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/Analysis/Normed/Group/Seminorm.lean
|
NonarchAddGroupNorm.sup_apply
|
[] |
[
963,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffAt.ccosh
|
[] |
[
492,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
LinearIsometryEquiv.comp_fderivWithin
|
[] |
[
346,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_one
|
[] |
[
560,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean
|
MeasureTheory.Integrable.withDensityᵥ_eq_iff
|
[] |
[
164,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Algebra/Homology/ShortExact/Preadditive.lean
|
CategoryTheory.Split.leftSplit
|
[] |
[
133,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM_Option_eq
|
[
{
"state_after": "α✝ : Type u_1\np : α✝ → Prop\n⊢ ∀ {x : Option α✝}, SatisfiesM p x → ∀ (a : α✝), x = some a → p a",
"state_before": "α✝ : Type u_1\np : α✝ → Prop\nx : Option α✝\n⊢ SatisfiesM p x → ∀ (a : α✝), x = some a → p a",
"tactic": "revert x"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np : α✝ → Prop\n⊢ ∀ {x : Option α✝}, SatisfiesM p x → ∀ (a : α✝), x = some a → p a",
"tactic": "intro | some _, ⟨some ⟨_, h⟩, rfl⟩, _, rfl => exact h"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np : α✝ → Prop\nx✝³ : Option α✝\nx✝² : SatisfiesM p x✝³\nx✝¹ : α✝\nx✝ : x✝³ = some x✝¹\nval✝ : α✝\nh : p val✝\n⊢ p { val := val✝, property := h }.val",
"tactic": "exact h"
}
] |
[
189,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
187,
9
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.denom_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\np : K[X]\n⊢ denom (↑(algebraMap K[X] (RatFunc K)) p) = 1",
"tactic": "convert denom_div p one_ne_zero <;> simp"
}
] |
[
1222,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.coe_toLinearMap
|
[] |
[
297,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.modMonomial_add_divMonomial_single
|
[] |
[
210,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.subtypeDomain_apply
|
[] |
[
1038,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1037,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_max
|
[] |
[
917,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
917,
1
] |
Mathlib/Data/MvPolynomial/Cardinal.lean
|
MvPolynomial.cardinal_mk_eq_max_lift
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u\nR : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ max (lift (#σ →₀ ℕ)) (lift (#R)) = max (max (lift (#R)) (lift (#σ))) ℵ₀",
"tactic": "rw [mk_finsupp_nat, max_assoc, lift_max, lift_aleph0, max_comm]"
}
] |
[
40,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.map_congr
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\ny : β\n⊢ y ∈ map f s ↔ y ∈ map g s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\n⊢ map f s = map g s",
"tactic": "ext y"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\ny : β\n⊢ (∃ a, a ∈ s ∧ f a = y) ↔ ∃ a, a ∈ s ∧ g a = y",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\ny : β\n⊢ y ∈ map f s ↔ y ∈ map g s",
"tactic": "simp only [mem_map]"
},
{
"state_after": "case h.mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\nw : α\nhw : w ∈ s\n⊢ ∃ a, a ∈ s ∧ f a = g w",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\ny : β\n⊢ (∃ a, a ∈ s ∧ g a = y) → ∃ a, a ∈ s ∧ f a = y",
"tactic": "rintro ⟨w, hw, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\nw : α\nhw : w ∈ s\n⊢ ∃ a, a ∈ s ∧ f a = g w",
"tactic": "exact ⟨w, hw, by simp [hw, h]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.39407\nf g : α → β\ns : Sym2 α\nh : ∀ (x : α), x ∈ s → f x = g x\nw : α\nhw : w ∈ s\n⊢ f w = g w",
"tactic": "simp [hw, h]"
}
] |
[
411,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_self'
|
[] |
[
394,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.stalkPushforward.id
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\n⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\n⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)"
},
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\nj : (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n j ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n j ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\n⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "ext1 j"
},
{
"state_after": "case w.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\nj : OpenNhds ((forget TopCat).map (𝟙 X) x)\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := j } ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := j } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\nj : (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n j ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n j ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "induction' j using Opposite.rec with j"
},
{
"state_after": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\ncarrier✝ : Set ↑X\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"state_before": "case w.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\nj : OpenNhds ((forget TopCat).map (𝟙 X) x)\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := j } ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := j } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "rcases j with ⟨⟨_, _⟩, _⟩"
},
{
"state_after": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\ncarrier✝ : Set ↑X\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n colimit.ι\n (Functor.op (OpenNhds.map (𝟙 X) x) ⋙\n ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (Functor.op (OpenNhds.inclusion x))).obj ℱ)\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (Functor.op (OpenNhds.inclusion x))).obj ℱ)\n (Functor.op (OpenNhds.map (𝟙 X) x)) =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"state_before": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\ncarrier✝ : Set ↑X\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (𝟙 X) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "erw [colimit.ι_map_assoc]"
},
{
"state_after": "no goals",
"state_before": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\ncarrier✝ : Set ↑X\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n colimit.ι\n (Functor.op (OpenNhds.map (𝟙 X) x) ⋙\n ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (Functor.op (OpenNhds.inclusion x))).obj ℱ)\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (Functor.op (OpenNhds.inclusion x))).obj ℱ)\n (Functor.op (OpenNhds.map (𝟙 X) x)) =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (𝟙 X) x)))).obj\n (𝟙 X _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n (stalkFunctor C x).map (Pushforward.id ℱ).hom",
"tactic": "simpa [stalkFunctor, stalkPushforward] using by rfl"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nx : ↑X\ncarrier✝ : Set ↑X\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ ℱ)\n { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }.op =\n colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ ℱ)\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } }",
"tactic": "rfl"
}
] |
[
