file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.mem_closure_pair
|
[
{
"state_after": "M : Type ?u.264602\nA✝ : Type ?u.264605\nB : Type ?u.264608\nA : Type u_1\ninst✝ : CommMonoid A\na b c : A\n⊢ (∃ y, y ∈ closure {a} ∧ ∃ z, z ∈ closure {b} ∧ y * z = c) ↔ ∃ m n, a ^ m * b ^ n = c",
"state_before": "M : Type ?u.264602\nA✝ : Type ?u.264605\nB : Type ?u.264608\nA : Type u_1\ninst✝ : CommMonoid A\na b c : A\n⊢ c ∈ closure {a, b} ↔ ∃ m n, a ^ m * b ^ n = c",
"tactic": "rw [← Set.singleton_union, Submonoid.closure_union, mem_sup]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.264602\nA✝ : Type ?u.264605\nB : Type ?u.264608\nA : Type u_1\ninst✝ : CommMonoid A\na b c : A\n⊢ (∃ y, y ∈ closure {a} ∧ ∃ z, z ∈ closure {b} ∧ y * z = c) ↔ ∃ m n, a ^ m * b ^ n = c",
"tactic": "simp_rw [mem_closure_singleton, exists_exists_eq_and]"
}
] |
[
689,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Std/Logic.lean
|
Or.comm
|
[] |
[
261,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
261,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.symm_symm
|
[
{
"state_after": "case mk\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ntoFun✝ : α → Option β\ninvFun✝ : β → Option α\ninv✝ : ∀ (a : α) (b : β), a ∈ invFun✝ b ↔ b ∈ toFun✝ a\n⊢ PEquiv.symm (PEquiv.symm { toFun := toFun✝, invFun := invFun✝, inv := inv✝ }) =\n { toFun := toFun✝, invFun := invFun✝, inv := inv✝ }",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\n⊢ PEquiv.symm (PEquiv.symm f) = f",
"tactic": "cases f"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ntoFun✝ : α → Option β\ninvFun✝ : β → Option α\ninv✝ : ∀ (a : α) (b : β), a ∈ invFun✝ b ↔ b ∈ toFun✝ a\n⊢ PEquiv.symm (PEquiv.symm { toFun := toFun✝, invFun := invFun✝, inv := inv✝ }) =\n { toFun := toFun✝, invFun := invFun✝, inv := inv✝ }",
"tactic": "rfl"
}
] |
[
138,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
one_lt_sq_iff_one_lt_abs
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.270901\nA : Type ?u.270904\nG : Type ?u.270907\nM : Type ?u.270910\nR : Type u_1\ninst✝ : LinearOrderedRing R\nx✝ y x : R\n⊢ 1 < x ^ 2 ↔ 1 < abs x",
"tactic": "simpa only [one_pow, abs_one] using @sq_lt_sq _ _ 1 x"
}
] |
[
746,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/CategoryTheory/Limits/Fubini.lean
|
CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_inv_π_π
|
[
{
"state_after": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ ((limitIsoLimitCurryCompLim G).inv ≫\n (HasLimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor ⋙ G))).inv ≫\n (limitIsoLimitCurryCompLim (Prod.swap K J ⋙ G)).hom) ≫\n limit.π (curry.obj (Prod.swap K J ⋙ G) ⋙ lim) k ≫ limit.π ((curry.obj (Prod.swap K J ⋙ G)).obj k) j =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"state_before": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ (limitCurrySwapCompLimIsoLimitCurryCompLim G).inv ≫\n limit.π (curry.obj (Prod.swap K J ⋙ G) ⋙ lim) k ≫ limit.π ((curry.obj (Prod.swap K J ⋙ G)).obj k) j =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"tactic": "dsimp [limitCurrySwapCompLimIsoLimitCurryCompLim]"
},
{
"state_after": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ (limitIsoLimitCurryCompLim G).inv ≫\n limit.π G ((Prod.braiding K J).functor.obj (k, j)) ≫ (𝟙 ((Prod.braiding K J).functor ⋙ G)).app (k, j) =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"state_before": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ ((limitIsoLimitCurryCompLim G).inv ≫\n (HasLimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor ⋙ G))).inv ≫\n (limitIsoLimitCurryCompLim (Prod.swap K J ⋙ G)).hom) ≫\n limit.π (curry.obj (Prod.swap K J ⋙ G) ⋙ lim) k ≫ limit.π ((curry.obj (Prod.swap K J ⋙ G)).obj k) j =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"tactic": "simp only [Iso.refl_hom, Prod.braiding_counitIso_hom_app, Limits.HasLimit.isoOfEquivalence_inv_π,\n Iso.refl_inv, limitIsoLimitCurryCompLim_hom_π_π, eqToIso_refl, Category.assoc]"
},
{
"state_after": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ (limitIsoLimitCurryCompLim G).inv ≫\n limit.π G ((Prod.braiding K J).functor.obj (k, j)) ≫ 𝟙 (((Prod.braiding K J).functor ⋙ G).obj (k, j)) =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"state_before": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ (limitIsoLimitCurryCompLim G).inv ≫\n limit.π G ((Prod.braiding K J).functor.obj (k, j)) ≫ (𝟙 ((Prod.braiding K J).functor ⋙ G)).app (k, j) =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"tactic": "erw [NatTrans.id_app]"
},
{
"state_after": "no goals",
"state_before": "J K : Type v\ninst✝³ : SmallCategory J\ninst✝² : SmallCategory K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝ : HasLimits C\nj : J\nk : K\n⊢ (limitIsoLimitCurryCompLim G).inv ≫\n limit.π G ((Prod.braiding K J).functor.obj (k, j)) ≫ 𝟙 (((Prod.braiding K J).functor ⋙ G).obj (k, j)) =\n limit.π (curry.obj G ⋙ lim) j ≫ limit.π ((curry.obj G).obj j) k",
"tactic": "simp"
}
] |
[
343,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
lt_div_iff_of_neg
|
[] |
[
745,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
744,
1
] |
Mathlib/Data/List/Basic.lean
|
List.head?_map
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.44944\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nl : List α\n⊢ head? (map f l) = Option.map f (head? l)",
"tactic": "cases l <;> rfl"
}
] |
[
933,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
1
] |
Mathlib/Analysis/SpecialFunctions/Sqrt.lean
|
DifferentiableWithinAt.sqrt
|
[] |
[
135,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
algebraicIndependent_image
|
[] |
[
205,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.inductionOn'
|
[] |
[
670,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
11
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.coe_comp_continuousOrderHom
|
[] |
[
319,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.le_div_iff_mul_le
|
[] |
[
805,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_set_le
|
[] |
[
294,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.homEquiv_leftAdjointUniq_hom_app
|
[
{
"state_after": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ ↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x)) =\n ↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ ↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x) = adj2.unit.app x",
"tactic": "apply (adj1.homEquiv _ _).symm.injective"
},
{
"state_after": "case a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ (↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op =\n (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op",
"state_before": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ ↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x)) =\n ↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)",
"tactic": "apply Quiver.Hom.op_inj"
},
{
"state_after": "case a.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op =\n coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op",
