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/Topology/Order/Basic.lean
|
Ioc_mem_nhdsWithin_Iio
|
[] |
[
479,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Topology/Partial.lean
|
ptendsto'_nhds
|
[] |
[
47,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.coe_toList
|
[] |
[
3383,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3382,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Term.realize_restrictVarLeft
|
[
{
"state_after": "case var\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\na : α ⊕ γ\nh : ↑(varFinsetLeft (var a)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (var a) (Set.inclusion h)) = realize (Sum.elim v xs) (var a)\n\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L (α ⊕ γ)\nih :\n ∀ (a : Fin l✝) (h : ↑(varFinsetLeft (_ts✝ a)) ⊆ s),\n realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (_ts✝ a) (Set.inclusion h)) =\n realize (Sum.elim v xs) (_ts✝ a)\nh : ↑(varFinsetLeft (func _f✝ _ts✝)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (func _f✝ _ts✝) (Set.inclusion h)) =\n realize (Sum.elim v xs) (func _f✝ _ts✝)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft t (Set.inclusion h)) = realize (Sum.elim v xs) t",
"tactic": "induction' t with a _ _ _ ih"
},
{
"state_after": "no goals",
"state_before": "case var\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\na : α ⊕ γ\nh : ↑(varFinsetLeft (var a)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (var a) (Set.inclusion h)) = realize (Sum.elim v xs) (var a)",
"tactic": "cases a <;> rfl"
},
{
"state_after": "case func\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝¹ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L (α ⊕ γ)\nih :\n ∀ (a : Fin l✝) (h : ↑(varFinsetLeft (_ts✝ a)) ⊆ s),\n realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (_ts✝ a) (Set.inclusion h)) =\n realize (Sum.elim v xs) (_ts✝ a)\nh✝ : ↑(Finset.biUnion Finset.univ fun i => varFinsetLeft (_ts✝ i)) ⊆ s\nh : ∀ (i : Fin l✝), i ∈ ↑Finset.univ → ↑(varFinsetLeft (_ts✝ i)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (func _f✝ _ts✝) (Set.inclusion h✝)) =\n realize (Sum.elim v xs) (func _f✝ _ts✝)",
"state_before": "case func\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L (α ⊕ γ)\nih :\n ∀ (a : Fin l✝) (h : ↑(varFinsetLeft (_ts✝ a)) ⊆ s),\n realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (_ts✝ a) (Set.inclusion h)) =\n realize (Sum.elim v xs) (_ts✝ a)\nh : ↑(varFinsetLeft (func _f✝ _ts✝)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (func _f✝ _ts✝) (Set.inclusion h)) =\n realize (Sum.elim v xs) (func _f✝ _ts✝)",
"tactic": "simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case func\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.12423\nP : Type ?u.12426\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nγ : Type u_1\nt : Term L (α ⊕ γ)\ns : Set α\nh✝¹ : ↑(varFinsetLeft t) ⊆ s\nv : α → M\nxs : γ → M\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L (α ⊕ γ)\nih :\n ∀ (a : Fin l✝) (h : ↑(varFinsetLeft (_ts✝ a)) ⊆ s),\n realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (_ts✝ a) (Set.inclusion h)) =\n realize (Sum.elim v xs) (_ts✝ a)\nh✝ : ↑(Finset.biUnion Finset.univ fun i => varFinsetLeft (_ts✝ i)) ⊆ s\nh : ∀ (i : Fin l✝), i ∈ ↑Finset.univ → ↑(varFinsetLeft (_ts✝ i)) ⊆ s\n⊢ realize (Sum.elim (v ∘ Subtype.val) xs) (restrictVarLeft (func _f✝ _ts✝) (Set.inclusion h✝)) =\n realize (Sum.elim v xs) (func _f✝ _ts✝)",
"tactic": "exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))"
}
] |
[
158,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.pred_toNat
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ toNat (↑(n + 1) - 1) = toNat ↑(n + 1) - 1",
"tactic": "simp [ofNat_add]"
}
] |
[
1396,
18
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1393,
9
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
IsUnit.aestronglyMeasurable_const_smul_iff
|
[] |
[
1766,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1763,
8
] |
Mathlib/Algebra/Lie/Abelian.lean
|
LieModule.mem_ker
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nx : L\n⊢ x ∈ LieModule.ker R L M ↔ ∀ (m : M), ⁅x, m⁆ = 0",
"tactic": "simp only [LieModule.ker, LieHom.mem_ker, LinearMap.ext_iff, LinearMap.zero_apply,\n toEndomorphism_apply_apply]"
}
] |
[
118,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
11
] |
Mathlib/Order/Antisymmetrization.lean
|
Antisymmetrization.ind
|
[] |
[
117,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
11
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.sInf_toSubsemiring
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Set (Subalgebra R A)\n⊢ ↑(sInf S).toSubsemiring = ↑(sInf (Subalgebra.toSubsemiring '' S))",
"tactic": "simp"
}
] |
[
872,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
870,
1
] |
Mathlib/Data/Real/Pi/Bounds.lean
|
Real.pi_gt_three
|
[
{
"state_after": "no goals",
"state_before": "⊢ 3 < π",
"tactic": "pi_lower_bound [23/16]"
}
] |
[
170,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Logic/Lemmas.lean
|
ite_ite_distrib_right
|
[] |
[
71,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
Cycle.formPerm_eq_self_of_not_mem
|
[
{
"state_after": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nx : α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhx : ¬x ∈ Quot.mk Setoid.r a✝\n⊢ ↑(formPerm (Quot.mk Setoid.r a✝) h) x = x",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s' s : Cycle α\nh : Nodup s\nx : α\nhx : ¬x ∈ s\n⊢ ↑(formPerm s h) x = x",
"tactic": "induction s using Quot.inductionOn"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nx : α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhx : ¬x ∈ Quot.mk Setoid.r a✝\n⊢ ↑(formPerm (Quot.mk Setoid.r a✝) h) x = x",
"tactic": "simpa using List.formPerm_eq_self_of_not_mem _ _ hx"
}
] |
[
180,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.count_map
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.414392\nβ✝ : Type ?u.414395\nγ : Type ?u.414398\ninst✝¹ : DecidableEq α✝\nα : Type u_1\nβ : Type u_2\nf : α → β\ns : Multiset α\ninst✝ : DecidableEq β\nb : β\n⊢ count b (map f s) = ↑card (filter (fun a => b = f a) s)",
"tactic": "simp [Bool.beq_eq_decide_eq, eq_comm, count, countp_map]"
}
] |
[
2535,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2533,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.EventuallyEq.preimage
|
[] |
[
1563,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1561,
1
] |
Mathlib/CategoryTheory/Linear/LinearFunctor.lean
|
CategoryTheory.Functor.map_smul
|
[] |
[
52,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.monic_prod_of_monic
|
[] |
[
276,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.coe_sSup_of_directedOn
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ (s : Subsemiring R) (_ : s ∈ S), ↑s",
"tactic": "simp [mem_sSup_of_directedOn Sne hS]"
}
] |
[
1115,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1113,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_add
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ (Finset.sup (support (p + q)) fun s => ↑toMultiset s) ≤\n (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ degrees (p + q) ≤ degrees p ⊔ degrees q",
"tactic": "simp_rw [degrees_def]"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ (Finset.sup (support (p + q)) fun s => ↑toMultiset s) ≤\n (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "refine' Finset.sup_le fun b hb => _"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nthis : b ∈ p.support ∪ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "have := Finsupp.support_add hb"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nthis : b ∈ p.support ∨ b ∈ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nthis : b ∈ p.support ∪ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "rw [Finset.mem_union] at this"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nh : b ∈ p.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s\n\ncase inr\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nh : b ∈ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nthis : b ∈ p.support ∨ b ∈ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "cases' this with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nh : b ∈ p.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "exact le_sup_of_le_left (Finset.le_sup h)"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.31525\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ support (p + q)\nh : b ∈ q.support\n⊢ ↑toMultiset b ≤ (Finset.sup (support p) fun s => ↑toMultiset s) ⊔ Finset.sup (support q) fun s => ↑toMultiset s",
"tactic": "exact le_sup_of_le_right (Finset.le_sup h)"
}
] |
[
148,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.inv_apply_apply
|
[] |
[
204,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.symm_symm
|
[] |
[
758,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_indicator
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ (⨆ (x : { i // ↑i ≤ fun a => indicator s f a }), SimpleFunc.lintegral (↑x) μ) =\n ⨆ (x : { i // ↑i ≤ fun a => f a }), SimpleFunc.lintegral (restrict (↑x) s) μ",
"state_before": "α : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (a : α), indicator s f a ∂μ) = ∫⁻ (a : α) in s, f a ∂μ",
"tactic": "simp only [lintegral, ← restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype']"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\n⊢ ∃ i', SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤ SimpleFunc.lintegral (restrict (↑i') s) μ\n\ncase a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\n⊢ ∃ i', SimpleFunc.lintegral (restrict (↑{ val := φ, property := hφ }) s) μ ≤ SimpleFunc.lintegral (↑i') μ",
"state_before": "α : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ (⨆ (x : { i // ↑i ≤ fun a => indicator s f a }), SimpleFunc.lintegral (↑x) μ) =\n ⨆ (x : { i // ↑i ≤ fun a => f a }), SimpleFunc.lintegral (restrict (↑x) s) μ",
"tactic": "apply le_antisymm <;> refine' iSup_mono' (Subtype.forall.2 fun φ hφ => _)"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\n⊢ SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤\n SimpleFunc.lintegral (restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) μ",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\n⊢ ∃ i', SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤ SimpleFunc.lintegral (restrict (↑i') s) μ",
"tactic": "refine' ⟨⟨φ, le_trans hφ (indicator_le_self _ _)⟩, _⟩"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\n⊢ SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤\n SimpleFunc.lintegral (restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) μ",
