ S ^ (n + 2) → a⁻¹ * b ∈ S ^ (n + 1)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))\n fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)"
},
{
"state_after": "case h₂\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\n⊢ {a} * S ^ n = S ^ (n + 1)\n\ncase neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\n⊢ ∀ (b : G), b ∈ S ^ (n + 2) → a⁻¹ * b ∈ S ^ (n + 1)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\n⊢ ∀ (b : G), b ∈ S ^ (n + 2) → a⁻¹ * b ∈ S ^ (n + 1)",
"tactic": "replace h₂ : {a} * S ^ n = S ^ (n + 1)"
},
{
"state_after": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\n⊢ ∀ (b : G), b ∈ {a} * S ^ (n + 1) → a⁻¹ * b ∈ S ^ (n + 1)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\n⊢ ∀ (b : G), b ∈ S ^ (n + 2) → a⁻¹ * b ∈ S ^ (n + 1)",
"tactic": "rw [pow_succ', ← h₂, mul_assoc, ← pow_succ', h₂]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\nb c : G\nhb : b ∈ {a}\nhc : c ∈ S ^ (n + 1)\n⊢ a⁻¹ * (fun x x_1 => x * x_1) b c ∈ S ^ (n + 1)",
"state_before": "case neg.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\n⊢ ∀ (b : G), b ∈ {a} * S ^ (n + 1) → a⁻¹ * b ∈ S ^ (n + 1)",
"tactic": "rintro _ ⟨b, c, hb, hc, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nh₂ : {a} * S ^ n = S ^ (n + 1)\nb c : G\nhb : b ∈ {a}\nhc : c ∈ S ^ (n + 1)\n⊢ a⁻¹ * (fun x x_1 => x * x_1) b c ∈ S ^ (n + 1)",
"tactic": "rwa [Set.mem_singleton_iff.mp hb, inv_mul_cancel_left]"
},
{
"state_after": "case pos\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : S = ∅\nk : ℕ\nhk : Fintype.card G ≤ k\n⊢ ↑(S ^ k) ≃ ↑(S ^ Fintype.card G)",
"state_before": "case pos\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : S = ∅\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"tactic": "refine' fun k hk ↦ Fintype.card_congr _"
},
{
"state_after": "no goals",
"state_before": "case pos\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : S = ∅\nk : ℕ\nhk : Fintype.card G ≤ k\n⊢ ↑(S ^ k) ≃ ↑(S ^ Fintype.card G)",
"tactic": "rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]"
},
{
"state_after": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na✝ : G\nha : a✝ ∈ S\na : G\ns t : Set G\nx✝¹ : Fintype ↑s\nx✝ : Fintype ↑t\nh : ∀ (b : G), b ∈ s → a * b ∈ t\n⊢ Function.Injective fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) }",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\n⊢ ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t",
"tactic": "refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _"
},
{
"state_after": "case mk.mk\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na✝ : G\nha : a✝ ∈ S\na : G\ns t : Set G\nx✝¹ : Fintype ↑s\nx✝ : Fintype ↑t\nh : ∀ (b : G), b ∈ s → a * b ∈ t\nb : G\nhb : b ∈ s\nc : G\nhc : c ∈ s\nhbc :\n (fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) })\n { val := b, property := hb } =\n (fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) })\n { val := c, property := hc }\n⊢ { val := b, property := hb } = { val := c, property := hc }",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na✝ : G\nha : a✝ ∈ S\na : G\ns t : Set G\nx✝¹ : Fintype ↑s\nx✝ : Fintype ↑t\nh : ∀ (b : G), b ∈ s → a * b ∈ t\n⊢ Function.Injective fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) }",
"tactic": "rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na✝ : G\nha : a✝ ∈ S\na : G\ns t : Set G\nx✝¹ : Fintype ↑s\nx✝ : Fintype ↑t\nh : ∀ (b : G), b ∈ s → a * b ∈ t\nb : G\nhb : b ∈ s\nc : G\nhc : c ∈ s\nhbc :\n (fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) })\n { val := b, property := hb } =\n (fun x =>\n match x with\n | { val := b, property := hb } => { val := a * b, property := (_ : a * b ∈ t) })\n { val := c, property := hc }\n⊢ { val := b, property := hb } = { val := c, property := hc }",
"tactic": "exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))"
},
{
"state_after": "case h₂\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ {a} * S ^ n = S ^ (n + 1)",
"state_before": "case h₂\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\n⊢ {a} * S ^ n = S ^ (n + 1)",
"tactic": "have : Fintype (Set.singleton a * S ^ n) := by\n classical!\n apply fintypeMul"
},
{
"state_after": "case h₂.refine'_1\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ {a} * S ^ n ⊆ S ^ (n + 1)\n\ncase h₂.refine'_2\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ Fintype.card ↑(S ^ n) ≤ Fintype.card ↑({a} * S ^ n)",
"state_before": "case h₂\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ {a} * S ^ n = S ^ (n + 1)",
"tactic": "refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)"
},
{
"state_after": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nem✝ : (a : Prop) → Decidable a\n⊢ Fintype ↑(Set.singleton a * S ^ n)",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\n⊢ Fintype ↑(Set.singleton a * S ^ n)",
"tactic": "classical!"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nem✝ : (a : Prop) → Decidable a\n⊢ Fintype ↑(Set.singleton a * S ^ n)",
"tactic": "apply fintypeMul"
},
{
"state_after": "no goals",
"state_before": "case h₂.refine'_1\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ {a} * S ^ n ⊆ S ^ (n + 1)",