"state_before": "case a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ (↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op =\n (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op",
"tactic": "apply coyoneda.map_injective"
},
{
"state_after": "case a.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\nx✝ : D\na✝ : (coyoneda.obj (F'.obj x).op).obj x✝\n⊢ (coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm\n (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op).app\n x✝ a✝ =\n (coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op).app x✝ a✝",
"state_before": "case a.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\n⊢ coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op =\n coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op",
"tactic": "ext"
},
{
"state_after": "case a.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\nx✝ : D\na✝ : (coyoneda.obj (F'.obj x).op).obj x✝\n⊢ (coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm\n (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op).app\n x✝ a✝ =\n (coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op).app x✝ a✝",
"state_before": "case a.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\nx✝ : D\na✝ : (coyoneda.obj (F'.obj x).op).obj x✝\n⊢ (coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm\n (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op).app\n x✝ a✝ =\n (coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op).app x✝ a✝",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case a.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : C\nx✝ : D\na✝ : (coyoneda.obj (F'.obj x).op).obj x✝\n⊢ (coyoneda.map\n (↑(homEquiv adj1 x (F'.obj x)).symm\n (↑(homEquiv adj1 x (F'.obj x)) ((leftAdjointUniq adj1 adj2).hom.app x))).op).app\n x✝ a✝ =\n (coyoneda.map (↑(homEquiv adj1 x (F'.obj x)).symm (adj2.unit.app x)).op).app x✝ a✝",
"tactic": "simp [leftAdjointUniq, leftAdjointsCoyonedaEquiv]"
}
] |
[
141,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.compl_ne_eq_singleton
|
[] |
[
1705,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1704,
1
] |
Std/Data/List/Lemmas.lean
|
List.length_take_of_le
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\nα✝ : Type u_1\nl : List α✝\nh : n ≤ length l\n⊢ length (take n l) = n",
"tactic": "simp [Nat.min_eq_left h]"
}
] |
[
699,
100
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
699,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_eq_pow_card
|
[] |
[
1447,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1446,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mk'_spec'
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.767746\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x ∈ S }\n⊢ ↑(toMap f) ↑y * mk' f x y = ↑(toMap f) x",
"tactic": "rw [mul_comm, mk'_spec]"
}
] |
[
748,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
748,
1
] |
Mathlib/RingTheory/EisensteinCriterion.lean
|
Polynomial.EisensteinCriterionAux.le_natDegree_of_map_eq_mul_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nP : Ideal R\nhP : Ideal.IsPrime P\nq : R[X]\nc : (R ⧸ P)[X]\nhq : map (mk P) q = c * X ^ n\nhc0 : degree c = 0\n⊢ ↑n = degree (map (mk P) q)",
"tactic": "rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one, Nat.cast_withBot]"
}
] |
[
64,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsFiltered.of_isRightAdjoint
|
[] |
[
324,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/CategoryTheory/EssentialImage.lean
|
CategoryTheory.Functor.obj_mem_essImage
|
[] |
[
79,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.add_right_cancel_iff
|
[] |
[
65,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
64,
11
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.tendsto_symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.30205\nδ : Type ?u.30208\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nx : α\nhx : x ∈ e.source\n⊢ Tendsto (↑(LocalHomeomorph.symm e)) (𝓝 (↑e x)) (𝓝 x)",
"tactic": "simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx)"
}
] |
[
363,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_dual
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ dual (dual (node s l x r)) = node s l x r",
"tactic": "rw [dual, dual, dual_dual l, dual_dual r]"
}
] |
[
155,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Topology/Constructions.lean
|
Subtype.dense_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.158230\nδ : Type ?u.158233\nε : Type ?u.158236\nζ : Type ?u.158239\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\np : α → Prop\ns : Set α\nt : Set ↑s\n⊢ (∀ (x : α) (h : x ∈ s), ↑{ val := x, property := h } ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.158230\nδ : Type ?u.158233\nε : Type ?u.158236\nζ : Type ?u.158239\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\np : α → Prop\ns : Set α\nt : Set ↑s\n⊢ Dense t ↔ s ⊆ closure (val '' t)",
"tactic": "rw [inducing_subtype_val.dense_iff, SetCoe.forall]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.158230\nδ : Type ?u.158233\nε : Type ?u.158236\nζ : Type ?u.158239\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\np : α → Prop\ns : Set α\nt : Set ↑s\n⊢ (∀ (x : α) (h : x ∈ s), ↑{ val := x, property := h } ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)",
"tactic": "rfl"
}
] |
[
1063,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1061,
1
] |
Mathlib/LinearAlgebra/Orientation.lean
|
Basis.orientation_isEmpty
|
[
{
"state_after": "R : Type u_1\ninst✝⁸ : StrictOrderedCommRing R\nM : Type u_3\nN : Type ?u.117323\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nι : Type u_2\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nb : Basis ι R M\n⊢ rayOfNeZero R (det b) (_ : det b ≠ 0) = positiveOrientation",
"state_before": "R : Type u_1\ninst✝⁸ : StrictOrderedCommRing R\nM : Type u_3\nN : Type ?u.117323\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nι : Type u_2\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nb : Basis ι R M\n⊢ Basis.orientation b = positiveOrientation",
"tactic": "rw [Basis.orientation]"
},
{
"state_after": "case e_v\nR : Type u_1\ninst✝⁸ : StrictOrderedCommRing R\nM : Type u_3\nN : Type ?u.117323\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nι : Type u_2\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nb : Basis ι R M\n⊢ det b = ↑AlternatingMap.constLinearEquivOfIsEmpty 1",
"state_before": "R : Type u_1\ninst✝⁸ : StrictOrderedCommRing R\nM : Type u_3\nN : Type ?u.117323\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nι : Type u_2\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nb : Basis ι R M\n⊢ rayOfNeZero R (det b) (_ : det b ≠ 0) = positiveOrientation",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_v\nR : Type u_1\ninst✝⁸ : StrictOrderedCommRing R\nM : Type u_3\nN : Type ?u.117323\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\nι : Type u_2\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nb : Basis ι R M\n⊢ det b = ↑AlternatingMap.constLinearEquivOfIsEmpty 1",
