file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.gcd_add_self_left
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ gcd (m + n) n = gcd m n",
"tactic": "rw [gcd_comm, gcd_add_self_right, gcd_comm]"
}
] |
[
82,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.mul_eq_one_comm
|
[
{
"state_after": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis : Invertible (det B) := detInvertibleOfLeftInverse B A h\n⊢ B ⬝ A = 1",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\n⊢ B ⬝ A = 1",
"tactic": "letI : Invertible B.det := detInvertibleOfLeftInverse _ _ h"
},
{
"state_after": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis✝ : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n⊢ B ⬝ A = 1",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis : Invertible (det B) := detInvertibleOfLeftInverse B A h\n⊢ B ⬝ A = 1",
"tactic": "letI : Invertible B := invertibleOfDetInvertible B"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis✝ : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n⊢ B ⬝ A = 1",
"tactic": "calc\n B ⬝ A = B ⬝ A ⬝ (B ⬝ ⅟ B) := by rw [Matrix.mul_invOf_self, Matrix.mul_one]\n _ = B ⬝ (A ⬝ B ⬝ ⅟ B) := by simp only [Matrix.mul_assoc]\n _ = B ⬝ ⅟ B := by rw [h, Matrix.one_mul]\n _ = 1 := Matrix.mul_invOf_self B"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis✝ : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n⊢ B ⬝ A = B ⬝ A ⬝ (B ⬝ ⅟B)",
"tactic": "rw [Matrix.mul_invOf_self, Matrix.mul_one]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis✝ : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n⊢ B ⬝ A ⬝ (B ⬝ ⅟B) = B ⬝ (A ⬝ B ⬝ ⅟B)",
"tactic": "simp only [Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.57544\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\nh : A ⬝ B = 1\nthis✝ : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n⊢ B ⬝ (A ⬝ B ⬝ ⅟B) = B ⬝ ⅟B",
"tactic": "rw [h, Matrix.one_mul]"
}
] |
[
161,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
limsSup_eq_of_le_nhds
|
[] |
[
194,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Multiset/Dedup.lean
|
Multiset.subset_dedup'
|
[] |
[
75,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Int/Interval.lean
|
Int.card_fintype_Ico
|
[
{
"state_after": "no goals",
"state_before": "a b : ℤ\n⊢ Fintype.card ↑(Set.Ico a b) = toNat (b - a)",
"tactic": "rw [← card_Ico, Fintype.card_ofFinset]"
}
] |
[
153,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.continuousOn_mul
|
[
{
"state_after": "ι : Type ?u.6487579\n𝕜 : Type ?u.6487582\nE : Type ?u.6487585\nF : Type ?u.6487588\nA : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf✝ g✝ : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → A\nhf : IntegrableOn f (Ι a b)\nhg : ContinuousOn g [[a, b]]\n⊢ IntegrableOn (fun x => g x * f x) (Ι a b)",
"state_before": "ι : Type ?u.6487579\n𝕜 : Type ?u.6487582\nE : Type ?u.6487585\nF : Type ?u.6487588\nA : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf✝ g✝ : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → A\nhf : IntervalIntegrable f μ a b\nhg : ContinuousOn g [[a, b]]\n⊢ IntervalIntegrable (fun x => g x * f x) μ a b",
"tactic": "rw [intervalIntegrable_iff] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6487579\n𝕜 : Type ?u.6487582\nE : Type ?u.6487585\nF : Type ?u.6487588\nA : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf✝ g✝ : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nf g : ℝ → A\nhf : IntegrableOn f (Ι a b)\nhg : ContinuousOn g [[a, b]]\n⊢ IntegrableOn (fun x => g x * f x) (Ι a b)",
"tactic": "exact hf.continuousOn_mul_of_subset hg isCompact_uIcc measurableSet_Ioc Ioc_subset_Icc_self"
}
] |
[
273,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.coe_map
|
[] |
[
300,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
src/lean/Init/Core.lean
|
Setoid.symm
|
[] |
[
1098,
17
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
1097,
1
] |
Mathlib/Data/Part.lean
|
Part.right_dom_of_append_dom
|
[] |
[
810,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
810,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_iff_subset_of_fully_disjoint_closed
|
[
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : Disjoint u v\n⊢ s ⊆ u ∨ s ⊆ v\n\ncase refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nhs : IsClosed s\n⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v",
"tactic": "refine isPreconnected_iff_subset_of_disjoint_closed.trans ⟨?_, ?_⟩ <;> intro H u v hu hv hss huv"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∩ s ∨ s ⊆ v ∩ s\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "have H1 := H (u ∩ s) (v ∩ s)"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∧ s ⊆ s ∨ s ⊆ v ∧ s ⊆ s\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∩ s ∨ s ⊆ v ∩ s\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "rw [subset_inter_iff, subset_inter_iff] at H1"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∧ s ⊆ s ∨ s ⊆ v ∧ s ⊆ s\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "simp only [Subset.refl, and_true] at H1"
},
{
"state_after": "case refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ u ∩ s ∪ v ∩ s\n\ncase refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ Disjoint (u ∩ s) (v ∩ s)",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "apply H1 (hu.inter hs) (hv.inter hs)"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : Disjoint u v\n⊢ s ∩ (u ∩ v) = ∅",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : Disjoint u v\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "apply H u v hu hv hss"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : Disjoint u v\n⊢ s ∩ (u ∩ v) = ∅",
"tactic": "rw [huv.inter_eq, inter_empty]"
},
{
"state_after": "case refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ (u ∪ v) ∩ s",
"state_before": "case refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ u ∩ s ∪ v ∩ s",
"tactic": "rw [← inter_distrib_right]"
},
{
"state_after": "no goals",
"state_before": "case refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ s ⊆ (u ∪ v) ∩ s",
"tactic": "exact subset_inter hss Subset.rfl"
},
{
"state_after": "no goals",
"state_before": "case refine_2.a\nα : Type u\nβ : Type v\nι : Type ?u.105502\nπ : ι → Type ?u.105507\ninst✝ : TopologicalSpace α\ns✝ t u✝ v✝ s : Set α\nhs : IsClosed s\nH : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhss : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nH1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v\n⊢ Disjoint (u ∩ s) (v ∩ s)",
"tactic": "rwa [disjoint_iff_inter_eq_empty, ← inter_inter_distrib_right, inter_comm]"
}
] |
[
986,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
974,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean
|
CategoryTheory.Limits.PreservesCokernel.iso_inv
|
[
{
"state_after": "C : Type u₁\ninst✝⁷ : Category C\ninst✝⁶ : HasZeroMorphisms C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝³ : Functor.PreservesZeroMorphisms G\nX Y Z : C\nf : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = 0\ninst✝² : HasCokernel f\ninst✝¹ : HasCokernel (G.map f)\ninst✝ : PreservesColimit (parallelPair f 0) G\n⊢ cokernel.π (G.map f) ≫ (iso G f).inv = cokernel.π (G.map f) ≫ cokernelComparison f G",
"state_before": "C : Type u₁\ninst✝⁷ : Category C\ninst✝⁶ : HasZeroMorphisms C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝³ : Functor.PreservesZeroMorphisms G\nX Y Z : C\nf : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = 0\ninst✝² : HasCokernel f\ninst✝¹ : HasCokernel (G.map f)\ninst✝ : PreservesColimit (parallelPair f 0) G\n⊢ (iso G f).inv = cokernelComparison f G",
"tactic": "rw [← cancel_epi (cokernel.π _)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁷ : Category C\ninst✝⁶ : HasZeroMorphisms C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝³ : Functor.PreservesZeroMorphisms G\nX Y Z : C\nf : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = 0\ninst✝² : HasCokernel f\ninst✝¹ : HasCokernel (G.map f)\ninst✝ : PreservesColimit (parallelPair f 0) G\n⊢ cokernel.π (G.map f) ≫ (iso G f).inv = cokernel.π (G.map f) ≫ cokernelComparison f G",
"tactic": "simp [PreservesCokernel.iso]"
}
] |
[
214,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/RingTheory/Algebraic.lean
|
isAlgebraic_of_larger_base
|
[] |
[
245,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Order/BoundedOrder.lean
|
top_eq_true
|
[] |
[
920,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_eq_setToFun
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.812425\n𝕜 : Type ?u.812428\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\n⊢ (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) =\n setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) f",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.812425\n𝕜 : Type ?u.812428\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\n⊢ (∫ (a : α), f a ∂μ) = setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) f",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.812425\n𝕜 : Type ?u.812428\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\n⊢ (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) =\n setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) f",
"tactic": "rfl"
}
] |
[
826,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.smul_mem_of_zero_mem
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.214738\nβ : Type ?u.214741\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ t • x ∈ s",
"tactic": "simpa using hs.add_smul_mem zero_mem (by simpa using hx) ht"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.214738\nβ : Type ?u.214741\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ 0 + x ∈ s",
