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/MeasureTheory/Function/LocallyIntegrable.lean
|
ContinuousOn.locallyIntegrableOn
|
[] |
[
260,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.setToFun_eq_setToL1
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1320608\nG : Type ?u.1320611\n𝕜 : Type ?u.1320614\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : { x // x ∈ Lp E 1 }\n⊢ setToFun μ T hT ↑↑f = ↑(setToL1 hT) f",
"tactic": "rw [setToFun_eq hT (L1.integrable_coeFn f), Integrable.toL1_coeFn]"
}
] |
[
1289,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1287,
1
] |
Mathlib/LinearAlgebra/FreeModule/Basic.lean
|
Module.Free.of_equiv'
|
[] |
[
133,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
frobenius_pthRoot
|
[] |
[
67,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.320231\nγ : Type ?u.320234\nδ : Type ?u.320237\nι : Type ?u.320240\nR : Type ?u.320243\nR' : Type ?u.320246\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nS : Set (Set α)\nhc : Set.Countable S\nhs : ⋃₀ S = univ\n⊢ (⋃ (i : Set α) (_ : i ∈ S), i) = univ",
"tactic": "rwa [← sUnion_eq_biUnion]"
}
] |
[
1913,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1911,
1
] |
Mathlib/Order/Filter/Bases.lean
|
FilterBasis.generate
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ generate B.sets ≤ FilterBasis.filter B\n\ncase a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ FilterBasis.filter B ≤ generate B.sets",
"state_before": "α : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ generate B.sets = FilterBasis.filter B",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\nU : Set α\nU_in : U ∈ FilterBasis.filter B\n⊢ U ∈ generate B.sets",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ generate B.sets ≤ FilterBasis.filter B",
"tactic": "intro U U_in"
},
{
"state_after": "case a.intro.intro\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\nU : Set α\nU_in : U ∈ FilterBasis.filter B\nV : Set α\nV_in : V ∈ B\nh : V ⊆ U\n⊢ U ∈ generate B.sets",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\nU : Set α\nU_in : U ∈ FilterBasis.filter B\n⊢ U ∈ generate B.sets",
"tactic": "rcases B.mem_filter_iff.mp U_in with ⟨V, V_in, h⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\nU : Set α\nU_in : U ∈ FilterBasis.filter B\nV : Set α\nV_in : V ∈ B\nh : V ⊆ U\n⊢ U ∈ generate B.sets",
"tactic": "exact GenerateSets.superset (GenerateSets.basic V_in) h"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ B.sets ⊆ (FilterBasis.filter B).sets",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ FilterBasis.filter B ≤ generate B.sets",
"tactic": "rw [le_generate_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.8511\nγ : Type ?u.8514\nι : Sort ?u.8517\nι' : Sort ?u.8520\nB : FilterBasis α\n⊢ B.sets ⊆ (FilterBasis.filter B).sets",
"tactic": "apply mem_filter_of_mem"
}
] |
[
202,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
11
] |
Mathlib/Data/List/Rotate.lean
|
List.mem_cyclicPermutations_self
|
[
{
"state_after": "case nil\nα : Type u\nl l' : List α\n⊢ [] ∈ cyclicPermutations []\n\ncase cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ x :: l ∈ cyclicPermutations (x :: l)",
"state_before": "α : Type u\nl✝ l' l : List α\n⊢ l ∈ cyclicPermutations l",
"tactic": "cases' l with x l"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nl l' : List α\n⊢ [] ∈ cyclicPermutations []",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ ∃ n h, nthLe (cyclicPermutations (x :: l)) n h = x :: l",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ x :: l ∈ cyclicPermutations (x :: l)",
"tactic": "rw [mem_iff_nthLe]"
},
{
"state_after": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ nthLe (cyclicPermutations (x :: l)) 0 (_ : 0 < length (cyclicPermutations (x :: l))) = x :: l",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ ∃ n h, nthLe (cyclicPermutations (x :: l)) n h = x :: l",
"tactic": "refine' ⟨0, by simp, _⟩"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ nthLe (cyclicPermutations (x :: l)) 0 (_ : 0 < length (cyclicPermutations (x :: l))) = x :: l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ 0 < length (cyclicPermutations (x :: l))",
"tactic": "simp"
}
] |
[
616,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finSumFinEquiv_symm_last
|
[] |
[
355,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.congr_left
|
[] |
[
341,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.ext
|
[] |
[
153,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.denote_combine
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\np₁ p₂ : Poly\n⊢ denote ctx (combine p₁ p₂) = denote ctx p₁ + denote ctx p₂",
"tactic": "simp [combine, denote_combineAux]"
}
] |
[
521,
36
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
520,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.mul_coe_eq_smul
|
[] |
[
1070,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1069,
1
] |
Mathlib/MeasureTheory/Measure/MutuallySingular.lean
|
MeasureTheory.Measure.MutuallySingular.smul_nnreal
|
[] |
[
126,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.bot_toAddSubmonoid
|
[] |
[
61,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.eta
|
[] |
[
181,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
11
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.ndunion_le_union
|
[] |
[
195,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.extend_eq
|
[] |
[
167,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/CategoryTheory/Category/Grpd.lean
|
CategoryTheory.Grpd.id_to_functor
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
pairwise_disjoint_fiber
|
[] |
[
397,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean
|
CategoryTheory.solutionSetCondition_of_isRightAdjoint
|
[
{
"state_after": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\n⊢ ∃ ι B f, ∀ (X : D) (h : A ⟶ G.obj X), ∃ i g, f i ≫ G.map g = h",
"state_before": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\n⊢ SolutionSetCondition G",
"tactic": "intro A"
},
{
"state_after": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\n⊢ ∀ (X : D) (h : A ⟶ G.obj X), ∃ i g, (fun x => (Adjunction.ofRightAdjoint G).unit.app A) i ≫ G.map g = h",
"state_before": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\n⊢ ∃ ι B f, ∀ (X : D) (h : A ⟶ G.obj X), ∃ i g, f i ≫ G.map g = h",
"tactic": "refine'\n ⟨PUnit, fun _ => (leftAdjoint G).obj A, fun _ => (Adjunction.ofRightAdjoint G).unit.app A, _⟩"
},
{
"state_after": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\nB : D\nh : A ⟶ G.obj B\n⊢ ∃ i g, (fun x => (Adjunction.ofRightAdjoint G).unit.app A) i ≫ G.map g = h",
"state_before": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\n⊢ ∀ (X : D) (h : A ⟶ G.obj X), ∃ i g, (fun x => (Adjunction.ofRightAdjoint G).unit.app A) i ≫ G.map g = h",
"tactic": "intro B h"
},
{
"state_after": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\nB : D\nh : A ⟶ G.obj B\n⊢ (fun x => (Adjunction.ofRightAdjoint G).unit.app A) PUnit.unit ≫\n G.map (↑(Adjunction.homEquiv (Adjunction.ofRightAdjoint G) A B).symm h) =\n h",
"state_before": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\nB : D\nh : A ⟶ G.obj B\n⊢ ∃ i g, (fun x => (Adjunction.ofRightAdjoint G).unit.app A) i ≫ G.map g = h",
"tactic": "refine' ⟨PUnit.unit, ((Adjunction.ofRightAdjoint G).homEquiv _ _).symm h, _⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v\nC : Type u\ninst✝² : Category C\nD : Type u\ninst✝¹ : Category D\nG : D ⥤ C\ninst✝ : IsRightAdjoint G\nA : C\nB : D\nh : A ⟶ G.obj B\n⊢ (fun x => (Adjunction.ofRightAdjoint G).unit.app A) PUnit.unit ≫\n G.map (↑(Adjunction.homEquiv (Adjunction.ofRightAdjoint G) A B).symm h) =\n h",