"tactic": "exact b.det_isEmpty"
}
] |
[
176,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.tsub_lt_self
|
[
{
"state_after": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh₁ : 0 < a\nh₂ : 0 < b\nh : a - b = a\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh₁ : 0 < a\nh₂ : 0 < b\n⊢ a - b < a",
"tactic": "refine' tsub_le_self.lt_of_ne fun h => _"
},
{
"state_after": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh₁ : b < a\nh₂ : 0 < b\nh : a - b = a\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh₁ : 0 < a\nh₂ : 0 < b\nh : a - b = a\n⊢ False",
"tactic": "rw [← h, tsub_pos_iff_lt] at h₁"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nh₁ : b < a\nh₂ : 0 < b\nh : a - b = a\n⊢ False",
"tactic": "exact h₂.not_le (ha.add_le_iff_nonpos_left.1 <| add_le_of_le_tsub_left_of_le h₁.le h.ge)"
}
] |
[
449,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
11
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.coe_ofNormedAddCommGroupDiscrete
|
[] |
[
903,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
901,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
UpperSet.coe_div
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.const_mul'
|
[] |
[
1084,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1082,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.domDomCongr_eq_iff
|
[] |
[
766,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
locallyCompactSpace_of_hasBasis
|
[] |
[
1103,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1098,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.not_injOn_infinite_finite_image
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis : Finite ↑(f '' s)\n⊢ ¬InjOn f s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\n⊢ ¬InjOn f s",
"tactic": "have : Finite (f '' s) := finite_coe_iff.mpr h_fin"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\n⊢ ¬InjOn f s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis : Finite ↑(f '' s)\n⊢ ¬InjOn f s",
"tactic": "have : Infinite s := infinite_coe_iff.mpr h_inf"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬Injective (codRestrict (restrict s f) (f '' s) (_ : ∀ (x : ↑s), ∃ a, a ∈ s ∧ f a = restrict s f x))\n⊢ ¬InjOn f s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\n⊢ ¬InjOn f s",
"tactic": "have h := not_injective_infinite_finite\n ((f '' s).codRestrict (s.restrict f) fun x => ⟨x, x.property, rfl⟩)"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬¬InjOn f s\n⊢ Injective (codRestrict (restrict s f) (f '' s) (_ : ∀ (x : ↑s), ∃ a, a ∈ s ∧ f a = restrict s f x))",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬Injective (codRestrict (restrict s f) (f '' s) (_ : ∀ (x : ↑s), ∃ a, a ∈ s ∧ f a = restrict s f x))\n⊢ ¬InjOn f s",
"tactic": "contrapose! h"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬¬InjOn f s\n⊢ InjOn f s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬¬InjOn f s\n⊢ Injective (codRestrict (restrict s f) (f '' s) (_ : ∀ (x : ↑s), ∃ a, a ∈ s ∧ f a = restrict s f x))",
"tactic": "rw [injective_codRestrict, ← injOn_iff_injective]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set α\nh_inf : Set.Infinite s\nh_fin : Set.Finite (f '' s)\nthis✝ : Finite ↑(f '' s)\nthis : Infinite ↑s\nh : ¬¬InjOn f s\n⊢ InjOn f s",
"tactic": "rwa [not_not] at h"
}
] |
[
1399,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1390,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
Convex.set_average_mem
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.367774\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhs : Convex ℝ s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nthis : Fact (↑↑μ t < ⊤)\n⊢ (⨍ (x : α) in t, f x ∂μ) ∈ s",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.367774\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhs : Convex ℝ s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\n⊢ (⨍ (x : α) in t, f x ∂μ) ∈ s",
"tactic": "haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.367774\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhs : Convex ℝ s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nthis : Fact (↑↑μ t < ⊤)\n⊢ Measure.restrict μ t ≠ 0",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.367774\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhs : Convex ℝ s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nthis : Fact (↑↑μ t < ⊤)\n⊢ (⨍ (x : α) in t, f x ∂μ) ∈ s",
"tactic": "refine' hs.average_mem hsc _ hfs hfi"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.367774\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhs : Convex ℝ s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nthis : Fact (↑↑μ t < ⊤)\n⊢ Measure.restrict μ t ≠ 0",
"tactic": "rwa [Ne.def, restrict_eq_zero]"
}
] |
[
104,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.not_even_iff
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ ¬Even n ↔ n % 2 = 1",
"tactic": "rw [even_iff, emod_two_ne_zero]"
}
] |
[
50,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_compl_le_add_iff
|
[] |
[
3077,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3074,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_sub_int
|
[
{
"state_after": "no goals",
"state_before": "x✝ y✝ z x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n",
"tactic": "simpa using rpow_add_int hx y (-n)"
}
] |
[
338,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
Set.Countable.exists_pos_hasSum_le
|
[
{
"state_after": "α : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε",
"state_before": "α : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε",
"tactic": "haveI := hs.toEncodable"
},
{
"state_after": "case mk.intro.intro.intro\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε",
"state_before": "α : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε",
"tactic": "rcases posSumOfEncodable hε s with ⟨f, hf0, ⟨c, hfc, hcε⟩⟩"
},
{
"state_after": "case mk.intro.intro.intro.refine'_1\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\n⊢ 0 < (fun i => if h : i ∈ s then f { val := i, property := h } else 1) i\n\ncase mk.intro.intro.intro.refine'_2\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\n⊢ HasSum (fun i => (fun i => if h : i ∈ s then f { val := i, property := h } else 1) ↑i) c",
"state_before": "case mk.intro.intro.intro\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε",
"tactic": "refine' ⟨fun i => if h : i ∈ s then f ⟨i, h⟩ else 1, fun i => _, ⟨c, _, hcε⟩⟩"
},
{
"state_after": "case mk.intro.intro.intro.refine'_1\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\n⊢ 0 < if h : i ∈ s then f { val := i, property := h } else 1",
"state_before": "case mk.intro.intro.intro.refine'_1\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\n⊢ 0 < (fun i => if h : i ∈ s then f { val := i, property := h } else 1) i",
"tactic": "conv_rhs => simp"
},
{
"state_after": "case mk.intro.intro.intro.refine'_1.inl\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\nh✝ : i ∈ s\n⊢ 0 < f { val := i, property := h✝ }\n\ncase mk.intro.intro.intro.refine'_1.inr\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\nh✝ : ¬i ∈ s\n⊢ 0 < 1",
"state_before": "case mk.intro.intro.intro.refine'_1\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\n⊢ 0 < if h : i ∈ s then f { val := i, property := h } else 1",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.refine'_1.inl\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\nh✝ : i ∈ s\n⊢ 0 < f { val := i, property := h✝ }\n\ncase mk.intro.intro.intro.refine'_1.inr\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\ni : ι\nh✝ : ¬i ∈ s\n⊢ 0 < 1",
"tactic": "exacts [hf0 _, zero_lt_one]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.refine'_2\nα : Type ?u.472618\nβ : Type ?u.472621\nι✝ : Type ?u.472624\nι : Type u_1\ns : Set ι\nhs : Set.Countable s\nε : ℝ\nhε : 0 < ε\nthis : Encodable ↑s\nf : ↑s → ℝ\nhf0 : ∀ (i : ↑s), 0 < f i\nc : ℝ\nhfc : HasSum f c\nhcε : c ≤ ε\n⊢ HasSum (fun i => (fun i => if h : i ∈ s then f { val := i, property := h } else 1) ↑i) c",
"tactic": "simpa only [Subtype.coe_prop, dif_pos, Subtype.coe_eta]"
}
] |
[
474,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.eq_top_of_forall_nnreal_le
|
[] |
[
1683,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1682,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_prod_symm
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.2456513\nβ : Type u_2\nβ' : Type ?u.2456519\nγ : Type ?u.2456522\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2456787\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf : α × β → E\nhf : Integrable f\n⊢ (∫ (z : β × α), f (Prod.swap z) ∂Measure.prod ν μ) = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν",
"state_before": "α : Type u_1\nα' : Type ?u.2456513\nβ : Type u_2\nβ' : Type ?u.2456519\nγ : Type ?u.2456522\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2456787\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf : α × β → E\nhf : Integrable f\n⊢ (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν",
"tactic": "simp_rw [← integral_prod_swap f hf.aestronglyMeasurable]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.2456513\nβ : Type u_2\nβ' : Type ?u.2456519\nγ : Type ?u.2456522\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2456787\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf : α × β → E\nhf : Integrable f\n⊢ (∫ (z : β × α), f (Prod.swap z) ∂Measure.prod ν μ) = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν",
"tactic": "exact integral_prod _ hf.swap"
}
] |
[
483,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.piCongr_apply_apply
|
[
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ (_ : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a) ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ↑(piCongr h₁ h₂) f (↑h₁ a) = ↑(h₂ a) (f a)",
"tactic": "change Eq.ndrec _ _ = _"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\nhZa : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a\n⊢ hZa ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ (_ : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a) ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"tactic": "generalize_proofs hZa"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ∀ (hZa : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a), hZa ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\nhZa : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a\n⊢ hZa ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"tactic": "revert hZa"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ∀ (hZa : ↑h₁.symm.symm a = ↑h₁ a), hZa ▸ ↑(piCongrRight h₂) f a = ↑(h₂ a) (f a)",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ∀ (hZa : ↑h₁.symm.symm (↑h₁.symm (↑h₁ a)) = ↑h₁ a), hZa ▸ ↑(piCongrRight h₂) f (↑h₁.symm (↑h₁ a)) = ↑(h₂ a) (f a)",
"tactic": "rw [h₁.symm_apply_apply a]"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ↑(piCongrRight h₂) f a = ↑(h₂ a) (f a)",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ∀ (hZa : ↑h₁.symm.symm a = ↑h₁ a), hZa ▸ ↑(piCongrRight h₂) f a = ↑(h₂ a) (f a)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\nW : α → Sort w\nZ : β → Sort z\nh₁ : α ≃ β\nh₂ : (a : α) → W a ≃ Z (↑h₁ a)\nf : (a : α) → W a\na : α\n⊢ ↑(piCongrRight h₂) f a = ↑(h₂ a) (f a)",
"tactic": "rfl"
}
] |
[
1801,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1796,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
IsNoetherian.induction
|
[] |
[
361,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.pointwise_smul_le_iff₀
|
[] |
[
173,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.le_mul_withDensity
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "let ν := μ.withDensity (v.limRatioMeas hρ)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "let f := v.limRatioMeas hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have f_meas : Measurable f := v.limRatioMeas_measurable hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by\n refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))\n exact v.measure_limRatioMeas_zero hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by\n apply le_trans (le_of_eq _) (zero_le _)\n apply hρ\n rw [← nonpos_iff_eq_zero]\n exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nC : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have C :\n ∀ n : ℤ,\n ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤\n (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by\n intro n\n let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))\n have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)\n simp only [M, withDensity_apply, coe_nnreal_smul_apply]\n calc\n ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by\n rw [← ENNReal.coe_zpow t_ne_zero']\n apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ\n intro x hx\n apply hx.2.2.trans_le (le_of_eq _)\n rw [ENNReal.coe_zpow t_ne_zero']\n _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by\n simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by\n apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))\n rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]\n exact mul_le_mul_left' hx.2.1 _\n _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nC : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))\n⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "calc\n ρ s =\n ρ (s ∩ f ⁻¹' {0}) + ρ (s ∩ f ⁻¹' {∞}) +\n ∑' n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=\n measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ρ f_meas hs ht\n _ ≤\n (t • ν) (s ∩ f ⁻¹' {0}) + (t • ν) (s ∩ f ⁻¹' {∞}) +\n ∑' n : ℤ, (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=\n (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C))\n _ = (t • ν) s :=\n (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow (t • ν) f_meas hs ht).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑t ≠ 0",
"tactic": "simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ρ (f ⁻¹' {0}) = 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})",
"tactic": "refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ρ (f ⁻¹' {0}) = 0",
"tactic": "exact v.measure_limRatioMeas_zero hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})",
"tactic": "apply le_trans (le_of_eq _) (zero_le _)"
},
{
"state_after": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0",
"tactic": "apply hρ"
},
{
"state_after": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0",
"state_before": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0",
"tactic": "rw [← nonpos_iff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0",
"tactic": "exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\n⊢ ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤\n ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "simp only [M, withDensity_apply, coe_nnreal_smul_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤\n ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ",
"tactic": "calc\n ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by\n rw [← ENNReal.coe_zpow t_ne_zero']\n apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ\n intro x hx\n apply hx.2.2.trans_le (le_of_eq _)\n rw [ENNReal.coe_zpow t_ne_zero']\n _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by\n simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by\n apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))\n rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]\n exact mul_le_mul_left' hx.2.1 _\n _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I)",
"tactic": "rw [← ENNReal.coe_zpow t_ne_zero']"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ s ∩ f ⁻¹' I ⊆ {x | limRatioMeas v hρ x < ↑(t ^ (n + 1))}",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I)",
"tactic": "apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ x ∈ {x | limRatioMeas v hρ x < ↑(t ^ (n + 1))}",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ s ∩ f ⁻¹' I ⊆ {x | limRatioMeas v hρ x < ↑(t ^ (n + 1))}",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ x ∈ {x | limRatioMeas v hρ x < ↑(t ^ (n + 1))}",
"tactic": "apply hx.2.2.trans_le (le_of_eq _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1))",
"tactic": "rw [ENNReal.coe_zpow t_ne_zero']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ",
"tactic": "simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t ^ (n + 1) ≤ ↑t * f x",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ (∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ) ≤ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t * f x ∂μ",
"tactic": "apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t * ↑t ^ n ≤ ↑t * f x",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t ^ (n + 1) ≤ ↑t * f x",
"tactic": "rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4937781\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})\nB : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑t * ↑t ^ n ≤ ↑t * f x",
"tactic": "exact mul_le_mul_left' hx.2.1 _"
}
] |
[
665,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/Data/Int/GCD.lean
|
Int.gcd_mul_left
|
[
{
"state_after": "i j k : ℤ\n⊢ Nat.gcd (natAbs i * natAbs j) (natAbs i * natAbs k) = natAbs i * Nat.gcd (natAbs j) (natAbs k)",
"state_before": "i j k : ℤ\n⊢ gcd (i * j) (i * k) = natAbs i * gcd j k",
"tactic": "rw [Int.gcd, Int.gcd, natAbs_mul, natAbs_mul]"
},
{
"state_after": "no goals",
"state_before": "i j k : ℤ\n⊢ Nat.gcd (natAbs i * natAbs j) (natAbs i * natAbs k) = natAbs i * Nat.gcd (natAbs j) (natAbs k)",
"tactic": "apply Nat.gcd_mul_left"
}
] |
[
303,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.Homotopy.trans_apply
|
[
{
"state_after": "case inr\nF✝ : Type ?u.203878\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ f₂ : C(X, Y)\nF : Homotopy f₀ f₁\nG : Homotopy f₁ f₂\nx : ↑I × X\nh✝ : ¬↑x.fst ≤ 1 / 2\n⊢ ↑(↑(curry G) { val := 2 * ↑x.fst - 1, property := ?inr.hx }) x.snd =\n ↑G ({ val := 2 * ↑x.fst - 1, property := (_ : 2 * ↑x.fst - 1 ∈ I) }, x.snd)\n\ncase inr.hx\nF✝ : Type ?u.203878\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ f₂ : C(X, Y)\nF : Homotopy f₀ f₁\nG : Homotopy f₁ f₂\nx : ↑I × X\nh✝ : ¬↑x.fst ≤ 1 / 2\n⊢ 2 * ↑x.fst - 1 ∈ Set.Icc 0 1",
"state_before": "case inr\nF✝ : Type ?u.203878\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ f₂ : C(X, Y)\nF : Homotopy f₀ f₁\nG : Homotopy f₁ f₂\nx : ↑I × X\nh✝ : ¬↑x.fst ≤ 1 / 2\n⊢ ↑(↑(extend G) (2 * ↑x.fst - 1)) x.snd = ↑G ({ val := 2 * ↑x.fst - 1, property := (_ : 2 * ↑x.fst - 1 ∈ I) }, x.snd)",
"tactic": "rw [extend, ContinuousMap.coe_IccExtend, Set.IccExtend_of_mem]"
},
{
"state_after": "no goals",
"state_before": "case inr\nF✝ : Type ?u.203878\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ f₂ : C(X, Y)\nF : Homotopy f₀ f₁\nG : Homotopy f₁ f₂\nx : ↑I × X\nh✝ : ¬↑x.fst ≤ 1 / 2\n⊢ ↑(↑(curry G) { val := 2 * ↑x.fst - 1, property := ?inr.hx }) x.snd =\n ↑G ({ val := 2 * ↑x.fst - 1, property := (_ : 2 * ↑x.fst - 1 ∈ I) }, x.snd)\n\ncase inr.hx\nF✝ : Type ?u.203878\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ f₂ : C(X, Y)\nF : Homotopy f₀ f₁\nG : Homotopy f₁ f₂\nx : ↑I × X\nh✝ : ¬↑x.fst ≤ 1 / 2\n⊢ 2 * ↑x.fst - 1 ∈ Set.Icc 0 1",
"tactic": "rfl"
}
] |
[
262,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
QuotientAddGroup.btw_coe_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nn : ℤ\nhp' : Fact (0 < p)\nx₁ x₂ x₃ : α\n⊢ btw ↑x₁ ↑x₂ ↑x₃ ↔ toIcoMod (_ : 0 < p) x₁ x₂ ≤ toIocMod (_ : 0 < p) x₁ x₃",
"tactic": "rw [btw_coe_iff', toIocMod_sub_eq_sub, toIcoMod_sub_eq_sub, zero_add, sub_le_sub_iff_right]"
}
] |
[
941,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.LeftInverse.rightInverse_of_surjective
|
[] |
[
376,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.mem_algebraMapSubmonoid_of_mem
|
[] |
[
558,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.adjoin_eq_top_of_gen_mem_adjoin
|
[
{
"state_after": "R : Type u_1\nS : Type u_2\nT : Type ?u.717680\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.717983\nB✝ : Type ?u.717986\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B✝\ninst✝² : IsDomain B✝\ninst✝¹ : Algebra A B✝\nK : Type ?u.718408\ninst✝ : Field K\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ adjoin R {x}\n⊢ adjoin R {B.gen} ≤ adjoin R {x}",
"state_before": "R : Type u_1\nS : Type u_2\nT : Type ?u.717680\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.717983\nB✝ : Type ?u.717986\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B✝\ninst✝² : IsDomain B✝\ninst✝¹ : Algebra A B✝\nK : Type ?u.718408\ninst✝ : Field K\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ adjoin R {x}\n⊢ adjoin R {x} = ⊤",
"tactic": "rw [_root_.eq_top_iff, ← B.adjoin_gen_eq_top]"
},
{
"state_after": "R : Type u_1\nS : Type u_2\nT : Type ?u.717680\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.717983\nB✝ : Type ?u.717986\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B✝\ninst✝² : IsDomain B✝\ninst✝¹ : Algebra A B✝\nK : Type ?u.718408\ninst✝ : Field K\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ adjoin R {x}\n⊢ {B.gen} ⊆ ↑(adjoin R {x})",
"state_before": "R : Type u_1\nS : Type u_2\nT : Type ?u.717680\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.717983\nB✝ : Type ?u.717986\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B✝\ninst✝² : IsDomain B✝\ninst✝¹ : Algebra A B✝\nK : Type ?u.718408\ninst✝ : Field K\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ adjoin R {x}\n⊢ adjoin R {B.gen} ≤ adjoin R {x}",
"tactic": "refine' adjoin_le _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\nT : Type ?u.717680\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.717983\nB✝ : Type ?u.717986\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B✝\ninst✝² : IsDomain B✝\ninst✝¹ : Algebra A B✝\nK : Type ?u.718408\ninst✝ : Field K\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ adjoin R {x}\n⊢ {B.gen} ⊆ ↑(adjoin R {x})",
"tactic": "simp [hx]"
}
] |
[
516,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Order/Filter/CountableInter.lean
|
eventually_countable_forall
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nα : Type u_2\nβ : Type ?u.1140\nl : Filter α\ninst✝¹ : CountableInterFilter l\ninst✝ : Countable ι\np : α → ι → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p x i) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p x i",
"tactic": "simpa only [Filter.Eventually, setOf_forall] using\n @countable_iInter_mem _ _ l _ _ fun i => { x | p x i }"
}
] |
[
67,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.eq_C_of_natDegree_le_zero
|
[] |
[
1117,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.zero_mul
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.110255\nα : Type u_1\nβ : Type ?u.110261\nγ : Type ?u.110264\ninst✝ : MulZeroClass α\ns t : Set α\nhs : Set.Nonempty s\n⊢ 0 ⊆ 0 * s",
"tactic": "simpa [mem_mul] using hs"
}
] |
[
1149,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1148,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.prod_eq_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.308406\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.308415\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type u_2\ninst✝¹ : Group N\nP : Type ?u.308442\ninst✝ : Group P\nH : Subgroup G\nK : Subgroup N\n⊢ prod H K = ⊥ ↔ H = ⊥ ∧ K = ⊥",
"tactic": "simpa only [← Subgroup.toSubmonoid_eq] using Submonoid.prod_eq_bot_iff"
}
] |
[
1779,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1778,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split
|
[
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ coeff (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots P))) 0 = (-1) ^ natDegree P * Multiset.prod (roots P)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ coeff P 0 = (-1) ^ natDegree P * Multiset.prod (roots P)",
"tactic": "nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (eval 0 ∘ fun a => X - ↑C a) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ coeff (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots P))) 0 = (-1) ^ natDegree P * Multiset.prod (roots P)",
"tactic": "rw [coeff_zero_eq_eval_zero, eval_multiset_prod, Multiset.map_map]"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (fun x => -x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (eval 0 ∘ fun a => X - ↑C a) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"tactic": "simp_rw [Function.comp_apply, eval_sub, eval_X, zero_sub, eval_C]"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (fun x => -1 * x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (fun x => -x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"tactic": "conv_lhs =>\n congr\n congr\n ext\n rw [neg_eq_neg_one_mul]"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n⊢ Multiset.prod (Multiset.map (fun x => -1 * x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)",
"tactic": "simp only [splits_iff_card_roots.1 hP, neg_mul, one_mul, Multiset.prod_map_neg]"
}
] |
[
473,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.trans_sbtw_right
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.292206\nP : Type u_3\nP' : Type ?u.292212\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\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Sbtw R x y z\n⊢ Sbtw R w y z",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.292206\nP : Type u_3\nP' : Type ?u.292212\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\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Sbtw R x y z\n⊢ Sbtw R w y z",
"tactic": "rw [wbtw_comm] at *"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.292206\nP : Type u_3\nP' : Type ?u.292212\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\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Sbtw R z y x\n⊢ Sbtw R z y w",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.292206\nP : Type u_3\nP' : Type ?u.292212\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\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Sbtw R x y z\n⊢ Sbtw R w y z",
"tactic": "rw [sbtw_comm] at *"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.292206\nP : Type u_3\nP' : Type ?u.292212\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\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Sbtw R z y x\n⊢ Sbtw R z y w",
"tactic": "exact h₁.trans_sbtw_left h₂"
}
] |
[
492,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isGLB_empty
|
[] |
[
892,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformContinuous.continuous
|
[] |
[
1338,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1336,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofRealClm_apply
|
[] |
[
1015,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_univ_seven
|
[
{
"state_after": "α : Type ?u.28368\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ f (↑castSucc 0) * f (↑castSucc 1) * f (↑castSucc 2) * f (↑castSucc 3) * f (↑castSucc 4) * f (↑castSucc 5) *\n f (last 6) =\n f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6",
"state_before": "α : Type ?u.28368\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ ∏ i : Fin 7, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6",
"tactic": "rw [prod_univ_castSucc, prod_univ_six]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.28368\nβ : Type u_1\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ f (↑castSucc 0) * f (↑castSucc 1) * f (↑castSucc 2) * f (↑castSucc 3) * f (↑castSucc 4) * f (↑castSucc 5) *\n f (last 6) =\n f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6",
"tactic": "rfl"
}
] |
[
151,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.norm_smulRightL_apply
|
[] |
[