"tactic": "exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl"
},
{
"state_after": "no goals",
"state_before": "case h₂.refine'_2\nF : Type ?u.35675\nα : Type ?u.35678\nβ : Type ?u.35681\nγ : Type ?u.35684\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\nhG : 0 < Fintype.card G\nhS : ¬S = ∅\na : G\nha : a ∈ S\nkey :\n ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],\n (∀ (b : G), b ∈ s → a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t\nmono : Monotone fun n => Fintype.card ↑(S ^ n)\nn : ℕ\nh : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))\nthis : Fintype ↑(Set.singleton a * S ^ n)\n⊢ Fintype.card ↑(S ^ n) ≤ Fintype.card ↑({a} * S ^ n)",
"tactic": "convert key a (S ^ n) ({a} * S ^ n) fun b hb ↦ Set.mul_mem_mul (Set.mem_singleton a) hb"
}
] |
[
203,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.balancedSz_down
|
[] |
[
217,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.one_comp
|
[] |
[
69,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Topology/LocallyConstant/Algebra.lean
|
LocallyConstant.coe_inv
|
[] |
[
46,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.AEDisjoint.preimage
|
[] |
[
2674,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2672,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_neg
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.285603\nE : Type u_3\nEₗ : Type ?u.285609\nF : Type u_4\nFₗ : Type ?u.285615\nG : Type ?u.285618\nGₗ : Type ?u.285621\n𝓕 : Type ?u.285624\ninst✝¹⁵ : SeminormedAddCommGroup E\ninst✝¹⁴ : SeminormedAddCommGroup Eₗ\ninst✝¹³ : SeminormedAddCommGroup F\ninst✝¹² : SeminormedAddCommGroup Fₗ\ninst✝¹¹ : SeminormedAddCommGroup G\ninst✝¹⁰ : SeminormedAddCommGroup Gₗ\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : NontriviallyNormedField 𝕜₃\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 Eₗ\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜 Fₗ\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : E →SL[σ₁₂] F\n⊢ ‖-f‖ = ‖f‖",
"tactic": "simp only [norm_def, neg_apply, norm_neg]"
}
] |
[
187,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.unbounded_of_tendsto_atTop'
|
[] |
[
1708,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1706,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.union_add_distrib
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.179823\nγ : Type ?u.179826\ninst✝ : DecidableEq α\ns✝ t✝ u✝ : Multiset α\na b : α\ns t u : Multiset α\n⊢ s + u - (t + u) = s - t",
"tactic": "rw [add_comm t, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]"
}
] |
[
1848,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1846,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiag_sub
|
[] |
[
601,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
eq_of_zero_eq_one
|
[] |
[
136,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Set/Pointwise/BigOperators.lean
|
Set.list_prod_mem_list_prod
|
[
{
"state_after": "case nil\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nhg : ∀ (i : ι), i ∈ [] → g i ∈ f i\n⊢ List.prod (List.map g []) ∈ List.prod (List.map f [])\n\ncase cons\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nh : ι\ntl : List ι\nih : (∀ (i : ι), i ∈ tl → g i ∈ f i) → List.prod (List.map g tl) ∈ List.prod (List.map f tl)\nhg : ∀ (i : ι), i ∈ h :: tl → g i ∈ f i\n⊢ List.prod (List.map g (h :: tl)) ∈ List.prod (List.map f (h :: tl))",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg : ∀ (i : ι), i ∈ t → g i ∈ f i\n⊢ List.prod (List.map g t) ∈ List.prod (List.map f t)",
"tactic": "induction' t with h tl ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nhg : ∀ (i : ι), i ∈ [] → g i ∈ f i\n⊢ List.prod (List.map g []) ∈ List.prod (List.map f [])",
"tactic": "simp_rw [List.map_nil, List.prod_nil, Set.mem_one]"
},
{
"state_after": "case cons\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nh : ι\ntl : List ι\nih : (∀ (i : ι), i ∈ tl → g i ∈ f i) → List.prod (List.map g tl) ∈ List.prod (List.map f tl)\nhg : ∀ (i : ι), i ∈ h :: tl → g i ∈ f i\n⊢ g h * List.prod (List.map g tl) ∈ f h * List.prod (List.map f tl)",
"state_before": "case cons\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nh : ι\ntl : List ι\nih : (∀ (i : ι), i ∈ tl → g i ∈ f i) → List.prod (List.map g tl) ∈ List.prod (List.map f tl)\nhg : ∀ (i : ι), i ∈ h :: tl → g i ∈ f i\n⊢ List.prod (List.map g (h :: tl)) ∈ List.prod (List.map f (h :: tl))",
"tactic": "simp_rw [List.map_cons, List.prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type u_1\nα : Type u_2\nβ : Type ?u.67083\nF : Type ?u.67086\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nt : List ι\nf : ι → Set α\ng : ι → α\nhg✝ : ∀ (i : ι), i ∈ t → g i ∈ f i\nh : ι\ntl : List ι\nih : (∀ (i : ι), i ∈ tl → g i ∈ f i) → List.prod (List.map g tl) ∈ List.prod (List.map f tl)\nhg : ∀ (i : ι), i ∈ h :: tl → g i ∈ f i\n⊢ g h * List.prod (List.map g tl) ∈ f h * List.prod (List.map f tl)",
"tactic": "exact mul_mem_mul (hg h <| List.mem_cons_self _ _)\n (ih fun i hi ↦ hg i <| List.mem_cons_of_mem _ hi)"
}
] |
[
105,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_half
|
[
{
