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/BigOperators/Finprod.lean
|
finprod_mem_iUnion
|
[
{
"state_after": "case intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nt : ι → Set α\nh : Pairwise (Disjoint on t)\nht : ∀ (i : ι), Set.Finite (t i)\nval✝ : Fintype ι\n⊢ (∏ᶠ (a : α) (_ : a ∈ ⋃ (i : ι), t i), f a) = ∏ᶠ (i : ι) (a : α) (_ : a ∈ t i), f a",
"state_before": "α : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nt : ι → Set α\nh : Pairwise (Disjoint on t)\nht : ∀ (i : ι), Set.Finite (t i)\n⊢ (∏ᶠ (a : α) (_ : a ∈ ⋃ (i : ι), t i), f a) = ∏ᶠ (i : ι) (a : α) (_ : a ∈ t i), f a",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ (∏ᶠ (a : α) (_ : a ∈ ⋃ (i : ι), (fun i => ↑(t i)) i), f a) = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a",
"state_before": "case intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nt : ι → Set α\nh : Pairwise (Disjoint on t)\nht : ∀ (i : ι), Set.Finite (t i)\nval✝ : Fintype ι\n⊢ (∏ᶠ (a : α) (_ : a ∈ ⋃ (i : ι), t i), f a) = ∏ᶠ (i : ι) (a : α) (_ : a ∈ t i), f a",
"tactic": "lift t to ι → Finset α using ht"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ ∏ x : ι, ∏ i in t x, f i = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a\n\ncase intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ PairwiseDisjoint ↑Finset.univ fun x => t x",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ (∏ᶠ (a : α) (_ : a ∈ ⋃ (i : ι), (fun i => ↑(t i)) i), f a) = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a",
"tactic": "rw [← biUnion_univ, ← Finset.coe_univ, ← Finset.coe_biUnion, finprod_mem_coe_finset,\n Finset.prod_biUnion]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ ∏ x : ι, ∏ i in t x, f i = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a",
"tactic": "simp only [finprod_mem_coe_finset, finprod_eq_prod_of_fintype]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.365704\nι : Type u_1\nG : Type ?u.365710\nM : Type u_3\nN : Type ?u.365716\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf g : α → M\na b : α\ns t✝ : Set α\ninst✝ : Finite ι\nval✝ : Fintype ι\nt : ι → Finset α\nh : Pairwise (Disjoint on fun i => ↑(t i))\n⊢ PairwiseDisjoint ↑Finset.univ fun x => t x",
"tactic": "exact fun x _ y _ hxy => Finset.disjoint_coe.1 (h hxy)"
}
] |
[
1065,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1057,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_insert
|
[] |
[
339,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
TensorProduct.Algebra.smul_def
|
[] |
[
1208,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
borel_eq_top_of_countable
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\n⊢ MeasurableSet (⋃ (x : α) (_ : x ∈ s), {x})",
"state_before": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\n⊢ borel α = ⊤",
"tactic": "refine' top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\n⊢ ∀ (b : α), b ∈ s → MeasurableSet {b}",
"state_before": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\n⊢ MeasurableSet (⋃ (x : α) (_ : x ∈ s), {x})",
"tactic": "apply MeasurableSet.biUnion s.to_countable"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ MeasurableSet {x}",
"state_before": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\n⊢ ∀ (b : α), b ∈ s → MeasurableSet {b}",
"tactic": "intro x _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ MeasurableSet ({x}ᶜ)",
"state_before": "α : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ MeasurableSet {x}",
"tactic": "apply MeasurableSet.of_compl"
},
{
"state_after": "case h.a\nα : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ {x}ᶜ ∈ {s | IsOpen s}",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ MeasurableSet ({x}ᶜ)",
"tactic": "apply GenerateMeasurable.basic"
},
{
"state_after": "no goals",
"state_before": "case h.a\nα : Type u_1\nβ : Type ?u.26837\nγ : Type ?u.26840\nγ₂ : Type ?u.26843\nδ : Type ?u.26846\nι : Sort y\ns✝ t u : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : Countable α\ns : Set α\nx✝ : MeasurableSet s\nx : α\na✝ : x ∈ s\n⊢ {x}ᶜ ∈ {s | IsOpen s}",
"tactic": "exact isClosed_singleton.isOpen_compl"
}
] |
[
77,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivAt.csin
|
[] |
[
214,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.coprod_neBot_left
|
[] |
[
509,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.nat_coe_snd
|
[] |
[
280,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
9
] |
Mathlib/Data/Set/Pointwise/BigOperators.lean
|
Set.list_prod_singleton
|
[] |
[
124,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
uniformity_basis_edist_inv_two_pow
|
[] |
[
263,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasStrictDerivAt_exp_smul_const'
|
[] |
[
419,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.kernelSubobject_zero
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : HasKernel f\nA B : C\n⊢ IsIso (kernel.ι 0)",
"tactic": "infer_instance"
}
] |
[
194,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_eq_of_eq_add
|
[] |
[
1124,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1123,
11
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
CategoryTheory.Presheaf.isSheaf_iff_isLimit
|
[] |
[
177,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Algebra/Homology/Additive.lean
|
ChainComplex.single₀MapHomologicalComplex_inv_app_succ
|
[] |
[
392,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_add
|
[] |
[
377,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.isRotated_nil_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl l' : List α\n⊢ [] ~r l ↔ [] = l",
"tactic": "rw [isRotated_comm, isRotated_nil_iff, eq_comm]"
}
] |
[
497,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.BlankExtends.trans
|
[
{
"state_after": "case intro.intro\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ : List Γ\ni j : ℕ\n⊢ BlankExtends l₁ (l₁ ++ List.replicate i default ++ List.replicate j default)",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\n⊢ BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃",
"tactic": "rintro ⟨i, rfl⟩ ⟨j, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ : List Γ\ni j : ℕ\n⊢ BlankExtends l₁ (l₁ ++ List.replicate i default ++ List.replicate j default)",
"tactic": "exact ⟨i + j, by simp [List.replicate_add]⟩"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nl₁ : List Γ\ni j : ℕ\n⊢ l₁ ++ List.replicate i default ++ List.replicate j default = l₁ ++ List.replicate (i + j) default",
"tactic": "simp [List.replicate_add]"
}
] |
[
90,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.swap_right_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.218711\nP : Type u_3\nP' : Type ?u.218717\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n⊢ Wbtw R x z y ↔ y = z",