"tactic": "refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : x ∈ s\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x\n\ncase neg\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : ¬x ∈ s\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"tactic": "by_cases hx : x ∈ s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : x ∈ s\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"tactic": "simp [hx, hs, le_refl]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : ¬x ∈ s\n⊢ (fun a => indicator s f a) x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : ¬x ∈ s\n⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"tactic": "apply le_trans (hφ x)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => indicator s f a\nx : α\nhx : ¬x ∈ s\n⊢ (fun a => indicator s f a) x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x",
"tactic": "simp [hx, hs, le_refl]"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nx : α\n⊢ ↑(restrict φ s) x ≤ (fun a => indicator s f a) x",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\n⊢ ∃ i', SimpleFunc.lintegral (restrict (↑{ val := φ, property := hφ }) s) μ ≤ SimpleFunc.lintegral (↑i') μ",
"tactic": "refine' ⟨⟨φ.restrict s, fun x => _⟩, le_rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.948903\nγ : Type ?u.948906\nδ : Type ?u.948909\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nx : α\n⊢ ↑(restrict φ s) x ≤ (fun a => indicator s f a) x",
"tactic": "simp [hφ x, hs, indicator_le_indicator]"
}
] |
[
781,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim
|
[
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ↑(trim m) s",
"tactic": "simp only [trim_eq_iInf]"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t",
"tactic": "set ms := ⨅ (t : Set α) (st : s ⊆ t) (ht : MeasurableSet t), m t"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms\n\ncase neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "by_cases hs : ms = ∞"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "simp only [hs]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ∀ (i : Set α), s ⊆ i → MeasurableSet i → ↑m i = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤",
"tactic": "simp only [iInf_eq_top] at hs"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ∀ (i : Set α), s ⊆ i → MeasurableSet i → ↑m i = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤",
"tactic": "exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by\n intro r hs\n have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs\n rcases this with ⟨t, hmt, hin, hlt⟩\n exists t"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nthis : ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by\n intro n\n refine' this _ (ENNReal.lt_add_right hs _)\n simp"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nthis : ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "choose t hsub hm hm' using this"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms",
"tactic": "refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 (ms + 0))\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"tactic": "have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=\n tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 (ms + 0))\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"tactic": "rw [add_zero] at this"
},
{
"state_after": "case neg.refine'_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\nn : ℕ\n⊢ ↑m (⋂ (n : ℕ), t n) ≤ ms + (↑n)⁻¹\n\ncase neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ ms ≤ ↑m (⋂ (n : ℕ), t n)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ ↑m (⋂ (n : ℕ), t n) = ms",
"tactic": "refine' le_antisymm (ge_of_tendsto' this fun n => _) _"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\n⊢ ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"tactic": "intro r hs"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\nthis : ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"tactic": "have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\nt : Set α\nhmt : MeasurableSet t\nhin : s ⊆ t\nhlt : ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\nthis : ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"tactic": "rcases this with ⟨t, hmt, hin, hlt⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\nt : Set α\nhmt : MeasurableSet t\nhin : s ⊆ t\nhlt : ↑m t < r\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r",
"tactic": "exists t"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs✝ : ¬ms = ⊤\nr : ℝ≥0∞\nhs : r > ms\n⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r",
"tactic": "simpa [iInf_lt_iff] using hs"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nn : ℕ\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\n⊢ ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nn : ℕ\n⊢ (↑n)⁻¹ ≠ 0",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nn : ℕ\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹",
"tactic": "refine' this _ (ENNReal.lt_add_right hs _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nn : ℕ\n⊢ (↑n)⁻¹ ≠ 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\nn : ℕ\n⊢ ↑m (⋂ (n : ℕ), t n) ≤ ms + (↑n)⁻¹",
"tactic": "exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le"
},
{
"state_after": "case neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ (⨅ (_ : s ⊆ ⋂ (n : ℕ), t n) (_ : MeasurableSet (⋂ (n : ℕ), t n)), ↑m (⋂ (n : ℕ), t n)) ≤ ↑m (⋂ (n : ℕ), t n)",
"state_before": "case neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ ms ≤ ↑m (⋂ (n : ℕ), t n)",
"tactic": "refine' iInf_le_of_le (⋂ n, t n) _"
},
{
"state_after": "case neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ (⨅ (_ : MeasurableSet (⋂ (n : ℕ), t n)), ↑m (⋂ (n : ℕ), t n)) ≤ ↑m (⋂ (n : ℕ), t n)",
"state_before": "case neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ (⨅ (_ : s ⊆ ⋂ (n : ℕ), t n) (_ : MeasurableSet (⋂ (n : ℕ), t n)), ↑m (⋂ (n : ℕ), t n)) ≤ ↑m (⋂ (n : ℕ), t n)",
"tactic": "refine' iInf_le_of_le (subset_iInter hsub) _"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), ↑m t\nhs : ¬ms = ⊤\nthis✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹\nthis : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms)\n⊢ (⨅ (_ : MeasurableSet (⋂ (n : ℕ), t n)), ↑m (⋂ (n : ℕ), t n)) ≤ ↑m (⋂ (n : ℕ), t n)",
"tactic": "refine' iInf_le _ (MeasurableSet.iInter hm)"
}
] |
[
1715,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1690,
1
] |
Mathlib/Analysis/NormedSpace/Star/Basic.lean
|
CstarRing.mul_star_self_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.44392\nE : Type u_1\nα : Type ?u.44398\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CstarRing E\nx : E\n⊢ x * x⋆ = 0 ↔ x = 0",
"tactic": "simpa only [star_eq_zero, star_star] using @star_mul_self_eq_zero_iff _ _ _ _ (star x)"
}
] |
[
149,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.deleteEdges_adj
|
[] |
[
1112,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1110,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.biproduct.lift_eq
|
[
{
"state_after": "case w\nC : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nf : J → C\ninst✝ : HasBiproduct f\nT : C\ng : (j : J) → T ⟶ f j\n⊢ ∀ (j : J), lift g ≫ π (fun b => f b) j = (∑ j : J, g j ≫ ι f j) ≫ π (fun b => f b) j",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nf : J → C\ninst✝ : HasBiproduct f\nT : C\ng : (j : J) → T ⟶ f j\n⊢ lift g = ∑ j : J, g j ≫ ι f j",
"tactic": "apply biproduct.hom_ext"
},
{
"state_after": "case w\nC : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nf : J → C\ninst✝ : HasBiproduct f\nT : C\ng : (j : J) → T ⟶ f j\nj : J\n⊢ lift g ≫ π (fun b => f b) j = (∑ j : J, g j ≫ ι f j) ≫ π (fun b => f b) j",
"state_before": "case w\nC : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nf : J → C\ninst✝ : HasBiproduct f\nT : C\ng : (j : J) → T ⟶ f j\n⊢ ∀ (j : J), lift g ≫ π (fun b => f b) j = (∑ j : J, g j ≫ ι f j) ≫ π (fun b => f b) j",
"tactic": "intro j"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nf : J → C\ninst✝ : HasBiproduct f\nT : C\ng : (j : J) → T ⟶ f j\nj : J\n⊢ lift g ≫ π (fun b => f b) j = (∑ j : J, g j ≫ ι f j) ≫ π (fun b => f b) j",
"tactic": "simp only [sum_comp, biproduct.ι_π, comp_dite, biproduct.lift_π, Category.assoc, comp_zero,\n Finset.sum_dite_eq', Finset.mem_univ, eqToHom_refl, Category.comp_id, if_true]"
}
] |
[
219,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_ofReal_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.826990\nβ : Type ?u.826993\na✝ b c✝ d : ℝ≥0∞\nr p q : ℝ≥0\nc : ℝ\na : ℝ≥0∞\nh : 0 ≤ c\n⊢ ENNReal.toReal (ENNReal.ofReal c * a) = c * ENNReal.toReal a",
"tactic": "rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal h]"
}
] |
[
2255,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2253,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.disjoint_iff
|
[] |
[
1518,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1517,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.subgroupOf_bot_eq_top
|
[] |
[
1657,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1656,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.mul_toAddSubmonoid
|
[
{
"state_after": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R A\n⊢ (map₂ (LinearMap.mul R A) M N).toAddSubmonoid =\n ⨆ (s : { x // x ∈ M.toAddSubmonoid }), AddSubmonoid.map (AddMonoidHom.mulLeft ↑s) N.toAddSubmonoid",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R A\n⊢ (M * N).toAddSubmonoid = M.toAddSubmonoid * N.toAddSubmonoid",
"tactic": "dsimp [HMul.hMul, Mul.mul]"
},
{
"state_after": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R A\n⊢ (⨆ (i : { x // x ∈ M }), (map (↑(LinearMap.mul R A) ↑i) N).toAddSubmonoid) =\n ⨆ (s : { x // x ∈ M.toAddSubmonoid }), AddSubmonoid.map (AddMonoidHom.mulLeft ↑s) N.toAddSubmonoid",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R A\n⊢ (map₂ (LinearMap.mul R A) M N).toAddSubmonoid =\n ⨆ (s : { x // x ∈ M.toAddSubmonoid }), AddSubmonoid.map (AddMonoidHom.mulLeft ↑s) N.toAddSubmonoid",
"tactic": "rw [map₂, iSup_toAddSubmonoid]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R A\n⊢ (⨆ (i : { x // x ∈ M }), (map (↑(LinearMap.mul R A) ↑i) N).toAddSubmonoid) =\n ⨆ (s : { x // x ∈ M.toAddSubmonoid }), AddSubmonoid.map (AddMonoidHom.mulLeft ↑s) N.toAddSubmonoid",
"tactic": "rfl"
}
] |
[
177,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.image2_iInter_subset_right
