file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_max_of_nonneg
|
[] |
[
969,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
968,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.proj_out
|
[] |
[
467,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
lt_of_lt_mul_of_le_one_of_nonneg_left
|
[] |
[
920,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.add_pos_right
|
[
{
"state_after": "n m : Nat\nh : 0 < n\n⊢ 0 < n + m",
"state_before": "n m : Nat\nh : 0 < n\n⊢ 0 < m + n",
"tactic": "rw [Nat.add_comm]"
},
{
"state_after": "no goals",
"state_before": "n m : Nat\nh : 0 < n\n⊢ 0 < n + m",
"tactic": "exact add_pos_left h m"
}
] |
[
100,
25
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
98,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_neg
|
[] |
[
427,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.sub_period
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.81127\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddGroup α\nh1 : Periodic f c₁\nh2 : Periodic f c₂\nx : α\n⊢ f (x + (c₁ - c₂)) = f x",
"tactic": "rw [sub_eq_add_neg, ← add_assoc, h2.neg, h1]"
}
] |
[
179,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.coe_ndinter
|
[] |
[
224,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Topology/Algebra/Field.lean
|
Subfield.le_topologicalClosure
|
[] |
[
66,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean
|
MvPolynomial.mem_symmetricSubalgebra
|
[] |
[
98,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Filter.coprod_cofinite
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.3352\nα : Type u_1\nβ : Type u_2\nl : Filter α\ns : Set (α × β)\n⊢ sᶜ ∈ Filter.coprod cofinite cofinite ↔ sᶜ ∈ cofinite",
"tactic": "simp only [compl_mem_coprod, mem_cofinite, compl_compl, finite_image_fst_and_snd_iff]"
}
] |
[
120,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
MulSalemSpencer.mul_left₀
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.95828\nα : Type u_1\nβ : Type ?u.95834\n𝕜 : Type ?u.95837\nE : Type ?u.95840\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"state_before": "F : Type ?u.95828\nα : Type u_1\nβ : Type ?u.95834\n𝕜 : Type ?u.95837\nE : Type ?u.95840\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\n⊢ MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)",
"tactic": "rintro _ _ _ ⟨b, hb, rfl⟩ ⟨c, hc, rfl⟩ ⟨d, hd, rfl⟩ h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.95828\nα : Type u_1\nβ : Type ?u.95834\n𝕜 : Type ?u.95837\nE : Type ?u.95840\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : a * a * (b * c) = a * a * (d * d)\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.95828\nα : Type u_1\nβ : Type ?u.95834\n𝕜 : Type ?u.95837\nE : Type ?u.95840\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"tactic": "rw [mul_mul_mul_comm, mul_mul_mul_comm a d] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.95828\nα : Type u_1\nβ : Type ?u.95834\n𝕜 : Type ?u.95837\nE : Type ?u.95840\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : a * a * (b * c) = a * a * (d * d)\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"tactic": "rw [hs hb hc hd (mul_left_cancel₀ (mul_ne_zero ha ha) h)]"
}
] |
[
241,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.mem_inf
|
[] |
[
141,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinFilter.bicomp_mono
|
[] |
[
374,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.preimage_symm_proj_inter
|
[
{
"state_after": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\n⊢ (x, y) ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ ↔ (x, y) ∈ (s ∩ e.baseSet) ×ˢ univ",
"state_before": "ι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx : Z\ns : Set B\n⊢ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ = (s ∩ e.baseSet) ×ˢ univ",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\n⊢ x ∈ e.baseSet → (proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s)",
"state_before": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\n⊢ (x, y) ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ ↔ (x, y) ∈ (s ∩ e.baseSet) ×ˢ univ",
"tactic": "suffices x ∈ e.baseSet → (proj (e.toLocalEquiv.symm (x, y)) ∈ s ↔ x ∈ s) by\n simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]"
},
{
"state_after": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\nh : x ∈ e.baseSet\n⊢ proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s",
"state_before": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\n⊢ x ∈ e.baseSet → (proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\nh : x ∈ e.baseSet\n⊢ proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s",
"tactic": "rw [e.proj_symm_apply' h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.9734\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9745\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\ns : Set B\nx : B\ny : F\nthis : x ∈ e.baseSet → (proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s)\n⊢ (x, y) ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ ↔ (x, y) ∈ (s ∩ e.baseSet) ×ˢ univ",
"tactic": "simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]"
}
] |
[
198,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
GeneralizedContinuedFraction.of_s_head
|
[
{
"state_after": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nh : fract v ≠ 0\n⊢ Stream'.Seq.get? (of v).s 0 = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nh : fract v ≠ 0\n⊢ Stream'.Seq.head (of v).s = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }",
"tactic": "change (of v).s.get? 0 = _"
},
{
"state_after": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nh : fract v ≠ 0\n⊢ (match some (IntFractPair.of (IntFractPair.of v).fr⁻¹), some ∘ fun p => { a := 1, b := ↑p.b } with\n | none, x => none\n | some a, b => b a) =\n some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nh : fract v ≠ 0\n⊢ Stream'.Seq.get? (of v).s 0 = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }",
"tactic": "rw [of_s_head_aux, stream_succ_of_some (stream_zero v) h, Option.bind]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nh : fract v ≠ 0\n⊢ (match some (IntFractPair.of (IntFractPair.of v).fr⁻¹), some ∘ fun p => { a := 1, b := ↑p.b } with\n | none, x => none\n | some a, b => b a) =\n some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }",
"tactic": "rfl"
}
] |
[
280,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
|
AlgebraicTopology.DoldKan.PInfty_f
|
[] |
[
75,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
nndist_left_lineMap
|
[] |
[
114,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Order/CompleteLattice.lean
|
toDual_iInf
|
[] |
[
450,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Data/Real/GoldenRatio.lean
|
gold_mul_goldConj
|
[
{
"state_after": "⊢ (1 + sqrt 5) * (1 - sqrt 5) = -(2 * 2)",
"state_before": "⊢ φ * ψ = -1",
"tactic": "field_simp"
},
{
"state_after": "⊢ 1 ^ 2 - sqrt 5 ^ 2 = -(2 * 2)",
"state_before": "⊢ (1 + sqrt 5) * (1 - sqrt 5) = -(2 * 2)",
"tactic": "rw [← sq_sub_sq]"
},
{
"state_after": "no goals",
"state_before": "⊢ 1 ^ 2 - sqrt 5 ^ 2 = -(2 * 2)",
"tactic": "norm_num"
}
] |
[
66,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
Subalgebra.star_mono
|
[] |
[
371,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.neg_equiv_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ (-x ≈ 0) ↔ (x ≈ 0)",
"tactic": "rw [neg_equiv_iff, neg_zero]"
}
] |
[
1396,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1396,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.abs_ofReal
|
[
{
"state_after": "no goals",
"state_before": "r : ℝ\n⊢ ↑abs ↑r = Abs.abs r",
"tactic": "simp [Complex.abs, normSq_ofReal, Real.sqrt_mul_self_eq_abs]"
}
] |
[
955,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
954,
1
] |
Mathlib/GroupTheory/GroupAction/Quotient.lean
|
MulAction.stabilizer_quotient
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ inst✝² : Group α\ninst✝¹ : MulAction α β\nx : β\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx✝ : G\n⊢ x✝ ∈ stabilizer G ↑1 ↔ x✝ ∈ H",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ inst✝² : Group α\ninst✝¹ : MulAction α β\nx : β\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ stabilizer G ↑1 = H",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ inst✝² : Group α\ninst✝¹ : MulAction α β\nx : β\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx✝ : G\n⊢ x✝ ∈ stabilizer G ↑1 ↔ x✝ ∈ H",
"tactic": "simp [QuotientGroup.eq]"
}
] |
[
232,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
RingEquiv.localRing
|
[] |
[
543,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.sum_le_iSup_lift
|
[
{
"state_after": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ sum fun x => iSup f",
"state_before": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ lift (#ι) * iSup f",
"tactic": "rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]"
},
{
"state_after": "no goals",
"state_before": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ sum fun x => iSup f",
"tactic": "exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)"
}
] |
[
994,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
991,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.out_empty_iff_eq_zero
|
[] |
[
272,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
IsLUB.ciSup_set_eq
|
[] |
[
542,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.toBlocks_fromBlocks₁₁
|
[] |
[
110,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.prodAssoc_prod
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\n⊢ ∀ (s : Set α),\n s ∈ {s | MeasurableSet s} →\n ∀ (t : Set (β × γ)),\n t ∈ image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t} →\n ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ t) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) t",
"state_before": "α : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\n⊢ map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ) = Measure.prod μ (Measure.prod ν τ)",
"tactic": "refine'\n (prod_eq_generateFrom generateFrom_measurableSet generateFrom_prod isPiSystem_measurableSet\n isPiSystem_prod μ.toFiniteSpanningSetsIn\n (ν.toFiniteSpanningSetsIn.prod τ.toFiniteSpanningSetsIn) _).symm"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nt : Set β\nu : Set γ\nht : t ∈ {s | MeasurableSet s}\nhu : u ∈ {t | MeasurableSet t}\n⊢ ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) ((fun x x_1 => x ×ˢ x_1) t u)",
"state_before": "α : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\n⊢ ∀ (s : Set α),\n s ∈ {s | MeasurableSet s} →\n ∀ (t : Set (β × γ)),\n t ∈ image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t} →\n ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ t) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) t",
