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/Algebra/Field/Basic.lean
|
toDual_rat_cast
|
[] |
[
382,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.mulVecLin_one
|
[
{
"state_after": "case h.h.h\nR : Type u_2\ninst✝² : CommSemiring R\nk : Type ?u.293203\nl : Type ?u.293206\nm : Type ?u.293209\nn : Type u_1\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\ni✝ x✝ : n\n⊢ ↑(comp (mulVecLin 1) (single i✝)) 1 x✝ = ↑(comp LinearMap.id (single i✝)) 1 x✝",
"state_before": "R : Type u_2\ninst✝² : CommSemiring R\nk : Type ?u.293203\nl : Type ?u.293206\nm : Type ?u.293209\nn : Type u_1\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\n⊢ mulVecLin 1 = LinearMap.id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nR : Type u_2\ninst✝² : CommSemiring R\nk : Type ?u.293203\nl : Type ?u.293206\nm : Type ?u.293209\nn : Type u_1\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\ni✝ x✝ : n\n⊢ ↑(comp (mulVecLin 1) (single i✝)) 1 x✝ = ↑(comp LinearMap.id (single i✝)) 1 x✝",
"tactic": "simp [Matrix.one_apply, Pi.single_apply]"
}
] |
[
251,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Polynomial/Reverse.lean
|
Polynomial.reverse_zero
|
[] |
[
264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Analysis/Convex/Extrema.lean
|
IsMinOn.of_isLocalMinOn_of_convexOn_Icc
|
[
{
"state_after": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ c ∈ {x | (fun x => f a ≤ f x) x}",
"state_before": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\n⊢ IsMinOn f (Icc a b) a",
"tactic": "rintro c hc"
},
{
"state_after": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ f a ≤ f c",
"state_before": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ c ∈ {x | (fun x => f a ≤ f x) x}",
"tactic": "dsimp only [mem_setOf_eq]"
},
{
"state_after": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ f a ≤ f c",
"state_before": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ f a ≤ f c",
"tactic": "rw [IsLocalMinOn, nhdsWithin_Icc_eq_nhdsWithin_Ici a_lt_b] at h_local_min"
},
{
"state_after": "case inl\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nhc : a ∈ Icc a b\n⊢ f a ≤ f a\n\ncase inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\n⊢ f a ≤ f c",
"state_before": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\n⊢ f a ≤ f c",
"tactic": "rcases hc.1.eq_or_lt with (rfl | a_lt_c)"
},
{
"state_after": "case inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\n⊢ f a ≤ f c",
"state_before": "case inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\n⊢ f a ≤ f c",
"tactic": "have H₁ : ∀ᶠ y in 𝓝[>] a, f a ≤ f y :=\n h_local_min.filter_mono (nhdsWithin_mono _ Ioi_subset_Ici_self)"
},
{
"state_after": "case inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\n⊢ f a ≤ f c",
"state_before": "case inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\n⊢ f a ≤ f c",
"tactic": "have H₂ : ∀ᶠ y in 𝓝[>] a, y ∈ Ioc a c := Ioc_mem_nhdsWithin_Ioi (left_mem_Ico.2 a_lt_c)"
},
{
"state_after": "case inr.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\ny : ℝ\nhfy : f a ≤ f y\nhy_ac : y ∈ Ioc a c\n⊢ f a ≤ f c",
"state_before": "case inr\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\n⊢ f a ≤ f c",
"tactic": "rcases(H₁.and H₂).exists with ⟨y, hfy, hy_ac⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ f a ≤ f c",
"state_before": "case inr.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\ny : ℝ\nhfy : f a ≤ f y\nhy_ac : y ∈ Ioc a c\n⊢ f a ≤ f c",
"tactic": "rcases(Convex.mem_Ioc a_lt_c).mp hy_ac with ⟨ya, yc, ya₀, yc₀, yac, rfl⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\nthis : ya • f a + yc • f a ≤ ya • f a + yc • f c\n⊢ f a ≤ f c\n\ncase this\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ ya • f a + yc • f a ≤ ya • f a + yc • f c",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ f a ≤ f c",
"tactic": "suffices : ya • f a + yc • f a ≤ ya • f a + yc • f c"
},
{
"state_after": "case this\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ ya • f a + yc • f a ≤ ya • f a + yc • f c",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\nthis : ya • f a + yc • f a ≤ ya • f a + yc • f c\n⊢ f a ≤ f c\n\ncase this\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ ya • f a + yc • f a ≤ ya • f a + yc • f c",
"tactic": "exact (smul_le_smul_iff_of_pos yc₀).1 (le_of_add_le_add_left this)"
},
{
"state_after": "no goals",
"state_before": "case this\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ ya • f a + yc • f a ≤ ya • f a + yc • f c",
"tactic": "calc\n ya • f a + yc • f a = f a := by rw [← add_smul, yac, one_smul]\n _ ≤ f (ya * a + yc * c) := hfy\n _ ≤ ya • f a + yc • f c := h_conv.2 (left_mem_Icc.2 a_lt_b.le) hc ya₀ yc₀.le yac"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nhc : a ∈ Icc a b\n⊢ f a ≤ f a",
"tactic": "exact le_rfl"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.1907\nβ : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module ℝ β\ninst✝ : OrderedSMul ℝ β\ns : Set E\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[Ici a] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc a b\na_lt_c : a < c\nH₁ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, f a ≤ f y\nH₂ : ∀ᶠ (y : ℝ) in 𝓝[Ioi a] a, y ∈ Ioc a c\nya yc : ℝ\nya₀ : 0 ≤ ya\nyc₀ : 0 < yc\nyac : ya + yc = 1\nhfy : f a ≤ f (ya * a + yc * c)\nhy_ac : ya * a + yc * c ∈ Ioc a c\n⊢ ya • f a + yc • f a = f a",
"tactic": "rw [← add_smul, yac, one_smul]"
}
] |
[
51,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.one_lt_exp_iff
|
[
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\n⊢ 1 < exp x ↔ 0 < x",
"tactic": "rw [← exp_zero, exp_lt_exp]"
}
] |
[
1543,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1543,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
summable_geometric_two_encode
|
[] |
[
210,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.le_trim
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3497107\nγ : Type ?u.3497110\nδ : Type ?u.3497113\nι : Type ?u.3497116\nR : Type ?u.3497119\nR' : Type ?u.3497122\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\n⊢ ↑↑μ s ≤ ↑↑(OuterMeasure.toMeasure ↑μ (_ : m ≤ OuterMeasure.caratheodory ↑μ)) s",
"state_before": "α : Type u_1\nβ : Type ?u.3497107\nγ : Type ?u.3497110\nδ : Type ?u.3497113\nι : Type ?u.3497116\nR : Type ?u.3497119\nR' : Type ?u.3497122\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\n⊢ ↑↑μ s ≤ ↑↑(trim μ hm) s",
"tactic": "simp_rw [Measure.trim]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3497107\nγ : Type ?u.3497110\nδ : Type ?u.3497113\nι : Type ?u.3497116\nR : Type ?u.3497119\nR' : Type ?u.3497122\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\n⊢ ↑↑μ s ≤ ↑↑(OuterMeasure.toMeasure ↑μ (_ : m ≤ OuterMeasure.caratheodory ↑μ)) s",
"tactic": "exact @le_toMeasure_apply _ m _ _ _"
}
] |
[
4377,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4375,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.einfsep_ne_top_iff
|
[] |
[
281,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.edgeSet_injective
|
[] |
[
504,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.self_trans_symm
|
[] |
[
371,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Probability/CondCount.lean
|
ProbabilityTheory.pred_true_of_condCount_eq_one
|
[
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\nhsf : Set.Finite s\n⊢ s ⊆ t",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\n⊢ s ⊆ t",
"tactic": "have hsf := finite_of_condCount_ne_zero (by rw [h]; exact one_ne_zero)"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑Measure.count (s ∩ t) * (↑↑Measure.count s)⁻¹ = 1\nhsf : Set.Finite s\n⊢ s ⊆ t",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\nhsf : Set.Finite s\n⊢ s ⊆ t",
"tactic": "rw [condCount, cond_apply _ hsf.measurableSet, mul_comm] at h"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : ↑↑Measure.count (s ∩ t) = (↑↑Measure.count s)⁻¹⁻¹\n⊢ s ⊆ t",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑Measure.count (s ∩ t) * (↑↑Measure.count s)⁻¹ = 1\nhsf : Set.Finite s\n⊢ s ⊆ t",
"tactic": "replace h := ENNReal.eq_inv_of_mul_eq_one_left h"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ s ⊆ t",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : ↑↑Measure.count (s ∩ t) = (↑↑Measure.count s)⁻¹⁻¹\n⊢ s ⊆ t",
"tactic": "rw [inv_inv, Measure.count_apply_finite _ hsf, Measure.count_apply_finite _ (hsf.inter_of_left _),\n Nat.cast_inj] at h"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ s ∩ t = s",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ s ⊆ t",
"tactic": "suffices s ∩ t = s by exact this ▸ fun x hx => hx.2"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ Set.Finite.toFinset (_ : Set.Finite (s ∩ t)) = Set.Finite.toFinset hsf",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ s ∩ t = s",
"tactic": "rw [← @Set.Finite.toFinset_inj _ _ _ (hsf.inter_of_left _) hsf]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\n⊢ Set.Finite.toFinset (_ : Set.Finite (s ∩ t)) = Set.Finite.toFinset hsf",
"tactic": "exact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono <| s.inter_subset_left t) h.ge"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\n⊢ 1 ≠ 0",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\n⊢ ↑↑(condCount ?m.112707) ?m.112708 ≠ 0",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nh : ↑↑(condCount s) t = 1\n⊢ 1 ≠ 0",
"tactic": "exact one_ne_zero"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s ∩ t))) = Finset.card (Set.Finite.toFinset hsf)\nthis : s ∩ t = s\n⊢ s ⊆ t",
"tactic": "exact this ▸ fun x hx => hx.2"
}
] |
[
128,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.Finite_of_ncard_ne_zero
|
[] |
[
121,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Order/Sub/Basic.lean