"state_after": "case intro\nα : Type ?u.327971\nβ : Type ?u.327974\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ ↑a - ↑a / 2 = ↑a / 2",
"state_before": "α : Type ?u.327971\nβ : Type ?u.327974\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a ≠ ⊤\n⊢ a - a / 2 = a / 2",
"tactic": "lift a to ℝ≥0 using h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.327971\nβ : Type ?u.327974\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ ↑a - ↑a / 2 = ↑a / 2",
"tactic": "exact sub_eq_of_add_eq (mul_ne_top coe_ne_top <| by simp) (ENNReal.add_halves a)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.327971\nβ : Type ?u.327974\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\n⊢ 2⁻¹ ≠ ⊤",
"tactic": "simp"
}
] |
[
1764,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1762,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsIntegralClosure.isFractionRing_of_algebraic
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.445070\ninst✝¹⁶ : CommRing R\nM : Submonoid R\nS : Type ?u.445277\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Type ?u.445533\ninst✝¹³ : CommRing P\nA : Type u_1\nK : Type ?u.445542\ninst✝¹² : CommRing A\ninst✝¹¹ : IsDomain A\nL : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra A L\ninst✝⁶ : IsFractionRing A K\nC : Type u_3\ninst✝⁵ : CommRing C\ninst✝⁴ : IsDomain C\ninst✝³ : Algebra C L\ninst✝² : IsIntegralClosure C A L\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A C L\nalg : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ↑(algebraMap A L) x = 0 → x = 0\nz : L\nx : { x // x ∈ integralClosure A L }\ny : A\nhy : y ≠ 0\nhxy : z * ↑(algebraMap A L) y = ↑x\nh : ↑(algebraMap A C) y = 0\n⊢ ↑(algebraMap A L) y = 0",
"tactic": "rw [IsScalarTower.algebraMap_apply A C L, h, RingHom.map_zero]"
},
{
"state_after": "R : Type ?u.445070\ninst✝¹⁶ : CommRing R\nM : Submonoid R\nS : Type ?u.445277\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Type ?u.445533\ninst✝¹³ : CommRing P\nA : Type u_1\nK : Type ?u.445542\ninst✝¹² : CommRing A\ninst✝¹¹ : IsDomain A\nL : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra A L\ninst✝⁶ : IsFractionRing A K\nC : Type u_3\ninst✝⁵ : CommRing C\ninst✝⁴ : IsDomain C\ninst✝³ : Algebra C L\ninst✝² : IsIntegralClosure C A L\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A C L\nalg : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ↑(algebraMap A L) x = 0 → x = 0\nz : L\nx : { x // x ∈ integralClosure A L }\ny : A\nhy : y ≠ 0\nhxy : z * ↑(algebraMap A L) y = ↑x\n⊢ z * ↑(algebraMap C L) (↑(algebraMap A C) y) = ↑(algebraMap C L) (mk' C ↑x (_ : ↑x ∈ integralClosure A L))",
"state_before": "R : Type ?u.445070\ninst✝¹⁶ : CommRing R\nM : Submonoid R\nS : Type ?u.445277\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Type ?u.445533\ninst✝¹³ : CommRing P\nA : Type u_1\nK : Type ?u.445542\ninst✝¹² : CommRing A\ninst✝¹¹ : IsDomain A\nL : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra A L\ninst✝⁶ : IsFractionRing A K\nC : Type u_3\ninst✝⁵ : CommRing C\ninst✝⁴ : IsDomain C\ninst✝³ : Algebra C L\ninst✝² : IsIntegralClosure C A L\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A C L\nalg : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ↑(algebraMap A L) x = 0 → x = 0\nz : L\nx : { x // x ∈ integralClosure A L }\ny : A\nhy : y ≠ 0\nhxy : z * ↑(algebraMap A L) y = ↑x\n⊢ z *\n ↑(algebraMap C L)\n ↑(mk' C ↑x (_ : ↑x ∈ integralClosure A L),\n { val := ↑(algebraMap A C) y, property := (_ : ↑(algebraMap A C) y ∈ nonZeroDivisors C) }).snd =\n ↑(algebraMap C L)\n (mk' C ↑x (_ : ↑x ∈ integralClosure A L),\n { val := ↑(algebraMap A C) y, property := (_ : ↑(algebraMap A C) y ∈ nonZeroDivisors C) }).fst",
"tactic": "simp only"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.445070\ninst✝¹⁶ : CommRing R\nM : Submonoid R\nS : Type ?u.445277\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Type ?u.445533\ninst✝¹³ : CommRing P\nA : Type u_1\nK : Type ?u.445542\ninst✝¹² : CommRing A\ninst✝¹¹ : IsDomain A\nL : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra A L\ninst✝⁶ : IsFractionRing A K\nC : Type u_3\ninst✝⁵ : CommRing C\ninst✝⁴ : IsDomain C\ninst✝³ : Algebra C L\ninst✝² : IsIntegralClosure C A L\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A C L\nalg : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ↑(algebraMap A L) x = 0 → x = 0\nz : L\nx : { x // x ∈ integralClosure A L }\ny : A\nhy : y ≠ 0\nhxy : z * ↑(algebraMap A L) y = ↑x\n⊢ z * ↑(algebraMap C L) (↑(algebraMap A C) y) = ↑(algebraMap C L) (mk' C ↑x (_ : ↑x ∈ integralClosure A L))",
"tactic": "rw [algebraMap_mk', ← IsScalarTower.algebraMap_apply A C L, hxy]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.445070\ninst✝¹⁶ : CommRing R\nM : Submonoid R\nS : Type ?u.445277\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\nP : Type ?u.445533\ninst✝¹³ : CommRing P\nA : Type u_1\nK : Type ?u.445542\ninst✝¹² : CommRing A\ninst✝¹¹ : IsDomain A\nL : Type u_2\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra A L\ninst✝⁶ : IsFractionRing A K\nC : Type u_3\ninst✝⁵ : CommRing C\ninst✝⁴ : IsDomain C\ninst✝³ : Algebra C L\ninst✝² : IsIntegralClosure C A L\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A C L\nalg : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ↑(algebraMap A L) x = 0 → x = 0\nx y : C\nh : ↑(algebraMap C L) x = ↑(algebraMap C L) y\n⊢ ↑1 * x = ↑1 * y",