"tactic": "rw [← wbtw_swap_right_iff R x, and_iff_right h]"
}
] |
[
396,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Std/Data/Int/DivMod.lean
|
Int.ediv_mul_cancel
|
[] |
[
704,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
703,
11
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.stalk_mono_of_mono
|
[] |
[
510,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.congr'
|
[] |
[
304,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.testBit_lnot
|
[
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ testBit (lnot ↑n) k = !testBit (↑n) k",
"tactic": "simp [lnot, testBit]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ testBit (lnot -[n+1]) k = !testBit -[n+1] k",
"tactic": "simp [lnot, testBit]"
}
] |
[
367,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_product_right
|
[] |
[
437,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.map_sub
|
[] |
[
741,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_mul_const_Ioi
|
[] |
[
519,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.sUnion_empty
|
[
{
"state_after": "case a\nz✝ : ZFSet\n⊢ z✝ ∈ ⋃₀ ∅ ↔ z✝ ∈ ∅",
"state_before": "⊢ ⋃₀ ∅ = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nz✝ : ZFSet\n⊢ z✝ ∈ ⋃₀ ∅ ↔ z✝ ∈ ∅",
"tactic": "simp"
}
] |
[
1070,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1068,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.sizeOf_eq_sizeOf
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : SizeOf α\nl₁ l₂ : List α\nh : l₁ ~ l₂\n⊢ sizeOf l₁ = sizeOf l₂",
"tactic": "induction h with | nil => rfl\n| cons _ _ h_sz₁₂ => simp [h_sz₁₂]\n| swap => simp [add_left_comm]\n| trans _ _ h_sz₁₂ h_sz₂₃ => simp [h_sz₁₂, h_sz₂₃]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : SizeOf α\nl₁ l₂ : List α\n⊢ sizeOf [] = sizeOf []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : SizeOf α\nl₁ l₂ : List α\nx✝ : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nh_sz₁₂ : sizeOf l₁✝ = sizeOf l₂✝\n⊢ sizeOf (x✝ :: l₁✝) = sizeOf (x✝ :: l₂✝)",
"tactic": "simp [h_sz₁₂]"
},
{
"state_after": "no goals",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : SizeOf α\nl₁ l₂ : List α\nx✝ y✝ : α\nl✝ : List α\n⊢ sizeOf (y✝ :: x✝ :: l✝) = sizeOf (x✝ :: y✝ :: l✝)",
"tactic": "simp [add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : SizeOf α\nl₁ l₂ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nh_sz₁₂ : sizeOf l₁✝ = sizeOf l₂✝\nh_sz₂₃ : sizeOf l₂✝ = sizeOf l₃✝\n⊢ sizeOf l₁✝ = sizeOf l₃✝",
"tactic": "simp [h_sz₁₂, h_sz₂₃]"
}
] |
[
322,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.pi_caratheodory
|
[
{
"state_after": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\n⊢ ∀ (i : ι),\n MeasurableSpace.comap (fun b => b i) ((fun a => inst✝ a) i) ≤\n OuterMeasure.caratheodory (OuterMeasure.pi fun i => ↑(μ i))",
"state_before": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\n⊢ MeasurableSpace.pi ≤ OuterMeasure.caratheodory (OuterMeasure.pi fun i => ↑(μ i))",
"tactic": "refine' iSup_le _"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((a : ι) → α a)\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"state_before": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\n⊢ ∀ (i : ι),\n MeasurableSpace.comap (fun b => b i) ((fun a => inst✝ a) i) ≤\n OuterMeasure.caratheodory (OuterMeasure.pi fun i => ↑(μ i))",
"tactic": "intro i s hs"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((a : ι) → α a)\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"state_before": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((a : ι) → α a)\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"tactic": "rw [MeasurableSpace.comap] at hs"
},
{
"state_after": "case intro.intro\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\n⊢ MeasurableSet ((fun b => b i) ⁻¹' s)",
"state_before": "ι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((a : ι) → α a)\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"tactic": "rcases hs with ⟨s, hs, rfl⟩"
},
{
"state_after": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\n⊢ ∀ (t : Set ((a : ι) → α a)),\n piPremeasure (fun i => ↑(μ i)) (t ∩ (fun b => b i) ⁻¹' s) +\n piPremeasure (fun i => ↑(μ i)) (t \\ (fun b => b i) ⁻¹' s) ≤\n piPremeasure (fun i => ↑(μ i)) t",
"state_before": "case intro.intro\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\n⊢ MeasurableSet ((fun b => b i) ⁻¹' s)",
"tactic": "apply boundedBy_caratheodory"
},
{
"state_after": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ piPremeasure (fun i => ↑(μ i)) (t ∩ (fun b => b i) ⁻¹' s) +\n piPremeasure (fun i => ↑(μ i)) (t \\ (fun b => b i) ⁻¹' s) ≤\n piPremeasure (fun i => ↑(μ i)) t",
"state_before": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\n⊢ ∀ (t : Set ((a : ι) → α a)),\n piPremeasure (fun i => ↑(μ i)) (t ∩ (fun b => b i) ⁻¹' s) +\n piPremeasure (fun i => ↑(μ i)) (t \\ (fun b => b i) ⁻¹' s) ≤\n piPremeasure (fun i => ↑(μ i)) t",
"tactic": "intro t"
},
{
"state_after": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∏ x : ι, ↑↑(μ x) (eval x '' (t ∩ (fun b => b i) ⁻¹' s)) + ∏ x : ι, ↑↑(μ x) (eval x '' (t \\ (fun b => b i) ⁻¹' s)) ≤\n ∏ x : ι, ↑↑(μ x) (eval x '' t)",
"state_before": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ piPremeasure (fun i => ↑(μ i)) (t ∩ (fun b => b i) ⁻¹' s) +\n piPremeasure (fun i => ↑(μ i)) (t \\ (fun b => b i) ⁻¹' s) ≤\n piPremeasure (fun i => ↑(μ i)) t",
"tactic": "simp_rw [piPremeasure]"
},
{
"state_after": "case intro.intro.hs.refine'_1\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ↑↑(μ i) (eval i '' (t ∩ (fun b => b i) ⁻¹' s)) + ↑↑(μ i) (eval i '' (t \\ (fun b => b i) ⁻¹' s)) ≤\n ↑↑(μ i) (eval i '' t)\n\ncase intro.intro.hs.refine'_2\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∀ (j : ι), j ∈ Finset.univ → j ≠ i → ↑↑(μ j) (eval j '' (t ∩ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)\n\ncase intro.intro.hs.refine'_3\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∀ (j : ι), j ∈ Finset.univ → j ≠ i → ↑↑(μ j) (eval j '' (t \\ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"state_before": "case intro.intro.hs\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∏ x : ι, ↑↑(μ x) (eval x '' (t ∩ (fun b => b i) ⁻¹' s)) + ∏ x : ι, ↑↑(μ x) (eval x '' (t \\ (fun b => b i) ⁻¹' s)) ≤\n ∏ x : ι, ↑↑(μ x) (eval x '' t)",
"tactic": "refine' Finset.prod_add_prod_le' (Finset.mem_univ i) _ _ _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hs.refine'_1\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ↑↑(μ i) (eval i '' (t ∩ (fun b => b i) ⁻¹' s)) + ↑↑(μ i) (eval i '' (t \\ (fun b => b i) ⁻¹' s)) ≤\n ↑↑(μ i) (eval i '' t)",