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort ?u.252102\nι₂ : Sort ?u.252105\nκ : ι → Sort ?u.252110\nκ₁ : ι → Sort ?u.252115\nκ₂ : ι → Sort ?u.252120\nκ' : ι' → Sort ?u.252125\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : Set α\nt : ι → Set β\n⊢ ∀ (x : α), x ∈ s → ∀ (y : β), (∀ (i : ι), y ∈ t i) → ∀ (i : ι), f x y ∈ image2 f s (t i)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort ?u.252102\nι₂ : Sort ?u.252105\nκ : ι → Sort ?u.252110\nκ₁ : ι → Sort ?u.252115\nκ₂ : ι → Sort ?u.252120\nκ' : ι' → Sort ?u.252125\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : Set α\nt : ι → Set β\n⊢ image2 f s (⋂ (i : ι), t i) ⊆ ⋂ (i : ι), image2 f s (t i)",
"tactic": "simp_rw [image2_subset_iff, mem_iInter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort ?u.252102\nι₂ : Sort ?u.252105\nκ : ι → Sort ?u.252110\nκ₁ : ι → Sort ?u.252115\nκ₂ : ι → Sort ?u.252120\nκ' : ι' → Sort ?u.252125\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : Set α\nt : ι → Set β\n⊢ ∀ (x : α), x ∈ s → ∀ (y : β), (∀ (i : ι), y ∈ t i) → ∀ (i : ι), f x y ∈ image2 f s (t i)",
"tactic": "exact fun x hx y hy i => mem_image2_of_mem hx (hy _)"
}
] |
[
1906,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1903,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
div_sub_div_same
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.39149\nβ : Type ?u.39152\nK : Type u_1\ninst✝ : DivisionRing K\na✝ b✝ c✝ d a b c : K\n⊢ a / c - b / c = (a - b) / c",
"tactic": "rw [sub_eq_add_neg, ← neg_div, div_add_div_same, sub_eq_add_neg]"
}
] |
[
153,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_pi_over_two_pow_succ
|
[
{
"state_after": "x : ℝ\nn : ℕ\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 = sin (π / 2 ^ (n + 2))",
"state_before": "x : ℝ\nn : ℕ\n⊢ sin (π / 2 ^ (n + 2)) = sqrt (2 - sqrtTwoAddSeries 0 n) / 2",
"tactic": "symm"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ sin (π / 2 ^ (n + 2)) * 2 = sqrt (2 - sqrtTwoAddSeries 0 n)\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "x : ℝ\nn : ℕ\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 = sin (π / 2 ^ (n + 2))",
"tactic": "rw [div_eq_iff_mul_eq]"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) = sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "x : ℝ\nn : ℕ\n⊢ sin (π / 2 ^ (n + 2)) * 2 = sqrt (2 - sqrtTwoAddSeries 0 n)\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "symm"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ 1 / 2 * 2 ^ 2 - sqrtTwoAddSeries 0 n / 4 * 2 ^ 2 = 2 - sqrtTwoAddSeries 0 n\n\ncase hx\nx : ℝ\nn : ℕ\n⊢ 0 ≤ 2 - sqrtTwoAddSeries 0 n\n\ncase hy\nx : ℝ\nn : ℕ\n⊢ 0 ≤ sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "x : ℝ\nn : ℕ\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) = sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "rw [sqrt_eq_iff_sq_eq, mul_pow, sin_sq_pi_over_two_pow_succ, sub_mul]"
},
{
"state_after": "case hy.a\nx : ℝ\nn : ℕ\n⊢ 0 < sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy\nx : ℝ\nn : ℕ\n⊢ 0 ≤ sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply le_of_lt"
},
{
"state_after": "case hy.a.ha\nx : ℝ\nn : ℕ\n⊢ 0 < sin (π / 2 ^ (n + 2))\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a\nx : ℝ\nn : ℕ\n⊢ 0 < sin (π / 2 ^ (n + 2)) * 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply mul_pos"
},
{
"state_after": "case hy.a.ha.h0x\nx : ℝ\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)\n\ncase hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ π / 2 ^ (n + 2) < π\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha\nx : ℝ\nn : ℕ\n⊢ 0 < sin (π / 2 ^ (n + 2))\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply sin_pos_of_pos_of_lt_pi"
},
{
"state_after": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ π / 2 ^ (n + 2) < π / 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ π / 2 ^ (n + 2) < π\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "refine' lt_of_lt_of_le _ (le_of_eq (div_one _))"
},
{
"state_after": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ 1 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ π / 2 ^ (n + 2) < π / 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "rw [div_lt_div_left]"
},
{
"state_after": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ 2 ^ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ 1 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "refine' lt_of_le_of_lt (le_of_eq (pow_zero 2).symm) _"
},
{
"state_after": "case hy.a.ha.hxp.h\nx : ℝ\nn : ℕ\n⊢ 1 < 2\n\ncase hy.a.ha.hxp.h2\nx : ℝ\nn : ℕ\n⊢ 0 < n + 2\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp\nx : ℝ\nn : ℕ\n⊢ 2 ^ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply pow_lt_pow"
},
{
"state_after": "case hy.a.ha.hxp.h2\nx : ℝ\nn : ℕ\n⊢ 0 < n + 2\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp.h\nx : ℝ\nn : ℕ\n⊢ 1 < 2\n\ncase hy.a.ha.hxp.h2\nx : ℝ\nn : ℕ\n⊢ 0 < n + 2\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "norm_num"
},
{
"state_after": "case hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp.h2\nx : ℝ\nn : ℕ\n⊢ 0 < n + 2\n\ncase hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply Nat.succ_pos"
},
{
"state_after": "case hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp.ha\nx : ℝ\nn : ℕ\n⊢ 0 < π\n\ncase hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply pi_pos"
},
{
"state_after": "case hy.a.ha.hxp.hb.H\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"state_before": "case hy.a.ha.hxp.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "apply pow_pos"
},
{
"state_after": "no goals",
"state_before": "case hy.a.ha.hxp.hb.H\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\ncase hy.a.ha.hxp.hc\nx : ℝ\nn : ℕ\n⊢ 0 < 1\n\ncase hy.a.hb\nx : ℝ\nn : ℕ\n⊢ 0 < 2\n\nx : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "all_goals norm_num"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℕ\n⊢ 1 / 2 * 2 ^ 2 - sqrtTwoAddSeries 0 n / 4 * 2 ^ 2 = 2 - sqrtTwoAddSeries 0 n",
"tactic": "congr <;> norm_num"
},
{
"state_after": "case hx\nx : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n ≤ 2",
"state_before": "case hx\nx : ℝ\nn : ℕ\n⊢ 0 ≤ 2 - sqrtTwoAddSeries 0 n",
"tactic": "rw [sub_nonneg]"
},
{
"state_after": "case hx.a\nx : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n < 2",
"state_before": "case hx\nx : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n ≤ 2",
"tactic": "apply le_of_lt"
},
{
"state_after": "no goals",
"state_before": "case hx.a\nx : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n < 2",
"tactic": "apply sqrtTwoAddSeries_lt_two"
},
{
"state_after": "case hy.a.ha.h0x\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case hy.a.ha.h0x\nx : ℝ\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)",
"tactic": "apply div_pos pi_pos"
},
{
"state_after": "case hy.a.ha.h0x.H\nx : ℝ\nn : ℕ\n⊢ 0 < 2",
"state_before": "case hy.a.ha.h0x\nx : ℝ\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply pow_pos"
},
{
"state_after": "no goals",
"state_before": "case hy.a.ha.h0x.H\nx : ℝ\nn : ℕ\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℕ\n⊢ 2 ≠ 0",
"tactic": "norm_num"
}
] |
[
777,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
762,
1
] |
Std/Logic.lean
|
not_and_of_not_or_not
|
[] |
[
342,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
342,
1
] |
Mathlib/MeasureTheory/Group/Integration.lean
|
MeasureTheory.Integrable.comp_div_left
|
[] |
[
176,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_same
|
[] |
[
807,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_prod
|
[
{
"state_after": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ (c : E) ⦃s : Set (α × β)⦄,\n MeasurableSet s →\n ↑↑(Measure.prod μ ν) s < ⊤ →\n (∫ (z : α × β), indicator s (fun x => c) z ∂Measure.prod μ ν) =\n ∫ (x : α), ∫ (y : β), indicator s (fun x => c) (x, y) ∂ν ∂μ\n\ncase h_add\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ ⦃f g : α × β → E⦄,\n Disjoint (support f) (support g) →\n Integrable f →\n Integrable g →\n ((∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ) →\n ((∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ) →\n (∫ (z : α × β), (f + g) z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ\n\ncase h_closed\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ IsClosed {f | (∫ (z : α × β), ↑↑f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ}\n\ncase h_ae\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ ⦃f g : α × β → E⦄,\n f =ᶠ[ae (Measure.prod μ ν)] g →\n Integrable f →\n ((∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ) →\n (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ",
"state_before": "α : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ (f : α × β → E), Integrable f → (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ",
"tactic": "apply Integrable.induction"
},
{
"state_after": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nc : E\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : ↑↑(Measure.prod μ ν) s < ⊤\n⊢ (∫ (z : α × β), indicator s (fun x => c) z ∂Measure.prod μ ν) =\n ∫ (x : α), ∫ (y : β), indicator s (fun x => c) (x, y) ∂ν ∂μ",
"state_before": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ (c : E) ⦃s : Set (α × β)⦄,\n MeasurableSet s →\n ↑↑(Measure.prod μ ν) s < ⊤ →\n (∫ (z : α × β), indicator s (fun x => c) z ∂Measure.prod μ ν) =\n ∫ (x : α), ∫ (y : β), indicator s (fun x => c) (x, y) ∂ν ∂μ",
"tactic": "intro c s hs h2s"
},
{
"state_after": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nc : E\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : ↑↑(Measure.prod μ ν) s < ⊤\n⊢ ENNReal.toReal (↑↑(Measure.prod μ ν) s) • c = ENNReal.toReal (∫⁻ (a : α), ↑↑ν (Prod.mk a ⁻¹' s) ∂μ) • c",
"state_before": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nc : E\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : ↑↑(Measure.prod μ ν) s < ⊤\n⊢ (∫ (z : α × β), indicator s (fun x => c) z ∂Measure.prod μ ν) =\n ∫ (x : α), ∫ (y : β), indicator s (fun x => c) (x, y) ∂ν ∂μ",
"tactic": "simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp,\n integral_indicator (measurable_prod_mk_left hs), set_integral_const, integral_smul_const,\n integral_toReal (measurable_measure_prod_mk_left hs).aemeasurable\n (ae_measure_lt_top hs h2s.ne)]"
},
{
"state_after": "no goals",
"state_before": "case h_ind\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nc : E\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : ↑↑(Measure.prod μ ν) s < ⊤\n⊢ ENNReal.toReal (↑↑(Measure.prod μ ν) s) • c = ENNReal.toReal (∫⁻ (a : α), ↑↑ν (Prod.mk a ⁻¹' s) ∂μ) • c",
"tactic": "rw [prod_apply hs]"
},
{