"tactic": "simpa using algebraMap_injective C A L h"
}
] |
[
342,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nnnorm_inv'
|
[] |
[
935,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.curryFinFinset_symm_apply_const
|
[] |
[
1887,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1884,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.CartesianClosed.curry_id_eq_coev
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nA✝ B X✝ X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A✝\nA X : C\ninst✝ : Exponentiable A\n⊢ (exp.coev A).app ((𝟭 C).obj X) ≫ 𝟙 (A ⟹ A ⨯ (𝟭 C).obj X) = (exp.coev A).app X",
"state_before": "C : Type u\ninst✝³ : Category C\nA✝ B X✝ X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A✝\nA X : C\ninst✝ : Exponentiable A\n⊢ curry (𝟙 (A ⨯ (𝟭 C).obj X)) = (exp.coev A).app X",
"tactic": "rw [curry_eq, (exp A).map_id (A ⨯ _)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nA✝ B X✝ X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A✝\nA X : C\ninst✝ : Exponentiable A\n⊢ (exp.coev A).app ((𝟭 C).obj X) ≫ 𝟙 (A ⟹ A ⨯ (𝟭 C).obj X) = (exp.coev A).app X",
"tactic": "apply comp_id"
}
] |
[
237,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.normalize_zero
|
[
{
"state_after": "d : Nat\nnz : d ≠ 0\n⊢ mk' 0 1 = 0",
"state_before": "d : Nat\nnz : d ≠ 0\n⊢ normalize 0 d = 0",
"tactic": "simp [normalize, Int.zero_div, Int.natAbs_zero, Nat.div_self (Nat.pos_of_ne_zero nz)]"
},
{
"state_after": "no goals",
"state_before": "d : Nat\nnz : d ≠ 0\n⊢ mk' 0 1 = 0",
"tactic": "rfl"
}
] |
[
29,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
28,
9
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.map_id
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns : Stream' (Option α)\nal : IsSeq s\n⊢ ↑(map id { val := s, property := al }) = ↑{ val := s, property := al }",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Stream' (Option α)\nal : IsSeq s\n⊢ map id { val := s, property := al } = { val := s, property := al }",
"tactic": "apply Subtype.eq"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns : Stream' (Option α)\nal : IsSeq s\n⊢ Stream'.map (Option.map id) s = s",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns : Stream' (Option α)\nal : IsSeq s\n⊢ ↑(map id { val := s, property := al }) = ↑{ val := s, property := al }",
"tactic": "dsimp [map]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns : Stream' (Option α)\nal : IsSeq s\n⊢ Stream'.map (Option.map id) s = s",
"tactic": "rw [Option.map_id, Stream'.map_id]"
}
] |
[
707,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
704,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.out
|
[] |
[
516,
17
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
515,
1
] |
Mathlib/Data/Nat/GCD/BigOperators.lean
|
Nat.coprime_prod_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nx : ℕ\ns : ι → ℕ\nt : Finset ι\n⊢ (fun y => coprime x y) 1",
"tactic": "simp"
}
] |
[
33,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.inv_coe_int_num
|
[
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ (↑a)⁻¹.num = Int.sign a",
"tactic": "induction a using Int.induction_on <;>\n simp [← Int.negSucc_coe', Int.negSucc_coe, -neg_add_rev, Rat.inv_neg, Int.ofNat_add_one_out,\n -Nat.cast_succ, inv_coe_nat_num_of_pos, -Int.cast_negSucc, @eq_comm ℤ 1,\n Int.sign_eq_one_of_pos, ofInt_eq_cast]"
}
] |
[
286,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
stronglyMeasurable_id
|
[] |
[
643,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.Monic.ne_zero_of_ne
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np✝ q : R[X]\nι : Type ?u.646647\nh : 0 ≠ 1\np : R[X]\nhp : Monic p\ninst✝ : Nontrivial R\n⊢ p ≠ 0",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.646647\nh : 0 ≠ 1\np : R[X]\nhp : Monic p\n⊢ p ≠ 0",
"tactic": "nontriviality R"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np✝ q : R[X]\nι : Type ?u.646647\nh : 0 ≠ 1\np : R[X]\nhp : Monic p\ninst✝ : Nontrivial R\n⊢ p ≠ 0",
"tactic": "exact hp.ne_zero"
}
] |
[
853,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
851,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.prime_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq : Fact (Nat.Prime q)\nhpq : p ≠ q\n⊢ ↑q ≠ 0",
"tactic": "rwa [← Nat.cast_zero, Ne.def, eq_iff_modEq_nat, Nat.modEq_zero_iff_dvd, ←\n hp.1.coprime_iff_not_dvd, Nat.coprime_primes hp.1 hq.1]"
}
] |
[
1044,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
powers.one_mem
|
[] |
[
138,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime_self
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ coprime n n ↔ n = 1",
"tactic": "simp [coprime]"
}
] |
[
365,
82
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
365,
9
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
SimpleGraph.IsUniform.symm
|
[