|
AddHom.le_map_tsub
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁷ : Preorder α\ninst✝⁶ : Add α\ninst✝⁵ : Sub α\ninst✝⁴ : OrderedSub α\na✝ b✝ c d : α\ninst✝³ : Preorder β\ninst✝² : Add β\ninst✝¹ : Sub β\ninst✝ : OrderedSub β\nf : AddHom α β\nhf : Monotone ↑f\na b : α\n⊢ ↑f a ≤ ↑f (a - b + b)",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁷ : Preorder α\ninst✝⁶ : Add α\ninst✝⁵ : Sub α\ninst✝⁴ : OrderedSub α\na✝ b✝ c d : α\ninst✝³ : Preorder β\ninst✝² : Add β\ninst✝¹ : Sub β\ninst✝ : OrderedSub β\nf : AddHom α β\nhf : Monotone ↑f\na b : α\n⊢ ↑f a - ↑f b ≤ ↑f (a - b)",
"tactic": "rw [tsub_le_iff_right, ← f.map_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁷ : Preorder α\ninst✝⁶ : Add α\ninst✝⁵ : Sub α\ninst✝⁴ : OrderedSub α\na✝ b✝ c d : α\ninst✝³ : Preorder β\ninst✝² : Add β\ninst✝¹ : Sub β\ninst✝ : OrderedSub β\nf : AddHom α β\nhf : Monotone ↑f\na b : α\n⊢ ↑f a ≤ ↑f (a - b + b)",
"tactic": "exact hf le_tsub_add"
}
] |
[
31,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
div_pow_le
|
[] |
[
92,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsClopen.preimage
|
[] |
[
1617,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1615,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.compl_inter_self
|
[] |
[
1655,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1654,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_self_inter
|
[] |
[
2040,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2039,
1
] |
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.ext_functor
|
[
{
"state_after": "case h_obj\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\n⊢ ∀ (X : Paths V), F.obj X = G.obj X\n\ncase h_map\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\n⊢ autoParam\n (∀ (X Y : Paths V) (f : X ⟶ Y),\n F.map f = eqToHom (_ : ?F.obj X = ?G.obj X) ≫ G.map f ≫ eqToHom (_ : G.obj Y = F.obj Y))\n _auto✝",
"state_before": "V : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\n⊢ F = G",
"tactic": "fapply Functor.ext"
},
{
"state_after": "case h_obj\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX : Paths V\n⊢ F.obj X = G.obj X",
"state_before": "case h_obj\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\n⊢ ∀ (X : Paths V), F.obj X = G.obj X",
"tactic": "intro X"
},
{
"state_after": "no goals",
"state_before": "case h_obj\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX : Paths V\n⊢ F.obj X = G.obj X",
"tactic": "rw [h_obj]"
},
{
"state_after": "case h_map\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\nf : X ⟶ Y\n⊢ F.map f = eqToHom (_ : F.obj X = G.obj X) ≫ G.map f ≫ eqToHom (_ : G.obj Y = F.obj Y)",
"state_before": "case h_map\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\n⊢ autoParam\n (∀ (X Y : Paths V) (f : X ⟶ Y),\n F.map f = eqToHom (_ : F.obj X = G.obj X) ≫ G.map f ≫ eqToHom (_ : G.obj Y = F.obj Y))\n _auto✝",
"tactic": "intro X Y f"
},
{
"state_after": "case h_map.nil\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\n⊢ F.map Quiver.Path.nil = eqToHom (_ : F.obj X = G.obj X) ≫ G.map Quiver.Path.nil ≫ eqToHom (_ : G.obj X = F.obj X)\n\ncase h_map.cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' ⟶ Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')\n⊢ F.map (Quiver.Path.cons g e) =\n eqToHom (_ : F.obj X = G.obj X) ≫ G.map (Quiver.Path.cons g e) ≫ eqToHom (_ : G.obj Z' = F.obj Z')",
"state_before": "case h_map\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\nf : X ⟶ Y\n⊢ F.map f = eqToHom (_ : F.obj X = G.obj X) ≫ G.map f ≫ eqToHom (_ : G.obj Y = F.obj Y)",
"tactic": "induction' f with Y' Z' g e ih"
},
{
"state_after": "no goals",
"state_before": "case h_map.nil\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\n⊢ F.map Quiver.Path.nil = eqToHom (_ : F.obj X = G.obj X) ≫ G.map Quiver.Path.nil ≫ eqToHom (_ : G.obj X = F.obj X)",
"tactic": "erw [F.map_id, G.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]"
},
{
"state_after": "case h_map.cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' ⟶ Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')\n⊢ (eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')) ≫\n eqToHom (_ : F.obj Y' = G.obj Y') ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj Z' = F.obj Z') =\n eqToHom (_ : F.obj X = G.obj X) ≫ (G.map g ≫ G.map (Quiver.Hom.toPath e)) ≫ eqToHom (_ : G.obj Z' = F.obj Z')",
"state_before": "case h_map.cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' ⟶ Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')\n⊢ F.map (Quiver.Path.cons g e) =\n eqToHom (_ : F.obj X = G.obj X) ≫ G.map (Quiver.Path.cons g e) ≫ eqToHom (_ : G.obj Z' = F.obj Z')",
"tactic": "erw [F.map_comp g (Quiver.Hom.toPath e), G.map_comp g (Quiver.Hom.toPath e), ih, h]"
},
{
"state_after": "no goals",
"state_before": "case h_map.cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh :\n ∀ (a b : V) (e : a ⟶ b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' ⟶ Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')\n⊢ (eqToHom (_ : F.obj X = G.obj X) ≫ G.map g ≫ eqToHom (_ : G.obj Y' = F.obj Y')) ≫\n eqToHom (_ : F.obj Y' = G.obj Y') ≫ G.map (Quiver.Hom.toPath e) ≫ eqToHom (_ : G.obj Z' = F.obj Z') =\n eqToHom (_ : F.obj X = G.obj X) ≫ (G.map g ≫ G.map (Quiver.Hom.toPath e)) ≫ eqToHom (_ : G.obj Z' = F.obj Z')",
"tactic": "simp only [Category.id_comp, eqToHom_refl, eqToHom_trans_assoc, Category.assoc]"
}
] |
[
138,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.roots_map_of_injective_of_card_eq_natDegree
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ↑f\nhroots : ↑Multiset.card (roots p) = natDegree p\n⊢ ↑Multiset.card (roots (map f p)) ≤ ↑Multiset.card (Multiset.map (↑f) (roots p))",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ↑f\nhroots : ↑Multiset.card (roots p) = natDegree p\n⊢ Multiset.map (↑f) (roots p) = roots (map f p)",
"tactic": "apply Multiset.eq_of_le_of_card_le (map_roots_le_of_injective p hf)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ↑f\nhroots : ↑Multiset.card (roots p) = natDegree p\n⊢ ↑Multiset.card (roots (map f p)) ≤ ↑Multiset.card (Multiset.map (↑f) (roots p))",
"tactic": "simpa only [Multiset.card_map, hroots] using (card_roots' _).trans (natDegree_map_le f p)"
}
] |
[
1213,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1209,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toMul_add
|
[] |
[
164,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.tensorHom
|
[] |
[
506,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Topology/Basic.lean
|
all_mem_nhds_filter
|
[] |
[
1022,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_eq_zero_iff_of_nonneg_ae
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ) = 0 ↔ f =ᵐ[μ] 0",
"state_before": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (∫ (x : α), f x ∂μ) = 0 ↔ f =ᵐ[μ] 0",
"tactic": "simp_rw [integral_eq_lintegral_of_nonneg_ae hf hfi.1, ENNReal.toReal_eq_zero_iff,\n ← ENNReal.not_lt_top, ← hasFiniteIntegral_iff_ofReal hf, hfi.2, not_true, or_false_iff]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (fun a => ENNReal.ofReal (f a)) =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0\n\nα : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ AEMeasurable fun a => ENNReal.ofReal (f a)",
"state_before": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ) = 0 ↔ f =ᵐ[μ] 0",
"tactic": "rw [lintegral_eq_zero_iff']"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (∀ᵐ (x : α) ∂μ, ENNReal.ofReal (f x) = OfNat.ofNat 0 x) ↔ ∀ᵐ (x : α) ∂μ, f x ≤ OfNat.ofNat 0 x",
"state_before": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (fun a => ENNReal.ofReal (f a)) =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0",
"tactic": "rw [← hf.le_iff_eq, Filter.EventuallyEq, Filter.EventuallyLE]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (∀ᵐ (x : α) ∂μ, ENNReal.ofReal (f x) = OfNat.ofNat 0 x) ↔ ∀ᵐ (x : α) ∂μ, f x ≤ OfNat.ofNat 0 x",
"tactic": "simp only [Pi.zero_apply, ofReal_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1106298\nF : Type ?u.1106301\n𝕜 : Type ?u.1106304\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1108995\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ AEMeasurable fun a => ENNReal.ofReal (f a)",
"tactic": "exact (ENNReal.measurable_ofReal.comp_aemeasurable hfi.1.aemeasurable)"
}
] |
[
1230,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1222,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.Nonempty.zero_smul
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.85988\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85997\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : SMulWithZero α β\ns : Set α\nt : Set β\nht : Set.Nonempty t\n⊢ 0 ⊆ 0 • t",
"tactic": "simpa [mem_smul] using ht"
}
] |
[
806,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
805,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Poly.add_denote
|
[
{
"state_after": "no goals",
"state_before": "case zero\nctx : Context\np₁✝ p₂✝ p₁ p₂ : Poly\n⊢ denote ctx (add.go zero p₁ p₂) = denote ctx p₁ + denote ctx p₂",
"tactic": "simp! [append_denote]"
},
{
"state_after": "case succ\nctx : Context\np₁✝ p₂✝ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁ p₂ : Poly\n⊢ denote ctx\n (match p₁, p₂ with\n | p₁, [] => p₁\n | [], p₁ => p₁\n | (k₁, m₁) :: p₁, (k₂, m₂) :: p₂ =>\n bif decide (m₁ < m₂) then (k₁, m₁) :: add.go n✝ p₁ ((k₂, m₂) :: p₂)\n else\n bif decide (m₂ < m₁) then (k₂, m₂) :: add.go n✝ ((k₁, m₁) :: p₁) p₂\n else bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n denote ctx p₁ + denote ctx p₂",
"state_before": "case succ\nctx : Context\np₁✝ p₂✝ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁ p₂ : Poly\n⊢ denote ctx (add.go (succ n✝) p₁ p₂) = denote ctx p₁ + denote ctx p₂",
"tactic": "simp!"