"state_after": "case h_add\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\ni_f : Integrable f\ni_g : Integrable g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\nhg : (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), (f + g) z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ",
"state_before": "case h_add\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ ⦃f g : α × β → E⦄,\n Disjoint (support f) (support g) →\n Integrable f →\n Integrable g →\n ((∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ) →\n ((∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ) →\n (∫ (z : α × β), (f + g) z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ",
"tactic": "rintro f g - i_f i_g hf hg"
},
{
"state_after": "no goals",
"state_before": "case h_add\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\ni_f : Integrable f\ni_g : Integrable g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\nhg : (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), (f + g) z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ",
"tactic": "simp_rw [integral_add' i_f i_g, integral_integral_add' i_f i_g, hf, hg]"
},
{
"state_after": "no goals",
"state_before": "case h_closed\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ IsClosed {f | (∫ (z : α × β), ↑↑f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ}",
"tactic": "exact isClosed_eq continuous_integral continuous_integral_integral"
},
{
"state_after": "case h_ae\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ",
"state_before": "case h_ae\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ ⦃f g : α × β → E⦄,\n f =ᶠ[ae (Measure.prod μ ν)] g →\n Integrable f →\n ((∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ) →\n (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ",
"tactic": "rintro f g hfg - hf"
},
{
"state_after": "case h.e'_2\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (z : α × β), f z ∂Measure.prod μ ν\n\ncase h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ",
"state_before": "case h_ae\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ",
"tactic": "convert hf using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (z : α × β), g z ∂Measure.prod μ ν) = ∫ (z : α × β), f z ∂Measure.prod μ ν",
"tactic": "exact integral_congr_ae hfg.symm"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (fun x => ∫ (y : β), g (x, y) ∂ν) =ᶠ[ae μ] fun x => ∫ (y : β), f (x, y) ∂ν",
"state_before": "case h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ",
"tactic": "refine' integral_congr_ae _"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ ∀ᵐ (x : α) ∂μ,\n (∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)) → (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"state_before": "case h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ (fun x => ∫ (y : β), g (x, y) ∂ν) =ᶠ[ae μ] fun x => ∫ (y : β), f (x, y) ∂ν",
"tactic": "refine' (ae_ae_of_ae_prod hfg).mp _"
},
{
"state_after": "case h.e'_3.hp\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ ∀ (x : α),\n (∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)) → (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"state_before": "case h.e'_3\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ ∀ᵐ (x : α) ∂μ,\n (∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)) → (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"tactic": "apply eventually_of_forall"
},
{
"state_after": "case h.e'_3.hp\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\nx : α\nhfgx : ∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)\n⊢ (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"state_before": "case h.e'_3.hp\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\n⊢ ∀ (x : α),\n (∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)) → (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"tactic": "intro x hfgx"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.hp\nα : Type u_1\nα' : Type ?u.2445862\nβ : Type u_2\nβ' : Type ?u.2445868\nγ : Type ?u.2445871\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2446136\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nhfg : f =ᶠ[ae (Measure.prod μ ν)] g\nhf : (∫ (z : α × β), f z ∂Measure.prod μ ν) = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ\nx : α\nhfgx : ∀ᵐ (y : β) ∂ν, f (x, y) = g (x, y)\n⊢ (fun x => ∫ (y : β), g (x, y) ∂ν) x = (fun x => ∫ (y : β), f (x, y) ∂ν) x",
"tactic": "exact integral_congr_ae (ae_eq_symm hfgx)"
}
] |
[
475,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) ∈ measurableLE μ ν\n\ncase succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) ∈ measurableLE μ ν",
"state_before": "α : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nn : ℕ\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ n), f k x) ∈ measurableLE μ ν",
"tactic": "induction' n with m hm"
},
{
"state_after": "case zero.refine'_1\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ Measurable fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x\n\ncase zero.refine'_2\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ ∀ (A : Set α), MeasurableSet A → (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) x ∂μ) ≤ ↑↑ν A",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) ∈ measurableLE μ ν",
"tactic": "refine' ⟨_, _⟩"
},
{
"state_after": "no goals",
"state_before": "case zero.refine'_1\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ Measurable fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x",
"tactic": "simp [(hf 0).1]"
},
{
"state_after": "case zero.refine'_2\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) x ∂μ) ≤ ↑↑ν A",
"state_before": "case zero.refine'_2\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\n⊢ ∀ (A : Set α), MeasurableSet A → (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) x ∂μ) ≤ ↑↑ν A",
"tactic": "intro A hA"
},
{
"state_after": "no goals",
"state_before": "case zero.refine'_2\nα : Type u_1\nβ : Type ?u.104521\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.zero), f k x) x ∂μ) ≤ ↑↑ν A",
"tactic": "simp [(hf 0).2 A hA]"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) ∈ measurableLE μ ν",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) ∈ measurableLE μ ν",
"tactic": "have :\n (fun a : α => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a =>\n f m.succ a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a :=\n funext fun _ => iSup_succ_eq_sup _ _ _"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) x ∂μ) ≤ ↑↑ν A",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\n⊢ (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) ∈ measurableLE μ ν",
"tactic": "refine' ⟨measurable_iSup fun n => Measurable.iSup_Prop _ (hf n).1, fun A hA => _⟩"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a ∂μ) ≤ ↑↑ν A",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (fun x => ⨆ (k : ℕ) (_ : k ≤ Nat.succ m), f k x) x ∂μ) ≤ ↑↑ν A",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.104521\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), f n ∈ measurableLE μ ν\nm : ℕ\nhm : (fun x => ⨆ (k : ℕ) (_ : k ≤ m), f k x) ∈ measurableLE μ ν\nthis : (fun a => ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a) = fun a => f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, f (Nat.succ m) a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a ∂μ) ≤ ↑↑ν A",
"tactic": "exact (sup_mem_measurableLE (hf m.succ) hm).2 A hA"
}
] |
[
512,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/Data/Sign.lean
|
sign_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedRing α\na b x y : α\n⊢ ↑sign (x * y) = ↑sign x * ↑sign y",
"tactic": "rcases lt_trichotomy x 0 with (hx | hx | hx) <;> rcases lt_trichotomy y 0 with (hy | hy | hy) <;>\n simp [hx, hy, mul_pos_of_neg_of_neg, mul_neg_of_neg_of_pos, mul_neg_of_pos_of_neg]"
}
] |
[
403,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
coe_affineSpan
|
[] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Data/List/Lattice.lean
|
List.cons_bagInter_of_neg
|
[
{
"state_after": "case nil\nα : Type u_1\nl l₁✝ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nh : ¬a ∈ []\n⊢ List.bagInter (a :: l₁) [] = List.bagInter l₁ []\n\ncase cons\nα : Type u_1\nl l₁✝ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nhead✝ : α\ntail✝ : List α\nh : ¬a ∈ head✝ :: tail✝\n⊢ List.bagInter (a :: l₁) (head✝ :: tail✝) = List.bagInter l₁ (head✝ :: tail✝)",
"state_before": "α : Type u_1\nl l₁✝ l₂ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nh : ¬a ∈ l₂\n⊢ List.bagInter (a :: l₁) l₂ = List.bagInter l₁ l₂",
"tactic": "cases l₂"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl l₁✝ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nhead✝ : α\ntail✝ : List α\nh : ¬a ∈ head✝ :: tail✝\n⊢ List.bagInter (a :: l₁) (head✝ :: tail✝) = List.bagInter l₁ (head✝ :: tail✝)",
"tactic": "simp only [erase_of_not_mem h, List.bagInter, if_neg (mt mem_of_elem_eq_true h)]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl l₁✝ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nh : ¬a ∈ []\n⊢ List.bagInter (a :: l₁) [] = List.bagInter l₁ []",
"tactic": "simp only [bagInter_nil]"
}
] |
[
228,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.UnifIntegrable.add
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\n⊢ UnifIntegrable (f + g) p μ",
"tactic": "intro ε hε"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "have hε2 : 0 < ε / 2 := half_pos hε"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, fun i s hs hμs => _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s (f i) + Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "simp_rw [Pi.add_apply, Set.indicator_add']"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s (f i) + Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' (snorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans _"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\nhε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "have hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by\n rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\nhε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤\n ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\nhε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "rw [hε_halves]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\nhε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)\n⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤\n ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)",