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns✝ t✝ : Finset α\na b : α\ns t : Finset α\nh : IsUniform G ε s t\nt' : Finset α\nht' : t' ⊆ t\ns' : Finset α\nhs' : s' ⊆ s\nht : ↑(card t) * ε ≤ ↑(card t')\nhs : ↑(card s) * ε ≤ ↑(card s')\n⊢ abs (↑(edgeDensity G s' t') - ↑(edgeDensity G s t)) < ε",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns✝ t✝ : Finset α\na b : α\ns t : Finset α\nh : IsUniform G ε s t\nt' : Finset α\nht' : t' ⊆ t\ns' : Finset α\nhs' : s' ⊆ s\nht : ↑(card t) * ε ≤ ↑(card t')\nhs : ↑(card s) * ε ≤ ↑(card s')\n⊢ abs (↑(edgeDensity G t' s') - ↑(edgeDensity G t s)) < ε",
"tactic": "rw [edgeDensity_comm _ t', edgeDensity_comm _ t]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns✝ t✝ : Finset α\na b : α\ns t : Finset α\nh : IsUniform G ε s t\nt' : Finset α\nht' : t' ⊆ t\ns' : Finset α\nhs' : s' ⊆ s\nht : ↑(card t) * ε ≤ ↑(card t')\nhs : ↑(card s) * ε ≤ ↑(card s')\n⊢ abs (↑(edgeDensity G s' t') - ↑(edgeDensity G s t)) < ε",
"tactic": "exact h hs' ht' hs ht"
}
] |
[
73,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.coe_toMultiset
|
[] |
[
694,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
693,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.37832\ninst✝ : DecidableEq ι\nx y z : M\nf : M →ₗ[R] N\nh : SameRay R x y\nhx : x = 0\n⊢ ↑f x = 0",
"tactic": "rw [hx, map_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.37832\ninst✝ : DecidableEq ι\nx y z : M\nf : M →ₗ[R] N\nh : SameRay R x y\nhy : y = 0\n⊢ ↑f y = 0",
"tactic": "rw [hy, map_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.37832\ninst✝ : DecidableEq ι\nx y z : M\nf : M →ₗ[R] N\nh✝ : SameRay R x y\nx✝ : ∃ r₁ r₂, 0 < r₁ ∧ 0 < r₂ ∧ r₁ • x = r₂ • y\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh : r₁ • x = r₂ • y\n⊢ r₁ • ↑f x = r₂ • ↑f y",
"tactic": "rw [← f.map_smul, ← f.map_smul, h]"
}
] |
[
168,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycleOn.exists_pow_eq'
|
[
{
"state_after": "case intro\nι : Type ?u.1917955\nα : Type u_1\nβ : Type ?u.1917961\nf g : Perm α\nt : Set α\na b x y : α\ns : Finset α\nhs : Set.Finite ↑s\nhf : IsCycleOn f ↑s\nha : a ∈ ↑s\nhb : b ∈ ↑s\n⊢ ∃ n, ↑(f ^ n) a = b",
"state_before": "ι : Type ?u.1917955\nα : Type u_1\nβ : Type ?u.1917961\nf g : Perm α\ns t : Set α\na b x y : α\nhs : Set.Finite s\nhf : IsCycleOn f s\nha : a ∈ s\nhb : b ∈ s\n⊢ ∃ n, ↑(f ^ n) a = b",
"tactic": "lift s to Finset α using id hs"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.1917955\nα : Type u_1\nβ : Type ?u.1917961\nf g : Perm α\nt : Set α\na b x y : α\ns : Finset α\nhs : Set.Finite ↑s\nhf : IsCycleOn f ↑s\nha : a ∈ ↑s\nhb : b ∈ ↑s\nn : ℕ\nhn : ↑(f ^ n) a = b\n⊢ ∃ n, ↑(f ^ n) a = b",
"state_before": "case intro\nι : Type ?u.1917955\nα : Type u_1\nβ : Type ?u.1917961\nf g : Perm α\nt : Set α\na b x y : α\ns : Finset α\nhs : Set.Finite ↑s\nhf : IsCycleOn f ↑s\nha : a ∈ ↑s\nhb : b ∈ ↑s\n⊢ ∃ n, ↑(f ^ n) a = b",
"tactic": "obtain ⟨n, -, hn⟩ := hf.exists_pow_eq ha hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type ?u.1917955\nα : Type u_1\nβ : Type ?u.1917961\nf g : Perm α\nt : Set α\na b x y : α\ns : Finset α\nhs : Set.Finite ↑s\nhf : IsCycleOn f ↑s\nha : a ∈ ↑s\nhb : b ∈ ↑s\nn : ℕ\nhn : ↑(f ^ n) a = b\n⊢ ∃ n, ↑(f ^ n) a = b",
"tactic": "exact ⟨n, hn⟩"
}
] |
[
914,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
910,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.one_imJ
|
[] |
[
199,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
9
] |
Mathlib/CategoryTheory/Sums/Basic.lean
|
CategoryTheory.sum_comp_inr
|
[] |
[
77,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_le_sdiff_left
|
[] |
[
670,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_ball_lt_top
|
[] |
[
3960,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3958,
1
] |
Mathlib/CategoryTheory/Sites/Subsheaf.lean
|
CategoryTheory.GrothendieckTopology.imagePresheaf_comp_le
|
[] |
[
385,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Topology/MetricSpace/MetricSeparated.lean
|
IsMetricSeparated.union_right
|
[] |
[
100,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.property_index
|
[] |
[
305,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
Polynomial.Chebyshev.T_eq_X_mul_T_sub_pol_U
|
[
{
"state_after": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * X - 1 = X * X - (1 - X ^ 2) * 1",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ T R (0 + 2) = X * T R (0 + 1) - (1 - X ^ 2) * U R 0",
"tactic": "simp only [T, U]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * X - 1 = X * X - (1 - X ^ 2) * 1",
"tactic": "ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * (2 * X * X - 1) - X = X * (2 * X * X - 1) - (1 - X ^ 2) * (2 * X)",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ T R (1 + 2) = X * T R (1 + 1) - (1 - X ^ 2) * U R 1",
"tactic": "simp only [T, U]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * (2 * X * X - 1) - X = X * (2 * X * X - 1) - (1 - X ^ 2) * (2 * X)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ 2 * X * T R (n + 2 + 1) - T R (n + 2) =\n 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n)",
