file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.submonoidMap_surjective
|
[
{
"state_after": "case mk.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.149443\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.149464\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.149488\nmc : MonoidHomClass F M N\nf : M →* N\nM' : Submonoid M\nx : M\nhx : x ∈ ↑M'\n⊢ ∃ a, ↑(submonoidMap f M') a = { val := ↑f x, property := (_ : ∃ a, a ∈ ↑M' ∧ ↑f a = ↑f x) }",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.149443\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.149464\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.149488\nmc : MonoidHomClass F M N\nf : M →* N\nM' : Submonoid M\n⊢ Function.Surjective ↑(submonoidMap f M')",
"tactic": "rintro ⟨_, x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type ?u.149443\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.149464\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.149488\nmc : MonoidHomClass F M N\nf : M →* N\nM' : Submonoid M\nx : M\nhx : x ∈ ↑M'\n⊢ ∃ a, ↑(submonoidMap f M') a = { val := ↑f x, property := (_ : ∃ a, a ∈ ↑M' ∧ ↑f a = ↑f x) }",
"tactic": "exact ⟨⟨x, hx⟩, rfl⟩"
}
] |
[
1277,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1274,
1
] |
Mathlib/Combinatorics/Configuration.lean
|
Configuration.ProjectivePlane.card_lines
|
[] |
[
523,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
522,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeRing.coe_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nx : FreeRing α\n⊢ ↑(-x) = -↑x",
"tactic": "rw [castFreeCommRing, map_neg]"
}
] |
[
345,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
11
] |
Mathlib/Algebra/Module/LinearMap.lean
|
AddMonoidHom.toIntLinearMap_injective
|
[
{
"state_after": "R : Type ?u.355190\nR₁ : Type ?u.355193\nR₂ : Type ?u.355196\nR₃ : Type ?u.355199\nk : Type ?u.355202\nS : Type ?u.355205\nS₃ : Type ?u.355208\nT : Type ?u.355211\nM : Type u_1\nM₁ : Type ?u.355217\nM₂ : Type u_2\nM₃ : Type ?u.355223\nN₁ : Type ?u.355226\nN₂ : Type ?u.355229\nN₃ : Type ?u.355232\nι : Type ?u.355235\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\nf g : M →+ M₂\nh : toIntLinearMap f = toIntLinearMap g\n⊢ f = g",
"state_before": "R : Type ?u.355190\nR₁ : Type ?u.355193\nR₂ : Type ?u.355196\nR₃ : Type ?u.355199\nk : Type ?u.355202\nS : Type ?u.355205\nS₃ : Type ?u.355208\nT : Type ?u.355211\nM : Type u_1\nM₁ : Type ?u.355217\nM₂ : Type u_2\nM₃ : Type ?u.355223\nN₁ : Type ?u.355226\nN₂ : Type ?u.355229\nN₃ : Type ?u.355232\nι : Type ?u.355235\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\n⊢ Injective toIntLinearMap",
"tactic": "intro f g h"
},
{
"state_after": "case h\nR : Type ?u.355190\nR₁ : Type ?u.355193\nR₂ : Type ?u.355196\nR₃ : Type ?u.355199\nk : Type ?u.355202\nS : Type ?u.355205\nS₃ : Type ?u.355208\nT : Type ?u.355211\nM : Type u_1\nM₁ : Type ?u.355217\nM₂ : Type u_2\nM₃ : Type ?u.355223\nN₁ : Type ?u.355226\nN₂ : Type ?u.355229\nN₃ : Type ?u.355232\nι : Type ?u.355235\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\nf g : M →+ M₂\nh : toIntLinearMap f = toIntLinearMap g\nx : M\n⊢ ↑f x = ↑g x",
"state_before": "R : Type ?u.355190\nR₁ : Type ?u.355193\nR₂ : Type ?u.355196\nR₃ : Type ?u.355199\nk : Type ?u.355202\nS : Type ?u.355205\nS₃ : Type ?u.355208\nT : Type ?u.355211\nM : Type u_1\nM₁ : Type ?u.355217\nM₂ : Type u_2\nM₃ : Type ?u.355223\nN₁ : Type ?u.355226\nN₂ : Type ?u.355229\nN₃ : Type ?u.355232\nι : Type ?u.355235\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\nf g : M →+ M₂\nh : toIntLinearMap f = toIntLinearMap g\n⊢ f = g",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.355190\nR₁ : Type ?u.355193\nR₂ : Type ?u.355196\nR₃ : Type ?u.355199\nk : Type ?u.355202\nS : Type ?u.355205\nS₃ : Type ?u.355208\nT : Type ?u.355211\nM : Type u_1\nM₁ : Type ?u.355217\nM₂ : Type u_2\nM₃ : Type ?u.355223\nN₁ : Type ?u.355226\nN₂ : Type ?u.355229\nN₃ : Type ?u.355232\nι : Type ?u.355235\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\nf g : M →+ M₂\nh : toIntLinearMap f = toIntLinearMap g\nx : M\n⊢ ↑f x = ↑g x",
"tactic": "exact LinearMap.congr_fun h x"
}
] |
[
768,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.Nonempty.exists_eq_singleton_or_nontrivial
|
[] |
[
812,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
810,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.compl_filter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13578\nγ : Type ?u.13581\ninst✝³ : Fintype α\ns t : Finset α\ninst✝² : DecidableEq α\na : α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : (x : α) → Decidable ¬p x\n⊢ ∀ (a : α), a ∈ filter p univᶜ ↔ a ∈ filter (fun x => ¬p x) univ",
"tactic": "simp"
}
] |
[
241,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_toSubmonoid
|
[] |
[
528,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.neLocus_neg_neg
|
[] |
[
142,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderOnWith.continuousOn
|
[] |
[
149,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
11
] |
Mathlib/Algebra/Lie/DirectSum.lean
|
DirectSum.bracket_apply
|
[] |
[
125,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/List/Basic.lean
|
List.map₂Left_eq_zipWith
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas : List α\nx✝ : length [] ≤ length []\n⊢ map₂Left f [] [] = zipWith (fun a b => f a (some b)) [] []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas : List α\nhead✝ : β\ntail✝ : List β\nx✝ : length [] ≤ length (head✝ :: tail✝)\n⊢ map₂Left f [] (head✝ :: tail✝) = zipWith (fun a b => f a (some b)) [] (head✝ :: tail✝)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas✝ : List α\na : α\nas : List α\nh : length (a :: as) ≤ length []\n⊢ map₂Left f (a :: as) [] = zipWith (fun a b => f a (some b)) (a :: as) []",
"tactic": "simp at h"
},
{
"state_after": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas✝ : List α\na : α\nas : List α\nb : β\nbs : List β\nh : length as ≤ length bs\n⊢ map₂Left f (a :: as) (b :: bs) = zipWith (fun a b => f a (some b)) (a :: as) (b :: bs)",
"state_before": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas✝ : List α\na : α\nas : List α\nb : β\nbs : List β\nh : length (a :: as) ≤ length (b :: bs)\n⊢ map₂Left f (a :: as) (b :: bs) = zipWith (fun a b => f a (some b)) (a :: as) (b :: bs)",
"tactic": "simp [Nat.succ_le_succ_iff] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.466133\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → Option β → γ\nas✝ : List α\na : α\nas : List α\nb : β\nbs : List β\nh : length as ≤ length bs\n⊢ map₂Left f (a :: as) (b :: bs) = zipWith (fun a b => f a (some b)) (a :: as) (b :: bs)",
"tactic": "simp [h, map₂Left_eq_zipWith]"
}
] |
[
4124,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4116,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
DifferentiableAt.arctan
|
[] |
[
201,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Topology/StoneCech.lean
|
ultrafilter_isOpen_basic
|
[] |
[
57,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
le_compl_iff_disjoint_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.156709\nα : Type u_1\nβ : Type ?u.156715\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ a ≤ bᶜ ↔ Disjoint a b",
"tactic": "rw [← himp_bot, le_himp_iff, disjoint_iff_inf_le]"
}
] |
[
820,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivWithinAt.add
|
[] |
[
66,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
8
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.minimalPeriod_eq_minimalPeriod_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf fa : α → α\nfb : β → β\nx y✝ : α\nm n : ℕ\ng : β → β\ny : β\n⊢ minimalPeriod f x = minimalPeriod g y ↔ ∀ (n : ℕ), IsPeriodicPt f n x ↔ IsPeriodicPt g n y",
"tactic": "simp_rw [isPeriodicPt_iff_minimalPeriod_dvd, dvd_right_iff_eq]"
}
] |
[
412,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.dom_of_le_some
|
[] |
[
217,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.compl_map
|
[] |
[
1059,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.sigmaUncurry_smul
|
[] |
[
1576,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1572,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.iInf_nonneg
|
[] |
[
905,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.destruct_nil
|
[] |
[
239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
GroupCat.mono_iff_injective
|
[] |
[
96,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorPrebundle.continuous_totalSpaceMk
|
[] |
[
961,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
959,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM2to1.addBottom_head_fst
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.605156\ninst✝¹ : Inhabited Λ\nσ : Type ?u.605162\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ (ListBlank.head (addBottom L)).fst = true",
"tactic": "rw [addBottom, ListBlank.head_cons]"
}
] |
[
2400,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2399,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.injective
|
[] |
[
187,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
11
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
stronglyMeasurable_derivWithin_Ioi
|
[
{
"state_after": "F : Type u_1\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\ninst✝¹ : CompleteSpace F\ninst✝ : SecondCountableTopology F\nthis✝¹ : MeasurableSpace F := borel F\nthis✝ : BorelSpace F\n⊢ StronglyMeasurable fun x => derivWithin f (Ioi x) x",
"state_before": "F : Type u_1\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\ninst✝¹ : CompleteSpace F\ninst✝ : SecondCountableTopology F\n⊢ StronglyMeasurable fun x => derivWithin f (Ioi x) x",
"tactic": "borelize F"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\ninst✝¹ : CompleteSpace F\ninst✝ : SecondCountableTopology F\nthis✝¹ : MeasurableSpace F := borel F\nthis✝ : BorelSpace F\n⊢ StronglyMeasurable fun x => derivWithin f (Ioi x) x",
"tactic": "exact (measurable_derivWithin_Ioi f).stronglyMeasurable"
}
] |
[
820,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
1
] |
Mathlib/Data/Sigma/Interval.lean
|
Sigma.card_Icc
|
[] |
[
64,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.trans_source
|
[] |
[
710,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_le_one'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type ?u.18790\nβ : Type ?u.18793\nM : Type ?u.18796\nN : Type u_2\nG : Type ?u.18802\nk : Type ?u.18805\nR : Type ?u.18808\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf g : ι → N\ns t : Finset ι\nh : ∀ (i : ι), i ∈ s → f i ≤ 1\n⊢ ∏ i in s, 1 = 1",
"tactic": "rw [prod_const_one]"
}
] |
[
155,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
toZ_neg
|
[
{
"state_after": "case refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ toZ i0 i ≤ 0\n\ncase refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ toZ i0 i ≠ 0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ toZ i0 i < 0",
"tactic": "refine' lt_of_le_of_ne _ _"
},
{
"state_after": "case refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ 0 ≤ ↑(Nat.find (_ : ∃ n, (pred^[n]) i0 = i))",
"state_before": "case refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ toZ i0 i ≤ 0",
"tactic": "rw [toZ_of_lt hi, neg_nonpos]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ 0 ≤ ↑(Nat.find (_ : ∃ n, (pred^[n]) i0 = i))",
"tactic": "exact Nat.cast_nonneg _"
},
{
"state_after": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\nh : toZ i0 i = 0\n⊢ False",
"state_before": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\n⊢ toZ i0 i ≠ 0",
"tactic": "by_contra h"
},
{
"state_after": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\nh : toZ i0 i = 0\nh_eq : (pred^[Int.toNat (-toZ i0 i)]) i0 = i\n⊢ False",
"state_before": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\nh : toZ i0 i = 0\n⊢ False",
"tactic": "have h_eq := iterate_pred_toZ i hi"
},
{
"state_after": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : (pred^[Int.toNat (-0)]) i0 < i0\nh : toZ i0 i = 0\nh_eq : (pred^[Int.toNat (-toZ i0 i)]) i0 = i\n⊢ False",
"state_before": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i < i0\nh : toZ i0 i = 0\nh_eq : (pred^[Int.toNat (-toZ i0 i)]) i0 = i\n⊢ False",
"tactic": "rw [← h_eq, h] at hi"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : (pred^[Int.toNat (-0)]) i0 < i0\nh : toZ i0 i = 0\nh_eq : (pred^[Int.toNat (-toZ i0 i)]) i0 = i\n⊢ False",
"tactic": "simp only [neg_zero, Int.toNat_zero, Function.iterate_zero, id.def, lt_self_iff_false] at hi"
}
] |
[
238,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
Ideal.nonarchimedean
|
[] |
[
92,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.conj_re
|
[] |
[
508,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/CategoryTheory/Linear/LinearFunctor.lean
|
CategoryTheory.Functor.coe_mapLinearMap
|
[] |
[
68,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.drop_append1'
|
[] |
[
122,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_map_inv'
|
[] |
[
308,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.zero_apply
|
[] |
[
1822,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1821,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.xor_comm
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ (a b : Bool), xor a b = xor b a",
"tactic": "decide"
}
] |
[
263,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
Real.geom_mean_le_arith_mean4_weighted
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nw₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ\nhw₁ : 0 ≤ w₁\nhw₂ : 0 ≤ w₂\nhw₃ : 0 ≤ w₃\nhw₄ : 0 ≤ w₄\nhp₁ : 0 ≤ p₁\nhp₂ : 0 ≤ p₂\nhp₃ : 0 ≤ p₃\nhp₄ : 0 ≤ p₄\nhw : w₁ + w₂ + w₃ + w₄ = 1\n⊢ ↑({ val := w₁, property := hw₁ } + { val := w₂, property := hw₂ } + { val := w₃, property := hw₃ } +\n { val := w₄, property := hw₄ }) =\n ↑1",
"tactic": "assumption"
}
] |
[
240,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_map_measure
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.857469\nδ : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → δ\ng : δ → β\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f\n⊢ Memℒp g 1 ↔ Memℒp (g ∘ f) 1",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.857469\nδ : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → δ\ng : δ → β\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f\n⊢ Integrable g ↔ Integrable (g ∘ f)",
"tactic": "simp_rw [← memℒp_one_iff_integrable]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.857469\nδ : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → δ\ng : δ → β\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f\n⊢ Memℒp g 1 ↔ Memℒp (g ∘ f) 1",
"tactic": "exact memℒp_map_measure_iff hg hf"
}
] |
[
600,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
596,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.distortion_bot
|
[] |
[
709,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.flip_apply
|
[] |
[
799,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.eq_zero_or_one_of_isField
|
[] |
[
1233,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toPartENat_surjective
|
[] |
[
1880,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1879,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
not_bddBelow_Iio
|
[] |
[
356,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiable_const_sub_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.569259\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.569354\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ (Differentiable 𝕜 fun y => c - f y) ↔ Differentiable 𝕜 f",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
654,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Algebra/Order/Module.lean
|
eq_of_smul_eq_smul_of_neg_of_le
|
[
{
"state_after": "k : Type u_2\nM : Type u_1\nN : Type ?u.17203\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhab : -c • a = -c • b\nhc : c < 0\nh : a ≤ b\n⊢ a = b",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.17203\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhab : c • a = c • b\nhc : c < 0\nh : a ≤ b\n⊢ a = b",
"tactic": "rw [← neg_neg c, neg_smul, neg_smul (-c), neg_inj] at hab"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.17203\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhab : -c • a = -c • b\nhc : c < 0\nh : a ≤ b\n⊢ a = b",
"tactic": "exact eq_of_smul_eq_smul_of_pos_of_le hab (neg_pos_of_neg hc) h"
}
] |
[
68,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Init/Set.lean
|
Set.ext
|
[] |
[
53,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Monotone.isBoundedUnder_le_comp
|
[
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) → IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "refine' ⟨_, fun h => h.isBoundedUnder (α := β) hg⟩"
},
{
"state_after": "case intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) → IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "rintro ⟨c, hc⟩"
},
{
"state_after": "case intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"state_before": "case intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "rw [eventually_map] at hc"
},
{
"state_after": "case intro.intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\nb : β\nhb : ∀ (a : β), a ≥ b → c < g a\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"state_before": "case intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "obtain ⟨b, hb⟩ : ∃ b, ∀ a ≥ b, c < g a := eventually_atTop.1 (hg'.eventually_gt_atTop c)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.197009\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c\nb : β\nhb : ∀ (a : β), a ≥ b → c < g a\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "exact ⟨b, hc.mono fun x hx => not_lt.1 fun h => (hb _ h.le).not_le hx⟩"
}
] |
[
1184,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1178,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.univ_pi_ite
|
[
{
"state_after": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ (x✝ ∈ pi univ fun i => if i ∈ s then t i else univ) ↔ x✝ ∈ pi s t",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\n⊢ (pi univ fun i => if i ∈ s then t i else univ) = pi s t",
"tactic": "ext"
},
{
"state_after": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ (∀ (i : ι), x✝ i ∈ if i ∈ s then t i else univ) ↔ x✝ ∈ pi s t",
"state_before": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ (x✝ ∈ pi univ fun i => if i ∈ s then t i else univ) ↔ x✝ ∈ pi s t",
"tactic": "simp_rw [mem_univ_pi]"
},
{
"state_after": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\ni : ι\n⊢ (x✝ i ∈ if i ∈ s then t i else univ) ↔ i ∈ s → x✝ i ∈ t i",
"state_before": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\n⊢ (∀ (i : ι), x✝ i ∈ if i ∈ s then t i else univ) ↔ x✝ ∈ pi s t",
"tactic": "refine' forall_congr' fun i => _"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.166768\ns✝ s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ : ι\ns : Set ι\ninst✝ : DecidablePred fun x => x ∈ s\nt : (i : ι) → Set (α i)\nx✝ : (i : ι) → α i\ni : ι\n⊢ (x✝ i ∈ if i ∈ s then t i else univ) ↔ i ∈ s → x✝ i ∈ t i",
"tactic": "split_ifs with h <;> simp [h]"
}
] |
[
874,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/GroupTheory/FreeProduct.lean
|
FreeProduct.NeWord.singleton_last
|
[] |
[
602,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.mem_sUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nA : α → PSet\ny : PSet\nx✝ : y ∈ ⋃₀ mk α A\na : Type (mk α A)\nc : Type (Func (mk α A) a)\ne : Equiv y (Func (A a) c)\nthis : Func (A a) c ∈ mk (Type (A a)) (Func (A a))\n⊢ Func (A a) c ∈ A (?m.27126 α A y x✝ a c e this)",
"tactic": "rwa [eta] at this"
},
{
"state_after": "α : Type u\nA : α → PSet\ny : PSet\nx✝ : ∃ z, z ∈ mk α A ∧ y ∈ z\nβ : Type u\nB : β → PSet\na : Type (mk α A)\ne : Equiv (mk β B) (mk (Type (A a)) (Func (A a)))\nb : Type (mk β B)\nyb : Equiv y (Func (mk β B) b)\n⊢ y ∈ ⋃₀ mk α A",
"state_before": "α : Type u\nA : α → PSet\ny : PSet\nx✝ : ∃ z, z ∈ mk α A ∧ y ∈ z\nβ : Type u\nB : β → PSet\na : Type (mk α A)\ne : Equiv (mk β B) (A a)\nb : Type (mk β B)\nyb : Equiv y (Func (mk β B) b)\n⊢ y ∈ ⋃₀ mk α A",
"tactic": "rw [← eta (A a)] at e"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nA : α → PSet\ny : PSet\nx✝ : ∃ z, z ∈ mk α A ∧ y ∈ z\nβ : Type u\nB : β → PSet\na : Type (mk α A)\ne : Equiv (mk β B) (mk (Type (A a)) (Func (A a)))\nb : Type (mk β B)\nyb : Equiv y (Func (mk β B) b)\n⊢ y ∈ ⋃₀ mk α A",
"tactic": "exact\n let ⟨βt, _⟩ := e\n let ⟨c, bc⟩ := βt b\n ⟨⟨a, c⟩, yb.trans bc⟩"
}
] |
[
493,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.one_def
|
[] |
[
96,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Analysis/LocallyConvex/Polar.lean
|
LinearMap.polar_iUnion
|
[] |
[
92,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.dlookup_nil
|
[] |
[
179,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.negSucc_coe'
|
[
{
"state_after": "n : Nat\n⊢ -[n+1] = -(↑n + 1)",
"state_before": "n : Nat\n⊢ -[n+1] = -↑n - 1",
"tactic": "rw [Int.sub_eq_add_neg, ← Int.neg_add]"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ -[n+1] = -(↑n + 1)",
"tactic": "rfl"
}
] |
[
502,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
501,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.coe_multilinearMap_injective
|
[] |
[
168,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Topology/MetricSpace/Kuratowski.lean
|
KuratowskiEmbedding.embeddingOfSubset_dist_le
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ ‖↑(embeddingOfSubset x a - embeddingOfSubset x b) n‖ ≤ dist a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\n⊢ dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b",
"tactic": "refine' lp.norm_le_of_forall_le dist_nonneg fun n => _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ ‖dist a (x n) - dist (x 0) (x n) - (dist b (x n) - dist (x 0) (x n))‖ ≤ dist a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ ‖↑(embeddingOfSubset x a - embeddingOfSubset x b) n‖ ≤ dist a b",
"tactic": "simp only [lp.coeFn_sub, Pi.sub_apply, embeddingOfSubset_coe, Real.dist_eq]"
},
{
"state_after": "case h.e'_3.h.e'_1\nα : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ dist a (x n) - dist (x 0) (x n) - (dist b (x n) - dist (x 0) (x n)) = dist a (x n) - dist b (x n)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ ‖dist a (x n) - dist (x 0) (x n) - (dist b (x n) - dist (x 0) (x n))‖ ≤ dist a b",
"tactic": "convert abs_dist_sub_le a b (x n) using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_1\nα : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn✝ : ℕ\nC : ℝ\ninst✝ : MetricSpace α\nx : ℕ → α\na✝ b✝ a b : α\nn : ℕ\n⊢ dist a (x n) - dist (x 0) (x n) - (dist b (x n) - dist (x 0) (x n)) = dist a (x n) - dist b (x n)",
"tactic": "ring"
}
] |
[
63,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Algebra/CharP/Algebra.lean
|
IsFractionRing.charP_of_isFractionRing
|
[] |
[
139,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_of_not_ge
|
[] |
[
337,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Computability/Halting.lean
|
ComputablePred.computable_iff
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nσ : Type ?u.282635\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → Bool\nh : Computable f\n⊢ ComputablePred fun a => f a = true",
"state_before": "α : Type u_1\nσ : Type ?u.282635\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\np : α → Prop\n⊢ (∃ f, Computable f ∧ p = fun a => f a = true) → ComputablePred p",
"tactic": "rintro ⟨f, h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nσ : Type ?u.282635\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → Bool\nh : Computable f\n⊢ ComputablePred fun a => f a = true",
"tactic": "exact ⟨by infer_instance, by simpa using h⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type ?u.282635\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → Bool\nh : Computable f\n⊢ DecidablePred fun a => f a = true",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type ?u.282635\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → Bool\nh : Computable f\n⊢ Computable fun a => decide ((fun a => f a = true) a)",
"tactic": "simpa using h"
}
] |
[
178,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Order/BoundedOrder.lean
|
Subtype.mk_eq_bot_iff
|
[] |
[
794,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.continuousAt_log_iff
|
[
{
"state_after": "x y : ℝ\n⊢ ContinuousAt log x → x ≠ 0",
"state_before": "x y : ℝ\n⊢ ContinuousAt log x ↔ x ≠ 0",
"tactic": "refine' ⟨_, continuousAt_log⟩"
},
{
"state_after": "y : ℝ\nh : ContinuousAt log 0\n⊢ False",
"state_before": "x y : ℝ\n⊢ ContinuousAt log x → x ≠ 0",
"tactic": "rintro h rfl"
},
{
"state_after": "no goals",
"state_before": "y : ℝ\nh : ContinuousAt log 0\n⊢ False",
"tactic": "exact not_tendsto_nhds_of_tendsto_atBot tendsto_log_nhdsWithin_zero _\n (h.tendsto.mono_left inf_le_left)"
}
] |
[
329,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.ker_eq_bot'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.1204225\nR₂ : Type u_3\nR₃ : Type ?u.1204231\nR₄ : Type ?u.1204234\nS : Type ?u.1204237\nK : Type ?u.1204240\nK₂ : Type ?u.1204243\nM : Type u_1\nM' : Type ?u.1204249\nM₁ : Type ?u.1204252\nM₂ : Type u_4\nM₃ : Type ?u.1204258\nM₄ : Type ?u.1204261\nN : Type ?u.1204264\nN₂ : Type ?u.1204267\nι : Type ?u.1204270\nV : Type ?u.1204273\nV₂ : Type ?u.1204276\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring R₃\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_5\nsc : SemilinearMapClass F τ₁₂ M M₂\nf : F\n⊢ ker f = ⊥ ↔ ∀ (m : M), ↑f m = 0 → m = 0",
"tactic": "simpa [disjoint_iff_inf_le] using @disjoint_ker _ _ _ _ _ _ _ _ _ _ _ _ _ f ⊤"
}
] |
[
1359,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1358,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_right_injective
|
[] |
[
191,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.weightedHomogeneousComponent_zero
|
[
{
"state_after": "case a\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\n⊢ coeff d (↑(weightedHomogeneousComponent w 0) φ) = coeff d (↑C (coeff 0 φ))",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\n⊢ ↑(weightedHomogeneousComponent w 0) φ = ↑C (coeff 0 φ)",
"tactic": "ext1 d"
},
{
"state_after": "case a.inl\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\n⊢ coeff 0 (↑(weightedHomogeneousComponent w 0) φ) = coeff 0 (↑C (coeff 0 φ))\n\ncase a.inr\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ coeff d (↑(weightedHomogeneousComponent w 0) φ) = coeff d (↑C (coeff 0 φ))",
"state_before": "case a\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\n⊢ coeff d (↑(weightedHomogeneousComponent w 0) φ) = coeff d (↑C (coeff 0 φ))",
"tactic": "rcases Classical.em (d = 0) with (rfl | hd)"
},
{
"state_after": "no goals",
"state_before": "case a.inl\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\n⊢ coeff 0 (↑(weightedHomogeneousComponent w 0) φ) = coeff 0 (↑C (coeff 0 φ))",
"tactic": "simp only [coeff_weightedHomogeneousComponent, if_pos, map_zero, coeff_zero_C]"
},
{
"state_after": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ¬↑(weightedDegree' w) d = 0",
"state_before": "case a.inr\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ coeff d (↑(weightedHomogeneousComponent w 0) φ) = coeff d (↑C (coeff 0 φ))",
"tactic": "rw [coeff_weightedHomogeneousComponent, if_neg, coeff_C, if_neg (Ne.symm hd)]"
},
{
"state_after": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"state_before": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ¬↑(weightedDegree' w) d = 0",
"tactic": "simp only [weightedDegree', LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum,\n sum_eq_zero_iff, Finsupp.mem_support_iff, Ne.def, smul_eq_zero, not_forall, not_or,\n and_self_left, exists_prop]"
},
{
"state_after": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ∃ x, ¬↑d x = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"state_before": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"tactic": "simp only [FunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, not_forall] at hd"
},
{
"state_after": "case a.inr.hnc.intro\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\ni : σ\nhi : ¬↑d i = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"state_before": "case a.inr.hnc\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\nhd : ∃ x, ¬↑d x = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"tactic": "obtain ⟨i, hi⟩ := hd"
},
{
"state_after": "no goals",
"state_before": "case a.inr.hnc.intro\nR : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : CanonicallyOrderedAddMonoid M\nw : σ → M\nφ : MvPolynomial σ R\ninst✝ : NoZeroSMulDivisors ℕ M\nhw : ∀ (i : σ), w i ≠ 0\nd : σ →₀ ℕ\ni : σ\nhi : ¬↑d i = 0\n⊢ ∃ x, ¬↑d x = 0 ∧ ¬w x = 0",
"tactic": "exact ⟨i, hi, hw i⟩"
}
] |
[
465,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Std/Logic.lean
|
Or.imp_left
|
[] |
[
251,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
251,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
AlgebraicTopology.DoldKan.Γ₀.splitting_map_eq_id
|
[
{
"state_after": "case h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝¹ : Discrete (Splitting.IndexSet Δ✝)\nA✝ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝ : Discrete (Splitting.IndexSet Δ.op)\nA : Splitting.IndexSet Δ.op\n⊢ colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op))\n { as := A } ≫\n Splitting.map (obj K) (fun n => Sigma.ι (Obj.summand K [n].op) (Splitting.IndexSet.id [n].op)) Δ.op =\n colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op))\n { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op)",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ : SimplexCategoryᵒᵖ\nx✝ : Discrete (Splitting.IndexSet Δ)\nA : Splitting.IndexSet Δ\n⊢ colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ))\n { as := A } ≫\n Splitting.map (obj K) (fun n => Sigma.ι (Obj.summand K [n].op) (Splitting.IndexSet.id [n].op)) Δ =\n colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ))\n { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ)",
"tactic": "induction' Δ using Opposite.rec' with Δ"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op))\n { as := A } ≫\n Splitting.map (obj K) (fun n => Sigma.ι (Obj.summand K [n].op) (Splitting.IndexSet.id [n].op)) [n].op =\n colimit.ι\n (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op))\n { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"state_before": "case h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝¹ : Discrete (Splitting.IndexSet Δ✝)\nA✝ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝ : Discrete (Splitting.IndexSet Δ.op)\nA : Splitting.IndexSet Δ.op\n⊢ colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op))\n { as := A } ≫\n Splitting.map (obj K) (fun n => Sigma.ι (Obj.summand K [n].op) (Splitting.IndexSet.id [n].op)) Δ.op =\n colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op))\n { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) Δ.op)",
"tactic": "induction' Δ using SimplexCategory.rec with n"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ (colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n Sigma.desc fun A =>\n Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K (Splitting.IndexSet.e A).op) =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"state_before": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ colimit.ι (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op))\n { as := A } ≫\n Splitting.map (obj K) (fun n => Sigma.ι (Obj.summand K [n].op) (Splitting.IndexSet.id [n].op)) [n].op =\n colimit.ι\n (Discrete.functor (Splitting.summand (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op))\n { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"tactic": "dsimp [Splitting.map]"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K (Splitting.IndexSet.e A).op =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"state_before": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ (colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n Sigma.desc fun A =>\n Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K (Splitting.IndexSet.e A).op) =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"tactic": "simp only [colimit.ι_desc, Cofan.mk_ι_app, Γ₀.obj_map]"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ Sigma.ι (Obj.summand K [n].op)\n (Splitting.IndexSet.mk\n (Splitting.IndexSet.e\n { fst := A.fst, snd := { val := Splitting.IndexSet.e A, property := (_ : Epi ↑A.snd) } })) =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst))\n { as := { fst := A.fst, snd := { val := Splitting.IndexSet.e A, property := (_ : Epi ↑A.snd) } } }",
"state_before": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K (Splitting.IndexSet.e A).op =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst)) { as := A } ≫\n 𝟙 (Splitting.coprod (fun n => Obj.summand K [n].op (Splitting.IndexSet.id [n].op)) [n].op)",
"tactic": "erw [Γ₀.Obj.map_on_summand₀ K (SimplicialObject.Splitting.IndexSet.id A.1)\n (show A.e ≫ 𝟙 _ = A.e.op.unop ≫ 𝟙 _ by rfl),\n Γ₀.Obj.Termwise.mapMono_id, A.ext', id_comp, comp_id]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ Sigma.ι (Obj.summand K [n].op)\n (Splitting.IndexSet.mk\n (Splitting.IndexSet.e\n { fst := A.fst, snd := { val := Splitting.IndexSet.e A, property := (_ : Epi ↑A.snd) } })) =\n colimit.ι (Discrete.functor fun A => Obj.summand K A.fst (Splitting.IndexSet.id A.fst))\n { as := { fst := A.fst, snd := { val := Splitting.IndexSet.e A, property := (_ : Epi ↑A.snd) } } }",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ✝¹ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nΔ✝ : SimplexCategoryᵒᵖ\nx✝² : Discrete (Splitting.IndexSet Δ✝)\nA✝¹ : Splitting.IndexSet Δ✝\nΔ : SimplexCategory\nx✝¹ : Discrete (Splitting.IndexSet Δ.op)\nA✝ : Splitting.IndexSet Δ.op\nn : ℕ\nx✝ : Discrete (Splitting.IndexSet [n].op)\nA : Splitting.IndexSet [n].op\n⊢ Splitting.IndexSet.e A ≫ 𝟙 A.fst.unop = (Splitting.IndexSet.e A).op.unop ≫ 𝟙 A.fst.unop.op.unop",
"tactic": "rfl"
}
] |
[
247,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.rel_add_right
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.451251\nδ : Type ?u.451254\nr : α → β → Prop\np : γ → δ → Prop\nas : Multiset α\nbs₀ bs₁ : Multiset β\n⊢ (∃ bs₀_1 bs₁_1, Rel (flip r) bs₀ bs₀_1 ∧ Rel (flip r) bs₁ bs₁_1 ∧ as = bs₀_1 + bs₁_1) ↔\n ∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.451251\nδ : Type ?u.451254\nr : α → β → Prop\np : γ → δ → Prop\nas : Multiset α\nbs₀ bs₁ : Multiset β\n⊢ Rel r as (bs₀ + bs₁) ↔ ∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁",
"tactic": "rw [← rel_flip, rel_add_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.451251\nδ : Type ?u.451254\nr : α → β → Prop\np : γ → δ → Prop\nas : Multiset α\nbs₀ bs₁ : Multiset β\n⊢ (∃ bs₀_1 bs₁_1, Rel (flip r) bs₀ bs₀_1 ∧ Rel (flip r) bs₁ bs₁_1 ∧ as = bs₀_1 + bs₁_1) ↔\n ∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁",
"tactic": "simp [rel_flip]"
}
] |
[
2762,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2760,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.lt_of_lt_floor
|
[] |
[
223,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
AffineIndependent.map'
|
[
{
"state_after": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh : IsEmpty ι\n⊢ AffineIndependent k (↑f ∘ p)\n\ncase inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh : Nonempty ι\n⊢ AffineIndependent k (↑f ∘ p)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "cases' isEmpty_or_nonempty ι with h h"
},
{
"state_after": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\n⊢ AffineIndependent k (↑f ∘ p)",
"state_before": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh : Nonempty ι\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "obtain ⟨i⟩ := h"
},
{
"state_after": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\n⊢ AffineIndependent k (↑f ∘ p)",
"state_before": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "rw [affineIndependent_iff_linearIndependent_vsub k p i] at hai"
},
{
"state_after": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\n⊢ LinearIndependent k fun i_1 => ↑f.linear (p ↑i_1 -ᵥ p i)",
"state_before": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "simp_rw [affineIndependent_iff_linearIndependent_vsub k (f ∘ p) i, Function.comp_apply, ←\n f.linearMap_vsub]"
},
{
"state_after": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\nhf' : LinearMap.ker f.linear = ⊥\n⊢ LinearIndependent k fun i_1 => ↑f.linear (p ↑i_1 -ᵥ p i)",
"state_before": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\n⊢ LinearIndependent k fun i_1 => ↑f.linear (p ↑i_1 -ᵥ p i)",
"tactic": "have hf' : LinearMap.ker f.linear = ⊥ := by rwa [LinearMap.ker_eq_bot, f.linear_injective_iff]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\nhf' : LinearMap.ker f.linear = ⊥\n⊢ LinearIndependent k fun i_1 => ↑f.linear (p ↑i_1 -ᵥ p i)",
"tactic": "exact LinearIndependent.map' hai f.linear hf'"
},
{
"state_after": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh this : IsEmpty ι\n⊢ AffineIndependent k (↑f ∘ p)",
"state_before": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh : IsEmpty ι\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "haveI := h"
},
{
"state_after": "no goals",
"state_before": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nhai : AffineIndependent k p\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\nh this : IsEmpty ι\n⊢ AffineIndependent k (↑f ∘ p)",
"tactic": "apply affineIndependent_of_subsingleton"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nι : Type u_4\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\np : ι → P\nf : P →ᵃ[k] P₂\nhf : Injective ↑f\ni : ι\nhai : LinearIndependent k fun i_1 => p ↑i_1 -ᵥ p i\n⊢ LinearMap.ker f.linear = ⊥",
"tactic": "rwa [LinearMap.ker_eq_bot, f.linear_injective_iff]"
}
] |
[
389,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.ofFin_toFin
|
[
{
"state_after": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.ofNat n ↑(toFin v) = v",
"state_before": "n : ℕ\nv : Bitvec n\n⊢ ofFin (toFin v) = v",
"tactic": "dsimp [ofFin]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.ofNat n ↑(toFin v) = v",
"tactic": "rw [toFin_val, ofNat_toNat]"
}
] |
[
173,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Data/Matrix/Rank.lean
|
Matrix.rank_le_card_height
|
[
{
"state_after": "l : Type ?u.223607\nm : Type u_2\nn : Type u_3\no : Type ?u.223616\nR : Type u_1\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis : Module.Finite R (m → R)\n⊢ rank A ≤ Fintype.card m",
"state_before": "l : Type ?u.223607\nm : Type u_2\nn : Type u_3\no : Type ?u.223616\nR : Type u_1\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\n⊢ rank A ≤ Fintype.card m",
"tactic": "haveI : Module.Finite R (m → R) := Module.Finite.pi"
},
{
"state_after": "l : Type ?u.223607\nm : Type u_2\nn : Type u_3\no : Type ?u.223616\nR : Type u_1\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis✝ : Module.Finite R (m → R)\nthis : Module.Free R (m → R)\n⊢ rank A ≤ Fintype.card m",
"state_before": "l : Type ?u.223607\nm : Type u_2\nn : Type u_3\no : Type ?u.223616\nR : Type u_1\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis : Module.Finite R (m → R)\n⊢ rank A ≤ Fintype.card m",
"tactic": "haveI : Module.Free R (m → R) := Module.Free.pi _ _"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.223607\nm : Type u_2\nn : Type u_3\no : Type ?u.223616\nR : Type u_1\nm_fin : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis✝ : Module.Finite R (m → R)\nthis : Module.Free R (m → R)\n⊢ rank A ≤ Fintype.card m",
"tactic": "exact (Submodule.finrank_le _).trans (finrank_pi R).le"
}
] |
[
154,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/Order/Basic.lean
|
frontier_Ioi
|
[] |
[
2350,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2349,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi
|
[
{
"state_after": "case mp\nθ ψ : ℝ\n⊢ sin θ = sin ψ → ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π\n\ncase mpr\nθ ψ : ℝ\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π → sin θ = sin ψ",
"state_before": "θ ψ : ℝ\n⊢ sin θ = sin ψ ↔ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "constructor"
},
{
"state_after": "case mp\nθ ψ : ℝ\nHsin : sin θ = sin ψ\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"state_before": "case mp\nθ ψ : ℝ\n⊢ sin θ = sin ψ → ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "intro Hsin"
},
{
"state_after": "case mp\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"state_before": "case mp\nθ ψ : ℝ\nHsin : sin θ = sin ψ\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin"
},
{
"state_after": "case mp.inl\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = ↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π\n\ncase mp.inr\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = -↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"state_before": "case mp\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "cases' cos_eq_iff_coe_eq_or_eq_neg.mp Hsin with h h"
},
{
"state_after": "case mp.inr.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = -↑(π / 2 - ψ)\n⊢ ↑θ + ↑ψ = ↑π",
"state_before": "case mp.inr\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = -↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "right"
},
{
"state_after": "case mp.inr.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑π = ↑θ + ↑ψ\n⊢ ↑θ + ↑ψ = ↑π",
"state_before": "case mp.inr.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = -↑(π / 2 - ψ)\n⊢ ↑θ + ↑ψ = ↑π",
"tactic": "rw [coe_sub, coe_sub, eq_neg_iff_add_eq_zero, add_sub, sub_add_eq_add_sub, ← coe_add,\n add_halves, sub_sub, sub_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case mp.inr.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑π = ↑θ + ↑ψ\n⊢ ↑θ + ↑ψ = ↑π",
"tactic": "exact h.symm"
},
{
"state_after": "case mp.inl.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = ↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ",
"state_before": "case mp.inl\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = ↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"tactic": "left"
},
{
"state_after": "case mp.inl.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2) - ↑θ = ↑(π / 2) - ↑ψ\n⊢ ↑θ = ↑ψ",
"state_before": "case mp.inl.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2 - θ) = ↑(π / 2 - ψ)\n⊢ ↑θ = ↑ψ",
"tactic": "rw [coe_sub, coe_sub] at h"
},
{
"state_after": "no goals",
"state_before": "case mp.inl.h\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\nh : ↑(π / 2) - ↑θ = ↑(π / 2) - ↑ψ\n⊢ ↑θ = ↑ψ",
"tactic": "exact sub_right_inj.1 h"
},
{
"state_after": "case mpr\nθ ψ : ℝ\n⊢ ((∃ k, θ - ψ = 2 * π * ↑k) ∨ ∃ k, θ - (π - ψ) = 2 * π * ↑k) → sin θ = sin ψ",
"state_before": "case mpr\nθ ψ : ℝ\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π → sin θ = sin ψ",
"tactic": "rw [angle_eq_iff_two_pi_dvd_sub, ← eq_sub_iff_add_eq, ← coe_sub, angle_eq_iff_two_pi_dvd_sub]"
},
{
"state_after": "case mpr.inl.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - ψ = 2 * π * ↑k\n⊢ sin θ = sin ψ\n\ncase mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\n⊢ sin θ = sin ψ",
"state_before": "case mpr\nθ ψ : ℝ\n⊢ ((∃ k, θ - ψ = 2 * π * ↑k) ∨ ∃ k, θ - (π - ψ) = 2 * π * ↑k) → sin θ = sin ψ",
"tactic": "rintro (⟨k, H⟩ | ⟨k, H⟩)"
},
{
"state_after": "case mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\n⊢ sin θ = sin ψ",
"state_before": "case mpr.inl.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - ψ = 2 * π * ↑k\n⊢ sin θ = sin ψ\n\ncase mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\n⊢ sin θ = sin ψ",
"tactic": "rw [← sub_eq_zero, sin_sub_sin, H, mul_assoc 2 π k, mul_div_cancel_left _ (two_ne_zero' ℝ),\n mul_comm π _, sin_int_mul_pi, MulZeroClass.mul_zero, MulZeroClass.zero_mul]"
},
{
"state_after": "case mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\nH' : θ + ψ = 2 * ↑k * π + π\n⊢ sin θ = sin ψ",
"state_before": "case mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\n⊢ sin θ = sin ψ",
"tactic": "have H' : θ + ψ = 2 * k * π + π := by\n rwa [← sub_add, sub_add_eq_add_sub, sub_eq_iff_eq_add, mul_assoc, mul_comm π _, ←\n mul_assoc] at H"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.intro\nθ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\nH' : θ + ψ = 2 * ↑k * π + π\n⊢ sin θ = sin ψ",
"tactic": "rw [← sub_eq_zero, sin_sub_sin, H', add_div, mul_assoc 2 _ π,\n mul_div_cancel_left _ (two_ne_zero' ℝ), cos_add_pi_div_two, sin_int_mul_pi, neg_zero,\n MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "θ ψ : ℝ\nk : ℤ\nH : θ - (π - ψ) = 2 * π * ↑k\n⊢ θ + ψ = 2 * ↑k * π + π",
"tactic": "rwa [← sub_add, sub_add_eq_add_sub, sub_eq_iff_eq_add, mul_assoc, mul_comm π _, ←\n mul_assoc] at H"
}
] |
[
289,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/CategoryTheory/Extensive.lean
|
CategoryTheory.mapPair_equifibered
|
[
{
"state_after": "case mk.left.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } })\n (α.app { as := WalkingPair.left }) (α.app { as := { as := WalkingPair.left }.as })\n (F'.map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } })\n\ncase mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })\n (α.app { as := WalkingPair.right }) (α.app { as := { as := WalkingPair.right }.as })\n (F'.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ NatTrans.Equifibered α",
"tactic": "rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.left.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } })\n (α.app { as := WalkingPair.left }) (α.app { as := { as := WalkingPair.left }.as })\n (F'.map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } })\n\ncase mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })\n (α.app { as := WalkingPair.right }) (α.app { as := { as := WalkingPair.right }.as })\n (F'.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })",
"tactic": "all_goals\n dsimp; simp only [Discrete.functor_map_id]\n exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩"
},
{
"state_after": "case mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : WalkingPair.right = WalkingPair.right) } })\n (α.app { as := WalkingPair.right }) (α.app { as := WalkingPair.right })\n (F'.map { down := { down := (_ : WalkingPair.right = WalkingPair.right) } })",
"state_before": "case mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })\n (α.app { as := WalkingPair.right }) (α.app { as := { as := WalkingPair.right }.as })\n (F'.map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } })",
"tactic": "dsimp"
},
{
"state_after": "case mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (𝟙 (F.obj { as := WalkingPair.right })) (α.app { as := WalkingPair.right })\n (α.app { as := WalkingPair.right }) (𝟙 (F'.obj { as := WalkingPair.right }))",
"state_before": "case mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (F.map { down := { down := (_ : WalkingPair.right = WalkingPair.right) } })\n (α.app { as := WalkingPair.right }) (α.app { as := WalkingPair.right })\n (F'.map { down := { down := (_ : WalkingPair.right = WalkingPair.right) } })",
"tactic": "simp only [Discrete.functor_map_id]"
},
{
"state_after": "no goals",
"state_before": "case mk.right.mk.up.up\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ IsPullback (𝟙 (F.obj { as := WalkingPair.right })) (α.app { as := WalkingPair.right })\n (α.app { as := WalkingPair.right }) (𝟙 (F'.obj { as := WalkingPair.right }))",
"tactic": "exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nX Y : C\nF F' : Discrete WalkingPair ⥤ C\nα : F ⟶ F'\n⊢ 𝟙 (F.obj { as := WalkingPair.right }) ≫ α.app { as := WalkingPair.right } =\n α.app { as := WalkingPair.right } ≫ 𝟙 (F'.obj { as := WalkingPair.right })",
"tactic": "simp only [Category.comp_id, Category.id_comp]"
}
] |
[
157,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.destruct_think
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ Computation.destruct (destruct (think s)) = Sum.inr (destruct s)",
"tactic": "simp [destruct, think, Computation.rmap]"
}
] |
[
639,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetLen_coe'
|
[] |
[
237,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.hasSum_integral_of_dominated_convergence
|
[
{
"state_after": "ι✝ : Type ?u.17150684\n𝕜 : Type ?u.17150687\nE : Type u_2\nF✝ : Type ?u.17150693\nA : Type ?u.17150696\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nι : Type u_1\ninst✝ : Countable ι\nF : ι → ℝ → E\nbound : ι → ℝ → ℝ\nhF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (Measure.restrict μ (Ι a b))\nh_bound : ∀ (n : ι), ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ‖F n x‖ ≤ bound n x\nbound_summable : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), Summable fun n => bound n x\nbound_integrable : IntegrableOn (fun t => ∑' (n : ι), bound n t) (Ι a b)\nh_lim : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), HasSum (fun n => F n x) (f x)\n⊢ HasSum (fun n => (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F n x ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ)",
"state_before": "ι✝ : Type ?u.17150684\n𝕜 : Type ?u.17150687\nE : Type u_2\nF✝ : Type ?u.17150693\nA : Type ?u.17150696\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nι : Type u_1\ninst✝ : Countable ι\nF : ι → ℝ → E\nbound : ι → ℝ → ℝ\nhF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (Measure.restrict μ (Ι a b))\nh_bound : ∀ (n : ι), ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F n t‖ ≤ bound n t\nbound_summable : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → Summable fun n => bound n t\nbound_integrable : IntervalIntegrable (fun t => ∑' (n : ι), bound n t) μ a b\nh_lim : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasSum (fun n => F n t) (f t)\n⊢ HasSum (fun n => ∫ (t : ℝ) in a..b, F n t ∂μ) (∫ (t : ℝ) in a..b, f t ∂μ)",
"tactic": "simp only [intervalIntegrable_iff, intervalIntegral_eq_integral_uIoc, ←\n ae_restrict_iff' (α := ℝ) (μ := μ) measurableSet_uIoc] at *"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.17150684\n𝕜 : Type ?u.17150687\nE : Type u_2\nF✝ : Type ?u.17150693\nA : Type ?u.17150696\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nι : Type u_1\ninst✝ : Countable ι\nF : ι → ℝ → E\nbound : ι → ℝ → ℝ\nhF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (Measure.restrict μ (Ι a b))\nh_bound : ∀ (n : ι), ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ‖F n x‖ ≤ bound n x\nbound_summable : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), Summable fun n => bound n x\nbound_integrable : IntegrableOn (fun t => ∑' (n : ι), bound n t) (Ι a b)\nh_lim : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), HasSum (fun n => F n x) (f x)\n⊢ HasSum (fun n => (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F n x ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ)",
"tactic": "exact\n (hasSum_integral_of_dominated_convergence bound hF_meas h_bound bound_summable bound_integrable\n h_lim).const_smul\n _"
}
] |
[
1039,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1027,
8
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Trivialization.symmₗ_linearMapAt
|
[] |
[
264,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.one_mod
|
[] |
[
498,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.basisDivisor_self
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\n⊢ basisDivisor x x = 0",
"tactic": "simp only [basisDivisor, sub_self, inv_zero, map_zero, MulZeroClass.zero_mul]"
}
] |
[
133,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.lt_coe_iff
|
[
{
"state_after": "no goals",
"state_before": "x : PartENat\nn : ℕ\n⊢ x < ↑n ↔ ∃ h, Part.get x h < n",
"tactic": "simp only [lt_def, forall_prop_of_true, get_natCast', dom_natCast]"
}
] |
[
297,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Combinatorics/Configuration.lean
|
Configuration.HasLines.existsUnique_line
|
[] |
[
121,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
BilinForm.nondegenerate_toMatrix_iff
|
[] |
[
589,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ball_half_subset
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.44437\nι : Type ?u.44440\ninst✝ : PseudoMetricSpace α\nx y✝ z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ny : α\nh : y ∈ ball x (ε / 2)\n⊢ dist y x ≤ ε / 2",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.44437\nι : Type ?u.44440\ninst✝ : PseudoMetricSpace α\nx y✝ z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ny : α\nh : y ∈ ball x (ε / 2)\n⊢ dist y x ≤ ε - ε / 2",
"tactic": "rw [sub_self_div_two]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.44437\nι : Type ?u.44440\ninst✝ : PseudoMetricSpace α\nx y✝ z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ny : α\nh : y ∈ ball x (ε / 2)\n⊢ dist y x ≤ ε / 2",
"tactic": "exact le_of_lt h"
}
] |
[
666,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean
|
CategoryTheory.NatTrans.id_app
|
[] |
[
75,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.isCobounded_univ
|
[] |
[
187,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.smul_of_tower_mem
|
[] |
[
243,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.eventually_ne_atBot
|
[] |
[
226,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.toPrenex_isPrenex
|
[] |
[
867,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm'_add_le
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ≤ ∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ^ (1 / q) ≤\n (∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ) ^ (1 / q)",
"tactic": "refine' ENNReal.rpow_le_rpow _ (by simp [le_trans zero_le_one hq1] : 0 ≤ 1 / q)"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\na : α\n⊢ ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ≤ ∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ",
"tactic": "refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ (le_trans zero_le_one hq1)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\na : α\n⊢ ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a",
"tactic": "simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2325640\nG : Type ?u.2325643\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhq1 : 1 ≤ q\n⊢ 0 ≤ 1 / q",
"tactic": "simp [le_trans zero_le_one hq1]"
}
] |
[
754,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.baseSet_at
|
[] |
[
681,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/RingTheory/Localization/Module.lean
|
Basis.localizationLocalization_repr_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing Rₛ\ninst✝⁸ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nA : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nAₛ : Type u_5\ninst✝⁵ : CommRing Aₛ\ninst✝⁴ : Algebra A Aₛ\ninst✝³ : Algebra Rₛ Aₛ\ninst✝² : Algebra R Aₛ\ninst✝¹ : IsScalarTower R Rₛ Aₛ\ninst✝ : IsScalarTower R A Aₛ\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ\nι : Type u_1\nb : Basis ι R A\nx : A\ni : ι\n⊢ ↑(↑(localizationLocalization Rₛ S Aₛ b).repr (↑(algebraMap A Aₛ) x)) i =\n ↑(↑(localizationLocalization Rₛ S Aₛ b).repr\n (Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(algebraMap A Aₛ) (↑b j)))\n i",
"tactic": "simp_rw [IsScalarTower.algebraMap_smul, Algebra.smul_def,\n IsScalarTower.algebraMap_apply R A Aₛ, ← _root_.map_mul, ← map_finsupp_sum, ←\n Algebra.smul_def, ← Finsupp.total_apply, Basis.total_repr]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing Rₛ\ninst✝⁸ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nA : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nAₛ : Type u_5\ninst✝⁵ : CommRing Aₛ\ninst✝⁴ : Algebra A Aₛ\ninst✝³ : Algebra Rₛ Aₛ\ninst✝² : Algebra R Aₛ\ninst✝¹ : IsScalarTower R Rₛ Aₛ\ninst✝ : IsScalarTower R A Aₛ\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ\nι : Type u_1\nb : Basis ι R A\nx : A\ni : ι\n⊢ ↑(↑(localizationLocalization Rₛ S Aₛ b).repr\n (Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(algebraMap A Aₛ) (↑b j)))\n i =\n Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i",
"tactic": "simp_rw [← b.localizationLocalization_apply Rₛ S Aₛ, map_finsupp_sum, LinearEquiv.map_smul,\n Basis.repr_self, Finsupp.sum_apply, Finsupp.smul_apply]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing Rₛ\ninst✝⁸ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nA : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nAₛ : Type u_5\ninst✝⁵ : CommRing Aₛ\ninst✝⁴ : Algebra A Aₛ\ninst✝³ : Algebra Rₛ Aₛ\ninst✝² : Algebra R Aₛ\ninst✝¹ : IsScalarTower R Rₛ Aₛ\ninst✝ : IsScalarTower R A Aₛ\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ\nι : Type u_1\nb : Basis ι R A\nx : A\ni j : ι\nx✝ : j ∈ (↑b.repr x).support\nhj : j ≠ i\n⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) j (↑(↑b.repr x) j) = 0",
"tactic": "simp [hj]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing Rₛ\ninst✝⁸ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nA : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nAₛ : Type u_5\ninst✝⁵ : CommRing Aₛ\ninst✝⁴ : Algebra A Aₛ\ninst✝³ : Algebra Rₛ Aₛ\ninst✝² : Algebra R Aₛ\ninst✝¹ : IsScalarTower R Rₛ Aₛ\ninst✝ : IsScalarTower R A Aₛ\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ\nι : Type u_1\nb : Basis ι R A\nx : A\ni : ι\nhi : ¬i ∈ (↑b.repr x).support\n⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) i (↑(↑b.repr x) i) = 0",
"tactic": "simp [Finsupp.not_mem_support_iff.mp hi]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing Rₛ\ninst✝⁸ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nA : Type u_3\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nAₛ : Type u_5\ninst✝⁵ : CommRing Aₛ\ninst✝⁴ : Algebra A Aₛ\ninst✝³ : Algebra Rₛ Aₛ\ninst✝² : Algebra R Aₛ\ninst✝¹ : IsScalarTower R Rₛ Aₛ\ninst✝ : IsScalarTower R A Aₛ\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ\nι : Type u_1\nb : Basis ι R A\nx : A\ni : ι\n⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) i (↑(↑b.repr x) i) = ↑(algebraMap R Rₛ) (↑(↑b.repr x) i)",
"tactic": "simp [Algebra.smul_def]"
}
] |
[
143,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mk'_eq_of_eq
|
[] |
[
827,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
825,
1
] |
Std/Data/AssocList.lean
|
Std.AssocList.mapKey_toList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nδ : Type u_2\nβ : Type u_3\nf : α → δ\nl : AssocList α β\n⊢ toList (mapKey f l) =\n List.map\n (fun x =>\n match x with\n | (a, b) => (f a, b))\n (toList l)",
"tactic": "induction l <;> simp [*]"
}
] |
[
87,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
85,
9
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.zero_tprodCoeff
|
[] |
[
150,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.top_map
|
[] |
[
157,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Logic/Basic.lean
|
or_of_or_of_imp_right
|
[] |
[
342,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/Topology/Connected.lean
|
locallyConnectedSpace_iff_connectedComponentIn_open
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ LocallyConnectedSpace α → ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)\n\ncase mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ (∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)) → LocallyConnectedSpace α",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ LocallyConnectedSpace α ↔ ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : LocallyConnectedSpace α\n⊢ ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ LocallyConnectedSpace α → ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : LocallyConnectedSpace α\n⊢ ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)",
"tactic": "exact fun F hF x _ => hF.connectedComponentIn"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)\n⊢ LocallyConnectedSpace α",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ (∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)) → LocallyConnectedSpace α",
"tactic": "intro h"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)\n⊢ ∀ (x : α) (U : Set α), U ∈ 𝓝 x → ∃ V, V ⊆ U ∧ IsOpen V ∧ x ∈ V ∧ IsConnected V",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)\n⊢ LocallyConnectedSpace α",
"tactic": "rw [locallyConnectedSpace_iff_open_connected_subsets]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.115042\nπ : ι → Type ?u.115047\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (F : Set α), IsOpen F → ∀ (x : α), x ∈ F → IsOpen (connectedComponentIn F x)\n⊢ ∀ (x : α) (U : Set α), U ∈ 𝓝 x → ∃ V, V ⊆ U ∧ IsOpen V ∧ x ∈ V ∧ IsConnected V",
"tactic": "refine' fun x U hU =>\n ⟨connectedComponentIn (interior U) x,\n (connectedComponentIn_subset _ _).trans interior_subset, h _ isOpen_interior x _,\n mem_connectedComponentIn _, isConnected_connectedComponentIn_iff.mpr _⟩ <;>\n exact mem_interior_iff_mem_nhds.mpr hU"
}
] |
[
1180,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
1
] |
Mathlib/MeasureTheory/Integral/Average.lean
|
MeasureTheory.average_neg
|
[] |
[
89,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
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