"tactic": "exact add_le_add (hfδ₁ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_left _ _))))\n (hgδ₂ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_right _ _))))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nhg : UnifIntegrable g p μ\nhp : 1 ≤ p\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ\nhε : 0 < ε\nhε2 : 0 < ε / 2\nδ₁ : ℝ\nhδ₁_pos : 0 < δ₁\nhfδ₁ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)\nδ₂ : ℝ\nhδ₂_pos : 0 < δ₂\nhgδ₂ :\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)\n⊢ ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)",
"tactic": "rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]"
}
] |
[
125,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
11
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.FinStronglyMeasurable.inf
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.280281\nι : Type ?u.280284\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => FinStronglyMeasurable.approx hf n ⊓ FinStronglyMeasurable.approx hg n) n)) < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.280281\nι : Type ?u.280284\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\n⊢ FinStronglyMeasurable (f ⊓ g) μ",
"tactic": "refine'\n ⟨fun n => hf.approx n ⊓ hg.approx n, fun n => _, fun x =>\n (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.280281\nι : Type ?u.280284\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ\n ((support fun x => ↑(FinStronglyMeasurable.approx hf n) x) ∪\n support fun x => ↑(FinStronglyMeasurable.approx hg n) x) <\n ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.280281\nι : Type ?u.280284\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => FinStronglyMeasurable.approx hf n ⊓ FinStronglyMeasurable.approx hg n) n)) < ⊤",
"tactic": "refine' (measure_mono (support_inf _ _)).trans_lt _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.280281\nι : Type ?u.280284\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ\n ((support fun x => ↑(FinStronglyMeasurable.approx hf n) x) ∪\n support fun x => ↑(FinStronglyMeasurable.approx hg n) x) <\n ⊤",
"tactic": "exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩"
}
] |
[
1125,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1119,
11
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.stabilizer_of_subMul.submonoid
|
[
{
"state_after": "case h\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\nx✝ : R\n⊢ x✝ ∈ MulAction.Stabilizer.submonoid R m ↔ x✝ ∈ MulAction.Stabilizer.submonoid R ↑m",
"state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\n⊢ MulAction.Stabilizer.submonoid R m = MulAction.Stabilizer.submonoid R ↑m",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\nx✝ : R\n⊢ x✝ ∈ MulAction.Stabilizer.submonoid R m ↔ x✝ ∈ MulAction.Stabilizer.submonoid R ↑m",
"tactic": "simp only [MulAction.mem_stabilizer_submonoid_iff, ← SubMulAction.val_smul, SetLike.coe_eq_coe]"
}
] |
[
310,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Data/List/Basic.lean
|
List.getLast?_isNone
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.34045\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\n⊢ Option.isNone (getLast? []) = true ↔ [] = []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.34045\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\n⊢ Option.isNone (getLast? [a]) = true ↔ [a] = []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.34045\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ Option.isNone (getLast? (a :: b :: l)) = true ↔ a :: b :: l = []",
"tactic": "simp [@getLast?_isNone (b :: l)]"
}
] |
[
779,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.coe_one
|
[] |
[
685,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.kerase_cons_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\ns : Sigma β\nl : List (Sigma β)\nh : a = s.fst\n⊢ kerase a (s :: l) = l",
"tactic": "simp [kerase, h]"
}
] |
[
405,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderOnWith.ediam_image_le_of_le
|
[] |
[
159,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.le_span_singleton_mul_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.230454\ninst✝ : CommSemiring R\nI✝ J✝ K L : Ideal R\nx : R\nI J : Ideal R\n⊢ (∀ {zI : R}, zI ∈ I → zI ∈ span {x} * J) ↔ ∀ (zI : R), zI ∈ I → ∃ zJ, zJ ∈ J ∧ x * zJ = zI",
"tactic": "simp only [mem_span_singleton_mul]"
}
] |
[
549,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Topology/Separation.lean
|
exists_nhds_disjoint_closure
|
[] |
[
1007,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiable_neg_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.427827\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.427922\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : Differentiable 𝕜 fun y => -f y\n⊢ Differentiable 𝕜 f",
"tactic": "simpa only [neg_neg] using h.neg"
}
] |
[
452,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.tail_add
|
[] |
[
482,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.coe_toSupHom
|
[] |
[
1047,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.toList_empty
|
[] |
[
3370,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3369,
1
] |
Std/Data/List/Lemmas.lean
|
List.get?_len_le
|
[] |
[
511,
71
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
509,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.I_mul_re
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.2411518\ninst✝ : IsROrC K\nz : K\n⊢ ↑re (I * z) = -↑im z",
"tactic": "simp only [I_re, zero_sub, I_im', zero_mul, mul_re]"
}
] |
[
329,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.not
|
[] |
[
716,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
11
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.lt_succ_iff
|
[] |
[
337,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Topology/Separation.lean
|
continuousAt_of_tendsto_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space β\nf : α → β\na : α\nb : β\nh : Tendsto f (𝓝 a) (𝓝 b)\n⊢ Tendsto f (𝓝 a) (𝓝 (f a))",
"tactic": "rwa [eq_of_tendsto_nhds h]"
}
] |
[
758,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.shiftl'_add
|
[] |
[
321,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto
|
[
{
"state_after": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.153372\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.153372\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))",
"tactic": "rw [tendsto_iff_forall_toWeakDualBCNN_tendsto]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\ninst✝⁶ : MeasurableSpace Ω\nR : Type ?u.153372\ninst✝⁵ : SMul R ℝ≥0\ninst✝⁴ : SMul R ℝ≥0∞\ninst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_1\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\n⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔\n ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))",
"tactic": "simp_rw [toWeakDualBCNN_apply _ _, ← testAgainstNN_coe_eq, ENNReal.tendsto_coe,\n ENNReal.toNNReal_coe]"
}
] |
[
552,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.coe_nat_re
|
[] |
[
278,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
MonoidAlgebra.support_gen_of_gen'
|
[
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\n⊢ (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) =\n ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\n⊢ adjoin R (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) = ⊤",
"tactic": "suffices (of R M '' ⋃ f ∈ S, (f.support : Set M)) = ⋃ f ∈ S, of R M '' (f.support : Set M) by\n rw [this]\n exact support_gen_of_gen hS"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\n⊢ (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) =\n ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support",
"tactic": "simp only [Set.image_iUnion]"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\nthis :\n (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) =\n ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support\n⊢ adjoin R (⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support) = ⊤",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\nthis :\n (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) =\n ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support\n⊢ adjoin R (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) = ⊤",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Monoid M\nS : Set (MonoidAlgebra R M)\nhS : adjoin R S = ⊤\nthis :\n (↑(of R M) '' ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑f.support) =\n ⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support\n⊢ adjoin R (⋃ (f : MonoidAlgebra R M) (_ : f ∈ S), ↑(of R M) '' ↑f.support) = ⊤",
"tactic": "exact support_gen_of_gen hS"
}
] |
[
530,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsUpperSet.ordConnected
|
[] |
[
238,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_eq_self_of_superset
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3809\nι : Type ?u.3812\nM : Type u_2\nN : Type ?u.3818\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nh1 : mulSupport f ⊆ s\nh2 : s ⊆ t\n⊢ mulSupport f ⊆ t",
"state_before": "α : Type u_1\nβ : Type ?u.3809\nι : Type ?u.3812\nM : Type u_2\nN : Type ?u.3818\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nh1 : mulIndicator s f = f\nh2 : s ⊆ t\n⊢ mulIndicator t f = f",
"tactic": "rw [mulIndicator_eq_self] at h1⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3809\nι : Type ?u.3812\nM : Type u_2\nN : Type ?u.3818\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nh1 : mulSupport f ⊆ s\nh2 : s ⊆ t\n⊢ mulSupport f ⊆ t",
"tactic": "exact Subset.trans h1 h2"
}
] |
[
118,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.le_iterate_pos_iff
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m ≤ (↑f^[n]) x ↔ x + ↑m ≤ ↑f x",
"tactic": "simpa only [not_lt] using not_congr (f.iterate_pos_lt_iff hn)"
}
] |
[
608,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/Data/MvPolynomial/Monad.lean
|
MvPolynomial.eval₂Hom_eq_bind₂
|
[] |
[
121,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/List/Chain.lean
|
List.exists_chain_of_relationReflTransGen
|
[
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\n⊢ ∃ l, Chain r b l ∧ getLast (b :: l) (_ : b :: l ≠ []) = b\n\ncase refine'_2\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\n⊢ ∀ {a c : α},\n r a c →\n Relation.ReflTransGen r c b →\n (∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b) →\n ∃ l, Chain r a l ∧ getLast (a :: l) (_ : a :: l ≠ []) = b",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\n⊢ ∃ l, Chain r a l ∧ getLast (a :: l) (_ : a :: l ≠ []) = b",
"tactic": "refine' Relation.ReflTransGen.head_induction_on h _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\n⊢ ∃ l, Chain r b l ∧ getLast (b :: l) (_ : b :: l ≠ []) = b",