186,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_sub_const_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) ⁻¹' Iic b = Iic (b + a)",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
187,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.norm_sum
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3676885\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl : (i : ι) → G i\ns : Finset ι\nthis : ↑‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ↑‖l i‖ ^ 2\n⊢ ‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ‖l i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3676885\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl : (i : ι) → G i\ns : Finset ι\n⊢ ‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ‖l i‖ ^ 2",
"tactic": "have : ((‖∑ i in s, V i (l i)‖ : ℝ) : 𝕜) ^ 2 = ∑ i in s, ((‖l i‖ : ℝ) : 𝕜) ^ 2 := by\n simp only [← inner_self_eq_norm_sq_to_K, hV.inner_sum]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3676885\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl : (i : ι) → G i\ns : Finset ι\nthis : ↑‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ↑‖l i‖ ^ 2\n⊢ ‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ‖l i‖ ^ 2",
"tactic": "exact_mod_cast this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3676885\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl : (i : ι) → G i\ns : Finset ι\n⊢ ↑‖∑ i in s, ↑(V i) (l i)‖ ^ 2 = ∑ i in s, ↑‖l i‖ ^ 2",
"tactic": "simp only [← inner_self_eq_norm_sq_to_K, hV.inner_sum]"
}
] |
[
2058,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2054,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiag'_conjTranspose
|
[] |
[
828,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.isComplement_top_left
|
[
{
"state_after": "case refine'_1\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\n⊢ Set.Nonempty S\n\ncase refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : G\nha : a ∈ S\nb : G\nhb : b ∈ S\n⊢ a = b\n\ncase refine'_3\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ (∃ g, S = {g}) → IsComplement ⊤ S",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ IsComplement ⊤ S ↔ ∃ g, S = {g}",
"tactic": "refine'\n ⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨_, fun a ha b hb => _⟩, _⟩"
},
{
"state_after": "case refine'_1.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : ↑⊤ × ↑S\nh✝ : (fun x => ↑x.fst * ↑x.snd) a = 1\n⊢ Set.Nonempty S",
"state_before": "case refine'_1\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\n⊢ Set.Nonempty S",
"tactic": "obtain ⟨a, _⟩ := h.2 1"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : ↑⊤ × ↑S\nh✝ : (fun x => ↑x.fst * ↑x.snd) a = 1\n⊢ Set.Nonempty S",
"tactic": "exact ⟨a.2.1, a.2.2⟩"
},
{
"state_after": "case refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : G\nha : a ∈ S\nb : G\nhb : b ∈ S\nthis :\n ({ val := a⁻¹, property := (_ : a⁻¹ ∈ ⊤) }, { val := a, property := ha }) =\n ({ val := b⁻¹, property := (_ : b⁻¹ ∈ ⊤) }, { val := b, property := hb })\n⊢ a = b",
"state_before": "case refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : G\nha : a ∈ S\nb : G\nhb : b ∈ S\n⊢ a = b",
"tactic": "have : (⟨⟨_, mem_top a⁻¹⟩, ⟨a, ha⟩⟩ : (⊤ : Set G) × S) = ⟨⟨_, mem_top b⁻¹⟩, ⟨b, hb⟩⟩ :=\n h.1 ((inv_mul_self a).trans (inv_mul_self b).symm)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement ⊤ S\na : G\nha : a ∈ S\nb : G\nhb : b ∈ S\nthis :\n ({ val := a⁻¹, property := (_ : a⁻¹ ∈ ⊤) }, { val := a, property := ha }) =\n ({ val := b⁻¹, property := (_ : b⁻¹ ∈ ⊤) }, { val := b, property := hb })\n⊢ a = b",
"tactic": "exact Subtype.ext_iff.mp (Prod.ext_iff.mp this).2"
},
{
"state_after": "case refine'_3.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nT : Set G\ng : G\n⊢ IsComplement ⊤ {g}",
"state_before": "case refine'_3\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ (∃ g, S = {g}) → IsComplement ⊤ S",
"tactic": "rintro ⟨g, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nT : Set G\ng : G\n⊢ IsComplement ⊤ {g}",
"tactic": "exact isComplement_top_singleton"
}
] |
[
156,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
Abelianization.map_of
|
[] |
[
183,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.lift.unique
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.606732\ninst✝¹³ : Monoid R'\nR'' : Type ?u.606738\ninst✝¹² : Semiring R''\nM : Type u_3\nN : Type u_2\nP : Type u_4\nQ : Type ?u.606753\nS : Type ?u.606756\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\nf : M →ₗ[R] N →ₗ[R] P\ng : M ⊗[R] N →ₗ[R] P\nH : ∀ (x : M) (y : N), ↑g (x ⊗ₜ[R] y) = ↑(↑f x) y\nm : M\nn : N\n⊢ ↑g (m ⊗ₜ[R] n) = ↑(lift f) (m ⊗ₜ[R] n)",
"tactic": "rw [H, lift.tmul]"
}
] |
[
496,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_self_nonneg
|
[] |
[
578,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.op
|
[] |
[
645,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Topology/Algebra/Module/CharacterSpace.lean
|
WeakDual.CharacterSpace.apply_mem_spectrum
|
[] |
[
196,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/RingTheory/Localization/Ideal.lean
|
IsLocalization.mem_map_algebraMap_iff
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ z ∈ Ideal.map (algebraMap R S) I → ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) ↑x.fst\n\ncase mpr\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ (∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) ↑x.fst) → z ∈ Ideal.map (algebraMap R S) I",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ z ∈ Ideal.map (algebraMap R S) I ↔ ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) ↑x.fst",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ z ∈ Ideal.map (algebraMap R S) I → z ∈ IsLocalization.map_ideal M S I",
"state_before": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ z ∈ Ideal.map (algebraMap R S) I → ∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) ↑x.fst",
"tactic": "change _ → z ∈ map_ideal M S I"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\nhz : z ∈ ↑(algebraMap R S) '' ↑I\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"state_before": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ z ∈ Ideal.map (algebraMap R S) I → z ∈ IsLocalization.map_ideal M S I",
"tactic": "refine' fun h => Ideal.mem_sInf.1 h fun z hz => _"
},
{
"state_after": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"state_before": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\nhz : z ∈ ↑(algebraMap R S) '' ↑I\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"tactic": "obtain ⟨y, hy⟩ := hz"
},
{
"state_after": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\nZ : { x // x ∈ I } := { val := y, property := (_ : y ∈ ↑I) }\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"state_before": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"tactic": "let Z : { x // x ∈ I } := ⟨y, hy.left⟩"
},
{
"state_after": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\nZ : { x // x ∈ I } := { val := y, property := (_ : y ∈ ↑I) }\n⊢ z * ↑(algebraMap R S) ↑(Z, 1).snd = ↑(algebraMap R S) ↑(Z, 1).fst",
