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/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.IsHomogeneous.iInf₂
|
[] |
[
301,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.coconePointsIsoOfNatIso_hom_desc
|
[
{
"state_after": "no goals",
"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✝ : Cocone F✝\nF G : J ⥤ C\ns : Cocone F\nr t : Cocone G\nP : IsColimit s\nQ : IsColimit t\nw : F ≅ G\n⊢ ∀ (j : J), s.ι.app j ≫ (coconePointsIsoOfNatIso P Q w).hom ≫ desc Q r = s.ι.app j ≫ map P r w.hom",
"tactic": "simp"
}
] |
[
844,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.mem_normalizedFactors_eq_of_associated
|
[
{
"state_after": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\na b c : α\nha : a ∈ normalizedFactors c\nhb : b ∈ normalizedFactors c\nh : a ~ᵤ b\n⊢ a ∣ b ∧ b ∣ a",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\na b c : α\nha : a ∈ normalizedFactors c\nhb : b ∈ normalizedFactors c\nh : a ~ᵤ b\n⊢ a = b",
"tactic": "rw [← normalize_normalized_factor a ha, ← normalize_normalized_factor b hb,\n normalize_eq_normalize_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\na b c : α\nha : a ∈ normalizedFactors c\nhb : b ∈ normalizedFactors c\nh : a ~ᵤ b\n⊢ a ∣ b ∧ b ∣ a",
"tactic": "exact Associated.dvd_dvd h"
}
] |
[
774,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
Interval.length_sub_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.392141\nα : Type u_1\ninst✝ : OrderedAddCommGroup α\ns t : Interval α\na : α\n⊢ length (s - t) ≤ length s + length t",
"tactic": "simpa [sub_eq_add_neg] using length_add_le s (-t)"
}
] |
[
717,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.of_smul
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc : ℝ\nhf : Integrable I l (c • f) vol\nhc : c ≠ 0\n⊢ Integrable I l f vol",
"tactic": "simpa [inv_smul_smul₀ hc] using hf.smul c⁻¹"
}
] |
[
362,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean
|
Polynomial.coeff_sub_eq_neg_right_of_lt
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Ring R\np q : R[X]\ndf : natDegree p < n\n⊢ coeff (p - q) n = -coeff q n",
"tactic": "rwa [sub_eq_add_neg, coeff_add_eq_right_of_lt, coeff_neg]"
}
] |
[
316,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.coe_bot
|
[] |
[
670,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/CategoryTheory/Sites/CoverLifting.lean
|
CategoryTheory.RanIsSheafOfCoverLifting.getSection_commute
|
[
{
"state_after": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\n⊢ IsAmalgamation (pulledbackFamily ℱ S x Z) (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right)",
"state_before": "C D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\n⊢ getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right = getSection hu ℱ hS hx Z",
"tactic": "apply getSection_is_unique"
},
{
"state_after": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\n⊢ IsAmalgamation (pulledbackFamily ℱ S x Z) (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right)",
"tactic": "intro V' fV' hV'"
},
{
"state_after": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'",
"tactic": "have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by\n convert f.w\n erw [Category.id_comp]"
},
{
"state_after": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'✝",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'",
"tactic": "rw [eq] at hV'"
},
{
"state_after": "case h.e'_2\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map (fV' ≫ f.right.unop).op (getSection hu ℱ hS hx Y)\n\ncase h.e'_3\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ pulledbackFamily ℱ S x Z fV' hV'✝ = pulledbackFamily ℱ S x Y (fV' ≫ f.right.unop) ?H\n\ncase H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' ≫ f.right.unop)",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n pulledbackFamily ℱ S x Z fV' hV'✝",
"tactic": "convert getSection_isAmalgamation hu ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1"
},
{
"state_after": "case h.e'_2.h\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\ne_1✝ :\n ((Functor.fromPUnit U.op).obj Z.left ⟶ (Functor.op G).obj Z.right) =\n ((Functor.fromPUnit U.op).obj Y.left ⟶ (Functor.op G).obj Z.right)\n⊢ Z.hom = (Functor.fromPUnit U.op).map f.left ≫ Z.hom",
"state_before": "C D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\n⊢ Z.hom = Y.hom ≫ (G.map f.right.unop).op",
"tactic": "convert f.w"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\ne_1✝ :\n ((Functor.fromPUnit U.op).obj Z.left ⟶ (Functor.op G).obj Z.right) =\n ((Functor.fromPUnit U.op).obj Y.left ⟶ (Functor.op G).obj Z.right)\n⊢ Z.hom = (Functor.fromPUnit U.op).map f.left ≫ Z.hom",
"tactic": "erw [Category.id_comp]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map fV'.op (getSection hu ℱ hS hx Y ≫ ℱ.val.map f.right) =\n ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj X.op).map (fV' ≫ f.right.unop).op (getSection hu ℱ hS hx Y)",
"tactic": "aesop_cat"
},
{
"state_after": "case h.e'_3\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ x (G.map fV' ≫ Z.hom.unop) hV'✝ ≫ ((Ran.adjunction A (Functor.op G)).counit.app ℱ.val).app V'.op =\n x (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop) ?H ≫ ((Ran.adjunction A (Functor.op G)).counit.app ℱ.val).app V'.op\n\ncase H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' ≫ f.right.unop)\n\ncase H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' ≫ f.right.unop)",
"state_before": "case h.e'_3\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ pulledbackFamily ℱ S x Z fV' hV'✝ = pulledbackFamily ℱ S x Y (fV' ≫ f.right.unop) ?H",
"tactic": "rw [pulledbackFamily_apply, pulledbackFamily_apply]"
},
{
"state_after": "case h.e'_3.e_a.e_f\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ G.map fV' ≫ Z.hom.unop = G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop",
"state_before": "case h.e'_3\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ x (G.map fV' ≫ Z.hom.unop) hV'✝ ≫ ((Ran.adjunction A (Functor.op G)).counit.app ℱ.val).app V'.op =\n x (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop) ?H ≫ ((Ran.adjunction A (Functor.op G)).counit.app ℱ.val).app V'.op",
"tactic": "congr 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.e_a.e_f\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ G.map fV' ≫ Z.hom.unop = G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop",
"tactic": "simp [eq]"
},
{
"state_after": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ S.arrows (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop)",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' ≫ f.right.unop)",
"tactic": "change S (G.map _ ≫ Y.hom.unop)"
},
{
"state_after": "no goals",
"state_before": "case H\nC D : Type u\ninst✝³ : Category C\ninst✝² : Category D\nA : Type w\ninst✝¹ : Category A\ninst✝ : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nhu : CoverLifting J K G\nℱ : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S ∈ GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran (Functor.op G)).obj ℱ.val ⋙ coyoneda.obj X.op) S.arrows\nhx : Compatible x\nY Z : StructuredArrow U.op (Functor.op G)\nf : Y ⟶ Z\nV' : C\nfV' : V' ⟶ Z.right.unop\nhV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom ≫ (G.map f.right.unop).op\n⊢ S.arrows (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop)",
"tactic": "simpa only [Functor.map_comp, Category.assoc] using hV'"
}
] |
[
201,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Topology/Separation.lean
|
t1Space_iff_disjoint_pure_nhds
|
[] |
[
512,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
CochainComplex.prev
|
[] |
[
147,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.T_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nm n : ℤ\n⊢ T (m + n) = T m * T n",
"tactic": "simp [T, single_mul_single]"
}
] |
[
180,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_union
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.87559\nγ : Type ?u.87562\ns t : Set α\ninst✝³ : DecidableEq α\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ∪ t)\na✝ : α\n⊢ a✝ ∈ toFinset (s ∪ t) ↔ a✝ ∈ toFinset s ∪ toFinset t",
"state_before": "α : Type u_1\nβ : Type ?u.87559\nγ : Type ?u.87562\ns t : Set α\ninst✝³ : DecidableEq α\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ∪ t)\n⊢ toFinset (s ∪ t) = toFinset s ∪ toFinset t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.87559\nγ : Type ?u.87562\ns t : Set α\ninst✝³ : DecidableEq α\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ∪ t)\na✝ : α\n⊢ a✝ ∈ toFinset (s ∪ t) ↔ a✝ ∈ toFinset s ∪ toFinset t",
"tactic": "simp"
}
] |
[
705,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
ContinuousLinearMap.measurable_apply₂
|
[] |
[
95,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_bit1
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.145698\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\na : G\nn : ℤ\n⊢ a ^ bit1 n = a ^ n * a ^ n * a",
"tactic": "rw [bit1, zpow_add, zpow_bit0, zpow_one]"
}
] |
[
266,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.SignedMeasure.rnDeriv_add
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.653242\nm : MeasurableSpace α\nμ✝ ν : Measure α\ns✝ t✝ s t : SignedMeasure α\nμ : Measure α\ninst✝² : HaveLebesgueDecomposition s μ\ninst✝¹ : HaveLebesgueDecomposition t μ\ninst✝ : HaveLebesgueDecomposition (s + t) μ\n⊢ withDensityᵥ μ (rnDeriv (s + t) μ) = withDensityᵥ μ (rnDeriv s μ + rnDeriv t μ)",
"state_before": "α : Type u_1\nβ : Type ?u.653242\nm : MeasurableSpace α\nμ✝ ν : Measure α\ns✝ t✝ s t : SignedMeasure α\nμ : Measure α\ninst✝² : HaveLebesgueDecomposition s μ\ninst✝¹ : HaveLebesgueDecomposition t μ\ninst✝ : HaveLebesgueDecomposition (s + t) μ\n⊢ rnDeriv (s + t) μ =ᶠ[ae μ] rnDeriv s μ + rnDeriv t μ",
"tactic": "refine'\n Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)\n ((integrable_rnDeriv _ _).add (integrable_rnDeriv _ _)) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.653242\nm : MeasurableSpace α\nμ✝ ν : Measure α\ns✝ t✝ s t : SignedMeasure α\nμ : Measure α\ninst✝² : HaveLebesgueDecomposition s μ\ninst✝¹ : HaveLebesgueDecomposition t μ\ninst✝ : HaveLebesgueDecomposition (s + t) μ\n⊢ withDensityᵥ μ (rnDeriv (s + t) μ) = withDensityᵥ μ (rnDeriv s μ + rnDeriv t μ)",