1881,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1880,
1
] |
Mathlib/Data/Part.lean
|
Part.append_mem_append
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.74613\nγ : Type ?u.74616\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ++ a = ma ++ mb",
"state_before": "α : Type u_1\nβ : Type ?u.74613\nγ : Type ?u.74616\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ma ++ mb ∈ a ++ b",
"tactic": "simp [append_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.74613\nγ : Type ?u.74616\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ++ a = ma ++ mb",
"tactic": "aesop"
}
] |
[
804,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.relabelAux_sum_inl
|
[
{
"state_after": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.87752\nP : Type ?u.87755\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.87783\nn k : ℕ\nx : α ⊕ Fin k\n⊢ relabelAux Sum.inl k x = Sum.map id (↑(natAdd n)) x",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.87752\nP : Type ?u.87755\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.87783\nn k : ℕ\n⊢ relabelAux Sum.inl k = Sum.map id ↑(natAdd n)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.87752\nP : Type ?u.87755\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.87783\nn k : ℕ\nval✝ : Fin k\n⊢ relabelAux Sum.inl k (Sum.inr val✝) = Sum.map id (↑(natAdd n)) (Sum.inr val✝)",
"tactic": "simp [relabelAux]"
}
] |
[
589,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
586,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocDiv_zsmul_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIocDiv hp a (m • p + b) = m + toIocDiv hp a b",
"tactic": "rw [add_comm, toIocDiv_add_zsmul, add_comm]"
}
] |
[
272,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.card_support_swap_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nhx : ↑f x ≠ x\nh : support f ⊆ support (swap x (↑f x) * f)\n⊢ ¬↑(swap x (↑f x) * f) x ≠ x",
"tactic": "simp"
}
] |
[
612,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Order/Lattice.lean
|
inf_right_idem
|
[] |
[
520,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/CategoryTheory/Bicategory/Functor.lean
|
CategoryTheory.Pseudofunctor.to_oplax_obj
|
[] |
[
466,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.fract_add_zspan
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm v : E\nh : v ∈ span ℤ (Set.range ↑b)\n⊢ fract b (m + v) = fract b m",
"tactic": "rw [add_comm, fract_zspan_add b m h]"
}
] |
[
143,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
BddBelow.inter_of_left
|
[] |
[
405,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_lt_cos_of_nonneg_of_le_pi
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx✝ : x ≤ π / 2\nhy : π / 2 ≤ y\nhx : x < π / 2\n⊢ y ≤ π + π / 2",
"tactic": "linarith [pi_pos]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx✝ : x ≤ π / 2\nhy : π / 2 ≤ y\nhx : x < π / 2\n⊢ -(π / 2) < x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx✝ : x ≤ π / 2\nhy : π / 2 ≤ y\nhx : x = π / 2\n⊢ π / 2 < y",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx✝ : x ≤ π / 2\nhy : π / 2 ≤ y\nhx : x = π / 2\n⊢ y < π + π / 2",
"tactic": "linarith [pi_pos]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx✝ : x ≤ π / 2\nhy : π / 2 ≤ y\nhx : x = π / 2\n⊢ 0 = cos x",
"tactic": "rw [hx, cos_pi_div_two]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx : π / 2 ≤ x\nhy : y ≤ π / 2\n⊢ cos y < cos x",
"tactic": "linarith"
},
{
"state_after": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx : π / 2 ≤ x\nhy : π / 2 ≤ y\n⊢ cos (π - x) < cos (π - y)",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx : π / 2 ≤ x\nhy : π / 2 ≤ y\n⊢ -cos x < -cos y",
"tactic": "rw [← cos_pi_sub, ← cos_pi_sub]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : 0 ≤ x\nhy₂ : y ≤ π\nhxy : x < y\nhx : π / 2 ≤ x\nhy : π / 2 ≤ y\n⊢ cos (π - x) < cos (π - y)",
"tactic": "apply cos_lt_cos_of_nonneg_of_le_pi_div_two <;>\nlinarith"
}
] |
[
583,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.summable
|
[] |
[
308,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
11
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.mem_div_iff_forall_mul_mem
|
[] |
[
690,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
GaloisConnection.u_ciInf
|
[] |
[
1310,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1308,
1
] |
Mathlib/Order/BoundedOrder.lean
|
not_isMax_iff_ne_top
|
[] |
[
144,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_lt_of_le_of_lt_one'
|
[] |
[
752,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
750,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.sub_of_right
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ Monic (p - q)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\n⊢ Monic (p - q)",
"tactic": "have : (-q).coeff (-q).natDegree = 1 := by\n rw [natDegree_neg, coeff_neg, show q.coeff q.natDegree = -1 from hq, neg_neg]"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ Monic (p + -q)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ Monic (p - q)",
"tactic": "rw [sub_eq_add_neg]"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ degree p < degree (-q)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ Monic (p + -q)",
"tactic": "apply Monic.add_of_right this"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\nthis : coeff (-q) (natDegree (-q)) = 1\n⊢ degree p < degree (-q)",
"tactic": "rwa [degree_neg]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Ring R\np✝ p q : R[X]\nhq : Polynomial.leadingCoeff q = -1\nhpq : degree p < degree q\n⊢ coeff (-q) (natDegree (-q)) = 1",
"tactic": "rw [natDegree_neg, coeff_neg, show q.coeff q.natDegree = -1 from hq, neg_neg]"
}
] |
[
416,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_add_pi
|
[] |
[
389,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
inv_closedBall
|
[] |
[
100,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.Equiv.exists_left
|
[] |
[
163,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.iSup_empty
|
[] |
[
941,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_image_iff
|
[] |
[
238,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Multiset.mem_Iio
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.34554\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\nb x : α\n⊢ x ∈ Iio b ↔ x < b",
"tactic": "rw [Iio, ← Finset.mem_def, Finset.mem_Iio]"
}
] |
[
616,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.normSq_apply
|
[] |
[
462,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.infinite_of_exists_right
|
[] |
[
2385,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2382,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.id_f
|
[] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_finset_biUnion_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.116671\nι✝ : Type ?u.116674\nM : Type u_3\nN : Type ?u.116680\ninst✝ : CommMonoid M\nι : Type u_1\nI : Finset ι\ns : ι → Set α\nf : α → M\nh : ∀ (i : ι), i ∈ I → ∀ (j : ι), j ∈ I → i ≠ j → Disjoint (s i) (s j)\nx : α\n⊢ mulIndicator (⋃ (i : ι) (_ : i ∈ I), s i) f x = ∏ i in I, mulIndicator (s i) f x",