"state_before": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\nZ : { x // x ∈ I } := { val := y, property := (_ : y ∈ ↑I) }\n⊢ z ∈ ↑(IsLocalization.map_ideal M S I)",
"tactic": "use ⟨Z, 1⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ ↑(algebraMap R S) y = z\nZ : { x // x ∈ I } := { val := y, property := (_ : y ∈ ↑I) }\n⊢ z * ↑(algebraMap R S) ↑(Z, 1).snd = ↑(algebraMap R S) ↑(Z, 1).fst",
"tactic": "simp [hy.right]"
},
{
"state_after": "case mpr.intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\na : { x // x ∈ I }\ns : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑(a, s).snd = ↑(algebraMap R S) ↑(a, s).fst\n⊢ z ∈ Ideal.map (algebraMap R S) I",
"state_before": "case mpr\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\n⊢ (∃ x, z * ↑(algebraMap R S) ↑x.snd = ↑(algebraMap R S) ↑x.fst) → z ∈ Ideal.map (algebraMap R S) I",
"tactic": "rintro ⟨⟨a, s⟩, h⟩"
},
{
"state_after": "case mpr.intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\na : { x // x ∈ I }\ns : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑(a, s).snd = ↑(algebraMap R S) ↑(a, s).fst\n⊢ z * ↑(algebraMap R S) ↑s ∈ Ideal.map (algebraMap R S) I",
"state_before": "case mpr.intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\na : { x // x ∈ I }\ns : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑(a, s).snd = ↑(algebraMap R S) ↑(a, s).fst\n⊢ z ∈ Ideal.map (algebraMap R S) I",
"tactic": "rw [← Ideal.unit_mul_mem_iff_mem _ (map_units S s), mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz : S\na : { x // x ∈ I }\ns : { x // x ∈ M }\nh : z * ↑(algebraMap R S) ↑(a, s).snd = ↑(algebraMap R S) ↑(a, s).fst\n⊢ z * ↑(algebraMap R S) ↑s ∈ Ideal.map (algebraMap R S) I",
"tactic": "exact h.symm ▸ Ideal.mem_map_of_mem _ a.2"
}
] |
[
67,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
expMapCircle_sub_two_pi
|
[] |
[
106,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.le_add_one_iff
|
[] |
[
217,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/Set/Semiring.lean
|
Set.up_mul
|
[] |
[
164,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/CategoryTheory/Category/Preorder.lean
|
CategoryTheory.homOfLE_comp
|
[] |
[
85,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.erase_subset_iff_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.219160\nγ : Type ?u.219163\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nh : a ∈ t\n⊢ erase s a ⊆ t ↔ s ⊆ t",
"tactic": "rw [← subset_insert_iff, insert_eq_of_mem h]"
}
] |
[
2007,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2006,
1
] |
Mathlib/Data/List/Chain.lean
|
List.Pairwise.chain'
|
[] |
[
244,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.set_integral_eq
|
[
{
"state_after": "case pos\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ\n\ncase neg\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : ¬IntegrableOn f s\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"state_before": "G : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"tactic": "by_cases hfs : IntegrableOn f s μ"
},
{
"state_after": "case pos\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"state_before": "case pos\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"tactic": "have hft : IntegrableOn f t μ := by rwa [ht.integrableOn_iff hs hf]"
},
{
"state_after": "no goals",
"state_before": "case pos\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"tactic": "calc\n (∫ x in s, f x ∂μ) = ∑' g : G, ∫ x in s ∩ g • t, f x ∂μ := ht.set_integral_eq_tsum hfs\n _ = ∑' g : G, ∫ x in g • t ∩ s, f (g⁻¹ • x) ∂μ := by simp only [hf, inter_comm]\n _ = ∫ x in t, f x ∂μ := (hs.set_integral_eq_tsum' hft).symm"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\n⊢ IntegrableOn f t",
"tactic": "rwa [ht.integrableOn_iff hs hf]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n⊢ (∑' (g : G), ∫ (x : α) in s ∩ g • t, f x ∂μ) = ∑' (g : G), ∫ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ",
"tactic": "simp only [hf, inter_comm]"
},
{
"state_after": "case neg\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : ¬IntegrableOn f s\n⊢ ¬Integrable fun x => f x",
"state_before": "case neg\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : ¬IntegrableOn f s\n⊢ (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in t, f x ∂μ",
"tactic": "rw [integral_undef hfs, integral_undef]"
},
{
"state_after": "no goals",
"state_before": "case neg\nG : Type u_1\nH : Type ?u.410725\nα : Type u_2\nβ : Type ?u.410731\nE : Type u_3\ninst✝¹² : Group G\ninst✝¹¹ : Group H\ninst✝¹⁰ : MulAction G α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MulAction H β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nν : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → E\nhf : ∀ (g : G) (x : α), f (g • x) = f x\nhfs : ¬IntegrableOn f s\n⊢ ¬Integrable fun x => f x",
"tactic": "rwa [hs.integrableOn_iff ht hf] at hfs"
}
] |
[
466,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
11
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.ghostMap_apply
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coe_divNat
|
[] |
[
1622,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1621,
1
] |
Mathlib/Algebra/CharP/Quotient.lean
|
CharP.quotient
|
[] |
[
34,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
23,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
RingEquiv.map_det
|
[] |
[
337,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
not_mem_of_lt_csInf
|
[] |
[
599,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
IsScalarTower.algebraMap_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nx : R\n⊢ ↑(algebraMap R A) x = ↑(RingHom.comp (algebraMap S A) (algebraMap R S)) x",
"tactic": "simp_rw [RingHom.comp_apply, Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]"
}
] |
[
105,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_ne_zero
|
[] |
[
160,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Topology/Separation.lean
|
isOpen_ne_fun
|
[] |
[
1184,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1182,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.single_coeff_of_ne
|
[] |
[
166,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_empty_type
|
[] |
[
178,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
nilpotencyClass_le_of_ker_le_center
|
[
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ nilpotencyClass G ≤ nilpotencyClass H + 1",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\n⊢ nilpotencyClass G ≤ nilpotencyClass H + 1",
"tactic": "haveI : IsNilpotent G := isNilpotent_of_ker_le_center f hf1 hH"
},
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥) ≤ nilpotencyClass H + 1",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ nilpotencyClass G ≤ nilpotencyClass H + 1",
"tactic": "rw [← lowerCentralSeries_length_eq_nilpotencyClass]"
},
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ lowerCentralSeries G (nilpotencyClass H + 1) = ⊥",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥) ≤ nilpotencyClass H + 1",