"tactic": "exact ⟨[], Chain.nil, rfl⟩"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nih : ∃ l, Chain r d l ∧ getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ ∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\n⊢ ∀ {a c : α},\n r a c →\n Relation.ReflTransGen r c b →\n (∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b) →\n ∃ l, Chain r a l ∧ getLast (a :: l) (_ : a :: l ≠ []) = b",
"tactic": "intro c d e _ ih"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nl : List α\nhl₁ : Chain r d l\nhl₂ : getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ ∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nih : ∃ l, Chain r d l ∧ getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ ∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b",
"tactic": "obtain ⟨l, hl₁, hl₂⟩ := ih"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nl : List α\nhl₁ : Chain r d l\nhl₂ : getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ getLast (c :: d :: l) (_ : c :: d :: l ≠ []) = b",
"state_before": "case refine'_2.intro.intro\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nl : List α\nhl₁ : Chain r d l\nhl₂ : getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ ∃ l, Chain r c l ∧ getLast (c :: l) (_ : c :: l ≠ []) = b",
"tactic": "refine' ⟨d :: l, Chain.cons e hl₁, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na b : α\nh : Relation.ReflTransGen r a b\nc d : α\ne : r c d\nh✝ : Relation.ReflTransGen r d b\nl : List α\nhl₁ : Chain r d l\nhl₂ : getLast (d :: l) (_ : d :: l ≠ []) = b\n⊢ getLast (c :: d :: l) (_ : c :: d :: l ≠ []) = b",
"tactic": "rwa [getLast_cons_cons]"
}
] |
[
403,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/BinaryProducts.lean
|
hasBinaryCoproducts_of_hasInitial_and_pushouts
|
[] |
[
196,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/CategoryTheory/Sums/Basic.lean
|
CategoryTheory.Sum.swap_map_inr
|
[] |
[
137,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Logic/Hydra.lean
|
Acc.cutExpand
|
[
{
"state_after": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\n⊢ Acc (CutExpand r) {a}",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na : α\nhacc : Acc r a\n⊢ Acc (CutExpand r) {a}",
"tactic": "induction' hacc with a h ih"
},
{
"state_after": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\ns : Multiset α\n⊢ CutExpand r s {a} → Acc (CutExpand r) s",
"state_before": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\n⊢ Acc (CutExpand r) {a}",
"tactic": "refine' Acc.intro _ fun s ↦ _"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\ns : Multiset α\n⊢ CutExpand r s {a} → Acc (CutExpand r) s",
"tactic": "classical\nsimp only [cutExpand_iff, mem_singleton]\nrintro ⟨t, a, hr, rfl, rfl⟩\nrefine' acc_of_singleton fun a' ↦ _\nrw [erase_singleton, zero_add]\nexact ih a' ∘ hr a'"
},
{
"state_after": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\ns : Multiset α\n⊢ (∃ t a_1, (∀ (a' : α), a' ∈ t → r a' a_1) ∧ a_1 = a ∧ s = erase {a} a_1 + t) → Acc (CutExpand r) s",
"state_before": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\ns : Multiset α\n⊢ CutExpand r s {a} → Acc (CutExpand r) s",
"tactic": "simp only [cutExpand_iff, mem_singleton]"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\n⊢ Acc (CutExpand r) (erase {a} a + t)",
"state_before": "case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\ns : Multiset α\n⊢ (∃ t a_1, (∀ (a' : α), a' ∈ t → r a' a_1) ∧ a_1 = a ∧ s = erase {a} a_1 + t) → Acc (CutExpand r) s",
"tactic": "rintro ⟨t, a, hr, rfl, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\na' : α\n⊢ a' ∈ erase {a} a + t → Acc (CutExpand r) {a'}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\n⊢ Acc (CutExpand r) (erase {a} a + t)",
"tactic": "refine' acc_of_singleton fun a' ↦ _"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\na' : α\n⊢ a' ∈ t → Acc (CutExpand r) {a'}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\na' : α\n⊢ a' ∈ erase {a} a + t → Acc (CutExpand r) {a'}",
"tactic": "rw [erase_singleton, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\na✝ : α\nt : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\na' : α\n⊢ a' ∈ t → Acc (CutExpand r) {a'}",
"tactic": "exact ih a' ∘ hr a'"
}
] |
[
152,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
Hamming.toHamming_inj
|
[] |
[
340,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Algebra/IsPrimePow.lean
|
IsPrimePow.ne_zero
|
[] |
[
70,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
lebesgue_number_lemma_of_metric
|
[] |
[
2278,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2272,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.mono_null
|
[] |
[
105,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableAt.const_mul
|
[] |
[
485,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.ihom.ihom_adjunction_counit
|
[] |
[
111,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Bijective.existsUnique
|
[] |
[
251,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
11
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.eq_of_le_of_finrank_le
|
[] |
[
705,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
8
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.max_zero_left
|
[] |
[
1004,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1003,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_False
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.321046\nγ : Type ?u.321049\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns : Finset α\na : α\n⊢ a ∈ filter (fun x => False) s ↔ a ∈ ∅",
"tactic": "simp [mem_filter, and_false_iff]"
}
] |
[
2687,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2686,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_inter_le_ncard_left
|
[] |
[
306,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
Profinite.exists_clopen_of_cofiltered
|
[
{
"state_after": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\n⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\n\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have hT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis\n ((fun j ↦ {W : Set (F.obj j)| IsClopen W}) j)"
},
{
"state_after": "case compat\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\n⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\n\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j ↦ {W | IsClopen W}) j → (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈\n (fun j ↦ {W | IsClopen W}) i"
},
{
"state_after": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nhB :\n TopologicalSpace.IsTopologicalBasis\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, u} (F ⋙ Profinite.toTopCat)\n (Profinite.toTopCat.mapCone C) (isLimitOfPreserves _ hC) (fun j => {W | IsClopen W}) hT\n (fun i => isClopen_univ) (fun i U1 U2 hU1 hU2 => hU1.inter hU2) compat"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nhB :\n TopologicalSpace.IsTopologicalBasis\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nhB :\n TopologicalSpace.IsTopologicalBasis\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "obtain ⟨S, hS, h⟩ := hB.open_eq_sUnion hU.1"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nhB :\n TopologicalSpace.IsTopologicalBasis\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "clear hB"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "let j : S → J := fun s => (hS s.2).choose"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "let V : ∀ s : S, Set (F.obj (j s)) := fun s => (hS s.2).choose_spec.choose"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have hV : ∀ s : S, IsClopen (V s) ∧ s.1 = C.π.app (j s) ⁻¹' V s := fun s =>\n (hS s.2).choose_spec.choose_spec"
},
{
"state_after": "case hUo\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\n⊢ ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n\ncase intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have hUo : ∀ (i : ↑S), IsOpen ((fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)"
},
{
"state_after": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n\ncase intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have hsU : U ⊆ ⋃ (i : ↑S), (fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nthis : ∃ t, U ⊆ ⋃ (i : ↑S) (_ : i ∈ t), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "have := hU.2.isCompact.elim_finite_subcover (fun s : S => C.π.app (j s) ⁻¹' V s) hUo hsU"
},
{
"state_after": "case intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nthis : ∃ t, U ⊆ ⋃ (i : ↑S) (_ : i ∈ t), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "obtain ⟨G, hG⟩ := this"
},
{
"state_after": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists (G.image j)"
},
{
"state_after": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "let f : ∀ (s : S) (_ : s ∈ G), j0 ⟶ j s := fun s hs =>\n (hj0 (Finset.mem_image.mpr ⟨s, hs, rfl⟩)).some"
},
{
"state_after": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"state_before": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "let W : S → Set (F.obj j0) := fun s => if hs : s ∈ G then F.map (f s hs) ⁻¹' V s else Set.univ"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ IsClopen (⋃ (s : ↑S) (_ : s ∈ G), W s)\n\ncase intro.intro.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ U = (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"state_before": "case intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ ∃ j V x, U = (forget Profinite).map (C.π.app j) ⁻¹' V",
"tactic": "refine' ⟨j0, ⋃ (s : S) (_ : s ∈ G), W s, _, _⟩"
},
{
"state_after": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\ni : J\n⊢ TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) i)",
"state_before": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\n⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)",
"tactic": "intro i"
},
{
"state_after": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\ni : J\n⊢ TopologicalSpace.IsTopologicalBasis {W | IsClopen W}",
"state_before": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\ni : J\n⊢ TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) i)",
"tactic": "change TopologicalSpace.IsTopologicalBasis {W : Set (F.obj i) | IsClopen W}"
},
{
"state_after": "no goals",
"state_before": "case hT\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\ni : J\n⊢ TopologicalSpace.IsTopologicalBasis {W | IsClopen W}",
"tactic": "apply isTopologicalBasis_clopen"
},
{
"state_after": "case compat\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ni j : J\nf : i ⟶ j\nV : Set ↑((F ⋙ toTopCat).obj j)\nhV : IsClopen V\n⊢ (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i",