"tactic": "simpa using hx"
}
] |
[
472,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Algebra/Category/Ring/Adjunctions.lean
|
CommRingCat.free_obj_coe
|
[] |
[
47,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.comp_assoc
|
[] |
[
704,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldrM_append
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m β\nb : β\nl l' : List α\n⊢ foldrM f b (l ++ l') = do\n let init ← foldrM f b l'\n foldrM f init l",
"tactic": "induction l <;> simp [*]"
}
] |
[
194,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
192,
9
] |
Mathlib/Algebra/Category/ModuleCat/Images.lean
|
ModuleCat.image.lift_fac
|
[
{
"state_after": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑(lift F' ≫ F'.m) x = ↑(ι f) x",
"state_before": "R : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\n⊢ lift F' ≫ F'.m = ι f",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑(F'.e ≫ F'.m) ↑(Classical.indefiniteDescription (fun x_1 => ↑f x_1 = ↑x) (_ : ↑x ∈ LinearMap.range f)) = ↑(ι f) x",
"state_before": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑(lift F' ≫ F'.m) x = ↑(ι f) x",
"tactic": "change (F'.e ≫ F'.m) _ = _"
},
{
"state_after": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑x = ↑(ι f) x",
"state_before": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑(F'.e ≫ F'.m) ↑(Classical.indefiniteDescription (fun x_1 => ↑f x_1 = ↑x) (_ : ↑x ∈ LinearMap.range f)) = ↑(ι f) x",
"tactic": "rw [F'.fac, (Classical.indefiniteDescription _ x.2).2]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf : G ⟶ H\nF' : MonoFactorisation f\nx : ↑(image f)\n⊢ ↑x = ↑(ι f) x",
"tactic": "rfl"
}
] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
differentiableOn_congr
|
[] |
[
928,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
925,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
Real.diam_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.565531\nβ : Type ?u.565534\nγ : Type ?u.565537\ninst✝ : PseudoEMetricSpace α\na b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ioo a b) = b - a",
"tactic": "simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)]"
}
] |
[
1568,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1567,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.index_dvd_card
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH K L : Subgroup G\ninst✝ : Fintype G\n⊢ index H ∣ Fintype.card G",
"tactic": "classical exact ⟨Fintype.card H, H.index_mul_card.symm⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH K L : Subgroup G\ninst✝ : Fintype G\n⊢ index H ∣ Fintype.card G",
"tactic": "exact ⟨Fintype.card H, H.index_mul_card.symm⟩"
}
] |
[
373,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.infs_union_left
|
[] |
[
338,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/String/Basic.lean
|
String.toList_nonempty
|
[
{
"state_after": "case nil\nh : { data := [] } ≠ \"\"\n⊢ toList { data := [] } = head { data := [] } :: toList (drop { data := [] } 1)\n\ncase cons\nhead✝ : Char\ntail✝ : List Char\nh : { data := head✝ :: tail✝ } ≠ \"\"\n⊢ toList { data := head✝ :: tail✝ } = head { data := head✝ :: tail✝ } :: toList (drop { data := head✝ :: tail✝ } 1)",
"state_before": "s : List Char\nh : { data := s } ≠ \"\"\n⊢ toList { data := s } = head { data := s } :: toList (drop { data := s } 1)",
"tactic": "cases s"
},
{
"state_after": "no goals",
"state_before": "case nil\nh : { data := [] } ≠ \"\"\n⊢ toList { data := [] } = head { data := [] } :: toList (drop { data := [] } 1)",
"tactic": "simp only at h"
},
{
"state_after": "case cons\nc : Char\ncs : List Char\nh : { data := c :: cs } ≠ \"\"\n⊢ toList { data := c :: cs } = head { data := c :: cs } :: toList (drop { data := c :: cs } 1)",
"state_before": "case cons\nhead✝ : Char\ntail✝ : List Char\nh : { data := head✝ :: tail✝ } ≠ \"\"\n⊢ toList { data := head✝ :: tail✝ } = head { data := head✝ :: tail✝ } :: toList (drop { data := head✝ :: tail✝ } 1)",
"tactic": "rename_i c cs"
},
{
"state_after": "case cons\nc : Char\ncs : List Char\nh : { data := c :: cs } ≠ \"\"\n⊢ c = head { data := c :: cs } ∧ cs = (drop { data := c :: cs } 1).data",
"state_before": "case cons\nc : Char\ncs : List Char\nh : { data := c :: cs } ≠ \"\"\n⊢ toList { data := c :: cs } = head { data := c :: cs } :: toList (drop { data := c :: cs } 1)",
"tactic": "simp only [toList, List.cons.injEq]"
},
{
"state_after": "no goals",
"state_before": "case cons\nc : Char\ncs : List Char\nh : { data := c :: cs } ≠ \"\"\n⊢ c = head { data := c :: cs } ∧ cs = (drop { data := c :: cs } 1).data",
"tactic": "constructor <;> [rfl; simp [drop_eq]]"
}
] |
[
148,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.eq_of_not_mem_of_mem_insert
|
[] |
[
1129,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1128,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.mul_assoc
|
[
{
"state_after": "R : Type u\ninst✝⁴ : CommSemiring R\nA : Type v₁\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nB : Type v₂\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nx y z : A ⊗[R] B\na₁✝ a₂✝ a₃✝ : A\nb₁✝ b₂✝ b₃✝ : B\n⊢ ↑(↑mul (↑(↑mul (a₁✝ ⊗ₜ[R] b₁✝)) (a₂✝ ⊗ₜ[R] b₂✝))) (a₃✝ ⊗ₜ[R] b₃✝) =\n ↑(↑mul (a₁✝ ⊗ₜ[R] b₁✝)) (↑(↑mul (a₂✝ ⊗ₜ[R] b₂✝)) (a₃✝ ⊗ₜ[R] b₃✝))",
"state_before": "R : Type u\ninst✝⁴ : CommSemiring R\nA : Type v₁\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nB : Type v₂\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nx y z : A ⊗[R] B\n⊢ ∀ (a₁ a₂ a₃ : A) (b₁ b₂ b₃ : B),\n ↑(↑mul (↑(↑mul (a₁ ⊗ₜ[R] b₁)) (a₂ ⊗ₜ[R] b₂))) (a₃ ⊗ₜ[R] b₃) =\n ↑(↑mul (a₁ ⊗ₜ[R] b₁)) (↑(↑mul (a₂ ⊗ₜ[R] b₂)) (a₃ ⊗ₜ[R] b₃))",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : CommSemiring R\nA : Type v₁\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nB : Type v₂\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nx y z : A ⊗[R] B\na₁✝ a₂✝ a₃✝ : A\nb₁✝ b₂✝ b₃✝ : B\n⊢ ↑(↑mul (↑(↑mul (a₁✝ ⊗ₜ[R] b₁✝)) (a₂✝ ⊗ₜ[R] b₂✝))) (a₃✝ ⊗ₜ[R] b₃✝) =\n ↑(↑mul (a₁✝ ⊗ₜ[R] b₁✝)) (↑(↑mul (a₂✝ ⊗ₜ[R] b₂✝)) (a₃✝ ⊗ₜ[R] b₃✝))",
"tactic": "simp only [mul_apply, mul_assoc]"
}
] |
[
421,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
8
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.le_ndinter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s t u : Multiset α\n⊢ s ≤ ndinter t u ↔ s ≤ t ∧ s ⊆ u",
"tactic": "simp [ndinter, le_filter, subset_iff]"
}
] |
[
253,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
OrthonormalBasis.coe_toBasis
|
[
{
"state_after": "ι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\n⊢ ↑(Basis.ofEquivFun b.repr.toLinearEquiv) = ↑b",
"state_before": "ι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\n⊢ ↑(OrthonormalBasis.toBasis b) = ↑b",
"tactic": "rw [OrthonormalBasis.toBasis]"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nj : ι\n⊢ ↑(Basis.ofEquivFun b.repr.toLinearEquiv) j = ↑b j",
"state_before": "ι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\n⊢ ↑(Basis.ofEquivFun b.repr.toLinearEquiv) = ↑b",
"tactic": "ext j"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nj : ι\n⊢ ↑(Basis.ofEquivFun b.repr.toLinearEquiv) j = ↑b j",
"tactic": "classical\n rw [Basis.coe_ofEquivFun]\n congr"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nj : ι\n⊢ (fun i => ↑(LinearEquiv.symm b.repr.toLinearEquiv) (update 0 i 1)) j = ↑b j",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nj : ι\n⊢ ↑(Basis.ofEquivFun b.repr.toLinearEquiv) j = ↑b j",
"tactic": "rw [Basis.coe_ofEquivFun]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.788937\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.788966\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.788984\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.789004\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nj : ι\n⊢ (fun i => ↑(LinearEquiv.symm b.repr.toLinearEquiv) (update 0 i 1)) j = ↑b j",
"tactic": "congr"
}
] |
[
423,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
11
] |
Mathlib/AlgebraicGeometry/RingedSpace.lean
|
AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen
|
[
{
"state_after": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\n⊢ ∀ (x : { x // x ∈ basicOpen X f }),\n IsUnit (↑(germ X.presheaf x) (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f))",
"state_before": "X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\n⊢ IsUnit (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f)",
"tactic": "apply isUnit_of_isUnit_germ"
},
{
"state_after": "case h.mk.intro.intro\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : IsUnit (↑(germ X.presheaf x) f)\n⊢ IsUnit\n (↑(germ X.presheaf { val := ↑x, property := (_ : ∃ a, a ∈ {x | IsUnit (↑(germ X.presheaf x) f)} ∧ ↑a = ↑x) })\n (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f))",
"state_before": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\n⊢ ∀ (x : { x // x ∈ basicOpen X f }),\n IsUnit (↑(germ X.presheaf x) (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f))",
"tactic": "rintro ⟨_, ⟨x, (hx : IsUnit _), rfl⟩⟩"
},
{
"state_after": "case h.e'_3\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : IsUnit (↑(germ X.presheaf x) f)\n⊢ ↑(germ X.presheaf { val := ↑x, property := (_ : ∃ a, a ∈ {x | IsUnit (↑(germ X.presheaf x) f)} ∧ ↑a = ↑x) })\n (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f) =\n ↑(germ X.presheaf x) f",
"state_before": "case h.mk.intro.intro\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : IsUnit (↑(germ X.presheaf x) f)\n⊢ IsUnit\n (↑(germ X.presheaf { val := ↑x, property := (_ : ∃ a, a ∈ {x | IsUnit (↑(germ X.presheaf x) f)} ∧ ↑a = ↑x) })\n (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f))",
"tactic": "convert hx"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : IsUnit (↑(germ X.presheaf x) f)\n⊢ ↑(germ X.presheaf { val := ↑x, property := (_ : ∃ a, a ∈ {x | IsUnit (↑(germ X.presheaf x) f)} ∧ ↑a = ↑x) })\n (↑(X.presheaf.map (homOfLE (_ : basicOpen X f ≤ U)).op) f) =\n ↑(germ X.presheaf x) f",
"tactic": "convert X.presheaf.germ_res_apply _ _ _"
}
] |
[
193,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.mulEquivOfQuotient_symm_mk'
|
[] |
[
1717,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1715,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_sub_cancel
|
[] |
[
1189,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1188,
1
] |
Mathlib/Data/String/Basic.lean
|
String.asString_inv_toList
|
[] |
[
137,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Topology/List.lean
|
List.tendsto_insertNth'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\n⊢ Tendsto (fun p => insertNth (n + 1) p.fst p.snd) (𝓝 a ×ˢ 𝓝 []) (𝓝 (insertNth (n + 1) a []))",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\na' : α\nl : List α\nthis : 𝓝 a ×ˢ 𝓝 (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n⊢ Tendsto ((fun p => insertNth (n + 1) p.fst p.snd) ∘ fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n (𝓝 (insertNth (n + 1) a (a' :: l)))",
"state_before": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\na' : α\nl : List α\nthis : 𝓝 a ×ˢ 𝓝 (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n⊢ Tendsto (fun p => insertNth (n + 1) p.fst p.snd) (𝓝 a ×ˢ 𝓝 (a' :: l)) (𝓝 (insertNth (n + 1) a (a' :: l)))",
"tactic": "rw [this, tendsto_map'_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\na' : α\nl : List α\nthis : 𝓝 a ×ˢ 𝓝 (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n⊢ Tendsto ((fun p => insertNth (n + 1) p.fst p.snd) ∘ fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n (𝓝 (insertNth (n + 1) a (a' :: l)))",
"tactic": "exact\n (tendsto_fst.comp tendsto_snd).cons\n ((@tendsto_insertNth' _ n l).comp <| tendsto_fst.prod_mk <| tendsto_snd.comp tendsto_snd)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\na' : α\nl : List α\n⊢ (Seq.seq (Prod.mk <$> 𝓝 a) fun x => Seq.seq (cons <$> 𝓝 a') fun x => 𝓝 l) =\n (fun p => (p.fst, p.snd.fst :: p.snd.snd)) <$>\n Seq.seq (Prod.mk <$> 𝓝 a) fun x => Seq.seq (Prod.mk <$> 𝓝 a') fun x => 𝓝 l",
"state_before": "α : Type u_1\nβ : Type ?u.13544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\na : α\nn : ℕ\na' : α\nl : List α\n⊢ 𝓝 a ×ˢ 𝓝 (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)",
"tactic": "simp only [nhds_cons, Filter.prod_eq, ← Filter.map_def, ← Filter.seq_eq_filter_seq]"
}
] |
[
138,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.IsTrail.takeUntil
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\np : Walk G v w\nhc : IsTrail p\nh : u ∈ support p\n⊢ IsTrail (append (takeUntil p u h) (?m.255284 hc h))",
"tactic": "rwa [← take_spec _ h] at hc"
}
] |
[
1183,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1181,
11
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
volume_regionBetween_eq_lintegral
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ofReal ((g - f) y) ∂μ",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ofReal ((g - f) y) ∂μ",
"tactic": "have h₁ :\n (fun y => ENNReal.ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y =>\n ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ofReal ((g - f) y) ∂μ",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ofReal ((g - f) y) ∂μ",
"tactic": "have h₂ :\n (μ.restrict s).prod volume (regionBetween f g s) =\n (μ.restrict s).prod volume\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) := by\n apply measure_congr\n apply EventuallyEq.rfl.inter\n exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp₂ _ EventuallyEq.rfl).inter\n (EventuallyEq.rfl.comp₂ _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) =\n ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ∫⁻ (y : α) in s, ofReal ((g - f) y) ∂μ",
"tactic": "rw [lintegral_congr_ae h₁, ←\n volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]"
},
{
"state_after": "case h.e'_2\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s)\n\ncase h.e'_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) =\n ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"tactic": "convert h₂ using 1"
},
{
"state_after": "case H\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ regionBetween f g s =ᶠ[ae (Measure.prod (Measure.restrict μ s) volume)]\n regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"tactic": "apply measure_congr"
},
{
"state_after": "case H\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ (fun p => Ioo (f p.fst) (g p.fst) p.snd) =ᶠ[ae (Measure.prod (Measure.restrict μ s) volume)] fun p =>\n Ioo (AEMeasurable.mk f hf p.fst) (AEMeasurable.mk g hg p.fst) p.snd",
"state_before": "case H\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ regionBetween f g s =ᶠ[ae (Measure.prod (Measure.restrict μ s) volume)]\n regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s",
"tactic": "apply EventuallyEq.rfl.inter"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n⊢ (fun p => Ioo (f p.fst) (g p.fst) p.snd) =ᶠ[ae (Measure.prod (Measure.restrict μ s) volume)] fun p =>\n Ioo (AEMeasurable.mk f hf p.fst) (AEMeasurable.mk g hg p.fst) p.snd",
"tactic": "exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp₂ _ EventuallyEq.rfl).inter\n (EventuallyEq.rfl.comp₂ _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)"
},
{
"state_after": "case h.e'_2\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) =\n ↑↑(Measure.restrict (Measure.prod μ volume) (s ×ˢ univ)) (regionBetween f g s)",
"state_before": "case h.e'_2\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) = ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s)",
"tactic": "rw [Measure.restrict_prod_eq_prod_univ]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween f g s) =\n ↑↑(Measure.restrict (Measure.prod μ volume) (s ×ˢ univ)) (regionBetween f g s)",
"tactic": "exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm"
},
{
"state_after": "case h.e'_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n ↑↑(Measure.restrict (Measure.prod μ volume) (s ×ˢ univ))\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"state_before": "case h.e'_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"tactic": "rw [Measure.restrict_prod_eq_prod_univ]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh₁ :\n (fun y => ofReal ((g - f) y)) =ᶠ[ae (Measure.restrict μ s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh₂ :\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween f g s) =\n ↑↑(Measure.prod (Measure.restrict μ s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n⊢ ↑↑(Measure.prod μ volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n ↑↑(Measure.restrict (Measure.prod μ volume) (s ×ˢ univ))\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)",
"tactic": "exact\n (Measure.restrict_eq_self _\n (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm"
}
] |
[
580,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_sdiff_div_prod_sdiff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.856258\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommGroup β\ninst✝ : DecidableEq α\n⊢ (∏ x in s₂ \\ s₁, f x) / ∏ x in s₁ \\ s₂, f x = (∏ x in s₂, f x) / ∏ x in s₁, f x",
"tactic": "simp [← Finset.prod_sdiff (@inf_le_left _ _ s₁ s₂), ← Finset.prod_sdiff (@inf_le_right _ _ s₁ s₂)]"
}
] |
[
1839,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1837,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_subset_iff
|
[] |
[
109,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/Derivation/Basic.lean
|
Derivation.map_sum
|
[] |
[
133,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
LinearMap.mkContinuous₂_norm_le'
|
[] |
[
763,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
Mathlib/Order/Closure.lean
|
ClosureOperator.closure_is_closed
|
[] |
[
187,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.mul_X_divMonomial
|
[] |
[
179,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMin.comp_antitone
|
[] |
[
233,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
8
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ContinuousHom.apply_mono
|
[] |
[
619,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_pi_const_le
|
[] |
[
2020,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2019,
1
] |
Mathlib/Topology/Constructions.lean
|
Embedding.codRestrict
|
[] |
[
1120,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1118,
1
] |
Mathlib/GroupTheory/Exponent.lean
|
Monoid.exponent_pos_of_exists
|
[
{
"state_after": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ 0 < exponent G",
"state_before": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ 0 < exponent G",
"tactic": "have h : ∃ n, 0 < n ∧ ∀ g : G, g ^ n = 1 := ⟨n, hpos, hG⟩"
},
{
"state_after": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ 0 < Nat.find ?hc\n\ncase hc\nG : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ ExponentExists G",
"state_before": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ 0 < exponent G",
"tactic": "rw [exponent, dif_pos]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ 0 < Nat.find ?hc\n\ncase hc\nG : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\nh : ∃ n, 0 < n ∧ ∀ (g : G), g ^ n = 1\n⊢ ExponentExists G",
"tactic": "exact (Nat.find_spec h).1"
}
] |
[
125,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.map'_mk''
|
[] |
[
750,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
748,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.truncate_zero_zero
|
[
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.473740\nY : Type ?u.473743\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.473758\nγ✝ : Path x y\nX : Type ?u.473807\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\n⊢ extend γ (min 0 0) = a",
"tactic": "rw [min_self, γ.extend_zero]"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.473740\nY : Type ?u.473743\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.473758\nγ✝ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\n⊢ truncate γ 0 0 = cast (refl a) (_ : extend γ (min 0 0) = a) (_ : extend γ 0 = a)",
"tactic": "convert γ.truncate_self 0"
}
] |
[
696,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/RingTheory/IntegrallyClosed.lean
|
isIntegrallyClosed_iff
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ IsIntegrallyClosed R ↔ ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\n⊢ IsIntegrallyClosed R ↔ ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"tactic": "let e : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ _ _"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ IsIntegrallyClosed R → ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\n\ncase mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ (∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x) → IsIntegrallyClosed R",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ IsIntegrallyClosed R ↔ ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"tactic": "constructor"
},
{
"state_after": "case mp.mk\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\n⊢ ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"state_before": "case mp\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ IsIntegrallyClosed R → ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"tactic": "rintro ⟨cl⟩"
},
{
"state_after": "case mp.mk\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\nx✝ : K\nhx : IsIntegral R x✝\n⊢ ∃ y, ↑(algebraMap R K) y = x✝",
"state_before": "case mp.mk\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\n⊢ ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x",
"tactic": "refine' fun hx => _"
},
{
"state_after": "case mp.mk.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\nx✝ : K\nhx : IsIntegral R x✝\ny : R\nhy : ↑(algebraMap R (FractionRing R)) y = ↑e x✝\n⊢ ∃ y, ↑(algebraMap R K) y = x✝",
"state_before": "case mp.mk\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\nx✝ : K\nhx : IsIntegral R x✝\n⊢ ∃ y, ↑(algebraMap R K) y = x✝",
"tactic": "obtain ⟨y, hy⟩ := cl ((isIntegral_algEquiv e).mpr hx)"
},
{
"state_after": "no goals",
"state_before": "case mp.mk.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : FractionRing R}, IsIntegral R x → ∃ y, ↑(algebraMap R (FractionRing R)) y = x\nx✝ : K\nhx : IsIntegral R x✝\ny : R\nhy : ↑(algebraMap R (FractionRing R)) y = ↑e x✝\n⊢ ∃ y, ↑(algebraMap R K) y = x✝",
"tactic": "exact ⟨y, e.algebraMap_eq_apply.mp hy⟩"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\n⊢ IsIntegrallyClosed R",
"state_before": "case mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\n⊢ (∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x) → IsIntegrallyClosed R",
"tactic": "rintro cl"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\nx✝ : FractionRing R\nhx : IsIntegral R x✝\n⊢ ∃ y, ↑(algebraMap R (FractionRing R)) y = x✝",
"state_before": "case mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\n⊢ IsIntegrallyClosed R",
"tactic": "refine' ⟨fun hx => _⟩"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\nx✝ : FractionRing R\nhx : IsIntegral R x✝\ny : R\nhy : ↑(algebraMap R K) y = ↑(AlgEquiv.symm e) x✝\n⊢ ∃ y, ↑(algebraMap R (FractionRing R)) y = x✝",
"state_before": "case mpr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\nx✝ : FractionRing R\nhx : IsIntegral R x✝\n⊢ ∃ y, ↑(algebraMap R (FractionRing R)) y = x✝",
"tactic": "obtain ⟨y, hy⟩ := cl ((isIntegral_algEquiv e.symm).mpr hx)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\ne : K ≃ₐ[R] FractionRing R := IsLocalization.algEquiv R⁰ K (FractionRing R)\ncl : ∀ {x : K}, IsIntegral R x → ∃ y, ↑(algebraMap R K) y = x\nx✝ : FractionRing R\nhx : IsIntegral R x✝\ny : R\nhy : ↑(algebraMap R K) y = ↑(AlgEquiv.symm e) x✝\n⊢ ∃ y, ↑(algebraMap R (FractionRing R)) y = x✝",
"tactic": "exact ⟨y, e.symm.algebraMap_eq_apply.mp hy⟩"
}
] |
[
65,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/RepresentationTheory/Maschke.lean
|
MonoidAlgebra.exists_leftInverse_of_injective
|
[
{
"state_after": "case intro\nk : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\nφ : W →ₗ[k] V\nhφ : LinearMap.comp φ (↑k f) = LinearMap.id\n⊢ ∃ g, LinearMap.comp g f = LinearMap.id",
"state_before": "k : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\n⊢ ∃ g, LinearMap.comp g f = LinearMap.id",
"tactic": "obtain ⟨φ, hφ⟩ := (f.restrictScalars k).exists_leftInverse_of_injective <| by\n simp only [hf, Submodule.restrictScalars_bot, LinearMap.ker_restrictScalars]"
},
{
"state_after": "case intro\nk : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\nφ : W →ₗ[k] V\nhφ : LinearMap.comp φ (↑k f) = LinearMap.id\n⊢ ∀ (x : V), ↑(LinearMap.comp (LinearMap.equivariantProjection G φ) f) x = ↑LinearMap.id x",
"state_before": "case intro\nk : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\nφ : W →ₗ[k] V\nhφ : LinearMap.comp φ (↑k f) = LinearMap.id\n⊢ ∃ g, LinearMap.comp g f = LinearMap.id",
"tactic": "refine ⟨φ.equivariantProjection G, FunLike.ext _ _ ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nk : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\nφ : W →ₗ[k] V\nhφ : LinearMap.comp φ (↑k f) = LinearMap.id\n⊢ ∀ (x : V), ↑(LinearMap.comp (LinearMap.equivariantProjection G φ) f) x = ↑LinearMap.id x",
"tactic": "exact φ.equivariantProjection_condition G _ <| FunLike.congr_fun hφ"
},
{
"state_after": "no goals",
"state_before": "k : Type u\ninst✝¹¹ : Field k\nG : Type u\ninst✝¹⁰ : Fintype G\ninst✝⁹ : Invertible ↑(Fintype.card G)\ninst✝⁸ : Group G\nV : Type u\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nf : V →ₗ[MonoidAlgebra k G] W\nhf : LinearMap.ker f = ⊥\n⊢ LinearMap.ker (↑k f) = ⊥",
"tactic": "simp only [hf, Submodule.restrictScalars_bot, LinearMap.ker_restrictScalars]"
}
] |
[
162,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.disjoint_union_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nx y z : Finmap β\n⊢ Disjoint x (y ∪ z) ↔ Disjoint x y ∧ Disjoint x z",
"tactic": "rw [Disjoint.symm_iff, disjoint_union_left, Disjoint.symm_iff _ x, Disjoint.symm_iff _ x]"
}
] |
[
679,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Control/Basic.lean
|
seq_bind_eq
|
[] |
[
109,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformFun.mono
|
[] |
[
359,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
11
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Ordered.memP_iff_find?
|
[
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nht : Ordered cmp t\nH : MemP cut t\n⊢ ∃ x, x ∈ find? cut t",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nht : Ordered cmp t\n⊢ MemP cut t ↔ ∃ x, x ∈ find? cut t",
"tactic": "refine ⟨fun H => ?_, fun ⟨x, h⟩ => find?_some_memP h⟩"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nht : Ordered cmp nil\nH : MemP cut nil\n⊢ False",
"tactic": "cases H"
},
{
"state_after": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ ∃ x,\n (match cut v✝ with\n | Ordering.lt => find? cut l\n | Ordering.gt => find? cut r\n | Ordering.eq => some v✝) =\n some x",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\n⊢ ∃ x,\n (match cut v✝ with\n | Ordering.lt => find? cut l\n | Ordering.gt => find? cut r\n | Ordering.eq => some v✝) =\n some x",
"tactic": "let ⟨lx, xr, hl, hr⟩ := ht"
},
{
"state_after": "case node.h_1\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ ∃ x, find? cut l = some x\n\ncase node.h_2\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ ∃ x, find? cut r = some x\n\ncase node.h_3\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ x, some v✝ = some x",
"state_before": "case node\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\n⊢ ∃ x,\n (match cut v✝ with\n | Ordering.lt => find? cut l\n | Ordering.gt => find? cut r\n | Ordering.eq => some v✝) =\n some x",
"tactic": "split"
},
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\n⊢ MemP cut l",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\n⊢ ∃ x, find? cut l = some x",
"tactic": "refine ihl hl ?_"
},
{
"state_after": "case inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nev' : cut v✝ = Ordering.eq\n⊢ MemP cut l\n\ncase inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) l\n⊢ MemP cut l\n\ncase inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) r\n⊢ MemP cut l",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\n⊢ MemP cut l",
"tactic": "rcases H with ev' | hx | hx"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nev' : cut v✝ = Ordering.eq\n⊢ MemP cut l",
"tactic": "cases ev.symm.trans ev'"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) l\n⊢ MemP cut l",
"tactic": "exact hx"
},
{
"state_after": "case inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) r\nz : α\nhz : z ∈ r\nez : cut z = Ordering.eq\n⊢ MemP cut l",
"state_before": "case inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) r\n⊢ MemP cut l",
"tactic": "have ⟨z, hz, ez⟩ := Any_def.1 hx"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.lt\nhx : Any (fun x => cut x = Ordering.eq) r\nz : α\nhz : z ∈ r\nez : cut z = Ordering.eq\n⊢ MemP cut l",
"tactic": "cases ez.symm.trans <| IsCut.lt_trans (All_def.1 xr _ hz).1 ev"
},
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\n⊢ MemP cut r",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\n⊢ ∃ x, find? cut r = some x",
"tactic": "refine ihr hr ?_"
},
{
"state_after": "case inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nev' : cut v✝ = Ordering.eq\n⊢ MemP cut r\n\ncase inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) l\n⊢ MemP cut r\n\ncase inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) r\n⊢ MemP cut r",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\n⊢ MemP cut r",
"tactic": "rcases H with ev' | hx | hx"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nev' : cut v✝ = Ordering.eq\n⊢ MemP cut r",
"tactic": "cases ev.symm.trans ev'"
},
{
"state_after": "case inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) l\nz : α\nhz : z ∈ l\nez : cut z = Ordering.eq\n⊢ MemP cut r",
"state_before": "case inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) l\n⊢ MemP cut r",
"tactic": "have ⟨z, hz, ez⟩ := Any_def.1 hx"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) l\nz : α\nhz : z ∈ l\nez : cut z = Ordering.eq\n⊢ MemP cut r",
"tactic": "cases ez.symm.trans <| IsCut.gt_trans (All_def.1 lx _ hz).1 ev"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nev : cut v✝ = Ordering.gt\nhx : Any (fun x => cut x = Ordering.eq) r\n⊢ MemP cut r",
"tactic": "exact hx"
},
{
"state_after": "no goals",
"state_before": "case node.h_3\nα : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\ninst✝¹ : TransCmp cmp\ninst✝ : IsCut cmp cut\nc✝ : RBColor\nl : RBNode α\nv✝ : α\nr : RBNode α\nihl : Ordered cmp l → MemP cut l → ∃ x, x ∈ find? cut l\nihr : Ordered cmp r → MemP cut r → ∃ x, x ∈ find? cut r\nht : Ordered cmp (node c✝ l v✝ r)\nH : MemP cut (node c✝ l v✝ r)\nlx : All (fun x => cmpLT cmp x v✝) l\nxr : All (fun x => cmpLT cmp v✝ x) r\nhl : Ordered cmp l\nhr : Ordered cmp r\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ x, some v✝ = some x",
"tactic": "exact ⟨_, rfl⟩"
}
] |
[
188,
21
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
166,
1
] |
Mathlib/Order/Cover.lean
|
AntisymmRel.wcovby
|
[] |
[
72,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Subgroup.fg_iff
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.68895\nN : Type ?u.68898\ninst✝³ : Monoid M\ninst✝² : AddMonoid N\nG : Type u_1\nH : Type ?u.68910\ninst✝¹ : Group G\ninst✝ : AddGroup H\nP : Subgroup G\nx✝ : ∃ S, closure S = P ∧ Set.Finite S\nS : Set G\nhS : closure S = P\nhf : Set.Finite S\n⊢ closure ↑(Set.Finite.toFinset hf) = P",
"tactic": "simp [hS]"
}
] |
[
243,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Closeds.gc
|
[] |
[
75,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
|
CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁴ : Category D\nE : Type w₂\ninst✝³ : Category E\nF : D ⥤ E\ninst✝² : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝¹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\nX : C\nW : (Cover J X)ᵒᵖ\ni : Cover.Arrow W.unop\n⊢ (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) ⋙ F)) ≪≫\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom.app\n W ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i =\n F.map (Multiequalizer.ι (Cover.index W.unop P) i)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁴ : Category D\nE : Type w₂\ninst✝³ : Category E\nF : D ⥤ E\ninst✝² : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝¹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\nX : C\nW : (Cover J X)ᵒᵖ\ni : Cover.Arrow W.unop\n⊢ (diagramCompIso J F P X).hom.app W ≫ Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i =\n F.map (Multiequalizer.ι (Cover.index W.unop P) i)",
"tactic": "delta diagramCompIso"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁴ : Category D\nE : Type w₂\ninst✝³ : Category E\nF : D ⥤ E\ninst✝² : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝¹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\nX : C\nW : (Cover J X)ᵒᵖ\ni : Cover.Arrow W.unop\n⊢ ((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) ⋙ F))).hom ≫\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i =\n F.map (Multiequalizer.ι (Cover.index W.unop P) i)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁴ : Category D\nE : Type w₂\ninst✝³ : Category E\nF : D ⥤ E\ninst✝² : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝¹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\nX : C\nW : (Cover J X)ᵒᵖ\ni : Cover.Arrow W.unop\n⊢ (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) ⋙ F)) ≪≫\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom.app\n W ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i =\n F.map (Multiequalizer.ι (Cover.index W.unop P) i)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁴ : Category D\nE : Type w₂\ninst✝³ : Category E\nF : D ⥤ E\ninst✝² : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝¹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\nX : C\nW : (Cover J X)ᵒᵖ\ni : Cover.Arrow W.unop\n⊢ ((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) ⋙ F))).hom ≫\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i =\n F.map (Multiequalizer.ι (Cover.index W.unop P) i)",
"tactic": "simp"
}
] |
[
69,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean
|
IsCompactOperator.neg
|
[] |
[
230,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.star_im
|
[] |
[
674,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
ULift.nndist_up_up
|
[] |
[
1729,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1729,
1
] |
Mathlib/Data/Finset/MulAntidiagonal.lean
|
Finset.support_mulAntidiagonal_subset_mul
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsPwo s\nht : Set.IsPwo t\na✝ : α\nu : Set α\nhu : Set.IsPwo u\nx : α × α\na : α\nx✝ : a ∈ {a | Finset.Nonempty (mulAntidiagonal hs ht a)}\nb : α × α\nhb : b.fst ∈ s ∧ b.snd ∈ t ∧ b.fst * b.snd = a\n⊢ a ∈ s * t",
"state_before": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsPwo s\nht : Set.IsPwo t\na✝ : α\nu : Set α\nhu : Set.IsPwo u\nx : α × α\na : α\nx✝ : a ∈ {a | Finset.Nonempty (mulAntidiagonal hs ht a)}\nb : α × α\nhb : b ∈ mulAntidiagonal hs ht a\n⊢ a ∈ s * t",
"tactic": "rw [mem_mulAntidiagonal] at hb"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsPwo s\nht : Set.IsPwo t\na✝ : α\nu : Set α\nhu : Set.IsPwo u\nx : α × α\na : α\nx✝ : a ∈ {a | Finset.Nonempty (mulAntidiagonal hs ht a)}\nb : α × α\nhb : b.fst ∈ s ∧ b.snd ∈ t ∧ b.fst * b.snd = a\n⊢ a ∈ s * t",
"tactic": "exact ⟨b.1, b.2, hb⟩"
}
] |
[
107,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_eq_add_fract_mul
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ b - ⌊(b - a) / p⌋ • p = a + ((b - a) / p - ↑⌊(b - a) / p⌋) * p",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoMod hp a b = a + Int.fract ((b - a) / p) * p",
"tactic": "rw [toIcoMod, toIcoDiv_eq_floor, Int.fract]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ b - ↑⌊(b - a) / p⌋ * p = a + (b - a - p * ↑⌊(b - a) / p⌋)",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ b - ⌊(b - a) / p⌋ • p = a + ((b - a) / p - ↑⌊(b - a) / p⌋) * p",
"tactic": "field_simp [hp.ne.symm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ b - ↑⌊(b - a) / p⌋ * p = a + (b - a - p * ↑⌊(b - a) / p⌋)",
"tactic": "ring"
}
] |
[
1022,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1018,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33924\nδ : Type ?u.33927\ninst✝ : MeasurableSpace α\nf : α →ₛ β\nμ : Measure α\n⊢ ∑ y in SimpleFunc.range f, ↑↑μ (↑f ⁻¹' {y}) = ↑↑μ univ",
"tactic": "rw [f.sum_measure_preimage_singleton, coe_range, preimage_range]"
}
] |
[
232,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.Cover.Arrow.to_middle_condition
|
[] |
[
655,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.coe_iSup_of_directed
|
[
{
"state_after": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\n⊢ ↑(iSup S) ⊆ ⋃ (i : ι), ↑(S i)",
"state_before": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\n⊢ ↑(iSup S) = ⋃ (i : ι), ↑(S i)",
"tactic": "refine' Subset.antisymm _ (iUnion_subset <| le_iSup S)"
},
{
"state_after": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\n⊢ ↑(span R (⋃ (i : ι), ↑(S i))) ⊆ ⋃ (i : ι), ↑(S i)",
"state_before": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\n⊢ ↑(iSup S) ⊆ ⋃ (i : ι), ↑(S i)",
"tactic": "suffices (span R (⋃ i, (S i : Set M)) : Set M) ⊆ ⋃ i : ι, ↑(S i) by\n simpa only [span_iUnion, span_eq] using this"
},
{
"state_after": "case refine'_1\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∃ i, 0 ∈ ↑(S i)\n\ncase refine'_2\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∀ (x y : M) (x_1 : ι), x ∈ ↑(S x_1) → ∀ (x_2 : ι), y ∈ ↑(S x_2) → ∃ i, x + y ∈ ↑(S i)\n\ncase refine'_3\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∀ (a : R) (x : M) (x_1 : ι), x ∈ ↑(S x_1) → ∃ i, a • x ∈ ↑(S i)",
"state_before": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\n⊢ ↑(span R (⋃ (i : ι), ↑(S i))) ⊆ ⋃ (i : ι), ↑(S i)",
"tactic": "refine' fun x hx => span_induction hx (fun _ => id) _ _ _ <;> simp only [mem_iUnion, exists_imp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nthis : ↑(span R (⋃ (i : ι), ↑(S i))) ⊆ ⋃ (i : ι), ↑(S i)\n⊢ ↑(iSup S) ⊆ ⋃ (i : ι), ↑(S i)",
"tactic": "simpa only [span_iUnion, span_eq] using this"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∃ i, 0 ∈ ↑(S i)",
"tactic": "exact hι.elim fun i => ⟨i, (S i).zero_mem⟩"
},
{
"state_after": "case refine'_2\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝¹ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx✝ : M\nhx : x✝ ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\nx y : M\ni : ι\nhi : x ∈ ↑(S i)\nj : ι\nhj : y ∈ ↑(S j)\n⊢ ∃ i, x + y ∈ ↑(S i)",
"state_before": "case refine'_2\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∀ (x y : M) (x_1 : ι), x ∈ ↑(S x_1) → ∀ (x_2 : ι), y ∈ ↑(S x_2) → ∃ i, x + y ∈ ↑(S i)",
"tactic": "intro x y i hi j hj"
},
{
"state_after": "case refine'_2.intro.intro\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝¹ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx✝ : M\nhx : x✝ ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\nx y : M\ni : ι\nhi : x ∈ ↑(S i)\nj : ι\nhj : y ∈ ↑(S j)\nk : ι\nik : S i ≤ S k\njk : S j ≤ S k\n⊢ ∃ i, x + y ∈ ↑(S i)",
"state_before": "case refine'_2\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝¹ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx✝ : M\nhx : x✝ ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\nx y : M\ni : ι\nhi : x ∈ ↑(S i)\nj : ι\nhj : y ∈ ↑(S j)\n⊢ ∃ i, x + y ∈ ↑(S i)",
"tactic": "rcases H i j with ⟨k, ik, jk⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝¹ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx✝ : M\nhx : x✝ ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\nx y : M\ni : ι\nhi : x ∈ ↑(S i)\nj : ι\nhj : y ∈ ↑(S j)\nk : ι\nik : S i ≤ S k\njk : S j ≤ S k\n⊢ ∃ i, x + y ∈ ↑(S i)",
"tactic": "exact ⟨k, add_mem (ik hi) (jk hj)⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nR : Type u_2\nR₂ : Type ?u.86611\nK : Type ?u.86614\nM : Type u_3\nM₂ : Type ?u.86620\nV : Type ?u.86623\nS✝ : Type ?u.86626\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Submodule R M\nH : Directed (fun x x_1 => x ≤ x_1) S\nx : M\nhx : x ∈ ↑(span R (⋃ (i : ι), ↑(S i)))\n⊢ ∀ (a : R) (x : M) (x_1 : ι), x ∈ ↑(S x_1) → ∃ i, a • x ∈ ↑(S i)",
"tactic": "exact fun a x i hi => ⟨i, smul_mem _ a hi⟩"
}
] |
[
307,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.coe_injective
|
[] |
[
832,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
AddMonoidAlgebra.of'_mul_modOf
|
[
{
"state_after": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = ↑0 g'",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\n⊢ of' k G g * x %ᵒᶠ g = 0",
"tactic": "refine Finsupp.ext fun g' => ?_"
},
{
"state_after": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = 0",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = ↑0 g'",
"tactic": "rw [Finsupp.zero_apply]"
},
{
"state_after": "case inl.intro\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\nd : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) (g + d) = 0\n\ncase inr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\nh : ¬∃ d, g' = g + d\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = 0",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = 0",
"tactic": "obtain ⟨d, rfl⟩ | h := em (∃ d, g' = g + d)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\nd : G\n⊢ ↑(of' k G g * x %ᵒᶠ g) (g + d) = 0",
"tactic": "rw [modOf_apply_self_add]"
},
{
"state_after": "no goals",
"state_before": "case inr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\ng' : G\nh : ¬∃ d, g' = g + d\n⊢ ↑(of' k G g * x %ᵒᶠ g) g' = 0",
"tactic": "rw [modOf_apply_of_not_exists_add _ _ _ h, of'_apply, single_mul_apply_of_not_exists_add _ _ h]"
}
] |
[
162,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Data/List/OfFn.lean
|
List.ofFn_eq_nil_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nn : ℕ\nf : Fin n → α\n⊢ ofFn f = [] ↔ n = 0",
"tactic": "cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero]"
}
] |
[
135,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_smul_left
|
[] |
[
480,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.disjoint_nhdsSet_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.17404\nπ : ι → Type ?u.17409\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\n⊢ Disjoint l (𝓝ˢ s) ↔ ∀ (x : α), x ∈ s → Disjoint l (𝓝 x)",
"tactic": "simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left"
}
] |
[
235,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_add_lt_sub
|
[] |
[
483,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
476,
11
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_himp_right
|
[] |
[
608,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.map_map
|
[
{
"state_after": "case a.h\nX : Type u_1\nY✝ : Type ?u.306794\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y✝\nx y z : X\nι : Type ?u.306809\nγ✝ γ : Path x y\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nZ : Type u_3\ninst✝ : TopologicalSpace Z\nf : X → Y\nhf : Continuous f\ng : Y → Z\nhg : Continuous g\nx✝ : ↑I\n⊢ ↑(map (map γ hf) hg) x✝ = ↑(map γ (_ : Continuous (g ∘ f))) x✝",
"state_before": "X : Type u_1\nY✝ : Type ?u.306794\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y✝\nx y z : X\nι : Type ?u.306809\nγ✝ γ : Path x y\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nZ : Type u_3\ninst✝ : TopologicalSpace Z\nf : X → Y\nhf : Continuous f\ng : Y → Z\nhg : Continuous g\n⊢ map (map γ hf) hg = map γ (_ : Continuous (g ∘ f))",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nX : Type u_1\nY✝ : Type ?u.306794\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y✝\nx y z : X\nι : Type ?u.306809\nγ✝ γ : Path x y\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nZ : Type u_3\ninst✝ : TopologicalSpace Z\nf : X → Y\nhf : Continuous f\ng : Y → Z\nhg : Continuous g\nx✝ : ↑I\n⊢ ↑(map (map γ hf) hg) x✝ = ↑(map γ (_ : Continuous (g ∘ f))) x✝",
"tactic": "rfl"
}
] |
[
457,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Algebra/Hom/GroupInstances.lean
|
AddMonoidHom.coe_flip_mul
|
[] |
[
323,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.preimage_comap_zeroLocus_aux
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.109078\ninst✝ : CommRing S'\nf : R →+* S\ns : Set R\nx : PrimeSpectrum S\n⊢ x ∈\n (fun y => { asIdeal := Ideal.comap f y.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.comap f y.asIdeal)) }) ⁻¹'\n zeroLocus s ↔\n x ∈ zeroLocus (↑f '' s)",
"state_before": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.109078\ninst✝ : CommRing S'\nf : R →+* S\ns : Set R\n⊢ (fun y => { asIdeal := Ideal.comap f y.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.comap f y.asIdeal)) }) ⁻¹'\n zeroLocus s =\n zeroLocus (↑f '' s)",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.109078\ninst✝ : CommRing S'\nf : R →+* S\ns : Set R\nx : PrimeSpectrum S\n⊢ x ∈\n (fun y => { asIdeal := Ideal.comap f y.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.comap f y.asIdeal)) }) ⁻¹'\n zeroLocus s ↔\n s ⊆ (fun a => ↑f a) ⁻¹' ↑x.asIdeal",
"state_before": "case h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.109078\ninst✝ : CommRing S'\nf : R →+* S\ns : Set R\nx : PrimeSpectrum S\n⊢ x ∈\n (fun y => { asIdeal := Ideal.comap f y.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.comap f y.asIdeal)) }) ⁻¹'\n zeroLocus s ↔\n x ∈ zeroLocus (↑f '' s)",
"tactic": "simp only [mem_zeroLocus, Set.image_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.109078\ninst✝ : CommRing S'\nf : R →+* S\ns : Set R\nx : PrimeSpectrum S\n⊢ x ∈\n (fun y => { asIdeal := Ideal.comap f y.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.comap f y.asIdeal)) }) ⁻¹'\n zeroLocus s ↔\n s ⊆ (fun a => ↑f a) ⁻¹' ↑x.asIdeal",
"tactic": "rfl"
}
] |
[
583,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.coe_toHomeomorph
|
[] |
[
466,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.mem_commonNeighbors
|
[] |
[
1017,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono_id
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\n⊢ (if h : Δ = Δ then eqToHom (_ : HomologicalComplex.X K (len Δ) = HomologicalComplex.X K (len Δ))\n else if h : Isδ₀ (𝟙 Δ) then HomologicalComplex.d K (len Δ) (len Δ) else 0) =\n 𝟙 (HomologicalComplex.X K (len Δ))",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\n⊢ mapMono K (𝟙 Δ) = 𝟙 (HomologicalComplex.X K (len Δ))",
"tactic": "unfold mapMono"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\n⊢ (if h : Δ = Δ then eqToHom (_ : HomologicalComplex.X K (len Δ) = HomologicalComplex.X K (len Δ))\n else if h : Isδ₀ (𝟙 Δ) then HomologicalComplex.d K (len Δ) (len Δ) else 0) =\n 𝟙 (HomologicalComplex.X K (len Δ))",
"tactic": "simp only [eq_self_iff_true, eqToHom_refl, dite_eq_ite, if_true]"
}
] |
[
107,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.single_trans_of_eq_none
|
[] |
[
405,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Formula.realize_bot
|
[] |
[
622,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.coe_sumCoords_eq_finsum
|
[
{
"state_after": "case h\nι : Type u_3\nι' : Type ?u.157559\nR : Type u_1\nR₂ : Type ?u.157565\nK : Type ?u.157568\nM : Type u_2\nM' : Type ?u.157574\nM'' : Type ?u.157577\nV : Type u\nV' : Type ?u.157582\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx m : M\n⊢ ↑(sumCoords b) m = ∑ᶠ (i : ι), ↑(coord b i) m",
"state_before": "ι : Type u_3\nι' : Type ?u.157559\nR : Type u_1\nR₂ : Type ?u.157565\nK : Type ?u.157568\nM : Type u_2\nM' : Type ?u.157574\nM'' : Type ?u.157577\nV : Type u\nV' : Type ?u.157582\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx : M\n⊢ ↑(sumCoords b) = fun m => ∑ᶠ (i : ι), ↑(coord b i) m",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_3\nι' : Type ?u.157559\nR : Type u_1\nR₂ : Type ?u.157565\nK : Type ?u.157568\nM : Type u_2\nM' : Type ?u.157574\nM'' : Type ?u.157577\nV : Type u\nV' : Type ?u.157582\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx m : M\n⊢ ↑(sumCoords b) m = ∑ᶠ (i : ι), ↑(coord b i) m",
"tactic": "simp only [Basis.sumCoords, Basis.coord, Finsupp.lapply_apply, LinearMap.id_coe,\n LinearEquiv.coe_coe, Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp,\n finsum_eq_sum _ (b.repr m).finite_support, Finsupp.sum, Finset.finite_toSet_toFinset, id.def,\n Finsupp.fun_support_eq]"