"tactic": "rintro s hs _ ⟨t, u, ht, hu, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\ns : Set α\nhs : MeasurableSet s\nt : Set β\nu : Set γ\nht : MeasurableSet t\nhu : MeasurableSet u\n⊢ ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) ((fun x x_1 => x ×ˢ x_1) t u)",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nt : Set β\nu : Set γ\nht : t ∈ {s | MeasurableSet s}\nhu : u ∈ {t | MeasurableSet t}\n⊢ ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) ((fun x x_1 => x ×ˢ x_1) t u)",
"tactic": "rw [mem_setOf_eq] at hs ht hu"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.4458428\nβ : Type u_3\nβ' : Type ?u.4458434\nγ : Type u_1\nE : Type ?u.4458440\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite τ\ns : Set α\nhs : MeasurableSet s\nt : Set β\nu : Set γ\nht : MeasurableSet t\nhu : MeasurableSet u\n⊢ ↑↑(map (↑MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod μ ν) τ)) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) =\n ↑↑μ s * ↑↑(Measure.prod ν τ) ((fun x x_1 => x ×ˢ x_1) t u)",
"tactic": "simp_rw [map_apply (MeasurableEquiv.measurable _) (hs.prod (ht.prod hu)),\n MeasurableEquiv.prodAssoc, MeasurableEquiv.coe_mk, Equiv.prod_assoc_preimage, prod_prod,\n mul_assoc]"
}
] |
[
536,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_inter_of_mem
|
[] |
[
1583,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1582,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.int_coe_zmod_eq_iff
|
[
{
"state_after": "case mp\np : ℕ\nn : ℤ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z → ∃ k, n = ↑(val z) + ↑p * k\n\ncase mpr\np : ℕ\nn : ℤ\nz : ZMod p\ninst✝ : NeZero p\n⊢ (∃ k, n = ↑(val z) + ↑p * k) → ↑n = z",
"state_before": "p : ℕ\nn : ℤ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z ↔ ∃ k, n = ↑(val z) + ↑p * k",
"tactic": "constructor"
},
{
"state_after": "case mp\np : ℕ\nn : ℤ\ninst✝ : NeZero p\n⊢ ∃ k, n = ↑(val ↑n) + ↑p * k",
"state_before": "case mp\np : ℕ\nn : ℤ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z → ∃ k, n = ↑(val z) + ↑p * k",
"tactic": "rintro rfl"
},
{
"state_after": "case mp\np : ℕ\nn : ℤ\ninst✝ : NeZero p\n⊢ n = ↑(val ↑n) + ↑p * (n / ↑p)",
"state_before": "case mp\np : ℕ\nn : ℤ\ninst✝ : NeZero p\n⊢ ∃ k, n = ↑(val ↑n) + ↑p * k",
"tactic": "refine' ⟨n / p, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp\np : ℕ\nn : ℤ\ninst✝ : NeZero p\n⊢ n = ↑(val ↑n) + ↑p * (n / ↑p)",
"tactic": "rw [val_int_cast, Int.emod_add_ediv]"
},
{
"state_after": "case mpr.intro\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℤ\n⊢ ↑(↑(val z) + ↑p * k) = z",
"state_before": "case mpr\np : ℕ\nn : ℤ\nz : ZMod p\ninst✝ : NeZero p\n⊢ (∃ k, n = ↑(val z) + ↑p * k) → ↑n = z",
"tactic": "rintro ⟨k, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℤ\n⊢ ↑(↑(val z) + ↑p * k) = z",
"tactic": "rw [Int.cast_add, Int.cast_mul, Int.cast_ofNat, Int.cast_ofNat, nat_cast_val,\n ZMod.nat_cast_self, MulZeroClass.zero_mul, add_zero, cast_id]"
}
] |
[
548,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Topology/Inseparable.lean
|
Inseparable.symm
|
[] |
[
354,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
8
] |
Mathlib/Analysis/NormedSpace/MazurUlam.lean
|
IsometryEquiv.to_real_linear_equiv_apply
|
[] |
[
139,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/Sequences.lean
|
IsSeqClosed.preimage
|
[] |
[
214,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Topology/Maps.lean
|
Inducing.nhdsSet_eq_comap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1466\nδ : Type ?u.1469\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nhf : Inducing f\ns : Set α\n⊢ 𝓝ˢ s = comap f (𝓝ˢ (f '' s))",
"tactic": "simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]"
}
] |
[
101,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.vsub_apply
|
[] |
[
338,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_diagonal
|
[
{
"state_after": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type ?u.116901\no : Type ?u.116904\np : Type ?u.116907\nq : Type ?u.116910\nm' : o → Type ?u.116915\nn' : o → Type ?u.116920\np' : o → Type ?u.116925\nR : Type ?u.116928\nS : Type ?u.116931\nα : Type u_3\nβ : Type ?u.116937\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd₁ : l → α\nd₂ : m → α\ni j : l ⊕ m\n⊢ fromBlocks (diagonal d₁) 0 0 (diagonal d₂) i j = diagonal (Sum.elim d₁ d₂) i j",
"state_before": "l : Type u_1\nm : Type u_2\nn : Type ?u.116901\no : Type ?u.116904\np : Type ?u.116907\nq : Type ?u.116910\nm' : o → Type ?u.116915\nn' : o → Type ?u.116920\np' : o → Type ?u.116925\nR : Type ?u.116928\nS : Type ?u.116931\nα : Type u_3\nβ : Type ?u.116937\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd₁ : l → α\nd₂ : m → α\n⊢ fromBlocks (diagonal d₁) 0 0 (diagonal d₂) = diagonal (Sum.elim d₁ d₂)",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type ?u.116901\no : Type ?u.116904\np : Type ?u.116907\nq : Type ?u.116910\nm' : o → Type ?u.116915\nn' : o → Type ?u.116920\np' : o → Type ?u.116925\nR : Type ?u.116928\nS : Type ?u.116931\nα : Type u_3\nβ : Type ?u.116937\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd₁ : l → α\nd₂ : m → α\ni j : l ⊕ m\n⊢ fromBlocks (diagonal d₁) 0 0 (diagonal d₂) i j = diagonal (Sum.elim d₁ d₂) i j",
"tactic": "rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> simp [diagonal]"
}
] |
[
303,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.map₂_mk
|
[] |
[
269,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
exists_square_le'
|
[
{
"state_after": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\nha : a < 1\n⊢ ∃ b, b * b ≤ a\n\ncase inr\nα : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\nha : 1 ≤ a\n⊢ ∃ b, b * b ≤ a",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\n⊢ ∃ b, b * b ≤ a",
"tactic": "obtain ha | ha := lt_or_le a 1"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\nha : a < 1\n⊢ ∃ b, b * b ≤ a",
"tactic": "exact ⟨a, (mul_lt_of_lt_one_right a0 ha).le⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\nha : 1 ≤ a\n⊢ ∃ b, b * b ≤ a",
"tactic": "exact ⟨1, by rwa [mul_one]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulOneClass α\ninst✝² : Zero α\ninst✝¹ : LinearOrder α\ninst✝ : PosMulStrictMono α\na0 : 0 < a\nha : 1 ≤ a\n⊢ 1 * 1 ≤ a",
"tactic": "rwa [mul_one]"
}
] |
[
970,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
967,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_deriv_comp_smul_deriv'
|
[
{
"state_after": "ι : Type ?u.1918101\n𝕜 : Type ?u.1918104\nE : Type u_1\nF : Type ?u.1918110\nA : Type ?u.1918113\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng'✝ g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng g' : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ (x : ℝ), x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)) → HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n⊢ g (f b) - g (f a) = (g ∘ f) b - (g ∘ f) a\n\nι : Type ?u.1918101\n𝕜 : Type ?u.1918104\nE : Type u_1\nF : Type ?u.1918110\nA : Type ?u.1918113\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng'✝ g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng g' : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ (x : ℝ), x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)) → HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n⊢ [[f a, f b]] ⊆ f '' [[a, b]]",
"state_before": "ι : Type ?u.1918101\n𝕜 : Type ?u.1918104\nE : Type u_1\nF : Type ?u.1918110\nA : Type ?u.1918113\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng'✝ g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng g' : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ (x : ℝ), x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)) → HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n⊢ (∫ (x : ℝ) in a..b, f' x • (g' ∘ f) x) = (g ∘ f) b - (g ∘ f) a",
"tactic": "rw [integral_comp_smul_deriv'' hf hff' hf' hg',\n integral_eq_sub_of_hasDeriv_right hg hgg' (hg'.mono _).intervalIntegrable]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1918101\n𝕜 : Type ?u.1918104\nE : Type u_1\nF : Type ?u.1918110\nA : Type ?u.1918113\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng'✝ g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng g' : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ (x : ℝ), x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)) → HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n⊢ g (f b) - g (f a) = (g ∘ f) b - (g ∘ f) a\n\nι : Type ?u.1918101\n𝕜 : Type ?u.1918104\nE : Type u_1\nF : Type ?u.1918110\nA : Type ?u.1918113\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng'✝ g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng g' : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ (x : ℝ), x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)) → HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f '' [[a, b]])\n⊢ [[f a, f b]] ⊆ f '' [[a, b]]",
"tactic": "exacts [rfl, intermediate_value_uIcc hf]"
}
] |
[
1442,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1434,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocMod_sub_eq_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c✝ : α\nn : ℤ\na b c : α\n⊢ toIocMod hp a (b - c) = toIocMod hp (a + c) b - c",
"tactic": "simp_rw [toIocMod, toIocDiv_sub_eq_toIocDiv_add, sub_right_comm]"
}
] |
[
537,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
MeasureTheory.analyticSet_empty
|
[
{
"state_after": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type ?u.465\n⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅",
"state_before": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type ?u.465\n⊢ AnalyticSet ∅",
"tactic": "rw [AnalyticSet]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type ?u.465\n⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅",
"tactic": "exact Or.inl rfl"
}
] |
[
86,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Data/Finset/Option.lean
|
Finset.eraseNone_union
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11560\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\na✝ : α\n⊢ a✝ ∈ ↑eraseNone (s ∪ t) ↔ a✝ ∈ ↑eraseNone s ∪ ↑eraseNone t",
"state_before": "α : Type u_1\nβ : Type ?u.11560\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\n⊢ ↑eraseNone (s ∪ t) = ↑eraseNone s ∪ ↑eraseNone t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11560\ninst✝¹ : DecidableEq (Option α)\ninst✝ : DecidableEq α\ns t : Finset (Option α)\na✝ : α\n⊢ a✝ ∈ ↑eraseNone (s ∪ t) ↔ a✝ ∈ ↑eraseNone s ∪ ↑eraseNone t",
"tactic": "simp"
}
] |
[
123,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.coe_pow
|
[] |
[
735,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.of_null
|
[] |
[
123,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
src/lean/Init/Control/StateCps.lean
|
StateCpsT.runK_get