"state_before": "case compat\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\n⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i",
"tactic": "rintro i j f V (hV : IsClopen _)"
},
{
"state_after": "no goals",
"state_before": "case compat\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ni j : J\nf : i ⟶ j\nV : Set ↑((F ⋙ toTopCat).obj j)\nhV : IsClopen V\n⊢ (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i",
"tactic": "exact ⟨hV.1.preimage ((F ⋙ toTopCat).map f).continuous,\n hV.2.preimage ((F ⋙ toTopCat).map f).continuous⟩"
},
{
"state_after": "case hUo\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\ns : ↑S\n⊢ IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s)",
"state_before": "case hUo\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\n⊢ ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)",
"tactic": "intro s"
},
{
"state_after": "no goals",
"state_before": "case hUo\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\ns : ↑S\n⊢ IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s)",
"tactic": "exact (hV s).1.1.preimage (C.π.app (j s)).continuous"
},
{
"state_after": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ U ⊆\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"state_before": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i",
"tactic": "dsimp only"
},
{
"state_after": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ ⋃₀ S ⊆\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"state_before": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ U ⊆\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"tactic": "rw [h]"
},
{
"state_after": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"state_before": "case hsU\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\n⊢ ⋃₀ S ⊆\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"tactic": "rintro x ⟨T, hT, hx⟩"
},
{
"state_after": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n (fun i =>\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V))\n { val := T, property := hT }",
"state_before": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n ⋃ (i : ↑S),\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"tactic": "refine' ⟨_, ⟨⟨T, hT⟩, rfl⟩, _⟩"
},
{
"state_after": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n T ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n T =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n T ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"state_before": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n (fun i =>\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V))\n { val := T, property := hT }",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case hsU.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT✝ : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nx : ↑C.pt.toCompHaus.toTop\nT : Set ↑(toTopCat.mapCone C).pt\nhT : T ∈ S\nhx : x ∈ T\n⊢ x ∈\n (forget Profinite).map\n (C.π.app\n (Exists.choose\n (_ :\n T ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈ {W | IsClopen W} ∧\n T =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n T ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)",
"tactic": "rwa [← (hV ⟨T, hT⟩).2]"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ ∀ (i : ↑S), i ∈ G → IsClopen (W i)",
"state_before": "case intro.intro.intro.intro.refine'_1\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ IsClopen (⋃ (s : ↑S) (_ : s ∈ G), W s)",
"tactic": "apply isClopen_biUnion_finset"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen (W s)",
"state_before": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ ∀ (i : ↑S), i ∈ G → IsClopen (W i)",
"tactic": "intro s hs"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen\n (if hs : s ∈ G then\n (forget Profinite).map\n (F.map\n (Nonempty.some\n (_ :\n Nonempty\n (j0 ⟶\n Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n IsClopen V ∧\n ↑s =\n (forget TopCat).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ)",
"state_before": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen (W s)",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen\n ((forget Profinite).map\n (F.map\n (Nonempty.some\n (_ :\n Nonempty\n (j0 ⟶\n Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n IsClopen V ∧\n ↑s =\n (forget TopCat).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V))",
"state_before": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen\n (if hs : s ∈ G then\n (forget Profinite).map\n (F.map\n (Nonempty.some\n (_ :\n Nonempty\n (j0 ⟶\n Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n IsClopen V ∧\n ↑s =\n (forget TopCat).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ)",
"tactic": "rw [dif_pos hs]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\ns : ↑S\nhs : s ∈ G\n⊢ IsClopen\n ((forget Profinite).map\n (F.map\n (Nonempty.some\n (_ :\n Nonempty\n (j0 ⟶\n Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n IsClopen V ∧\n ↑s =\n (forget TopCat).map\n (C.π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V))",
"tactic": "exact ⟨(hV s).1.1.preimage (F.map _).continuous, (hV s).1.2.preimage (F.map _).continuous⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ x ∈ U ↔ x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"state_before": "case intro.intro.intro.intro.refine'_2\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\n⊢ U = (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"tactic": "ext x"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ x ∈ U → x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s\n\ncase intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ (x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s) → x ∈ U",
"state_before": "case intro.intro.intro.intro.refine'_2.h\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ x ∈ U ↔ x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"tactic": "constructor"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\n⊢ x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ x ∈ U → x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"tactic": "intro hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\n⊢ ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\n⊢ x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s",
"tactic": "simp_rw [Set.preimage_iUnion, Set.mem_iUnion]"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\ns : ↑S\nhs : s ∈ G\nhh : x ∈ (fun h => (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) hs\n⊢ ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mp\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\n⊢ ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"tactic": "obtain ⟨_, ⟨s, rfl⟩, _, ⟨hs, rfl⟩, hh⟩ := hG hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\ns : ↑S\nhs : s ∈ G\nhh : x ∈ (fun h => (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) hs\n⊢ x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\ns : ↑S\nhs : s ∈ G\nhh : x ∈ (fun h => (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) hs\n⊢ ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"tactic": "refine' ⟨s, hs, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ U\ns : ↑S\nhs : s ∈ G\nhh : x ∈ (fun h => (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) hs\n⊢ x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ",
"tactic": "rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w]"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s\n⊢ x ∈ U",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\n⊢ (x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s) → x ∈ U",
"tactic": "intro hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx :\n ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ U",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx : x ∈ (forget Profinite).map (C.π.app j0) ⁻¹' ⋃ (s : ↑S) (_ : s ∈ G), W s\n⊢ x ∈ U",
"tactic": "simp_rw [Set.preimage_iUnion, Set.mem_iUnion] at hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ U",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\nhx :\n ∃ i i_1,\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : i ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑i =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑i ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ U",
"tactic": "obtain ⟨s, hs, hx⟩ := hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ ⋃₀ S",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ U",
"tactic": "rw [h]"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ ↑s",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ ⋃₀ S",
"tactic": "refine' ⟨s.1, s.2, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ ↑s",
"tactic": "rw [(hV s).2]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2.h.mpr.intro.intro\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nU : Set ↑C.pt.toCompHaus.toTop\nhC : IsLimit C\nhU : IsClopen U\nhT : ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)\ncompat :\n ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)),\n V ∈ (fun j_1 => {W | IsClopen W}) j →\n (forget TopCat).map ((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i\nS : Set (Set ↑(toTopCat.mapCone C).pt)\nhS : S ⊆ {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}\nh : U = ⋃₀ S\nj : ↑S → J :=\n fun s =>\n Exists.choose\n (_ :\n ↑s ∈\n {U | ∃ j V, V ∈ (fun j => {W | IsClopen W}) j ∧ U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})\nV : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=\n fun s =>\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\nhV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = (forget Profinite).map (C.π.app (j s)) ⁻¹' V s\nhUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)\nhsU : U ⊆ ⋃ (i : ↑S), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nG : Finset ↑S\nhG : U ⊆ ⋃ (i : ↑S) (_ : i ∈ G), (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i\nj0 : J\nhj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)\nf : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))\nW : ↑S → Set ↑(F.obj j0).toCompHaus.toTop :=\n fun s => if hs : s ∈ G then (forget Profinite).map (F.map (f s hs)) ⁻¹' V s else Set.univ\nx : ↑C.pt.toCompHaus.toTop\ns : ↑S\nhs : s ∈ G\nhx :\n x ∈\n (forget Profinite).map (C.π.app j0) ⁻¹'\n if h : s ∈ G then\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹'\n Exists.choose\n (_ :\n ∃ V,\n V ∈\n (fun j => {W | IsClopen W})\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧\n ↑s =\n (forget TopCat).map\n ((toTopCat.mapCone C).π.app\n (Exists.choose\n (_ :\n ↑s ∈\n {U |\n ∃ j V,\n V ∈ (fun j => {W | IsClopen W}) j ∧\n U = (forget TopCat).map ((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'\n V)\n else Set.univ\n⊢ x ∈ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s",
