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/Data/Multiset/Basic.lean
|
Multiset.induction_on'
|
[] |
[
441,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Data/ULift.lean
|
ULift.up_bijective
|
[] |
[
109,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.fp_bfamily_unbounded
|
[
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ ∀ (i : Ordinal) (j : i < o), nfpBFamily o f a ∈ fixedPoints (f i j)",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ nfpBFamily o f a ∈ ⋂ (i : Ordinal) (hi : i < o), fixedPoints (f i hi)",
"tactic": "rw [Set.mem_iInter₂]"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ ∀ (i : Ordinal) (j : i < o), nfpBFamily o f a ∈ fixedPoints (f i j)",
"tactic": "exact fun i hi => nfpBFamily_fp (H i hi) _"
}
] |
[
349,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderivWithin_fst
|
[] |
[
225,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Filter.Tendsto.atBot_zsmul_neg_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedAddCommGroup R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x => f x • r) l atTop",
"tactic": "simpa using hf.atBot_zsmul_const (neg_pos.2 hr)"
}
] |
[
250,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.eq_top_iff
|
[
{
"state_after": "F : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : S = ⊤\nx : A\n⊢ x ∈ ⊤",
"state_before": "F : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : S = ⊤\nx : A\n⊢ x ∈ S",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : S = ⊤\nx : A\n⊢ x ∈ ⊤",
"tactic": "exact mem_top"
},
{
"state_after": "case h\nF : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : ∀ (x : A), x ∈ S\nx : A\n⊢ x ∈ S ↔ x ∈ ⊤",
"state_before": "F : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : ∀ (x : A), x ∈ S\n⊢ S = ⊤",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.755056\nR : Type u_1\nA : Type u_2\nB : Type ?u.755065\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : StarSubalgebra R A\nh : ∀ (x : A), x ∈ S\nx : A\n⊢ x ∈ S ↔ x ∈ ⊤",
"tactic": "exact ⟨fun _ => mem_top, fun _ => h x⟩"
}
] |
[
715,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Dfinsupp.mapRange.linearEquiv_refl
|
[] |
[
242,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Tactic/Abel.lean
|
Mathlib.Tactic.Abel.term_atomg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\nx : α\n⊢ x = termg 1 x 0",
"tactic": "simp [termg]"
}
] |
[
236,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
tendsto_fract_left'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.22476\nγ : Type ?u.22479\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : TopologicalAddGroup α\nn : ℤ\n⊢ Tendsto fract (𝓝[Iio ↑n] ↑n) (𝓝 (↑n - (↑n - 1)))",
"state_before": "α : Type u_1\nβ : Type ?u.22476\nγ : Type ?u.22479\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : TopologicalAddGroup α\nn : ℤ\n⊢ Tendsto fract (𝓝[Iio ↑n] ↑n) (𝓝 1)",
"tactic": "rw [← sub_sub_cancel (n : α) 1]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.22476\nγ : Type ?u.22479\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : TopologicalAddGroup α\nn : ℤ\n⊢ Tendsto (fun x => ↑⌊x⌋) (𝓝[Iio ↑n] ↑n) (𝓝 (↑n - 1))",
"state_before": "α : Type u_1\nβ : Type ?u.22476\nγ : Type ?u.22479\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : TopologicalAddGroup α\nn : ℤ\n⊢ Tendsto fract (𝓝[Iio ↑n] ↑n) (𝓝 (↑n - (↑n - 1)))",
"tactic": "refine (tendsto_id.mono_left nhdsWithin_le_nhds).sub ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22476\nγ : Type ?u.22479\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : TopologicalAddGroup α\nn : ℤ\n⊢ Tendsto (fun x => ↑⌊x⌋) (𝓝[Iio ↑n] ↑n) (𝓝 (↑n - 1))",
"tactic": "exact tendsto_floor_left' n"
}
] |
[
171,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.inter_smul_union_subset_union
|
[] |
[
1407,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1406,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.hasDerivWithinAt_arccos_Iic
|
[] |
[
161,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_apply_ne
|
[] |
[
371,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_true
|
[] |
[
1300,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1299,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.modEq_and_modEq_iff_modEq_mul
|
[
{
"state_after": "m✝ n✝ a✝ b✝ c d a b m n : ℕ\nhmn : coprime m n\nh : m ∣ Int.natAbs (↑b - ↑a) ∧ n ∣ Int.natAbs (↑b - ↑a)\n⊢ a ≡ b [MOD m * n]",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℕ\nhmn : coprime m n\nh : a ≡ b [MOD m] ∧ a ≡ b [MOD n]\n⊢ a ≡ b [MOD m * n]",
"tactic": "rw [Nat.modEq_iff_dvd, Nat.modEq_iff_dvd, ← Int.dvd_natAbs, Int.coe_nat_dvd, ← Int.dvd_natAbs,\n Int.coe_nat_dvd] at h"
},
{
"state_after": "m✝ n✝ a✝ b✝ c d a b m n : ℕ\nhmn : coprime m n\nh : m ∣ Int.natAbs (↑b - ↑a) ∧ n ∣ Int.natAbs (↑b - ↑a)\n⊢ m * n ∣ Int.natAbs (↑b - ↑a)",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℕ\nhmn : coprime m n\nh : m ∣ Int.natAbs (↑b - ↑a) ∧ n ∣ Int.natAbs (↑b - ↑a)\n⊢ a ≡ b [MOD m * n]",
"tactic": "rw [Nat.modEq_iff_dvd, ← Int.dvd_natAbs, Int.coe_nat_dvd]"
},
{
"state_after": "no goals",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℕ\nhmn : coprime m n\nh : m ∣ Int.natAbs (↑b - ↑a) ∧ n ∣ Int.natAbs (↑b - ↑a)\n⊢ m * n ∣ Int.natAbs (↑b - ↑a)",
"tactic": "exact hmn.mul_dvd_of_dvd_of_dvd h.1 h.2"
}
] |
[
388,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
summable_iff_of_summable_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.406920\nδ : Type ?u.406923\ninst✝² : AddCommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\nhfg : Summable fun b => f b - g b\nhf : Summable f\n⊢ Summable fun b => g b - f b",
"tactic": "simpa only [neg_sub] using hfg.neg"
}
] |
[
835,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_singleton
|
[] |
[
682,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/Algebra/Associated.lean
|
dvd_dvd_iff_associated
|
[] |
[
564,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Finset.isPwo_bUnion
|
[] |
[
606,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
frobenius_nat_cast
|
[] |
[
406,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.of_near
|
[] |
[
671,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullbackDiagonalMapIso_hom_fst
|
[
{
"state_after": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (Iso.mk (lift (snd ≫ fst) (snd ≫ snd) (_ : (snd ≫ fst) ≫ i₁ = (snd ≫ snd) ≫ i₂))\n (lift (fst ≫ i₁ ≫ fst)\n (map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd))\n (_ :\n (fst ≫ i₁ ≫ fst) ≫ diagonal f =\n map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd) ≫\n map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (_ : (i₁ ≫ snd) ≫ i = (i₁ ≫ fst) ≫ f)\n (_ : (i₂ ≫ snd) ≫ i = (i₂ ≫ fst) ≫ f)))).hom ≫\n fst =\n snd ≫ fst",
"state_before": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (pullbackDiagonalMapIso f i i₁ i₂).hom ≫ fst = snd ≫ fst",