|
[] |
[
53,
123
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
53,
9
] |
Mathlib/Data/Int/ModEq.lean
|
Int.add_modEq_left
|
[
{
"state_after": "no goals",
"state_before": "m n a b c d : ℤ\n⊢ n ∣ n + a - a",
"tactic": "simp"
}
] |
[
263,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_integral
|
[] |
[
489,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/Data/List/Basic.lean
|
List.foldlM_nil
|
[] |
[
2790,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2789,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_span_le
|
[
{
"state_after": "case intro.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.566078\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\ns b : Set V\nhb : b ⊆ s\nhsab : span K b = span K s\nhlib : LinearIndependent K Subtype.val\n⊢ Module.rank K { x // x ∈ span K s } ≤ (#↑s)",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.566078\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\ns : Set V\n⊢ Module.rank K { x // x ∈ span K s } ≤ (#↑s)",
"tactic": "obtain ⟨b, hb, hsab, hlib⟩ := exists_linearIndependent K s"
},
{
"state_after": "case h.e'_3\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.566078\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\ns b : Set V\nhb : b ⊆ s\nhsab : span K b = span K s\nhlib : LinearIndependent K Subtype.val\n⊢ Module.rank K { x // x ∈ span K s } = (#↑b)",
"state_before": "case intro.intro.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.566078\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\ns b : Set V\nhb : b ⊆ s\nhsab : span K b = span K s\nhlib : LinearIndependent K Subtype.val\n⊢ Module.rank K { x // x ∈ span K s } ≤ (#↑s)",
"tactic": "convert Cardinal.mk_le_mk_of_subset hb"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.566078\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\ns b : Set V\nhb : b ⊆ s\nhsab : span K b = span K s\nhlib : LinearIndependent K Subtype.val\n⊢ Module.rank K { x // x ∈ span K s } = (#↑b)",
"tactic": "rw [← hsab, rank_span_set hlib]"
}
] |
[
1072,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1069,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_trunc
|
[
{
"state_after": "α : Type ?u.74019\nβ : Type ?u.74022\nM : Type u_1\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (↑(natAdd a) j) = 1\n⊢ ∏ i : Fin a, f (↑(castAdd b) i) = ∏ i : Fin a, f (↑(castLE (_ : a ≤ a + b)) i)",
"state_before": "α : Type ?u.74019\nβ : Type ?u.74022\nM : Type u_1\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (↑(natAdd a) j) = 1\n⊢ ∏ i : Fin (a + b), f i = ∏ i : Fin a, f (↑(castLE (_ : a ≤ a + b)) i)",
"tactic": "rw [prod_univ_add, Fintype.prod_eq_one _ hf, mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.74019\nβ : Type ?u.74022\nM : Type u_1\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (↑(natAdd a) j) = 1\n⊢ ∏ i : Fin a, f (↑(castAdd b) i) = ∏ i : Fin a, f (↑(castLE (_ : a ≤ a + b)) i)",
"tactic": "rfl"
}
] |
[
211,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.one_le
|
[] |
[
909,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.sublists'Aux_eq_array_foldl
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ sublists'Aux a r₁ r₂ =\n Array.toList (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁) 0 (Array.size (toArray r₁)))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\n⊢ ∀ (r₁ r₂ : List (List α)),\n sublists'Aux a r₁ r₂ =\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁) 0 (Array.size (toArray r₁)))",
"tactic": "intro r₁ r₂"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁).data)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ sublists'Aux a r₁ r₂ =\n Array.toList (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁) 0 (Array.size (toArray r₁)))",
"tactic": "rw [sublists'Aux, Array.foldl_eq_foldl_data]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\nthis :\n foldl (fun r l => r ++ [a :: l]) (Array.toList (toArray r₂)) r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) r₁)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁).data)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁).data)",
"tactic": "have := List.foldl_hom Array.toList (fun r l => r.push (a :: l))\n (fun r l => r ++ [a :: l]) r₁ r₂.toArray (by simp)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\nthis :\n foldl (fun r l => r ++ [a :: l]) (Array.toList (toArray r₂)) r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) r₁)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ =\n Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r₂) (toArray r₁).data)",
"tactic": "simpa using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ ∀ (x : Array (List α)) (y : List α),\n (fun r l => r ++ [a :: l]) (Array.toList x) y = Array.toList ((fun r l => Array.push r (a :: l)) x y)",
"tactic": "simp"
}
] |
[
61,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.preimage_Ioc
|
[
{
"state_after": "case h\nF : Type ?u.240835\nα : Type u_1\nβ : Type ?u.240841\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ioc a b ↔ x✝ ∈ Ioc ⌊a⌋ ⌊b⌋",
"state_before": "F : Type ?u.240835\nα : Type u_1\nβ : Type ?u.240841\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ Int.cast ⁻¹' Ioc a b = Ioc ⌊a⌋ ⌊b⌋",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.240835\nα : Type u_1\nβ : Type ?u.240841\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ioc a b ↔ x✝ ∈ Ioc ⌊a⌋ ⌊b⌋",
"tactic": "simp [floor_lt, le_floor]"
}
] |
[
1282,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.TransGen.head_induction_on
|
[
{
"state_after": "case single\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b c d a b✝ : α\na✝ : r a b✝\nP : (a : α) → TransGen r a b✝ → Prop\nbase : ∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)\n⊢ P a (_ : TransGen r a b✝)\n\ncase tail\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝² b c d a b✝ c✝ : α\na✝¹ : TransGen r a b✝\na✝ : r b✝ c✝\na_ih✝ :\n ∀ {P : (a : α) → TransGen r a b✝ → Prop},\n (∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)) → P a a✝¹\nP : (a : α) → TransGen r a c✝ → Prop\nbase : ∀ {a : α} (h : r a c✝), P a (_ : TransGen r a c✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c c✝), P c h → P a (_ : TransGen r a c✝)\n⊢ P a (_ : TransGen r a c✝)",
"state_before": "α : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝ b c d : α\nP : (a : α) → TransGen r a b → Prop\na : α\nh : TransGen r a b\nbase : ∀ {a : α} (h : r a b), P a (_ : TransGen r a b)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)\n⊢ P a h",
"tactic": "induction h"
},
{
"state_after": "case tail\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝² b c d a b✝ c✝ : α\na✝¹ : TransGen r a b✝\na✝ : r b✝ c✝\na_ih✝ :\n ∀ {P : (a : α) → TransGen r a b✝ → Prop},\n (∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)) → P a a✝¹\nP : (a : α) → TransGen r a c✝ → Prop\nbase : ∀ {a : α} (h : r a c✝), P a (_ : TransGen r a c✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c c✝), P c h → P a (_ : TransGen r a c✝)\n⊢ P a (_ : TransGen r a c✝)",
"state_before": "case single\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b c d a b✝ : α\na✝ : r a b✝\nP : (a : α) → TransGen r a b✝ → Prop\nbase : ∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)\n⊢ P a (_ : TransGen r a b✝)\n\ncase tail\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝² b c d a b✝ c✝ : α\na✝¹ : TransGen r a b✝\na✝ : r b✝ c✝\na_ih✝ :\n ∀ {P : (a : α) → TransGen r a b✝ → Prop},\n (∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)) → P a a✝¹\nP : (a : α) → TransGen r a c✝ → Prop\nbase : ∀ {a : α} (h : r a c✝), P a (_ : TransGen r a c✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c c✝), P c h → P a (_ : TransGen r a c✝)\n⊢ P a (_ : TransGen r a c✝)",
"tactic": "case single a h => exact base h"
},
{
"state_after": "no goals",
"state_before": "case tail\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝² b c d a b✝ c✝ : α\na✝¹ : TransGen r a b✝\na✝ : r b✝ c✝\na_ih✝ :\n ∀ {P : (a : α) → TransGen r a b✝ → Prop},\n (∀ {a : α} (h : r a b✝), P a (_ : TransGen r a b✝)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b✝), P c h → P a (_ : TransGen r a b✝)) → P a a✝¹\nP : (a : α) → TransGen r a c✝ → Prop\nbase : ∀ {a : α} (h : r a c✝), P a (_ : TransGen r a c✝)\nih : ∀ {a c : α} (h' : r a c) (h : TransGen r c c✝), P c h → P a (_ : TransGen r a c✝)\n⊢ P a (_ : TransGen r a c✝)",
"tactic": "case tail b c _ hbc h_ih =>\nrefine @h_ih (λ {a : α} (hab : @TransGen α r a b) => P a (TransGen.tail hab hbc)) ?_ ?_;\nexact fun h ↦ ih h (single hbc) (base hbc)\nexact fun hab hbc ↦ ih hab _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b c d a✝ a : α\nh : r a✝ a\nP : (a_1 : α) → TransGen r a_1 a → Prop\nbase : ∀ {a_1 : α} (h : r a_1 a), P a_1 (_ : TransGen r a_1 a)\nih : ∀ {a_1 c : α} (h' : r a_1 c) (h : TransGen r c a), P c h → P a_1 (_ : TransGen r a_1 a)\n⊢ P a✝ (_ : TransGen r a✝ a)",
"tactic": "exact base h"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a : α} (h : r a b), (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)\n\ncase refine_2\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 b),\n (fun {a} hab => P a (_ : TransGen r a c)) h → (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)",
"state_before": "α : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ P a (_ : TransGen r a c)",
"tactic": "refine @h_ih (λ {a : α} (hab : @TransGen α r a b) => P a (TransGen.tail hab hbc)) ?_ ?_"
},
{
"state_after": "case refine_2\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 b),\n (fun {a} hab => P a (_ : TransGen r a c)) h → (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)",
"state_before": "case refine_1\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a : α} (h : r a b), (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)\n\ncase refine_2\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 b),\n (fun {a} hab => P a (_ : TransGen r a c)) h → (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)",
"tactic": "exact fun h ↦ ih h (single hbc) (base hbc)"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\nβ : Type ?u.19922\nγ : Type ?u.19925\nδ : Type ?u.19928\nr : α → α → Prop\na✝¹ b✝ c✝ d a b c : α\na✝ : TransGen r a b\nhbc : r b c\nh_ih :\n ∀ {P : (a : α) → TransGen r a b → Prop},\n (∀ {a : α} (h : r a b), P a (_ : TransGen r a b)) →\n (∀ {a c : α} (h' : r a c) (h : TransGen r c b), P c h → P a (_ : TransGen r a b)) → P a a✝\nP : (a : α) → TransGen r a c → Prop\nbase : ∀ {a : α} (h : r a c), P a (_ : TransGen r a c)\nih : ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 c), P c_1 h → P a (_ : TransGen r a c)\n⊢ ∀ {a c_1 : α} (h' : r a c_1) (h : TransGen r c_1 b),\n (fun {a} hab => P a (_ : TransGen r a c)) h → (fun {a} hab => P a (_ : TransGen r a c)) (_ : TransGen r a b)",