"tactic": "refine @Nat.find_min' _ (Classical.decPred _) _ _ ?_"
},
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Subgroup.map f (lowerCentralSeries G (nilpotencyClass H)) = ⊥",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ lowerCentralSeries G (nilpotencyClass H + 1) = ⊥",
"tactic": "refine lowerCentralSeries_succ_eq_bot (le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1)"
},
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Subgroup.map f (lowerCentralSeries G (nilpotencyClass H)) ≤ ⊥",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Subgroup.map f (lowerCentralSeries G (nilpotencyClass H)) = ⊥",
"tactic": "apply eq_bot_iff.mpr"
},
{
"state_after": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ lowerCentralSeries H (nilpotencyClass H) ≤ ⊥",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ Subgroup.map f (lowerCentralSeries G (nilpotencyClass H)) ≤ ⊥",
"tactic": "apply le_trans (lowerCentralSeries.map f _)"
},
{
"state_after": "no goals",
"state_before": "G : Type u_2\ninst✝² : Group G\nH✝ : Subgroup G\ninst✝¹ : Normal H✝\nH : Type u_1\ninst✝ : Group H\nf : G →* H\nhf1 : MonoidHom.ker f ≤ center G\nhH : Group.IsNilpotent H\nthis : Group.IsNilpotent G\n⊢ lowerCentralSeries H (nilpotencyClass H) ≤ ⊥",
"tactic": "simp only [lowerCentralSeries_nilpotencyClass, le_bot_iff]"
}
] |
[
539,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.ker_apply_eq_preimage
|
[] |
[
892,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/Init/Algebra/Order.lean
|
le_of_lt
|
[] |
[
115,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.FinMeasAdditive.add
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.16086\nF : Type ?u.16089\nF' : Type ?u.16092\nG : Type ?u.16095\n𝕜 : Type ?u.16098\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝ : AddCommMonoid β\nT T' : Set α → β\nhT : FinMeasAdditive μ T\nhT' : FinMeasAdditive μ T'\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : ↑↑μ s ≠ ⊤\nhμt : ↑↑μ t ≠ ⊤\nhst : s ∩ t = ∅\n⊢ (T + T') (s ∪ t) = (T + T') s + (T + T') t",
"state_before": "α : Type u_1\nE : Type ?u.16086\nF : Type ?u.16089\nF' : Type ?u.16092\nG : Type ?u.16095\n𝕜 : Type ?u.16098\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝ : AddCommMonoid β\nT T' : Set α → β\nhT : FinMeasAdditive μ T\nhT' : FinMeasAdditive μ T'\n⊢ FinMeasAdditive μ (T + T')",
"tactic": "intro s t hs ht hμs hμt hst"
},
{
"state_after": "α : Type u_1\nE : Type ?u.16086\nF : Type ?u.16089\nF' : Type ?u.16092\nG : Type ?u.16095\n𝕜 : Type ?u.16098\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝ : AddCommMonoid β\nT T' : Set α → β\nhT : FinMeasAdditive μ T\nhT' : FinMeasAdditive μ T'\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : ↑↑μ s ≠ ⊤\nhμt : ↑↑μ t ≠ ⊤\nhst : s ∩ t = ∅\n⊢ T s + T t + (T' s + T' t) = T s + T' s + (T t + T' t)",
"state_before": "α : Type u_1\nE : Type ?u.16086\nF : Type ?u.16089\nF' : Type ?u.16092\nG : Type ?u.16095\n𝕜 : Type ?u.16098\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝ : AddCommMonoid β\nT T' : Set α → β\nhT : FinMeasAdditive μ T\nhT' : FinMeasAdditive μ T'\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : ↑↑μ s ≠ ⊤\nhμt : ↑↑μ t ≠ ⊤\nhst : s ∩ t = ∅\n⊢ (T + T') (s ∪ t) = (T + T') s + (T + T') t",
"tactic": "simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.16086\nF : Type ?u.16089\nF' : Type ?u.16092\nG : Type ?u.16095\n𝕜 : Type ?u.16098\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝ : AddCommMonoid β\nT T' : Set α → β\nhT : FinMeasAdditive μ T\nhT' : FinMeasAdditive μ T'\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : ↑↑μ s ≠ ⊤\nhμt : ↑↑μ t ≠ ⊤\nhst : s ∩ t = ∅\n⊢ T s + T t + (T' s + T' t) = T s + T' s + (T t + T' t)",
"tactic": "abel"
}
] |
[
113,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : 0 ≤ r\n⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s\n\ncase neg\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s",
"state_before": "α : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\n⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s",
"tactic": "by_cases hr : 0 ≤ r"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : 0 ≤ r\n⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s",
"tactic": "exact toJordanDecomposition_smul_real_nonneg s r hr"
},
{
"state_after": "case neg.posPart\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ (toJordanDecomposition (r • s)).posPart = (r • toJordanDecomposition s).posPart\n\ncase neg.negPart\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ (toJordanDecomposition (r • s)).negPart = (r • toJordanDecomposition s).negPart",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ toJordanDecomposition (r • s) = r • toJordanDecomposition s",
"tactic": "ext1"
},
{
"state_after": "case neg.posPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r\n\ncase neg.posPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"state_before": "case neg.posPart\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ (toJordanDecomposition (r • s)).posPart = (r • toJordanDecomposition s).posPart",
"tactic": "rw [real_smul_posPart_neg _ _ (not_le.1 hr),\n show r • s = -(-r • s) by rw [neg_smul, neg_neg], toJordanDecomposition_neg, neg_posPart,\n toJordanDecomposition_smul_real_nonneg, ← smul_negPart, real_smul_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case neg.posPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r\n\ncase neg.posPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"tactic": "all_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ r • s = -(-r • s)",
"tactic": "rw [neg_smul, neg_neg]"
},
{
"state_after": "no goals",
"state_before": "case neg.posPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"tactic": "exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))"
},
{
"state_after": "case neg.negPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r\n\ncase neg.negPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"state_before": "case neg.negPart\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ (toJordanDecomposition (r • s)).negPart = (r • toJordanDecomposition s).negPart",
"tactic": "rw [real_smul_negPart_neg _ _ (not_le.1 hr),\n show r • s = -(-r • s) by rw [neg_smul, neg_neg], toJordanDecomposition_neg, neg_negPart,\n toJordanDecomposition_smul_real_nonneg, ← smul_posPart, real_smul_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case neg.negPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r\n\ncase neg.negPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"tactic": "all_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ r • s = -(-r • s)",
"tactic": "rw [neg_smul, neg_neg]"
},
{
"state_after": "no goals",
"state_before": "case neg.negPart.hr\nα : Type u_1\nβ : Type ?u.79282\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : ¬0 ≤ r\n⊢ 0 ≤ -r",
"tactic": "exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))"
}
] |
[
490,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_lt_div_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.67571\nα : Type u_1\nβ : Type ?u.67577\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nb0 : 0 < b\nd0 : 0 < d\n⊢ a / b < c / d ↔ a * d < c * b",
"tactic": "rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]"
}
] |
[
382,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.eqToIso_refl
|
[] |
[
140,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.triangle_assoc_comp_right
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ (α_ f (𝟙 b) g).inv ≫ (ρ_ f).hom ▷ g = f ◁ (λ_ g).hom",
"tactic": "rw [← triangle, inv_hom_id_assoc]"
}
] |
[
328,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_diff_assoc
|
[] |
[
1856,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1855,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.Equiv.ext
|
[] |
[
240,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.zero_apply
|
[] |
[
264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Order/Basic.lean
|
not_lt_iff_eq_or_lt
|
[] |
[
490,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.strongEpi_of_strongEpiMonoFactorisation
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX Y : C\nf : X ⟶ Y\nF : StrongEpiMonoFactorisation f\nF' : MonoFactorisation f\nhF' : IsImage F'\n⊢ StrongEpi (F.e ≫ (IsImage.isoExt (StrongEpiMonoFactorisation.toMonoIsImage F) hF').hom)",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf : X ⟶ Y\nF : StrongEpiMonoFactorisation f\nF' : MonoFactorisation f\nhF' : IsImage F'\n⊢ StrongEpi F'.e",
"tactic": "rw [← IsImage.e_isoExt_hom F.toMonoIsImage hF']"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf : X ⟶ Y\nF : StrongEpiMonoFactorisation f\nF' : MonoFactorisation f\nhF' : IsImage F'\n⊢ StrongEpi (F.e ≫ (IsImage.isoExt (StrongEpiMonoFactorisation.toMonoIsImage F) hF').hom)",
"tactic": "apply strongEpi_comp"
}
] |
[
955,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
951,
1
] |
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
Besicovitch.TauPackage.monotone_iUnionUpTo
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\ni j : Ordinal\nhij : i ≤ j\n⊢ iUnionUpTo p i ≤ iUnionUpTo p j",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\n⊢ Monotone (iUnionUpTo p)",
"tactic": "intro i j hij"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\ni j : Ordinal\nhij : i ≤ j\n⊢ (⋃ (j : { j // j < i }),\n ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j))) ≤\n ⋃ (j_1 : { j_1 // j_1 < j }),\n ball (BallPackage.c p.toBallPackage (index p ↑j_1)) (BallPackage.r p.toBallPackage (index p ↑j_1))",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\ni j : Ordinal\nhij : i ≤ j\n⊢ iUnionUpTo p i ≤ iUnionUpTo p j",
"tactic": "simp only [iUnionUpTo]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\ni j : Ordinal\nhij : i ≤ j\n⊢ (⋃ (j : { j // j < i }),\n ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j))) ≤\n ⋃ (j_1 : { j_1 // j_1 < j }),\n ball (BallPackage.c p.toBallPackage (index p ↑j_1)) (BallPackage.r p.toBallPackage (index p ↑j_1))",
"tactic": "exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩"
}
] |
[
267,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/RingTheory/Adjoin/FG.lean
|
Subalgebra.FG.map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhs✝ : FG S\ns : Finset A\nhs : Algebra.adjoin R ↑s = S\n⊢ Algebra.adjoin R ↑(Finset.image (↑f) s) = Subalgebra.map f S",
"tactic": "rw [Finset.coe_image, Algebra.adjoin_image, hs]"
}
] |
[
150,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.candidatesBOfCandidates_mem
|
[] |
[
217,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Analysis/Calculus/DiffContOnCl.lean
|
DiffContOnCl.add_const
|
[] |
[
98,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsLimit.OfNatIso.coneOfHom_fac
|
[
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\n⊢ { pt := Y, π := h.hom.app Y.op { down := f } } = { pt := Y, π := (const J).map f ≫ h.hom.app X.op { down := 𝟙 X } }",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\n⊢ coneOfHom h f = Cone.extend (limitCone h) f",
"tactic": "dsimp [coneOfHom, limitCone, Cone.extend]"
},
{
"state_after": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\n⊢ { pt := Y, π := h.hom.app Y.op { down := f } } = { pt := Y, π := (const J).map f ≫ h.hom.app X.op { down := 𝟙 X } }",
"tactic": "congr with j"
},
{
"state_after": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt :\n ((yoneda.obj X ⋙ uliftFunctor).map f.op ≫ h.hom.app Y.op) { down := 𝟙 X } =\n (h.hom.app X.op ≫ (Functor.cones F).map f.op) { down := 𝟙 X }\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"state_before": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"tactic": "have t := congrFun (h.hom.naturality f.op) ⟨𝟙 X⟩"
},
{
"state_after": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f ≫ 𝟙 X } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"state_before": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt :\n ((yoneda.obj X ⋙ uliftFunctor).map f.op ≫ h.hom.app Y.op) { down := 𝟙 X } =\n (h.hom.app X.op ≫ (Functor.cones F).map f.op) { down := 𝟙 X }\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"tactic": "dsimp at t"
},
{
"state_after": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"state_before": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f ≫ 𝟙 X } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"tactic": "simp only [comp_id] at t"
},
{
"state_after": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ ((Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })).app j =\n ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"state_before": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ (h.hom.app Y.op { down := f }).app j = ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"tactic": "rw [congrFun (congrArg NatTrans.app t) j]"
},
{
"state_after": "no goals",
"state_before": "case e_π.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\nY : C\nf : Y ⟶ X\nj : J\nt : h.hom.app Y.op { down := f } = (Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })\n⊢ ((Functor.cones F).map f.op (h.hom.app X.op { down := 𝟙 X })).app j =\n ((const J).map f ≫ h.hom.app X.op { down := 𝟙 X }).app j",
"tactic": "rfl"
}
] |
[
516,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.le_prod_map_fst_snd
|
[] |
[
364,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/MeasureTheory/Measure/Complex.lean
|
MeasureTheory.SignedMeasure.re_toComplexMeasure
|
[] |
[
82,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.lift_id
|
[] |
[
1055,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1054,
1
] |
Mathlib/Data/List/Basic.lean
|
List.take_add
|
[
{
"state_after": "case h.e'_2\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n) l = take (m + n) (take m l ++ drop m l)\n\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l)",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n) l = take m l ++ take n (drop m l)",
"tactic": "convert_to take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l)"
},
{
"state_after": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n - length (take m l)) (drop m l) = take n (drop m l)\n\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ length (take m l) ≤ m + n",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l)",
"tactic": "rw [take_append_eq_append_take, take_all_of_le, append_right_inj]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n) l = take (m + n) (take m l ++ drop m l)",
"tactic": "rw [take_append_drop]"
},
{
"state_after": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ min (m + n - min m (length l)) (length l - m) = min n (length l - m)",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ take (m + n - length (take m l)) (drop m l) = take n (drop m l)",
"tactic": "simp only [take_eq_take, length_take, length_drop]"
},
{
"state_after": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ min (m + n - min m (length l)) (length l - m) = min n (length l - m)",
"tactic": "generalize l.length = k"
},
{
"state_after": "case pos\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : m ≤ k\n⊢ min (m + n - min m k) (k - m) = min n (k - m)\n\ncase neg\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : ¬m ≤ k\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"tactic": "by_cases h : m ≤ k"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : m ≤ k\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"tactic": "simp [min_eq_left_iff.mpr h]"
},
{
"state_after": "case neg\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : k < m\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"state_before": "case neg\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : ¬m ≤ k\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"tactic": "push_neg at h"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n k : ℕ\nh : k < m\n⊢ min (m + n - min m k) (k - m) = min n (k - m)",
"tactic": "simp [Nat.sub_eq_zero_of_le (le_of_lt h)]"
},
{
"state_after": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ length (take m l) ≤ m\n\nι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ m ≤ m + n",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ length (take m l) ≤ m + n",
"tactic": "trans m"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ length (take m l) ≤ m",
"tactic": "apply length_take_le"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.185033\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nm n : ℕ\n⊢ m ≤ m + n",
"tactic": "simp"
}
] |
[
2067,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2056,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.invOn
|
[] |
[
248,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
11
] |
Mathlib/Data/List/Count.lean
|
List.count_cons_of_ne
|
[
{
"state_after": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na b : α\nh : a ≠ b\nl : List α\n⊢ a = b → False",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na b : α\nh : a ≠ b\nl : List α\n⊢ count a (b :: l) = count a l",
"tactic": "simp [count_cons']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na b : α\nh : a ≠ b\nl : List α\n⊢ a = b → False",
"tactic": "exact h"
}
] |
[
198,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
HasFTaylorSeriesUpTo.hasFDerivAt
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhn : 1 ≤ n\nx : E\n⊢ HasFDerivWithinAt f (↑(continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) univ x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhn : 1 ≤ n\nx : E\n⊢ HasFDerivAt f (↑(continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) x",
"tactic": "rw [← hasFDerivWithinAt_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhn : 1 ≤ n\nx : E\n⊢ HasFDerivWithinAt f (↑(continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) univ x",
"tactic": "exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _)"
}
] |
[
1290,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1287,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.pointwise_smul_le_pointwise_smul_iff
|
[] |
[
123,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_int_mul_pi
|
[] |
[
1173,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1172,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
Int.associated_iff_natAbs
|
[
{
"state_after": "a b : ℤ\n⊢ Associated (natAbs a) (natAbs b) ↔ natAbs a = natAbs b",
"state_before": "a b : ℤ\n⊢ Associated a b ↔ natAbs a = natAbs b",
"tactic": "rw [← dvd_dvd_iff_associated, ← Int.natAbs_dvd_natAbs, ← Int.natAbs_dvd_natAbs,\n dvd_dvd_iff_associated]"
},
{
"state_after": "no goals",
"state_before": "a b : ℤ\n⊢ Associated (natAbs a) (natAbs b) ↔ natAbs a = natAbs b",
"tactic": "exact associated_iff_eq"
}
] |
[
376,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSubOfPoint_map
|
[
{
"state_after": "k : Type u_4\nV : Type u_3\nP : Type u_5\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type u_1\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\nb : P\n⊢ ∑ i in map e s₂, w i • (p i -ᵥ b) = ∑ i in s₂, (w ∘ ↑e) i • ((p ∘ ↑e) i -ᵥ b)",
"state_before": "k : Type u_4\nV : Type u_3\nP : Type u_5\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type u_1\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\nb : P\n⊢ ↑(weightedVSubOfPoint (map e s₂) p b) w = ↑(weightedVSubOfPoint s₂ (p ∘ ↑e) b) (w ∘ ↑e)",
"tactic": "simp_rw [weightedVSubOfPoint_apply]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_4\nV : Type u_3\nP : Type u_5\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type u_1\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\nb : P\n⊢ ∑ i in map e s₂, w i • (p i -ᵥ b) = ∑ i in s₂, (w ∘ ↑e) i • ((p ∘ ↑e) i -ᵥ b)",
"tactic": "exact Finset.sum_map _ _ _"
}
] |
[
183,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.tan_add_nat_mul_pi
|
[] |
[
1327,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1326,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.22119\nM : Type u_3\nN : Type ?u.22125\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → M\ng : α → β\nhg : Injective g\nx : α\n⊢ mulIndicator (g '' s) f (g x) = mulIndicator s (f ∘ g) x",
"tactic": "rw [← mulIndicator_comp_right, preimage_image_eq _ hg]"
}
] |
[
270,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.indicator_one_inj
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.150393\nι : Type ?u.150396\nM : Type u_2\nN : Type ?u.150402\ninst✝¹ : MulZeroOneClass M\ninst✝ : Nontrivial M\nU V : Set α\nh : indicator U 1 = indicator V 1\nx✝ : α\n⊢ x✝ ∈ U ↔ x✝ ∈ V",
"state_before": "α : Type u_1\nβ : Type ?u.150393\nι : Type ?u.150396\nM : Type u_2\nN : Type ?u.150402\ninst✝¹ : MulZeroOneClass M\ninst✝ : Nontrivial M\nU V : Set α\nh : indicator U 1 = indicator V 1\n⊢ U = V",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.150393\nι : Type ?u.150396\nM : Type u_2\nN : Type ?u.150402\ninst✝¹ : MulZeroOneClass M\ninst✝ : Nontrivial M\nU V : Set α\nh : indicator U 1 = indicator V 1\nx✝ : α\n⊢ x✝ ∈ U ↔ x✝ ∈ V",
"tactic": "simp_rw [← indicator_eq_one_iff_mem M, h]"
}
] |
[
743,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
741,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_tower_top_of_isIntegral
|
[
{
"state_after": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap R B) x p = 0\n⊢ IsIntegral A x",
"state_before": "R : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\nh : IsIntegral R x\n⊢ IsIntegral A x",
"tactic": "rcases h with ⟨p, ⟨hp, hp'⟩⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap R B) x p = 0\n⊢ eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0",
"state_before": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap R B) x p = 0\n⊢ IsIntegral A x",
"tactic": "refine' ⟨p.map (algebraMap R A), ⟨hp.map (algebraMap R A), _⟩⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n⊢ eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0",
"state_before": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap R B) x p = 0\n⊢ eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0",
"tactic": "rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\nA : Type u_3\nB : Type u_2\nS : Type ?u.1578175\nT : Type ?u.1578178\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n⊢ eval₂ (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0",
"tactic": "exact hp'"
}
] |
[
1078,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1074,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.isCountablyGenerated_bot
|
[] |
[
1188,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1187,
1
] |
Mathlib/LinearAlgebra/Orientation.lean
|
Orientation.map_eq_det_inv_smul
|
[] |
[
365,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/SetTheory/Ordinal/Topology.lean
|
Ordinal.mem_closure_iff_sup
|
[
{
"state_after": "no goals",
"state_before": "s : Set Ordinal\na : Ordinal\n⊢ (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) ↔ ∃ ι x f, (∀ (i : ι), f i ∈ s) ∧ sup f = a",
"tactic": "simp only [exists_prop]"
}
] |
[
132,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Probability/Integration.lean
|
ProbabilityTheory.IndepFun.integrable_left_of_integrable_mul
|
[
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\n⊢ HasFiniteIntegral X",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\n⊢ Integrable X",
"tactic": "refine' ⟨hX, _⟩"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\n⊢ HasFiniteIntegral X",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\n⊢ HasFiniteIntegral X",
"tactic": "have I : (∫⁻ ω, ‖Y ω‖₊ ∂μ) ≠ 0 := fun H ↦ by\n have I : (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.ennnorm).1 H\n apply h'Y\n filter_upwards [I] with ω hω\n simpa using hω"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\n⊢ HasFiniteIntegral X",
"tactic": "refine lt_top_iff_ne_top.2 fun H => ?_"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\n⊢ False",
"tactic": "have J : IndepFun (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) μ := by\n have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal\n apply IndepFun.comp hXY M M"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω * Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\n⊢ False",
"tactic": "have A : (∫⁻ ω, ‖X ω * Y ω‖₊ ∂μ) < ∞ := h'XY.2"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ * ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω * Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"tactic": "simp only [nnnorm_mul, ENNReal.coe_mul] at A"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (⊤ * ∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ * ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"tactic": "rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.ennnorm hY.ennnorm J, H] at A"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (⊤ * ∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"tactic": "simp only [ENNReal.top_mul I] at A"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\n⊢ False",
"tactic": "have I : (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.ennnorm).1 H"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\n⊢ Y =ᵐ[μ] 0",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\n⊢ False",
"tactic": "apply h'Y"
},
{
"state_after": "case h\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\nω : Ω\nhω : ↑‖Y ω‖₊ = OfNat.ofNat 0 ω\n⊢ Y ω = OfNat.ofNat 0 ω",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\n⊢ Y =ᵐ[μ] 0",
"tactic": "filter_upwards [I] with ω hω"
},
{
"state_after": "no goals",
"state_before": "case h\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖Y ω‖₊) =ᵐ[μ] 0\nω : Ω\nhω : ↑‖Y ω‖₊ = OfNat.ofNat 0 ω\n⊢ Y ω = OfNat.ofNat 0 ω",
"tactic": "simpa using hω"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nM : Measurable fun x => ↑‖x‖₊\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"tactic": "have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'Y : ¬Y =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖Y ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖X a‖₊ ∂μ) = ⊤\nM : Measurable fun x => ↑‖x‖₊\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"tactic": "apply IndepFun.comp hXY M M"
}
] |
[
177,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_ge_of_forall_exists_gt
|
[] |
[
62,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.isCobounded_ge_of_top
|
[] |
[
239,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Topology/LocalExtr.lean
|
isLocalMin_const
|
[] |
[
203,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.SummableFamily.sub_apply
|
[] |
[
1563,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1562,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM_StateRefT_eq
|
[
{
"state_after": "no goals",
"state_before": "m : Type → Type\nω σ α✝ : Type\np : α✝ → Prop\nx : StateRefT' ω σ m α✝\ninst✝ : Monad m\n⊢ SatisfiesM p x ↔ ∀ (s : ST.Ref ω σ), SatisfiesM p (x s)",
"tactic": "simp"
}
] |
[
200,
80
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
199,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_lt_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.126138\nf : α → β\ns t : Multiset α\nh : s < t\nH : map f t ≤ map f s\n⊢ ↑card t ≤ ↑card s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.126138\nf : α → β\ns t : Multiset α\nh : s < t\n⊢ map f s < map f t",
"tactic": "refine' (map_le_map h.le).lt_of_not_le fun H => h.ne <| eq_of_le_of_card_le h.le _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.126138\nf : α → β\ns t : Multiset α\nh : s < t\nH : map f t ≤ map f s\n⊢ ↑card (map f t) ≤ ↑card (map f s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.126138\nf : α → β\ns t : Multiset α\nh : s < t\nH : map f t ≤ map f s\n⊢ ↑card t ≤ ↑card s",
"tactic": "rw [← s.card_map f, ← t.card_map f]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.126138\nf : α → β\ns t : Multiset α\nh : s < t\nH : map f t ≤ map f s\n⊢ ↑card (map f t) ≤ ↑card (map f s)",
"tactic": "exact card_le_of_le H"
}
] |
[
1316,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1313,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.27932\nι' : Sort ?u.27935\nf✝ : α → β\ns t : Set α\nβ' : Type u_1\nf : β → γ\ng : α → β\nf' : α → β'\ng' : β' → γ\nh_comm : ∀ (a : α), f (g a) = g' (f' a)\n⊢ f '' (g '' s) = g' '' (f' '' s)",
"tactic": "simp_rw [image_image, h_comm]"
}
] |
[
285,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
Metric.Bounded.measure_lt_top
|
[] |
[
3949,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3945,
1
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
|
ZMod.χ₈'_int_eq_χ₄_mul_χ₈
|
[
{
"state_after": "a : ℤ\n⊢ ↑χ₈' ↑a = ↑χ₄ ↑↑a * ↑χ₈ ↑a",
"state_before": "a : ℤ\n⊢ ↑χ₈' ↑a = ↑χ₄ ↑a * ↑χ₈ ↑a",
"tactic": "rw [← @cast_int_cast 8 (ZMod 4) _ 4 _ (by norm_num) a]"
},
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ ↑χ₈' ↑a = ↑χ₄ ↑↑a * ↑χ₈ ↑a",
"tactic": "exact χ₈'_eq_χ₄_mul_χ₈ a"
},
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ 4 ∣ 8",
"tactic": "norm_num"
}
] |
[
221,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
NNReal.geom_mean_le_arith_mean3_weighted
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nw₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0\n⊢ w₁ + w₂ + w₃ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃",
"tactic": "simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,\n Fintype.univ_of_isEmpty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,\n mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]"
}
] |
[
204,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Data/List/AList.lean
|
AList.keys_empty
|
[] |
[
127,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Std/Data/List/Lemmas.lean
|
List.eq_or_ne_mem_of_mem
|
[] |
[
82,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
81,
1
] |
Mathlib/Data/Sum/Order.lean
|
WithTop.orderIsoSumLexPUnit_symm_inr
|
[] |
[
795,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.toEquiv_eq_coe
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
IsCompact.exists_sInf_image_eq
|
[] |
[
222,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.addVal_add
|
[] |
[
479,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_apply
|
[
{
"state_after": "case mk\nl : Type ?u.196229\nm : Type ?u.196232\nn : Type ?u.196235\no : Type u_4\np : Type ?u.196241\nq : Type ?u.196244\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.196259\nR : Type ?u.196262\nS : Type ?u.196265\nα : Type u_3\nβ : Type ?u.196271\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\njk : (i : o) × n' i\nfst✝ : o\nsnd✝ : m' fst✝\n⊢ blockDiagonal' M { fst := fst✝, snd := snd✝ } jk =\n if h : { fst := fst✝, snd := snd✝ }.fst = jk.fst then\n M { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd\n (cast (_ : n' jk.fst = n' { fst := fst✝, snd := snd✝ }.fst) jk.snd)\n else 0",
"state_before": "l : Type ?u.196229\nm : Type ?u.196232\nn : Type ?u.196235\no : Type u_4\np : Type ?u.196241\nq : Type ?u.196244\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.196259\nR : Type ?u.196262\nS : Type ?u.196265\nα : Type u_3\nβ : Type ?u.196271\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nik : (i : o) × m' i\njk : (i : o) × n' i\n⊢ blockDiagonal' M ik jk = if h : ik.fst = jk.fst then M ik.fst ik.snd (cast (_ : n' jk.fst = n' ik.fst) jk.snd) else 0",
"tactic": "cases ik"
},
{
"state_after": "case mk.mk\nl : Type ?u.196229\nm : Type ?u.196232\nn : Type ?u.196235\no : Type u_4\np : Type ?u.196241\nq : Type ?u.196244\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.196259\nR : Type ?u.196262\nS : Type ?u.196265\nα : Type u_3\nβ : Type ?u.196271\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nfst✝¹ : o\nsnd✝¹ : m' fst✝¹\nfst✝ : o\nsnd✝ : n' fst✝\n⊢ blockDiagonal' M { fst := fst✝¹, snd := snd✝¹ } { fst := fst✝, snd := snd✝ } =\n if h : { fst := fst✝¹, snd := snd✝¹ }.fst = { fst := fst✝, snd := snd✝ }.fst then\n M { fst := fst✝¹, snd := snd✝¹ }.fst { fst := fst✝¹, snd := snd✝¹ }.snd\n (cast (_ : n' { fst := fst✝, snd := snd✝ }.fst = n' { fst := fst✝¹, snd := snd✝¹ }.fst)\n { fst := fst✝, snd := snd✝ }.snd)\n else 0",
"state_before": "case mk\nl : Type ?u.196229\nm : Type ?u.196232\nn : Type ?u.196235\no : Type u_4\np : Type ?u.196241\nq : Type ?u.196244\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.196259\nR : Type ?u.196262\nS : Type ?u.196265\nα : Type u_3\nβ : Type ?u.196271\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\njk : (i : o) × n' i\nfst✝ : o\nsnd✝ : m' fst✝\n⊢ blockDiagonal' M { fst := fst✝, snd := snd✝ } jk =\n if h : { fst := fst✝, snd := snd✝ }.fst = jk.fst then\n M { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd\n (cast (_ : n' jk.fst = n' { fst := fst✝, snd := snd✝ }.fst) jk.snd)\n else 0",
"tactic": "cases jk"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nl : Type ?u.196229\nm : Type ?u.196232\nn : Type ?u.196235\no : Type u_4\np : Type ?u.196241\nq : Type ?u.196244\nm' : o → Type u_1\nn' : o → Type u_2\np' : o → Type ?u.196259\nR : Type ?u.196262\nS : Type ?u.196265\nα : Type u_3\nβ : Type ?u.196271\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : (i : o) → Matrix (m' i) (n' i) α\nfst✝¹ : o\nsnd✝¹ : m' fst✝¹\nfst✝ : o\nsnd✝ : n' fst✝\n⊢ blockDiagonal' M { fst := fst✝¹, snd := snd✝¹ } { fst := fst✝, snd := snd✝ } =\n if h : { fst := fst✝¹, snd := snd✝¹ }.fst = { fst := fst✝, snd := snd✝ }.fst then\n M { fst := fst✝¹, snd := snd✝¹ }.fst { fst := fst✝¹, snd := snd✝¹ }.snd\n (cast (_ : n' { fst := fst✝, snd := snd✝ }.fst = n' { fst := fst✝¹, snd := snd✝¹ }.fst)\n { fst := fst✝, snd := snd✝ }.snd)\n else 0",
"tactic": "rfl"
}
] |
[
654,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.MemLeftTransversals.inv_toFun_mul_mem
|
[] |
[
375,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieEquiv.to_linearEquiv_mk
|
[] |
[
566,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_comap
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter β\nm : α → β\n⊢ f ⊓ 𝓟 (range m) ≤ map m (comap m f)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter β\nm : α → β\n⊢ map m (comap m f) = f ⊓ 𝓟 (range m)",
"tactic": "refine' le_antisymm (le_inf map_comap_le <| le_principal_iff.2 range_mem_map) _"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\n⊢ t' ∈ f ⊓ 𝓟 (range m)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter β\nm : α → β\n⊢ f ⊓ 𝓟 (range m) ≤ map m (comap m f)",
"tactic": "rintro t' ⟨t, ht, sub⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\n⊢ t ⊆ {x | x ∈ range m → x ∈ t'}",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\n⊢ t' ∈ f ⊓ 𝓟 (range m)",
"tactic": "refine' mem_inf_principal.2 (mem_of_superset ht _)"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\nx : α\nhxt : m x ∈ t\n⊢ m x ∈ t'",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\n⊢ t ⊆ {x | x ∈ range m → x ∈ t'}",
"tactic": "rintro _ hxt ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.256455\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt✝ : Set β\nf : Filter β\nm : α → β\nt' t : Set β\nht : t ∈ f\nsub : m ⁻¹' t ⊆ m ⁻¹' t'\nx : α\nhxt : m x ∈ t\n⊢ m x ∈ t'",
"tactic": "exact sub hxt"
}
] |
[
2252,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2247,
1
] |
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