},
{
"state_after": "case succ.h_3\nctx : Context\np₁ p₂ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝¹ p₂✝¹ : Poly\nk₁✝ : Nat\nm₁✝ : Mon\np₁✝ : List (Nat × Mon)\nk₂✝ : Nat\nm₂✝ : Mon\np₂✝ : List (Nat × Mon)\n⊢ denote ctx\n (bif decide (m₁✝ < m₂✝) then (k₁✝, m₁✝) :: add.go n✝ p₁✝ ((k₂✝, m₂✝) :: p₂✝)\n else\n bif decide (m₂✝ < m₁✝) then (k₂✝, m₂✝) :: add.go n✝ ((k₁✝, m₁✝) :: p₁✝) p₂✝\n else bif k₁✝ + k₂✝ == 0 then add.go n✝ p₁✝ p₂✝ else (k₁✝ + k₂✝, m₁✝) :: add.go n✝ p₁✝ p₂✝) =\n k₁✝ * Mon.denote ctx m₁✝ + denote ctx p₁✝ + (k₂✝ * Mon.denote ctx m₂✝ + denote ctx p₂✝)",
"state_before": "case succ\nctx : Context\np₁✝ p₂✝ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁ p₂ : Poly\n⊢ denote ctx\n (match p₁, p₂ with\n | p₁, [] => p₁\n | [], p₁ => p₁\n | (k₁, m₁) :: p₁, (k₂, m₂) :: p₂ =>\n bif decide (m₁ < m₂) then (k₁, m₁) :: add.go n✝ p₁ ((k₂, m₂) :: p₂)\n else\n bif decide (m₂ < m₁) then (k₂, m₂) :: add.go n✝ ((k₁, m₁) :: p₁) p₂\n else bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n denote ctx p₁ + denote ctx p₂",
"tactic": "split <;> simp!"
},
{
"state_after": "case inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\n⊢ denote ctx\n (bif decide (m₂ < m₁) then (k₂, m₂) :: add.go n✝ ((k₁, m₁) :: p₁) p₂\n else bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (denote ctx p₁ + (k₂ * Mon.denote ctx m₂ + denote ctx p₂))",
"state_before": "ctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\n⊢ denote ctx\n (bif decide (m₁ < m₂) then (k₁, m₁) :: add.go n✝ p₁ ((k₂, m₂) :: p₂)\n else\n bif decide (m₂ < m₁) then (k₂, m₂) :: add.go n✝ ((k₁, m₁) :: p₁) p₂\n else bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + denote ctx p₁ + (k₂ * Mon.denote ctx m₂ + denote ctx p₂)",
"tactic": "by_cases hlt : m₁ < m₂ <;> simp! [hlt, Nat.add_assoc, ih]"
},
{
"state_after": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\nhgt : ¬m₂ < m₁\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₂ + (denote ctx p₁ + denote ctx p₂))",
"state_before": "case inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\n⊢ denote ctx\n (bif decide (m₂ < m₁) then (k₂, m₂) :: add.go n✝ ((k₁, m₁) :: p₁) p₂\n else bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (denote ctx p₁ + (k₂ * Mon.denote ctx m₂ + denote ctx p₂))",
"tactic": "by_cases hgt : m₂ < m₁ <;> simp! [hgt, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, ih]"
},
{
"state_after": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\nhgt : ¬m₂ < m₁\nthis : m₁ = m₂\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₂ + (denote ctx p₁ + denote ctx p₂))",
"state_before": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\nhgt : ¬m₂ < m₁\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₂ + (denote ctx p₁ + denote ctx p₂))",
"tactic": "have : m₁ = m₂ := List.le_antisymm hgt hlt"
},
{
"state_after": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))",
"state_before": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\nm₂ : Mon\np₂ : List (Nat × Mon)\nhlt : ¬m₁ < m₂\nhgt : ¬m₂ < m₁\nthis : m₁ = m₂\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₂ + (denote ctx p₁ + denote ctx p₂))",
"tactic": "subst m₂"
},
{
"state_after": "case inr.inr.inl\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\nheq : (k₁ + k₂ == 0) = true\n⊢ denote ctx p₁ + denote ctx p₂ = k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))\n\ncase inr.inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\nheq : ¬(k₁ + k₂ == 0) = true\n⊢ (k₁ + k₂) * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))",
"state_before": "case inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\n⊢ denote ctx (bif k₁ + k₂ == 0 then add.go n✝ p₁ p₂ else (k₁ + k₂, m₁) :: add.go n✝ p₁ p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))",
"tactic": "by_cases heq : k₁ + k₂ == 0 <;> simp! [heq, ih]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\nheq : (k₁ + k₂ == 0) = true\n⊢ denote ctx p₁ + denote ctx p₂ = k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))",
"tactic": "simp [← Nat.add_assoc, ← Nat.right_distrib, eq_of_beq heq]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr\nctx : Context\np₁✝¹ p₂✝¹ : Poly\nn✝ : Nat\nih : ∀ (p₁ p₂ : Poly), denote ctx (add.go n✝ p₁ p₂) = denote ctx p₁ + denote ctx p₂\np₁✝ p₂✝ : Poly\nk₁ : Nat\nm₁ : Mon\np₁ : List (Nat × Mon)\nk₂ : Nat\np₂ : List (Nat × Mon)\nhlt hgt : ¬m₁ < m₁\nheq : ¬(k₁ + k₂ == 0) = true\n⊢ (k₁ + k₂) * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂) =\n k₁ * Mon.denote ctx m₁ + (k₂ * Mon.denote ctx m₁ + (denote ctx p₁ + denote ctx p₂))",
"tactic": "simp [Nat.right_distrib, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]"
}
] |
[
143,
83
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
127,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Prod.mk_vadd_mk
|
[] |
[
316,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.prod_map_comap_prod'
|
[] |
[
1219,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1216,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_X_mul'
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ (if Finsupp.single s 1 ≤ m then 1 * coeff (m - Finsupp.single s 1) p else 0) =\n if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ coeff m (X s * p) = if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"tactic": "refine' (coeff_monomial_mul' _ _ _ _).trans _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ (if Finsupp.single s 1 ≤ m then 1 * coeff (m - Finsupp.single s 1) p else 0) =\n if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"tactic": "simp_rw [Finsupp.single_le_iff, Finsupp.mem_support_iff, Nat.succ_le_iff, pos_iff_ne_zero,\n one_mul]"
}
] |
[
804,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsLocalization.coeffIntegerNormalization_of_not_mem_support
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.3630\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nh : coeff p i = 0\n⊢ coeffIntegerNormalization M p i = 0",
"tactic": "simp only [coeffIntegerNormalization, h, mem_support_iff, eq_self_iff_true, not_true, Ne.def,\n dif_neg, not_false_iff]"
}
] |
[
62,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.coprime_pow_primes
|
[] |
[
680,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
678,
1
] |
src/lean/Init/WF.lean
|
Nat.strongInductionOn
|
[] |
[
173,
28
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
169,
11
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_neg_iff
|
[] |
[
178,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Polynomial/Derivative.lean
|
Polynomial.derivative_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑derivative (X ^ n) = ↑C ↑n * X ^ (n - 1)",
"tactic": "convert derivative_C_mul_X_pow (1 : R) n <;> simp"
}
] |
[
115,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.range_top_iff_surjective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Subring R\nf : R →+* S\n⊢ ↑(range f) = ↑⊤ ↔ Set.range ↑f = Set.univ",
"tactic": "rw [coe_range, coe_top]"
}
] |
[
1214,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1212,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.mkUnits_eq
|
[] |
[
478,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.ModEq.cancel_right_div_gcd
|
[
{
"state_after": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"state_before": "m n a b c d : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"tactic": "letI d := gcd m c"
},
{
"state_after": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"state_before": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"tactic": "have hmd := gcd_dvd_left m c"
},
{
"state_after": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"state_before": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"tactic": "have hcd := gcd_dvd_right m c"
},
{
"state_after": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑(gcd m c) ∣ b - a",
"state_before": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : a * c ≡ b * c [ZMOD m]\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ a ≡ b [ZMOD m / ↑(gcd m c)]",
"tactic": "rw [modEq_iff_dvd] at h ⊢"
},
{
"state_after": "case refine_1\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑d ∣ c / ↑d * (b - a)\n\ncase refine_2\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ gcd (m / ↑(gcd m c)) (c / ↑d) = 1",
"state_before": "m n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑(gcd m c) ∣ b - a",
"tactic": "refine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d ∣ c / d * (b - a)) ?_"
},
{
"state_after": "case refine_1\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑d ∣ (b * c - a * c) / ↑(gcd m c)",
"state_before": "case refine_1\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑d ∣ c / ↑d * (b - a)",
"tactic": "rw [mul_comm, ← Int.mul_ediv_assoc (b - a) hcd, sub_mul]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ m / ↑d ∣ (b * c - a * c) / ↑(gcd m c)",
"tactic": "exact Int.ediv_dvd_ediv hmd h"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nm n a b c d✝ : ℤ\nhm : 0 < m\nh : m ∣ b * c - a * c\nd : ℕ := gcd m c\nhmd : ↑(gcd m c) ∣ m\nhcd : ↑(gcd m c) ∣ c\n⊢ gcd (m / ↑(gcd m c)) (c / ↑d) = 1",
"tactic": "rw [gcd_div hmd hcd, natAbs_ofNat, Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')]"
}
] |
[
236,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_inf
|
[] |
[
606,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.subset_image₂
|
[
{
"state_after": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ ∅ ⊆ image₂ f s' t'\n\ncase h₂\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∀ {a : γ} {s_1 : Finset γ},\n a ∈ u →\n s_1 ⊆ u →\n ¬a ∈ s_1 →\n (∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ s_1 ⊆ image₂ f s' t') →\n ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a s_1 ⊆ image₂ f s' t'",
"state_before": "α : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t'",
"tactic": "apply @Finset.induction_on' γ _ _ u"
},
{
"state_after": "case h₂.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"state_before": "case h₂\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∀ {a : γ} {s_1 : Finset γ},\n a ∈ u →\n s_1 ⊆ u →\n ¬a ∈ s_1 →\n (∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ s_1 ⊆ image₂ f s' t') →\n ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a s_1 ⊆ image₂ f s' t'",
"tactic": "rintro a u ha _ _ ⟨s', t', hs, hs', h⟩"
},
{
"state_after": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"state_before": "case h₂.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"tactic": "obtain ⟨x, y, hx, hy, ha⟩ := hu ha"
},
{
"state_after": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis : DecidableEq α\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"state_before": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"tactic": "haveI := Classical.decEq α"
},
{
"state_after": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"state_before": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis : DecidableEq α\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"tactic": "haveI := Classical.decEq β"
},
{
"state_after": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ ↑(insert x s') ⊆ s ∧ ↑(insert y t') ⊆ t ∧ insert a u ⊆ image₂ f (insert x s') (insert y t')",
"state_before": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ insert a u ⊆ image₂ f s' t'",
"tactic": "refine' ⟨insert x s', insert y t', _⟩"
},
{
"state_after": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ (x ∈ s ∧ ↑s' ⊆ s) ∧ (y ∈ t ∧ ↑t' ⊆ t) ∧ insert a u ⊆ image₂ f (insert x s') (insert y t')",
"state_before": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ ↑(insert x s') ⊆ s ∧ ↑(insert y t') ⊆ t ∧ insert a u ⊆ image₂ f (insert x s') (insert y t')",
"tactic": "simp_rw [coe_insert, Set.insert_subset]"
},
{
"state_after": "no goals",
"state_before": "case h₂.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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✝² a' : α\nb b' : β\nc : γ\ns : Set α\nt : Set β\nhu : ↑u✝ ⊆ image2 f s t\na : γ\nu : Finset γ\nha✝ : a ∈ u✝\na✝¹ : u ⊆ u✝\na✝ : ¬a ∈ u\ns' : Finset α\nt' : Finset β\nhs : ↑s' ⊆ s\nhs' : ↑t' ⊆ t\nh : u ⊆ image₂ f s' t'\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\nha : f x y = a\nthis✝ : DecidableEq α\nthis : DecidableEq β\n⊢ (x ∈ s ∧ ↑s' ⊆ s) ∧ (y ∈ t ∧ ↑t' ⊆ t) ∧ insert a u ⊆ image₂ f (insert x s') (insert y t')",
"tactic": "exact\n ⟨⟨hx, hs⟩, ⟨hy, hs'⟩,\n insert_subset.2\n ⟨mem_image₂.2 ⟨x, y, mem_insert_self _ _, mem_insert_self _ _, ha⟩,\n h.trans <| image₂_subset (subset_insert _ _) <| subset_insert _ _⟩⟩"
},
{
"state_after": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∃ t', ↑∅ ⊆ s ∧ ↑t' ⊆ t ∧ ∅ ⊆ image₂ f ∅ t'",
"state_before": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ ∅ ⊆ image₂ f s' t'",
"tactic": "use ∅"
},
{
"state_after": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ↑∅ ⊆ s ∧ ↑∅ ⊆ t ∧ ∅ ⊆ image₂ f ∅ ∅",
"state_before": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∃ t', ↑∅ ⊆ s ∧ ↑t' ⊆ t ∧ ∅ ⊆ image₂ f ∅ t'",
"tactic": "use ∅"
},
{
"state_after": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∅ ⊆ s ∧ ∅ ⊆ t ∧ ∅ ⊆ image₂ f ∅ ∅",
"state_before": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ↑∅ ⊆ s ∧ ↑∅ ⊆ t ∧ ∅ ⊆ image₂ f ∅ ∅",
"tactic": "simp only [coe_empty]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nα : Type u_1\nα' : Type ?u.47258\nβ : Type u_2\nβ' : Type ?u.47264\nγ : Type u_3\nγ' : Type ?u.47270\nδ : Type ?u.47273\nδ' : Type ?u.47276\nε : Type ?u.47279\nε' : Type ?u.47282\nζ : Type ?u.47285\nζ' : Type ?u.47288\nν : Type ?u.47291\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\ns : Set α\nt : Set β\nhu : ↑u ⊆ image2 f s t\n⊢ ∅ ⊆ s ∧ ∅ ⊆ t ∧ ∅ ⊆ image₂ f ∅ ∅",
"tactic": "exact ⟨Set.empty_subset _, Set.empty_subset _, empty_subset _⟩"
}
] |
[
252,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.eqvGen_mono
|
[] |
[
268,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Rel.lean
|
Rel.dom_inv
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1450\nr : Rel α β\nx : β\n⊢ x ∈ dom (inv r) ↔ x ∈ codom r",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1450\nr : Rel α β\n⊢ dom (inv r) = codom r",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1450\nr : Rel α β\nx : β\n⊢ x ∈ dom (inv r) ↔ x ∈ codom r",
"tactic": "rfl"
}
] |
[
88,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Fork.app_one_eq_ι_comp_left
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\ns : Fork f g\n⊢ s.π.app one = ι s ≫ f",
"tactic": "rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left]"
}
] |
[
341,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.toNonUnitalRingHom_commutes
|
[] |
[
702,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.argmin_mem
|
[] |
[
158,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.r.isEquiv
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝ : M × { x // x ∈ S }\nm : M\ns : { x // x ∈ S }\n⊢ 1 • (m, s).snd • (m, s).fst = 1 • (m, s).snd • (m, s).fst",
"tactic": "rw [one_smul]"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × { x // x ∈ S }\nm1 : M\ns1 : { x // x ∈ S }\nm2 : M\ns2 : { x // x ∈ S }\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x ∈ S }\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : { x // x ∈ S }\nhu1 : u1 • (m2, s2).snd • (m1, s1).fst = u1 • (m1, s1).snd • (m2, s2).fst\nu2 : { x // x ∈ S }\nhu2 : u2 • (m3, s3).snd • (m2, s2).fst = u2 • (m2, s2).snd • (m3, s3).fst\n⊢ (u1 * u2 * s2) • (m3, s3).snd • (m1, s1).fst = (u1 * u2 * s2) • (m1, s1).snd • (m3, s3).fst",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × { x // x ∈ S }\nm1 : M\ns1 : { x // x ∈ S }\nm2 : M\ns2 : { x // x ∈ S }\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x ∈ S }\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : { x // x ∈ S }\nhu1 : u1 • (m2, s2).snd • (m1, s1).fst = u1 • (m1, s1).snd • (m2, s2).fst\nu2 : { x // x ∈ S }\nhu2 : u2 • (m3, s3).snd • (m2, s2).fst = u2 • (m2, s2).snd • (m3, s3).fst\n⊢ r S M (m1, s1) (m3, s3)",
"tactic": "use u1 * u2 * s2"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × { x // x ∈ S }\nm1 : M\ns1 : { x // x ∈ S }\nm2 : M\ns2 : { x // x ∈ S }\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x ∈ S }\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : { x // x ∈ S }\nhu1 : u1 • (m2, s2).snd • (m1, s1).fst = u1 • (m1, s1).snd • (m2, s2).fst\nu2 : { x // x ∈ S }\nhu2 : u2 • (m3, s3).snd • (m2, s2).fst = u2 • (m2, s2).snd • (m3, s3).fst\nhu1' : (s1 * (s3 * (u1 * u2))) • m2 = (s2 * (s3 * (u1 * u2))) • m1\nhu2' : (s1 * (s2 * (u1 * u2))) • m3 = (s1 * (s3 * (u1 * u2))) • m2\n⊢ (s2 * (s3 * (u1 * u2))) • m1 = (s1 * (s2 * (u1 * u2))) • m3",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × { x // x ∈ S }\nm1 : M\ns1 : { x // x ∈ S }\nm2 : M\ns2 : { x // x ∈ S }\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x ∈ S }\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : { x // x ∈ S }\nhu1 : u1 • (m2, s2).snd • (m1, s1).fst = u1 • (m1, s1).snd • (m2, s2).fst\nu2 : { x // x ∈ S }\nhu2 : u2 • (m3, s3).snd • (m2, s2).fst = u2 • (m2, s2).snd • (m3, s3).fst\nhu1' :\n (fun x x_1 => x • x_1) (u2 * s3) (u1 • (m1, s1).snd • (m2, s2).fst) =\n (fun x x_1 => x • x_1) (u2 * s3) (u1 • (m2, s2).snd • (m1, s1).fst)\nhu2' :\n (fun x x_1 => x • x_1) (u1 * s1) (u2 • (m2, s2).snd • (m3, s3).fst) =\n (fun x x_1 => x • x_1) (u1 * s1) (u2 • (m3, s3).snd • (m2, s2).fst)\n⊢ (u1 * u2 * s2) • (m3, s3).snd • (m1, s1).fst = (u1 * u2 * s2) • (m1, s1).snd • (m3, s3).fst",
"tactic": "simp only [← mul_smul, smul_assoc, mul_assoc, mul_comm, mul_left_comm] at hu1' hu2'⊢"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × { x // x ∈ S }\nm1 : M\ns1 : { x // x ∈ S }\nm2 : M\ns2 : { x // x ∈ S }\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x ∈ S }\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : { x // x ∈ S }\nhu1 : u1 • (m2, s2).snd • (m1, s1).fst = u1 • (m1, s1).snd • (m2, s2).fst\nu2 : { x // x ∈ S }\nhu2 : u2 • (m3, s3).snd • (m2, s2).fst = u2 • (m2, s2).snd • (m3, s3).fst\nhu1' : (s1 * (s3 * (u1 * u2))) • m2 = (s2 * (s3 * (u1 * u2))) • m1\nhu2' : (s1 * (s2 * (u1 * u2))) • m3 = (s1 * (s3 * (u1 * u2))) • m2\n⊢ (s2 * (s3 * (u1 * u2))) • m1 = (s1 * (s2 * (u1 * u2))) • m3",
"tactic": "rw [hu2', hu1']"
}
] |
[
68,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.nilpotencyLength_eq_succ_iff
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ nilpotencyLength R L M = k + 1 ↔ lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\n⊢ nilpotencyLength R L M = k + 1 ↔ lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥",
"tactic": "let s := {k | lowerCentralSeries R L M k = ⊥}"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ nilpotencyLength R L M = k + 1 ↔ lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥",
"tactic": "change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"tactic": "have hs : ∀ k₁ k₂, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by\n rintro k₁ k₂ h₁₂ (h₁ : lowerCentralSeries R L M k₁ = ⊥)\n exact eq_bot_iff.mpr (h₁ ▸ antitone_lowerCentralSeries R L M h₁₂)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"tactic": "exact Nat.sInf_upward_closed_eq_succ_iff hs k"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : lowerCentralSeries R L M k₁ = ⊥\n⊢ k₂ ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s",
"tactic": "rintro k₁ k₂ h₁₂ (h₁ : lowerCentralSeries R L M k₁ = ⊥)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : lowerCentralSeries R L M k₁ = ⊥\n⊢ k₂ ∈ s",
"tactic": "exact eq_bot_iff.mpr (h₁ ▸ antitone_lowerCentralSeries R L M h₁₂)"
}
] |
[
283,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.list_prod_inv_reverse
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.356623\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (List.prod [])⁻¹ = List.prod (List.map Inv.inv (List.reverse []))",
"tactic": "rw [List.reverse_nil, List.map_nil, List.prod_nil, inv_one]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.356623\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nXs : List (Matrix n n α)\n⊢ (List.prod (A :: Xs))⁻¹ = List.prod (List.map Inv.inv (List.reverse (A :: Xs)))",
"tactic": "rw [List.reverse_cons', List.map_concat, List.prod_concat, List.prod_cons, Matrix.mul_eq_mul,\n Matrix.mul_eq_mul, mul_inv_rev, list_prod_inv_reverse Xs]"
}
] |
[
599,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.degree_toSubring
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.91195\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\n⊢ degree (toSubring p T hp) = degree p",
"tactic": "simp [degree]"
}
] |
[
379,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Order/Synonym.lean
|
OrderDual.forall
|
[] |
[
136,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
11
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.All.lowerBound?
|
[] |
[
244,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
243,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.isHomogeneous_X
|
[
{
"state_after": "case hn\nσ : Type u_1\nτ : Type ?u.153374\nR : Type u_2\nS : Type ?u.153380\ninst✝ : CommSemiring R\ni : σ\n⊢ ∑ i_1 in (Finsupp.single i 1).support, ↑(Finsupp.single i 1) i_1 = 1",
"state_before": "σ : Type u_1\nτ : Type ?u.153374\nR : Type u_2\nS : Type ?u.153380\ninst✝ : CommSemiring R\ni : σ\n⊢ IsHomogeneous (X i) 1",
"tactic": "apply isHomogeneous_monomial"
},
{
"state_after": "case hn\nσ : Type u_1\nτ : Type ?u.153374\nR : Type u_2\nS : Type ?u.153380\ninst✝ : CommSemiring R\ni : σ\n⊢ ↑(Finsupp.single i 1) i = 1",
"state_before": "case hn\nσ : Type u_1\nτ : Type ?u.153374\nR : Type u_2\nS : Type ?u.153380\ninst✝ : CommSemiring R\ni : σ\n⊢ ∑ i_1 in (Finsupp.single i 1).support, ↑(Finsupp.single i 1) i_1 = 1",
"tactic": "rw [Finsupp.support_single_ne_zero _ one_ne_zero, Finset.sum_singleton]"
},
{
"state_after": "no goals",
"state_before": "case hn\nσ : Type u_1\nτ : Type ?u.153374\nR : Type u_2\nS : Type ?u.153380\ninst✝ : CommSemiring R\ni : σ\n⊢ ↑(Finsupp.single i 1) i = 1",
"tactic": "exact Finsupp.single_eq_same"
}
] |
[
159,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.countable_biInf_principal_eq_seq_iInf
|
[] |
[
1045,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1043,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.linearMap_ext
|
[] |
[
404,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.snd_apply
|
[] |
[
84,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Option/NAry.lean
|
Option.map₂_coe_coe
|
[] |
[
57,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_one
|
[] |
[
972,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
971,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
UniformInducing.mk'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\nh : ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ ∃ t, t ∈ 𝓤 β ∧ ∀ (x y : α), (f x, f y) ∈ t → (x, y) ∈ s\n⊢ comap (fun x => (f x.fst, f x.snd)) (𝓤 β) = 𝓤 α",
"tactic": "simp [eq_comm, Filter.ext_iff, subset_def, h]"
}
] |
[
64,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_eq_right_iff
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type ?u.478404\nP2 : Type ?u.478407\nV3 : Type ?u.478410\nP3 : Type ?u.478413\nV4 : Type ?u.478416\nP4 : Type ?u.478419\ninst✝¹³ : Ring k\ninst✝¹² : AddCommGroup V1\ninst✝¹¹ : Module k V1\ninst✝¹⁰ : AffineSpace V1 P1\ninst✝⁹ : AddCommGroup V2\ninst✝⁸ : Module k V2\ninst✝⁷ : AffineSpace V2 P2\ninst✝⁶ : AddCommGroup V3\ninst✝⁵ : Module k V3\ninst✝⁴ : AffineSpace V3 P3\ninst✝³ : AddCommGroup V4\ninst✝² : Module k V4\ninst✝¹ : AffineSpace V4 P4\ninst✝ : NoZeroSMulDivisors k V1\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap p₀ p₁) c = p₁ ↔ p₀ = p₁ ∨ c = 1",
"tactic": "rw [← @lineMap_eq_lineMap_iff k V1, lineMap_apply_one]"
}
] |
[
587,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
585,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.IsZero.of_mono_zero
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Mono 0\n⊢ 𝟙 X ≫ 0 = 0 ≫ 0",
"tactic": "simp"
}
] |
[
198,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/LinearAlgebra/AdicCompletion.lean
|
adicCompletion.eval_comp_of
|
[] |
[
249,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.measurePreserving_div_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.367252\nG : Type u_1\nH : Type ?u.367258\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableSpace H\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\nμ : Measure G\ninst✝ : IsMulRightInvariant μ\ng : G\n⊢ MeasurePreserving fun x => x / g",
"tactic": "simp_rw [div_eq_mul_inv, measurePreserving_mul_right μ g⁻¹]"
}
] |
[
270,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_zero_eq_one'
|
[] |
[
920,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.cospanCompIso_hom_app_one
|
[] |
[
313,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.coe_erase