"tactic": "exact fun hab hbc ↦ ih hab _"
}
] |
[
399,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_predAbove_succ
|
[
{
"state_after": "case inl\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)\n\ncase inr\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"state_before": "n✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"tactic": "obtain h₁ | h₂ := lt_or_le (castSucc a) b"
},
{
"state_after": "case inl.h\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ ↑castSucc (succ a) < succ b\n\ncase inl.h\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ ↑castSucc (succ a) < succ b",
"state_before": "case inl\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"tactic": "rw [Fin.predAbove_above _ _ h₁, Fin.succ_pred, Fin.predAbove_above, Fin.pred_succ]"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ ↑castSucc (succ a) < succ b\n\ncase inl.h\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₁ : ↑castSucc a < b\n⊢ ↑castSucc (succ a) < succ b",
"tactic": "simpa only [lt_iff_val_lt_val, coe_castSucc, val_succ, add_lt_add_iff_right] using\n h₁"
},
{
"state_after": "case inr.zero\nn m : ℕ\na : Fin zero\nb : Fin (zero + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)\n\ncase inr.succ\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"state_before": "case inr\nn✝ m n : ℕ\na : Fin n\nb : Fin (n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"tactic": "cases' n with n"
},
{
"state_after": "case inr.zero.h\nn m : ℕ\na : Fin zero\nb : Fin (zero + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ False",
"state_before": "case inr.zero\nn m : ℕ\na : Fin zero\nb : Fin (zero + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case inr.zero.h\nn m : ℕ\na : Fin zero\nb : Fin (zero + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ False",
"tactic": "exact not_lt_zero' a.is_lt"
},
{
"state_after": "case inr.succ\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ castPred (succ b) = succ (castPred b)",
"state_before": "case inr.succ\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ predAbove (succ a) (succ b) = succ (predAbove a b)",
"tactic": "rw [Fin.predAbove_below a b h₂,\n Fin.predAbove_below a.succ b.succ\n (by\n simpa only [le_iff_val_le_val, val_succ, coe_castSucc, add_le_add_iff_right] using h₂)]"
},
{
"state_after": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"state_before": "case inr.succ\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ castPred (succ b) = succ (castPred b)",
"tactic": "ext"
},
{
"state_after": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : ↑b < n + 1\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"state_before": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"tactic": "have h₀ : (b : ℕ) < n + 1 := by\n simp only [le_iff_val_le_val, coe_castSucc] at h₂\n simpa only [lt_succ_iff] using h₂.trans a.is_le"
},
{
"state_after": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : ↑b < n + 1\nh₁ : ↑(succ b) < n + 2\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"state_before": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : ↑b < n + 1\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"tactic": "have h₁ : (b.succ : ℕ) < n + 2 := by\n rw [← Nat.succ_lt_succ_iff] at h₀\n simpa only [val_succ] using h₀"
},
{
"state_after": "no goals",
"state_before": "case inr.succ.h\nn✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : ↑b < n + 1\nh₁ : ↑(succ b) < n + 2\n⊢ ↑(castPred (succ b)) = ↑(succ (castPred b))",
"tactic": "simp only [coe_castPred b h₀, coe_castPred b.succ h₁, val_succ]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ succ b ≤ ↑castSucc (succ a)",
"tactic": "simpa only [le_iff_val_le_val, val_succ, coe_castSucc, add_le_add_iff_right] using h₂"
},
{
"state_after": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : ↑b ≤ ↑a\n⊢ ↑b < n + 1",
"state_before": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\n⊢ ↑b < n + 1",
"tactic": "simp only [le_iff_val_le_val, coe_castSucc] at h₂"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : ↑b ≤ ↑a\n⊢ ↑b < n + 1",
"tactic": "simpa only [lt_succ_iff] using h₂.trans a.is_le"
},
{
"state_after": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : Nat.succ ↑b < Nat.succ (n + 1)\n⊢ ↑(succ b) < n + 2",
"state_before": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : ↑b < n + 1\n⊢ ↑(succ b) < n + 2",
"tactic": "rw [← Nat.succ_lt_succ_iff] at h₀"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\na : Fin (Nat.succ n)\nb : Fin (Nat.succ n + 1)\nh₂ : b ≤ ↑castSucc a\nh₀ : Nat.succ ↑b < Nat.succ (n + 1)\n⊢ ↑(succ b) < n + 2",
"tactic": "simpa only [val_succ] using h₀"
}
] |
[
2448,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2428,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_iff_subset_of_disjoint_closed
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : IsPreconnected s\n⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\n\ncase mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\n⊢ IsPreconnected s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\n⊢ IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : IsPreconnected s\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : IsPreconnected s\n⊢ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v",
"tactic": "intro u v hu hv hs huv"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh :\n ∀ (t t' : Set α),\n IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : IsPreconnected s\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "rw [isPreconnected_closed_iff] at h"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\n⊢ s ⊆ u ∨ s ⊆ v",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh :\n ∀ (t t' : Set α),\n IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "specialize h u v hu hv hs"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : ¬s ⊆ u ∧ ¬s ⊆ v\n⊢ s ∩ (u ∩ v) ≠ ∅",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhuv : s ∩ (u ∩ v) = ∅\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\n⊢ s ⊆ u ∨ s ⊆ v",
"tactic": "contrapose! huv"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : ¬s ⊆ u ∧ ¬s ⊆ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : ¬s ⊆ u ∧ ¬s ⊆ v\n⊢ s ∩ (u ∩ v) ≠ ∅",
"tactic": "rw [← nonempty_iff_ne_empty]"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : ¬s ⊆ u ∧ ¬s ⊆ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "simp [not_subset] at huv"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nhuv : (∃ a, a ∈ s ∧ ¬a ∈ u) ∧ ∃ a, a ∈ s ∧ ¬a ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\nhxv : x ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\nhxv : x ∈ v\nhyu : y ∈ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\nhxv : x ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "have hyu : y ∈ u := or_iff_not_imp_right.mp (hs hys) hyv"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nh : Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\nx : α\nhxs : x ∈ s\nhxu : ¬x ∈ u\ny : α\nhys : y ∈ s\nhyv : ¬y ∈ v\nhxv : x ∈ v\nhyu : y ∈ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "exact h ⟨y, hys, hyu⟩ ⟨x, hxs, hxv⟩"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\n⊢ ∀ (t t' : Set α),\n IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\n⊢ IsPreconnected s",
"tactic": "rw [isPreconnected_closed_iff]"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u v : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\n⊢ ∀ (t t' : Set α),\n IsClosed t → IsClosed t' → s ⊆ t ∪ t' → Set.Nonempty (s ∩ t) → Set.Nonempty (s ∩ t') → Set.Nonempty (s ∩ (t ∩ t'))",
"tactic": "intro u v hu hv hs hsu hsv"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\n⊢ s ∩ (u ∩ v) ≠ ∅",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "rw [nonempty_iff_ne_empty]"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : s ∩ (u ∩ v) = ∅\n⊢ False",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\n⊢ s ∩ (u ∩ v) ≠ ∅",
"tactic": "intro H"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : s ∩ (u ∩ v) = ∅\nh : s ⊆ u ∨ s ⊆ v\n⊢ False",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ : Set α\nh : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : s ∩ (u ∩ v) = ∅\n⊢ False",
"tactic": "specialize h u v hu hv hs H"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nh : s ⊆ u ∨ s ⊆ v\nH : ¬False\n⊢ ¬s ∩ (u ∩ v) = ∅",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : s ∩ (u ∩ v) = ∅\nh : s ⊆ u ∨ s ⊆ v\n⊢ False",
"tactic": "contrapose H"
},
{
"state_after": "case mpr.a\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nh : s ⊆ u ∨ s ⊆ v\nH : ¬False\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nh : s ⊆ u ∨ s ⊆ v\nH : ¬False\n⊢ ¬s ∩ (u ∩ v) = ∅",
"tactic": "apply Nonempty.ne_empty"
},
{
"state_after": "case mpr.a.inl\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))\n\ncase mpr.a.inr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mpr.a\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nh : s ⊆ u ∨ s ⊆ v\nH : ¬False\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "cases' h with h h"
},
{
"state_after": "case mpr.a.inl.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nH : ¬False\nh : s ⊆ u\nx : α\nhxs : x ∈ s\nhxv : x ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mpr.a.inl\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "rcases hsv with ⟨x, hxs, hxv⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.a.inl.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nH : ¬False\nh : s ⊆ u\nx : α\nhxs : x ∈ s\nhxv : x ∈ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "exact ⟨x, hxs, ⟨h hxs, hxv⟩⟩"
},
{
"state_after": "case mpr.a.inr.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"state_before": "case mpr.a.inr\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsu : Set.Nonempty (s ∩ u)\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ v\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "rcases hsu with ⟨x, hxs, hxu⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.a.inr.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.102865\nπ : ι → Type ?u.102870\ninst✝ : TopologicalSpace α\ns t u✝ v✝ u v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhs : s ⊆ u ∪ v\nhsv : Set.Nonempty (s ∩ v)\nH : ¬False\nh : s ⊆ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\n⊢ Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩"
}
] |
[
969,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
944,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsGLB.isLUB_of_tendsto
|
[] |
[
2083,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2080,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_eq
|
[] |
[