}
] |
[
242,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.mul_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : CommSemiring β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\nc : β\nz : ℕ\n⊢ Tendsto (fun a => k a ^ z * (fun n => f n * c) a) l (𝓝 0)",
"tactic": "simpa only [← mul_assoc, MulZeroClass.zero_mul] using Tendsto.mul_const c (hf z)"
}
] |
[
99,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.nonneg_smul
|
[
{
"state_after": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\n⊢ Nonneg (↑↑n * a)",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\n⊢ Nonneg (↑n * a)",
"tactic": "rw [← Int.cast_ofNat n]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\n⊢ Nonneg (↑↑n * a)",
"tactic": "exact\n match a, nonneg_cases ha, ha with\n | _, ⟨x, y, Or.inl rfl⟩, _ => by rw [smul_val]; trivial\n | _, ⟨x, y, Or.inr <| Or.inl rfl⟩, ha => by\n rw [smul_val]; simpa using nonnegg_pos_neg.2 (sqLe_smul n <| nonnegg_pos_neg.1 ha)\n | _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ha => by\n rw [smul_val]; simpa using nonnegg_neg_pos.2 (sqLe_smul n <| nonnegg_neg_pos.1 ha)"
},
{
"state_after": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\nx y : ℕ\nx✝ : Nonneg { re := ↑x, im := ↑y }\n⊢ Nonneg { re := ↑n * ↑x, im := ↑n * ↑y }",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\nx y : ℕ\nx✝ : Nonneg { re := ↑x, im := ↑y }\n⊢ Nonneg (↑↑n * { re := ↑x, im := ↑y })",
"tactic": "rw [smul_val]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : Nonneg a\nx y : ℕ\nx✝ : Nonneg { re := ↑x, im := ↑y }\n⊢ Nonneg { re := ↑n * ↑x, im := ↑n * ↑y }",
"tactic": "trivial"
},
{
"state_after": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := ↑x, im := -↑y }\n⊢ Nonneg { re := ↑n * ↑x, im := ↑n * -↑y }",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := ↑x, im := -↑y }\n⊢ Nonneg (↑↑n * { re := ↑x, im := -↑y })",
"tactic": "rw [smul_val]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := ↑x, im := -↑y }\n⊢ Nonneg { re := ↑n * ↑x, im := ↑n * -↑y }",
"tactic": "simpa using nonnegg_pos_neg.2 (sqLe_smul n <| nonnegg_pos_neg.1 ha)"
},
{
"state_after": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := -↑x, im := ↑y }\n⊢ Nonneg { re := ↑n * -↑x, im := ↑n * ↑y }",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := -↑x, im := ↑y }\n⊢ Nonneg (↑↑n * { re := -↑x, im := ↑y })",
"tactic": "rw [smul_val]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : Nonneg a\nx y : ℕ\nha : Nonneg { re := -↑x, im := ↑y }\n⊢ Nonneg { re := ↑n * -↑x, im := ↑n * ↑y }",
"tactic": "simpa using nonnegg_neg_pos.2 (sqLe_smul n <| nonnegg_neg_pos.1 ha)"
}
] |
[
805,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
797,
1
] |
Mathlib/Data/List/TFAE.lean
|
List.tfae_cons_of_mem
|
[
{
"state_after": "no goals",
"state_before": "a b : Prop\nl : List Prop\nh : b ∈ l\nH : TFAE (a :: l)\n⊢ a ∈ a :: l",
"tactic": "simp"
},
{
"state_after": "case intro.head.head\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\n⊢ a ↔ a\n\ncase intro.head.tail\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\nq : Prop\nhq : Mem q l\n⊢ a ↔ q\n\ncase intro.tail.head\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\np : Prop\nhp : Mem p l\n⊢ p ↔ a\n\ncase intro.tail.tail\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\np : Prop\nhp : Mem p l\nq : Prop\nhq : Mem q l\n⊢ p ↔ q",
"state_before": "a b : Prop\nl : List Prop\nh : b ∈ l\n⊢ (a ↔ b) ∧ TFAE l → TFAE (a :: l)",
"tactic": "rintro ⟨ab, H⟩ p (_ | ⟨_, hp⟩) q (_ | ⟨_, hq⟩)"
},
{
"state_after": "no goals",
"state_before": "case intro.head.head\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\n⊢ a ↔ a",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.head.tail\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\nq : Prop\nhq : Mem q l\n⊢ a ↔ q",
"tactic": "exact ab.trans (H _ h _ hq)"
},
{
"state_after": "no goals",
"state_before": "case intro.tail.head\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\np : Prop\nhp : Mem p l\n⊢ p ↔ a",
"tactic": "exact (ab.trans (H _ h _ hp)).symm"
},
{
"state_after": "no goals",
"state_before": "case intro.tail.tail\na b : Prop\nl : List Prop\nh : b ∈ l\nab : a ↔ b\nH : TFAE l\np : Prop\nhp : Mem p l\nq : Prop\nhq : Mem q l\n⊢ p ↔ q",
"tactic": "exact H _ hp _ hq"
}
] |
[
48,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
LinearMap.le_rank_iff_exists_linearIndependent_finset
|
[
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x) ↔\n ∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ ↑n ≤ rank f ↔ ∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "simp only [le_rank_iff_exists_linearIndependent, Cardinal.lift_natCast, Cardinal.lift_eq_nat_iff,\n Cardinal.mk_set_eq_nat_iff_finset]"
},
{
"state_after": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x) →\n ∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x\n\ncase mpr\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x) →\n ∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x) ↔\n ∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nf : V →ₗ[K] V'\nt : Finset V\nsi : LinearIndependent K fun x => ↑f ↑x\n⊢ ∃ s, Finset.card s = Finset.card t ∧ LinearIndependent K fun x => ↑f ↑x",
"state_before": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x) →\n ∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nf : V →ₗ[K] V'\nt : Finset V\nsi : LinearIndependent K fun x => ↑f ↑x\n⊢ ∃ s, Finset.card s = Finset.card t ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "exact ⟨t, rfl, si⟩"
},
{
"state_after": "case mpr.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nf : V →ₗ[K] V'\ns : Finset V\nsi : LinearIndependent K fun x => ↑f ↑x\n⊢ ∃ s_1, (∃ t, ↑t = s_1 ∧ Finset.card t = Finset.card s) ∧ LinearIndependent K fun x => ↑f ↑x",
"state_before": "case mpr\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nn : ℕ\nf : V →ₗ[K] V'\n⊢ (∃ s, Finset.card s = n ∧ LinearIndependent K fun x => ↑f ↑x) →\n ∃ s, (∃ t, ↑t = s ∧ Finset.card t = n) ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "rintro ⟨s, rfl, si⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1195191\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nf : V →ₗ[K] V'\ns : Finset V\nsi : LinearIndependent K fun x => ↑f ↑x\n⊢ ∃ s_1, (∃ t, ↑t = s_1 ∧ Finset.card t = Finset.card s) ∧ LinearIndependent K fun x => ↑f ↑x",
"tactic": "exact ⟨s, ⟨s, rfl, rfl⟩, si⟩"
}
] |
[
1419,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1411,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart
|
[] |
[
314,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM2to1.trStmts₁_run
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_4\ninst✝¹ : Inhabited Λ\nσ : Type u_3\ninst✝ : Inhabited σ\nk : K\ns : StAct k\nq : Stmt₂\n⊢ trStmts₁ (stRun s q) = {go k s q, ret q} ∪ trStmts₁ q",
"tactic": "cases s <;> simp only [trStmts₁]"
}
] |
[
2548,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2546,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearMap.pi_comp
|
[] |
[
77,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_smul
|
[
{
"state_after": "case h\nR : Type ?u.955175\nR₁ : Type ?u.955178\nR₂ : Type ?u.955181\nR₃ : Type ?u.955184\nR₄ : Type ?u.955187\nS : Type ?u.955190\nK : Type u_1\nK₂ : Type ?u.955196\nM : Type ?u.955199\nM' : Type ?u.955202\nM₁ : Type ?u.955205\nM₂ : Type ?u.955208\nM₃ : Type ?u.955211\nM₄ : Type ?u.955214\nN : Type ?u.955217\nN₂ : Type ?u.955220\nι : Type ?u.955223\nV : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\np : Submodule K V₂\na : K\nh : a ≠ 0\nb : V\n⊢ b ∈ comap (a • f) p ↔ b ∈ comap f p",
"state_before": "R : Type ?u.955175\nR₁ : Type ?u.955178\nR₂ : Type ?u.955181\nR₃ : Type ?u.955184\nR₄ : Type ?u.955187\nS : Type ?u.955190\nK : Type u_1\nK₂ : Type ?u.955196\nM : Type ?u.955199\nM' : Type ?u.955202\nM₁ : Type ?u.955205\nM₂ : Type ?u.955208\nM₃ : Type ?u.955211\nM₄ : Type ?u.955214\nN : Type ?u.955217\nN₂ : Type ?u.955220\nι : Type ?u.955223\nV : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\np : Submodule K V₂\na : K\nh : a ≠ 0\n⊢ comap (a • f) p = comap f p",
"tactic": "ext b"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.955175\nR₁ : Type ?u.955178\nR₂ : Type ?u.955181\nR₃ : Type ?u.955184\nR₄ : Type ?u.955187\nS : Type ?u.955190\nK : Type u_1\nK₂ : Type ?u.955196\nM : Type ?u.955199\nM' : Type ?u.955202\nM₁ : Type ?u.955205\nM₂ : Type ?u.955208\nM₃ : Type ?u.955211\nM₄ : Type ?u.955214\nN : Type ?u.955217\nN₂ : Type ?u.955220\nι : Type ?u.955223\nV : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\np : Submodule K V₂\na : K\nh : a ≠ 0\nb : V\n⊢ b ∈ comap (a • f) p ↔ b ∈ comap f p",
"tactic": "simp only [Submodule.mem_comap, p.smul_mem_iff h, LinearMap.smul_apply]"
}
] |
[
1074,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1072,
1
] |
src/lean/Init/Control/Lawful.lean
|
bind_congr
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα β : Type u_1\ninst✝ : Bind m\nx : m α\nf g : α → m β\nh : ∀ (a : α), f a = g a\n⊢ x >>= f = x >>= g",
"tactic": "simp [funext h]"
}
] |
[
69,
18
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
68,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isLUB_univ
|
[] |
[
816,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
815,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffWithinAt.prod
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ContDiffWithinAt 𝕜 n (fun x => (f x, g x)) s x",
"tactic": "intro m hm"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑m) f p u\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"tactic": "rcases hf m hm with ⟨u, hu, p, hp⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑m) f p u\nv : Set E\nhv : v ∈ 𝓝[insert x s] x\nq : E → FormalMultilinearSeries 𝕜 E G\nhq : HasFTaylorSeriesUpToOn (↑m) g q v\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑m) f p u\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"tactic": "rcases hg m hm with ⟨v, hv, q, hq⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.686689\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf : E → F\ng : E → G\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nm : ℕ\nhm : ↑m ≤ n\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑m) f p u\nv : Set E\nhv : v ∈ 𝓝[insert x s] x\nq : E → FormalMultilinearSeries 𝕜 E G\nhq : HasFTaylorSeriesUpToOn (↑m) g q v\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) (fun x => (f x, g x)) p u",
"tactic": "exact\n ⟨u ∩ v, Filter.inter_mem hu hv, _,\n (hp.mono (inter_subset_left u v)).prod (hq.mono (inter_subset_right u v))⟩"
}
] |
[
527,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.toLin'_injective
|
[] |
[
216,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.IsMaximal.coprime_of_ne
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\nM M' : Ideal α\nhM : IsMaximal M\nhM' : IsMaximal M'\nh : M ⊔ M' ≠ ⊤\n⊢ M = M'",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\nM M' : Ideal α\nhM : IsMaximal M\nhM' : IsMaximal M'\nhne : M ≠ M'\n⊢ M ⊔ M' = ⊤",
"tactic": "contrapose! hne with h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\nM M' : Ideal α\nhM : IsMaximal M\nhM' : IsMaximal M'\nh : M ⊔ M' ≠ ⊤\n⊢ M = M'",
"tactic": "exact hM.eq_of_le hM'.ne_top (le_sup_left.trans_eq (hM'.eq_of_le h le_sup_right).symm)"
}
] |
[
328,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
|
MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable
|
[
{
"state_after": "case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \\ s → HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a\n\ncase inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \\ s → HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \\ s → HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"tactic": "cases' le_total a b with hab hab"
},
{
"state_after": "case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : a ≤ b\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo a b \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"state_before": "case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \\ s → HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"tactic": "simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : a ≤ b\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo a b \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"tactic": "exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\nHc : ContinuousOn f (Set.Icc b a)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo b a \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \\ s → HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"tactic": "simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\nHc : ContinuousOn f (Set.Icc b a)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo b a \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in b..a, f' x) = f a - f b",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\nHc : ContinuousOn f (Set.Icc b a)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo b a \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in a..b, f' x) = f b - f a",
"tactic": "rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b ≤ a\nHc : ContinuousOn f (Set.Icc b a)\nHd : ∀ (x : ℝ), x ∈ Set.Ioo b a \\ s → HasDerivAt f (f' x) x\n⊢ (∫ (x : ℝ) in b..a, f' x) = f a - f b",
"tactic": "exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi.symm"
}
] |
[
430,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
Convex.subset_toCone
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.172872\nG : Type ?u.172875\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nx✝ x : E\nhx : x ∈ s\n⊢ 1 • x ∈ s",
"tactic": "rwa [one_smul]"
}
] |
[
673,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.mem_prod_principal
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\n⊢ (∃ t₁, t₁ ∈ f ∧ ∃ t₂, t₂ ∈ 𝓟 t ∧ t₁ ×ˢ t₂ ⊆ s) ↔ ∃ t_1, t_1 ∈ f ∧ t_1 ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\n⊢ s ∈ f ×ˢ 𝓟 t ↔ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s} ∈ f",
"tactic": "rw [← @exists_mem_subset_iff _ f, mem_prod_iff]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\n⊢ (∃ t₂, t₂ ∈ 𝓟 t ∧ u ×ˢ t₂ ⊆ s) → u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nh : u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}\n⊢ u ×ˢ t ⊆ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\n⊢ (∃ t₁, t₁ ∈ f ∧ ∃ t₂, t₂ ∈ 𝓟 t ∧ t₁ ×ˢ t₂ ⊆ s) ↔ ∃ t_1, t_1 ∈ f ∧ t_1 ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}",
"tactic": "refine' exists_congr fun u => Iff.rfl.and ⟨_, fun h => ⟨t, mem_principal_self t, _⟩⟩"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nv : Set β\nv_in : v ∈ 𝓟 t\nhv : u ×ˢ v ⊆ s\na : α\na_in : a ∈ u\nb : β\nb_in : b ∈ t\n⊢ (a, b) ∈ s",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\n⊢ (∃ t₂, t₂ ∈ 𝓟 t ∧ u ×ˢ t₂ ⊆ s) → u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}",
"tactic": "rintro ⟨v, v_in, hv⟩ a a_in b b_in"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nv : Set β\nv_in : v ∈ 𝓟 t\nhv : u ×ˢ v ⊆ s\na : α\na_in : a ∈ u\nb : β\nb_in : b ∈ t\n⊢ (a, b) ∈ s",
"tactic": "exact hv (mk_mem_prod a_in <| v_in b_in)"
},
{
"state_after": "case refine'_2.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nh : u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}\nx : α\ny : β\nhx : (x, y).fst ∈ u\nhy : (x, y).snd ∈ t\n⊢ (x, y) ∈ s",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nh : u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}\n⊢ u ×ˢ t ⊆ s",
"tactic": "rintro ⟨x, y⟩ ⟨hx, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2996\nδ : Type ?u.2999\nι : Sort ?u.3002\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\nt : Set β\nu : Set α\nh : u ⊆ {a | ∀ (b : β), b ∈ t → (a, b) ∈ s}\nx : α\ny : β\nhx : (x, y).fst ∈ u\nhy : (x, y).snd ∈ t\n⊢ (x, y) ∈ s",
"tactic": "exact h hx y hy"
}
] |
[
97,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_lt_add
|
[
{
"state_after": "case intro\nα : Type ?u.138661\nβ : Type ?u.138664\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na : ℝ≥0\nac : ↑a < c\n⊢ ↑a + b < c + d",
"state_before": "α : Type ?u.138661\nβ : Type ?u.138664\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nac : a < c\nbd : b < d\n⊢ a + b < c + d",
"tactic": "lift a to ℝ≥0 using ac.ne_top"
},
{
"state_after": "case intro.intro\nα : Type ?u.138661\nβ : Type ?u.138664\nc d : ℝ≥0∞\nr p q a : ℝ≥0\nac : ↑a < c\nb : ℝ≥0\nbd : ↑b < d\n⊢ ↑a + ↑b < c + d",
"state_before": "case intro\nα : Type ?u.138661\nβ : Type ?u.138664\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na : ℝ≥0\nac : ↑a < c\n⊢ ↑a + b < c + d",
"tactic": "lift b to ℝ≥0 using bd.ne_top"
},
{
"state_after": "case intro.intro.none\nα : Type ?u.138661\nβ : Type ?u.138664\nd : ℝ≥0∞\nr p q a b : ℝ≥0\nbd : ↑b < d\nac : ↑a < none\n⊢ ↑a + ↑b < none + d\n\ncase intro.intro.some\nα : Type ?u.138661\nβ : Type ?u.138664\nd : ℝ≥0∞\nr p q a b : ℝ≥0\nbd : ↑b < d\nval✝ : ℝ≥0\nac : ↑a < Option.some val✝\n⊢ ↑a + ↑b < Option.some val✝ + d",
"state_before": "case intro.intro\nα : Type ?u.138661\nβ : Type ?u.138664\nc d : ℝ≥0∞\nr p q a : ℝ≥0\nac : ↑a < c\nb : ℝ≥0\nbd : ↑b < d\n⊢ ↑a + ↑b < c + d",
"tactic": "cases c"
},
{
"state_after": "case intro.intro.some.none\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝ : ℝ≥0\nac : ↑a < Option.some val✝\nbd : ↑b < none\n⊢ ↑a + ↑b < Option.some val✝ + none\n\ncase intro.intro.some.some\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝¹ : ℝ≥0\nac : ↑a < Option.some val✝¹\nval✝ : ℝ≥0\nbd : ↑b < Option.some val✝\n⊢ ↑a + ↑b < Option.some val✝¹ + Option.some val✝",
"state_before": "case intro.intro.some\nα : Type ?u.138661\nβ : Type ?u.138664\nd : ℝ≥0∞\nr p q a b : ℝ≥0\nbd : ↑b < d\nval✝ : ℝ≥0\nac : ↑a < Option.some val✝\n⊢ ↑a + ↑b < Option.some val✝ + d",
"tactic": "cases d"
},
{
"state_after": "case intro.intro.some.some\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝¹ val✝ : ℝ≥0\nac : a < val✝¹\nbd : b < val✝\n⊢ a + b < val✝¹ + val✝",
"state_before": "case intro.intro.some.some\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝¹ : ℝ≥0\nac : ↑a < Option.some val✝¹\nval✝ : ℝ≥0\nbd : ↑b < Option.some val✝\n⊢ ↑a + ↑b < Option.some val✝¹ + Option.some val✝",
"tactic": "simp only [← coe_add, some_eq_coe, coe_lt_coe] at *"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.some.some\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝¹ val✝ : ℝ≥0\nac : a < val✝¹\nbd : b < val✝\n⊢ a + b < val✝¹ + val✝",
"tactic": "exact add_lt_add ac bd"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.none\nα : Type ?u.138661\nβ : Type ?u.138664\nd : ℝ≥0∞\nr p q a b : ℝ≥0\nbd : ↑b < d\nac : ↑a < none\n⊢ ↑a + ↑b < none + d",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.some.none\nα : Type ?u.138661\nβ : Type ?u.138664\nr p q a b val✝ : ℝ≥0\nac : ↑a < Option.some val✝\nbd : ↑b < none\n⊢ ↑a + ↑b < Option.some val✝ + none",
"tactic": "simp"
}
] |
[
909,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
903,
11
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.image_Icc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : α\n⊢ ↑e '' Icc a b = Icc (↑e a) (↑e b)",
"tactic": "rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]"
}
] |
[
106,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
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