"tactic": "simp [image_inter_preimage, image_diff_preimage, measure_inter_add_diff _ hs, le_refl]"
},
{
"state_after": "case intro.intro.hs.refine'_2\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ ↑↑(μ j) (eval j '' (t ∩ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"state_before": "case intro.intro.hs.refine'_2\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∀ (j : ι), j ∈ Finset.univ → j ≠ i → ↑↑(μ j) (eval j '' (t ∩ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"tactic": "rintro j - _"
},
{
"state_after": "case intro.intro.hs.refine'_2.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ eval j '' (t ∩ (fun b => b i) ⁻¹' s) ⊆ eval j '' t",
"state_before": "case intro.intro.hs.refine'_2\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ ↑↑(μ j) (eval j '' (t ∩ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"tactic": "apply mono'"
},
{
"state_after": "case intro.intro.hs.refine'_2.h.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ t ∩ (fun b => b i) ⁻¹' s ⊆ t",
"state_before": "case intro.intro.hs.refine'_2.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ eval j '' (t ∩ (fun b => b i) ⁻¹' s) ⊆ eval j '' t",
"tactic": "apply image_subset"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hs.refine'_2.h.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ t ∩ (fun b => b i) ⁻¹' s ⊆ t",
"tactic": "apply inter_subset_left"
},
{
"state_after": "case intro.intro.hs.refine'_3\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ ↑↑(μ j) (eval j '' (t \\ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"state_before": "case intro.intro.hs.refine'_3\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\n⊢ ∀ (j : ι), j ∈ Finset.univ → j ≠ i → ↑↑(μ j) (eval j '' (t \\ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"tactic": "rintro j - _"
},
{
"state_after": "case intro.intro.hs.refine'_3.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ eval j '' (t \\ (fun b => b i) ⁻¹' s) ⊆ eval j '' t",
"state_before": "case intro.intro.hs.refine'_3\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ ↑↑(μ j) (eval j '' (t \\ (fun b => b i) ⁻¹' s)) ≤ ↑↑(μ j) (eval j '' t)",
"tactic": "apply mono'"
},
{
"state_after": "case intro.intro.hs.refine'_3.h.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ t \\ (fun b => b i) ⁻¹' s ⊆ t",
"state_before": "case intro.intro.hs.refine'_3.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ eval j '' (t \\ (fun b => b i) ⁻¹' s) ⊆ eval j '' t",
"tactic": "apply image_subset"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hs.refine'_3.h.h\nι : Type u_1\nι' : Type ?u.2502507\nα : ι → Type u_2\ninst✝¹ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((fun i => α i) i)\nhs : MeasurableSet s\nt : Set ((a : ι) → α a)\nj : ι\na✝ : j ≠ i\n⊢ t \\ (fun b => b i) ⁻¹' s ⊆ t",
"tactic": "apply diff_subset"
}
] |
[
301,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Analysis/VonNeumannAlgebra/Basic.lean
|
VonNeumannAlgebra.mem_commutant_iff
|
[] |
[
141,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Std/Data/List/Lemmas.lean
|
List.diff_nil
|
[] |
[
1492,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1492,
9
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.sin_sub_sin
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(sin x - sin y) = ↑(2 * sin ((x - y) / 2) * cos ((x + y) / 2))",
"tactic": "simp [sin_sub_sin]"
}
] |
[
1214,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
8
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.mem_comap
|
[] |
[
283,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.diff
|
[] |
[
806,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
805,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.indep_of_indep_of_le_right
|
[] |
[
199,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupNorm.apply_one
|
[] |
[
887,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
886,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.refl_symm
|
[] |
[
374,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/AlgebraicGeometry/PresheafedSpace.lean
|
AlgebraicGeometry.PresheafedSpace.mk_coe
|
[] |
[
83,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.inf_prod
|
[] |
[
1688,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1687,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Icc.coe_eq_one
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ x = 1 ↔ ↑x = 1",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ ↑x = 1 ↔ x = 1",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ x = 1 ↔ ↑x = 1",
"tactic": "exact Subtype.ext_iff"
}
] |
[
92,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.set_lintegral_max
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1032041\nγ : Type ?u.1032044\nδ : Type ?u.1032047\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\n⊢ MeasurableSet {x | g x < f x}\n\nα : Type u_1\nβ : Type ?u.1032041\nγ : Type ?u.1032044\nδ : Type ?u.1032047\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\n⊢ MeasurableSet {x | f x ≤ g x}",
"state_before": "α : Type u_1\nβ : Type ?u.1032041\nγ : Type ?u.1032044\nδ : Type ?u.1032047\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\n⊢ (∫⁻ (x : α) in s, max (f x) (g x) ∂μ) =\n (∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ) + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ",
"tactic": "rw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1032041\nγ : Type ?u.1032044\nδ : Type ?u.1032047\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\n⊢ MeasurableSet {x | g x < f x}\n\nα : Type u_1\nβ : Type ?u.1032041\nγ : Type ?u.1032044\nδ : Type ?u.1032047\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\n⊢ MeasurableSet {x | f x ≤ g x}",
"tactic": "exacts [measurableSet_lt hg hf, measurableSet_le hf hg]"
}
] |
[
1257,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1253,
1
] |
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
SymplecticGroup.coe_J
|
[] |
[
132,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.congrArg_mpr_hom_left
|
[
{
"state_after": "case refl\nC : Type u₁\ninst✝ : Category C\nX Z : C\nq : X ⟶ Z\n⊢ Eq.mpr (_ : (X ⟶ Z) = (X ⟶ Z)) q = eqToHom (_ : X = X) ≫ q",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y Z : C\np : X = Y\nq : Y ⟶ Z\n⊢ Eq.mpr (_ : (X ⟶ Z) = (Y ⟶ Z)) q = eqToHom p ≫ q",
"tactic": "cases p"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u₁\ninst✝ : Category C\nX Z : C\nq : X ⟶ Z\n⊢ Eq.mpr (_ : (X ⟶ Z) = (X ⟶ Z)) q = eqToHom (_ : X = X) ≫ q",
"tactic": "simp"
}
] |
[
94,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.inducing