1446,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1445,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
|
Matrix.represents_iff
|
[] |
[
102,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.IsPartition.nonempty_boxes
|
[] |
[
753,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
751,
1
] |
Mathlib/Algebra/Group/Units.lean
|
inv_eq_one_divp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b c : α\nu : αˣ\n⊢ ↑u⁻¹ = 1 /ₚ u",
"tactic": "rw [one_divp]"
}
] |
[
509,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.eventually_mul_div_cancel
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.617198\nE : Type ?u.617201\nF : Type ?u.617204\nG : Type ?u.617207\nE' : Type ?u.617210\nF' : Type ?u.617213\nG' : Type ?u.617216\nE'' : Type ?u.617219\nF'' : Type ?u.617222\nG'' : Type ?u.617225\nR : Type ?u.617228\nR' : Type ?u.617231\n𝕜 : Type u_2\n𝕜' : Type ?u.617237\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nu v : α → 𝕜\nh : IsBigOWith c l u v\ny : α\nhy : ‖u y‖ ≤ c * ‖v y‖\nhv : v y = 0\n⊢ u y = 0",
"tactic": "simpa [hv] using hy"
}
] |
[
1905,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1904,
1
] |
Mathlib/Algebra/Jordan/Basic.lean
|
aux2
|
[
{
"state_after": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L a, ↑L (c * c)⁆ + ⁅↑L a, 2 • ↑L (a * b)⁆ + ⁅↑L a, 2 • ↑L (c * a)⁆ + ⁅↑L a, 2 • ↑L (b * c)⁆ +\n (⁅↑L b, ↑L (a * a)⁆ + ⁅↑L b, ↑L (c * c)⁆ + ⁅↑L b, 2 • ↑L (a * b)⁆ + ⁅↑L b, 2 • ↑L (c * a)⁆ +\n ⁅↑L b, 2 • ↑L (b * c)⁆) +\n (⁅↑L c, ↑L (a * a)⁆ + ⁅↑L c, ↑L (b * b)⁆ + ⁅↑L c, 2 • ↑L (a * b)⁆ + ⁅↑L c, 2 • ↑L (c * a)⁆ +\n ⁅↑L c, 2 • ↑L (b * c)⁆) =\n ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L b, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (a * b)⁆ + ⁅↑L b, ↑L (a * b)⁆) +\n (⁅↑L a, ↑L (c * c)⁆ + ⁅↑L c, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (c * a)⁆ + ⁅↑L c, ↑L (c * a)⁆)) +\n (⁅↑L b, ↑L (c * c)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • (⁅↑L b, ↑L (b * c)⁆ + ⁅↑L c, ↑L (b * c)⁆)) +\n (2 • ⁅↑L a, ↑L (b * c)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆)",
"state_before": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅↑L a, ↑L (a * a)⁆ + ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L a, ↑L (c * c)⁆ + ⁅↑L a, 2 • ↑L (a * b)⁆ + ⁅↑L a, 2 • ↑L (c * a)⁆ +\n ⁅↑L a, 2 • ↑L (b * c)⁆ +\n (⁅↑L b, ↑L (a * a)⁆ + ⁅↑L b, ↑L (b * b)⁆ + ⁅↑L b, ↑L (c * c)⁆ + ⁅↑L b, 2 • ↑L (a * b)⁆ +\n ⁅↑L b, 2 • ↑L (c * a)⁆ +\n ⁅↑L b, 2 • ↑L (b * c)⁆) +\n (⁅↑L c, ↑L (a * a)⁆ + ⁅↑L c, ↑L (b * b)⁆ + ⁅↑L c, ↑L (c * c)⁆ + ⁅↑L c, 2 • ↑L (a * b)⁆ + ⁅↑L c, 2 • ↑L (c * a)⁆ +\n ⁅↑L c, 2 • ↑L (b * c)⁆) =\n ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L b, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (a * b)⁆ + ⁅↑L b, ↑L (a * b)⁆) +\n (⁅↑L a, ↑L (c * c)⁆ + ⁅↑L c, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (c * a)⁆ + ⁅↑L c, ↑L (c * a)⁆)) +\n (⁅↑L b, ↑L (c * c)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • (⁅↑L b, ↑L (b * c)⁆ + ⁅↑L c, ↑L (b * c)⁆)) +\n (2 • ⁅↑L a, ↑L (b * c)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆)",
"tactic": "rw [(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq,\n (commute_lmul_lmul_sq c).lie_eq, zero_add, add_zero, add_zero]"
},
{
"state_after": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L a, ↑L (c * c)⁆ + 2 • ⁅↑L a, ↑L (a * b)⁆ + 2 • ⁅↑L a, ↑L (c * a)⁆ + 2 • ⁅↑L a, ↑L (b * c)⁆ +\n (⁅↑L b, ↑L (a * a)⁆ + ⁅↑L b, ↑L (c * c)⁆ + 2 • ⁅↑L b, ↑L (a * b)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ +\n 2 • ⁅↑L b, ↑L (b * c)⁆) +\n (⁅↑L c, ↑L (a * a)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆ + 2 • ⁅↑L c, ↑L (c * a)⁆ +\n 2 • ⁅↑L c, ↑L (b * c)⁆) =\n ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L b, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (a * b)⁆ + ⁅↑L b, ↑L (a * b)⁆) +\n (⁅↑L a, ↑L (c * c)⁆ + ⁅↑L c, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (c * a)⁆ + ⁅↑L c, ↑L (c * a)⁆)) +\n (⁅↑L b, ↑L (c * c)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • (⁅↑L b, ↑L (b * c)⁆ + ⁅↑L c, ↑L (b * c)⁆)) +\n (2 • ⁅↑L a, ↑L (b * c)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆)",
"state_before": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L a, ↑L (c * c)⁆ + ⁅↑L a, 2 • ↑L (a * b)⁆ + ⁅↑L a, 2 • ↑L (c * a)⁆ + ⁅↑L a, 2 • ↑L (b * c)⁆ +\n (⁅↑L b, ↑L (a * a)⁆ + ⁅↑L b, ↑L (c * c)⁆ + ⁅↑L b, 2 • ↑L (a * b)⁆ + ⁅↑L b, 2 • ↑L (c * a)⁆ +\n ⁅↑L b, 2 • ↑L (b * c)⁆) +\n (⁅↑L c, ↑L (a * a)⁆ + ⁅↑L c, ↑L (b * b)⁆ + ⁅↑L c, 2 • ↑L (a * b)⁆ + ⁅↑L c, 2 • ↑L (c * a)⁆ +\n ⁅↑L c, 2 • ↑L (b * c)⁆) =\n ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L b, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (a * b)⁆ + ⁅↑L b, ↑L (a * b)⁆) +\n (⁅↑L a, ↑L (c * c)⁆ + ⁅↑L c, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (c * a)⁆ + ⁅↑L c, ↑L (c * a)⁆)) +\n (⁅↑L b, ↑L (c * c)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • (⁅↑L b, ↑L (b * c)⁆ + ⁅↑L c, ↑L (b * c)⁆)) +\n (2 • ⁅↑L a, ↑L (b * c)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆)",
"tactic": "simp only [lie_nsmul]"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L a, ↑L (c * c)⁆ + 2 • ⁅↑L a, ↑L (a * b)⁆ + 2 • ⁅↑L a, ↑L (c * a)⁆ + 2 • ⁅↑L a, ↑L (b * c)⁆ +\n (⁅↑L b, ↑L (a * a)⁆ + ⁅↑L b, ↑L (c * c)⁆ + 2 • ⁅↑L b, ↑L (a * b)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ +\n 2 • ⁅↑L b, ↑L (b * c)⁆) +\n (⁅↑L c, ↑L (a * a)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆ + 2 • ⁅↑L c, ↑L (c * a)⁆ +\n 2 • ⁅↑L c, ↑L (b * c)⁆) =\n ⁅↑L a, ↑L (b * b)⁆ + ⁅↑L b, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (a * b)⁆ + ⁅↑L b, ↑L (a * b)⁆) +\n (⁅↑L a, ↑L (c * c)⁆ + ⁅↑L c, ↑L (a * a)⁆ + 2 • (⁅↑L a, ↑L (c * a)⁆ + ⁅↑L c, ↑L (c * a)⁆)) +\n (⁅↑L b, ↑L (c * c)⁆ + ⁅↑L c, ↑L (b * b)⁆ + 2 • (⁅↑L b, ↑L (b * c)⁆ + ⁅↑L c, ↑L (b * c)⁆)) +\n (2 • ⁅↑L a, ↑L (b * c)⁆ + 2 • ⁅↑L b, ↑L (c * a)⁆ + 2 • ⁅↑L c, ↑L (a * b)⁆)",
"tactic": "abel"
}
] |
[
230,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
9
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto
|
[
{
"state_after": "Ω : Type u_2\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\n⊢ Tendsto (toFiniteMeasure ∘ μs) F (𝓝 (toFiniteMeasure μ)) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑f ω) ∂↑(μs i)) F (𝓝 (∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ))",
"state_before": "Ω : Type u_2\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\n⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑f ω) ∂↑(μs i)) F (𝓝 (∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ))",
"tactic": "rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\n⊢ Tendsto (toFiniteMeasure ∘ μs) F (𝓝 (toFiniteMeasure μ)) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑f ω) ∂↑(μs i)) F (𝓝 (∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ))",
"tactic": "exact FiniteMeasure.tendsto_iff_forall_lintegral_tendsto"
}
] |
[
282,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Std/Data/List/Lemmas.lean
|
List.isPrefix.isInfix
|
[] |
[
1568,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1568,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
ModelWithCorners.closedEmbedding
|
[] |
[
298,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
11
] |
Mathlib/Data/List/Duplicate.lean
|
List.duplicate_iff_two_le_count
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nx : α\ninst✝ : DecidableEq α\n⊢ x ∈+ l ↔ 2 ≤ count x l",
"tactic": "simp [duplicate_iff_sublist, le_count_iff_replicate_sublist]"
}
] |
[
146,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/Hom/Equiv/Units/Basic.lean
|
Units.coe_mapEquiv
|
[] |
[
58,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.op_act_op_eq
|
[] |
[
278,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one
|
[
{
"state_after": "k : Type u_1\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.38533\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\nh : ∑ i in s, w i = 1\nb₁ b₂ : P\n⊢ b₁ -ᵥ b₂ + ∑ x in s, (w x • (p x -ᵥ b₁) - w x • (p x -ᵥ b₂)) = 0",
"state_before": "k : Type u_1\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.38533\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\nh : ∑ i in s, w i = 1\nb₁ b₂ : P\n⊢ ↑(weightedVSubOfPoint s p b₁) w +ᵥ b₁ = ↑(weightedVSubOfPoint s p b₂) w +ᵥ b₂",
"tactic": "erw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, ← @vsub_eq_zero_iff_eq V,\n vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, ← add_sub_assoc, add_comm, add_sub_assoc, ←\n sum_sub_distrib]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_4\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.38533\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\nh : ∑ i in s, w i = 1\nb₁ b₂ : P\n⊢ b₁ -ᵥ b₂ + ∑ x in s, w x • (b₂ -ᵥ b₁) = 0",
"tactic": "rw [← sum_smul, h, one_smul, vsub_add_vsub_cancel, vsub_self]"
}
] |
[
144,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Logic/Embedding/Set.lean
|
subtypeOrEquiv_symm_inl
|
[] |
[
143,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/CategoryTheory/Abelian/Exact.lean
|
CategoryTheory.Abelian.IsEquivalence.exact_iff
|
[
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nD : Type u₁\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : IsEquivalence F\n⊢ F.map f ≫ F.map g = 0 ∧ kernel.ι (F.map g) ≫ cokernel.π (F.map f) = 0 ↔\n F.map f ≫ F.map g = 0 ∧\n kernelComparison g F ≫ kernel.ι (F.map g) ≫ cokernel.π (F.map f) ≫ cokernelComparison f F = 0",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nD : Type u₁\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : IsEquivalence F\n⊢ Exact (F.map f) (F.map g) ↔ Exact f g",
"tactic": "simp only [exact_iff, ← F.map_eq_zero_iff, F.map_comp, Category.assoc, ←\n kernelComparison_comp_ι g F, ← π_comp_cokernelComparison f F]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nD : Type u₁\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : IsEquivalence F\n⊢ F.map f ≫ F.map g = 0 ∧ kernel.ι (F.map g) ≫ cokernel.π (F.map f) = 0 ↔\n F.map f ≫ F.map g = 0 ∧\n kernelComparison g F ≫ kernel.ι (F.map g) ≫ cokernel.π (F.map f) ≫ cokernelComparison f F = 0",
"tactic": "rw [IsIso.comp_left_eq_zero (kernelComparison g F), ← Category.assoc,\n IsIso.comp_right_eq_zero _ (cokernelComparison f F)]"
}
] |
[
114,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
8
] |
Mathlib/RingTheory/IntegralClosure.lean
|
map_isIntegral
|
[
{
"state_after": "case intro\nR : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P ∧ eval₂ (algebraMap R B) b P = 0\n⊢ IsIntegral R (↑f b)",
"state_before": "R : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nhb : IsIntegral R b\n⊢ IsIntegral R (↑f b)",
"tactic": "obtain ⟨P, hP⟩ := hb"
},
{
"state_after": "case intro\nR : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P ∧ eval₂ (algebraMap R B) b P = 0\n⊢ eval₂ (algebraMap R ((fun x => C) b)) (↑f b) P = 0",