"tactic": "simp only [T_eq_X_mul_T_sub_pol_U]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n) =\n X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.22740\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) =\n X * T R (n + 2 + 1) - (1 - X ^ 2) * U R (n + 2)",
"tactic": "rw [T_add_two _ (n + 1), U_add_two]"
}
] |
[
153,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.inclusion_self
|
[] |
[
1051,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.SameCycle.cycleOf_apply
|
[] |
[
1001,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
MulSemiringActionHom.comp_apply
|
[] |
[
562,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.perm_dlookup
|
[
{
"state_after": "case a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\nnd₁ : NodupKeys l₁\nnd₂ : NodupKeys l₂\np : l₁ ~ l₂\nb : β a\n⊢ b ∈ dlookup a l₁ ↔ b ∈ dlookup a l₂",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\nnd₁ : NodupKeys l₁\nnd₂ : NodupKeys l₂\np : l₁ ~ l₂\n⊢ dlookup a l₁ = dlookup a l₂",
"tactic": "ext b"
},
{
"state_after": "case a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\nnd₁ : NodupKeys l₁\nnd₂ : NodupKeys l₂\np : l₁ ~ l₂\nb : β a\n⊢ { fst := a, snd := b } ∈ l₁ ↔ { fst := a, snd := b } ∈ l₂",
"state_before": "case a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\nnd₁ : NodupKeys l₁\nnd₂ : NodupKeys l₂\np : l₁ ~ l₂\nb : β a\n⊢ b ∈ dlookup a l₁ ↔ b ∈ dlookup a l₂",
"tactic": "simp only [mem_dlookup_iff nd₁, mem_dlookup_iff nd₂]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\nnd₁ : NodupKeys l₁\nnd₂ : NodupKeys l₂\np : l₁ ~ l₂\nb : β a\n⊢ { fst := a, snd := b } ∈ l₁ ↔ { fst := a, snd := b } ∈ l₂",
"tactic": "exact p.mem_iff"
}
] |
[
241,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Std/Data/List/Lemmas.lean
|
List.modify_get?_length
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nl : List α\n⊢ length (modifyHead f l) = length l",
"tactic": "cases l <;> rfl"
}
] |
[
754,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
753,
9
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_lt_one_div
|
[] |
[
478,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.map_coeSubmodule
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑(map f I) = ↑f '' ↑I\n⊢ ↑f '' ↑I = ↑↑f '' ↑↑I",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑(map f I) = ↑f '' ↑I\n⊢ ↑(map f I) = Submodule.map ↑f ↑I",
"tactic": "rw [SetLike.ext'_iff, LieSubmodule.coe_toSubmodule, h, Submodule.map_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑(map f I) = ↑f '' ↑I\n⊢ ↑f '' ↑I = ↑↑f '' ↑↑I",
"tactic": "rfl"
}
] |
[
809,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sUnion_iUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.178747\nγ : Type ?u.178750\nι : Sort u_2\nι' : Sort ?u.178756\nι₂ : Sort ?u.178759\nκ : ι → Sort ?u.178764\nκ₁ : ι → Sort ?u.178769\nκ₂ : ι → Sort ?u.178774\nκ' : ι' → Sort ?u.178779\ns : ι → Set (Set α)\n⊢ (⋃₀ ⋃ (i : ι), s i) = ⋃ (i : ι), ⋃₀ s i",
"tactic": "simp only [sUnion_eq_biUnion, biUnion_iUnion]"
}
] |
[
1368,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1367,
1
] |
src/lean/Init/Core.lean
|
eqRec_heq
|
[] |
[
645,
25
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
644,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.div_bot
|
[] |
[
459,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.image_subset_infs_right
|
[] |
[
279,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Logic/Nonempty.lean
|
nonempty_psigma
|
[] |
[
55,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderivWithin_pi
|
[] |
[
457,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
|
BoxIntegral.Box.measurableSet_Icc
|
[] |
[
66,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.equivQuotientZPowOfEquiv_refl
|
[
{
"state_after": "case h\nG : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nφ : G →* H\nA B C : Type u\ninst✝² : CommGroup A\ninst✝¹ : CommGroup B\ninst✝ : CommGroup C\nf : A →* B\ng : B →* A\ne : A ≃* B\nd : B ≃* C\nn : ℤ\nx : A ⧸ range (zpowGroupHom n)\n⊢ ↑(MulEquiv.refl (A ⧸ range (zpowGroupHom n))) x = ↑(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) x",
"state_before": "G : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nφ : G →* H\nA B C : Type u\ninst✝² : CommGroup A\ninst✝¹ : CommGroup B\ninst✝ : CommGroup C\nf : A →* B\ng : B →* A\ne : A ≃* B\nd : B ≃* C\nn : ℤ\n⊢ MulEquiv.refl (A ⧸ range (zpowGroupHom n)) = equivQuotientZPowOfEquiv (MulEquiv.refl A) n",
"tactic": "ext x"
},
{
"state_after": "case h\nG : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nφ : G →* H\nA B C : Type u\ninst✝² : CommGroup A\ninst✝¹ : CommGroup B\ninst✝ : CommGroup C\nf : A →* B\ng : B →* A\ne : A ≃* B\nd : B ≃* C\nn : ℤ\nx : A ⧸ range (zpowGroupHom n)\n⊢ ↑(MulEquiv.refl (A ⧸ range (zpowGroupHom n))) (Quotient.mk'' (Quotient.out' x)) =\n ↑(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) (Quotient.mk'' (Quotient.out' x))",
"state_before": "case h\nG : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nφ : G →* H\nA B C : Type u\ninst✝² : CommGroup A\ninst✝¹ : CommGroup B\ninst✝ : CommGroup C\nf : A →* B\ng : B →* A\ne : A ≃* B\nd : B ≃* C\nn : ℤ\nx : A ⧸ range (zpowGroupHom n)\n⊢ ↑(MulEquiv.refl (A ⧸ range (zpowGroupHom n))) x = ↑(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) x",
"tactic": "rw [← Quotient.out_eq' x]"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nφ : G →* H\nA B C : Type u\ninst✝² : CommGroup A\ninst✝¹ : CommGroup B\ninst✝ : CommGroup C\nf : A →* B\ng : B →* A\ne : A ≃* B\nd : B ≃* C\nn : ℤ\nx : A ⧸ range (zpowGroupHom n)\n⊢ ↑(MulEquiv.refl (A ⧸ range (zpowGroupHom n))) (Quotient.mk'' (Quotient.out' x)) =\n ↑(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) (Quotient.mk'' (Quotient.out' x))",
"tactic": "rfl"
}
] |
[
547,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_le
|
[] |
[
514,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.map_inl
|
[
{
"state_after": "no goals",
"state_before": "N : Type u_3\nG : Type u_4\nH : Type ?u.240686\ninst✝⁴ : Group N\ninst✝³ : Group G\ninst✝² : Group H\nφ : G →* MulAut N\nN₁ : Type u_1\nG₁ : Type u_2\ninst✝¹ : Group N₁\ninst✝ : Group G₁\nφ₁ : G₁ →* MulAut N₁\nf₁ : N →* N₁\nf₂ : G →* G₁\nh : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁\nn : N\n⊢ ↑(map f₁ f₂ h) (↑inl n) = ↑inl (↑f₁ n)",
"tactic": "simp [map]"
}
] |
[
296,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.pre_map
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nA B X X' Y Y' Z : C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : Exponentiable A\nA₁ A₂ A₃ : C\ninst✝² : Exponentiable A₁\ninst✝¹ : Exponentiable A₂\ninst✝ : Exponentiable A₃\nf : A₁ ⟶ A₂\ng : A₂ ⟶ A₃\n⊢ pre (f ≫ g) = pre g ≫ pre f",
"tactic": "rw [pre, pre, pre, transferNatTransSelf_comp, prod.functor.map_comp]"
}
] |
[
307,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.order_mul
|
[
{
"state_after": "case a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ order (x * y) ≤ order x + order y\n\ncase a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ order x + order y ≤ order (x * y)",
"state_before": "Γ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ order (x * y) = order x + order y",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a.h\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ coeff (x * y) (order x + order y) ≠ 0",
"state_before": "case a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ order (x * y) ≤ order x + order y",
"tactic": "apply order_le_of_coeff_ne_zero"
},
{
"state_after": "case a.h\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ coeff x (order x) * coeff y (order y) ≠ 0",
"state_before": "case a.h\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ coeff (x * y) (order x + order y) ≠ 0",
"tactic": "rw [mul_coeff_order_add_order x y]"
},
{
"state_after": "no goals",
"state_before": "case a.h\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ coeff x (order x) * coeff y (order y) ≠ 0",
"tactic": "exact mul_ne_zero (coeff_order_ne_zero hx) (coeff_order_ne_zero hy)"
},
{
"state_after": "case a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ Set.IsWf.min (_ : Set.IsWf (support x + support y)) (_ : Set.Nonempty (support x + support y)) ≤\n Set.IsWf.min (_ : Set.IsWf (support (x * y))) (_ : Set.Nonempty (support (x * y)))",
"state_before": "case a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ order x + order y ≤ order (x * y)",
"tactic": "rw [order_of_ne hx, order_of_ne hy, order_of_ne (mul_ne_zero hx hy), ← Set.IsWf.min_add]"
},
{
"state_after": "no goals",
"state_before": "case a\nΓ✝ : Type ?u.1293056\nR : Type u_2\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_1\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : HahnSeries Γ R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ Set.IsWf.min (_ : Set.IsWf (support x + support y)) (_ : Set.Nonempty (support x + support y)) ≤\n Set.IsWf.min (_ : Set.IsWf (support (x * y))) (_ : Set.Nonempty (support (x * y)))",
"tactic": "exact Set.IsWf.min_le_min_of_subset support_mul_subset_add_support"
}
] |
[
917,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.sumAddHom_apply
|
[
{
"state_after": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ↑(sumAddHom φ) { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } =\n sum { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } fun x => ↑(φ x)",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : Π₀ (i : ι), β i\n⊢ ↑(sumAddHom φ) f = sum f fun x => ↑(φ x)",
"tactic": "rcases f with ⟨f, s, hf⟩"
},
{
"state_after": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ∑ i in Multiset.toFinset ↑{ val := s, property := hf },\n ↑(φ i) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n ∑ i in\n Finset.filter (fun i => ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i ≠ 0)\n (Multiset.toFinset ↑{ val := s, property := hf }),\n (fun x => ↑(φ x)) i (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)",
"state_before": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ↑(sumAddHom φ) { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } =\n sum { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } fun x => ↑(φ x)",
"tactic": "change (∑ i in _, _) = ∑ i in Finset.filter _ _, _"
},
{