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.57703\nδ : Type ?u.57706\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\n⊢ Inducing (↑(Homeomorph.symm h) ∘ ↑h)",
"tactic": "simp only [symm_comp_self, inducing_id]"
}
] |
[
229,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
11
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.eval_gcd_eq_zero
|
[] |
[
334,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift_mono
|
[] |
[
101,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.IsColimit.homIso_natural
|
[] |
[
588,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Odd.of_dvd_nat
|
[] |
[
367,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
|
intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\n⊢ IntervalIntegrable (F' x₀) μ a b ∧\n HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x t‖ ≤ bound t\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x t) (F' x t) x\n⊢ IntervalIntegrable (F' x₀) μ a b ∧\n HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀",
"tactic": "rw [← ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF_int : IntervalIntegrable (F x₀) μ a b\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nbound_integrable : IntervalIntegrable bound μ a b\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\n⊢ IntervalIntegrable (F' x₀) μ a b ∧\n HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀",
"tactic": "simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧\n HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀",
"tactic": "simp only [intervalIntegral_eq_integral_uIoc]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\nthis : Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : ℝ) in Ι a b, F n a ∂μ) (∫ (a : ℝ) in Ι a b, F' x₀ a ∂μ) x₀\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧\n HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧\n HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀",
"tactic": "have := hasDerivAt_integral_of_dominated_loc_of_deriv_le ε_pos hF_meas hF_int hF'_meas h_bound\n bound_integrable h_diff"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nμ : MeasureTheory.Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.684895\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))\nhF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))\nh_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x\nh_diff :\n ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x₀) (Ι a b)\nbound_integrable : IntegrableOn bound (Ι a b)\nthis : Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : ℝ) in Ι a b, F n a ∂μ) (∫ (a : ℝ) in Ι a b, F' x₀ a ∂μ) x₀\n⊢ IntegrableOn (F' x₀) (Ι a b) ∧\n HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)\n ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀",
"tactic": "exact ⟨this.1, this.2.const_smul _⟩"
}
] |
[
115,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
8
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_const_id_atTop
|
[] |
[
1885,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1884,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
Continuous.piecewise
|
[] |
[
1211,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1208,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullback.condition
|
[] |
[
1214,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1212,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
mulSalemSpencer_mul_left_iff₀
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.103189\nα : Type u_1\nβ : Type ?u.103195\n𝕜 : Type ?u.103198\nE : Type ?u.103201\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nha : a ≠ 0\nhs : MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)\nb c d : α\nhb : b ∈ s\nhc : c ∈ s\nhd : d ∈ s\nh : b * c = d * d\n⊢ a * b * (a * c) = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d",
"tactic": "rw [mul_mul_mul_comm, h, mul_mul_mul_comm]"
}
] |
[
257,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
TopHom.cancel_right
|
[] |
[
288,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Down.mem_compression
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ (∃ a_1, a_1 ∈ 𝒜 ∧ erase a_1 a = s) ∧ ¬s ∈ 𝒜 ↔ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ insert a s ∈ 𝒜 ∧ ¬s ∈ 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝓓 a 𝒜 ↔ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ ¬s ∈ 𝒜 ∧ insert a s ∈ 𝒜",
"tactic": "simp_rw [compression, mem_disjUnion, mem_filter, mem_image, and_comm (a := (¬ s ∈ 𝒜))]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nhs : ¬s ∈ 𝒜\n⊢ (∃ a_1, a_1 ∈ 𝒜 ∧ erase a_1 a = s) → insert a s ∈ 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ (∃ a_1, a_1 ∈ 𝒜 ∧ erase a_1 a = s) ∧ ¬s ∈ 𝒜 ↔ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ insert a s ∈ 𝒜 ∧ ¬s ∈ 𝒜",
"tactic": "refine'\n or_congr_right\n (and_congr_left fun hs =>\n ⟨_, fun h => ⟨_, h, erase_insert <| insert_ne_self.1 <| ne_of_mem_of_not_mem h hs⟩⟩)"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\nt : Finset α\nht : t ∈ 𝒜\nhs : ¬erase t a ∈ 𝒜\n⊢ insert a (erase t a) ∈ 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nhs : ¬s ∈ 𝒜\n⊢ (∃ a_1, a_1 ∈ 𝒜 ∧ erase a_1 a = s) → insert a s ∈ 𝒜",
"tactic": "rintro ⟨t, ht, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\nt : Finset α\nht : t ∈ 𝒜\nhs : ¬erase t a ∈ 𝒜\n⊢ insert a (erase t a) ∈ 𝒜",
"tactic": "rwa [insert_erase (erase_ne_self.1 (ne_of_mem_of_not_mem ht hs).symm)]"
}
] |
[
174,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/Normed/Ring/Seminorm.lean
|
RingNorm.apply_one
|
[] |
[
221,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.pushoutIsoUnopPullback_inv_fst
|
[
{
"state_after": "case a\nC : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : X ⟶ Y\ninst✝¹ : HasPushout f g\ninst✝ : HasPullback f.op g.op\n⊢ ((pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst).unop = pushout.inl.op.unop",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : X ⟶ Y\ninst✝¹ : HasPushout f g\ninst✝ : HasPullback f.op g.op\n⊢ (pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst = pushout.inl.op",
"tactic": "apply Quiver.Hom.unop_inj"
},
{
"state_after": "case a\nC : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : X ⟶ Y\ninst✝¹ : HasPushout f g\ninst✝ : HasPullback f.op g.op\n⊢ pullback.fst.unop ≫ (pushoutIsoUnopPullback f g).inv = pushout.inl",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : X ⟶ Y\ninst✝¹ : HasPushout f g\ninst✝ : HasPullback f.op g.op\n⊢ ((pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst).unop = pushout.inl.op.unop",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : X ⟶ Y\ninst✝¹ : HasPushout f g\ninst✝ : HasPullback f.op g.op\n⊢ pullback.fst.unop ≫ (pushoutIsoUnopPullback f g).inv = pushout.inl",