"tactic": "delta pullbackDiagonalMapIso"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (Iso.mk (lift (snd ≫ fst) (snd ≫ snd) (_ : (snd ≫ fst) ≫ i₁ = (snd ≫ snd) ≫ i₂))\n (lift (fst ≫ i₁ ≫ fst)\n (map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd))\n (_ :\n (fst ≫ i₁ ≫ fst) ≫ diagonal f =\n map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd) ≫\n map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (_ : (i₁ ≫ snd) ≫ i = (i₁ ≫ fst) ≫ f)\n (_ : (i₂ ≫ snd) ≫ i = (i₂ ≫ fst) ≫ f)))).hom ≫\n fst =\n snd ≫ fst",
"tactic": "simp"
}
] |
[
158,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.mk_ker
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nx✝ : R\n⊢ x✝ ∈ RingHom.ker (Quotient.mk I) ↔ x✝ ∈ I",
"state_before": "R : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\n⊢ RingHom.ker (Quotient.mk I) = I",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nx✝ : R\n⊢ x✝ ∈ RingHom.ker (Quotient.mk I) ↔ x✝ ∈ I",
"tactic": "rw [RingHom.ker, mem_comap, @Submodule.mem_bot _ _ _ _ Semiring.toModule _,\n Quotient.eq_zero_iff_mem]"
}
] |
[
106,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Combinatorics/Derangements/Basic.lean
|
mem_derangements_iff_fixedPoints_eq_empty
|
[] |
[
44,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
strictMono_of_deriv_pos
|
[] |
[
908,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.lfp_lfp
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑lfp (comp lfp h) = ↑lfp (onDiag h)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\n⊢ ↑lfp (comp lfp h) = ↑lfp (onDiag h)",
"tactic": "let a := lfp (lfp.comp h)"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑(comp lfp h) (↑lfp (onDiag h)) ≤ ↑lfp (onDiag h)\n\ncase refine'_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑(onDiag h) (↑lfp (comp lfp h)) = ↑lfp (comp lfp h)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑lfp (comp lfp h) = ↑lfp (onDiag h)",
"tactic": "refine' (lfp_le _ _).antisymm (lfp_le _ (Eq.le _))"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\nha : (↑lfp ∘ ↑h) a = a\n⊢ ↑(onDiag h) (↑lfp (comp lfp h)) = ↑lfp (comp lfp h)",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑(onDiag h) (↑lfp (comp lfp h)) = ↑lfp (comp lfp h)",
"tactic": "have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\nha : (↑lfp ∘ ↑h) a = a\n⊢ ↑(onDiag h) (↑lfp (comp lfp h)) = ↑lfp (comp lfp h)",
"tactic": "calc\n h a a = h a (lfp (h a)) := congr_arg (h a) ha.symm\n _ = lfp (h a) := (h a).map_lfp\n _ = a := ha"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : β →o α\ng : α →o β\nh : α →o α →o α\na : α := ↑lfp (comp lfp h)\n⊢ ↑(comp lfp h) (↑lfp (onDiag h)) ≤ ↑lfp (onDiag h)",
"tactic": "exact lfp_le _ h.onDiag.map_lfp.le"
}
] |
[
177,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/Chain.lean
|
Flag.chain_le
|
[] |
[
333,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_ne_self
|
[] |
[
1108,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Dvd.dvd.modEq_zero_int
|
[] |
[
90,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.iIndepFun.indepFun
|
[] |
[
304,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.coe_comp_sup_hom
|
[] |
[
1124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1122,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Ioc_mem_nhdsWithin_Iio'
|
[] |
[
484,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.inf_subgroupOf_right
|
[] |
[
1676,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1675,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_norm_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.121013\nE : Type u_2\nF : Type ?u.121019\nG : Type ?u.121022\nE' : Type ?u.121025\nF' : Type u_3\nG' : Type ?u.121031\nE'' : Type ?u.121034\nF'' : Type ?u.121037\nG'' : Type ?u.121040\nR : Type ?u.121043\nR' : Type ?u.121046\n𝕜 : Type ?u.121049\n𝕜' : Type ?u.121052\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x => ‖g' x‖) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'",
"state_before": "α : Type u_1\nβ : Type ?u.121013\nE : Type u_2\nF : Type ?u.121019\nG : Type ?u.121022\nE' : Type ?u.121025\nF' : Type u_3\nG' : Type ?u.121031\nE'' : Type ?u.121034\nF'' : Type ?u.121037\nG'' : Type ?u.121040\nR : Type ?u.121043\nR' : Type ?u.121046\n𝕜 : Type ?u.121049\n𝕜' : Type ?u.121052\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (f =o[l] fun x => ‖g' x‖) ↔ f =o[l] g'",
"tactic": "simp only [IsLittleO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.121013\nE : Type u_2\nF : Type ?u.121019\nG : Type ?u.121022\nE' : Type ?u.121025\nF' : Type u_3\nG' : Type ?u.121031\nE'' : Type ?u.121034\nF'' : Type ?u.121037\nG'' : Type ?u.121040\nR : Type ?u.121043\nR' : Type ?u.121046\n𝕜 : Type ?u.121049\n𝕜' : Type ?u.121052\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x => ‖g' x‖) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'",
"tactic": "exact forall₂_congr fun _ _ => isBigOWith_norm_right"
}
] |
[
735,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
733,
1
] |
Mathlib/Logic/Basic.lean
|
BAll.imp_right
|
[] |
[
1047,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.tprod_nil
|
[] |
[
239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/List/Sort.lean
|
List.orderedInsert_count
|
[
{
"state_after": "α : Type uu\nr : α → α → Prop\ninst✝¹ : DecidableRel r\ninst✝ : DecidableEq α\nL : List α\na b : α\n⊢ (if a = b then Nat.succ (count a L) else count a L) = count a L + if a = b then 1 else 0",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝¹ : DecidableRel r\ninst✝ : DecidableEq α\nL : List α\na b : α\n⊢ count a (orderedInsert r b L) = count a L + if a = b then 1 else 0",
"tactic": "rw [(L.perm_orderedInsert r b).count_eq, count_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝¹ : DecidableRel r\ninst✝ : DecidableEq α\nL : List α\na b : α\n⊢ (if a = b then Nat.succ (count a L) else count a L) = count a L + if a = b then 1 else 0",
"tactic": "split_ifs <;> simp only [Nat.succ_eq_add_one, add_zero]"
}
] |
[
240,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.IsNormal.le_iff_deriv
|
[
{
"state_after": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na : Ordinal\n⊢ f a ≤ a ↔ ∃ o, derivFamily (fun x => f) o = a",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na : Ordinal\n⊢ f a ≤ a ↔ ∃ o, deriv f o = a",
"tactic": "unfold deriv"
},
{
"state_after": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na : Ordinal\n⊢ f a ≤ a ↔ Unit → f a ≤ a",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na : Ordinal\n⊢ f a ≤ a ↔ ∃ o, derivFamily (fun x => f) o = a",
"tactic": "rw [← le_iff_derivFamily fun _ : Unit => H]"
},
{
"state_after": "no goals",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na : Ordinal\n⊢ f a ≤ a ↔ Unit → f a ≤ a",
"tactic": "exact ⟨fun h _ => h, fun h => h Unit.unit⟩"
}
] |
[
545,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.norm_smulRightL
|
[] |
[