"state_before": "case intro\nR : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P ∧ eval₂ (algebraMap R B) b P = 0\n⊢ IsIntegral R (↑f b)",
"tactic": "refine' ⟨P, hP.1, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P ∧ eval₂ (algebraMap R B) b P = 0\n⊢ eval₂ (algebraMap R ((fun x => C) b)) (↑f b) P = 0",
"tactic": "rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp,\n aeval_algHom_apply, aeval_map_algebraMap, aeval_def, hP.2, _root_.map_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_4\nA : Type u_5\nB✝ : Type ?u.113227\nS : Type ?u.113230\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B✝\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B✝\nf✝ : R →+* S\nB : Type u_1\nC : Type u_2\nF : Type u_3\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R C\ninst✝³ : IsScalarTower R A B\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower R A C\nb : B\ninst✝ : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P ∧ eval₂ (algebraMap R B) b P = 0\n⊢ ↑(aeval (↑f b)) P = ↑(aeval (↑f b)) (Polynomial.map (algebraMap R A) P)",
"tactic": "simp"
}
] |
[
133,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.pair_injective
|
[
{
"state_after": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z ∈ pair x y ↔ z ∈ pair x' y'\n⊢ x = x' ∧ y = y'",
"state_before": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\n⊢ x = x' ∧ y = y'",
"tactic": "have ae := ext_iff.1 H"
},
{
"state_after": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\n⊢ x = x' ∧ y = y'",
"state_before": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z ∈ pair x y ↔ z ∈ pair x' y'\n⊢ x = x' ∧ y = y'",
"tactic": "simp only [pair, mem_pair] at ae"
},
{
"state_after": "case inl\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyx : {x, y} = {x}\n⊢ x = x ∧ y = y'\n\ncase inr\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyy' : {x, y} = {x, y'}\n⊢ x = x ∧ y = y'",
"state_before": "x y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\n⊢ x = x ∧ y = y'",
"tactic": "obtain xyx | xyy' := (ae {x, y}).1 (by simp)"
},
{
"state_after": "case inl\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x'}\n⊢ x = x'\n\ncase inr\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x', y'}\n⊢ x = x'",
"state_before": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\n⊢ x = x'",
"tactic": "cases' (ae {x}).1 (by simp) with h h"
},
{
"state_after": "no goals",
"state_before": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\n⊢ {x} = {x} ∨ {x} = {x, y}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inl\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x'}\n⊢ x = x'",
"tactic": "exact singleton_injective h"
},
{
"state_after": "case inr\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x', y'}\nm : x' ∈ {x}\n⊢ x = x'",
"state_before": "case inr\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x', y'}\n⊢ x = x'",
"tactic": "have m : x' ∈ ({x} : ZFSet) := by simp [h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x', y'}\nm : x' ∈ {x}\n⊢ x = x'",
"tactic": "rw [mem_singleton.mp m]"
},
{
"state_after": "no goals",
"state_before": "x x' y y' : ZFSet\nH : pair x y = pair x' y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\nh : {x} = {x', y'}\n⊢ x' ∈ {x}",
"tactic": "simp [h]"
},
{
"state_after": "x y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\n⊢ x = y'",
"state_before": "x y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\n⊢ x = y → y = y'",
"tactic": "rintro rfl"
},
{
"state_after": "case inl\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'x : {x, y'} = {x}\n⊢ x = y'\n\ncase inr\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'xx : {x, y'} = {x, x}\n⊢ x = y'",
"state_before": "x y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\n⊢ x = y'",
"tactic": "cases' (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true_iff]) with xy'x xy'xx"
},
{
"state_after": "no goals",
"state_before": "x y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\n⊢ {x, y'} = {x} ∨ {x, y'} = {x, y'}",
"tactic": "simp only [eq_self_iff_true, or_true_iff]"
},
{
"state_after": "case inl\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'x : {x, y'} = {x}\n⊢ y' = x ∨ y' = y'",
"state_before": "case inl\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'x : {x, y'} = {x}\n⊢ x = y'",
"tactic": "rw [eq_comm, ← mem_singleton, ← xy'x, mem_pair]"
},
{
"state_after": "no goals",
"state_before": "case inl\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'x : {x, y'} = {x}\n⊢ y' = x ∨ y' = y'",
"tactic": "exact Or.inr rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nx y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'xx : {x, y'} = {x, x}\n⊢ x = y'",
"tactic": "simpa [eq_comm] using (ext_iff.1 xy'xx y').1 (by simp)"
},
{
"state_after": "no goals",
"state_before": "x y' : ZFSet\nH : pair x x = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, x} ↔ z = {x} ∨ z = {x, y'}\nxy'xx : {x, y'} = {x, x}\n⊢ y' ∈ {x, y'}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\n⊢ {x, y} = {x} ∨ {x, y} = {x, y}",
"tactic": "simp"
},
{
"state_after": "case inl\ny y' : ZFSet\nH : pair y y = pair y y'\nae : ∀ (z : ZFSet), z = {y} ∨ z = {y, y} ↔ z = {y} ∨ z = {y, y'}\nhe : y = y → y = y'\nxyx : {y, y} = {y}\n⊢ y = y ∧ y = y'",
"state_before": "case inl\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyx : {x, y} = {x}\n⊢ x = x ∧ y = y'",
"tactic": "obtain rfl := mem_singleton.mp ((ext_iff.1 xyx y).1 <| by simp)"
},
{
"state_after": "no goals",
"state_before": "case inl\ny y' : ZFSet\nH : pair y y = pair y y'\nae : ∀ (z : ZFSet), z = {y} ∨ z = {y, y} ↔ z = {y} ∨ z = {y, y'}\nhe : y = y → y = y'\nxyx : {y, y} = {y}\n⊢ y = y ∧ y = y'",
"tactic": "simp [he rfl]"
},
{
"state_after": "no goals",
"state_before": "x y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyx : {x, y} = {x}\n⊢ y ∈ {x, y}",
"tactic": "simp"
},
{
"state_after": "case inr.inl\ny y' : ZFSet\nH : pair y y = pair y y'\nae : ∀ (z : ZFSet), z = {y} ∨ z = {y, y} ↔ z = {y} ∨ z = {y, y'}\nhe : y = y → y = y'\nxyy' : {y, y} = {y, y'}\n⊢ y = y ∧ y = y'\n\ncase inr.inr\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyy' : {x, y} = {x, y'}\nyy' : y = y'\n⊢ x = x ∧ y = y'",
"state_before": "case inr\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyy' : {x, y} = {x, y'}\n⊢ x = x ∧ y = y'",
"tactic": "obtain rfl | yy' := mem_pair.mp ((ext_iff.1 xyy' y).1 <| by simp)"
},
{
"state_after": "no goals",
"state_before": "x y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyy' : {x, y} = {x, y'}\n⊢ y ∈ {x, y}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\ny y' : ZFSet\nH : pair y y = pair y y'\nae : ∀ (z : ZFSet), z = {y} ∨ z = {y, y} ↔ z = {y} ∨ z = {y, y'}\nhe : y = y → y = y'\nxyy' : {y, y} = {y, y'}\n⊢ y = y ∧ y = y'",
"tactic": "simp [he rfl]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nx y y' : ZFSet\nH : pair x y = pair x y'\nae : ∀ (z : ZFSet), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\nhe : x = y → y = y'\nxyy' : {x, y} = {x, y'}\nyy' : y = y'\n⊢ x = x ∧ y = y'",
"tactic": "simp [yy']"
}
] |
[
1330,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1311,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieAlgebra.nilpotent_of_nilpotent_quotient
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nh₂ : IsNilpotent R (L ⧸ I)\n⊢ LieModule.IsNilpotent R L (L ⧸ I)",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nh₂ : IsNilpotent R (L ⧸ I)\n⊢ IsNilpotent R L",
"tactic": "suffices LieModule.IsNilpotent R L (L ⧸ I) by\n exact LieModule.nilpotentOfNilpotentQuotient R L L h₁ this"
},
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nk : ℕ\nhk : lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥\n⊢ LieModule.IsNilpotent R L (L ⧸ I)",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nh₂ : IsNilpotent R (L ⧸ I)\n⊢ LieModule.IsNilpotent R L (L ⧸ I)",
"tactic": "obtain ⟨k, hk⟩ := h₂"
},
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nk : ℕ\nhk : lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥\n⊢ lowerCentralSeries R L (L ⧸ I) k = ⊥",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nk : ℕ\nhk : lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥\n⊢ LieModule.IsNilpotent R L (L ⧸ I)",
"tactic": "use k"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nk : ℕ\nhk : lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥\n⊢ lowerCentralSeries R L (L ⧸ I) k = ⊥",
"tactic": "simp [← LieSubmodule.coe_toSubmodule_eq_iff, coe_lowerCentralSeries_ideal_quot_eq, hk]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI : LieIdeal R L\nh₁ : I ≤ center R L\nh₂ : IsNilpotent R (L ⧸ I)\nthis : LieModule.IsNilpotent R L (L ⧸ I)\n⊢ IsNilpotent R L",
"tactic": "exact LieModule.nilpotentOfNilpotentQuotient R L L h₁ this"
}
] |
[
595,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
589,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
MonotoneOn.quasilinearOn
|
[] |
[
193,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean
|
AlgebraicTopology.DoldKan.MorphComponents.id_φ
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ (id X n).a +\n ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫\n SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n HomologicalComplex.Hom.f (P (n + 1)) (n + 1) + HomologicalComplex.Hom.f (Q (n + 1)) (n + 1)",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ φ (id X n) = 𝟙 (X.obj [n + 1].op)",
"tactic": "simp only [← P_add_Q_f (n + 1) (n + 1), φ]"
},
{
"state_after": "case e_a\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ (id X n).a = HomologicalComplex.Hom.f (P (n + 1)) (n + 1)\n\ncase e_a\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫ SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n HomologicalComplex.Hom.f (Q (n + 1)) (n + 1)",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ (id X n).a +\n ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫\n SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n HomologicalComplex.Hom.f (P (n + 1)) (n + 1) + HomologicalComplex.Hom.f (Q (n + 1)) (n + 1)",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_a\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫ SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n HomologicalComplex.Hom.f (Q (n + 1)) (n + 1)",
"tactic": ". exact Eq.trans (by congr ; simp) (decomposition_Q n (n + 1)).symm"
},
{
"state_after": "no goals",
"state_before": "case e_a\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ (id X n).a = HomologicalComplex.Hom.f (P (n + 1)) (n + 1)",
"tactic": "simp only [id, PInfty_f, P_f_idem]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫ SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n HomologicalComplex.Hom.f (Q (n + 1)) (n + 1)",