"tactic": "rw [← pushoutIsoUnopPullback_inl_hom, Category.assoc, Iso.hom_inv_id, Category.comp_id]"
}
] |
[
699,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
lowerCentralSeries_pi_of_finite
|
[
{
"state_after": "G : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\nn : ℕ\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\n⊢ lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n",
"state_before": "G : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\nn : ℕ\n⊢ lowerCentralSeries ((i : η) → Gs i) n = pi Set.univ fun i => lowerCentralSeries (Gs i) n",
"tactic": "let pi := fun f : ∀ i, Subgroup (Gs i) => Subgroup.pi Set.univ f"
},
{
"state_after": "case zero\nG : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\n⊢ lowerCentralSeries ((i : η) → Gs i) Nat.zero = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) Nat.zero\n\ncase succ\nG : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\nn : ℕ\nih : lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n\n⊢ lowerCentralSeries ((i : η) → Gs i) (Nat.succ n) =\n Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) (Nat.succ n)",
"state_before": "G : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\nn : ℕ\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\n⊢ lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nG : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\n⊢ lowerCentralSeries ((i : η) → Gs i) Nat.zero = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) Nat.zero",
"tactic": "simp [pi_top]"
},
{
"state_after": "no goals",
"state_before": "case succ\nG : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\nn : ℕ\nih : lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n\n⊢ lowerCentralSeries ((i : η) → Gs i) (Nat.succ n) =\n Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) (Nat.succ n)",
"tactic": "calc\n lowerCentralSeries (∀ i, Gs i) n.succ = ⁅lowerCentralSeries (∀ i, Gs i) n, ⊤⁆ := rfl\n _ = ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆ := by rw [ih]\n _ = ⁅pi fun i => lowerCentralSeries (Gs i) n, pi fun i => ⊤⁆ := by simp [pi_top]\n _ = pi fun i => ⁅lowerCentralSeries (Gs i) n, ⊤⁆ := (commutator_pi_pi_of_finite _ _)\n _ = pi fun i => lowerCentralSeries (Gs i) n.succ := rfl"
},
{
"state_after": "no goals",
"state_before": "G : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\nn : ℕ\nih : lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n\n⊢ ⁅lowerCentralSeries ((i : η) → Gs i) n, ⊤⁆ = ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆",
"tactic": "rw [ih]"
},
{
"state_after": "no goals",
"state_before": "G : Type ?u.343966\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nη : Type u_1\nGs : η → Type u_2\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : Finite η\npi : ((i : η) → Subgroup (Gs i)) → Subgroup ((i : η) → Gs i) := fun f => Subgroup.pi Set.univ f\nn : ℕ\nih : lowerCentralSeries ((i : η) → Gs i) n = Subgroup.pi Set.univ fun i => lowerCentralSeries (Gs i) n\n⊢ ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆ = ⁅pi fun i => lowerCentralSeries (Gs i) n, pi fun i => ⊤⁆",
"tactic": "simp [pi_top]"
}
] |
[
799,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/GroupTheory/Coset.lean
|
leftCoset_mem_leftCoset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\na : α\nha : a ∈ s\n⊢ ∀ (x : α), x ∈ a *l ↑s ↔ x ∈ ↑s",
"tactic": "simp [mem_leftCoset_iff, mul_mem_cancel_left (s.inv_mem ha)]"
}
] |
[
224,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/CompleteLattice.lean
|
Monotone.iInf_comp_eq
|
[] |
[
1001,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.bijOn
|
[] |
[
256,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
11
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.inf_dedup
|
[] |
[
162,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean
|
Set.subset_center_units
|
[] |
[
89,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Order/Filter/Extr.lean
|
isMinOn_dual_iff
|
[] |
[
234,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Order/BoundedOrder.lean
|
min_bot_left
|
[] |
[
835,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_update_same
|
[] |
[
798,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Tactic/CategoryTheory/Elementwise.lean
|
Tactic.Elementwise.forall_congr_forget_Type
|
[] |
[
46,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Topology/LocallyConstant/Algebra.lean
|
LocallyConstant.charFn_eq_one
|
[] |
[
115,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
HasFDerivWithinAt.cexp
|
[] |
[
130,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.Lex.lt_def
|
[] |
[
344,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_symm_eq_iff_map_eq
|
[
{
"state_after": "case mp\nG : Type u_2\nG' : Type ?u.263490\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.263499\ninst✝² : AddGroup A\nH✝ : Subgroup G\nk : Set G\nN : Type u_1\ninst✝¹ : Group N\nP : Type ?u.263526\ninst✝ : Group P\nH : Subgroup N\ne : G ≃* N\n⊢ map (↑e) (map (↑(MulEquiv.symm e)) H) = H\n\ncase mpr\nG : Type u_2\nG' : Type ?u.263490\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.263499\ninst✝² : AddGroup A\nH K : Subgroup G\nk : Set G\nN : Type u_1\ninst✝¹ : Group N\nP : Type ?u.263526\ninst✝ : Group P\ne : G ≃* N\n⊢ map (↑(MulEquiv.symm e)) (map (↑e) K) = K",
"state_before": "G : Type u_2\nG' : Type ?u.263490\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.263499\ninst✝² : AddGroup A\nH✝ K : Subgroup G\nk : Set G\nN : Type u_1\ninst✝¹ : Group N\nP : Type ?u.263526\ninst✝ : Group P\nH : Subgroup N\ne : G ≃* N\n⊢ map (↑(MulEquiv.symm e)) H = K ↔ map (↑e) K = H",
"tactic": "constructor <;> rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nG : Type u_2\nG' : Type ?u.263490\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.263499\ninst✝² : AddGroup A\nH✝ : Subgroup G\nk : Set G\nN : Type u_1\ninst✝¹ : Group N\nP : Type ?u.263526\ninst✝ : Group P\nH : Subgroup N\ne : G ≃* N\n⊢ map (↑e) (map (↑(MulEquiv.symm e)) H) = H",
"tactic": "rw [map_map, ← MulEquiv.coe_monoidHom_trans, MulEquiv.symm_trans_self,\n MulEquiv.coe_monoidHom_refl, map_id]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nG : Type u_2\nG' : Type ?u.263490\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.263499\ninst✝² : AddGroup A\nH K : Subgroup G\nk : Set G\nN : Type u_1\ninst✝¹ : Group N\nP : Type ?u.263526\ninst✝ : Group P\ne : G ≃* N\n⊢ map (↑(MulEquiv.symm e)) (map (↑e) K) = K",
"tactic": "rw [map_map, ← MulEquiv.coe_monoidHom_trans, MulEquiv.self_trans_symm,\n MulEquiv.coe_monoidHom_refl, map_id]"
}
] |
[
1491,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1485,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
deriv_cexp
|
[] |
[
113,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Std/Data/List/Lemmas.lean
|
List.suffix_of_suffix_length_le
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nl₁ l₃ l₂ : List α✝\nh₁ : l₁ <:+ l₃\nh₂ : l₂ <:+ l₃\nll : length l₁ ≤ length l₂\n⊢ length (reverse l₁) ≤ length (reverse l₂)",
"tactic": "simp [ll]"
}
] |
[
1674,
90
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1671,
1
] |
Mathlib/Data/Matrix/Rank.lean
|
Matrix.rank_le_height
|
[] |
[
159,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_list_sum
|
[] |
[
256,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.subset_zeroLocus_vanishingIdeal
|
[] |
[
236,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
Decidable.mul_lt_mul''
|
[
{
"state_after": "α : Type u\nβ : Type ?u.59742\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nh1 : a < c\nh2 : b < d\nh3 : 0 ≤ a\nh4 : 0 ≤ b\nb0 : 0 = b\n⊢ 0 < c * d",
"state_before": "α : Type u\nβ : Type ?u.59742\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nh1 : a < c\nh2 : b < d\nh3 : 0 ≤ a\nh4 : 0 ≤ b\nb0 : 0 = b\n⊢ a * b < c * d",
"tactic": "rw [← b0, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.59742\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nh1 : a < c\nh2 : b < d\nh3 : 0 ≤ a\nh4 : 0 ≤ b\nb0 : 0 = b\n⊢ 0 < c * d",
"tactic": "exact mul_pos (h3.trans_lt h1) (h4.trans_lt h2)"
}
] |
[
556,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
11
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_mapTermRel_id
|
[
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) falsum) v' xs ↔ Realize falsum v xs\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (equal t₁✝ t₂✝)) v' xs ↔ Realize (equal t₁✝ t₂✝) v xs\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (rel R✝ ts✝)) v' xs ↔ Realize (rel R✝ ts✝) v xs\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ {xs : Fin n✝ → M}, Realize (mapTermRel ft fr (fun x => id) f₁✝) v' xs ↔ Realize f₁✝ v xs\nih2 : ∀ {xs : Fin n✝ → M}, Realize (mapTermRel ft fr (fun x => id) f₂✝) v' xs ↔ Realize f₂✝ v xs\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (f₁✝ ⟹ f₂✝)) v' xs ↔ Realize (f₁✝ ⟹ f₂✝) v xs\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih : ∀ {xs : Fin (n✝ + 1) → M}, Realize (mapTermRel ft fr (fun x => id) f✝) v' xs ↔ Realize f✝ v xs\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (∀'f✝)) v' xs ↔ Realize (∀'f✝) v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nv' : β → M\nxs : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\n⊢ Realize (mapTermRel ft fr (fun x => id) φ) v' xs ↔ Realize φ v xs",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) falsum) v' xs ↔ Realize falsum v xs",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (equal t₁✝ t₂✝)) v' xs ↔ Realize (equal t₁✝ t₂✝) v xs",
"tactic": "simp [mapTermRel, Realize, h1]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (rel R✝ ts✝)) v' xs ↔ Realize (rel R✝ ts✝) v xs",
"tactic": "simp [mapTermRel, Realize, h1, h2]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ {xs : Fin n✝ → M}, Realize (mapTermRel ft fr (fun x => id) f₁✝) v' xs ↔ Realize f₁✝ v xs\nih2 : ∀ {xs : Fin n✝ → M}, Realize (mapTermRel ft fr (fun x => id) f₂✝) v' xs ↔ Realize f₂✝ v xs\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (f₁✝ ⟹ f₂✝)) v' xs ↔ Realize (f₁✝ ⟹ f₂✝) v xs",
"tactic": "simp [mapTermRel, Realize, ih1, ih2]"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.71406\nP : Type ?u.71409\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : Structure L' M\nft : (n : ℕ) → Term L (α ⊕ Fin n) → Term L' (β ⊕ Fin n)\nfr : (n : ℕ) → Relations L n → Relations L' n\nn : ℕ\nv : α → M\nv' : β → M\nxs✝ : Fin n → M\nh1 : ∀ (n : ℕ) (t : Term L (α ⊕ Fin n)) (xs : Fin n → M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : ∀ (n : ℕ) (R : Relations L n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih : ∀ {xs : Fin (n✝ + 1) → M}, Realize (mapTermRel ft fr (fun x => id) f✝) v' xs ↔ Realize f✝ v xs\nxs : Fin n✝ → M\n⊢ Realize (mapTermRel ft fr (fun x => id) (∀'f✝)) v' xs ↔ Realize (∀'f✝) v xs",
"tactic": "simp only [mapTermRel, Realize, ih, id.def]"
}
] |
[
379,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.hasBasis_nhds_of_ne_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.66801\nβ : Type ?u.66804\nγ : Type ?u.66807\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nxt : x ≠ ⊤\n⊢ HasBasis (𝓝 x) (fun x => 0 < x) fun ε => Icc (x - ε) (x + ε)",
"tactic": "simpa only [pos_iff_ne_zero] using hasBasis_nhds_of_ne_top' xt"
}
] |
[
254,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_foldr_inf
|
[
{
"state_after": "case nil\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.45572\nP : Type ?u.45575\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ []) v xs ↔ ∀ (φ : BoundedFormula L α n), φ ∈ [] → Realize φ v xs\n\ncase cons\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.45572\nP : Type ?u.45575\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ✝ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nv : α → M\nxs : Fin n → M\nφ : BoundedFormula L α n\nl : List (BoundedFormula L α n)\nih : Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l) v xs ↔ ∀ (φ : BoundedFormula L α n), φ ∈ l → Realize φ v xs\n⊢ Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ (φ :: l)) v xs ↔\n ∀ (φ_1 : BoundedFormula L α n), φ_1 ∈ φ :: l → Realize φ_1 v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.45572\nP : Type ?u.45575\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nl : List (BoundedFormula L α n)\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l) v xs ↔ ∀ (φ : BoundedFormula L α n), φ ∈ l → Realize φ v xs",
"tactic": "induction' l with φ l ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.45572\nP : Type ?u.45575\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ []) v xs ↔ ∀ (φ : BoundedFormula L α n), φ ∈ [] → Realize φ v xs",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.45572\nP : Type ?u.45575\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ✝ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nv : α → M\nxs : Fin n → M\nφ : BoundedFormula L α n\nl : List (BoundedFormula L α n)\nih : Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l) v xs ↔ ∀ (φ : BoundedFormula L α n), φ ∈ l → Realize φ v xs\n⊢ Realize (List.foldr (fun x x_1 => x ⊓ x_1) ⊤ (φ :: l)) v xs ↔\n ∀ (φ_1 : BoundedFormula L α n), φ_1 ∈ φ :: l → Realize φ_1 v xs",
"tactic": "simp [ih]"
}
] |
[