"tactic": "rw [← Adjunction.homEquiv_unit, Equiv.apply_symm_apply]"
}
] |
[
80,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.sup_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns₁ s₂ : Multiset α\n⊢ sup (s₁ + s₂) = fold (fun x x_1 => x ⊔ x_1) (⊥ ⊔ ⊥) (s₁ + s₂)",
"tactic": "simp [sup]"
}
] |
[
58,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph'_pos
|
[
{
"state_after": "no goals",
"state_before": "o : Ordinal\nho : 0 < o\n⊢ 0 < aleph' o",
"tactic": "rwa [← aleph'_zero, aleph'_lt]"
}
] |
[
297,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Antitone.tendsto_rightLim_within
|
[] |
[
350,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/MeasureTheory/Measure/MutuallySingular.lean
|
MeasureTheory.Measure.MutuallySingular.symm
|
[] |
[
67,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
|
[
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\n⊢ f =ᵐ[μ] 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nht : IsSeparable t\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\n⊢ f =ᵐ[μ] 0",
"tactic": "rcases ht with ⟨d, d_count, hd⟩"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\n⊢ f =ᵐ[μ] 0",
"tactic": "haveI : Encodable d := d_count.toEncodable"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝ : Encodable ↑d\nthis : ∀ (x : ↑d), ∃ g, ‖g‖ ≤ 1 ∧ ↑g ↑x = ↑‖↑x‖\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\n⊢ f =ᵐ[μ] 0",
"tactic": "have : ∀ x : d, ∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖(x : E)‖ :=\n fun x => exists_dual_vector'' 𝕜 (x : E)"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝ : Encodable ↑d\nthis : ∀ (x : ↑d), ∃ g, ‖g‖ ≤ 1 ∧ ↑g ↑x = ↑‖↑x‖\n⊢ f =ᵐ[μ] 0",
"tactic": "choose s hs using this"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\n⊢ f =ᵐ[μ] 0",
"tactic": "have A : ∀ a : E, a ∈ t → (∀ x, ⟪a, s x⟫ = (0 : 𝕜)) → a = 0 := by\n intro a hat ha\n contrapose! ha\n have a_pos : 0 < ‖a‖ := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n have a_mem : a ∈ closure d := hd hat\n obtain ⟨x, hx⟩ : ∃ x : d, dist a x < ‖a‖ / 2 := by\n rcases Metric.mem_closure_iff.1 a_mem (‖a‖ / 2) (half_pos a_pos) with ⟨x, h'x, hx⟩\n exact ⟨⟨x, h'x⟩, hx⟩\n use x\n have I : ‖a‖ / 2 < ‖(x : E)‖ := by\n have : ‖a‖ ≤ ‖(x : E)‖ + ‖a - x‖ := norm_le_insert' _ _\n have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx\n linarith\n intro h\n apply lt_irrefl ‖s x x‖\n calc\n ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n _ < ‖(x : E)‖ := I\n _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\n⊢ f =ᵐ[μ] 0",
"tactic": "have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\nhf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), ↑(s y) (f x) = 0\n⊢ f =ᵐ[μ] 0",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\n⊢ f =ᵐ[μ] 0",
"tactic": "have hf' : ∀ᵐ x ∂μ, ∀ y : d, ⟪f x, s y⟫ = (0 : 𝕜) := by rwa [ae_all_iff]"
},
{
"state_after": "case h\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\nhf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), ↑(s y) (f x) = 0\nx : α\nhx : ∀ (y : ↑d), ↑(s y) (f x) = 0\nh'x : f x ∈ t\n⊢ f x = OfNat.ofNat 0 x",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\nhf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), ↑(s y) (f x) = 0\n⊢ f =ᵐ[μ] 0",
"tactic": "filter_upwards [hf', h't] with x hx h'x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\nhf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), ↑(s y) (f x) = 0\nx : α\nhx : ∀ (y : ↑d), ↑(s y) (f x) = 0\nh'x : f x ∈ t\n⊢ f x = OfNat.ofNat 0 x",
"tactic": "exact A (f x) h'x hx"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : ∀ (x : ↑d), ↑(s x) a = 0\n⊢ a = 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\n⊢ ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0",
"tactic": "intro a hat ha"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\n⊢ ∃ x, ↑(s x) a ≠ 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : ∀ (x : ↑d), ↑(s x) a = 0\n⊢ a = 0",
"tactic": "contrapose! ha"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\n⊢ ∃ x, ↑(s x) a ≠ 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\n⊢ ∃ x, ↑(s x) a ≠ 0",
"tactic": "have a_pos : 0 < ‖a‖ := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\n⊢ ∃ x, ↑(s x) a ≠ 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\n⊢ ∃ x, ↑(s x) a ≠ 0",
"tactic": "have a_mem : a ∈ closure d := hd hat"
},
{
"state_after": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\n⊢ ∃ x, ↑(s x) a ≠ 0",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\n⊢ ∃ x, ↑(s x) a ≠ 0",
"tactic": "obtain ⟨x, hx⟩ : ∃ x : d, dist a x < ‖a‖ / 2 := by\n rcases Metric.mem_closure_iff.1 a_mem (‖a‖ / 2) (half_pos a_pos) with ⟨x, h'x, hx⟩\n exact ⟨⟨x, h'x⟩, hx⟩"
},
{
"state_after": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\n⊢ ↑(s x) a ≠ 0",
"state_before": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\n⊢ ∃ x, ↑(s x) a ≠ 0",
"tactic": "use x"
},
{
"state_after": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\n⊢ ↑(s x) a ≠ 0",
"state_before": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\n⊢ ↑(s x) a ≠ 0",
"tactic": "have I : ‖a‖ / 2 < ‖(x : E)‖ := by\n have : ‖a‖ ≤ ‖(x : E)‖ + ‖a - x‖ := norm_le_insert' _ _\n have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx\n linarith"
},
{
"state_after": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ False",
"state_before": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\n⊢ ↑(s x) a ≠ 0",
"tactic": "intro h"
},
{
"state_after": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑(s x) ↑x‖ < ‖↑(s x) ↑x‖",
"state_before": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ False",
"tactic": "apply lt_irrefl ‖s x x‖"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑(s x) ↑x‖ < ‖↑(s x) ↑x‖",
"tactic": "calc\n ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n _ < ‖(x : E)‖ := I\n _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\n⊢ 0 < ‖a‖",
"tactic": "simp only [ha, norm_pos_iff, Ne.def, not_false_iff]"
},
{
"state_after": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : E\nh'x : x ∈ d\nhx : dist a x < ‖a‖ / 2\n⊢ ∃ x, dist a ↑x < ‖a‖ / 2",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\n⊢ ∃ x, dist a ↑x < ‖a‖ / 2",
"tactic": "rcases Metric.mem_closure_iff.1 a_mem (‖a‖ / 2) (half_pos a_pos) with ⟨x, h'x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : E\nh'x : x ∈ d\nhx : dist a x < ‖a‖ / 2\n⊢ ∃ x, dist a ↑x < ‖a‖ / 2",
"tactic": "exact ⟨⟨x, h'x⟩, hx⟩"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝ : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nthis : ‖a‖ ≤ ‖↑x‖ + ‖a - ↑x‖\n⊢ ‖a‖ / 2 < ‖↑x‖",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\n⊢ ‖a‖ / 2 < ‖↑x‖",
"tactic": "have : ‖a‖ ≤ ‖(x : E)‖ + ‖a - x‖ := norm_le_insert' _ _"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝¹ : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nthis✝ : ‖a‖ ≤ ‖↑x‖ + ‖a - ↑x‖\nthis : ‖a - ↑x‖ < ‖a‖ / 2\n⊢ ‖a‖ / 2 < ‖↑x‖",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝ : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nthis : ‖a‖ ≤ ‖↑x‖ + ‖a - ↑x‖\n⊢ ‖a‖ / 2 < ‖↑x‖",
"tactic": "have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝¹ : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nthis✝ : ‖a‖ ≤ ‖↑x‖ + ‖a - ↑x‖\nthis : ‖a - ↑x‖ < ‖a‖ / 2\n⊢ ‖a‖ / 2 < ‖↑x‖",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis✝ : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nthis : ‖a‖ ≤ ‖↑x‖ + ‖a - ↑x‖\n⊢ ‖a - ↑x‖ < ‖a‖ / 2",