"tactic": "exact Eq.trans (by congr ; simp) (decomposition_Q n (n + 1)).symm"
},
{
"state_after": "case e_s\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ Finset.univ = Finset.filter (fun i => ↑i < n + 1) Finset.univ",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ ∑ x : Fin (n + 1),\n HomologicalComplex.Hom.f (P ↑x) (n + 1) ≫ SimplicialObject.δ X (Fin.succ (↑Fin.rev x)) ≫ b (id X n) (↑Fin.rev x) =\n ∑ i in Finset.filter (fun i => ↑i < n + 1) Finset.univ,\n HomologicalComplex.Hom.f (P ↑i) (n + 1) ≫\n SimplicialObject.δ X (Fin.succ (↑Fin.rev i)) ≫ SimplicialObject.σ X (↑Fin.rev i)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_s\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX X' : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\ng : X' ⟶ X\nh : Z ⟶ Z'\n⊢ Finset.univ = Finset.filter (fun i => ↑i < n + 1) Finset.univ",
"tactic": "simp"
}
] |
[
124,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_product_right'
|
[] |
[
691,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.trace_kroneckerTMul
|
[] |
[
530,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocMod_zsmul_add'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIocMod hp (m • p + a) b = m • p + toIocMod hp a b",
"tactic": "rw [add_comm, toIocMod_add_zsmul', add_comm]"
}
] |
[
447,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.ofList_subset
|
[
{
"state_after": "case nil\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nh : [] ⊆ l₂\n⊢ ofList [] ⊆ ofList l₂\n\ncase cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ :: tail✝ ⊆ l₂\n⊢ ofList (head✝ :: tail✝) ⊆ ofList l₂",
"state_before": "α : Type u_1\nl₁ l₂ : List (Lists α)\nh : l₁ ⊆ l₂\n⊢ ofList l₁ ⊆ ofList l₂",
"tactic": "induction' l₁ with _ _ l₁_ih"
},
{
"state_after": "case cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ :: tail✝ ⊆ l₂\n⊢ head✝ ∈ toList (ofList l₂)",
"state_before": "case cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ :: tail✝ ⊆ l₂\n⊢ ofList (head✝ :: tail✝) ⊆ ofList l₂",
"tactic": "refine' Subset.cons (Lists.Equiv.refl _) _ (l₁_ih (List.subset_of_cons_subset h))"
},
{
"state_after": "case cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ ∈ l₂ ∧ tail✝ ⊆ l₂\n⊢ head✝ ∈ toList (ofList l₂)",
"state_before": "case cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ :: tail✝ ⊆ l₂\n⊢ head✝ ∈ toList (ofList l₂)",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nhead✝ : Lists α\ntail✝ : List (Lists α)\nl₁_ih : tail✝ ⊆ l₂ → ofList tail✝ ⊆ ofList l₂\nh : head✝ ∈ l₂ ∧ tail✝ ⊆ l₂\n⊢ head✝ ∈ toList (ofList l₂)",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl₁ l₂ : List (Lists α)\nh✝ : l₁ ⊆ l₂\nh : [] ⊆ l₂\n⊢ ofList [] ⊆ ofList l₂",
"tactic": "exact Subset.nil"
}
] |
[
181,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean
|
IsFractionRing.num_mul_den_eq_num_mul_den_iff_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.148388\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.148594\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.148848\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : num A y * ↑(den A x) = num A x * ↑(den A y)\n⊢ x = y",
"tactic": "simpa only [mk'_num_den] using mk'_eq_of_eq' (S := K) h"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.148388\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.148594\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.148848\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : x = y\n⊢ num A y * ↑(den A x) = num A x * ↑(den A y)",
"tactic": "rw [h]"
}
] |
[
95,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.extract
|
[
{
"state_after": "case refl\nl m r : List Char\nit₂ : Iterator\nh₂ : ValidFor (List.reverse m ++ l) r it₂\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } it₂ = { data := m }",
"state_before": "l m r : List Char\nit₁ it₂ : Iterator\nh₁ : ValidFor l (m ++ r) it₁\nh₂ : ValidFor (List.reverse m ++ l) r it₂\n⊢ Iterator.extract it₁ it₂ = { data := m }",
"tactic": "cases h₁.out"
},
{
"state_after": "case refl.refl\nl m r : List Char\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh₂ :\n ValidFor (List.reverse m ++ l) r\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n⊢ Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } =\n { data := m }",
"state_before": "case refl\nl m r : List Char\nit₂ : Iterator\nh₂ : ValidFor (List.reverse m ++ l) r it₂\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } it₂ = { data := m }",
"tactic": "cases h₂.out"
},
{
"state_after": "case refl.refl\nl m r : List Char\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh₂ :\n ValidFor (List.reverse m ++ l) r\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n⊢ String.extract { data := List.reverse l ++ (m ++ r) } { byteIdx := utf8Len l } { byteIdx := utf8Len m + utf8Len l } =\n { data := m }",
"state_before": "case refl.refl\nl m r : List Char\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh₂ :\n ValidFor (List.reverse m ++ l) r\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n⊢ Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } =\n { data := m }",
"tactic": "simp [Iterator.extract, List.reverseAux_eq, Nat.not_lt.2 (Nat.le_add_left ..)]"
},
{
"state_after": "no goals",
"state_before": "case refl.refl\nl m r : List Char\nh₁ : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }\nh₂ :\n ValidFor (List.reverse m ++ l) r\n { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } }\n⊢ String.extract { data := List.reverse l ++ (m ++ r) } { byteIdx := utf8Len l } { byteIdx := utf8Len m + utf8Len l } =\n { data := m }",
"tactic": "simpa [Nat.add_comm] using extract_of_valid l.reverse m r"
}
] |
[
599,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
595,
1
] |
Mathlib/Topology/CompactOpen.lean
|
ContinuousMap.compactOpen_eq_sInf_induced
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ compactOpen ≤ ⨅ (s : Set α) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ (⨅ (s : Set α) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen) ≤ compactOpen",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ compactOpen = ⨅ (s : Set α) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ TopologicalSpace.generateFrom\n (⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}) ≤\n TopologicalSpace.generateFrom {m | ∃ s x u x, m = CompactOpen.gen s u}",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ (⨅ (s : Set α) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen) ≤ compactOpen",
"tactic": "simp only [← generateFrom_iUnion, induced_generateFrom_eq, ContinuousMap.compactOpen]"
},
{
"state_after": "case refine'_2.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ {m | ∃ s x u x, m = CompactOpen.gen s u} ⊆\n ⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ TopologicalSpace.generateFrom\n (⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}) ≤\n TopologicalSpace.generateFrom {m | ∃ s x u x, m = CompactOpen.gen s u}",
"tactic": "apply TopologicalSpace.generateFrom_anti"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ CompactOpen.gen s u ∈\n ⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"state_before": "case refine'_2.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ {m | ∃ s x u x, m = CompactOpen.gen s u} ⊆\n ⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"tactic": "rintro _ ⟨s, hs, u, hu, rfl⟩"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ ∃ i j, CompactOpen.gen s u ∈ preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"state_before": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ CompactOpen.gen s u ∈\n ⋃ (i : Set α) (_ : IsCompact i), preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"tactic": "rw [mem_iUnion₂]"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ restrict s ⁻¹' CompactOpen.gen univ u = CompactOpen.gen s u",
"state_before": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ ∃ i j, CompactOpen.gen s u ∈ preimage (restrict i) '' {m | ∃ s x u x, m = CompactOpen.gen s u}",
"tactic": "refine' ⟨s, hs, _, ⟨univ, isCompact_iff_isCompact_univ.mp hs, u, hu, rfl⟩, _⟩"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ f ∈ restrict s ⁻¹' CompactOpen.gen univ u ↔ f ∈ CompactOpen.gen s u",
"state_before": "case refine'_2.h.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\n⊢ restrict s ⁻¹' CompactOpen.gen univ u = CompactOpen.gen s u",
"tactic": "ext f"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ (fun a => (↑f ∘ Subtype.val) a) '' univ ⊆ u ↔ (fun a => ↑f a) '' s ⊆ u",
"state_before": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ f ∈ restrict s ⁻¹' CompactOpen.gen univ u ↔ f ∈ CompactOpen.gen s u",
"tactic": "simp only [CompactOpen.gen, mem_setOf_eq, mem_preimage, ContinuousMap.coe_restrict]"
},
{
"state_after": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ ↑f '' (Subtype.val '' univ) ⊆ u ↔ (fun a => ↑f a) '' s ⊆ u",
"state_before": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ (fun a => (↑f ∘ Subtype.val) a) '' univ ⊆ u ↔ (fun a => ↑f a) '' s ⊆ u",
"tactic": "rw [image_comp f ((↑) : s → α)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h.intro.intro.intro.intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsCompact s\nu : Set β\nhu : IsOpen u\nf : C(α, β)\n⊢ ↑f '' (Subtype.val '' univ) ⊆ u ↔ (fun a => ↑f a) '' s ⊆ u",
"tactic": "simp"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ ∀ (i : Set α), IsCompact i → compactOpen ≤ TopologicalSpace.induced (restrict i) compactOpen",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ compactOpen ≤ ⨅ (s : Set α) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen",
"tactic": "refine' le_iInf₂ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19379\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ ∀ (i : Set α), IsCompact i → compactOpen ≤ TopologicalSpace.induced (restrict i) compactOpen",
"tactic": "exact fun s _ => compactOpen_le_induced s"
}
] |
[
246,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_tensor
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ ((g ⊗ f) ≫ (α_ Z' Z Y').inv ⊗ 𝟙 Y) ≫ (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ ↑(tensorRightHomEquiv (X' ⊗ X) Y Y' (Z' ⊗ Z)).symm ((g ⊗ f) ≫ (α_ Z' Z Y').inv) =\n (α_ X' X Y).hom ≫ (g ⊗ ↑(tensorRightHomEquiv X Y Y' Z).symm f)",