295,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.BinaryBicone.binary_cofan_inl_toCocone
|
[] |
[
1040,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1039,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_toAddSubmonoid
|
[] |
[
489,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhds_basis_abs_sub_lt
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\na : α\n⊢ HasBasis (⨅ (r : α) (_ : r > 0), 𝓟 {b | abs (b - a) < r}) (fun ε => 0 < ε) fun ε => {b | abs (b - a) < ε}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\na : α\n⊢ HasBasis (𝓝 a) (fun ε => 0 < ε) fun ε => {b | abs (b - a) < ε}",
"tactic": "simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\na x : α\nhx : x > 0\ny : α\nhy : y > 0\n⊢ ∃ k, k > 0 ∧ {b | abs (b - a) < k} ⊆ {b | abs (b - a) < x} ∧ {b | abs (b - a) < k} ⊆ {b | abs (b - a) < y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\na : α\n⊢ HasBasis (⨅ (r : α) (_ : r > 0), 𝓟 {b | abs (b - a) < r}) (fun ε => 0 < ε) fun ε => {b | abs (b - a) < ε}",
"tactic": "refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrderedAddCommGroup α\ninst✝¹ : OrderTopology α\nl : Filter β\nf g : β → α\ninst✝ : NoMaxOrder α\na x : α\nhx : x > 0\ny : α\nhy : y > 0\n⊢ ∃ k, k > 0 ∧ {b | abs (b - a) < k} ⊆ {b | abs (b - a) < x} ∧ {b | abs (b - a) < k} ⊆ {b | abs (b - a) < y}",
"tactic": "exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩"
}
] |
[
1915,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1910,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_inter_union_subset
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.103900\nβ : Type u_2\nβ' : Type ?u.103906\nγ : Type ?u.103909\nγ' : Type ?u.103912\nδ : Type ?u.103915\nδ' : Type ?u.103918\nε : Type ?u.103921\nε' : Type ?u.103924\nζ : Type ?u.103927\nζ' : Type ?u.103930\nν : Type ?u.103933\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (↑s ∩ ↑t) (↑s ∪ ↑t) ⊆ image2 f ↑s ↑t",
"state_before": "α : Type u_1\nα' : Type ?u.103900\nβ : Type u_2\nβ' : Type ?u.103906\nγ : Type ?u.103909\nγ' : Type ?u.103912\nδ : Type ?u.103915\nδ' : Type ?u.103918\nε : Type ?u.103921\nε' : Type ?u.103924\nζ : Type ?u.103927\nζ' : Type ?u.103930\nν : Type ?u.103933\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ ↑(image₂ f (s ∩ t) (s ∪ t)) ⊆ ↑(image₂ f s t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.103900\nβ : Type u_2\nβ' : Type ?u.103906\nγ : Type ?u.103909\nγ' : Type ?u.103912\nδ : Type ?u.103915\nδ' : Type ?u.103918\nε : Type ?u.103921\nε' : Type ?u.103924\nζ : Type ?u.103927\nζ' : Type ?u.103930\nν : Type ?u.103933\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (↑s ∩ ↑t) (↑s ∪ ↑t) ⊆ image2 f ↑s ↑t",
"tactic": "exact image2_inter_union_subset hf"
}
] |
[
541,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.isTheta_const_smul_left
|
[] |
[
283,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_eq_coe_nat
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nn : ℕ\n⊢ ↑C ↑n = ↑n",
"tactic": "induction n <;> simp [Nat.succ_eq_add_one, *]"
}
] |
[
269,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Algebra/Lie/BaseChange.lean
|
LieAlgebra.ExtendScalars.bracket_lie_smul
|
[
{
"state_after": "case refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ⁅0, a • y⁆ = a • ⁅0, y⁆\n\ncase refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ∀ (x : A) (y_1 : L), ⁅x ⊗ₜ[R] y_1, a • y⁆ = a • ⁅x ⊗ₜ[R] y_1, y⁆\n\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ∀ (x y_1 : A ⊗[R] L), ⁅x, a • y⁆ = a • ⁅x, y⁆ → ⁅y_1, a • y⁆ = a • ⁅y_1, y⁆ → ⁅x + y_1, a • y⁆ = a • ⁅x + y_1, y⁆",
"state_before": "R : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ⁅x, a • y⁆ = a • ⁅x, y⁆",
"tactic": "refine' x.induction_on _ _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ⁅0, a • y⁆ = a • ⁅0, y⁆",
"tactic": "simp only [zero_lie, smul_zero]"
},
{
"state_after": "case refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, y⁆",
"state_before": "case refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ∀ (x : A) (y_1 : L), ⁅x ⊗ₜ[R] y_1, a • y⁆ = a • ⁅x ⊗ₜ[R] y_1, y⁆",
"tactic": "intro a₁ l₁"
},
{
"state_after": "case refine'_2.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • 0⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, 0⁆\n\ncase refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ∀ (x : A) (y : L), ⁅a₁ ⊗ₜ[R] l₁, a • x ⊗ₜ[R] y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x ⊗ₜ[R] y⁆\n\ncase refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ∀ (x y : A ⊗[R] L),\n ⁅a₁ ⊗ₜ[R] l₁, a • x⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x⁆ →\n ⁅a₁ ⊗ₜ[R] l₁, a • y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, y⁆ → ⁅a₁ ⊗ₜ[R] l₁, a • (x + y)⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x + y⁆",
"state_before": "case refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, y⁆",
"tactic": "refine' y.induction_on _ _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • 0⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, 0⁆",
"tactic": "simp only [lie_zero, smul_zero]"
},
{
"state_after": "case refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\na₂ : A\nl₂ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • a₂ ⊗ₜ[R] l₂⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, a₂ ⊗ₜ[R] l₂⁆",
"state_before": "case refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ∀ (x : A) (y : L), ⁅a₁ ⊗ₜ[R] l₁, a • x ⊗ₜ[R] y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x ⊗ₜ[R] y⁆",
"tactic": "intro a₂ l₂"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\na₂ : A\nl₂ : L\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • a₂ ⊗ₜ[R] l₂⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, a₂ ⊗ₜ[R] l₂⁆",
"tactic": "simp only [bracket_def, bracket', TensorProduct.smul_tmul', mul_left_comm a₁ a a₂,\n TensorProduct.curry_apply, LinearMap.mul'_apply, Algebra.id.smul_eq_mul,\n Function.comp_apply, LinearEquiv.coe_coe, LinearMap.coe_comp, TensorProduct.map_tmul,\n TensorProduct.tensorTensorTensorComm_tmul]"
},
{
"state_after": "case refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\nz₁ z₂ : A ⊗[R] L\nh₁ : ⁅a₁ ⊗ₜ[R] l₁, a • z₁⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₁⁆\nh₂ : ⁅a₁ ⊗ₜ[R] l₁, a • z₂⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₂⁆\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • (z₁ + z₂)⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₁ + z₂⁆",
"state_before": "case refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\n⊢ ∀ (x y : A ⊗[R] L),\n ⁅a₁ ⊗ₜ[R] l₁, a • x⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x⁆ →\n ⁅a₁ ⊗ₜ[R] l₁, a • y⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, y⁆ → ⁅a₁ ⊗ₜ[R] l₁, a • (x + y)⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, x + y⁆",
"tactic": "intro z₁ z₂ h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\na₁ : A\nl₁ : L\nz₁ z₂ : A ⊗[R] L\nh₁ : ⁅a₁ ⊗ₜ[R] l₁, a • z₁⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₁⁆\nh₂ : ⁅a₁ ⊗ₜ[R] l₁, a • z₂⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₂⁆\n⊢ ⁅a₁ ⊗ₜ[R] l₁, a • (z₁ + z₂)⁆ = a • ⁅a₁ ⊗ₜ[R] l₁, z₁ + z₂⁆",
"tactic": "simp only [h₁, h₂, smul_add, lie_add]"
},
{
"state_after": "case refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y z₁ z₂ : A ⊗[R] L\nh₁ : ⁅z₁, a • y⁆ = a • ⁅z₁, y⁆\nh₂ : ⁅z₂, a • y⁆ = a • ⁅z₂, y⁆\n⊢ ⁅z₁ + z₂, a • y⁆ = a • ⁅z₁ + z₂, y⁆",
"state_before": "case refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y : A ⊗[R] L\n⊢ ∀ (x y_1 : A ⊗[R] L), ⁅x, a • y⁆ = a • ⁅x, y⁆ → ⁅y_1, a • y⁆ = a • ⁅y_1, y⁆ → ⁅x + y_1, a • y⁆ = a • ⁅x + y_1, y⁆",
"tactic": "intro z₁ z₂ h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\na : A\nx y z₁ z₂ : A ⊗[R] L\nh₁ : ⁅z₁, a • y⁆ = a • ⁅z₁, y⁆\nh₂ : ⁅z₂, a • y⁆ = a • ⁅z₂, y⁆\n⊢ ⁅z₁ + z₂, a • y⁆ = a • ⁅z₁ + z₂, y⁆",
"tactic": "simp only [h₁, h₂, smul_add, add_lie]"
}
] |
[
131,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
9
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.FinMeasSupp.of_map
|
[] |
[
1215,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.left_inr
|
[] |
[
145,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.bound
|
[] |
[
294,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Logic/Basic.lean
|
dite_ne_left_iff
|
[
{
"state_after": "α : Sort u_1\nβ : Sort ?u.35535\nσ : α → Sort ?u.35531\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\n⊢ (∃ x, ¬B x = a) ↔ ∃ h, a ≠ B h",
"state_before": "α : Sort u_1\nβ : Sort ?u.35535\nσ : α → Sort ?u.35531\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\n⊢ dite P (fun x => a) B ≠ a ↔ ∃ h, a ≠ B h",
"tactic": "rw [Ne.def, dite_eq_left_iff, not_forall]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort ?u.35535\nσ : α → Sort ?u.35531\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\n⊢ (∃ x, ¬B x = a) ↔ ∃ h, a ≠ B h",
"tactic": "exact exists_congr fun h ↦ by rw [ne_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort ?u.35535\nσ : α → Sort ?u.35531\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\nh : ¬P\n⊢ ¬B h = a ↔ a ≠ B h",
"tactic": "rw [ne_comm]"
}
] |
[
1161,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1159,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_pi_pi_le
|
[] |
[
221,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_down
|
[
{
"state_after": "α : Type ?u.113662\nβ : Type ?u.113665\nγ : Type ?u.113668\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\nb : Ordinal\nh : b ≤ lift a\n⊢ card b ≤ card (lift a)",
"state_before": "α : Type ?u.113662\nβ : Type ?u.113665\nγ : Type ?u.113668\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\nb : Ordinal\nh : b ≤ lift a\n⊢ card b ≤ Cardinal.lift (card a)",
"tactic": "rw [lift_card]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.113662\nβ : Type ?u.113665\nγ : Type ?u.113668\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\nb : Ordinal\nh : b ≤ lift a\n⊢ card b ≤ card (lift a)",
"tactic": "exact card_le_card h"
}
] |
[
815,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Data/List/Infix.lean
|
List.prefix_rfl
|
[] |
[
60,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
|
AlgebraicTopology.DoldKan.QInfty_comp_PInfty
|
[
{
"state_after": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (QInfty ≫ PInfty) n = HomologicalComplex.Hom.f 0 n",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ QInfty ≫ PInfty = 0",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (QInfty ≫ PInfty) n = HomologicalComplex.Hom.f 0 n",
"tactic": "apply QInfty_f_comp_PInfty_f"
}
] |
[
157,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/MvPolynomial/PDeriv.lean
|
MvPolynomial.pderiv_eq_zero_of_not_mem_vars
|
[] |
[
113,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Dense.exists_le'
|
[
{
"state_after": "case pos\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : IsBot x\n⊢ ∃ y, y ∈ s ∧ y ≤ x\n\ncase neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ¬IsBot x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"tactic": "by_cases hx : IsBot x"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : IsBot x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"tactic": "exact ⟨x, hbot x hx, le_rfl⟩"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ∃ x_1, x_1 < x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ¬IsBot x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"tactic": "simp only [IsBot, not_forall, not_le] at hx"
},
{
"state_after": "case neg.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ∃ x_1, x_1 < x\ny : α\nhys : y ∈ s\nhy : y < x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ∃ x_1, x_1 < x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"tactic": "rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\ns : Set α\nhs : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nx : α\nhx : ∃ x_1, x_1 < x\ny : α\nhys : y ∈ s\nhy : y < x\n⊢ ∃ y, y ∈ s ∧ y ≤ x",
"tactic": "exact ⟨y, hys, hy.le⟩"
}
] |
[
781,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Algebra/Invertible.lean
|
mul_right_eq_iff_eq_mul_invOf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ a * c = b ↔ a = b * ⅟c",
"tactic": "rw [← mul_right_inj_of_invertible (c := ⅟c), mul_mul_invOf_self_cancel]"
}
] |
[
321,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.BoundedFormula.induction_on_exists_not
|
[] |
[
635,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.normalize.reduced'
|
[
{
"state_after": "num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = Nat.gcd (Int.natAbs num) den\n⊢ Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)",
"state_before": "num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = Nat.gcd (Int.natAbs num) den\n⊢ Nat.coprime (Int.natAbs (num / ↑g)) (den / g)",
"tactic": "rw [← Int.div_eq_ediv_of_dvd (e ▸ Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]"
},
{
"state_after": "no goals",
"state_before": "num : Int\nden g : Nat\nden_nz : den ≠ 0\ne : g = Nat.gcd (Int.natAbs num) den\n⊢ Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)",
"tactic": "exact normalize.reduced den_nz e"
}
] |
[
18,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
15,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.ResidueField.mapEquiv_trans
|
[] |
[
468,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
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