"tactic": "rwa [dist_eq_norm] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑(s x) ↑x‖ = ‖↑(s x) (↑x - a)‖",
"tactic": "simp only [h, sub_zero, ContinuousLinearMap.map_sub]"
},
{
"state_after": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑x - a‖ < ‖a‖ / 2",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ 1 * ‖↑x - a‖ < ‖a‖ / 2",
"tactic": "rw [one_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑x - a‖ < ‖a‖ / 2",
"tactic": "rwa [dist_eq_norm'] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\na : E\nhat : a ∈ t\nha : a ≠ 0\na_pos : 0 < ‖a‖\na_mem : a ∈ closure d\nx : ↑d\nhx : dist a ↑x < ‖a‖ / 2\nI : ‖a‖ / 2 < ‖↑x‖\nh : ↑(s x) a = 0\n⊢ ‖↑x‖ = ‖↑(s x) ↑x‖",
"tactic": "rw [(hs x).2, IsROrC.norm_coe_norm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_1\n𝕜 : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : IsROrC 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : Dual 𝕜 E), (fun x => ↑c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : Set.Countable d\nhd : t ⊆ closure d\nthis : Encodable ↑d\ns : ↑d → E →L[𝕜] 𝕜\nhs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ ↑(s x) ↑x = ↑‖↑x‖\nA : ∀ (a : E), a ∈ t → (∀ (x : ↑d), ↑(s x) a = 0) → a = 0\nhfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, ↑(s y) (f x) = 0\n⊢ ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), ↑(s y) (f x) = 0",
"tactic": "rwa [ae_all_iff]"
}
] |
[
110,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/RingTheory/Prime.lean
|
Prime.abs
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : LinearOrder α\np : α\nhp : Prime p\nh : Abs.abs p = p\n⊢ Prime p\n\ncase inr\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : LinearOrder α\np : α\nhp : Prime p\nh : Abs.abs p = -p\n⊢ Prime (-p)",
"state_before": "α : Type u_1\ninst✝¹ : CommRing α\ninst✝ : LinearOrder α\np : α\nhp : Prime p\n⊢ Prime (Abs.abs p)",
"tactic": "obtain h | h := abs_choice p <;> rw [h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : LinearOrder α\np : α\nhp : Prime p\nh : Abs.abs p = p\n⊢ Prime p",
"tactic": "exact hp"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : LinearOrder α\np : α\nhp : Prime p\nh : Abs.abs p = -p\n⊢ Prime (-p)",
"tactic": "exact hp.neg"
}
] |
[
77,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.mapRange_id
|
[
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nh : optParam (∀ (i : ι), id 0 = 0) (_ : ∀ (i : ι), id 0 = id 0)\ng : Π₀ (i : ι), β₁ i\ni✝ : ι\n⊢ ↑(mapRange (fun i => id) h g) i✝ = ↑g i✝",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nh : optParam (∀ (i : ι), id 0 = 0) (_ : ∀ (i : ι), id 0 = id 0)\ng : Π₀ (i : ι), β₁ i\n⊢ mapRange (fun i => id) h g = g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nh : optParam (∀ (i : ι), id 0 = 0) (_ : ∀ (i : ι), id 0 = id 0)\ng : Π₀ (i : ι), β₁ i\ni✝ : ι\n⊢ ↑(mapRange (fun i => id) h g) i✝ = ↑g i✝",
"tactic": "rfl"
}
] |
[
167,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Std/Logic.lean
|
Or.imp
|
[] |
[
249,
89
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
249,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.prod_eq_of_fintype
|
[
{
"state_after": "α✝ β α : Type u\nh : Fintype α\n⊢ ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)",
"state_before": "α✝ β α : Type u\nh : Fintype α\nf : α → Cardinal\n⊢ prod f = lift (∏ i : α, f i)",
"tactic": "revert f"
},
{
"state_after": "case refine'_1\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β),\n (∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)) → ∀ (f : β → Cardinal), prod f = lift (∏ i : β, f i)\n\ncase refine'_2\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (f : PEmpty → Cardinal), prod f = lift (∏ i : PEmpty, f i)\n\ncase refine'_3\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (α : Type u) [inst : Fintype α],\n (∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)) →\n ∀ (f : Option α → Cardinal), prod f = lift (∏ i : Option α, f i)",
"state_before": "α✝ β α : Type u\nh : Fintype α\n⊢ ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)",
"tactic": "refine' Fintype.induction_empty_option _ _ _ α (h_fintype := h)"
},
{
"state_after": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\n⊢ prod f = lift (∏ i : β, f i)",
"state_before": "case refine'_1\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β),\n (∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)) → ∀ (f : β → Cardinal), prod f = lift (∏ i : β, f i)",
"tactic": "intro α β hβ e h f"
},
{
"state_after": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\nthis : Fintype α := Fintype.ofEquiv β e.symm\n⊢ prod f = lift (∏ i : β, f i)",
"state_before": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\n⊢ prod f = lift (∏ i : β, f i)",
"tactic": "letI := Fintype.ofEquiv β e.symm"
},
{
"state_after": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\nthis : Fintype α := Fintype.ofEquiv β e.symm\n⊢ prod f = prod fun i => f (↑e i)",
"state_before": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\nthis : Fintype α := Fintype.ofEquiv β e.symm\n⊢ prod f = lift (∏ i : β, f i)",
"tactic": "rw [← e.prod_comp f, ← h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα✝¹ β✝ α✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : β → Cardinal\nthis : Fintype α := Fintype.ofEquiv β e.symm\n⊢ prod f = prod fun i => f (↑e i)",
"tactic": "exact mk_congr (e.piCongrLeft _).symm"
},
{
"state_after": "case refine'_2\nα✝ β α : Type u\nh : Fintype α\nf : PEmpty → Cardinal\n⊢ prod f = lift (∏ i : PEmpty, f i)",
"state_before": "case refine'_2\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (f : PEmpty → Cardinal), prod f = lift (∏ i : PEmpty, f i)",
"tactic": "intro f"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα✝ β α : Type u\nh : Fintype α\nf : PEmpty → Cardinal\n⊢ prod f = lift (∏ i : PEmpty, f i)",
"tactic": "rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one]"
},
{
"state_after": "case refine'_3\nα✝¹ β α✝ : Type u\nh✝ : Fintype α✝\nα : Type u\nhα : Fintype α\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : Option α → Cardinal\n⊢ prod f = lift (∏ i : Option α, f i)",
"state_before": "case refine'_3\nα✝ β α : Type u\nh : Fintype α\n⊢ ∀ (α : Type u) [inst : Fintype α],\n (∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)) →\n ∀ (f : Option α → Cardinal), prod f = lift (∏ i : Option α, f i)",
"tactic": "intro α hα h f"
},
{
"state_after": "case refine'_3\nα✝¹ β α✝ : Type u\nh✝ : Fintype α✝\nα : Type u\nhα : Fintype α\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : Option α → Cardinal\n⊢ (lift (f none) * prod fun i => lift (f (some i))) = lift (f none) * prod fun a => f (some a)",
"state_before": "case refine'_3\nα✝¹ β α✝ : Type u\nh✝ : Fintype α✝\nα : Type u\nhα : Fintype α\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : Option α → Cardinal\n⊢ prod f = lift (∏ i : Option α, f i)",
"tactic": "rw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, ←\n Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, ← h fun a => f (some a)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα✝¹ β α✝ : Type u\nh✝ : Fintype α✝\nα : Type u\nhα : Fintype α\nh : ∀ (f : α → Cardinal), prod f = lift (∏ i : α, f i)\nf : Option α → Cardinal\n⊢ (lift (f none) * prod fun i => lift (f (some i))) = lift (f none) * prod fun a => f (some a)",
"tactic": "simp only [lift_id]"
}
] |
[
1092,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1079,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
LinearEquiv.dualMap_refl
|
[
{
"state_after": "case h.h\nR : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nx✝¹ : Dual R M₁\nx✝ : M₁\n⊢ ↑(↑(dualMap (refl R M₁)) x✝¹) x✝ = ↑(↑(refl R (Dual R M₁)) x✝¹) x✝",
"state_before": "R : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\n⊢ dualMap (refl R M₁) = refl R (Dual R M₁)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u\ninst✝⁴ : CommSemiring R\nM₁ : Type v\nM₂ : Type v'\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M₁\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nx✝¹ : Dual R M₁\nx✝ : M₁\n⊢ ↑(↑(dualMap (refl R M₁)) x✝¹) x✝ = ↑(↑(refl R (Dual R M₁)) x✝¹) x✝",
"tactic": "rfl"
}
] |
[
259,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.iSup_eq_closure
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.29826\nA : Type ?u.29829\ninst✝¹ : MulOneClass M\ns : Set M\ninst✝ : AddZeroClass A\nt : Set A\nS : Submonoid M\nι : Sort u_1\np : ι → Submonoid M\n⊢ (⨆ (i : ι), p i) = closure (⋃ (i : ι), ↑(p i))",
"tactic": "simp_rw [Submonoid.closure_iUnion, Submonoid.closure_eq]"
}
] |
[
558,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Order/Antichain.lean
|
IsStrongAntichain.insert
|
[] |
[
336,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
11
] |
Mathlib/Data/List/Indexes.lean
|
List.mapIdxGo_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "α : Type u\nβ : Type v\n⊢ ∀ (f : ℕ → α → β) (l₁ l₂ : List α) (arr : Array β),\n mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "intros f l₁ l₂ arr"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nlen : ℕ\ne : length (l₁ ++ l₂) = len\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "generalize e : (l₁ ++ l₂).length = len"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\n⊢ ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nlen : ℕ\ne : length (l₁ ++ l₂) = len\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "revert l₁ l₂ arr"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\n\ncase succ\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\n⊢ ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "induction' len with len ih <;> intros l₁ l₂ arr h"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "have l₁_nil : l₁ = [] := by cases l₁; rfl; contradiction"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "have l₂_nil : l₂ = [] := by cases l₂; rfl; rw [List.length_append] at h; contradiction"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ mapIdx.go f ([] ++ []) arr = mapIdx.go f [] (toArray (mapIdx.go f [] arr))",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "rw [l₁_nil, l₂_nil]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ mapIdx.go f ([] ++ []) arr = mapIdx.go f [] (toArray (mapIdx.go f [] arr))",
"tactic": "simp only [mapIdx.go, Array.toList_eq, Array.toArray_data]"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nh : length ([] ++ l₂) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ l₁ = []",
"tactic": "cases l₁"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nh : length ([] ++ l₂) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₂ : List α\narr : Array β\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "contradiction"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nh : length (l₁ ++ []) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ l₂ = []",
"tactic": "cases l₂"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nh : length (l₁ ++ []) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length l₁ + length (head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rw [List.length_append] at h"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nl₁ : List α\narr : Array β\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length l₁ + length (head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "contradiction"
},
{
"state_after": "case succ.nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nh : length ([] ++ l₂) = Nat.succ len\n⊢ mapIdx.go f ([] ++ l₂) arr = mapIdx.go f l₂ (toArray (Array.toList arr))\n\ncase succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (List.append tail l₂) (Array.push arr (f (Array.size arr) head)) =\n mapIdx.go f l₂ (toArray (mapIdx.go f tail (Array.push arr (f (Array.size arr) head))))",
"state_before": "case succ\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₁ l₂ : List α\narr : Array β\nh : length (l₁ ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))",
"tactic": "cases' l₁ with head tail <;> simp only [mapIdx.go]"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nh : length ([] ++ l₂) = Nat.succ len\n⊢ mapIdx.go f ([] ++ l₂) arr = mapIdx.go f l₂ (toArray (Array.toList arr))",
"tactic": "simp only [nil_append, Array.toList_eq, Array.toArray_data]"
},
{
"state_after": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (tail ++ l₂) (Array.push arr (f (Array.size arr) head)) =\n mapIdx.go f l₂ (toArray (mapIdx.go f tail (Array.push arr (f (Array.size arr) head))))",
"state_before": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (List.append tail l₂) (Array.push arr (f (Array.size arr) head)) =\n mapIdx.go f l₂ (toArray (mapIdx.go f tail (Array.push arr (f (Array.size arr) head))))",
"tactic": "simp only [List.append_eq]"
},
{
"state_after": "case succ.cons.e\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ length (tail ++ l₂) = len",
"state_before": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ mapIdx.go f (tail ++ l₂) (Array.push arr (f (Array.size arr) head)) =\n mapIdx.go f l₂ (toArray (mapIdx.go f tail (Array.push arr (f (Array.size arr) head))))",
"tactic": "rw [ih]"
},
{
"state_after": "case succ.cons.e\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length tail + length l₂ = len\n⊢ length (tail ++ l₂) = len",
"state_before": "case succ.cons.e\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ length (tail ++ l₂) = len",
"tactic": "simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h"
},
{
"state_after": "no goals",
"state_before": "case succ.cons.e\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (l₁ l₂ : List α) (arr : Array β),\n length (l₁ ++ l₂) = len → mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (toArray (mapIdx.go f l₁ arr))\nl₂ : List α\narr : Array β\nhead : α\ntail : List α\nh : length tail + length l₂ = len\n⊢ length (tail ++ l₂) = len",
"tactic": "simp only [length_append, h]"
}
] |
[
119,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.vec2_dotProduct'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : AddCommMonoid α\ninst✝ : Mul α\na₀ a₁ b₀ b₁ : α\n⊢ ![a₀, a₁] ⬝ᵥ ![b₀, b₁] = a₀ * b₀ + a₁ * b₁",
"tactic": "rw [cons_dotProduct_cons, cons_dotProduct_cons, dotProduct_empty, add_zero]"
}
] |
[
491,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.map_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.346\nf : α → β\np : Pmf α\nb : β\n⊢ ↑(map f p) b = ∑' (a : α), if b = f a then ↑p a else 0",
"tactic": "simp [map]"
}
] |
[
53,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/Order/Monoid/TypeTags.lean
|
Multiplicative.toAdd_le
|
[] |
[
160,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.measure_support_lt_top_of_memℒp
|
[] |
[
415,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.irreducible_p
|
[] |
[
615,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.coe_injective
|
[] |
[
90,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Logic/Nontrivial.lean
|
not_nontrivial
|
[] |
[
123,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_Iic_iff
|
[] |
[
1190,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1189,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.induction_on'
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : RatFunc K → Prop\nx : FractionRing K[X]\nf : ∀ (p q : K[X]), q ≠ 0 → P (RatFunc.mk p q)\nx✝ : K[X] × { x // x ∈ K[X]⁰ }\np : K[X]\nq : { x // x ∈ K[X]⁰ }\n⊢ P { toFractionRing := Localization.mk (p, q).fst (p, q).snd }",
"tactic": "simpa only [mk_coe_def, Localization.mk_eq_mk'] using\n f p q (mem_nonZeroDivisors_iff_ne_zero.mp q.2)"
}
] |
[
292,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
11
] |
Mathlib/Order/LocallyFinite.lean
|
Finset.mem_Icc
|
[] |
[
328,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean
|
CategoryTheory.Limits.hasFiniteCoproducts_of_hasCoproducts
|
[] |
[
94,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.has_filtered_colimits_of_has_cofiltered_limits_op
|
[] |
[
329,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffAt_pi
|
[] |
[
1161,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1160,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.map_le_iff
|
[
{
"state_after": "case mono_iff\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.103518\nδ : Type ?u.103521\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\na b : WithTop α\nmono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b\n⊢ ∀ {a b : αᵒᵈ}, (↑OrderDual.toDual ∘ f ∘ ↑OrderDual.ofDual) a ≤ (↑OrderDual.toDual ∘ f ∘ ↑OrderDual.ofDual) b ↔ a ≤ b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103518\nδ : Type ?u.103521\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\na b : WithTop α\nmono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b\n⊢ map f a ≤ map f b ↔ a ≤ b",
"tactic": "erw [← toDual_le_toDual_iff, toDual_map, toDual_map, WithBot.map_le_iff, toDual_le_toDual_iff]"
},
{
"state_after": "no goals",
"state_before": "case mono_iff\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.103518\nδ : Type ?u.103521\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\na b : WithTop α\nmono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b\n⊢ ∀ {a b : αᵒᵈ}, (↑OrderDual.toDual ∘ f ∘ ↑OrderDual.ofDual) a ≤ (↑OrderDual.toDual ∘ f ∘ ↑OrderDual.ofDual) b ↔ a ≤ b",
"tactic": "simp [mono_iff]"
}
] |
[
1161,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1157,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode_zero
|
[] |
[
321,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_cons_of_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : α\nl : List α\np : α → Bool\nh : p a = true\n⊢ eraseP p (a :: l) = l",
"tactic": "simp [eraseP_cons, h]"
}
] |
[
932,
24
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
931,
9
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.coe_copy
|
[] |
[
124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.prodComparison_inv_natural
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁷ : Category C\nX Y : C\nD : Type u₂\ninst✝⁶ : Category D\nF : C ⥤ D\nA A' B B' : C\ninst✝⁵ : HasBinaryProduct A B\ninst✝⁴ : HasBinaryProduct A' B'\ninst✝³ : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝² : HasBinaryProduct (F.obj A') (F.obj B')\nf : A ⟶ A'\ng : B ⟶ B'\ninst✝¹ : IsIso (prodComparison F A B)\ninst✝ : IsIso (prodComparison F A' B')\n⊢ inv (prodComparison F A B) ≫ F.map (prod.map f g) = prod.map (F.map f) (F.map g) ≫ inv (prodComparison F A' B')",
"tactic": "rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, prodComparison_natural]"
}
] |
[
1304,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1300,
1
] |
Mathlib/Data/Sym/Card.lean
|
Sym2.card
|
[] |
[
213,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
11
] |
Mathlib/Algebra/Associated.lean
|
Prime.not_unit
|
[] |
[
40,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.abs_eq
|
[] |
[
918,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
917,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean
|
BoxIntegral.Prepartition.mem_splitCenter
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\n⊢ J ∈ splitCenter I ↔ ∃ s, Box.splitCenterBox I s = J",
"tactic": "simp [splitCenter]"
}
] |
[
58,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.toWeakDualBCNN_continuous
|
[] |
[
467,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Algebra/Algebra/Pi.lean
|
AlgEquiv.piCongrRight_trans
|
[] |
[
168,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.restrictScalars_mul
|
[] |
[
1549,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1543,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.isometry_extend
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.805244\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\nh : δ →ᵇ β\ng₁ g₂ : α →ᵇ β\n⊢ dist (extend f g₁ h) (extend f g₂ h) = dist g₁ g₂",
"tactic": "simp [dist_nonneg]"
}
] |
[
508,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.rel_foldr
|
[] |
[
288,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_eq_preimage_of_inverse
|
[] |
[
409,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
IsSelfAdjoint.mul_star_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type ?u.2929\ninst✝¹ : Semigroup R\ninst✝ : StarSemigroup R\nx : R\n⊢ IsSelfAdjoint (x * star x)",
"tactic": "simpa only [star_star] using star_mul_self (star x)"
}
] |
[
94,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Filter.map_neg_eq_comap_neg
|
[] |
[
1950,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1948,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
one_mem_commutatorSet
|
[] |
[
258,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.sigma_subset_preimage_fst
|
[] |
[
238,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic.monic
|
[
{
"state_after": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ Monic (map (Int.castRingHom R) (cyclotomic n ℤ))",
"state_before": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ Monic (cyclotomic n R)",
"tactic": "rw [← map_cyclotomic_int]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ Monic (map (Int.castRingHom R) (cyclotomic n ℤ))",
"tactic": "exact (int_cyclotomic_spec n).2.2.map _"
}
] |
[
324,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.iSup_sub
|
[] |
[
2405,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2403,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.getElem_mem_data
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ni : Nat\na : Array α\nh : i < size a\n⊢ a[i] ∈ a.data",
"tactic": "simp [getElem_eq_data_get, List.get_mem]"
}
] |
[
54,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
53,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
IsScalarTower.Algebra.ext
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S✝\ninst✝¹⁰ : Semiring A✝\ninst✝⁹ : Semiring B\ninst✝⁸ : Algebra R S✝\ninst✝⁷ : Algebra S✝ A✝\ninst✝⁶ : Algebra S✝ B\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B\ninst✝³ : IsScalarTower R S✝ A✝\ninst✝² : IsScalarTower R S✝ B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A",
"state_before": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S✝\ninst✝¹⁰ : Semiring A✝\ninst✝⁹ : Semiring B\ninst✝⁸ : Algebra R S✝\ninst✝⁷ : Algebra S✝ A✝\ninst✝⁶ : Algebra S✝ B\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B\ninst✝³ : IsScalarTower R S✝ A✝\ninst✝² : IsScalarTower R S✝ B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n⊢ A",
"tactic": "have I := h1"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S✝\ninst✝¹⁰ : Semiring A✝\ninst✝⁹ : Semiring B\ninst✝⁸ : Algebra R S✝\ninst✝⁷ : Algebra S✝ A✝\ninst✝⁶ : Algebra S✝ B\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B\ninst✝³ : IsScalarTower R S✝ A✝\ninst✝² : IsScalarTower R S✝ B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n⊢ A",
"tactic": "exact r • x"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nA✝ : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S✝\ninst✝¹⁰ : Semiring A✝\ninst✝⁹ : Semiring B\ninst✝⁸ : Algebra R S✝\ninst✝⁷ : Algebra S✝ A✝\ninst✝⁶ : Algebra S✝ B\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B\ninst✝³ : IsScalarTower R S✝ A✝\ninst✝² : IsScalarTower R S✝ B\nS : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nh :\n ∀ (r : S) (x : A),\n (let_fun I := h1;\n r • x) =\n r • x\nr : S\n⊢ ↑(algebraMap S A) r = ↑(algebraMap S A) r",
"tactic": "simpa only [@Algebra.smul_def _ _ _ _ h1, @Algebra.smul_def _ _ _ _ h2, mul_one] using h r 1"
}
] |
[
116,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_le_rpow_of_exponent_ge
|
[
{
"state_after": "case intro\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\n⊢ ↑x ^ y ≤ ↑x ^ z",
"state_before": "x : ℝ≥0∞\ny z : ℝ\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ x ^ y ≤ x ^ z",
"tactic": "lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx1 coe_lt_top)"
},
{
"state_after": "case pos\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z\n\ncase neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"state_before": "case intro\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\n⊢ ↑x ^ y ≤ ↑x ^ z",
"tactic": "by_cases h : x = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"tactic": "rcases lt_trichotomy y 0 with (Hy | Hy | Hy) <;>\nrcases lt_trichotomy z 0 with (Hz | Hz | Hz) <;>\nsimp [Hy, Hz, h, zero_rpow_of_neg, zero_rpow_of_pos, le_refl] <;>\nlinarith"
},
{
"state_after": "case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"state_before": "case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"tactic": "rw [coe_le_one_iff] at hx1"
},
{
"state_after": "no goals",
"state_before": "case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"tactic": "simp [coe_rpow_of_ne_zero h,\n NNReal.rpow_le_rpow_of_exponent_ge (bot_lt_iff_ne_bot.mpr h) hx1 hyz]"
}
] |
[
636,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.ind
|
[] |
[
114,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
11
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.quadratic_le_exp_of_nonneg
|
[
{
"state_after": "x✝ y x : ℝ\nhx : 0 ≤ x\n⊢ x ^ 2 / 2 = x ^ 2 / (1 + 1)",
"state_before": "x✝ y x : ℝ\nhx : 0 ≤ x\n⊢ 1 + x + x ^ 2 / 2 = ∑ i in range 3, x ^ i / ↑i !",
"tactic": "simp [Finset.sum_range_succ]"
},
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\nhx : 0 ≤ x\n⊢ x ^ 2 / 2 = x ^ 2 / (1 + 1)",
"tactic": "ring_nf"
}
] |
[
1477,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1474,
1
] |
Mathlib/Data/Int/GCD.lean
|
Int.lcm_dvd
|
[
{
"state_after": "i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(Nat.lcm (natAbs i) (natAbs j)) ∣ k",
"state_before": "i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(lcm i j) ∣ k",
"tactic": "rw [Int.lcm]"
},
{
"state_after": "i j k : ℤ\nhi : i ∣ k\nhj : j ∣ k\n⊢ ↑(Nat.lcm (natAbs i) (natAbs j)) ∣ k",
"state_before": "i j k : ℤ\n⊢ i ∣ k → j ∣ k → ↑(Nat.lcm (natAbs i) (natAbs j)) ∣ k",
"tactic": "intro hi hj"
},
{
"state_after": "no goals",
"state_before": "i j k : ℤ\nhi : i ∣ k\nhj : j ∣ k\n⊢ ↑(Nat.lcm (natAbs i) (natAbs j)) ∣ k",
"tactic": "exact coe_nat_dvd_left.mpr (Nat.lcm_dvd (natAbs_dvd_natAbs.mpr hi) (natAbs_dvd_natAbs.mpr hj))"
}
] |
[
498,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
IsLinearMap.map_zero
|
[] |
[
707,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.disjoint_iff_eq_or_eq
|
[] |
[
80,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.intersecting_insert
|
[] |
[
81,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.eq_ite
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3605350\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_4\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\ni j : ι\nv : G i\nw : G j\nh : i = j\n⊢ inner (↑(V i) v) (↑(V j) w) = inner (↑(V i) v) (↑(V j) w)\n\ncase inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3605350\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_4\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\ni j : ι\nv : G i\nw : G j\nh : ¬i = j\n⊢ inner (↑(V i) v) (↑(V j) w) = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3605350\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_4\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\ni j : ι\nv : G i\nw : G j\n⊢ inner (↑(V i) v) (↑(V j) w) = if i = j then inner (↑(V i) v) (↑(V j) w) else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3605350\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_4\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\ni j : ι\nv : G i\nw : G j\nh : i = j\n⊢ inner (↑(V i) v) (↑(V j) w) = inner (↑(V i) v) (↑(V j) w)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3605350\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_4\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\ni j : ι\nv : G i\nw : G j\nh : ¬i = j\n⊢ inner (↑(V i) v) (↑(V j) w) = 0",
"tactic": "exact hV h v w"
}
] |
[
2009,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2005,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.support_mk_subset
|
[] |
[
1108,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean
|
Doset.doset_union_leftCoset
|
[
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (x ∈ ⋃ (h : { x // x ∈ H }), leftCoset (↑h * a) ↑K) ↔ x ∈ doset a ↑H ↑K",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na : G\n⊢ (⋃ (h : { x // x ∈ H }), leftCoset (↑h * a) ↑K) = doset a ↑H ↑K",
"tactic": "ext x"
},
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K) ↔ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (x ∈ ⋃ (h : { x // x ∈ H }), leftCoset (↑h * a) ↑K) ↔ x ∈ doset a ↑H ↑K",
"tactic": "simp only [mem_leftCoset_iff, mul_inv_rev, Set.mem_iUnion, mem_doset]"
},
{
"state_after": "case h.mp\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K) → ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y\n\ncase h.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y) → ∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K) ↔ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\ny : { x // x ∈ H }\nh_h : a⁻¹ * (↑y)⁻¹ * x ∈ ↑K\n⊢ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y",
"state_before": "case h.mp\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K) → ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y",
"tactic": "rintro ⟨y, h_h⟩"
},
{
"state_after": "case h.mp.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\ny : { x // x ∈ H }\nh_h : a⁻¹ * (↑y)⁻¹ * x ∈ ↑K\n⊢ x = ↑y * a * (a⁻¹ * ↑y⁻¹ * x)",
"state_before": "case h.mp.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\ny : { x // x ∈ H }\nh_h : a⁻¹ * (↑y)⁻¹ * x ∈ ↑K\n⊢ ∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y",
"tactic": "refine' ⟨y, y.2, a⁻¹ * y⁻¹ * x, h_h, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\ny : { x // x ∈ H }\nh_h : a⁻¹ * (↑y)⁻¹ * x ∈ ↑K\n⊢ x = ↑y * a * (a⁻¹ * ↑y⁻¹ * x)",
"tactic": "simp only [← mul_assoc, one_mul, mul_right_inv, mul_inv_cancel_right, SubgroupClass.coe_inv]"
},
{
"state_after": "case h.mpr.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x✝ x : G\nhx : x ∈ ↑H\ny : G\nhy : y ∈ ↑K\nhxy : x✝ = x * a * y\n⊢ ∃ i, a⁻¹ * (↑i)⁻¹ * x✝ ∈ ↑K",
"state_before": "case h.mpr\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x : G\n⊢ (∃ x_1, x_1 ∈ ↑H ∧ ∃ y, y ∈ ↑K ∧ x = x_1 * a * y) → ∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K",
"tactic": "rintro ⟨x, hx, y, hy, hxy⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x✝ x : G\nhx : x ∈ ↑H\ny : G\nhy : y ∈ ↑K\nhxy : x✝ = x * a * y\n⊢ a⁻¹ * (↑{ val := x, property := hx })⁻¹ * x✝ ∈ ↑K",
"state_before": "case h.mpr.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x✝ x : G\nhx : x ∈ ↑H\ny : G\nhy : y ∈ ↑K\nhxy : x✝ = x * a * y\n⊢ ∃ i, a⁻¹ * (↑i)⁻¹ * x✝ ∈ ↑K",
"tactic": "refine' ⟨⟨x, hx⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.64712\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH K : Subgroup G\na x✝ x : G\nhx : x ∈ ↑H\ny : G\nhy : y ∈ ↑K\nhxy : x✝ = x * a * y\n⊢ a⁻¹ * (↑{ val := x, property := hx })⁻¹ * x✝ ∈ ↑K",
"tactic": "simp only [hxy, ← mul_assoc, hy, one_mul, mul_left_inv, Subgroup.coe_mk, inv_mul_cancel_right]"
}
] |
[
197,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Analysis/NormedSpace/Complemented.lean
|
Subspace.closedComplemented_iff_has_closed_compl
|
[] |
[
137,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MonoidHom.coprod_inl_inr
|
[] |
[
663,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/Logic/Basic.lean
|
Or.elim3
|
[] |
[
346,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.merge_aux
|
[
{
"state_after": "case nil\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ l o₂✝\nhr✝ : Valid' o₁✝ r✝ o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r✝) l\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ nil o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) nil\n⊢ Valid' o₁ (merge nil r) o₂ ∧ size (merge nil r) = size nil + size r\n\ncase node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ l o₂✝\nhr✝ : Valid' o₁✝ r✝ o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) r) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) r) = size (Ordnode.node ls ll lx lr) + size r",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl r : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) l\n⊢ Valid' o₁ (merge l r) o₂ ∧ size (merge l r) = size l + size r",
"tactic": "induction' l with ls ll lx lr _ IHlr generalizing o₁ o₂ r"
},
{
"state_after": "case node.nil\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ nil o₂\nsep : All (fun x => All (fun y => x < y) nil) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) nil) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) nil) = size (Ordnode.node ls ll lx lr) + size nil\n\ncase node.node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ l o₂✝\nhr✝ : Valid' o₁✝ r✝ o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) r) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) r) = size (Ordnode.node ls ll lx lr) + size r",
"tactic": "induction' r with rs rl rx rr IHrl _ generalizing o₁ o₂"
},
{
"state_after": "case node.node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁\n (if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr\n else\n if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))\n else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr))\n o₂ ∧\n size\n (if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr\n else\n if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))\n else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "rw [merge_node]"
},
{
"state_after": "case node.node.inl\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : delta * ls < rs\n⊢ Valid' o₁ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) o₂ ∧\n size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)\n\ncase node.node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)\n\ncase node.node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : ¬delta * rs < ls\n⊢ Valid' o₁ (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) o₂ ∧\n size (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁\n (if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr\n else\n if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))\n else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr))\n o₂ ∧\n size\n (if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr\n else\n if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))\n else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "split_ifs with h h_1"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ l o₂✝\nhr✝ : Valid' o₁✝ r✝ o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r✝) l\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ nil o₂\nhr : Valid' o₁ r o₂\nsep : All (fun x => All (fun y => x < y) r) nil\n⊢ Valid' o₁ (merge nil r) o₂ ∧ size (merge nil r) = size nil + size r",
"tactic": "exact ⟨hr, (zero_add _).symm⟩"
},
{
"state_after": "no goals",
"state_before": "case node.nil\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ nil o₂\nsep : All (fun x => All (fun y => x < y) nil) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ (merge (Ordnode.node ls ll lx lr) nil) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) nil) = size (Ordnode.node ls ll lx lr) + size nil",
"tactic": "exact ⟨hl, rfl⟩"
},
{
"state_after": "case node.node.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : delta * ls < rs\nv : Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) ↑rx\ne : size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\n⊢ Valid' o₁ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) o₂ ∧\n size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node.inl\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : delta * ls < rs\n⊢ Valid' o₁ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) o₂ ∧\n size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "cases'\n IHrl (hl.of_lt hr.1.1.to_nil <| sep.imp fun x h => h.2.1) hr.left\n (sep.imp fun x h => h.1) with\n v e"
},
{
"state_after": "no goals",
"state_before": "case node.node.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : delta * ls < rs\nv : Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) ↑rx\ne : size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\n⊢ Valid' o₁ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) o₂ ∧\n size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "exact Valid'.merge_aux₁ hl hr h v e"
},
{
"state_after": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "cases' IHlr hl.right (hr.of_gt hl.1.2.to_nil sep.2.1) sep.2.2 with v e"
},
{
"state_after": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\nthis :\n size (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) = rs + size (Ordnode.dual lr) →\n Valid' o₂ (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) o₁ ∧\n size (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) = rs + ls\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "have := Valid'.merge_aux₁ hr.dual hl.dual h_1 v.dual"
},
{
"state_after": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\nthis :\n size (merge lr (Ordnode.node rs rl rx rr)) = size lr + rs →\n Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) = ls + rs\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"state_before": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\nthis :\n size (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) = rs + size (Ordnode.dual lr) →\n Valid' o₂ (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) o₁ ∧\n size (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) = rs + ls\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "rw [size_dual, add_comm, size_dual, ← dual_balanceR, ← Valid'.dual_iff, size_dual,\n add_comm rs] at this"
},
{
"state_after": "no goals",
"state_before": "case node.node.inr.inl.intro\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : delta * rs < ls\nv : Valid' (↑lx) (merge lr (Ordnode.node rs rl rx rr)) o₂\ne : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)\nthis :\n size (merge lr (Ordnode.node rs rl rx rr)) = size lr + rs →\n Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) = ls + rs\n⊢ Valid' o₁ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) o₂ ∧\n size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "exact this e"
},
{
"state_after": "no goals",
"state_before": "case node.node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl r✝ : Ordnode α\no₁✝¹ : WithBot α\no₂✝¹ : WithTop α\nhl✝¹ : Valid' o₁✝¹ l o₂✝¹\nhr✝¹ : Valid' o₁✝¹ r✝ o₂✝¹\nsep✝¹ : All (fun x => All (fun y => x < y) r✝) l\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ ll o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (merge ll r) o₂ ∧ size (merge ll r) = size ll + size r\nIHlr :\n ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ lr o₂ →\n Valid' o₁ r o₂ →\n All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (merge lr r) o₂ ∧ size (merge lr r) = size lr + size r\nr : Ordnode α\no₁✝ : WithBot α\no₂✝ : WithTop α\nhl✝ : Valid' o₁✝ (Ordnode.node ls ll lx lr) o₂✝\nhr✝ : Valid' o₁✝ r o₂✝\nsep✝ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rl o₂ →\n All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rl) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl\nr_ih✝ :\n ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n Valid' o₁ rr o₂ →\n All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n Valid' o₁ (merge (Ordnode.node ls ll lx lr) rr) o₂ ∧\n size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\nh : ¬delta * ls < rs\nh_1 : ¬delta * rs < ls\n⊢ Valid' o₁ (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) o₂ ∧\n size (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =\n size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)",
"tactic": "refine' Valid'.glue_aux hl hr sep (Or.inr ⟨not_lt.1 h_1, not_lt.1 h⟩)"
}
] |
[
1497,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1479,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.subset_closure
|
[] |
[
270,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.constantCoeff_comp_C
|
[] |
[
501,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/LinearAlgebra/FinsuppVectorSpace.lean
|
Finset.sum_single_ite
|
[
{
"state_after": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ (Finset.sum Finset.univ fun x => Finsupp.single i a) = Finsupp.single i a\n\ncase w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ ∀ (x : n), x ∈ {i} → Finsupp.single x (if i = x then a else 0) = Finsupp.single i a\n\ncase w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ ∀ (x : n), ¬x ∈ {i} → Finsupp.single x (if i = x then a else 0) = 0",
"state_before": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ (Finset.sum Finset.univ fun x => Finsupp.single x (if i = x then a else 0)) = Finsupp.single i a",
"tactic": "rw [Finset.sum_congr_set {i} (fun x : n => Finsupp.single x (ite (i = x) a 0)) fun _ =>\n Finsupp.single i a]"
},
{
"state_after": "case w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\n⊢ Finsupp.single x (if i = x then a else 0) = 0",
"state_before": "case w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ ∀ (x : n), ¬x ∈ {i} → Finsupp.single x (if i = x then a else 0) = 0",
"tactic": "intro x hx"
},
{
"state_after": "case w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\nhx' : ¬i = x\n⊢ Finsupp.single x (if i = x then a else 0) = 0",
"state_before": "case w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\n⊢ Finsupp.single x (if i = x then a else 0) = 0",
"tactic": "have hx' : ¬i = x := by\n refine' ne_comm.mp _\n rwa [mem_singleton_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case w'\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\nhx' : ¬i = x\n⊢ Finsupp.single x (if i = x then a else 0) = 0",
"tactic": "simp [hx']"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ (Finset.sum Finset.univ fun x => Finsupp.single i a) = Finsupp.single i a",
"tactic": "simp"
},
{
"state_after": "case w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : x ∈ {i}\n⊢ Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
"state_before": "case w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni : n\n⊢ ∀ (x : n), x ∈ {i} → Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
"tactic": "intro x hx"
},
{
"state_after": "case w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : x = i\n⊢ Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
"state_before": "case w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : x ∈ {i}\n⊢ Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
"tactic": "rw [Set.mem_singleton_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case w\nR : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : x = i\n⊢ Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
"tactic": "simp [hx]"
},
{
"state_after": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\n⊢ x ≠ i",
"state_before": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\n⊢ ¬i = x",
"tactic": "refine' ne_comm.mp _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type ?u.93176\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\ni x : n\nhx : ¬x ∈ {i}\n⊢ x ≠ i",
"tactic": "rwa [mem_singleton_iff] at hx"
}
] |
[
167,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fin.image_castSucc
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.103387\nβ : Type ?u.103390\nγ : Type ?u.103393\nn : ℕ\n⊢ image (↑castSucc) univ = {last n}ᶜ",
"tactic": "rw [← Fin.succAbove_last, Fin.image_succAbove_univ]"
}
] |
[
827,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Dense.exists_between
|
[] |
[
791,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean
|
OrderIso.coe_toHomeomorph_symm
|
[] |
[
332,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isLeast_pair
|
[] |
[
1008,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
Monoid.subset_closure
|
[] |
[
312,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.le_of_op_norm_le
|
[] |
[
219,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.refl_apply
|
[] |
[
269,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.limNthHom_mul
|
[] |
[
606,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrixAlgEquiv_transpose_apply
|
[] |
[
753,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
751,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.roots_C_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nha : a ≠ 0\n⊢ roots (↑C a * p) = roots p",
"tactic": "by_cases hp : p = 0 <;>\n simp only [roots_mul, *, Ne.def, mul_eq_zero, C_eq_zero, or_self_iff, not_false_iff, roots_C,\n zero_add, MulZeroClass.mul_zero]"
}
] |
[
677,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/CategoryTheory/Endomorphism.lean
|
CategoryTheory.Aut.Aut_inv_def
|
[] |
[
157,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
MonovaryOn.dual
|
[] |
[
224,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Data/Sign.lean
|
SignType.self_eq_neg_iff
|
[
{
"state_after": "no goals",
"state_before": "a : SignType\n⊢ a = -a ↔ a = 0",
"tactic": "cases a <;> decide"
}
] |
[
225,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_bit1
|
[] |
[
287,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
11
] |
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