"tactic": "rw [← add_right_inj ((s + t).singularPart μ), singularPart_add_withDensity_rnDeriv_eq,\n withDensityᵥ_add (integrable_rnDeriv _ _) (integrable_rnDeriv _ _), singularPart_add,\n add_assoc, add_comm (t.singularPart μ), add_assoc, add_comm _ (t.singularPart μ),\n singularPart_add_withDensity_rnDeriv_eq, ← add_assoc,\n singularPart_add_withDensity_rnDeriv_eq]"
}
] |
[
1177,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
1
] |
Mathlib/Algebra/Hom/Aut.lean
|
AddAut.one_def
|
[] |
[
212,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Absorbs.mono_right
|
[] |
[
82,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/Calculus/Deriv/Linear.lean
|
ContinuousLinearMap.hasDerivAtFilter
|
[] |
[
53,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycle_swap_mul_aux₂
|
[
{
"state_after": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℤ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\n⊢ ∀ (n : ℤ) {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"tactic": "intro n"
},
{
"state_after": "case ofNat\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ Int.ofNat n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b\n\ncase negSucc\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ Int.negSucc n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℤ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"tactic": "induction' n with n n"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ Int.ofNat n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"tactic": "exact isCycle_swap_mul_aux₁ n"
},
{
"state_after": "case negSucc\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\n⊢ ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"state_before": "case negSucc\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf g : Perm α✝\nx y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\n⊢ ∀ {b x : α} {f : Perm α},\n ↑(swap x (↑f x) * f) b ≠ b → ↑(f ^ Int.negSucc n) (↑f x) = b → ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"tactic": "intro b x f hb h"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\n⊢ ∃ i, ↑((swap x (↑f x) * f) ^ i) (↑f x) = b",
"tactic": "exact if hfbx' : f x = b then ⟨0, hfbx'⟩\n else\n have : f b ≠ b ∧ b ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hb\n have hb : (swap x (f⁻¹ x) * f⁻¹) (f⁻¹ b) ≠ f⁻¹ b := by\n rw [mul_apply, swap_apply_def]\n split_ifs <;>\n simp only [inv_eq_iff_eq, Perm.mul_apply, zpow_negSucc,\n Ne.def, Perm.apply_inv_self] at\n * <;> tauto\n let ⟨i, hi⟩ :=\n isCycle_swap_mul_aux₁ n hb\n (show (f⁻¹ ^ n) (f⁻¹ x) = f⁻¹ b by\n rw [← zpow_ofNat, ← h, ← mul_apply, ← mul_apply, ← mul_apply, zpow_negSucc, ← inv_pow,\n pow_succ', mul_assoc, mul_assoc, inv_mul_self, mul_one, zpow_ofNat, ← pow_succ', ←\n pow_succ])\n have h : (swap x (f⁻¹ x) * f⁻¹) (f x) = f⁻¹ x := by\n rw [mul_apply, inv_apply_self, swap_apply_left]\n ⟨-i, by\n rw [← add_sub_cancel i 1, neg_sub, sub_eq_add_neg, zpow_add, zpow_one, zpow_neg,\n ← inv_zpow, mul_inv_rev, swap_inv, mul_swap_eq_swap_mul, inv_apply_self, swap_comm _ x,\n zpow_add, zpow_one, mul_apply, mul_apply (_ ^ i), h, hi, mul_apply, apply_inv_self,\n swap_apply_of_ne_of_ne this.2 (Ne.symm hfbx')]⟩"
},
{
"state_after": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\n⊢ (if ↑f⁻¹ (↑f⁻¹ b) = x then ↑f⁻¹ x else if ↑f⁻¹ (↑f⁻¹ b) = ↑f⁻¹ x then x else ↑f⁻¹ (↑f⁻¹ b)) ≠ ↑f⁻¹ b",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\n⊢ ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f⁻¹ b) ≠ ↑f⁻¹ b",
"tactic": "rw [mul_apply, swap_apply_def]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\n⊢ (if ↑f⁻¹ (↑f⁻¹ b) = x then ↑f⁻¹ x else if ↑f⁻¹ (↑f⁻¹ b) = ↑f⁻¹ x then x else ↑f⁻¹ (↑f⁻¹ b)) ≠ ↑f⁻¹ b",
"tactic": "split_ifs <;>\n simp only [inv_eq_iff_eq, Perm.mul_apply, zpow_negSucc,\n Ne.def, Perm.apply_inv_self] at\n * <;> tauto"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb✝ : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\nhb : ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f⁻¹ b) ≠ ↑f⁻¹ b\n⊢ ↑(f⁻¹ ^ n) (↑f⁻¹ x) = ↑f⁻¹ b",
"tactic": "rw [← zpow_ofNat, ← h, ← mul_apply, ← mul_apply, ← mul_apply, zpow_negSucc, ← inv_pow,\n pow_succ', mul_assoc, mul_assoc, inv_mul_self, mul_one, zpow_ofNat, ← pow_succ', ←\n pow_succ]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb✝ : ↑(swap x (↑f x) * f) b ≠ b\nh : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\nhb : ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f⁻¹ b) ≠ ↑f⁻¹ b\ni : ℤ\nhi : ↑((swap x (↑f⁻¹ x) * f⁻¹) ^ i) (↑f⁻¹ x) = ↑f⁻¹ b\n⊢ ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f x) = ↑f⁻¹ x",
"tactic": "rw [mul_apply, inv_apply_self, swap_apply_left]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.676454\nα✝ : Type ?u.676457\nβ : Type ?u.676460\nf✝ g : Perm α✝\nx✝ y : α✝\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nhb✝ : ↑(swap x (↑f x) * f) b ≠ b\nh✝ : ↑(f ^ Int.negSucc n) (↑f x) = b\nhfbx' : ¬↑f x = b\nthis : ↑f b ≠ b ∧ b ≠ x\nhb : ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f⁻¹ b) ≠ ↑f⁻¹ b\ni : ℤ\nhi : ↑((swap x (↑f⁻¹ x) * f⁻¹) ^ i) (↑f⁻¹ x) = ↑f⁻¹ b\nh : ↑(swap x (↑f⁻¹ x) * f⁻¹) (↑f x) = ↑f⁻¹ x\n⊢ ↑((swap x (↑f x) * f) ^ (-i)) (↑f x) = b",
"tactic": "rw [← add_sub_cancel i 1, neg_sub, sub_eq_add_neg, zpow_add, zpow_one, zpow_neg,\n ← inv_zpow, mul_inv_rev, swap_inv, mul_swap_eq_swap_mul, inv_apply_self, swap_comm _ x,\n zpow_add, zpow_one, mul_apply, mul_apply (_ ^ i), h, hi, mul_apply, apply_inv_self,\n swap_apply_of_ne_of_ne this.2 (Ne.symm hfbx')]"
}
] |
[
495,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Function.Embedding.nonempty_of_card_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.50772\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nh : Fintype.card α ≤ Fintype.card β\n⊢ Nonempty (α ↪ β)",
"tactic": "classical exact (truncOfCardLe h).nonempty"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.50772\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nh : Fintype.card α ≤ Fintype.card β\n⊢ Nonempty (α ↪ β)",
"tactic": "exact (truncOfCardLe h).nonempty"
}
] |
[
798,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
797,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.vars_one
|
[] |
[
369,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/NumberTheory/FermatPsp.lean
|
FermatPsp.psp_from_prime_gt_p
|
[
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\n⊢ p < (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1))",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\n⊢ p < FermatPsp.psp_from_prime b p",
"tactic": "unfold psp_from_prime"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\n⊢ p < A * ((b ^ p + 1) / (b + 1))",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\n⊢ p < (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1))",
"tactic": "set A := (b ^ p - 1) / (b - 1)"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\n⊢ p < A * B",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\n⊢ p < A * ((b ^ p + 1) / (b + 1))",
"tactic": "set B := (b ^ p + 1) / (b + 1)"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\n⊢ p < A * B",
"tactic": "rw [show A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) from\n AB_id_helper _ _ b_ge_two (p_prime.odd_of_ne_two p_gt_two.ne.symm)]"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"tactic": "have AB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1 := by\n simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\ncase h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * (b ^ 2 - 1) < b ^ (2 * p) - 1",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"tactic": "suffices h : p * (b ^ 2 - 1) < b ^ (2 * p) - 1"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < (b ^ 2) ^ p - 1",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * (b ^ 2 - 1) < b ^ (2 * p) - 1",
"tactic": "rw [Nat.mul_sub_left_distrib, mul_one, pow_mul]"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < (b ^ 2) ^ (p - 1 + 1) - 1",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < (b ^ 2) ^ p - 1",
"tactic": "conv_rhs => rw [← Nat.sub_add_cancel (show 1 ≤ p by linarith)]"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < b ^ 2 * (b ^ 2) ^ (p - 1) - 1",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < (b ^ 2) ^ (p - 1 + 1) - 1",
"tactic": "rw [pow_succ (b ^ 2)]"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 - p < b ^ 2 * (b ^ 2) ^ (p - 1) - 1\n\ncase h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 - p < b ^ 2 * (b ^ 2) ^ (p - 1) - 1",
"tactic": "suffices h : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n\ncase h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < (b ^ 2) ^ (p - 1)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)",
"tactic": "suffices h : p < (b ^ 2) ^ (p - 1)"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < b ^ (2 * p - 2)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < (b ^ 2) ^ (p - 1)",
"tactic": "rw [← pow_mul, Nat.mul_sub_left_distrib, mul_one]"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis : 2 ≤ 2 * p - 2\n⊢ p < b ^ (2 * p - 2)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ p < b ^ (2 * p - 2)",
"tactic": "have : 2 ≤ 2 * p - 2 := le_tsub_of_add_le_left (show 4 ≤ 2 * p by linarith)"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis✝ : 2 ≤ 2 * p - 2\nthis : 2 + p ≤ 2 * p\n⊢ p < b ^ (2 * p - 2)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis : 2 ≤ 2 * p - 2\n⊢ p < b ^ (2 * p - 2)",
"tactic": "have : 2 + p ≤ 2 * p := by linarith"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis✝¹ : 2 ≤ 2 * p - 2\nthis✝ : 2 + p ≤ 2 * p\nthis : p ≤ 2 * p - 2\n⊢ p < b ^ (2 * p - 2)",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis✝ : 2 ≤ 2 * p - 2\nthis : 2 + p ≤ 2 * p\n⊢ p < b ^ (2 * p - 2)",
"tactic": "have : p ≤ 2 * p - 2 := le_tsub_of_add_le_left this"
},
{
"state_after": "no goals",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis✝¹ : 2 ≤ 2 * p - 2\nthis✝ : 2 + p ≤ 2 * p\nthis : p ≤ 2 * p - 2\n⊢ p < b ^ (2 * p - 2)",
"tactic": "exact Nat.lt_of_le_of_lt this (pow_gt_exponent _ b_ge_two)"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\n⊢ b ^ 2 - 1 ∣ b ^ (2 * p) - 1",
"tactic": "simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"tactic": "have h₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1) :=\n Nat.div_lt_div_of_lt_of_dvd AB_dvd h"
},
{
"state_after": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nh₂ : 0 < b ^ 2 - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"tactic": "have h₂ : 0 < b ^ 2 - 1 := by\n linarith [show 3 ≤ b ^ 2 - 1 from le_tsub_of_add_le_left (show 4 ≤ b ^ 2 by nlinarith)]"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nh₂ : 0 < b ^ 2 - 1\n⊢ p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)",
"tactic": "rwa [Nat.mul_div_cancel _ h₂] at h₁"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n⊢ 0 < b ^ 2 - 1",
"tactic": "linarith [show 3 ≤ b ^ 2 - 1 from le_tsub_of_add_le_left (show 4 ≤ b ^ 2 by nlinarith)]"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh₁ : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n⊢ 4 ≤ b ^ 2",
"tactic": "nlinarith"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ 1 ≤ p",
"tactic": "linarith"
},
{
"state_after": "case h.h₁\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ b ^ 2 * (b ^ 2) ^ (p - 1) - 1 ≥ ?h.b\n\ncase h.h₂\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ ?h.b > p * b ^ 2 - p\n\ncase h.b\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ ℕ",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 - p < b ^ 2 * (b ^ 2) ^ (p - 1) - 1",
"tactic": "apply gt_of_ge_of_gt"
},
{
"state_after": "no goals",
"state_before": "case h.h₁\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ b ^ 2 * (b ^ 2) ^ (p - 1) - 1 ≥ ?h.b",
"tactic": "exact tsub_le_tsub_left (show 1 ≤ p by linarith) (b ^ 2 * (b ^ 2) ^ (p - 1))"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ 1 ≤ p",
"tactic": "linarith"
},
{
"state_after": "case h.h₂\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\nthis : p ≤ p * b ^ 2\n⊢ b ^ 2 * (b ^ 2) ^ (p - 1) - p > p * b ^ 2 - p",
"state_before": "case h.h₂\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ b ^ 2 * (b ^ 2) ^ (p - 1) - p > p * b ^ 2 - p",
"tactic": "have : p ≤ p * b ^ 2 := Nat.le_mul_of_pos_right (show 0 < b ^ 2 by nlinarith)"
},
{
"state_after": "no goals",
"state_before": "case h.h₂\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\nthis : p ≤ p * b ^ 2\n⊢ b ^ 2 * (b ^ 2) ^ (p - 1) - p > p * b ^ 2 - p",
"tactic": "exact tsub_lt_tsub_right_of_le this h"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)\n⊢ 0 < b ^ 2",
"tactic": "nlinarith"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 < b ^ 2 * (b ^ 2) ^ (p - 1)",
"tactic": "rw [mul_comm (b ^ 2)]"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis : 4 ≤ b ^ 2\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"tactic": "have : 4 ≤ b ^ 2 := by nlinarith"
},
{
"state_after": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis✝ : 4 ≤ b ^ 2\nthis : 0 < b ^ 2\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis : 4 ≤ b ^ 2\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"tactic": "have : 0 < b ^ 2 := by linarith"
},
{
"state_after": "no goals",
"state_before": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis✝ : 4 ≤ b ^ 2\nthis : 0 < b ^ 2\n⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2",
"tactic": "exact mul_lt_mul_of_pos_right h this"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n⊢ 4 ≤ b ^ 2",
"tactic": "nlinarith"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis : 4 ≤ b ^ 2\n⊢ 0 < b ^ 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\n⊢ 4 ≤ 2 * p",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Nat.Prime p\np_gt_two : 2 < p\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1\nthis : 2 ≤ 2 * p - 2\n⊢ 2 + p ≤ 2 * p",
"tactic": "linarith"
}
] |
[
348,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
9
] |
Std/Data/List/Lemmas.lean
|
List.singleton_disjoint
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\na : α✝\nl : List α✝\n⊢ Disjoint [a] l ↔ ¬a ∈ l",
"tactic": "simp [Disjoint]"
}
] |
[
1378,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1378,
14
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.map_perm
|
[
{
"state_after": "case a\nR : Type u_4\ninst✝¹⁶ : Semiring R\nM : Type u_3\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.456393\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.456423\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.456453\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_2\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_1\nι' : Type ?u.457232\nι'' : Type ?u.457235\nM₂ : Type ?u.457238\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type ?u.457268\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv✝ : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → M\n⊢ ↑g (v ∘ ↑1) = ↑Equiv.Perm.sign 1 • ↑g v\n\ncase a\nR : Type u_4\ninst✝¹⁶ : Semiring R\nM : Type u_3\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.456393\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.456423\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.456453\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_2\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_1\nι' : Type ?u.457232\nι'' : Type ?u.457235\nM₂ : Type ?u.457238\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type ?u.457268\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv✝ : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → M\ns : Equiv.Perm ι\nx y : ι\nhxy : x ≠ y\nhI : ↑g (v ∘ ↑s) = ↑Equiv.Perm.sign s • ↑g v\n⊢ ↑g (v ∘ ↑(s * Equiv.swap x y)) = ↑Equiv.Perm.sign (s * Equiv.swap x y) • ↑g v",
"state_before": "R : Type u_4\ninst✝¹⁶ : Semiring R\nM : Type u_3\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.456393\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.456423\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.456453\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_2\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_1\nι' : Type ?u.457232\nι'' : Type ?u.457235\nM₂ : Type ?u.457238\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type ?u.457268\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv✝ : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → M\nσ : Equiv.Perm ι\n⊢ ↑g (v ∘ ↑σ) = ↑Equiv.Perm.sign σ • ↑g v",
"tactic": "induction' σ using Equiv.Perm.swap_induction_on' with s x y hxy hI"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_4\ninst✝¹⁶ : Semiring R\nM : Type u_3\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.456393\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.456423\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.456453\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_2\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_1\nι' : Type ?u.457232\nι'' : Type ?u.457235\nM₂ : Type ?u.457238\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type ?u.457268\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv✝ : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → M\n⊢ ↑g (v ∘ ↑1) = ↑Equiv.Perm.sign 1 • ↑g v",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_4\ninst✝¹⁶ : Semiring R\nM : Type u_3\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.456393\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.456423\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.456453\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type u_2\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_1\nι' : Type ?u.457232\nι'' : Type ?u.457235\nM₂ : Type ?u.457238\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nM₃ : Type ?u.457268\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv✝ : ι → M\nv' : ι → M'\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → M\ns : Equiv.Perm ι\nx y : ι\nhxy : x ≠ y\nhI : ↑g (v ∘ ↑s) = ↑Equiv.Perm.sign s • ↑g v\n⊢ ↑g (v ∘ ↑(s * Equiv.swap x y)) = ↑Equiv.Perm.sign (s * Equiv.swap x y) • ↑g v",
"tactic": "simpa [← Function.comp.assoc, g.map_swap (v ∘ s) hxy,\n Equiv.Perm.sign_swap hxy, -Equiv.Perm.sign_swap'] using hI"
}
] |
[
714,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
Set.Subsingleton.strictConvex
|
[] |
[
128,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_pow_anti
|
[] |
[
648,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
647,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
holderOnWith_one
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.128620\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\nC : ℝ≥0\nf : X → Y\ns : Set X\n⊢ HolderOnWith C 1 f s ↔ LipschitzOnWith C f s",
"tactic": "simp only [HolderOnWith, LipschitzOnWith, NNReal.coe_one, ENNReal.rpow_one]"
}
] |
[
87,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
Algebra.FiniteType.isNoetherianRing
|
[
{
"state_after": "case intro\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ IsNoetherianRing S",
"state_before": "R✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\n⊢ IsNoetherianRing S",
"tactic": "obtain ⟨s, hs⟩ := h.1"
},
{
"state_after": "case intro.hf\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ Surjective ↑↑(MvPolynomial.aeval Subtype.val)",
"state_before": "case intro\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ IsNoetherianRing S",
"tactic": "apply\n isNoetherianRing_of_surjective (MvPolynomial s R) S\n (MvPolynomial.aeval (↑) : MvPolynomial s R →ₐ[R] S).toRingHom"
},
{
"state_after": "case intro.hf\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ ↑⊤ = Set.univ",
"state_before": "case intro.hf\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ Surjective ↑↑(MvPolynomial.aeval Subtype.val)",
"tactic": "erw [← Set.range_iff_surjective, ← AlgHom.coe_range, ←\n Algebra.adjoin_range_eq_range_aeval, Subtype.range_coe_subtype, Finset.setOf_mem, hs]"
},
{
"state_after": "no goals",
"state_before": "case intro.hf\nR✝ : Type ?u.65970\nA : Type u\nB : Type ?u.65961\nM : Type ?u.65964\nN : Type ?u.65967\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R✝ A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra R✝ B\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R✝ M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R✝ N\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nh : FiniteType R S\ninst✝ : IsNoetherianRing R\ns : Finset S\nhs : adjoin R ↑s = ⊤\n⊢ ↑⊤ = Set.univ",
"tactic": "rfl"
}
] |
[
179,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.cliqueFinset_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nG H : SimpleGraph α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel G.Adj\nn : ℕ\na b c : α\ns : Finset α\n⊢ cliqueFinset G n = ∅ ↔ CliqueFree G n",
"tactic": "simp_rw [CliqueFree, eq_empty_iff_forall_not_mem, mem_cliqueFinset_iff]"
}
] |
[
298,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.mkSpanSingleton_apply
|
[] |
[
187,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mod_eq_zero_of_dvd
|
[
{
"state_after": "m n : Nat\nH✝ : m ∣ n\nz : Nat\nH : n = m * z\n⊢ n % m = 0",
"state_before": "m n : Nat\nH : m ∣ n\n⊢ n % m = 0",
"tactic": "let ⟨z, H⟩ := H"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nH✝ : m ∣ n\nz : Nat\nH : n = m * z\n⊢ n % m = 0",
"tactic": "rw [H, mul_mod_right]"
}
] |
[
724,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
723,
1
] |
src/lean/Init/Control/Lawful.lean
|
ReaderT.run_monadLift
|
[] |
[
196,
73
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
195,
9
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.suppPreservation_iff_liftpPreservation
|
[
{
"state_after": "case mp\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\n⊢ LiftpPreservation\n\ncase mpr\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : LiftpPreservation\n⊢ SuppPreservation",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\n⊢ SuppPreservation ↔ LiftpPreservation",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"state_before": "case mp\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\n⊢ LiftpPreservation",
"tactic": "rintro α p ⟨a, f⟩"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : SuppPreservation\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"tactic": "have h' := h"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : SuppPreservation\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"tactic": "rw [suppPreservation_iff_uniform] at h'"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : SuppPreservation\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"tactic": "dsimp only [SuppPreservation, supp] at h"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ (∀ (u : α), u ∈ f '' univ → p u) ↔ ∀ (i : PFunctor.B (P F) a), p (f i)",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ Liftp p (abs { fst := a, snd := f }) ↔ Liftp p { fst := a, snd := f }",
"tactic": "rw [liftp_iff_of_isUniform h', supp_eq_of_isUniform h', PFunctor.liftp_iff']"
},
{
"state_after": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ (∀ (u : α) (x : PFunctor.B (P F) a), f x = u → p u) ↔ ∀ (i : PFunctor.B (P F) a), p (f i)",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ (∀ (u : α), u ∈ f '' univ → p u) ↔ ∀ (i : PFunctor.B (P F) a), p (f i)",
"tactic": "simp only [image_univ, mem_range, exists_imp]"
},
{
"state_after": "no goals",
"state_before": "case mp.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh :\n ∀ ⦃α : Type u⦄ (x : PFunctor.Obj (P F) α),\n {y | ∀ ⦃p : α → Prop⦄, Liftp p (abs x) → p y} = {y | ∀ ⦃p : α → Prop⦄, Liftp p x → p y}\nα : Type u\np : α → Prop\na : (P F).A\nf : PFunctor.B (P F) a → α\nh' : IsUniform\n⊢ (∀ (u : α) (x : PFunctor.B (P F) a), f x = u → p u) ↔ ∀ (i : PFunctor.B (P F) a), p (f i)",
"tactic": "constructor <;> intros <;> subst_vars <;> solve_by_elim"
},
{
"state_after": "case mpr.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : LiftpPreservation\nα : Type u\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"state_before": "case mpr\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : LiftpPreservation\n⊢ SuppPreservation",
"tactic": "rintro α ⟨a, f⟩"
},
{
"state_after": "case mpr.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : ∀ ⦃α : Type u⦄ (p : α → Prop) (x : PFunctor.Obj (P F) α), Liftp p (abs x) ↔ Liftp p x\nα : Type u\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"state_before": "case mpr.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : LiftpPreservation\nα : Type u\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"tactic": "simp only [LiftpPreservation] at h"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nh : ∀ ⦃α : Type u⦄ (p : α → Prop) (x : PFunctor.Obj (P F) α), Liftp p (abs x) ↔ Liftp p x\nα : Type u\na : (P F).A\nf : PFunctor.B (P F) a → α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"tactic": "simp only [supp, h]"
}
] |
[
724,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/CategoryTheory/Functor/Flat.lean
|
CategoryTheory.cofiltered_of_hasFiniteLimits
|
[] |
[
186,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.hasColimit_of_equivalence_comp
|
[
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasColimit (e.functor ⋙ F)\nthis : HasColimit (e.inverse ⋙ e.functor ⋙ F)\n⊢ HasColimit F",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasColimit (e.functor ⋙ F)\n⊢ HasColimit F",
"tactic": "haveI : HasColimit (e.inverse ⋙ e.functor ⋙ F) := Limits.hasColimit_equivalence_comp e.symm"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasColimit (e.functor ⋙ F)\nthis : HasColimit (e.inverse ⋙ e.functor ⋙ F)\n⊢ HasColimit F",
"tactic": "apply hasColimitOfIso (e.invFunIdAssoc F).symm"
}
] |
[
1081,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1079,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.map_equiv_top
|
[] |
[
956,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMapOn.continuousOn
|
[] |
[
104,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
11
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.subset
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\nht : t ⊆ s\na✝ : Fintype ↑s\n⊢ Set.Finite t",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nhs : Set.Finite s\nt : Set α\nht : t ⊆ s\n⊢ Set.Finite t",
"tactic": "cases hs"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\nht : t ⊆ s\na✝ : Fintype ↑s\nthis : Finite ↑t\n⊢ Set.Finite t",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\nht : t ⊆ s\na✝ : Fintype ↑s\n⊢ Set.Finite t",
"tactic": "haveI := Finite.Set.subset _ ht"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\nht : t ⊆ s\na✝ : Fintype ↑s\nthis : Finite ↑t\n⊢ Set.Finite t",
"tactic": "apply toFinite"
}
] |
[
763,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
760,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.mem_join
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2554\nγ : Type ?u.2557\nδ : Type ?u.2560\na : α\nS : Multiset (Multiset α)\n⊢ a ∈ join 0 ↔ ∃ s, s ∈ 0 ∧ a ∈ s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2554\nγ : Type ?u.2557\nδ : Type ?u.2560\na : α\nS : Multiset (Multiset α)\n⊢ ∀ ⦃a_1 : Multiset α⦄ {s : Multiset (Multiset α)},\n (a ∈ join s ↔ ∃ s_1, s_1 ∈ s ∧ a ∈ s_1) → (a ∈ join (a_1 ::ₘ s) ↔ ∃ s_1, s_1 ∈ a_1 ::ₘ s ∧ a ∈ s_1)",
"tactic": "simp (config := { contextual := true }) [or_and_right, exists_or]"
}
] |
[
72,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
HasFDerivAt.continuousAt
|
[] |
[
728,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.int_cast_im
|
[] |
[
460,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_le_iff_coeff_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.702057\nf : R[X]\nn : WithBot ℕ\n⊢ degree f ≤ n ↔ ∀ (m : ℕ), n < ↑m → coeff f m = 0",
"tactic": "simp only [degree, Finset.max, Finset.sup_le_iff, mem_support_iff, Ne.def, ← not_le,\n not_imp_comm, Nat.cast_withBot]"
}
] |
[
1082,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/Topology/Basic.lean
|
IsOpen.inter_frontier_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsOpen s\n⊢ s ∩ frontier s = ∅",
"tactic": "rw [hs.frontier_eq, inter_diff_self]"
}
] |
[
765,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
chartedSpaceSelf_atlas
|
[] |
[
531,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_basis_open
|
[] |
[
91,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Order/Chain.lean
|
chainClosure_empty
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.10878\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nthis : ChainClosure r (⋃₀ ∅)\n⊢ ChainClosure r ∅",
"state_before": "α : Type u_1\nβ : Type ?u.10878\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\n⊢ ChainClosure r ∅",
"tactic": "have : ChainClosure r (⋃₀∅) := ChainClosure.union fun a h => False.rec h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10878\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nthis : ChainClosure r (⋃₀ ∅)\n⊢ ChainClosure r ∅",
"tactic": "simpa using this"
}
] |
[
207,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.liminf_eq_iSup_iInf_of_nat'
|
[] |
[
763,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
762,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mem_graph_iff
|
[
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.1790\nγ : Type ?u.1793\nι : Type ?u.1796\nM : Type u_2\nM' : Type ?u.1802\nN : Type ?u.1805\nP : Type ?u.1808\nG : Type ?u.1811\nH : Type ?u.1814\nR : Type ?u.1817\nS : Type ?u.1820\ninst✝ : Zero M\nf : α →₀ M\nfst✝ : α\nsnd✝ : M\n⊢ (fst✝, snd✝) ∈ graph f ↔ ↑f (fst✝, snd✝).fst = (fst✝, snd✝).snd ∧ (fst✝, snd✝).snd ≠ 0",
"state_before": "α : Type u_1\nβ : Type ?u.1790\nγ : Type ?u.1793\nι : Type ?u.1796\nM : Type u_2\nM' : Type ?u.1802\nN : Type ?u.1805\nP : Type ?u.1808\nG : Type ?u.1811\nH : Type ?u.1814\nR : Type ?u.1817\nS : Type ?u.1820\ninst✝ : Zero M\nc : α × M\nf : α →₀ M\n⊢ c ∈ graph f ↔ ↑f c.fst = c.snd ∧ c.snd ≠ 0",
"tactic": "cases c"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.1790\nγ : Type ?u.1793\nι : Type ?u.1796\nM : Type u_2\nM' : Type ?u.1802\nN : Type ?u.1805\nP : Type ?u.1808\nG : Type ?u.1811\nH : Type ?u.1814\nR : Type ?u.1817\nS : Type ?u.1820\ninst✝ : Zero M\nf : α →₀ M\nfst✝ : α\nsnd✝ : M\n⊢ (fst✝, snd✝) ∈ graph f ↔ ↑f (fst✝, snd✝).fst = (fst✝, snd✝).snd ∧ (fst✝, snd✝).snd ≠ 0",
"tactic": "exact mk_mem_graph_iff"
}
] |
[
83,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_approx_succ
|
[
{
"state_after": "n : ℕ\nx a₁ b₁ : ℝ\nm : ℕ\ne₁ : n + 1 = m\na₂ b₂ : ℝ\ne : abs' (1 + x / ↑m * a₂ - a₁) ≤ b₁ - abs' x / ↑m * b₂\nh : abs' (exp x - expNear m x a₂) ≤ abs' x ^ m / ↑(Nat.factorial m) * b₂\n⊢ abs' x ^ m / ↑(Nat.factorial m) * b₂ + abs' (expNear m x a₂ - expNear n x a₁) ≤ abs' x ^ n / ↑(Nat.factorial n) * b₁",
"state_before": "n : ℕ\nx a₁ b₁ : ℝ\nm : ℕ\ne₁ : n + 1 = m\na₂ b₂ : ℝ\ne : abs' (1 + x / ↑m * a₂ - a₁) ≤ b₁ - abs' x / ↑m * b₂\nh : abs' (exp x - expNear m x a₂) ≤ abs' x ^ m / ↑(Nat.factorial m) * b₂\n⊢ abs' (exp x - expNear n x a₁) ≤ abs' x ^ n / ↑(Nat.factorial n) * b₁",
"tactic": "refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).trans _)"
},
{
"state_after": "n : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ + abs' (expNear (n + 1) x a₂ - expNear n x a₁) ≤\n abs' x ^ n / ↑(Nat.factorial n) * b₁",
"state_before": "n : ℕ\nx a₁ b₁ : ℝ\nm : ℕ\ne₁ : n + 1 = m\na₂ b₂ : ℝ\ne : abs' (1 + x / ↑m * a₂ - a₁) ≤ b₁ - abs' x / ↑m * b₂\nh : abs' (exp x - expNear m x a₂) ≤ abs' x ^ m / ↑(Nat.factorial m) * b₂\n⊢ abs' x ^ m / ↑(Nat.factorial m) * b₂ + abs' (expNear m x a₂ - expNear n x a₁) ≤ abs' x ^ n / ↑(Nat.factorial n) * b₁",
"tactic": "subst e₁"
},
{
"state_after": "n : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +\n abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ - a₁) ≤\n abs' x ^ n / ↑(Nat.factorial n) * b₁",
"state_before": "n : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ + abs' (expNear (n + 1) x a₂ - expNear n x a₁) ≤\n abs' x ^ n / ↑(Nat.factorial n) * b₁",
"tactic": "rw [expNear_succ, expNear_sub, abs_mul]"
},
{
"state_after": "case h.e'_3\nn : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +\n abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ - a₁) =\n abs' x ^ n / ↑(Nat.factorial n) * (abs' x / ↑(n + 1) * b₂ + abs' (1 + x / ↑(n + 1) * a₂ - a₁))\n\nn : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ 0 ≤ abs' x ^ n / ↑(Nat.factorial n)",
"state_before": "n : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +\n abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ - a₁) ≤\n abs' x ^ n / ↑(Nat.factorial n) * b₁",
"tactic": "convert mul_le_mul_of_nonneg_left (a := abs' x ^ n / ↑(Nat.factorial n))\n (le_sub_iff_add_le'.1 e) ?_ using 1"
},
{
"state_after": "case h.e'_3\nn : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ n * abs' x * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * b₂ =\n abs' x ^ n * (↑(Nat.factorial n))⁻¹ * (abs' x * (↑n + 1)⁻¹ * b₂)",
"state_before": "case h.e'_3\nn : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +\n abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ - a₁) =\n abs' x ^ n / ↑(Nat.factorial n) * (abs' x / ↑(n + 1) * b₂ + abs' (1 + x / ↑(n + 1) * a₂ - a₁))",
"tactic": "simp [mul_add, pow_succ', div_eq_mul_inv, abs_mul, abs_inv, ← pow_abs, mul_inv]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nn : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ abs' x ^ n * abs' x * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * b₂ =\n abs' x ^ n * (↑(Nat.factorial n))⁻¹ * (abs' x * (↑n + 1)⁻¹ * b₂)",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx a₁ b₁ a₂ b₂ : ℝ\ne : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂\nh : abs' (exp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂\n⊢ 0 ≤ abs' x ^ n / ↑(Nat.factorial n)",
"tactic": "simp [div_nonneg, abs_nonneg]"
}
] |
[
1785,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1775,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
min_lt_of_mul_lt_sq
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.151798\nA : Type ?u.151801\nG : Type ?u.151804\nM : Type u_1\nR : Type ?u.151810\ninst✝³ : Monoid M\ninst✝² : LinearOrder M\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : M\nh : a * b < c ^ 2\n⊢ min a b < c",
"tactic": "simpa using min_lt_max_of_mul_lt_mul (h.trans_eq <| pow_two _)"
}
] |
[
305,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
Algebra.lsmul_coe
|
[] |
[
57,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.add_le_of_le
|
[] |
[
760,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.biUnion_and
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.88687\nγ : Type ?u.88690\nι : Sort u_2\nι' : Sort u_3\nι₂ : Sort ?u.88699\nκ : ι → Sort ?u.88704\nκ₁ : ι → Sort ?u.88709\nκ₂ : ι → Sort ?u.88714\nκ' : ι' → Sort ?u.88719\np : ι → Prop\nq : ι → ι' → Prop\ns : (x : ι) → (y : ι') → p x ∧ q x y → Set α\n⊢ (⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h) = ⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y (_ : p x ∧ q x y)",
"tactic": "simp only [iUnion_and, @iUnion_comm _ ι']"
}
] |
[
810,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Analysis/Complex/Conformal.lean
|
isConformalMap_complex_linear
|
[
{
"state_after": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ IsConformalMap (restrictScalars ℝ map)",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\n⊢ IsConformalMap (restrictScalars ℝ map)",
"tactic": "have minor₁ : ‖map 1‖ ≠ 0 := by simpa only [ext_ring_iff, Ne.def, norm_eq_zero] using nonzero"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖\n\ncase refine'_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ restrictScalars ℝ map =\n ‖↑map 1‖ • LinearIsometry.toContinuousLinearMap { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := ?refine'_1 }",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ IsConformalMap (restrictScalars ℝ map)",
"tactic": "refine' ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), _⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\n⊢ ‖↑map 1‖ ≠ 0",
"tactic": "simpa only [ext_ring_iff, Ne.def, norm_eq_zero] using nonzero"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\n⊢ ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖",
"tactic": "intro x"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\n⊢ ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖",
"tactic": "simp only [LinearMap.smul_apply]"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖",
"tactic": "have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) (x • 1)‖ = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖",
"tactic": "nth_rw 1 [this]"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑↑map (x • 1)‖ = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) (x • 1)‖ = ‖x‖",
"tactic": "rw [LinearMap.coe_restrictScalars]"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖↑map 1‖⁻¹ * (‖x‖ * ‖↑map 1‖) = ‖x‖",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖↑map 1‖⁻¹ • ↑↑map (x • 1)‖ = ‖x‖",
"tactic": "simp only [map.coe_coe, map.map_smul, norm_smul, norm_inv, norm_norm]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖↑map 1‖⁻¹ * (‖x‖ * ‖↑map 1‖) = ‖x‖",
"tactic": "field_simp only [one_mul]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx : ℂ\n⊢ x = x • 1",
"tactic": "rw [smul_eq_mul, mul_one]"
},
{
"state_after": "case refine'_2.h\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx✝ : ℂ\n⊢ ↑(restrictScalars ℝ map) x✝ =\n ↑(‖↑map 1‖ •\n LinearIsometry.toContinuousLinearMap\n { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := (_ : ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖) })\n x✝",
"state_before": "case refine'_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\n⊢ restrictScalars ℝ map =\n ‖↑map 1‖ •\n LinearIsometry.toContinuousLinearMap\n { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := (_ : ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖) }",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nz : ℂ\ng : ℂ →L[ℝ] E\nf : ℂ → E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖↑map 1‖ ≠ 0\nx✝ : ℂ\n⊢ ↑(restrictScalars ℝ map) x✝ =\n ↑(‖↑map 1‖ •\n LinearIsometry.toContinuousLinearMap\n { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := (_ : ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖) })\n x✝",
"tactic": "rw [coe_restrictScalars', coe_smul', LinearIsometry.coe_toContinuousLinearMap,\n LinearIsometry.coe_mk, Pi.smul_apply, LinearMap.smul_apply, LinearMap.coe_restrictScalars,\n coe_coe, smul_inv_smul₀ minor₁]"
}
] |
[
67,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/LinearAlgebra/Matrix/Dual.lean
|
Matrix.toLin_transpose
|
[
{
"state_after": "case a\nK : Type u_3\nV₁ : Type u_4\nV₂ : Type u_5\nι₁ : Type u_1\nι₂ : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V₁\ninst✝⁶ : Module K V₁\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\ninst✝³ : Fintype ι₁\ninst✝² : Fintype ι₂\ninst✝¹ : DecidableEq ι₁\ninst✝ : DecidableEq ι₂\nB₁ : Basis ι₁ K V₁\nB₂ : Basis ι₂ K V₂\nM : Matrix ι₁ ι₂ K\n⊢ ↑(LinearMap.toMatrix (Basis.dualBasis B₁) (Basis.dualBasis B₂))\n (↑(toLin (Basis.dualBasis B₁) (Basis.dualBasis B₂)) Mᵀ) =\n ↑(LinearMap.toMatrix (Basis.dualBasis B₁) (Basis.dualBasis B₂)) (↑Module.Dual.transpose (↑(toLin B₂ B₁) M))",
"state_before": "K : Type u_3\nV₁ : Type u_4\nV₂ : Type u_5\nι₁ : Type u_1\nι₂ : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V₁\ninst✝⁶ : Module K V₁\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\ninst✝³ : Fintype ι₁\ninst✝² : Fintype ι₂\ninst✝¹ : DecidableEq ι₁\ninst✝ : DecidableEq ι₂\nB₁ : Basis ι₁ K V₁\nB₂ : Basis ι₂ K V₂\nM : Matrix ι₁ ι₂ K\n⊢ ↑(toLin (Basis.dualBasis B₁) (Basis.dualBasis B₂)) Mᵀ = ↑Module.Dual.transpose (↑(toLin B₂ B₁) M)",
"tactic": "apply (LinearMap.toMatrix B₁.dualBasis B₂.dualBasis).injective"
},
{
"state_after": "no goals",
"state_before": "case a\nK : Type u_3\nV₁ : Type u_4\nV₂ : Type u_5\nι₁ : Type u_1\nι₂ : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V₁\ninst✝⁶ : Module K V₁\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module K V₂\ninst✝³ : Fintype ι₁\ninst✝² : Fintype ι₂\ninst✝¹ : DecidableEq ι₁\ninst✝ : DecidableEq ι₂\nB₁ : Basis ι₁ K V₁\nB₂ : Basis ι₂ K V₂\nM : Matrix ι₁ ι₂ K\n⊢ ↑(LinearMap.toMatrix (Basis.dualBasis B₁) (Basis.dualBasis B₂))\n (↑(toLin (Basis.dualBasis B₁) (Basis.dualBasis B₂)) Mᵀ) =\n ↑(LinearMap.toMatrix (Basis.dualBasis B₁) (Basis.dualBasis B₂)) (↑Module.Dual.transpose (↑(toLin B₂ B₁) M))",
"tactic": "rw [LinearMap.toMatrix_toLin, LinearMap.toMatrix_transpose, LinearMap.toMatrix_toLin]"
}
] |
[
47,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.ret_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nf : α → WSeq β\n⊢ bind (ret a) f ~ʷ f a",
"tactic": "simp [bind]"
}
] |
[
1743,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1743,
1
] |
Mathlib/Algebra/Group/UniqueProds.lean
|
UniqueMul.mulHom_image_iff
|
[
{
"state_after": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\n⊢ UniqueMul A B a0 b0\n\ncase refine'_2\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\n⊢ UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)",
"state_before": "G : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\n⊢ UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0) ↔ UniqueMul A B a0 b0",
"tactic": "refine' ⟨fun h ↦ _, fun h ↦ _⟩"
},
{
"state_after": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0",
"state_before": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\n⊢ UniqueMul A B a0 b0",
"tactic": "intro a b ha hb ab"
},
{
"state_after": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : a * b = a0 * b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"state_before": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0",
"tactic": "rw [← hf.eq_iff, ← hf.eq_iff]"
},
{
"state_after": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"state_before": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : a * b = a0 * b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"tactic": "rw [← hf.eq_iff, map_mul, map_mul] at ab"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)\na b : G\nha : a ∈ A\nhb : b ∈ B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"tactic": "exact h (Finset.mem_image.mpr ⟨_, ha, rfl⟩) (Finset.mem_image.mpr ⟨_, hb, rfl⟩) ab"
},
{
"state_after": "case refine'_2\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na b : H\naA : a ∈ Finset.image (↑f) A\nbB : b ∈ Finset.image (↑f) B\nab : a * b = ↑f a0 * ↑f b0\n⊢ a = ↑f a0 ∧ b = ↑f b0",
"state_before": "case refine'_2\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\n⊢ UniqueMul (Finset.image (↑f) A) (Finset.image (↑f) B) (↑f a0) (↑f b0)",
"tactic": "intro a b aA bB ab"
},
{
"state_after": "case refine'_2.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\nb : H\nbB : b ∈ Finset.image (↑f) B\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nab : ↑f a * b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ b = ↑f b0",
"state_before": "case refine'_2\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na b : H\naA : a ∈ Finset.image (↑f) A\nbB : b ∈ Finset.image (↑f) B\nab : a * b = ↑f a0 * ↑f b0\n⊢ a = ↑f a0 ∧ b = ↑f b0",
"tactic": "obtain ⟨a, ha, rfl⟩ : ∃ a' ∈ A, f a' = a := Finset.mem_image.mp aA"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"state_before": "case refine'_2.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\nb : H\nbB : b ∈ Finset.image (↑f) B\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nab : ↑f a * b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ b = ↑f b0",
"tactic": "obtain ⟨b, hb, rfl⟩ : ∃ b' ∈ B, f b' = b := Finset.mem_image.mp bB"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ a = a0 ∧ b = b0",
"state_before": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0",
"tactic": "rw [hf.eq_iff, hf.eq_iff]"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0",
"state_before": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ a = a0 ∧ b = b0",
"tactic": "rw [← map_mul, ← map_mul, hf.eq_iff] at ab"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro\nG : Type u_2\nH : Type u_1\ninst✝² : Mul G\ninst✝¹ : Mul H\nA B : Finset G\na0 b0 : G\ninst✝ : DecidableEq H\nf : G →ₙ* H\nhf : Function.Injective ↑f\nh : UniqueMul A B a0 b0\na : G\nha : a ∈ A\naA : ↑f a ∈ Finset.image (↑f) A\nb : G\nhb : b ∈ B\nbB : ↑f b ∈ Finset.image (↑f) B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0",
"tactic": "exact h ha hb ab"
}
] |
[
143,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.dist_of_im_eq
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.13792\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz w : ℂ\nh : z.im = w.im\n⊢ dist z w = dist z.re w.re",
"tactic": "rw [dist_eq_re_im, h, sub_self, zero_pow two_pos, add_zero, Real.sqrt_sq_eq_abs, Real.dist_eq]"
}
] |
[
119,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.mk'_right
|
[] |
[
564,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.nhdsWithin_basis_closed_eball
|
[] |
[
645,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
643,
1
] |
Mathlib/Algebra/Lie/OfAssociative.lean
|
LinearEquiv.lieConj_apply
|
[] |
[
334,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Topology/Inseparable.lean
|
SeparationQuotient.continuousWithinAt_lift₂
|
[] |
[
619,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
9
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
ciSup_pos
|
[] |
[
861,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.isLittleO_one_exp_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : α → ℝ\n⊢ ((fun x => 1) =o[l] fun x => exp (f x)) ↔ Tendsto f l atTop",
"tactic": "simp only [← exp_zero, isLittleO_exp_comp_exp_comp, sub_zero]"
}
] |
[
393,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
sdiff_sdiff_sup_sdiff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.31125\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ z \\ (x \\ y ⊔ y \\ x) = (z \\ x ⊔ z ⊓ x ⊓ y) ⊓ (z \\ y ⊔ z ⊓ y ⊓ x)",
"tactic": "rw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.31125\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (z \\ x ⊔ z ⊓ x ⊓ y) ⊓ (z \\ y ⊔ z ⊓ y ⊓ x) = z ⊓ (z \\ x ⊔ y) ⊓ (z \\ y ⊔ z ⊓ y ⊓ x)",
"tactic": "rw [sup_inf_left, sup_comm, sup_inf_sdiff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.31125\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ z ⊓ (z \\ x ⊔ y) ⊓ (z \\ y ⊔ z ⊓ y ⊓ x) = z ⊓ (z \\ x ⊔ y) ⊓ (z ⊓ (z \\ y ⊔ x))",
"tactic": "rw [sup_inf_left, @sup_comm _ _ (z \\ y), sup_inf_sdiff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.31125\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ z ⊓ (z \\ x ⊔ y) ⊓ (z ⊓ (z \\ y ⊔ x)) = z ⊓ z ⊓ (z \\ x ⊔ y) ⊓ (z \\ y ⊔ x)",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.31125\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ z ⊓ z ⊓ (z \\ x ⊔ y) ⊓ (z \\ y ⊔ x) = z ⊓ (z \\ x ⊔ y) ⊓ (z \\ y ⊔ x)",
"tactic": "rw [inf_idem]"
}
] |
[
411,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.rec_add_one
|
[] |
[
385,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.Step.cons_cons_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\n⊢ ∀ {p : α × Bool}, Step (p :: L₁) (p :: L₂) ↔ Step L₁ L₂",
"tactic": "simp (config := { contextual := true }) [Step.cons_left_iff, iff_def, or_imp]"
}
] |
[
188,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.rootMultiplicity_add
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\n⊢ min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q)",
"tactic": "rw [le_rootMultiplicity_iff hzero]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\nhdivp : (X - ↑C a) ^ rootMultiplicity a p ∣ p\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"tactic": "have hdivp : (X - C a) ^ rootMultiplicity a p ∣ p := pow_rootMultiplicity_dvd p a"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\nhdivp : (X - ↑C a) ^ rootMultiplicity a p ∣ p\nhdivq : (X - ↑C a) ^ rootMultiplicity a q ∣ q\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\nhdivp : (X - ↑C a) ^ rootMultiplicity a p ∣ p\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"tactic": "have hdivq : (X - C a) ^ rootMultiplicity a q ∣ q := pow_rootMultiplicity_dvd q a"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\na : R\nhzero : p + q ≠ 0\nhdivp : (X - ↑C a) ^ rootMultiplicity a p ∣ p\nhdivq : (X - ↑C a) ^ rootMultiplicity a q ∣ q\n⊢ (X - ↑C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q",
"tactic": "exact min_pow_dvd_add hdivp hdivq"
}
] |
[
395,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
LinearMap.splittingOfFinsuppSurjective_splits
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑(↑(comp (comp f (splittingOfFinsuppSurjective f s)) (lsingle x)) 1) y = ↑(↑(comp id (lsingle x)) 1) y",
"state_before": "R : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\n⊢ comp f (splittingOfFinsuppSurjective f s) = id",
"tactic": "refine lhom_ext' fun x => ext_ring <| Finsupp.ext fun y => ?_"
},
{
"state_after": "R : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑(↑f (sum (Finsupp.single x 1) fun x r => r • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1))) y =\n ↑(Finsupp.single x 1) y",
"state_before": "R : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑(↑(comp (comp f (splittingOfFinsuppSurjective f s)) (lsingle x)) 1) y = ↑(↑(comp id (lsingle x)) 1) y",
"tactic": "dsimp [splittingOfFinsuppSurjective]"
},
{
"state_after": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑f (sum (Finsupp.single x 1) fun x r => r • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1)) = Finsupp.single x 1",
"state_before": "R : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑(↑f (sum (Finsupp.single x 1) fun x r => r • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1))) y =\n ↑(Finsupp.single x 1) y",
"tactic": "congr"
},
{
"state_after": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑f (Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1)) = Finsupp.single x 1\n\ncase e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ 0 • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1) = 0",
"state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑f (sum (Finsupp.single x 1) fun x r => r • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1)) = Finsupp.single x 1",
"tactic": "rw [sum_single_index, one_smul]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ ↑f (Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1)) = Finsupp.single x 1",
"tactic": "exact (s (Finsupp.single x 1)).choose_spec"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type ?u.770436\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_3\nf : M →ₗ[R] α →₀ R\ns : Surjective ↑f\nx y : α\n⊢ 0 • Exists.choose (_ : ∃ a, ↑f a = Finsupp.single x 1) = 0",
"tactic": "rw [zero_smul]"
}
] |
[
1229,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/Algebra/Order/Ring/Abs.lean
|
abs_mul
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b✝ c a b : α\n⊢ a * b = abs a * abs b ∨ a * b = -(abs a * abs b)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b✝ c a b : α\n⊢ abs (a * b) = abs a * abs b",
"tactic": "rw [abs_eq (mul_nonneg (abs_nonneg a) (abs_nonneg b))]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b✝ c a b : α\n⊢ a * b = abs a * abs b ∨ a * b = -(abs a * abs b)",
"tactic": "cases' le_total a 0 with ha ha <;> cases' le_total b 0 with hb hb <;>\n simp only [abs_of_nonpos, abs_of_nonneg, true_or_iff, or_true_iff, eq_self_iff_true, neg_mul,\n mul_neg, neg_neg, *]"
}
] |
[
40,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.554527\nP : Type u_3\nP' : Type ?u.554533\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : r₁ ≤ 0\nhr₂ : 0 ≤ r₂\n⊢ Wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x)",
"tactic": "convert wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (r₁ • v +ᵥ x) v (Left.nonneg_neg_iff.2 hr₁)\n (neg_le_sub_iff_le_add.2 ((le_add_iff_nonneg_left r₁).2 hr₂)) using 1 <;>\n simp [sub_smul, ← add_vadd]"
}
] |
[
799,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupHom.symm_dual_comp
|
[] |
[
1396,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1393,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.lift_comp
|
[
{
"state_after": "case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx✝ : M\n⊢ ↑(MonoidHom.comp (lift f hg) (toMap f)) x✝ = ↑g x✝",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\n⊢ MonoidHom.comp (lift f hg) (toMap f) = g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx✝ : M\n⊢ ↑(MonoidHom.comp (lift f hg) (toMap f)) x✝ = ↑g x✝",
"tactic": "exact f.lift_eq hg _"
}
] |
[
1023,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1023,
1
] |
Mathlib/GroupTheory/Submonoid/Inverses.lean
|
Submonoid.leftInvEquiv_mul
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : { x // x ∈ leftInv S }\n⊢ ↑(↑(leftInvEquiv S hS) x) * ↑x = 1",
"tactic": "simpa only [leftInvEquiv_apply, fromCommLeftInv] using fromLeftInv_mul S x"
}
] |
[
198,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.not_mem_erase_self
|
[
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : Finmap β\n⊢ a = a ∨ ¬a ∈ s",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : Finmap β\n⊢ ¬a ∈ erase a s",
"tactic": "rw [mem_erase, not_and_or, not_not]"
},
{
"state_after": "case h\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : Finmap β\n⊢ a = a",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : Finmap β\n⊢ a = a ∨ ¬a ∈ s",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : Finmap β\n⊢ a = a",
"tactic": "rfl"
}
] |
[
445,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_union_of_antitone
|
[] |
[
606,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.re_conj
|
[] |
[
238,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Logic/Equiv/Option.lean
|
Equiv.optionSubtype_symm_apply_apply_none
|
[] |
[
260,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Order/Lattice.lean
|
le_antisymm'
|
[] |
[
71,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.pointwise_smul_toSubsemiring
|
[] |
[
74,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_max
|
[] |
[
581,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Analysis/Calculus/FDerivAnalytic.lean
|
HasFPowerSeriesAt.hasFDerivAt
|
[] |
[
49,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/GroupTheory/Coset.lean
|
leftCoset_eq_iff
|
[
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x_1 ∈ x *l ↑s ↔ x_1 ∈ y *l ↑s) ↔ x⁻¹ * y ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ x *l ↑s = y *l ↑s ↔ x⁻¹ * y ∈ s",
"tactic": "rw [Set.ext_iff]"
},
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) ↔ x⁻¹ * y ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x_1 ∈ x *l ↑s ↔ x_1 ∈ y *l ↑s) ↔ x⁻¹ * y ∈ s",
"tactic": "simp_rw [mem_leftCoset_iff, SetLike.mem_coe]"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) → x⁻¹ * y ∈ s\n\ncase mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ x⁻¹ * y ∈ s → ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) ↔ x⁻¹ * y ∈ s",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ x⁻¹ * y ∈ s",
"state_before": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) → x⁻¹ * y ∈ s",
"tactic": "intro h"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ y⁻¹ * y ∈ s",
"state_before": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ x⁻¹ * y ∈ s",
"tactic": "apply (h y).mpr"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ 1 ∈ s",
"state_before": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ y⁻¹ * y ∈ s",
"tactic": "rw [mul_left_inv]"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s\n⊢ 1 ∈ s",
"tactic": "exact s.one_mem"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * z ∈ s ↔ y⁻¹ * z ∈ s",
"state_before": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ x⁻¹ * y ∈ s → ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s",
"tactic": "intro h z"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * y * y⁻¹ * z ∈ s ↔ y⁻¹ * z ∈ s",
"state_before": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * z ∈ s ↔ y⁻¹ * z ∈ s",
"tactic": "rw [← mul_inv_cancel_right x⁻¹ y]"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * y * (y⁻¹ * z) ∈ s ↔ y⁻¹ * z ∈ s",
"state_before": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * y * y⁻¹ * z ∈ s ↔ y⁻¹ * z ∈ s",
"tactic": "rw [mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\nh : x⁻¹ * y ∈ s\nz : α\n⊢ x⁻¹ * y * (y⁻¹ * z) ∈ s ↔ y⁻¹ * z ∈ s",
"tactic": "exact s.mul_mem_cancel_left h"
}
] |
[
283,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.coe_prod
|
[] |
[
347,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.nonempty_iso_refl
|
[] |
[
150,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_indicator
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1768900\nγ : Type ?u.1768903\nδ : Type ?u.1768906\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(withDensity μ (indicator s f)) t = ↑↑(withDensity (Measure.restrict μ s) f) t",
"state_before": "α : Type u_1\nβ : Type ?u.1768900\nγ : Type ?u.1768903\nδ : Type ?u.1768906\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\n⊢ withDensity μ (indicator s f) = withDensity (Measure.restrict μ s) f",
"tactic": "ext1 t ht"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1768900\nγ : Type ?u.1768903\nδ : Type ?u.1768906\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(withDensity μ (indicator s f)) t = ↑↑(withDensity (Measure.restrict μ s) f) t",
"tactic": "rw [withDensity_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, ←\n withDensity_apply _ ht]"
}
] |
[
1640,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1636,
1
] |
Mathlib/RingTheory/WittVector/Teichmuller.lean
|
WittVector.teichmuller_coeff_pos
|
[] |
[
120,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.id_comp
|
[] |
[
246,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Init/CcLemmas.lean
|
ne_of_eq_of_ne
|
[] |
[
119,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_neg
|
[] |
[
296,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.setToL1S_add_left'
|
[] |
[
742,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
tendsto_ceil_atTop
|
[] |
[
46,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Algebra/GCDMonoid/Finset.lean
|
Finset.gcd_singleton
|
[] |
[
179,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
LinearOrderedField.smul_Ioi
|
[
{
"state_after": "case h\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ x ∈ r • Ioi a ↔ x ∈ Ioi (r • a)",
"state_before": "α : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\n⊢ r • Ioi a = Ioi (r • a)",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, a < y ∧ r * y = x) ↔ r * a < x",
"state_before": "case h\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ x ∈ r • Ioi a ↔ x ∈ Ioi (r • a)",
"tactic": "simp only [mem_smul_set, smul_eq_mul, mem_Ioi]"
},
{
"state_after": "case h.mp\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, a < y ∧ r * y = x) → r * a < x\n\ncase h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ r * a < x → ∃ y, a < y ∧ r * y = x",
"state_before": "case h\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, a < y ∧ r * y = x) ↔ r * a < x",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\na_w : K\na_h_left : a < a_w\n⊢ r * a < r * a_w",
"state_before": "case h.mp\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, a < y ∧ r * y = x) → r * a < x",
"tactic": "rintro ⟨a_w, a_h_left, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\na_w : K\na_h_left : a < a_w\n⊢ r * a < r * a_w",
"tactic": "exact (mul_lt_mul_left hr).mpr a_h_left"
},
{
"state_after": "case h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ ∃ y, a < y ∧ r * y = x",
"state_before": "case h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ r * a < x → ∃ y, a < y ∧ r * y = x",
"tactic": "rintro h"
},
{
"state_after": "case h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ a < x / r ∧ r * (x / r) = x",
"state_before": "case h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ ∃ y, a < y ∧ r * y = x",
"tactic": "use x / r"
},
{
"state_after": "case h.mpr.left\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ a < x / r\n\ncase h.mpr.right\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ r * (x / r) = x",
"state_before": "case h.mpr\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ a < x / r ∧ r * (x / r) = x",
"tactic": "constructor"
},
{
"state_after": "case h.mpr.right\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ r * (x / r) = x",
"state_before": "case h.mpr.left\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ a < x / r\n\ncase h.mpr.right\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ r * (x / r) = x",
"tactic": "exact (lt_div_iff' hr).mpr h"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.right\nα : Type ?u.61362\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\nh : r * a < x\n⊢ r * (x / r) = x",
"tactic": "exact mul_div_cancel' _ (ne_of_gt hr)"
}
] |
[
254,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/NumberTheory/SumFourSquares.lean
|
Nat.sum_four_squares_of_two_mul_sum_four_squares
|
[
{
"state_after": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "have : ∀ f : Fin 4 → ZMod 2, f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 → ∃ i : Fin 4,\n f i ^ 2 + f (swap i 0 1) ^ 2 = 0 ∧ f (swap i 0 2) ^ 2 + f (swap i 0 3) ^ 2 = 0 := by\n decide"
},
{
"state_after": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "set f : Fin 4 → ℤ := ![a, b, c, d]"
},
{
"state_after": "case intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nhσ : ↑(f i) ^ 2 + ↑(f (↑(swap i 0) 1)) ^ 2 = 0 ∧ ↑(f (↑(swap i 0) 2)) ^ 2 + ↑(f (↑(swap i 0) 3)) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "obtain ⟨i, hσ⟩ := this (fun x => ↑(f x)) <| by\n rw [← @zero_mul (ZMod 2) _ m, ← show ((2 : ℤ) : ZMod 2) = 0 from rfl, ← Int.cast_mul, ← h]\n simp only [Int.cast_add, Int.cast_pow]\n rfl"
},
{
"state_after": "case intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "case intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nhσ : ↑(f i) ^ 2 + ↑(f (↑(swap i 0) 1)) ^ 2 = 0 ∧ ↑(f (↑(swap i 0) 2)) ^ 2 + ↑(f (↑(swap i 0) 3)) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "set σ := swap i 0"
},
{
"state_after": "case intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "case intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "obtain ⟨x, hx⟩ : (2 : ℤ) ∣ f (σ 0) ^ 2 + f (σ 1) ^ 2 :=\n (CharP.int_cast_eq_zero_iff (ZMod 2) 2 _).1 <| by\n simpa only [Int.cast_pow, Int.cast_add, Equiv.swap_apply_right, ZMod.pow_card] using hσ.1"
},
{
"state_after": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"state_before": "case intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "obtain ⟨y, hy⟩ : (2 : ℤ) ∣ f (σ 2) ^ 2 + f (σ 3) ^ 2 :=\n (CharP.int_cast_eq_zero_iff (ZMod 2) 2 _).1 <| by\n simpa only [Int.cast_pow, Int.cast_add, ZMod.pow_card] using hσ.2"
},
{
"state_after": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\n⊢ ((f (↑σ 0) - f (↑σ 1)) / 2) ^ 2 + ((f (↑σ 0) + f (↑σ 1)) / 2) ^ 2 + ((f (↑σ 2) - f (↑σ 3)) / 2) ^ 2 +\n ((f (↑σ 2) + f (↑σ 3)) / 2) ^ 2 =\n m",
"state_before": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\n⊢ ∃ w x y z, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m",
"tactic": "refine ⟨(f (σ 0) - f (σ 1)) / 2, (f (σ 0) + f (σ 1)) / 2, (f (σ 2) - f (σ 3)) / 2,\n (f (σ 2) + f (σ 3)) / 2, ?_⟩"
},
{
"state_after": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\n⊢ 2 * x + 2 * y = a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2",
"state_before": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\n⊢ ((f (↑σ 0) - f (↑σ 1)) / 2) ^ 2 + ((f (↑σ 0) + f (↑σ 1)) / 2) ^ 2 + ((f (↑σ 2) - f (↑σ 3)) / 2) ^ 2 +\n ((f (↑σ 2) + f (↑σ 3)) / 2) ^ 2 =\n m",
"tactic": "rw [← Int.sq_add_sq_of_two_mul_sq_add_sq hx.symm, add_assoc,\n ← Int.sq_add_sq_of_two_mul_sq_add_sq hy.symm, ← mul_right_inj' two_ne_zero, ← h, mul_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nm a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis✝ :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\ny : ℤ\nhy : f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2 = 2 * y\nthis : ∑ x : Fin 4, f (↑σ x) ^ 2 = ∑ x : Fin 4, f x ^ 2\n⊢ 2 * x + 2 * y = a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2",
"tactic": "simpa only [← hx, ← hy, Fin.sum_univ_four, add_assoc] using this"
},
{
"state_after": "no goals",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\n⊢ ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0",
"tactic": "decide"
},
{
"state_after": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ (fun x => ↑(f x)) 0 ^ 2 + (fun x => ↑(f x)) 1 ^ 2 + (fun x => ↑(f x)) 2 ^ 2 + (fun x => ↑(f x)) 3 ^ 2 =\n ↑(a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2)",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ (fun x => ↑(f x)) 0 ^ 2 + (fun x => ↑(f x)) 1 ^ 2 + (fun x => ↑(f x)) 2 ^ 2 + (fun x => ↑(f x)) 3 ^ 2 = 0",
"tactic": "rw [← @zero_mul (ZMod 2) _ m, ← show ((2 : ℤ) : ZMod 2) = 0 from rfl, ← Int.cast_mul, ← h]"
},
{
"state_after": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ ↑(Matrix.vecCons a ![b, c, d] 0) ^ 2 + ↑(Matrix.vecCons a ![b, c, d] 1) ^ 2 + ↑(Matrix.vecCons a ![b, c, d] 2) ^ 2 +\n ↑(Matrix.vecCons a ![b, c, d] 3) ^ 2 =\n ↑a ^ 2 + ↑b ^ 2 + ↑c ^ 2 + ↑d ^ 2",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ (fun x => ↑(f x)) 0 ^ 2 + (fun x => ↑(f x)) 1 ^ 2 + (fun x => ↑(f x)) 2 ^ 2 + (fun x => ↑(f x)) 3 ^ 2 =\n ↑(a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2)",
"tactic": "simp only [Int.cast_add, Int.cast_pow]"
},
{
"state_after": "no goals",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\n⊢ ↑(Matrix.vecCons a ![b, c, d] 0) ^ 2 + ↑(Matrix.vecCons a ![b, c, d] 1) ^ 2 + ↑(Matrix.vecCons a ![b, c, d] 2) ^ 2 +\n ↑(Matrix.vecCons a ![b, c, d] 3) ^ 2 =\n ↑a ^ 2 + ↑b ^ 2 + ↑c ^ 2 + ↑d ^ 2",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\n⊢ ↑(f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2) = 0",
"tactic": "simpa only [Int.cast_pow, Int.cast_add, Equiv.swap_apply_right, ZMod.pow_card] using hσ.1"
},
{
"state_after": "no goals",
"state_before": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f (↑(swap i 0) 1) ^ 2 = 0 ∧ f (↑(swap i 0) 2) ^ 2 + f (↑(swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nσ : Perm (Fin 4) := swap i 0\nhσ : ↑(f i) ^ 2 + ↑(f (↑σ 1)) ^ 2 = 0 ∧ ↑(f (↑σ 2)) ^ 2 + ↑(f (↑σ 3)) ^ 2 = 0\nx : ℤ\nhx : f (↑σ 0) ^ 2 + f (↑σ 1) ^ 2 = 2 * x\n⊢ ↑(f (↑σ 2) ^ 2 + f (↑σ 3) ^ 2) = 0",
"tactic": "simpa only [Int.cast_pow, Int.cast_add, ZMod.pow_card] using hσ.2"
}
] |
[
143,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
9
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.Memℒp.snormEssSup_indicator_norm_ge_eq_zero
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ ∃ M, snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\n⊢ ∃ M, snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0",
"tactic": "have hbdd : snormEssSup f μ < ∞ := hf.snorm_lt_top"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ snormEssSup (Set.indicator {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} f) μ = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ ∃ M, snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0",
"tactic": "refine' ⟨(snorm f ∞ μ + 1).toReal, _⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ snormEssSup f (Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}) = 0\n\nα : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ snormEssSup (Set.indicator {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} f) μ = 0",
"tactic": "rw [snormEssSup_indicator_eq_snormEssSup_restrict]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} = 0\n⊢ snormEssSup f (Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}) = 0\n\nα : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}",
"tactic": "rw [this, snormEssSup_measure_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}",
"tactic": "exact measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} = 0",
"tactic": "simp only [coe_nnnorm, snorm_exponent_top, Measure.restrict_eq_zero]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\n⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ≤ 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\n⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} = 0",
"tactic": "rw [← nonpos_iff_eq_zero]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\n⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ≤ 0",
"tactic": "refine' (measure_mono this).trans _"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\nhle : ∀ᵐ (x : α) ∂μ, ↑‖f x‖₊ ≤ snormEssSup f μ\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"tactic": "have hle := coe_nnnorm_ae_le_snormEssSup f μ"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\nhle : ↑↑μ {a | snormEssSup f μ < ↑‖f a‖₊} = 0\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\nhle : ∀ᵐ (x : α) ∂μ, ↑‖f x‖₊ ≤ snormEssSup f μ\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"tactic": "simp_rw [ae_iff, not_le] at hle"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nthis : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}\nhle : ↑↑μ {a | snormEssSup f μ < ↑‖f a‖₊} = 0\n⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0",
"tactic": "exact nonpos_iff_eq_zero.2 hle"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ x ∈ {x | snormEssSup f μ < ↑‖f x‖₊}",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\n⊢ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ ENNReal.toReal (snormEssSup f μ) < ‖f x‖",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ x ∈ {x | snormEssSup f μ < ↑‖f x‖₊}",
"tactic": "rw [Set.mem_setOf_eq, ← ENNReal.toReal_lt_toReal hbdd.ne ENNReal.coe_lt_top.ne,\n ENNReal.coe_toReal, coe_nnnorm]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ ENNReal.toReal (snormEssSup f μ) < ENNReal.toReal (snormEssSup f μ + 1)",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ ENNReal.toReal (snormEssSup f μ) < ‖f x‖",
"tactic": "refine' lt_of_lt_of_le _ hx"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ snormEssSup f μ < snormEssSup f μ + 1\n\nα : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ snormEssSup f μ + 1 ≠ ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ ENNReal.toReal (snormEssSup f μ) < ENNReal.toReal (snormEssSup f μ + 1)",
"tactic": "rw [ENNReal.toReal_lt_toReal hbdd.ne]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ snormEssSup f μ < snormEssSup f μ + 1",
"tactic": "exact ENNReal.lt_add_right hbdd.ne one_ne_zero"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.92822\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : Memℒp f ⊤\nhmeas : StronglyMeasurable f\nhbdd : snormEssSup f μ < ⊤\nx : α\nhx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}\n⊢ snormEssSup f μ + 1 ≠ ⊤",
"tactic": "exact (ENNReal.add_lt_top.2 ⟨hbdd, ENNReal.one_lt_top⟩).ne"
}
] |
[
257,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Multiset.toFinset_bind_dedup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.453885\ninst✝¹ : DecidableEq α\ns t : Multiset α\ninst✝ : DecidableEq β\nm : Multiset α\nf : α → Multiset β\n⊢ toFinset (bind (dedup m) f) = toFinset (bind m f)",
"tactic": "simp_rw [toFinset, dedup_bind_dedup]"
}
] |
[
3203,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3202,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.to_inf_right
|
[] |
[
281,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.fract_sub_self
|
[] |
[
869,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
868,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.odd_iff_not_even
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ Odd n ↔ ¬Even n",
"tactic": "rw [not_even_iff, odd_iff]"
}
] |
[
64,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
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