"tactic": "dsimp [tensorRightHomEquiv]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (((g ⊗ f) ⊗ 𝟙 Y) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫ (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ ((g ⊗ f) ≫ (α_ Z' Z Y').inv ⊗ 𝟙 Y) ≫ (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "simp only [comp_tensor_id]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (((α_ X' X Y).hom ≫ (g ⊗ f ⊗ 𝟙 Y) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (((g ⊗ f) ⊗ 𝟙 Y) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫ (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "simp only [associator_conjugation]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y')) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (((α_ X' X Y).hom ≫ (g ⊗ f ⊗ 𝟙 Y) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom ≫ (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y') ≫ (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "slice_lhs 2 2 => rw [← tensor_id_comp_id_tensor]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y')) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫\n (g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y')) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫ (g ⊗ (f ⊗ 𝟙 Y) ≫ (α_ Z Y' Y).hom ≫ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "conv_rhs => rw [← tensor_id_comp_id_tensor, id_tensor_comp, id_tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 X ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (α_ Z' Z (Y' ⊗ Y)).hom ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (α_ Z' Z (𝟙_ C)).inv) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫\n (g ⊗ 𝟙 X ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (𝟙 (Z' ⊗ Z) ⊗ ε_ Y Y')) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫\n (g ⊗ 𝟙 (X ⊗ Y)) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "simp only [← tensor_id, associator_conjugation]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (g ⊗ f ⊗ 𝟙 Y) ≫\n (α_ Z' (Z ⊗ Y') Y).inv ≫\n ((α_ Z' Z Y').inv ⊗ 𝟙 Y) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom ≫ (α_ Z' Z (Y' ⊗ Y)).hom ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (𝟙 Z' ⊗ (ρ_ Z).hom) =\n (g ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (𝟙 Z' ⊗ (ρ_ Z).hom)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (α_ X' X Y).hom ≫\n ((((((g ⊗ 𝟙 X ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y)) ≫ (α_ Z' (Z ⊗ Y') Y).inv) ≫ ((α_ Z' Z Y').inv ⊗ 𝟙 Y)) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom) ≫\n (α_ Z' Z (Y' ⊗ Y)).hom ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (α_ Z' Z (𝟙_ C)).inv) ≫\n (ρ_ (Z' ⊗ Z)).hom =\n (α_ X' X Y).hom ≫\n (g ⊗ 𝟙 X ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ (𝟙 Z ⊗ ε_ Y Y') ≫ (ρ_ Z).hom)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Z ⊗ Y'\ng : X' ⟶ Z'\n⊢ (g ⊗ f ⊗ 𝟙 Y) ≫\n (α_ Z' (Z ⊗ Y') Y).inv ≫\n ((α_ Z' Z Y').inv ⊗ 𝟙 Y) ≫\n (α_ (Z' ⊗ Z) Y' Y).hom ≫ (α_ Z' Z (Y' ⊗ Y)).hom ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (𝟙 Z' ⊗ (ρ_ Z).hom) =\n (g ⊗ f ⊗ 𝟙 Y) ≫ (𝟙 Z' ⊗ (α_ Z Y' Y).hom) ≫ (𝟙 Z' ⊗ 𝟙 Z ⊗ ε_ Y Y') ≫ (𝟙 Z' ⊗ (ρ_ Z).hom)",
"tactic": "coherence"
}
] |
[
467,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Std/Classes/BEq.lean
|
beq_eq_false_iff_ne
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na b : α\n⊢ (a == b) = false ↔ ¬(a == b) = true",
"state_before": "α : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na b : α\n⊢ (a == b) = false ↔ a ≠ b",
"tactic": "rw [ne_eq, ← beq_iff_eq a b]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na b : α\n⊢ (a == b) = false ↔ ¬(a == b) = true",
"tactic": "cases a == b <;> decide"
}
] |
[
23,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
20,
9
] |
Mathlib/GroupTheory/GroupAction/Sigma.lean
|
Sigma.FaithfulSMul'
|
[] |
[
71,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
11
] |
Mathlib/Data/Complex/Cardinality.lean
|
not_countable_complex
|
[
{
"state_after": "⊢ ℵ₀ < 𝔠",
"state_before": "⊢ ¬Set.Countable Set.univ",
"tactic": "rw [← le_aleph0_iff_set_countable, not_le, mk_univ_complex]"
},
{
"state_after": "no goals",
"state_before": "⊢ ℵ₀ < 𝔠",
"tactic": "apply cantor"
}
] |
[
40,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Data/List/Count.lean
|
List.filter_beq
|
[] |
[
297,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.mem_iff
|
[] |
[
316,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
hasDerivAt_zpow
|
[] |
[
69,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.card_powerset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Multiset α\n⊢ ∀ (a : List α), ↑card (powerset (Quotient.mk (isSetoid α) a)) = 2 ^ ↑card (Quotient.mk (isSetoid α) a)",
"tactic": "simp"
}
] |
[
124,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.tmul_ite
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝¹⁷ : CommSemiring R\nR' : Type ?u.449877\ninst✝¹⁶ : Monoid R'\nR'' : Type ?u.449883\ninst✝¹⁵ : Semiring R''\nM : Type u_1\nN : Type u_2\nP✝ : Type ?u.449895\nQ : Type ?u.449898\nS : Type ?u.449901\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : AddCommMonoid P✝\ninst✝¹¹ : AddCommMonoid Q\ninst✝¹⁰ : AddCommMonoid S\ninst✝⁹ : Module R M\ninst✝⁸ : Module R N\ninst✝⁷ : Module R P✝\ninst✝⁶ : Module R Q\ninst✝⁵ : Module R S\ninst✝⁴ : DistribMulAction R' M\ninst✝³ : Module R'' M\ninst✝² : SMulCommClass R R' M\ninst✝¹ : SMulCommClass R R'' M\nx₁ : M\nx₂ : N\nP : Prop\ninst✝ : Decidable P\n⊢ (x₁ ⊗ₜ[R] if P then x₂ else 0) = if P then x₁ ⊗ₜ[R] x₂ else 0",
"tactic": "split_ifs <;> simp"
}
] |
[
382,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Std/Data/Int/DivMod.lean
|
Int.sub_ediv_of_dvd_sub
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\nhcab : c ∣ a - b\n⊢ (a - b) / c = a / c - b / c",
"tactic": "rw [← Int.add_sub_cancel ((a-b) / c), ← Int.add_ediv_of_dvd_left hcab, Int.sub_add_cancel]"
}
] |
[
812,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
810,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
Ring.two_ne_zero
|
[
{
"state_after": "R✝ : Type ?u.950230\nR : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\n⊢ ¬(ringChar R = 1 ∨ ringChar R = 2)",
"state_before": "R✝ : Type ?u.950230\nR : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\n⊢ 2 ≠ 0",
"tactic": "rw [Ne.def, (by norm_cast : (2 : R) = (2 : ℕ)), ringChar.spec, Nat.dvd_prime Nat.prime_two]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.950230\nR : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\n⊢ ¬(ringChar R = 1 ∨ ringChar R = 2)",
"tactic": "exact mt (or_iff_left hR).mp CharP.ringChar_ne_one"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.950230\nR : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\n⊢ 2 = ↑2",
"tactic": "norm_cast"
}
] |
[
622,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
11
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.monic_geom_sum_X
|
[
{
"state_after": "R : Type u\nS : Type ?u.60633\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ Monic (∑ i in range n, X ^ i)",
"state_before": "R : Type u\nS : Type ?u.60633\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\n⊢ Monic (∑ i in range n, X ^ i)",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u\nS : Type ?u.60633\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ 0 < natDegree X",
"state_before": "R : Type u\nS : Type ?u.60633\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ Monic (∑ i in range n, X ^ i)",
"tactic": "apply monic_X.geom_sum _ hn"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.60633\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ 0 < natDegree X",
"tactic": "simp only [natDegree_X, zero_lt_one]"
}
] |
[
255,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.numerator_isUnit
|
[] |
[
376,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat
|
[
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n⊢ ∀ (t : Set α), ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n⊢ IsCaratheodory m (⋃ (i : ℕ), s i)",
"tactic": "apply (isCaratheodory_iff_le' m).mpr"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n⊢ ∀ (t : Set α), ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"tactic": "intro t"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"tactic": "have hp : m (t ∩ ⋃ i, s i) ≤ ⨆ n, m (t ∩ ⋃ i < n, s i) := by\n convert m.iUnion fun i => t ∩ s i using 1\n . simp [inter_iUnion]\n . simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"tactic": "refine' le_trans (add_le_add_right hp _) _"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ (⨆ (b : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < b), s i) + ↑m (t \\ ⋃ (i : ℕ), s i)) ≤ ↑m t",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) + ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m t",
"tactic": "rw [ENNReal.iSup_add]"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\nn : ℕ\n⊢ ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m (t \\ ⋃ (i : ℕ) (_ : i < n), s i)",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\n⊢ (⨆ (b : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < b), s i) + ↑m (t \\ ⋃ (i : ℕ), s i)) ≤ ↑m t",
"tactic": "refine'\n iSup_le fun n =>\n le_trans (add_le_add_left _ _) (ge_of_eq (isCaratheodory_iUnion_lt m (fun i _ => h i) _))"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\nn : ℕ\n⊢ (⋃ (i : ℕ) (_ : i < n), s i) ⊆ ⋃ (i : ℕ), s i",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\nn : ℕ\n⊢ ↑m (t \\ ⋃ (i : ℕ), s i) ≤ ↑m (t \\ ⋃ (i : ℕ) (_ : i < n), s i)",
"tactic": "refine' m.mono (diff_subset_diff_right _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nhp : ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)\nn : ℕ\n⊢ (⋃ (i : ℕ) (_ : i < n), s i) ⊆ ⋃ (i : ℕ), s i",
"tactic": "exact iUnion₂_subset fun i _ => subset_iUnion _ i"
},
{
"state_after": "case h.e'_3\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) = ↑m (⋃ (i : ℕ), t ∩ s i)\n\ncase h.e'_4\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) = ∑' (i : ℕ), ↑m (t ∩ s i)",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) ≤ ⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)",
"tactic": "convert m.iUnion fun i => t ∩ s i using 1"
},
{
"state_after": "case h.e'_4\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) = ∑' (i : ℕ), ↑m (t ∩ s i)",
"state_before": "case h.e'_3\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) = ↑m (⋃ (i : ℕ), t ∩ s i)\n\ncase h.e'_4\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) = ∑' (i : ℕ), ↑m (t ∩ s i)",
"tactic": ". simp [inter_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) = ∑' (i : ℕ), ↑m (t ∩ s i)",
"tactic": ". simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ↑m (t ∩ ⋃ (i : ℕ), s i) = ↑m (⋃ (i : ℕ), t ∩ s i)",
"tactic": "simp [inter_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ (⨆ (n : ℕ), ↑m (t ∩ ⋃ (i : ℕ) (_ : i < n), s i)) = ∑' (i : ℕ), ↑m (t ∩ s i)",
"tactic": "simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]"
}
] |
[
1020,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1006,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.map_le_gfp
|
[] |
[
132,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
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