1886,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1885,
1
] |
Mathlib/Data/PFun.lean
|
PFun.lift_injective
|
[] |
[
157,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.wOppSide_iff_exists_left
|
[
{
"state_after": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ WOppSide s x y → x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ (x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)) → WOppSide s x y",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ WOppSide s x y ↔ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x -ᵥ p₁' = 0\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nhr : r₁ • (x -ᵥ p₁') = r₂ • (p₂' -ᵥ y)\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"state_before": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ WOppSide s x y → x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "rintro ⟨p₁', hp₁', p₂', hp₂', h0 | h0 | ⟨r₁, r₂, hr₁, hr₂, hr⟩⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x = p₁'\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x -ᵥ p₁' = 0\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "rw [vsub_eq_zero_iff_eq] at h0"
},
{
"state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x = p₁'\n⊢ p₁' ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (p₁' -ᵥ p₁) (p₂ -ᵥ y)",
"state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x = p₁'\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "rw [h0]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : x = p₁'\n⊢ p₁' ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (p₁' -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "exact Or.inl hp₁'"
},
{
"state_after": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ SameRay R (x -ᵥ p₁) (p₂' -ᵥ y)",
"state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "refine' Or.inr ⟨p₂', hp₂', _⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ SameRay R (x -ᵥ p₁) 0",
"state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ SameRay R (x -ᵥ p₁) (p₂' -ᵥ y)",
"tactic": "rw [h0]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nh0 : p₂' -ᵥ y = 0\n⊢ SameRay R (x -ᵥ p₁) 0",
"tactic": "exact SameRay.zero_right _"
},
{
"state_after": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nhr : r₁ • (x -ᵥ p₁') = r₂ • (p₂' -ᵥ y)\n⊢ r₁ • (x -ᵥ p₁) = r₂ • ((-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂' -ᵥ y)",
"state_before": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nhr : r₁ • (x -ᵥ p₁') = r₂ • (p₂' -ᵥ y)\n⊢ x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "refine' Or.inr ⟨(-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂', s.smul_vsub_vadd_mem _ h hp₁' hp₂',\n Or.inr (Or.inr ⟨r₁, r₂, hr₁, hr₂, _⟩)⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\np₁' : P\nhp₁' : p₁' ∈ s\np₂' : P\nhp₂' : p₂' ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nhr : r₁ • (x -ᵥ p₁') = r₂ • (p₂' -ᵥ y)\n⊢ r₁ • (x -ᵥ p₁) = r₂ • ((-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂' -ᵥ y)",
"tactic": "rw [vadd_vsub_assoc, smul_add, ← hr, smul_smul, neg_div, mul_neg,\n mul_div_cancel' _ hr₂.ne.symm, neg_smul, neg_add_eq_sub, ← smul_sub,\n vsub_sub_vsub_cancel_right]"
},
{
"state_after": "case mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\nh' : x ∈ s\n⊢ WOppSide s x y\n\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\nh₁ : P\nh₂ : h₁ ∈ s\nh₃ : SameRay R (x -ᵥ p₁) (h₁ -ᵥ y)\n⊢ WOppSide s x y",
"state_before": "case mpr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ (x ∈ s ∨ ∃ p₂, p₂ ∈ s ∧ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)) → WOppSide s x y",
"tactic": "rintro (h' | ⟨h₁, h₂, h₃⟩)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\nh' : x ∈ s\n⊢ WOppSide s x y",
"tactic": "exact wOppSide_of_left_mem y h'"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.254312\nP : Type u_3\nP' : Type ?u.254318\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\nh₁ : P\nh₂ : h₁ ∈ s\nh₃ : SameRay R (x -ᵥ p₁) (h₁ -ᵥ y)\n⊢ WOppSide s x y",
"tactic": "exact ⟨p₁, h, h₁, h₂, h₃⟩"
}
] |
[
493,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.toL1_coeFn
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1309276\nδ : Type ?u.1309279\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : { x // x ∈ Lp β 1 }\nhf : Integrable ↑↑f\n⊢ toL1 (↑↑f) hf = f",
"tactic": "simp [Integrable.toL1]"
}
] |
[
1355,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1354,
1
] |
Mathlib/Data/Finsupp/Pointwise.lean
|
Finsupp.mul_apply
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
continuousAt_extChartAt
|
[] |
[
1095,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1094,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
one_add_mul_le_pow
|
[] |
[
759,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_mulIndicator_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.171403\nι : Type ?u.171406\nG : Type ?u.171409\nM : Type u_2\nN : Type ?u.171415\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\ns : Set α\nf : α → M\na : α\n⊢ (∏ᶠ (_ : a ∈ s), f a) = mulIndicator s f a",
"tactic": "classical convert finprod_eq_if (M := M) (p := a ∈ s) (x := f a)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.171403\nι : Type ?u.171406\nG : Type ?u.171409\nM : Type u_2\nN : Type ?u.171415\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\ns : Set α\nf : α → M\na : α\n⊢ (∏ᶠ (_ : a ∈ s), f a) = mulIndicator s f a",
"tactic": "convert finprod_eq_if (M := M) (p := a ∈ s) (x := f a)"
}
] |
[
365,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.pos_add_limZero
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nF : α\nF0 : F > 0\nhF : ∃ i, ∀ (j : ℕ), j ≥ i → F ≤ ↑f j\nH : LimZero g\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → F ≤ ↑f j ∧ abs (↑g j) < F / 2\nj : ℕ\nij : j ≥ i\nh₁ : F ≤ ↑f j\nh₂ : abs (↑g j) < F / 2\n⊢ F / 2 ≤ ↑(f + g) j",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nF : α\nF0 : F > 0\nhF : ∃ i, ∀ (j : ℕ), j ≥ i → F ≤ ↑f j\nH : LimZero g\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → F ≤ ↑f j ∧ abs (↑g j) < F / 2\nj : ℕ\nij : j ≥ i\n⊢ F / 2 ≤ ↑(f + g) j",
"tactic": "cases' h j ij with h₁ h₂"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nF : α\nF0 : F > 0\nhF : ∃ i, ∀ (j : ℕ), j ≥ i → F ≤ ↑f j\nH : LimZero g\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → F ≤ ↑f j ∧ abs (↑g j) < F / 2\nj : ℕ\nij : j ≥ i\nh₁ : F ≤ ↑f j\nh₂ : abs (↑g j) < F / 2\nthis : F + -(F / 2) ≤ ↑f j + ↑g j\n⊢ F / 2 ≤ ↑(f + g) j",
"state_before": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nF : α\nF0 : F > 0\nhF : ∃ i, ∀ (j : ℕ), j ≥ i → F ≤ ↑f j\nH : LimZero g\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → F ≤ ↑f j ∧ abs (↑g j) < F / 2\nj : ℕ\nij : j ≥ i\nh₁ : F ≤ ↑f j\nh₂ : abs (↑g j) < F / 2\n⊢ F / 2 ≤ ↑(f + g) j",
"tactic": "have := add_le_add h₁ (le_of_lt (abs_lt.1 h₂).1)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nF : α\nF0 : F > 0\nhF : ∃ i, ∀ (j : ℕ), j ≥ i → F ≤ ↑f j\nH : LimZero g\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → F ≤ ↑f j ∧ abs (↑g j) < F / 2\nj : ℕ\nij : j ≥ i\nh₁ : F ≤ ↑f j\nh₂ : abs (↑g j) < F / 2\nthis : F + -(F / 2) ≤ ↑f j + ↑g j\n⊢ F / 2 ≤ ↑(f + g) j",
"tactic": "rwa [← sub_eq_add_neg, sub_self_div_two] at this"
}
] |
[
691,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/Algebra/Module/Injective.lean
|
Module.injective_iff_injective_object
|
[] |
[
89,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/List/Basic.lean
|
List.exists_mem_cons_of_exists
|
[] |
[
293,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Init/CcLemmas.lean
|
imp_eq_true_of_eq
|
[] |
[
70,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean
|
Finpartition.energy_le_one
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ ↑(card P.parts * card P.parts - card P.parts) ≤ ↑(card P.parts ^ 2)",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ ↑(card (offDiag P.parts)) ≤ 1 * ↑(card P.parts ^ 2)",
"tactic": "rw [offDiag_card, one_mul]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ card P.parts * card P.parts - card P.parts ≤ card P.parts ^ 2",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ ↑(card P.parts * card P.parts - card P.parts) ≤ ↑(card P.parts ^ 2)",
"tactic": "norm_cast"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ card P.parts * card P.parts - card P.parts ≤ card P.parts * card P.parts",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ card P.parts * card P.parts - card P.parts ≤ card P.parts ^ 2",
"tactic": "rw [sq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ns : Finset α\nP : Finpartition s\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\n⊢ card P.parts * card P.parts - card P.parts ≤ card P.parts * card P.parts",
"tactic": "exact tsub_le_self"
}
] |
[
61,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aeval_endomorphism
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.2133490\nB' : Type ?u.2133493\na b : R\nn : ℕ\ninst✝⁴ : CommSemiring A'\ninst✝³ : Semiring B'\nM : Type u_1\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\nv : M\np : R[X]\n⊢ ↑(sum p fun e a => ↑(algebraMap R (M →ₗ[R] M)) a * f ^ e) v = sum p fun n b => b • ↑(f ^ n) v",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.2133490\nB' : Type ?u.2133493\na b : R\nn : ℕ\ninst✝⁴ : CommSemiring A'\ninst✝³ : Semiring B'\nM : Type u_1\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\nv : M\np : R[X]\n⊢ ↑(↑(aeval f) p) v = sum p fun n b => b • ↑(f ^ n) v",
"tactic": "rw [aeval_def, eval₂_eq_sum]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.2133490\nB' : Type ?u.2133493\na b : R\nn : ℕ\ninst✝⁴ : CommSemiring A'\ninst✝³ : Semiring B'\nM : Type u_1\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\nv : M\np : R[X]\n⊢ ↑(sum p fun e a => ↑(algebraMap R (M →ₗ[R] M)) a * f ^ e) v = sum p fun n b => b • ↑(f ^ n) v",
"tactic": "exact (LinearMap.applyₗ v).map_sum"
}
] |
[
516,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.eq_empty_of_isEmpty
|
[] |
[
593,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.mul_mem_mul_rev
|
[] |
[
452,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.map_span_le
|
[
{
"state_after": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.11034\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.11043\nS : Type ?u.11046\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\ns : Set M\nN : Submodule R₂ M₂\n⊢ s ⊆ ↑f ⁻¹' ↑N ↔ ∀ (m : M), m ∈ s → ↑f m ∈ N",
"state_before": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.11034\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.11043\nS : Type ?u.11046\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\ns : Set M\nN : Submodule R₂ M₂\n⊢ map f (span R s) ≤ N ↔ ∀ (m : M), m ∈ s → ↑f m ∈ N",
"tactic": "rw [f.map_span, span_le, Set.image_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.11034\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.11043\nS : Type ?u.11046\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\ns : Set M\nN : Submodule R₂ M₂\n⊢ s ⊆ ↑f ⁻¹' ↑N ↔ ∀ (m : M), m ∈ s → ↑f m ∈ N",
"tactic": "exact Iff.rfl"
}
] |
[
111,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Std/Data/Int/DivMod.lean
|
Int.ediv_eq_zero_of_lt
|
[] |
[
139,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
137,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.filter_apply_eq_zero_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.162293\nγ : Type ?u.162296\np : Pmf α\ns : Set α\nh : ∃ a, a ∈ s ∧ a ∈ support p\na : α\nha : ¬a ∈ s\n⊢ ↑(filter p s h) a = 0",
"tactic": "rw [filter_apply, Set.indicator_apply_eq_zero.mpr fun ha' => absurd ha' ha, MulZeroClass.zero_mul]"
}
] |
[
280,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Analysis/Calculus/Deriv/Polynomial.lean
|
Polynomial.hasStrictDerivAt_aeval
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\np : 𝕜[X]\nq : R[X]\nx : 𝕜\n⊢ HasStrictDerivAt (fun x => ↑(aeval x) q) (↑(aeval x) (↑derivative q)) x",
"tactic": "simpa only [aeval_def, eval₂_eq_eval_map, derivative_map] using\n (q.map (algebraMap R 𝕜)).hasStrictDerivAt x"
}
] |
[
80,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
11
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.ghostFun_pow
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.815357\nT : Type ?u.815360\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.815375\nβ : Type ?u.815378\nx y : 𝕎 R\nm : ℕ\n⊢ WittVector.ghostFun (x ^ m) = WittVector.ghostFun x ^ m",
"tactic": "ghost_fun_tac X 0 ^ m, ![x.coeff]"
}
] |
[
226,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
9
] |
Mathlib/Order/SymmDiff.lean
|
ofDual_symmDiff
|
[] |
[
235,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.le_prod_nonempty_of_submultiplicative
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.154312\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.154321\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\nh_mul : ∀ (a b : α), f (a * b) ≤ f a * f b\ns : Multiset α\nhs_nonempty : s ≠ ∅\n⊢ ∀ (a b : α), (fun x => True) a → (fun x => True) b → f (a * b) ≤ f a * f b",
"tactic": "simp [h_mul]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.154312\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.154321\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\nh_mul : ∀ (a b : α), f (a * b) ≤ f a * f b\ns : Multiset α\nhs_nonempty : s ≠ ∅\n⊢ ∀ (a b : α), (fun x => True) a → (fun x => True) b → (fun x => True) (a * b)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.154312\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.154321\ninst✝¹ : CommMonoid α\ninst✝ : OrderedCommMonoid β\nf : α → β\nh_mul : ∀ (a b : α), f (a * b) ≤ f a * f b\ns : Multiset α\nhs_nonempty : s ≠ ∅\n⊢ ∀ (a : α), a ∈ s → (fun x => True) a",
"tactic": "simp"
}
] |
[
513,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
Monotone.mul_const_of_nonpos
|
[] |
[
430,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_Ioi
|
[] |
[
146,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.Nonempty.inr
|
[] |
[
503,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
ContinuousLinearMap.comp_hasFPowerSeriesOnBall
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\ng : F →L[𝕜] G\nh : HasFPowerSeriesOnBall f p x r\ny✝ : E\nhy : y✝ ∈ EMetric.ball 0 r\n⊢ HasSum (fun n => ↑(compFormalMultilinearSeries g p n) fun x => y✝) ((↑g ∘ f) (x + y✝))",
"tactic": "simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,\n ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using\n g.hasSum (h.hasSum hy)"
}
] |
[
632,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
IsUnit.smul_sub_iff_sub_inv_smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : Group α\ninst✝⁴ : Monoid β\ninst✝³ : AddGroup β\ninst✝² : DistribMulAction α β\ninst✝¹ : IsScalarTower α β β\ninst✝ : SMulCommClass α β β\nr : α\na : β\n⊢ IsUnit (r • 1 - a) ↔ IsUnit (1 - r⁻¹ • a)",
"tactic": "rw [← isUnit_smul_iff r (1 - r⁻¹ • a), smul_sub, smul_inv_smul]"
}
] |
[
412,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean
|
pochhammer_zero
|
[] |
[
54,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/LinearAlgebra/Vandermonde.lean
|
Matrix.vandermonde_mul_vandermonde_transpose
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nv w : Fin n → R\ni j : Fin n\n⊢ (vandermonde v ⬝ (vandermonde w)ᵀ) i j = ∑ k : Fin n, (v i * w j) ^ ↑k",
"tactic": "simp only [vandermonde_apply, Matrix.mul_apply, Matrix.transpose_apply, mul_pow]"
}
] |
[
70,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.HasGroupoid.comp
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\n⊢ LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f' ∈ G₁",
"state_before": "H✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H✝\nf f' : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\n⊢ ∀ {e e' : LocalHomeomorph H₃ H₁}, e ∈ atlas H₁ H₃ → e' ∈ atlas H₁ H₃ → LocalHomeomorph.symm e ≫ₕ e' ∈ G₁",
"tactic": "rintro _ _ ⟨e, f, he, hf, rfl⟩ ⟨e', f', he', hf', rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\n⊢ ∀ (x : H₁),\n x ∈ (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f').toLocalEquiv.source →\n ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\n⊢ LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f' ∈ G₁",
"tactic": "apply G₁.locality"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx : x ∈ (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\n⊢ ∀ (x : H₁),\n x ∈ (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f').toLocalEquiv.source →\n ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx : x ∈ (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "simp only [mfld_simps] at hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "have hxs : x ∈ f.symm ⁻¹' (e.symm ≫ₕ e').source := by simp only [hx, mfld_simps]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "have hxs' : x ∈ f.target ∩ f.symm ⁻¹' ((e.symm ≫ₕ e').source ∩ e.symm ≫ₕ e' ⁻¹' f'.source) :=\n by simp only [hx, mfld_simps]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ :\n EqOn (↑f' ∘ ↑(LocalHomeomorph.symm e ≫ₕ e') ∘ ↑(LocalHomeomorph.symm f)) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhφ_dom : x ∈ φ.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "obtain ⟨φ, hφG₁, hφ, hφ_dom⟩ := LocalInvariantProp.liftPropOn_indep_chart\n (isLocalStructomorphWithinAt_localInvariantProp G₁) (G₁.subset_maximalAtlas hf)\n (G₁.subset_maximalAtlas hf') (H _ (G₂.compatible he he')) hxs' hxs"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ :\n EqOn (↑f' ∘ ↑(LocalHomeomorph.symm e ≫ₕ e') ∘ ↑(LocalHomeomorph.symm f)) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhφ_dom : x ∈ φ.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "simp_rw [← LocalHomeomorph.coe_trans, LocalHomeomorph.trans_assoc] at hφ"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm (e ≫ₕ f) ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "simp_rw [LocalHomeomorph.trans_symm_eq_symm_trans_symm, LocalHomeomorph.trans_assoc]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') s ∈ G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "have hs : IsOpen (f.symm ≫ₕ e.symm ≫ₕ e' ≫ₕ f').source :=\n (f.symm ≫ₕ e.symm ≫ₕ e' ≫ₕ f').open_source"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ x ∈ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')\n ((LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source) ∈\n G₁",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') s ∈ G₁",
"tactic": "refine' ⟨_, hs.inter φ.open_source, _, _⟩"
},
{
"state_after": "no goals",
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"tactic": "simp only [hx, mfld_simps]"
},
{
"state_after": "no goals",
"state_before": "H✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\n⊢ x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)",
"tactic": "simp only [hx, mfld_simps]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ x ∈ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source",
"tactic": "simp only [hx, hφ_dom, mfld_simps]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')\n ((LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source) ≈\n LocalHomeomorph.restr φ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')\n ((LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source) ∈\n G₁",
"tactic": "refine' G₁.eq_on_source (closedUnderRestriction' hφG₁ hs) _"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') φ.source ≈\n LocalHomeomorph.restr φ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')\n ((LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source) ≈\n LocalHomeomorph.restr φ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source",
"tactic": "rw [LocalHomeomorph.restr_source_inter]"
},
{
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"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ LocalHomeomorph.restr (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f') φ.source ≈\n LocalHomeomorph.restr φ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source",
"tactic": "refine' LocalHomeomorph.Set.EqOn.restr_eqOn_source (hφ.mono _)"
},
{
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"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nH✝ : Type ?u.72128\nM : Type ?u.72131\nH' : Type ?u.72134\nM' : Type ?u.72137\nX : Type ?u.72140\ninst✝¹⁴ : TopologicalSpace H✝\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H✝ M\ninst✝¹¹ : TopologicalSpace H'\ninst✝¹⁰ : TopologicalSpace M'\ninst✝⁹ : ChartedSpace H' M'\ninst✝⁸ : TopologicalSpace X\nG : StructureGroupoid H✝\nG' : StructureGroupoid H'\ne✝ e'✝ : LocalHomeomorph M H✝\nf✝ f'✝ : LocalHomeomorph M' H'\nP : (H✝ → H') → Set H✝ → H✝ → Prop\ng g' : M → M'\ns t : Set M\nx✝¹ : M\nQ : (H✝ → H✝) → Set H✝ → H✝ → Prop\nH₁ : Type u_2\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_1\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_3\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroupoid H₂\ninst✝ : HasGroupoid H₃ G₂\nH : ∀ (e : LocalHomeomorph H₂ H₂), e ∈ G₂ → LiftPropOn (IsLocalStructomorphWithinAt G₁) (↑e) e.source\nx✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃\ne : LocalHomeomorph H₃ H₂\nf : LocalHomeomorph H₂ H₁\nhe : e ∈ atlas H₂ H₃\nhf : f ∈ atlas H₁ H₂\ne' : LocalHomeomorph H₃ H₂\nf' : LocalHomeomorph H₂ H₁\nhe' : e' ∈ atlas H₂ H₃\nhf' : f' ∈ atlas H₁ H₂\nx : H₁\nhx :\n (x ∈ f.target ∧ ↑(LocalHomeomorph.symm f) x ∈ e.target) ∧\n ↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x) ∈ e'.source ∧\n ↑e' (↑(LocalHomeomorph.symm e) (↑(LocalHomeomorph.symm f) x)) ∈ f'.source\nhxs : x ∈ ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\nhxs' :\n x ∈\n f.target ∩\n ↑(LocalHomeomorph.symm f) ⁻¹'\n ((LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm e ≫ₕ e') ⁻¹' f'.source)\nφ : LocalHomeomorph H₁ H₁\nhφG₁ : φ ∈ G₁\nhφ_dom : x ∈ φ.source\nhφ :\n EqOn (↑(LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f')) (↑φ.toLocalEquiv)\n (↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source)\nhs : IsOpen (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source\n⊢ (LocalHomeomorph.symm f ≫ₕ LocalHomeomorph.symm e ≫ₕ e' ≫ₕ f').toLocalEquiv.source ∩ φ.source ⊆\n ↑(LocalHomeomorph.symm f) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ φ.source",
"tactic": "mfld_set_tac"
}
] |
[
682,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.mem_inf_iff
|
[] |
[
587,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
586,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.out_proj
|
[] |
[
456,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.HomotopyRel.fst_eq_snd
|
[] |
[
601,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.sameRay_pos_smul_left
|
[] |
[
151,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.mem_span_pow
|
[
{
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"tactic": "first | exact lt_of_le_of_ne (Nat.zero_le d) hd.symm | exact WithBot.bot_lt_coe d"
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{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_2\nS : Type u_1\nT : Type ?u.27902\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.28205\nB : Type ?u.28208\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : IsDomain B\ninst✝¹ : Algebra A B\nK : Type ?u.28630\ninst✝ : Field K\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nhy : y = ↑(aeval x) f\nhf : f = 0\nh : 0 < d\n⊢ ⊥ < ↑d",
"tactic": "exact WithBot.bot_lt_coe d"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_2\nS : Type u_1\nT : Type ?u.27902\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.28205\nB : Type ?u.28208\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nK : Type ?u.28630\ninst✝¹ : Field K\nx y : S\nd : ℕ\nf : R[X]\ninst✝ : IsDomain B\nhd : d ≠ 0\nh : natDegree f < d\nhy : y = ↑(aeval x) f\nhf : ¬f = 0\n⊢ ↑(natDegree f) < ↑d",
"tactic": "first | exact WithBot.coe_lt_coe.1 h | exact WithBot.coe_lt_coe.2 h"
},
{
"state_after": "case neg\nR : Type u_2\nS : Type u_1\nT : Type ?u.27902\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.28205\nB : Type ?u.28208\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nK : Type ?u.28630\ninst✝¹ : Field K\nx y : S\nd : ℕ\nf : R[X]\ninst✝ : IsDomain B\nhd : d ≠ 0\nh : natDegree f < d\nhy : y = ↑(aeval x) f\nhf : ¬f = 0\n⊢ ↑(natDegree f) < ↑d",
"state_before": "case neg\nR : Type u_2\nS : Type u_1\nT : Type ?u.27902\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.28205\nB : Type ?u.28208\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nK : Type ?u.28630\ninst✝¹ : Field K\nx y : S\nd : ℕ\nf : R[X]\ninst✝ : IsDomain B\nhd : d ≠ 0\nh : natDegree f < d\nhy : y = ↑(aeval x) f\nhf : ¬f = 0\n⊢ ↑(natDegree f) < ↑d",
"tactic": "exact WithBot.coe_lt_coe.1 h"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_2\nS : Type u_1\nT : Type ?u.27902\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring S\ninst✝⁵ : Algebra R S\nA : Type ?u.28205\nB : Type ?u.28208\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nK : Type ?u.28630\ninst✝¹ : Field K\nx y : S\nd : ℕ\nf : R[X]\ninst✝ : IsDomain B\nhd : d ≠ 0\nh : natDegree f < d\nhy : y = ↑(aeval x) f\nhf : ¬f = 0\n⊢ ↑(natDegree f) < ↑d",
"tactic": "exact WithBot.coe_lt_coe.2 h"
}
] |
[
119,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid.size_eq
|
[] |
[
1113,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1111,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.prehaar_empty
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : TopologicalSpace G\nK₀ : PositiveCompacts G\nU : Set G\n⊢ prehaar (↑K₀) U ⊥ = 0",
"tactic": "rw [prehaar, Compacts.coe_bot, index_empty, Nat.cast_zero, zero_div]"
}
] |
[
126,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
multiplesAddHom_symm_apply
|
[] |
[
1004,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.lt_succ_iff_lt_or_eq
|
[] |
[
363,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/CategoryTheory/NatIso.lean
|
CategoryTheory.NatIso.cancel_natIso_hom_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nα : F ≅ G\nX : D\nY : C\nf f' : X ⟶ F.obj Y\n⊢ f ≫ α.hom.app Y = f' ≫ α.hom.app Y ↔ f = f'",
"tactic": "simp only [cancel_mono, refl]"
}
] |
[
142,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
DirectSum.coe_decompose_mul_of_right_mem
|
[
{
"state_after": "case intro\nι : Type u_1\nR : Type ?u.386108\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\na : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = if i ≤ n then ↑(↑(↑(decompose 𝒜) a) (n - i)) * ↑b else 0",
"state_before": "ι : Type u_1\nR : Type ?u.386108\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\na b : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\nb_mem : b ∈ 𝒜 i\n⊢ ↑(↑(↑(decompose 𝒜) (a * b)) n) = if i ≤ n then ↑(↑(↑(decompose 𝒜) a) (n - i)) * b else 0",
"tactic": "lift b to 𝒜 i using b_mem"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nR : Type ?u.386108\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\na : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = if i ≤ n then ↑(↑(↑(decompose 𝒜) a) (n - i)) * ↑b else 0",
"tactic": "rw [decompose_mul, decompose_coe, coe_mul_of_apply]"
}
] |
[
336,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
SetLike.coe_list_dProd
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_2\nα : Type u_4\nS : Type u_3\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : GradedMonoid A\nfι : α → ι\nfA : (a : α) → { x // x ∈ A (fι a) }\nl : List α\n⊢ ↑(List.dProd l fι fA) = List.prod (List.map (fun a => ↑(fA a)) l)",
"tactic": "match l with\n| [] =>\n rw [List.dProd_nil, coe_gOne, List.map_nil, List.prod_nil]\n| head::tail =>\n rw [List.dProd_cons, coe_gMul, List.map_cons, List.prod_cons,\n SetLike.coe_list_dProd _ _ _ tail]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_2\nα : Type u_4\nS : Type u_3\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : GradedMonoid A\nfι : α → ι\nfA : (a : α) → { x // x ∈ A (fι a) }\nl : List α\n⊢ ↑(List.dProd [] fι fA) = List.prod (List.map (fun a => ↑(fA a)) [])",
"tactic": "rw [List.dProd_nil, coe_gOne, List.map_nil, List.prod_nil]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_2\nα : Type u_4\nS : Type u_3\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : GradedMonoid A\nfι : α → ι\nfA : (a : α) → { x // x ∈ A (fι a) }\nl : List α\nhead : α\ntail : List α\n⊢ ↑(List.dProd (head :: tail) fι fA) = List.prod (List.map (fun a => ↑(fA a)) (head :: tail))",
"tactic": "rw [List.dProd_cons, coe_gMul, List.map_cons, List.prod_cons,\n SetLike.coe_list_dProd _ _ _ tail]"
}
] |
[
658,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/CategoryTheory/Yoneda.lean
|
CategoryTheory.yonedaEquiv_naturality
|
[
{
"state_after": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ (f.app X.op ≫ F.map g.op) (𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ F.map g.op (↑yonedaEquiv f) = ↑yonedaEquiv (yoneda.map g ≫ f)",
"tactic": "change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g)"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ ((yoneda.obj X).map g.op ≫ f.app Y.op) (𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ (f.app X.op ≫ F.map g.op) (𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"tactic": "rw [← f.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ f.app Y.op (g ≫ 𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ ((yoneda.obj X).map g.op ≫ f.app Y.op) (𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y : C\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj X ⟶ F\ng : Y ⟶ X\n⊢ f.app Y.op (g ≫ 𝟙 X) = f.app Y.op (𝟙 Y ≫ g)",
"tactic": "simp"
}
] |
[
402,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime.gcd_both
|
[] |
[
347,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
346,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.inv_eq_iff_mul_eq_one
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1927198\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) ψ ≠ 0\n⊢ ψ⁻¹ = φ ↔ φ * ψ = 1",
"tactic": "rw [eq_comm, MvPowerSeries.eq_inv_iff_mul_eq_one h]"
}
] |
[
1005,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
11
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_of_le
|
[] |
[
121,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
NNReal.inner_le_Lp_mul_Lq_tsum
|
[
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i)\n⊢ (Summable fun i => f i * g i) ∧\n (∑' (i : ι), f i * g i) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\n⊢ (Summable fun i => f i * g i) ∧\n (∑' (i : ι), f i * g i) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by\n refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩\n rintro a ⟨s, rfl⟩\n exact H₁ s"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i)\nH₂ : Summable fun i => f i * g i\n⊢ (Summable fun i => f i * g i) ∧\n (∑' (i : ι), f i * g i) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i)\n⊢ (Summable fun i => f i * g i) ∧\n (∑' (i : ι), f i * g i) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i)\nH₂ : Summable fun i => f i * g i\n⊢ (Summable fun i => f i * g i) ∧\n (∑' (i : ι), f i * g i) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩"
},
{
"state_after": "ι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\n⊢ ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "intro s"
},
{
"state_after": "case refine'_1\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p)\n\ncase refine'_2\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ (∑ i in s, g i ^ q) ^ (1 / q) ≤ (∑' (i : ι), g i ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)"
},
{
"state_after": "case refine'_1\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, f i ^ p ≤ ∑' (i : ι), f i ^ p",
"state_before": "case refine'_1\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p)",
"tactic": "rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, f i ^ p ≤ ∑' (i : ι), f i ^ p",
"tactic": "exact sum_le_tsum _ (fun _ _ => zero_le _) hf"
},
{
"state_after": "case refine'_2\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, g i ^ q ≤ ∑' (i : ι), g i ^ q",
"state_before": "case refine'_2\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ (∑ i in s, g i ^ q) ^ (1 / q) ≤ (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset ι\n⊢ ∑ i in s, g i ^ q ≤ ∑' (i : ι), g i ^ q",
"tactic": "exact sum_le_tsum _ (fun _ _ => zero_le _) hg"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\n⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) ∈\n upperBounds (Set.range fun s => ∑ i in s, f i * g i)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\n⊢ BddAbove (Set.range fun s => ∑ i in s, f i * g i)",
"tactic": "refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩"
},
{
"state_after": "case intro\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\ns : Finset ι\n⊢ (fun s => ∑ i in s, f i * g i) s ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\n⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) ∈\n upperBounds (Set.range fun s => ∑ i in s, f i * g i)",
"tactic": "rintro a ⟨s, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u\ns✝ : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i in s, f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\ns : Finset ι\n⊢ (fun s => ∑ i in s, f i * g i) s ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"tactic": "exact H₁ s"
}
] |
[
401,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le_add_of_neg_le_sub_left
|
[] |
[
1015,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1014,
11
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
le_inv_mul_iff_mul_le
|
[
{
"state_after": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * b ≤ a * (a⁻¹ * c) ↔ a * b ≤ c",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ b ≤ a⁻¹ * c ↔ a * b ≤ c",
"tactic": "rw [← mul_le_mul_iff_left a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * b ≤ a * (a⁻¹ * c) ↔ a * b ≤ c",
"tactic": "simp"
}
] |
[
118,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.le_natDegree_of_coe_le_degree
|
[
{
"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.736582\nhdeg : ↑n ≤ degree p\n⊢ ↑n ≤ ↑(natDegree p)",
"tactic": "rwa [degree_eq_natDegree <| ne_zero_of_coe_le_degree hdeg] at hdeg"
}
] |
[
1131,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1128,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Ordered.toList_sorted
|
[] |
[
411,
16
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
410,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.recBotCoe_coe
|
[] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Topology/Perfect.lean
|
IsClosed.exists_nat_bool_injection_of_not_countable
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"tactic": "letI := upgradePolishSpace α"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\nD : Set α\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D ⊆ C\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"tactic": "obtain ⟨D, hD, Dnonempty, hDC⟩ := exists_perfect_nonempty_of_isClosed_of_not_countable hC hunc"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\nD : Set α\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D ⊆ C\nf : (ℕ → Bool) → α\nhfD : range f ⊆ D\nhf : Continuous f ∧ Function.Injective f\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\nD : Set α\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D ⊆ C\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"tactic": "obtain ⟨f, hfD, hf⟩ := hD.exists_nat_bool_injection Dnonempty"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nC : Set α\nhC : IsClosed C\nhunc : ¬Set.Countable C\nthis : UpgradedPolishSpace α := upgradePolishSpace α\nD : Set α\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D ⊆ C\nf : (ℕ → Bool) → α\nhfD : range f ⊆ D\nhf : Continuous f ∧ Function.Injective f\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Function.Injective f",
"tactic": "exact ⟨f, hfD.trans hDC, hf⟩"
}
] |
[
330,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Uniform.lean
|
Pmf.support_uniformOfFinset
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.5502\nγ : Type ?u.5505\ns : Finset α\nhs : Finset.Nonempty s\na✝ a : α\nha : a ∈ s\n⊢ ∀ (x : α), x ∈ support (uniformOfFinset s (_ : ∃ x, x ∈ s)) ↔ x ∈ ↑s",
"state_before": "α : Type u_1\nβ : Type ?u.5502\nγ : Type ?u.5505\ns : Finset α\nhs : Finset.Nonempty s\na : α\n⊢ ∀ (x : α), x ∈ support (uniformOfFinset s hs) ↔ x ∈ ↑s",
"tactic": "let ⟨a, ha⟩ := hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5502\nγ : Type ?u.5505\ns : Finset α\nhs : Finset.Nonempty s\na✝ a : α\nha : a ∈ s\n⊢ ∀ (x : α), x ∈ support (uniformOfFinset s (_ : ∃ x, x ∈ s)) ↔ x ∈ ↑s",
"tactic": "simp [mem_support_iff, Finset.ne_empty_of_mem ha]"
}
] |
[
71,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.coprime_add_mul_left_left
|
[
{
"state_after": "no goals",
"state_before": "m n k : ℕ\n⊢ coprime (m + n * k) n ↔ coprime m n",
"tactic": "rw [coprime, coprime, gcd_add_mul_left_left]"
}
] |
[
179,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
Right.neg_of_mul_neg_left
|
[] |
[
588,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
ContinuousOn.aestronglyMeasurable
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ AEStronglyMeasurable f (Measure.restrict μ s)",
"state_before": "α : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\n⊢ AEStronglyMeasurable f (Measure.restrict μ s)",
"tactic": "borelize β"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ IsSeparable (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ AEStronglyMeasurable f (Measure.restrict μ s)",
"tactic": "refine'\n aestronglyMeasurable_iff_aemeasurable_separable.2\n ⟨hf.aemeasurable hs, f '' s, _,\n mem_of_superset (self_mem_ae_restrict hs) (subset_preimage_image _ _)⟩"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\n⊢ IsSeparable (f '' s)\n\ncase inr\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology β\n⊢ IsSeparable (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ IsSeparable (f '' s)",
"tactic": "cases h.out"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\n⊢ IsSeparable (f '' s)",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\n⊢ IsSeparable (f '' s)",
"tactic": "let f' : s → β := s.restrict f"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\n⊢ IsSeparable (f '' s)",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\n⊢ IsSeparable (f '' s)",
"tactic": "have A : Continuous f' := continuousOn_iff_continuous_restrict.1 hf"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\n⊢ IsSeparable (f '' s)",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\n⊢ IsSeparable (f '' s)",
"tactic": "have B : IsSeparable (univ : Set s) := isSeparable_of_separableSpace _"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\n⊢ f '' s = f' '' univ",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\n⊢ IsSeparable (f '' s)",
"tactic": "convert IsSeparable.image B A using 1"
},
{
"state_after": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\nx : β\n⊢ x ∈ f '' s ↔ x ∈ f' '' univ",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\n⊢ f '' s = f' '' univ",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology α\nf' : ↑s → β := Set.restrict s f\nA : Continuous f'\nB : IsSeparable univ\nx : β\n⊢ x ∈ f '' s ↔ x ∈ f' '' univ",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\nE : Type ?u.1903104\nF : Type ?u.1903107\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : MeasureTheory.Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh✝ : SecondCountableTopology β\n⊢ IsSeparable (f '' s)",
"tactic": "exact isSeparable_of_separableSpace _"
}
] |
[
537,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/Data/Dfinsupp/Order.lean
|
Dfinsupp.subset_support_tsub
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝⁴ : (i : ι) → CanonicallyOrderedAddMonoid (α i)\ninst✝³ : (i : ι) → Sub (α i)\ninst✝² : ∀ (i : ι), OrderedSub (α i)\nf g : Π₀ (i : ι), α i\ni : ι\na b : α i\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)\n⊢ support f \\ support g ⊆ support (f - g)",
"tactic": "simp (config := { contextual := true }) [subset_iff]"
}
] |
[
244,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/AssocList.lean
|
Std.AssocList.erase_toList
|
[] |
[
183,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
182,
9
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
GradedRing.proj_apply
|
[] |
[
117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.order_le_of_coeff_ne_zero
|
[] |
[
242,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
eventually_nhdsWithin_pos_mul_left
|
[
{
"state_after": "𝕜 : Type u_1\nα : Type ?u.57235\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nx : 𝕜\nhx : 0 < x\np : 𝕜 → Prop\nh : ∀ᶠ (ε : 𝕜) in 𝓝[Ioi 0] 0, p ε\n⊢ ∀ᶠ (ε : 𝕜) in comap (fun x_1 => x * x_1) (𝓝[Ioi 0] 0), p (x * ε)",
"state_before": "𝕜 : Type u_1\nα : Type ?u.57235\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nx : 𝕜\nhx : 0 < x\np : 𝕜 → Prop\nh : ∀ᶠ (ε : 𝕜) in 𝓝[Ioi 0] 0, p ε\n⊢ ∀ᶠ (ε : 𝕜) in 𝓝[Ioi 0] 0, p (x * ε)",
"tactic": "rw [← nhdsWithin_pos_comap_mul_left hx]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nα : Type ?u.57235\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nx : 𝕜\nhx : 0 < x\np : 𝕜 → Prop\nh : ∀ᶠ (ε : 𝕜) in 𝓝[Ioi 0] 0, p ε\n⊢ ∀ᶠ (ε : 𝕜) in comap (fun x_1 => x * x_1) (𝓝[Ioi 0] 0), p (x * ε)",
"tactic": "exact h.comap fun ε => x * ε"
}
] |
[
238,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
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