"tactic": "rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w] at hx"
}
] |
[
124,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Data/TwoPointing.lean
|
TwoPointing.swap_snd
|
[] |
[
65,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.comp_sub
|
[] |
[
944,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
942,
1
] |
Mathlib/Topology/Inseparable.lean
|
specializes_of_eq
|
[] |
[
176,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.SummableFamily.hsum_add
|
[
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nα : Type u_3\ns t : SummableFamily Γ R α\ng : Γ\n⊢ coeff (hsum (s + t)) g = coeff (hsum s + hsum t) g",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nα : Type u_3\ns t : SummableFamily Γ R α\n⊢ hsum (s + t) = hsum s + hsum t",
"tactic": "ext g"
},
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nα : Type u_3\ns t : SummableFamily Γ R α\ng : Γ\n⊢ (∑ᶠ (i : α), coeff (toFun s i) g + coeff (toFun t i) g) =\n (∑ᶠ (i : α), coeff (toFun s i) g) + ∑ᶠ (i : α), coeff (toFun t i) g",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nα : Type u_3\ns t : SummableFamily Γ R α\ng : Γ\n⊢ coeff (hsum (s + t)) g = coeff (hsum s + hsum t) g",
"tactic": "simp only [hsum_coeff, add_coeff, add_apply]"
},
{
"state_after": "no goals",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nα : Type u_3\ns t : SummableFamily Γ R α\ng : Γ\n⊢ (∑ᶠ (i : α), coeff (toFun s i) g + coeff (toFun t i) g) =\n (∑ᶠ (i : α), coeff (toFun s i) g) + ∑ᶠ (i : α), coeff (toFun t i) g",
"tactic": "exact finsum_add_distrib (s.finite_co_support _) (t.finite_co_support _)"
}
] |
[
1525,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1522,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.16765\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm n : ℕ\n⊢ a ^ (m * n) = (a ^ n) ^ m",
"tactic": "rw [Nat.mul_comm, pow_mul]"
}
] |
[
136,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.sum_single
|
[] |
[
120,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
SemiconjBy.zpow_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.114118\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\na x y : G\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ ↑n) (y ^ ↑n)",
"tactic": "simp [zpow_ofNat, h.pow_right n]"
},
{
"state_after": "α : Type ?u.114118\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\na x y : G\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ (n + 1)) (y ^ (n + 1))",
"state_before": "α : Type ?u.114118\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\na x y : G\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ Int.negSucc n) (y ^ Int.negSucc n)",
"tactic": "simp only [zpow_negSucc, inv_right_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.114118\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\na x y : G\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ (n + 1)) (y ^ (n + 1))",
"tactic": "apply pow_right h"
}
] |
[
472,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
Differentiable.norm
|
[] |
[
252,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.toEnd
|
[
{
"state_after": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ ValidFor (List.reverse r ++ l) [] { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l + utf8Len r } }",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ ValidFor (List.reverse r ++ l) [] (Iterator.toEnd it)",
"tactic": "simp [Iterator.toEnd, h.toString]"
},
{
"state_after": "no goals",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ ValidFor (List.reverse r ++ l) [] { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l + utf8Len r } }",
"tactic": "exact .of_eq _ (by simp [List.reverseAux_eq]) (by simp [Nat.add_comm])"
},
{
"state_after": "no goals",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l + utf8Len r } }.s.data =\n List.reverseAux (List.reverse r ++ l) []",
"tactic": "simp [List.reverseAux_eq]"
},
{
"state_after": "no goals",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l + utf8Len r } }.i.byteIdx =\n utf8Len (List.reverse r ++ l)",
"tactic": "simp [Nat.add_comm]"
}
] |
[
589,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
587,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
BilinForm.polar_to_quadratic_form
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.369777\nR : Type u_1\nR₁ : Type ?u.369783\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : BilinForm R M\nx y : M\n⊢ polar (fun x => bilin B x x) x y = bilin B x y + bilin B y x",
"tactic": "simp only [add_assoc, add_sub_cancel', add_right, polar, add_left_inj, add_neg_cancel_left,\n add_left, sub_eq_add_neg _ (B y y), add_comm (B y x) _]"
}
] |
[
726,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.toOrderAddMonoidHom_eq_coe
|
[] |
[
211,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.neg_apply
|
[] |
[
1232,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
sum_card_orderOf_eq_card_pow_eq_one
|
[
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx✝ : ℕ\na✝² : x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\ny✝ : ℕ\na✝¹ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : x✝ ≠ y✝\n⊢ Disjoint (Finset.filter (fun x => orderOf x = x✝) Finset.univ) (Finset.filter (fun x => orderOf x = y✝) Finset.univ)",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\n⊢ ∀ (x : ℕ),\n x ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n)) →\n ∀ (y : ℕ),\n y ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n)) →\n x ≠ y →\n Disjoint (Finset.filter (fun x_1 => orderOf x_1 = x) Finset.univ)\n (Finset.filter (fun x => orderOf x = y) Finset.univ)",
"tactic": "intros"
},
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx✝ : ℕ\na✝² : x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\ny✝ : ℕ\na✝¹ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : x✝ ≠ y✝\n⊢ ∀ (x : G), x ∈ Finset.univ → orderOf x = x✝ → ¬orderOf x = y✝",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx✝ : ℕ\na✝² : x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\ny✝ : ℕ\na✝¹ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : x✝ ≠ y✝\n⊢ Disjoint (Finset.filter (fun x => orderOf x = x✝) Finset.univ) (Finset.filter (fun x => orderOf x = y✝) Finset.univ)",
"tactic": "apply Finset.disjoint_filter.2"
},
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\ny✝ : ℕ\na✝³ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\nx✝ : G\na✝² : x✝ ∈ Finset.univ\na✝¹ : orderOf x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : orderOf x✝ ≠ y✝\n⊢ ¬orderOf x✝ = y✝",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx✝ : ℕ\na✝² : x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\ny✝ : ℕ\na✝¹ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : x✝ ≠ y✝\n⊢ ∀ (x : G), x ∈ Finset.univ → orderOf x = x✝ → ¬orderOf x = y✝",
"tactic": "rintro _ _ rfl"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\ny✝ : ℕ\na✝³ : y✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\nx✝ : G\na✝² : x✝ ∈ Finset.univ\na✝¹ : orderOf x✝ ∈ Finset.filter (fun x => x ∣ n) (Finset.range (succ n))\na✝ : orderOf x✝ ≠ y✝\n⊢ ¬orderOf x✝ = y✝",
"tactic": "assumption"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\n⊢ (x ∈\n Finset.biUnion (Finset.filter (fun x => x ∣ n) (Finset.range (succ n))) fun m =>\n Finset.filter (fun x => orderOf x = m) Finset.univ) ↔\n x ∈ Finset.filter (fun x => x ^ n = 1) Finset.univ",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\n⊢ ∀ (a : G),\n (a ∈\n Finset.biUnion (Finset.filter (fun x => x ∣ n) (Finset.range (succ n))) fun m =>\n Finset.filter (fun x => orderOf x = m) Finset.univ) ↔\n a ∈ Finset.filter (fun x => x ^ n = 1) Finset.univ",
"tactic": "intro x"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\n⊢ orderOf x ≤ n ∧ orderOf x ∣ n ↔ x ^ n = 1",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\n⊢ (x ∈\n Finset.biUnion (Finset.filter (fun x => x ∣ n) (Finset.range (succ n))) fun m =>\n Finset.filter (fun x => orderOf x = m) Finset.univ) ↔\n x ∈ Finset.filter (fun x => x ^ n = 1) Finset.univ",
"tactic": "suffices orderOf x ≤ n ∧ orderOf x ∣ n ↔ x ^ n = 1 by simpa [Nat.lt_succ_iff]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\n⊢ orderOf x ≤ n ∧ orderOf x ∣ n ↔ x ^ n = 1",
"tactic": "exact\n ⟨fun h => by\n let ⟨m, hm⟩ := h.2\n rw [hm, pow_mul, pow_orderOf_eq_one, one_pow], fun h =>\n ⟨orderOf_le_of_pow_eq_one hn.bot_lt h, orderOf_dvd_of_pow_eq_one h⟩⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\nthis : orderOf x ≤ n ∧ orderOf x ∣ n ↔ x ^ n = 1\n⊢ (x ∈\n Finset.biUnion (Finset.filter (fun x => x ∣ n) (Finset.range (succ n))) fun m =>\n Finset.filter (fun x => orderOf x = m) Finset.univ) ↔\n x ∈ Finset.filter (fun x => x ^ n = 1) Finset.univ",
"tactic": "simpa [Nat.lt_succ_iff]"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m✝ : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\nh : orderOf x ≤ n ∧ orderOf x ∣ n\nm : ℕ\nhm : n = orderOf x * m\n⊢ x ^ n = 1",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\nh : orderOf x ≤ n ∧ orderOf x ∣ n\n⊢ x ^ n = 1",
"tactic": "let ⟨m, hm⟩ := h.2"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m✝ : ℕ\ninst✝² : Monoid G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\nx : G\nh : orderOf x ≤ n ∧ orderOf x ∣ n\nm : ℕ\nhm : n = orderOf x * m\n⊢ x ^ n = 1",
"tactic": "rw [hm, pow_mul, pow_orderOf_eq_one, one_pow]"
}
] |
[
672,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
|
MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
|
[
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nthis :\n ∀ (a₁ b₁ : ℝ),\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n a₁ ≤ b₁ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) +\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\nh₁ : ¬a₁ ≤ b₁\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\n\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "wlog h₁ : a₁ ≤ b₁ generalizing a₁ b₁"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nthis :\n ∀ (a₂ b₂ : ℝ),\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n a₂ ≤ b₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) +\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\nh₂ : ¬a₂ ≤ b₂\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\n\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₂ : a₂ ