file_path
stringlengths 11
79
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stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.map_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns t : Seq α\n⊢ IsBisimulation fun s1 s2 => ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns t : Seq α\n⊢ map f (append s t) = append (map f s) (map f t)",
"tactic": "apply\n eq_of_bisim (fun s1 s2 => ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)) _\n ⟨s, t, rfl, rfl⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns t : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\n⊢ BisimO (fun s1 s2 => ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)) (destruct s1) (destruct s2)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns t : Seq α\n⊢ IsBisimulation fun s1 s2 => ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)",
"tactic": "intro s1 s2 h"
},
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ match destruct (map f t), destruct (map f t) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' ∧ ∃ s_1 t, s = map f (append s_1 t) ∧ s' = append (map f s_1) (map f t)\n | x, x_1 => False\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ α →\n ∀ (s : Seq α),\n ∃ s_1 t_1,\n map f (append s t) = map f (append s_1 t_1) ∧ append (map f s) (map f t) = append (map f s_1) (map f t_1)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ BisimO (fun s1 s2 => ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)) (destruct (map f (append s t)))\n (destruct (append (map f s) (map f t)))",
"tactic": "apply recOn s <;> simp"
},
{
"state_after": "case h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ α → ∀ (s : Seq α), ∃ s_1 t, map f s = map f (append s_1 t) ∧ map f s = append (map f s_1) (map f t)",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ match destruct (map f t), destruct (map f t) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' ∧ ∃ s_1 t, s = map f (append s_1 t) ∧ s' = append (map f s_1) (map f t)\n | x, x_1 => False",
"tactic": "apply recOn t <;> simp"
},
{
"state_after": "case h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝¹ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t✝ : Seq α\nx✝ : α\nt : Seq α\n⊢ ∃ s t_1, map f t = map f (append s t_1) ∧ map f t = append (map f s) (map f t_1)",
"state_before": "case h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ α → ∀ (s : Seq α), ∃ s_1 t, map f s = map f (append s_1 t) ∧ map f s = append (map f s_1) (map f t)",
"tactic": "intro _ t"
},
{
"state_after": "no goals",
"state_before": "case h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝¹ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t✝ : Seq α\nx✝ : α\nt : Seq α\n⊢ ∃ s t_1, map f t = map f (append s t_1) ∧ map f t = append (map f s) (map f t_1)",
"tactic": "refine' ⟨nil, t, _, _⟩ <;> simp"
},
{
"state_after": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝¹ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns✝ t : Seq α\nx✝ : α\ns : Seq α\n⊢ ∃ s_1 t_1, map f (append s t) = map f (append s_1 t_1) ∧ append (map f s) (map f t) = append (map f s_1) (map f t_1)",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns t : Seq α\n⊢ α →\n ∀ (s : Seq α),\n ∃ s_1 t_1,\n map f (append s t) = map f (append s_1 t_1) ∧ append (map f s) (map f t) = append (map f s_1) (map f t_1)",
"tactic": "intro _ s"
},
{
"state_after": "no goals",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝¹ t✝ : Seq α\ns1 s2 : Seq β\nh : ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)\ns✝ t : Seq α\nx✝ : α\ns : Seq α\n⊢ ∃ s_1 t_1, map f (append s t) = map f (append s_1 t_1) ∧ append (map f s) (map f t) = append (map f s_1) (map f t_1)",
"tactic": "refine' ⟨s, t, rfl, rfl⟩"
}
] |
[
736,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.symm_single
|
[] |
[
356,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.not_ge_eq
|
[] |
[
725,
16
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
724,
1
] |
Mathlib/Data/List/OfFn.lean
|
List.nthLe_ofFn
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nn : ℕ\nf : Fin n → α\ni : Fin n\n⊢ nthLe (ofFn f) ↑i (_ : ↑i < length (ofFn f)) = f i",
"tactic": "simp [nthLe]"
}
] |
[
76,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.toAddSubmonoid_injective
|
[] |
[
163,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_symm
|
[
{
"state_after": "case h\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\nx✝ : P\n⊢ ↑(AffineIsometryEquiv.symm (reflection s)) x✝ = ↑(reflection s) x✝",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\n⊢ AffineIsometryEquiv.symm (reflection s) = reflection s",
"tactic": "ext"
},
{
"state_after": "case h\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\nx✝ : P\n⊢ ↑(reflection s) (↑(AffineIsometryEquiv.symm (reflection s)) x✝) = ↑(reflection s) (↑(reflection s) x✝)",
"state_before": "case h\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\nx✝ : P\n⊢ ↑(AffineIsometryEquiv.symm (reflection s)) x✝ = ↑(reflection s) x✝",
"tactic": "rw [← (reflection s).injective.eq_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\nx✝ : P\n⊢ ↑(reflection s) (↑(AffineIsometryEquiv.symm (reflection s)) x✝) = ↑(reflection s) (↑(reflection s) x✝)",
"tactic": "simp"
}
] |
[
573,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PreservesPushout.inr_iso_hom
|
[
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nW X Y Z : C\nh : X ⟶ Z\nk : Y ⟶ Z\nf : W ⟶ X\ng : W ⟶ Y\ncomm : f ≫ h = g ≫ k\ninst✝² : PreservesColimit (span f g) G\ninst✝¹ : HasPushout f g\ninst✝ : HasPushout (G.map f) (G.map g)\n⊢ pushout.inr ≫\n (IsColimit.coconePointUniqueUpToIso (colimit.isColimit (span (G.map f) (G.map g)))\n (isColimitOfHasPushoutOfPreservesColimit G f g)).hom =\n G.map pushout.inr",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nW X Y Z : C\nh : X ⟶ Z\nk : Y ⟶ Z\nf : W ⟶ X\ng : W ⟶ Y\ncomm : f ≫ h = g ≫ k\ninst✝² : PreservesColimit (span f g) G\ninst✝¹ : HasPushout f g\ninst✝ : HasPushout (G.map f) (G.map g)\n⊢ pushout.inr ≫ (iso G f g).hom = G.map pushout.inr",
"tactic": "delta PreservesPushout.iso"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nW X Y Z : C\nh : X ⟶ Z\nk : Y ⟶ Z\nf : W ⟶ X\ng : W ⟶ Y\ncomm : f ≫ h = g ≫ k\ninst✝² : PreservesColimit (span f g) G\ninst✝¹ : HasPushout f g\ninst✝ : HasPushout (G.map f) (G.map g)\n⊢ pushout.inr ≫\n (IsColimit.coconePointUniqueUpToIso (colimit.isColimit (span (G.map f) (G.map g)))\n (isColimitOfHasPushoutOfPreservesColimit G f g)).hom =\n G.map pushout.inr",
"tactic": "simp"
}
] |
[
237,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.coe_add
|
[] |
[