|
[] |
[
220,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_id
|
[] |
[
2067,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2066,
1
] |
Mathlib/CategoryTheory/Opposites.lean
|
CategoryTheory.unop_inv
|
[
{
"state_after": "case hom_inv_id\nC : Type u₁\ninst : Category C\nX Y : Cᵒᵖ\nf : X ⟶ Y\ninst_1 : IsIso f\n⊢ f.unop ≫ (inv f).unop = 𝟙 Y.unop",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nX Y : Cᵒᵖ\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ (inv f).unop = inv f.unop",
"tactic": "aesop_cat_nonterminal"
},
{
"state_after": "no goals",
"state_before": "case hom_inv_id\nC : Type u₁\ninst : Category C\nX Y : Cᵒᵖ\nf : X ⟶ Y\ninst_1 : IsIso f\n⊢ f.unop ≫ (inv f).unop = 𝟙 Y.unop",
"tactic": "rw [← unop_comp, IsIso.inv_hom_id, unop_id]"
}
] |
[
171,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_biUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.498019\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nb : β\n⊢ b ∈ Finset.biUnion s t ↔ ∃ a, a ∈ s ∧ b ∈ t a",
"tactic": "simp only [mem_def, biUnion_val, mem_dedup, mem_bind, exists_prop]"
}
] |
[
3541,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3540,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.weightedSMul_zero_measure
|
[
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.11618\nF : Type u_2\n𝕜 : Type ?u.11624\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nx✝ : Set α\n⊢ weightedSMul 0 x✝ = OfNat.ofNat 0 x✝",
"state_before": "α : Type u_1\nE : Type ?u.11618\nF : Type u_2\n𝕜 : Type ?u.11624\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\n⊢ weightedSMul 0 = 0",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nE : Type ?u.11618\nF : Type u_2\n𝕜 : Type ?u.11624\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nx✝ : Set α\n⊢ weightedSMul 0 x✝ = OfNat.ofNat 0 x✝",
"tactic": "simp [weightedSMul]"
}
] |
[
179,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.left_mem_affineSpan
|
[] |
[
764,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
IsRightRegular.ne_zero
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝¹ : MulZeroClass R\na b : R\ninst✝ : Nontrivial R\nra : IsRightRegular a\n⊢ a ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\nx y : R\nxy : x ≠ y\n⊢ False",
"state_before": "R : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\n⊢ False",
"tactic": "rcases exists_pair_ne R with ⟨x, y, xy⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\nx y : R\nxy : x ≠ y\n⊢ x * 0 = y * 0",
"state_before": "case intro.intro\nR : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\nx y : R\nxy : x ≠ y\n⊢ False",
"tactic": "refine' xy (ra (_ : x * 0 = y * 0))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝¹ : MulZeroClass R\nb : R\ninst✝ : Nontrivial R\nra : IsRightRegular 0\nx y : R\nxy : x ≠ y\n⊢ x * 0 = y * 0",
"tactic": "rw [mul_zero, mul_zero]"
}
] |
[
250,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.toDualProd_injective
|
[] |
[
79,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Analysis/Complex/RealDeriv.lean
|
HasDerivWithinAt.complexToReal_fderiv
|
[
{
"state_after": "no goals",
"state_before": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nE : Type ?u.137741\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\ns : Set ℂ\nf' x : ℂ\nh : HasDerivWithinAt f f' s x\n⊢ HasFDerivWithinAt f (f' • 1) s x",
"tactic": "simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivWithinAt.restrictScalars ℝ"
}
] |
[
133,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Algebra/Homology/ImageToKernel.lean
|
homology.map_comp
|
[
{
"state_after": "case p\nι : Type ?u.138713\nV : Type u\ninst✝¹⁹ : Category V\ninst✝¹⁸ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ninst✝¹⁷ : HasImage f\ng : B ⟶ C\ninst✝¹⁶ : HasKernel g\nw : f ≫ g = 0\nA' B' C' : V\nf' : A' ⟶ B'\ninst✝¹⁵ : HasImage f'\ng' : B' ⟶ C'\ninst✝¹⁴ : HasKernel g'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\ninst✝¹³ : HasImageMap α\nβ : Arrow.mk g ⟶ Arrow.mk g'\nA₁ B₁ C₁ : V\nf₁ : A₁ ⟶ B₁\ninst✝¹² : HasImage f₁\ng₁ : B₁ ⟶ C₁\ninst✝¹¹ : HasKernel g₁\nw₁ : f₁ ≫ g₁ = 0\nA₂ B₂ C₂ : V\nf₂ : A₂ ⟶ B₂\ninst✝¹⁰ : HasImage f₂\ng₂ : B₂ ⟶ C₂\ninst✝⁹ : HasKernel g₂\nw₂ : f₂ ≫ g₂ = 0\nA₃ B₃ C₃ : V\nf₃ : A₃ ⟶ B₃\ninst✝⁸ : HasImage f₃\ng₃ : B₃ ⟶ C₃\ninst✝⁷ : HasKernel g₃\nw₃ : f₃ ≫ g₃ = 0\nα₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂\ninst✝⁶ : HasImageMap α₁\nβ₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂\nα₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃\ninst✝⁵ : HasImageMap α₂\nβ₂ : Arrow.mk g₂ ⟶ Arrow.mk g₃\ninst✝⁴ : HasCokernel (imageToKernel f g w)\ninst✝³ : HasCokernel (imageToKernel f' g' w')\ninst✝² : HasCokernel (imageToKernel f₁ g₁ w₁)\ninst✝¹ : HasCokernel (imageToKernel f₂ g₂ w₂)\ninst✝ : HasCokernel (imageToKernel f₃ g₃ w₃)\np₁ : α₁.right = β₁.left\np₂ : α₂.right = β₂.left\n⊢ π f₁ g₁ w₁ ≫ map w₁ w₂ α₁ β₁ p₁ ≫ map w₂ w₃ α₂ β₂ p₂ =\n π f₁ g₁ w₁ ≫ map w₁ w₃ (α₁ ≫ α₂) (β₁ ≫ β₂) (_ : (α₁ ≫ α₂).right = (β₁ ≫ β₂).left)",
"state_before": "ι : Type ?u.138713\nV : Type u\ninst✝¹⁹ : Category V\ninst✝¹⁸ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ninst✝¹⁷ : HasImage f\ng : B ⟶ C\ninst✝¹⁶ : HasKernel g\nw : f ≫ g = 0\nA' B' C' : V\nf' : A' ⟶ B'\ninst✝¹⁵ : HasImage f'\ng' : B' ⟶ C'\ninst✝¹⁴ : HasKernel g'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\ninst✝¹³ : HasImageMap α\nβ : Arrow.mk g ⟶ Arrow.mk g'\nA₁ B₁ C₁ : V\nf₁ : A₁ ⟶ B₁\ninst✝¹² : HasImage f₁\ng₁ : B₁ ⟶ C₁\ninst✝¹¹ : HasKernel g₁\nw₁ : f₁ ≫ g₁ = 0\nA₂ B₂ C₂ : V\nf₂ : A₂ ⟶ B₂\ninst✝¹⁰ : HasImage f₂\ng₂ : B₂ ⟶ C₂\ninst✝⁹ : HasKernel g₂\nw₂ : f₂ ≫ g₂ = 0\nA₃ B₃ C₃ : V\nf₃ : A₃ ⟶ B₃\ninst✝⁸ : HasImage f₃\ng₃ : B₃ ⟶ C₃\ninst✝⁷ : HasKernel g₃\nw₃ : f₃ ≫ g₃ = 0\nα₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂\ninst✝⁶ : HasImageMap α₁\nβ₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂\nα₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃\ninst✝⁵ : HasImageMap α₂\nβ₂ : Arrow.mk g₂ ⟶ Arrow.mk g₃\ninst✝⁴ : HasCokernel (imageToKernel f g w)\ninst✝³ : HasCokernel (imageToKernel f' g' w')\ninst✝² : HasCokernel (imageToKernel f₁ g₁ w₁)\ninst✝¹ : HasCokernel (imageToKernel f₂ g₂ w₂)\ninst✝ : HasCokernel (imageToKernel f₃ g₃ w₃)\np₁ : α₁.right = β₁.left\np₂ : α₂.right = β₂.left\n⊢ map w₁ w₂ α₁ β₁ p₁ ≫ map w₂ w₃ α₂ β₂ p₂ = map w₁ w₃ (α₁ ≫ α₂) (β₁ ≫ β₂) (_ : (α₁ ≫ α₂).right = (β₁ ≫ β₂).left)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case p\nι : Type ?u.138713\nV : Type u\ninst✝¹⁹ : Category V\ninst✝¹⁸ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ninst✝¹⁷ : HasImage f\ng : B ⟶ C\ninst✝¹⁶ : HasKernel g\nw : f ≫ g = 0\nA' B' C' : V\nf' : A' ⟶ B'\ninst✝¹⁵ : HasImage f'\ng' : B' ⟶ C'\ninst✝¹⁴ : HasKernel g'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\ninst✝¹³ : HasImageMap α\nβ : Arrow.mk g ⟶ Arrow.mk g'\nA₁ B₁ C₁ : V\nf₁ : A₁ ⟶ B₁\ninst✝¹² : HasImage f₁\ng₁ : B₁ ⟶ C₁\ninst✝¹¹ : HasKernel g₁\nw₁ : f₁ ≫ g₁ = 0\nA₂ B₂ C₂ : V\nf₂ : A₂ ⟶ B₂\ninst✝¹⁰ : HasImage f₂\ng₂ : B₂ ⟶ C₂\ninst✝⁹ : HasKernel g₂\nw₂ : f₂ ≫ g₂ = 0\nA₃ B₃ C₃ : V\nf₃ : A₃ ⟶ B₃\ninst✝⁸ : HasImage f₃\ng₃ : B₃ ⟶ C₃\ninst✝⁷ : HasKernel g₃\nw₃ : f₃ ≫ g₃ = 0\nα₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂\ninst✝⁶ : HasImageMap α₁\nβ₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂\nα₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃\ninst✝⁵ : HasImageMap α₂\nβ₂ : Arrow.mk g₂ ⟶ Arrow.mk g₃\ninst✝⁴ : HasCokernel (imageToKernel f g w)\ninst✝³ : HasCokernel (imageToKernel f' g' w')\ninst✝² : HasCokernel (imageToKernel f₁ g₁ w₁)\ninst✝¹ : HasCokernel (imageToKernel f₂ g₂ w₂)\ninst✝ : HasCokernel (imageToKernel f₃ g₃ w₃)\np₁ : α₁.right = β₁.left\np₂ : α₂.right = β₂.left\n⊢ π f₁ g₁ w₁ ≫ map w₁ w₂ α₁ β₁ p₁ ≫ map w₂ w₃ α₂ β₂ p₂ =\n π f₁ g₁ w₁ ≫ map w₁ w₃ (α₁ ≫ α₂) (β₁ ≫ β₂) (_ : (α₁ ≫ α₂).right = (β₁ ≫ β₂).left)",
"tactic": "simp only [kernelSubobjectMap_comp, homology.π_map_assoc, homology.π_map, Category.assoc]"
}
] |
[
319,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.neg_ne_self_iff
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ -θ ≠ θ ↔ θ ≠ 0 ∧ θ ≠ ↑π",
"tactic": "rw [← not_or, ← neg_eq_self_iff.not]"
}
] |
[
226,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Topology/Maps.lean