"state_after": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ∀ (x : ι),\n x ∈ Multiset.toFinset ↑{ val := s, property := hf } →\n ↑(φ x) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x) =\n if ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x ≠ 0 then\n (fun x => ↑(φ x)) x (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x)\n else 0",
"state_before": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ∑ i in Multiset.toFinset ↑{ val := s, property := hf },\n ↑(φ i) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n ∑ i in\n Finset.filter (fun i => ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i ≠ 0)\n (Multiset.toFinset ↑{ val := s, property := hf }),\n (fun x => ↑(φ x)) i (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)",
"tactic": "rw [Finset.sum_filter, Finset.sum_congr rfl]"
},
{
"state_after": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset ↑{ val := s, property := hf }\n⊢ ↑(φ i) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n if ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i ≠ 0 then\n (fun x => ↑(φ x)) i (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)\n else 0",
"state_before": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\n⊢ ∀ (x : ι),\n x ∈ Multiset.toFinset ↑{ val := s, property := hf } →\n ↑(φ x) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x) =\n if ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x ≠ 0 then\n (fun x => ↑(φ x)) x (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x)\n else 0",
"tactic": "intro i _"
},
{
"state_after": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\n⊢ ↑(φ i) (f i) = if f i ≠ 0 then ↑(φ i) (f i) else 0",
"state_before": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset ↑{ val := s, property := hf }\n⊢ ↑(φ i) (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n if ↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i ≠ 0 then\n (fun x => ↑(φ x)) i (↑{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)\n else 0",
"tactic": "dsimp only [coe_mk', Subtype.coe_mk] at *"
},
{
"state_after": "case mk'.mk.mk.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : f i ≠ 0\n⊢ ↑(φ i) (f i) = ↑(φ i) (f i)\n\ncase mk'.mk.mk.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : ¬f i ≠ 0\n⊢ ↑(φ i) (f i) = 0",
"state_before": "case mk'.mk.mk\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\n⊢ ↑(φ i) (f i) = if f i ≠ 0 then ↑(φ i) (f i) else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "case mk'.mk.mk.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : ¬f i ≠ 0\n⊢ ↑(φ i) (f i) = 0",
"state_before": "case mk'.mk.mk.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : f i ≠ 0\n⊢ ↑(φ i) (f i) = ↑(φ i) (f i)\n\ncase mk'.mk.mk.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : ¬f i ≠ 0\n⊢ ↑(φ i) (f i) = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case mk'.mk.mk.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommMonoid γ\nφ : (i : ι) → β i →+ γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\ns : Multiset ι\nhf : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\na✝ : i ∈ Multiset.toFinset s\nh : ¬f i ≠ 0\n⊢ ↑(φ i) (f i) = 0",
"tactic": "rw [not_not.mp h, AddMonoidHom.map_zero]"
}
] |
[
1934,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1925,
1
] |
Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
|
integral_exp_Iic
|
[
{
"state_after": "c : ℝ\n⊢ Tendsto (fun i => ∫ (x : ℝ) in id i..c, exp x) atBot (𝓝 (exp c))",
"state_before": "c : ℝ\n⊢ (∫ (x : ℝ) in Iic c, exp x) = exp c",
"tactic": "refine'\n tendsto_nhds_unique\n (intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) _"
},
{
"state_after": "c : ℝ\n⊢ Tendsto (fun i => exp c - exp (id i)) atBot (𝓝 (exp c - 0))",
"state_before": "c : ℝ\n⊢ Tendsto (fun i => ∫ (x : ℝ) in id i..c, exp x) atBot (𝓝 (exp c))",
"tactic": "simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]"
},
{
"state_after": "no goals",
"state_before": "c : ℝ\n⊢ Tendsto (fun i => exp c - exp (id i)) atBot (𝓝 (exp c - 0))",
"tactic": "exact tendsto_exp_atBot.const_sub _"
},
{
"state_after": "no goals",
"state_before": "c : ℝ\n⊢ 𝓝 (exp c) = 𝓝 (exp c - 0)",
"tactic": "rw [sub_zero]"
}
] |
[
49,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Topology/Order/LowerTopology.lean
|
WithLowerTopology.toLower_ofLower
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.smul_im
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.2153398\ninst✝ : IsROrC K\nr : ℝ\nz : K\n⊢ ↑im (r • z) = r * ↑im z",
"tactic": "rw [real_smul_eq_coe_mul, ofReal_mul_im]"
}
] |
[
290,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.measurePreserving_prod_inv_mul
|
[] |
[
129,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Divisibility/Units.lean
|
Units.dvd_mul_left
|
[
{
"state_after": "α : Type u_1\ninst✝ : CommMonoid α\na b : α\nu : αˣ\n⊢ a ∣ b * ↑u ↔ a ∣ b",
"state_before": "α : Type u_1\ninst✝ : CommMonoid α\na b : α\nu : αˣ\n⊢ a ∣ ↑u * b ↔ a ∣ b",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CommMonoid α\na b : α\nu : αˣ\n⊢ a ∣ b * ↑u ↔ a ∣ b",
"tactic": "apply dvd_mul_right"
}
] |
[
57,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
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