198,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearEquiv.snd_comp_prodComm
|
[
{
"state_after": "no goals",
"state_before": "N : Type u_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.405020\nM₆ : Type ?u.405023\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\n⊢ LinearMap.comp (LinearMap.snd R N M) ↑(prodComm R M N) = LinearMap.fst R M N",
"tactic": "ext <;> simp"
}
] |
[
759,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.map_div_left_divisors
|
[
{
"state_after": "case a\nn : ℕ\n⊢ map (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))\n (map\n { toFun := fun d => (n / d, d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (n / d, d)) p₁ = (fun d => (n / d, d)) p₂ →\n ((fun d => (n / d, d)) p₁).snd = ((fun d => (n / d, d)) p₂).snd) }\n (divisors n)) =\n map (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) (divisorsAntidiagonal n)",
"state_before": "n : ℕ\n⊢ map\n { toFun := fun d => (n / d, d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (n / d, d)) p₁ = (fun d => (n / d, d)) p₂ →\n ((fun d => (n / d, d)) p₁).snd = ((fun d => (n / d, d)) p₂).snd) }\n (divisors n) =\n divisorsAntidiagonal n",
"tactic": "apply Finset.map_injective (Equiv.prodComm _ _).toEmbedding"
},
{
"state_after": "case a\nn : ℕ\n⊢ map\n (Function.Embedding.trans\n { toFun := fun d => (n / d, d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (n / d, d)) p₁ = (fun d => (n / d, d)) p₂ →\n ((fun d => (n / d, d)) p₁).snd = ((fun d => (n / d, d)) p₂).snd) }\n (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)))\n (divisors n) =\n map\n { toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n (divisors n)",
"state_before": "case a\nn : ℕ\n⊢ map (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))\n (map\n { toFun := fun d => (n / d, d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (n / d, d)) p₁ = (fun d => (n / d, d)) p₂ →\n ((fun d => (n / d, d)) p₁).snd = ((fun d => (n / d, d)) p₂).snd) }\n (divisors n)) =\n map (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) (divisorsAntidiagonal n)",
"tactic": "rw [map_swap_divisorsAntidiagonal, ← map_div_right_divisors, Finset.map_map]"
},
{
"state_after": "no goals",
"state_before": "case a\nn : ℕ\n⊢ map\n (Function.Embedding.trans\n { toFun := fun d => (n / d, d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (n / d, d)) p₁ = (fun d => (n / d, d)) p₂ →\n ((fun d => (n / d, d)) p₁).snd = ((fun d => (n / d, d)) p₂).snd) }\n (Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)))\n (divisors n) =\n map\n { toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n (divisors n)",
"tactic": "rfl"
}
] |
[
282,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
PseudoEpimorphism.coe_copy
|
[] |
[
143,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.map_sub
|
[] |
[
641,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.abs_toReal_eq_pi_div_two_iff
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ abs (toReal θ) = π / 2 ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)",
"tactic": "rw [abs_eq (div_nonneg Real.pi_pos.le two_pos.le), ← neg_div, toReal_eq_pi_div_two_iff,\n toReal_eq_neg_pi_div_two_iff]"
}
] |
[
660,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
TotallyBounded.image
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : s ⊆ ⋃ (y : α) (_ : y ∈ c), {x | (x, y) ∈ {p | (f p.fst, f p.snd) ∈ t}}\n⊢ s ⊆ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : s ⊆ ⋃ (y : α) (_ : y ∈ c), {x | (x, y) ∈ {p | (f p.fst, f p.snd) ∈ t}}\n⊢ f '' s ⊆ ⋃ (y : β) (_ : y ∈ f '' c), {x | (x, y) ∈ t}",
"tactic": "simp [image_subset_iff]"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\n⊢ s ⊆ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : s ⊆ ⋃ (y : α) (_ : y ∈ c), {x | (x, y) ∈ {p | (f p.fst, f p.snd) ∈ t}}\n⊢ s ⊆ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"tactic": "simp [subset_def] at hct"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\n⊢ s ⊆ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\nx : α\nhx : x ∈ s\n⊢ ∃ i, i ∈ c ∧ (f x, f i) ∈ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ c), {a | (f a, f i) ∈ t}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (β × β)\nht : t ∈ 𝓤 β\nthis : {p | (f p.fst, f p.snd) ∈ t} ∈ 𝓤 α\nc : Set α\nhfc : Set.Finite c\nhct : ∀ (x : α), x ∈ s → ∃ i, i ∈ c ∧ (f x, f i) ∈ t\nx : α\nhx : x ∈ s\n⊢ ∃ i, i ∈ c ∧ (f x, f i) ∈ t",
"tactic": "exact hct x hx"
}
] |
[
526,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.mul_adjugate_apply
|
[
{
"state_after": "no goals",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\ni j k : n\n⊢ A i k * adjugate A k j = ↑(cramer Aᵀ) (Pi.single k (A i k)) j",
"tactic": "erw [← smul_eq_mul, adjugate, of_apply, ← Pi.smul_apply, ← LinearMap.map_smul, ← Pi.single_smul',\n smul_eq_mul, mul_one]"
}
] |
[
289,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.projIcc_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7809\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx : α\nhx : x ∈ Icc a b\n⊢ projIcc a b h x = { val := x, property := hx }",
"tactic": "simp [projIcc, hx.1, hx.2]"
}
] |
[
74,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.ext_on_range
|
[] |
[
972,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
970,
1
] |
Mathlib/Data/Multiset/Functor.lean
|
Multiset.bind_def
|
[] |
[
78,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_symm_sum
|
[] |
[
175,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/LinearAlgebra/Orientation.lean
|
Orientation.map_symm
|
[] |
[
88,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/NormedSpace/LpEquiv.lean
|
coe_ringEquiv_lpBcf_symm
|
[] |
[
205,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_add_const_uIcc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ (fun x => x + a) ⁻¹' [[b, c]] = [[b - a, c - a]]",
"tactic": "simpa only [add_comm] using preimage_const_add_uIcc a b c"
}
] |
[
435,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.univ_mul
|
[] |
[
1264,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1262,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier
|
[] |
[
585,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.algebraMap_C
|
[] |
[
1401,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1400,
1
] |
Mathlib/Topology/Algebra/Star.lean
|
continuousAt_star
|
[] |
[
46,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.gcd_eq_gcd_ab
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nthis : EuclideanDomain.P a b (xgcdAux a 1 0 b 0 1)\n⊢ gcd a b = a * gcdA a b + b * gcdB a b",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ gcd a b = a * gcdA a b + b * gcdB a b",
"tactic": "have :=\n @xgcdAux_P _ _ _ a b a b 1 0 0 1 (by dsimp [P]; rw [mul_one, mul_zero, add_zero])\n (by dsimp [P]; rw [mul_one, mul_zero, zero_add])"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nthis : EuclideanDomain.P a b (xgcdAux a 1 0 b 0 1)\n⊢ gcd a b = a * gcdA a b + b * gcdB a b",
"tactic": "rwa [xgcdAux_val, xgcd_val] at this"
},
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ a = a * 1 + b * 0",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ EuclideanDomain.P a b (a, 1, 0)",
"tactic": "dsimp [P]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ a = a * 1 + b * 0",
"tactic": "rw [mul_one, mul_zero, add_zero]"
},
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = a * 0 + b * 1",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ EuclideanDomain.P a b (b, 0, 1)",
"tactic": "dsimp [P]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = a * 0 + b * 1",
"tactic": "rw [mul_one, mul_zero, zero_add]"
}
] |
[
232,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.coe_mul
|
[] |
[
434,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/CategoryTheory/Abelian/Transfer.lean
|
CategoryTheory.AbelianOfAdjunction.hasKernels
|
[
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ (F ⋙ G).map f ≫ i.hom.app Y✝ = (𝟭 C).map f\n⊢ HasKernel f",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ HasKernel f",
"tactic": "have := NatIso.naturality_1 i f"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasKernel f",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ (F ⋙ G).map f ≫ i.hom.app Y✝ = (𝟭 C).map f\n⊢ HasKernel f",
"tactic": "simp at this"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasKernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasKernel f",
"tactic": "rw [← this]"
},
{
"state_after": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\nthis : HasKernel (G.map (F.map f) ≫ i.hom.app Y✝)\n⊢ HasKernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasKernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"tactic": "haveI : HasKernel (G.map (F.map f) ≫ i.hom.app _) := Limits.hasKernel_comp_mono _ _"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\ninst✝ : PreservesFiniteLimits G\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\nthis : HasKernel (G.map (F.map f) ≫ i.hom.app Y✝)\n⊢ HasKernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"tactic": "apply Limits.hasKernel_iso_comp"
}
] |
[
66,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.coe_comp_orderMonoidHom
|
[] |
[
698,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.coe_submodule_le_coe_submodule
|
[] |
[
429,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.OrdConnected.preimage_anti
|
[] |
[
76,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.aleph0_toNat
|
[] |
[
1744,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1743,
1
] |
Mathlib/Topology/LocallyConstant/Algebra.lean
|
LocallyConstant.inv_apply
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.log_mono_right
|
[] |
[
194,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Order/Lattice.lean
|
inf_eq_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : SemilatticeInf α\na b c d : α\n⊢ a ⊓ b ≤ b ∧ b ≤ a ⊓ b ↔ b ≤ a",
"tactic": "simp [le_rfl]"
}
] |
[
439,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.SameCycle.symm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.30729\nα : Type u_1\nβ : Type ?u.30735\nf g : Perm α\np : α → Prop\nx y z : α\nx✝ : SameCycle f x y\ni : ℤ\nhi : ↑(f ^ i) x = y\n⊢ ↑(f ^ (-i)) y = x",
"tactic": "rw [zpow_neg, ← hi, inv_apply_self]"
}
] |
[
94,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Control/Functor.lean
|
Functor.Comp.comp_map
|
[
{
"state_after": "no goals",
"state_before": "F : Type u → Type w\nG : Type v → Type u\ninst✝³ : Functor F\ninst✝² : Functor G\ninst✝¹ : LawfulFunctor F\ninst✝ : LawfulFunctor G\nα β γ : Type v\ng' : α → β\nh : β → γ\nx : F (G α)\n⊢ Comp.map (h ∘ g') (mk x) = Comp.map h (Comp.map g' (mk x))",
"tactic": "simp [Comp.map, Comp.mk, Functor.map_comp_map, functor_norm]"
}
] |
[
217,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
11
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.eq_cons
|
[
{
"state_after": "n m : ℕ\nh : n < m\n⊢ [n] ++ Ico (succ n) m = n :: Ico (n + 1) m",
"state_before": "n m : ℕ\nh : n < m\n⊢ Ico n m = n :: Ico (n + 1) m",
"tactic": "rw [← append_consecutive (Nat.le_succ n) h, succ_singleton]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : n < m\n⊢ [n] ++ Ico (succ n) m = n :: Ico (n + 1) m",
"tactic": "rfl"
}
] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.list_prod_mem
|
[] |
[
329,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
8
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.NeBot.zero_div_nonneg
|
[] |
[
932,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_zero
|
[] |
[
275,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean
|
LieSubmodule.le_normalizer
|
[
{
"state_after": "R : Type u_2\nL : Type u_3\nM : Type u_1\nM' : Type ?u.9664\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nm : M\nhm : m ∈ N\n⊢ m ∈ normalizer N",
"state_before": "R : Type u_2\nL : Type u_3\nM : Type u_1\nM' : Type ?u.9664\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\n⊢ N ≤ normalizer N",
"tactic": "intro m hm"
},
{
"state_after": "R : Type u_2\nL : Type u_3\nM : Type u_1\nM' : Type ?u.9664\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nm : M\nhm : m ∈ N\n⊢ ∀ (x : L), ⁅x, m⁆ ∈ N",
"state_before": "R : Type u_2\nL : Type u_3\nM : Type u_1\nM' : Type ?u.9664\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nm : M\nhm : m ∈ N\n⊢ m ∈ normalizer N",
"tactic": "rw [mem_normalizer]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nL : Type u_3\nM : Type u_1\nM' : Type ?u.9664\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nm : M\nhm : m ∈ N\n⊢ ∀ (x : L), ⁅x, m⁆ ∈ N",
"tactic": "exact fun x => N.lie_mem hm"
}
] |
[
70,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.kerLift_injective
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝ : Group H\nφ : G →* H\na✝ b✝ : G ⧸ ker φ\na b : G\nh : ↑φ a = ↑φ b\n⊢ Setoid.r a b",
"tactic": "rw [leftRel_apply, mem_ker, φ.map_mul, ← h, φ.map_inv, inv_mul_self]"
}
] |
[
367,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.coe_prop
|
[] |
[
266,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.iterate_derivative_at_0_ne_zero
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬eval (↑(n - (ν - 1))) (pochhammer R ν) = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) ≠ 0",
"tactic": "simp only [Int.coe_nat_eq_zero, bernsteinPolynomial.iterate_derivative_at_0, Ne.def,\n Nat.cast_eq_zero]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬↑(eval (n - (ν - 1)) (pochhammer ℕ ν)) = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬eval (↑(n - (ν - 1))) (pochhammer R ν) = 0",
"tactic": "simp only [← pochhammer_eval_cast]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬eval (n - (ν - 1)) (pochhammer ℕ ν) = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬↑(eval (n - (ν - 1)) (pochhammer ℕ ν)) = 0",
"tactic": "norm_cast"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬eval (n - (ν - 1)) (pochhammer ℕ ν) = 0",
"tactic": "apply ne_of_gt"
},
{
"state_after": "case h.inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn : ℕ\nh : 0 ≤ n\n⊢ 0 < eval (n - (0 - 1)) (pochhammer ℕ 0)\n\ncase h.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\nh' : ν > 0\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"state_before": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"tactic": "obtain rfl | h' := Nat.eq_zero_or_pos ν"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn : ℕ\nh : 0 ≤ n\n⊢ 0 < eval (n - (0 - 1)) (pochhammer ℕ 0)",
"tactic": "simp"
},
{
"state_after": "case h.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : Nat.succ (Nat.pred ν) ≤ n\nh' : ν > 0\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"state_before": "case h.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\nh' : ν > 0\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"tactic": "rw [← Nat.succ_pred_eq_of_pos h'] at h"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : Nat.succ (Nat.pred ν) ≤ n\nh' : ν > 0\n⊢ 0 < eval (n - (ν - 1)) (pochhammer ℕ ν)",
"tactic": "exact pochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h))"
}
] |
[
212,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.disjSups_subset
|
[] |
[
455,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Algebra/CharP/Invertible.lean
|
not_ringChar_dvd_of_invertible
|
[
{
"state_after": "K : Type u_1\ninst✝¹ : Field K\nt : ℕ\ninst✝ : Invertible ↑t\n⊢ ↑t ≠ 0",
"state_before": "K : Type u_1\ninst✝¹ : Field K\nt : ℕ\ninst✝ : Invertible ↑t\n⊢ ¬ringChar K ∣ t",
"tactic": "rw [← ringChar.spec, ← Ne.def]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : Field K\nt : ℕ\ninst✝ : Invertible ↑t\n⊢ ↑t ≠ 0",
"tactic": "exact nonzero_of_invertible (t : K)"
}
] |
[
38,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.constr_comp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_4\nι' : Type ?u.538910\nR : Type u_1\nR₂ : Type ?u.538916\nK : Type ?u.538919\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.538928\nV : Type u\nV' : Type ?u.538933\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nS : Type u_5\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : M' →ₗ[R] M'\nv : ι → M'\ni : ι\n⊢ ↑(↑(constr b S) (↑f ∘ v)) (↑b i) = ↑(LinearMap.comp f (↑(constr b S) v)) (↑b i)",
"tactic": "simp only [Basis.constr_basis, LinearMap.comp_apply, Function.comp]"
}
] |
[
659,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/Data/Finset/Sort.lean
|
Finset.mem_sort
|
[] |
[
62,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.Perm.kunion_right
|
[
{
"state_after": "case nil\nα : Type u\nβ : α → Type v\nl✝ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ l : List (Sigma β)\n⊢ kunion [] l ~ kunion [] l\n\ncase cons\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ : Sigma β\nl₁✝ l₂✝ : List (Sigma β)\na✝ : l₁✝ ~ l₂✝\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\nl : List (Sigma β)\n⊢ kunion (x✝ :: l₁✝) l ~ kunion (x✝ :: l₂✝) l\n\ncase swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\np : l₁ ~ l₂\nl : List (Sigma β)\n⊢ kunion l₁ l ~ kunion l₂ l",
"tactic": "induction p generalizing l"
},
{
"state_after": "case cons\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ : Sigma β\nl₁✝ l₂✝ : List (Sigma β)\na✝ : l₁✝ ~ l₂✝\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\nl : List (Sigma β)\n⊢ kunion (x✝ :: l₁✝) l ~ kunion (x✝ :: l₂✝) l\n\ncase swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"state_before": "case nil\nα : Type u\nβ : α → Type v\nl✝ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ l : List (Sigma β)\n⊢ kunion [] l ~ kunion [] l\n\ncase cons\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ : Sigma β\nl₁✝ l₂✝ : List (Sigma β)\na✝ : l₁✝ ~ l₂✝\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\nl : List (Sigma β)\n⊢ kunion (x✝ :: l₁✝) l ~ kunion (x✝ :: l₂✝) l\n\ncase swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"tactic": "case nil => rfl"
},
{
"state_after": "case swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"state_before": "case cons\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ : Sigma β\nl₁✝ l₂✝ : List (Sigma β)\na✝ : l₁✝ ~ l₂✝\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\nl : List (Sigma β)\n⊢ kunion (x✝ :: l₁✝) l ~ kunion (x✝ :: l₂✝) l\n\ncase swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"tactic": "case cons hd tl₁ tl₂ _ ih =>\n simp [ih (List.kerase _ _), Perm.cons]"
},
{
"state_after": "case trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"state_before": "case swap\nα : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nx✝ y✝ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (y✝ :: x✝ :: l✝) l ~ kunion (x✝ :: y✝ :: l✝) l\n\ncase trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"tactic": "case swap s₁ s₂ l => simp [kerase_comm, Perm.swap]"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nl₁✝ l₂✝ l₃✝ : List (Sigma β)\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List (Sigma β)), kunion l₁✝ l ~ kunion l₂✝ l\na_ih✝ : ∀ (l : List (Sigma β)), kunion l₂✝ l ~ kunion l₃✝ l\nl : List (Sigma β)\n⊢ kunion l₁✝ l ~ kunion l₃✝ l",
"tactic": "case trans l₁ l₂ l₃ _ _ ih₁₂ ih₂₃ => exact Perm.trans (ih₁₂ l) (ih₂₃ l)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ l : List (Sigma β)\n⊢ kunion [] l ~ kunion [] l",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\nhd : Sigma β\ntl₁ tl₂ : List (Sigma β)\na✝ : tl₁ ~ tl₂\nih : ∀ (l : List (Sigma β)), kunion tl₁ l ~ kunion tl₂ l\nl : List (Sigma β)\n⊢ kunion (hd :: tl₁) l ~ kunion (hd :: tl₂) l",
"tactic": "simp [ih (List.kerase _ _), Perm.cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝¹ l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁ l₂ : List (Sigma β)\ns₁ s₂ : Sigma β\nl✝ : List (Sigma β)\nl : List (Sigma β)\n⊢ kunion (s₂ :: s₁ :: l✝) l ~ kunion (s₁ :: s₂ :: l✝) l",
"tactic": "simp [kerase_comm, Perm.swap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁✝¹ l₂✝¹ : List (Sigma β)\ninst✝ : DecidableEq α\nl₁✝ l₂✝ : List (Sigma β)\nl₁ l₂ l₃ : List (Sigma β)\na✝¹ : l₁ ~ l₂\na✝ : l₂ ~ l₃\nih₁₂ : ∀ (l : List (Sigma β)), kunion l₁ l ~ kunion l₂ l\nih₂₃ : ∀ (l : List (Sigma β)), kunion l₂ l ~ kunion l₃ l\nl : List (Sigma β)\n⊢ kunion l₁ l ~ kunion l₃ l",
"tactic": "exact Perm.trans (ih₁₂ l) (ih₂₃ l)"
}
] |
[
731,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_add_one'
|
[] |
[
430,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.exists_root_of_splits'
|
[
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : degree (map i f) ≠ 0\nhf0' : map i f = 0\n⊢ ∃ x, eval₂ i x f = 0",
"tactic": "simp [eval₂_eq_eval_map, hf0']"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\nf : K[X]\nhs : Splits i✝ f\nhf0 : degree (map i✝ f) ≠ 0\nhf0' : ¬map i✝ f = 0\ng : L[X]\nhg : Irreducible g ∧ g ∣ map i✝ f\nx : L\nhx : IsRoot g x\ni : L[X]\nhi : map i✝ f = g * i\n⊢ eval₂ i✝ x f = 0",
"tactic": "rw [← eval_map, hi, eval_mul, show _ = _ from hx, MulZeroClass.zero_mul]"
}
] |
[
184,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.coe_rangeRestrict
|
[] |
[
2626,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2625,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Dart.symm_involutive
|
[] |
[
767,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
continuousWithinAt_const_smul_iff
|
[] |
[
210,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kroneckerMap_smul_left
|
[] |
[
119,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Matrix.conjTranspose_tsum
|
[
{
"state_after": "case pos\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : Summable f\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ\n\ncase neg\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : ¬Summable f\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"state_before": "X : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"tactic": "by_cases hf : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : Summable f\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"tactic": "exact hf.hasSum.matrix_conjTranspose.tsum_eq.symm"
},
{
"state_after": "case neg\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => (f x)ᴴ\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"state_before": "case neg\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : ¬Summable f\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"tactic": "have hft := summable_matrix_conjTranspose.not.mpr hf"
},
{
"state_after": "no goals",
"state_before": "case neg\nX : Type u_4\nα : Type ?u.67432\nl : Type ?u.67435\nm : Type u_2\nn : Type u_3\np : Type ?u.67444\nS : Type ?u.67447\nR : Type u_1\nm' : l → Type ?u.67455\nn' : l → Type ?u.67460\ninst✝⁶ : Semiring α\ninst✝⁵ : AddCommMonoid R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Module α R\ninst✝² : StarAddMonoid R\ninst✝¹ : ContinuousStar R\ninst✝ : T2Space R\nf : X → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => (f x)ᴴ\n⊢ (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ",
"tactic": "rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, conjTranspose_zero]"
}
] |
[
340,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.candidates_dist_inr
|
[] |
[
147,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
9
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.mem_wf
|
[
{
"state_after": "a x : ZFSet\nIH : ∀ (y : ZFSet), y ∈ x → Acc (fun x x_1 => x ∈ x_1) ↑y\n⊢ ∀ (y : Class), y ∈ ↑x → Acc (fun x x_1 => x ∈ x_1) y",
"state_before": "⊢ ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x",
"tactic": "refine' fun a => ZFSet.inductionOn a fun x IH => ⟨_, _⟩"
},
{
"state_after": "case intro.intro\na x : ZFSet\nIH : ∀ (y : ZFSet), y ∈ x → Acc (fun x x_1 => x ∈ x_1) ↑y\nz : ZFSet\nhz : ↑x z\n⊢ Acc (fun x x_1 => x ∈ x_1) ↑z",
"state_before": "a x : ZFSet\nIH : ∀ (y : ZFSet), y ∈ x → Acc (fun x x_1 => x ∈ x_1) ↑y\n⊢ ∀ (y : Class), y ∈ ↑x → Acc (fun x x_1 => x ∈ x_1) y",
"tactic": "rintro A ⟨z, rfl, hz⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\na x : ZFSet\nIH : ∀ (y : ZFSet), y ∈ x → Acc (fun x x_1 => x ∈ x_1) ↑y\nz : ZFSet\nhz : ↑x z\n⊢ Acc (fun x x_1 => x ∈ x_1) ↑z",
"tactic": "exact IH z hz"
},
{
"state_after": "H : ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x\nA : Class\n⊢ ∀ (y : Class), y ∈ A → Acc (fun x x_1 => x ∈ x_1) y",
"state_before": "H : ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x\n⊢ ∀ (a : Class), Acc (fun x x_1 => x ∈ x_1) a",
"tactic": "refine' fun A => ⟨A, _⟩"
},
{
"state_after": "case intro.intro\nH : ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x\nA : Class\nx : ZFSet\nright✝ : A x\n⊢ Acc (fun x x_1 => x ∈ x_1) ↑x",
"state_before": "H : ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x\nA : Class\n⊢ ∀ (y : Class), y ∈ A → Acc (fun x x_1 => x ∈ x_1) y",
"tactic": "rintro B ⟨x, rfl, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nH : ∀ (x : ZFSet), Acc (fun x x_1 => x ∈ x_1) ↑x\nA : Class\nx : ZFSet\nright✝ : A x\n⊢ Acc (fun x x_1 => x ∈ x_1) ↑x",
"tactic": "exact H x"
}
] |
[
1539,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1531,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
summable_geometric_two'
|
[] |
[
254,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_apply_eq_div
|
[
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝⁴ : Field 𝕂\ninst✝³ : DivisionRing 𝔸\ninst✝² : Algebra 𝕂 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\nn : ℕ\n⊢ (↑(expSeries 𝕂 𝔸 n) fun x_1 => x) = x ^ n / ↑n !",
"tactic": "rw [div_eq_mul_inv, ← (Nat.cast_commute n ! (x ^ n)).inv_left₀.eq, ← smul_eq_mul,\n expSeries_apply_eq, inv_nat_cast_smul_eq 𝕂 𝔸]"
}
] |
[
172,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Finset.subtype_Ici_eq
|
[] |
[
1320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1319,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.balanceL_eq_balance'
|
[
{
"state_after": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ size l = 0 → size r ≤ 1\n\ncase H2\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ 1 ≤ size l → 1 ≤ size r → size r ≤ delta * size l",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ balanceL l x r = balance' l x r",
"tactic": "rw [← balance_eq_balance' hl hr sl sr, balanceL_eq_balance sl sr]"
},
{
"state_after": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\nl0 : size l = 0\n⊢ size r ≤ 1",
"state_before": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ size l = 0 → size r ≤ 1",
"tactic": "intro l0"
},
{
"state_after": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' 0 ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz 0 r'\nl0 : size l = 0\n⊢ size r ≤ 1",
"state_before": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\nl0 : size l = 0\n⊢ size r ≤ 1",
"tactic": "rw [l0] at H"
},
{
"state_after": "case H1.inl.intro.intro.inl.refl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl0 : size l = 0\nH : BalancedSz 0 (size r)\n⊢ size r ≤ 1\n\ncase H1.inr.intro.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl0 : size l = 0\nr' : ℕ\ne : Raised (size r) r'\nH : BalancedSz 0 r'\n⊢ size r ≤ 1",
"state_before": "case H1\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' 0 ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz 0 r'\nl0 : size l = 0\n⊢ size r ≤ 1",
"tactic": "rcases H with (⟨_, ⟨⟨⟩⟩ | ⟨⟨⟩⟩, H⟩ | ⟨r', e, H⟩)"
},
{
"state_after": "no goals",
"state_before": "case H1.inr.intro.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl0 : size l = 0\nr' : ℕ\ne : Raised (size r) r'\nH : BalancedSz 0 r'\n⊢ size r ≤ 1",
"tactic": "exact le_trans (raised_iff.1 e).1 (balancedSz_zero.1 H.symm)"
},
{
"state_after": "no goals",
"state_before": "case H1.inl.intro.intro.inl.refl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl0 : size l = 0\nH : BalancedSz 0 (size r)\n⊢ size r ≤ 1",
"tactic": "exact balancedSz_zero.1 H.symm"
},
{
"state_after": "case H2\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\n⊢ size r ≤ delta * size l",
"state_before": "case H2\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ 1 ≤ size l → 1 ≤ size r → size r ≤ delta * size l",
"tactic": "intro l1 _"
},
{
"state_after": "case H2.inl.intro.intro.inl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nl' : ℕ\ne : Raised l' (size l)\nH : l' + size r ≤ 1\n⊢ size r ≤ delta * size l\n\ncase H2.inl.intro.intro.inr.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nl' : ℕ\ne : Raised l' (size l)\nleft✝ : l' ≤ delta * size r\nH₂ : size r ≤ delta * l'\n⊢ size r ≤ delta * size l\n\ncase H2.inr.intro.intro.inl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\n⊢ size r ≤ delta * size l\n\ncase H2.inr.intro.intro.inr.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nleft✝ : size l ≤ delta * r'\nH₂ : r' ≤ delta * size l\n⊢ size r ≤ delta * size l",
"state_before": "case H2\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\n⊢ size r ≤ delta * size l",
"tactic": "rcases H with (⟨l', e, H | ⟨_, H₂⟩⟩ | ⟨r', e, H | ⟨_, H₂⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "case H2.inl.intro.intro.inl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nl' : ℕ\ne : Raised l' (size l)\nH : l' + size r ≤ 1\n⊢ size r ≤ delta * size l",
"tactic": "exact le_trans (le_trans (Nat.le_add_left _ _) H) (mul_pos (by decide) l1 : (0 : ℕ) < _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nl' : ℕ\ne : Raised l' (size l)\nH : l' + size r ≤ 1\n⊢ 0 < delta",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case H2.inl.intro.intro.inr.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nl' : ℕ\ne : Raised l' (size l)\nleft✝ : l' ≤ delta * size r\nH₂ : size r ≤ delta * l'\n⊢ size r ≤ delta * size l",
"tactic": "exact le_trans H₂ (Nat.mul_le_mul_left _ (raised_iff.1 e).1)"
},
{
"state_after": "case H2.inr.intro.intro.inl.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\nleft✝ : size r ≤ r'\nright✝ : r' ≤ size r + 1\n⊢ size r ≤ delta * size l",
"state_before": "case H2.inr.intro.intro.inl\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\n⊢ size r ≤ delta * size l",
"tactic": "cases raised_iff.1 e"
},
{
"state_after": "case H2.inr.intro.intro.inl.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\nleft✝ : size r ≤ r'\nright✝ : r' ≤ size r + 1\n⊢ size r ≤ 3 * size l",
"state_before": "case H2.inr.intro.intro.inl.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\nleft✝ : size r ≤ r'\nright✝ : r' ≤ size r + 1\n⊢ size r ≤ delta * size l",
"tactic": "unfold delta"
},
{
"state_after": "no goals",
"state_before": "case H2.inr.intro.intro.inl.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nH : size l + r' ≤ 1\nleft✝ : size r ≤ r'\nright✝ : r' ≤ size r + 1\n⊢ size r ≤ 3 * size l",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "case H2.inr.intro.intro.inr.intro\nα : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\nhl : Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nl1 : 1 ≤ size l\na✝ : 1 ≤ size r\nr' : ℕ\ne : Raised (size r) r'\nleft✝ : size l ≤ delta * r'\nH₂ : r' ≤ delta * size l\n⊢ size r ≤ delta * size l",
"tactic": "exact le_trans (raised_iff.1 e).1 H₂"
}
] |
[
834,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.kernelSubobjectMap_arrow
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝ : HasKernel f'\nsq : Arrow.mk f ⟶ Arrow.mk f'\n⊢ kernelSubobjectMap sq ≫ arrow (kernelSubobject f') = arrow (kernelSubobject f) ≫ sq.left",
"tactic": "simp [kernelSubobjectMap]"
}
] |
[
163,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Std/Data/BinomialHeap.lean
|
Std.BinomialHeapImp.Heap.WellFormed.singleton
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\na : α✝\nle : α✝ → α✝ → Bool\n⊢ 0 ≤ 0",
"tactic": "decide"
}
] |
[
338,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
338,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.induction_on_pi
|
[
{
"state_after": "α✝ : Sort ?u.35644\nβ : Sort ?u.35647\nι : Type u_1\nα : ι → Sort u_2\ns : (i : ι) → Setoid (α i)\np : ((i : ι) → Quotient (s i)) → Prop\nf : (i : ι) → Quotient (s i)\nh : ∀ (a : (i : ι) → α i), p fun i => Quotient.mk (s i) (a i)\n⊢ p fun i => Quotient.mk (s i) (out (f i))",
"state_before": "α✝ : Sort ?u.35644\nβ : Sort ?u.35647\nι : Type u_1\nα : ι → Sort u_2\ns : (i : ι) → Setoid (α i)\np : ((i : ι) → Quotient (s i)) → Prop\nf : (i : ι) → Quotient (s i)\nh : ∀ (a : (i : ι) → α i), p fun i => Quotient.mk (s i) (a i)\n⊢ p f",
"tactic": "rw [← (funext fun i ↦ Quotient.out_eq (f i) : (fun i ↦ ⟦(f i).out⟧) = f)]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Sort ?u.35644\nβ : Sort ?u.35647\nι : Type u_1\nα : ι → Sort u_2\ns : (i : ι) → Setoid (α i)\np : ((i : ι) → Quotient (s i)) → Prop\nf : (i : ι) → Quotient (s i)\nh : ∀ (a : (i : ι) → α i), p fun i => Quotient.mk (s i) (a i)\n⊢ p fun i => Quotient.mk (s i) (out (f i))",
"tactic": "apply h"
}
] |
[
436,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/RingTheory/DedekindDomain/PID.lean
|
Ideal.IsPrincipal.of_finite_maximals_of_isUnit
|
[] |
[
182,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_target
|
[] |
[
838,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Std/Logic.lean
|
and_congr_left
|
[] |
[
165,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
164,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.one_lt_succ_succ
|
[] |
[
190,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.add_coeff'
|
[] |
[
373,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_right_mono
|
[] |
[
979,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
979,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.image_rootSet
|
[
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K →ₐ[F] L\n⊢ ↑f '' rootSet p K = rootSet p L",
"tactic": "classical\n rw [rootSet, ← Finset.coe_image, ← Multiset.toFinset_map, ← f.coe_toRingHom,\n ← roots_map _ ((splits_id_iff_splits (algebraMap F K)).mpr h), map_map, f.comp_algebraMap,\n ← rootSet]"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K →ₐ[F] L\n⊢ ↑f '' rootSet p K = rootSet p L",
"tactic": "rw [rootSet, ← Finset.coe_image, ← Multiset.toFinset_map, ← f.coe_toRingHom,\n ← roots_map _ ((splits_id_iff_splits (algebraMap F K)).mpr h), map_map, f.comp_algebraMap,\n ← rootSet]"
}
] |
[
341,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.pentagon_aux
|
[
{
"state_after": "case H\nR : Type u\ninst✝⁸ : CommRing R\nW : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : AddCommMonoid X\ninst✝⁵ : AddCommMonoid Y\ninst✝⁴ : AddCommMonoid Z\ninst✝³ : Module R W\ninst✝² : Module R X\ninst✝¹ : Module R Y\ninst✝ : Module R Z\n⊢ ∀ (w : W) (x : X) (y : Y) (z : Z),\n ↑(LinearMap.comp (LinearMap.comp (map 1 ↑(assoc R X Y Z)) ↑(assoc R W (X ⊗[R] Y) Z)) (map (↑(assoc R W X Y)) 1))\n (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑(LinearMap.comp ↑(assoc R W X (Y ⊗[R] Z)) ↑(assoc R (W ⊗[R] X) Y Z)) (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z)",
"state_before": "R : Type u\ninst✝⁸ : CommRing R\nW : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : AddCommMonoid X\ninst✝⁵ : AddCommMonoid Y\ninst✝⁴ : AddCommMonoid Z\ninst✝³ : Module R W\ninst✝² : Module R X\ninst✝¹ : Module R Y\ninst✝ : Module R Z\n⊢ LinearMap.comp (LinearMap.comp (map 1 ↑(assoc R X Y Z)) ↑(assoc R W (X ⊗[R] Y) Z)) (map (↑(assoc R W X Y)) 1) =\n LinearMap.comp ↑(assoc R W X (Y ⊗[R] Z)) ↑(assoc R (W ⊗[R] X) Y Z)",
"tactic": "apply TensorProduct.ext_fourfold"
},
{
"state_after": "case H\nR : Type u\ninst✝⁸ : CommRing R\nW : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : AddCommMonoid X\ninst✝⁵ : AddCommMonoid Y\ninst✝⁴ : AddCommMonoid Z\ninst✝³ : Module R W\ninst✝² : Module R X\ninst✝¹ : Module R Y\ninst✝ : Module R Z\nw : W\nx : X\ny : Y\nz : Z\n⊢ ↑(LinearMap.comp (LinearMap.comp (map 1 ↑(assoc R X Y Z)) ↑(assoc R W (X ⊗[R] Y) Z)) (map (↑(assoc R W X Y)) 1))\n (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑(LinearMap.comp ↑(assoc R W X (Y ⊗[R] Z)) ↑(assoc R (W ⊗[R] X) Y Z)) (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z)",
"state_before": "case H\nR : Type u\ninst✝⁸ : CommRing R\nW : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : AddCommMonoid X\ninst✝⁵ : AddCommMonoid Y\ninst✝⁴ : AddCommMonoid Z\ninst✝³ : Module R W\ninst✝² : Module R X\ninst✝¹ : Module R Y\ninst✝ : Module R Z\n⊢ ∀ (w : W) (x : X) (y : Y) (z : Z),\n ↑(LinearMap.comp (LinearMap.comp (map 1 ↑(assoc R X Y Z)) ↑(assoc R W (X ⊗[R] Y) Z)) (map (↑(assoc R W X Y)) 1))\n (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑(LinearMap.comp ↑(assoc R W X (Y ⊗[R] Z)) ↑(assoc R (W ⊗[R] X) Y Z)) (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z)",
"tactic": "intro w x y z"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\ninst✝⁸ : CommRing R\nW : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : AddCommMonoid X\ninst✝⁵ : AddCommMonoid Y\ninst✝⁴ : AddCommMonoid Z\ninst✝³ : Module R W\ninst✝² : Module R X\ninst✝¹ : Module R Y\ninst✝ : Module R Z\nw : W\nx : X\ny : Y\nz : Z\n⊢ ↑(LinearMap.comp (LinearMap.comp (map 1 ↑(assoc R X Y Z)) ↑(assoc R W (X ⊗[R] Y) Z)) (map (↑(assoc R W X Y)) 1))\n (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑(LinearMap.comp ↑(assoc R W X (Y ⊗[R] Z)) ↑(assoc R (W ⊗[R] X) Y Z)) (((w ⊗ₜ[R] x) ⊗ₜ[R] y) ⊗ₜ[R] z)",
"tactic": "rfl"
}
] |
[
117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
9
] |
Mathlib/RingTheory/IntegralClosure.lean
|
is_integral_of_mem_closure''
|
[] |
[
630,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.factors_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\n⊢ factors 0 = 0",
"tactic": "simp [factors]"
}
] |
[
474,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinFilter.sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝ : OrderedAddCommGroup β\nf g : α → β\na : α\ns : Set α\nl : Filter α\nhf : IsMinFilter f l a\nhg : IsMaxFilter g l a\n⊢ IsMinFilter (fun x => f x - g x) l a",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] |
[
507,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
RingHom.injective
|
[] |
[
260,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
11
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.finrank_bot
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ finrank F { x // x ∈ ⊥ } = 1",
"tactic": "rw [finrank_eq_one_iff]"
}
] |
[
728,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Logic/Basic.lean
|
by_contradiction
|
[] |
[
216,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ico_filter_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.139276\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\n⊢ filter (fun x => c ≤ x) (Ico a b) = Ico (max a c) b",
"tactic": "cases le_total a c with\n| inl h => rw [Ico_filter_le_of_left_le h, max_eq_right h]\n| inr h => rw [Ico_filter_le_of_le_left h, max_eq_left h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.139276\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\n⊢ filter (fun x => c ≤ x) (Ico a b) = Ico (max a c) b",
"tactic": "rw [Ico_filter_le_of_left_le h, max_eq_right h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.139276\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : c ≤ a\n⊢ filter (fun x => c ≤ x) (Ico a b) = Ico (max a c) b",
"tactic": "rw [Ico_filter_le_of_le_left h, max_eq_left h]"
}
] |
[
798,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean
|
AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_eq
|
[
{
"state_after": "case w.w.h\nA : Type u_4\nA' : Type u_5\nB : Type u_1\nB' : Type u_6\ninst✝³ : Category A\ninst✝² : Category A'\ninst✝¹ : Category B\ninst✝ : Category B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ (equivalence₂ eB hF).counitIso.hom.app Y' = (equivalence₂CounitIso eB hF).hom.app Y'",
"state_before": "A : Type u_4\nA' : Type u_5\nB : Type u_1\nB' : Type u_6\ninst✝³ : Category A\ninst✝² : Category A'\ninst✝¹ : Category B\ninst✝ : Category B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\n⊢ (equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF",
"tactic": "ext Y'"
},
{
"state_after": "case w.w.h\nA : Type u_4\nA' : Type u_5\nB : Type u_1\nB' : Type u_6\ninst✝³ : Category A\ninst✝² : Category A'\ninst✝¹ : Category B\ninst✝ : Category B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ eB.inverse.map ((equivalence₁ hF).counitIso.hom.app (eB.functor.obj Y')) ≫ eB.unitIso.inv.app Y' =\n (equivalence₂CounitIso eB hF).hom.app Y'",
"state_before": "case w.w.h\nA : Type u_4\nA' : Type u_5\nB : Type u_1\nB' : Type u_6\ninst✝³ : Category A\ninst✝² : Category A'\ninst✝¹ : Category B\ninst✝ : Category B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ (equivalence₂ eB hF).counitIso.hom.app Y' = (equivalence₂CounitIso eB hF).hom.app Y'",
"tactic": "dsimp [equivalence₂, Iso.refl]"
},
{
"state_after": "no goals",
"state_before": "case w.w.h\nA : Type u_4\nA' : Type u_5\nB : Type u_1\nB' : Type u_6\ninst✝³ : Category A\ninst✝² : Category A'\ninst✝¹ : Category B\ninst✝ : Category B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor\nY' : B\n⊢ eB.inverse.map ((equivalence₁ hF).counitIso.hom.app (eB.functor.obj Y')) ≫ eB.unitIso.inv.app Y' =\n (equivalence₂CounitIso eB hF).hom.app Y'",
"tactic": "simp only [equivalence₁CounitIso_eq, equivalence₂CounitIso_hom_app,\n equivalence₁CounitIso_hom_app, Functor.map_comp, assoc]"
}
] |
[
148,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.kernelComparison_comp_ι
|
[] |
[
1106,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1104,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_subgroupOf
|
[] |
[
1627,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1626,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Cardinal.mk_quaternionAlgebra_of_infinite
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nc₁ c₂ : R\ninst✝ : Infinite R\n⊢ (#ℍ[R,c₁,c₂]) = (#R)",
"tactic": "rw [mk_quaternionAlgebra, pow_four]"
}
] |
[
1436,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1435,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.inv_eq_top
|
[] |
[
1403,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1403,
9
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.locallyIntegrable_const
|
[
{
"state_after": "X : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\n⊢ IntegrableAtFilter (fun x => c) (𝓝 x)",
"state_before": "X : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\n⊢ LocallyIntegrable fun x => c",
"tactic": "intro x"
},
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\nU : Set X\nhU : U ∈ 𝓝 x\nh'U : ↑↑μ U < ⊤\n⊢ IntegrableAtFilter (fun x => c) (𝓝 x)",
"state_before": "X : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\n⊢ IntegrableAtFilter (fun x => c) (𝓝 x)",
"tactic": "rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩"
},
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\nU : Set X\nhU : U ∈ 𝓝 x\nh'U : ↑↑μ U < ⊤\n⊢ IntegrableOn (fun x => c) U",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\nU : Set X\nhU : U ∈ 𝓝 x\nh'U : ↑↑μ U < ⊤\n⊢ IntegrableAtFilter (fun x => c) (𝓝 x)",
"tactic": "refine' ⟨U, hU, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.128412\nE : Type u_2\nR : Type ?u.128418\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : IsLocallyFiniteMeasure μ\nc : E\nx : X\nU : Set X\nhU : U ∈ 𝓝 x\nh'U : ↑↑μ U < ⊤\n⊢ IntegrableOn (fun x => c) U",
"tactic": "simp only [h'U, integrableOn_const, or_true_iff]"
}
] |
[
210,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean
|
SimpleGraph.isAdjMatrix_adjMatrix
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.28278\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Zero α\ninst✝ : One α\ni j : V\n⊢ adjMatrix α G i j = 0 ∨ adjMatrix α G i j = 1",
"tactic": "by_cases G.Adj i j <;> simp [h]"
}
] |
[
188,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
AlgHom.FinitePresentation.of_comp_finiteType
|
[] |
[
531,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
8
] |
Mathlib/Data/Nat/Pow.lean
|
Nat.pow_lt_iff_lt_left
|
[] |
[
143,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Order/BoundedOrder.lean
|
max_top_left
|
[] |
[
839,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
838,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.le_nhds_infty
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nf : Filter (OnePoint X)\n⊢ f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → some '' sᶜ ∪ {∞} ∈ f",
"tactic": "simp only [hasBasis_nhds_infty.ge_iff, and_imp]"
}
] |
[
351,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.setToL1S_const
|
[] |
[
826,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousLinearEquiv.compContinuousMultilinearMapL_symm
|
[] |
[
1004,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_lt_of_unbounded_le
|
[] |
[
107,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.MutuallySingular.neg_left
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\n⊢ -v ⟂ᵥ w",
"state_before": "α : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ⟂ᵥ w\n⊢ -v ⟂ᵥ w",
"tactic": "obtain ⟨u, hmu, hu₁, hu₂⟩ := h"
},
{
"state_after": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\ns : Set α\nhs : s ⊆ u\n⊢ ↑(-v) s = 0",
"state_before": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\n⊢ -v ⟂ᵥ w",
"tactic": "refine' ⟨u, hmu, fun s hs => _, hu₂⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\ns : Set α\nhs : s ⊆ u\n⊢ ↑v s = 0",
"state_before": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\ns : Set α\nhs : s ⊆ u\n⊢ ↑(-v) s = 0",
"tactic": "rw [neg_apply v s, neg_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.681547\nm : MeasurableSpace α\nL : Type ?u.681553\nM✝ : Type ?u.681556\nN : Type u_3\ninst✝⁸ : AddCommMonoid L\ninst✝⁷ : TopologicalSpace L\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : TopologicalSpace M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nv✝ v₁ v₂ : VectorMeasure α M✝\nw✝ w₁ w₂ : VectorMeasure α N\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑v t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0\ns : Set α\nhs : s ⊆ u\n⊢ ↑v s = 0",
"tactic": "exact hu₁ s hs"
}
] |
[
1251,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1246,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
intermediate_value_Ioo'
|
[] |
[
598,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.cast_addNat_right
|
[] |
[
1387,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1385,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.le_left_of_right_le'
|
[] |
[
720,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
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