"tactic": "rw [Set.mulIndicator_finset_biUnion I s h]"
}
] |
[
668,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Data/Pi/Lex.lean
|
Pi.toLex_apply
|
[] |
[
58,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.319127\nγ : Type ?u.319130\nδ : Type ?u.319133\nι : Type ?u.319136\nR : Type ?u.319139\nR' : Type ?u.319142\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : ∃ᵐ (a : α) ∂restrict μ s, a ∈ s\np : α → Prop\nhp : ∀ᵐ (x : α) ∂restrict μ s, p x\n⊢ ∃ x, x ∈ s ∧ p x",
"state_before": "α : Type u_1\nβ : Type ?u.319127\nγ : Type ?u.319130\nδ : Type ?u.319133\nι : Type ?u.319136\nR : Type ?u.319139\nR' : Type ?u.319142\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : ↑↑μ s ≠ 0\np : α → Prop\nhp : ∀ᵐ (x : α) ∂restrict μ s, p x\n⊢ ∃ x, x ∈ s ∧ p x",
"tactic": "rw [← μ.restrict_apply_self, ← frequently_ae_mem_iff] at hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.319127\nγ : Type ?u.319130\nδ : Type ?u.319133\nι : Type ?u.319136\nR : Type ?u.319139\nR' : Type ?u.319142\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : ∃ᵐ (a : α) ∂restrict μ s, a ∈ s\np : α → Prop\nhp : ∀ᵐ (x : α) ∂restrict μ s, p x\n⊢ ∃ x, x ∈ s ∧ p x",
"tactic": "exact (hs.and_eventually hp).exists"
}
] |
[
1883,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1880,
1
] |
Std/Data/List/Lemmas.lean
|
List.get?_modifyNth
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nn : Nat\nl : List α\n⊢ get? (modifyNth f n l) 0 = (fun a => if n = 0 then f a else a) <$> get? l 0",
"tactic": "cases l <;> cases n <;> rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nn n✝ : Nat\n⊢ get? (modifyNth f n []) (n✝ + 1) = (fun a => if n = n✝ + 1 then f a else a) <$> get? [] (n✝ + 1)",
"tactic": "cases n <;> rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nhead✝ : α\nl : List α\nm : Nat\n⊢ get? (modifyNth f 0 (head✝ :: l)) (m + 1) = (fun a => if 0 = m + 1 then f a else a) <$> get? (head✝ :: l) (m + 1)",
"tactic": "cases l.get? m <;> rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nn : Nat\na : α\nl : List α\nm : Nat\n⊢ (fun a => if n = m then f a else a) <$> get? l m =\n (fun a => if n + 1 = m + 1 then f a else a) <$> get? (a :: l) (m + 1)",
"tactic": "cases l.get? m <;> by_cases h : n = m <;>\n simp only [h, if_pos, if_true, if_false, Option.map, mt Nat.succ.inj, not_false_iff]"
}
] |
[
735,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
727,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
NonarchAddGroupSeminorm.coe_zero
|
[] |
[
565,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_le_add_iff_right
|
[] |
[
786,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
785,
11
] |
Mathlib/Analysis/Convex/Segment.lean
|
left_mem_segment
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.28879\nG : Type ?u.28882\nι : Type ?u.28885\nπ : ι → Type ?u.28890\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : MulActionWithZero 𝕜 E\nx y : E\n⊢ 1 • x + 0 • y = x",
"tactic": "rw [zero_smul, one_smul, add_zero]"
}
] |
[
118,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_add_toIocMod_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - a) = p",
"tactic": "rw [toIcoMod_zero_sub_comm, sub_add_cancel]"
}
] |
[
785,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.continuousWithinAt_of_dominated_interval
|
[] |
[
1084,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.copy_eq
|
[] |
[
555,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Data/Finset/Image.lean
|
Function.Semiconj.finset_image
|
[] |
[
432,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.lowerCentralSeries_zero
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Std/Data/String/Lemmas.lean
|
String.extract_of_valid
|
[
{
"state_after": "l m r : List Char\n⊢ (if utf8Len l ≥ utf8Len l + utf8Len m then \"\"\n else { data := extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } }) =\n { data := m }",
"state_before": "l m r : List Char\n⊢ extract { data := l ++ m ++ r } { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } = { data := m }",
"tactic": "simp only [extract]"
},
{
"state_after": "case inl\nl m r : List Char\nh✝ : utf8Len l ≥ utf8Len l + utf8Len m\n⊢ \"\" = { data := m }\n\ncase inr\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ { data := extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } } = { data := m }",
"state_before": "l m r : List Char\n⊢ (if utf8Len l ≥ utf8Len l + utf8Len m then \"\"\n else { data := extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } }) =\n { data := m }",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case inl\nl m r : List Char\nh✝ : utf8Len l ≥ utf8Len l + utf8Len m\n⊢ \"\" = { data := m }",
"tactic": "next h => rw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_le_left _ _ 0).1 h]"
},
{
"state_after": "no goals",
"state_before": "l m r : List Char\nh : utf8Len l ≥ utf8Len l + utf8Len m\n⊢ \"\" = { data := m }",
"tactic": "rw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_le_left _ _ 0).1 h]"
},
{
"state_after": "case inr.e_data\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } = m",
"state_before": "case inr\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ { data := extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } } = { data := m }",
"tactic": "congr"
},
{
"state_after": "case inr.e_data\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ extract.go₂ (m ++ r) { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } = m",
"state_before": "case inr.e_data\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ extract.go₁ (l ++ m ++ r) 0 { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } = m",
"tactic": "rw [List.append_assoc, extract.go₁_append_right _ _ _ _ _ (by rfl)]"
},
{
"state_after": "case inr.e_data.a\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ utf8Len l + utf8Len m = utf8Len m + utf8Len l",
"state_before": "case inr.e_data\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ extract.go₂ (m ++ r) { byteIdx := utf8Len l } { byteIdx := utf8Len l + utf8Len m } = m",
"tactic": "apply extract.go₂_append_left"
},
{
"state_after": "no goals",
"state_before": "case inr.e_data.a\nl m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ utf8Len l + utf8Len m = utf8Len m + utf8Len l",
"tactic": "apply Nat.add_comm"
},
{
"state_after": "no goals",
"state_before": "l m r : List Char\nh✝ : ¬utf8Len l ≥ utf8Len l + utf8Len m\n⊢ utf8Len l = utf8Len l + 0.byteIdx",
"tactic": "rfl"
}
] |
[
450,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
444,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measure_closure_of_null_frontier
|
[] |
[
429,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
concaveOn_iff_div
|
[] |
[
1059,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
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