|
quotientMap_iff_closed
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.136882\nδ : Type ?u.136885\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\n⊢ (∀ (x : Set β), IsOpen (xᶜ) ↔ IsOpen (f ⁻¹' xᶜ)) ↔ ∀ (s : Set β), IsClosed s ↔ IsClosed (f ⁻¹' s)",
"tactic": "simp only [isOpen_compl_iff, preimage_compl]"
}
] |
[
284,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mul_add_subset
|
[] |
[
1075,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1074,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_le_csSup
|
[] |
[
467,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Combinatorics/Derangements/Finite.lean
|
card_derangements_fin_add_two
|
[
{
"state_after": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh1 : ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)\n⊢ card ↑(derangements (Fin (n + 2))) =\n (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1)))",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\n⊢ card ↑(derangements (Fin (n + 2))) =\n (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1)))",
"tactic": "have h1 : ∀ a : Fin (n + 1), card ({a}ᶜ : Set (Fin (n + 1))) = card (Fin n) := by\n intro a\n simp only [Fintype.card_fin, Finset.card_fin, Fintype.card_ofFinset, Finset.filter_ne' _ a,\n Set.mem_compl_singleton_iff, Finset.card_erase_of_mem (Finset.mem_univ a),\n add_tsub_cancel_right]"
},
{
"state_after": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh1 : ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)\nh2 : card (Fin (n + 2)) = card (Option (Fin (n + 1)))\n⊢ card ↑(derangements (Fin (n + 2))) =\n (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1)))",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh1 : ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)\n⊢ card ↑(derangements (Fin (n + 2))) =\n (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1)))",
"tactic": "have h2 : card (Fin (n + 2)) = card (Option (Fin (n + 1))) := by simp only [card_fin, card_option]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh1 : ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)\nh2 : card (Fin (n + 2)) = card (Option (Fin (n + 1)))\n⊢ card ↑(derangements (Fin (n + 2))) =\n (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1)))",
"tactic": "simp only [card_derangements_invariant h2,\n card_congr\n (@derangementsRecursionEquiv (Fin (n + 1))\n _),card_sigma,\n card_sum, card_derangements_invariant (h1 _), Finset.sum_const, nsmul_eq_mul, Finset.card_fin,\n mul_add, Nat.cast_id]"
},
{
"state_after": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\na : Fin (n + 1)\n⊢ card ↑({a}ᶜ) = card (Fin n)",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\n⊢ ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\na : Fin (n + 1)\n⊢ card ↑({a}ᶜ) = card (Fin n)",
"tactic": "simp only [Fintype.card_fin, Finset.card_fin, Fintype.card_ofFinset, Finset.filter_ne' _ a,\n Set.mem_compl_singleton_iff, Finset.card_erase_of_mem (Finset.mem_univ a),\n add_tsub_cancel_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.2521\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh1 : ∀ (a : Fin (n + 1)), card ↑({a}ᶜ) = card (Fin n)\n⊢ card (Fin (n + 2)) = card (Option (Fin (n + 1)))",
"tactic": "simp only [card_fin, card_option]"
}
] |
[
66,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.find_spec
|
[
{
"state_after": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\n⊢ p i",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi : i ∈ find p\n⊢ p i",
"tactic": "rw [find] at hi"
},
{
"state_after": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i\n\ncase some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\nj : Fin n\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"state_before": "m n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\n⊢ p i",
"tactic": "cases' h : find fun i : Fin n ↦ p (i.castLT (Nat.lt_succ_of_lt i.2)) with j"
},
{
"state_after": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn none (if x : p (last n) then some (last n) else none) fun i => some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i",
"state_before": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i",
"tactic": "rw [h] at hi"
},
{
"state_after": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi : i ∈ if p (last n) then some (last n) else none\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i",
"state_before": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn none (if x : p (last n) then some (last n) else none) fun i => some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i",
"tactic": "dsimp at hi"
},
{
"state_after": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last n)\nhi : i ∈ some (last n)\n⊢ p i\n\ncase none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last n)\nhi : i ∈ none\n⊢ p i",
"state_before": "case none\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi : i ∈ if p (last n) then some (last n) else none\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\n⊢ p i",
"tactic": "split_ifs at hi with hl"
},
{
"state_after": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last n)\nhi : last n = i\n⊢ p i",
"state_before": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last n)\nhi : i ∈ some (last n)\n⊢ p i",
"tactic": "simp only [Option.mem_def, Option.some.injEq] at hi"
},
{
"state_after": "no goals",
"state_before": "case none.inl\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : p (last n)\nhi : last n = i\n⊢ p i",
"tactic": "exact hi ▸ hl"
},
{
"state_after": "no goals",
"state_before": "case none.inr\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = none\nhl : ¬p (last n)\nhi : i ∈ none\n⊢ p i",
"tactic": "exact (Option.not_mem_none _ hi).elim"
},
{
"state_after": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi :\n i ∈\n Option.casesOn (some j) (if x : p (last n) then some (last n) else none) fun i =>\n some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"state_before": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nhi :\n i ∈\n Option.casesOn (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) (if x : p (last n) then some (last n) else none)\n fun i => some (castLT i (_ : ↑i < Nat.succ n))\nj : Fin n\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"tactic": "rw [h] at hi"
},
{
"state_after": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi : i ∈ some (castLT j (_ : ↑j < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"state_before": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi :\n i ∈\n Option.casesOn (some j) (if x : p (last n) then some (last n) else none) fun i =>\n some (castLT i (_ : ↑i < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"tactic": "dsimp at hi"
},
{
"state_after": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi : i ∈ some (castLT j (_ : ↑j < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p (castLT j (_ : ↑j < Nat.succ n))",
"state_before": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi : i ∈ some (castLT j (_ : ↑j < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p i",
"tactic": "rw [← Option.some_inj.1 hi]"
},
{
"state_after": "no goals",
"state_before": "case some\nm n✝ n : ℕ\np : Fin (n + 1) → Prop\nI : DecidablePred p\ni : Fin (n + 1)\nj : Fin n\nhi : i ∈ some (castLT j (_ : ↑j < Nat.succ n))\nh : (find fun i => p (castLT i (_ : ↑i < Nat.succ n))) = some j\n⊢ p (castLT j (_ : ↑j < Nat.succ n))",
"tactic": "refine @find_spec n (fun i ↦ p (i.castLT (Nat.lt_succ_of_lt i.2))) _ _ h"
}
] |
[
864,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
849,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.add_one_le_iff
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ∞\nhm : m ≠ ⊤\n⊢ ¬IsMax m",
"tactic": "rwa [isMax_iff_eq_top]"
}
] |
[
193,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.edges_append
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\np' : Walk G v w\n⊢ edges (append p p') = edges p ++ edges p'",
"tactic": "simp [edges]"
}
] |
[
762,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.nnnorm_zero
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.407281\nG : Type ?u.407284\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ ENNReal.toNNReal (snorm (↑↑0) p μ) = 0",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.407281\nG : Type ?u.407284\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ ‖0‖₊ = 0",
"tactic": "rw [nnnorm_def]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.407281\nG : Type ?u.407284\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ ENNReal.toNNReal (snorm (↑0) p μ) = 0",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.407281\nG : Type ?u.407284\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ ENNReal.toNNReal (snorm (↑↑0) p μ) = 0",
"tactic": "change (snorm (⇑(0 : α →ₘ[μ] E)) p μ).toNNReal = 0"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.407281\nG : Type ?u.407284\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ ENNReal.toNNReal (snorm (↑0) p μ) = 0",
"tactic": "simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]"
}
] |
[
313,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.iterate_eq_of_map_eq_add_int
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : ↑f x = x + ↑m\nn : ℕ\n⊢ (↑f^[n]) x = x + ↑n * ↑m",
"tactic": "simpa only [nsmul_eq_mul, add_right_iterate] using (f.commute_add_int m).iterate_eq_of_map_eq n h"
}
] |
[
585,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Data/Quot.lean
|
true_equivalence
|
[] |
[
449,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
uniformity_eq_comap_nhds_one_swapped
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.164871\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ comap ((fun x => x.snd / x.fst) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.fst / x.snd) (𝓝 1)",
"state_before": "α : Type u_1\nβ : Type ?u.164871\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (fun x => x.fst / x.snd) (𝓝 1)",
"tactic": "rw [← comap_swap_uniformity, uniformity_eq_comap_nhds_one, comap_comap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.164871\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ comap ((fun x => x.snd / x.fst) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.fst / x.snd) (𝓝 1)",
"tactic": "rfl"
}
] |
[
286,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.sdiff_adj
|
[] |
[
291,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Module.DualBases.lc_def
|
[] |
[
691,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
690,
1
] |
Mathlib/Order/RelClasses.lean
|
ssubset_of_ne_of_subset
|
[] |
[
765,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.continuous'
|
[] |
[
1170,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
11
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.isCompl
|
[] |
[
1369,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1368,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.coe_prod
|
[
{
"state_after": "case empty\nK : Type u_3\nL : Type u_2\nL' : Type ?u.112045\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\n⊢ ↑(∏ i in ∅, f i) = ∏ i in ∅, ↑(f i)\n\ncase insert\nK : Type u_3\nL : Type u_2\nL' : Type ?u.112045\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nH : ↑(∏ i in s, f i) = ∏ i in s, ↑(f i)\n⊢ ↑(∏ i in insert i s, f i) = ∏ i in insert i s, ↑(f i)",
"state_before": "K : Type u_3\nL : Type u_2\nL' : Type ?u.112045\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\n⊢ ↑(∏ i : ι, f i) = ∏ i : ι, ↑(f i)",
"tactic": "induction' (Finset.univ : Finset ι) using Finset.induction_on with i s hi H"
},
{
"state_after": "no goals",
"state_before": "case empty\nK : Type u_3\nL : Type u_2\nL' : Type ?u.112045\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\n⊢ ↑(∏ i in ∅, f i) = ∏ i in ∅, ↑(f i)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case insert\nK : Type u_3\nL : Type u_2\nL' : Type ?u.112045\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nH : ↑(∏ i in s, f i) = ∏ i in s, ↑(f i)\n⊢ ↑(∏ i in insert i s, f i) = ∏ i in insert i s, ↑(f i)",
"tactic": "rw [Finset.prod_insert hi, MulMemClass.coe_mul, H, Finset.prod_insert hi]"
}
] |
[
362,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_add_eq_right_of_degree_lt
|
[
{
"state_after": "no goals",
"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.530544\nh : degree p < degree q\n⊢ degree (p + q) = degree q",
"tactic": "rw [add_comm, degree_add_eq_left_of_degree_lt h]"
}
] |
[
698,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
697,
1
] |
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
AlgebraicTopology.AlternatingFaceMapComplex.obj_X
|
[] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/List/Func.lean
|
List.Func.get_pointwise
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\n⊢ get k (pointwise f [] []) = f (get k []) (get k [])",
"tactic": "simp only [h1, get_nil, pointwise, get]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nb : β\ntail✝ : List β\n⊢ get 0 (pointwise f [] (b :: tail✝)) = f (get 0 []) (get 0 (b :: tail✝))",
"tactic": "simp only [get_pointwise, get_nil, pointwise, get, Nat.zero_eq, map]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\nb : β\nbs : List β\nthis : get k (map (f default) bs) = f default (get k bs)\n⊢ get (k + 1) (pointwise f [] (b :: bs)) = f (get (k + 1) []) (get (k + 1) (b :: bs))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\nb : β\nbs : List β\n⊢ get (k + 1) (pointwise f [] (b :: bs)) = f (get (k + 1) []) (get (k + 1) (b :: bs))",
"tactic": "have : get k (map (f default) bs) = f default (get k bs) := by\n simpa [nil_pointwise, get_nil] using get_pointwise h1 k [] bs"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\nb : β\nbs : List β\nthis : get k (map (f default) bs) = f default (get k bs)\n⊢ get (k + 1) (pointwise f [] (b :: bs)) = f (get (k + 1) []) (get (k + 1) (b :: bs))",
"tactic": "simpa [get, get_nil, pointwise, map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\nb : β\nbs : List β\n⊢ get k (map (f default) bs) = f default (get k bs)",
"tactic": "simpa [nil_pointwise, get_nil] using get_pointwise h1 k [] bs"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\na : α\ntail✝ : List α\n⊢ get 0 (pointwise f (a :: tail✝) []) = f (get 0 (a :: tail✝)) (get 0 [])",
"tactic": "simp only [get_pointwise, get_nil, pointwise, get, Nat.zero_eq, map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\na : α\nas : List α\n⊢ get (k + 1) (pointwise f (a :: as) []) = f (get (k + 1) (a :: as)) (get (k + 1) [])",
"tactic": "simpa [get, get_nil, pointwise, map, pointwise_nil, get_nil] using get_pointwise h1 k as []"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\na : α\ntail✝¹ : List α\nb : β\ntail✝ : List β\n⊢ get 0 (pointwise f (a :: tail✝¹) (b :: tail✝)) = f (get 0 (a :: tail✝¹)) (get 0 (b :: tail✝))",
"tactic": "simp only [pointwise, get]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Inhabited α\ninst✝¹ : Inhabited β\ninst✝ : Inhabited γ\nf : α → β → γ\nh1 : f default default = default\nk : ℕ\nhead✝¹ : α\nas : List α\nhead✝ : β\nbs : List β\n⊢ get (k + 1) (pointwise f (head✝¹ :: as) (head✝ :: bs)) = f (get (k + 1) (head✝¹ :: as)) (get (k + 1) (head✝ :: bs))",
"tactic": "simp only [get, Nat.add_eq, add_zero, get_pointwise h1 k as bs]"
}
] |
[
301,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_C_mul_X
|
[
{
"state_after": "no goals",
"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.618012\na : R\n⊢ leadingCoeff (↑C a * X) = a",
"tactic": "simpa only [pow_one] using leadingCoeff_C_mul_X_pow a 1"
}
] |
[
807,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
div_ball
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s / ball x δ = x⁻¹ • thickening δ s",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
238,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.Rel.symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.124\nγ : Type ?u.127\nx y : α × α\n⊢ Rel α x y → Rel α y x",
"tactic": "aesop (rule_sets [Sym2])"
}
] |
[
76,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.coe_ndinsert
|
[] |
[
39,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Algebra/Group/OrderSynonym.lean
|
ofDual_mul
|
[] |
[
119,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.coeFn_abs
|
[] |
[
1564,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1564,
1
] |
Std/Logic.lean
|
false_imp_iff
|
[] |
[
118,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
118,
1
] |
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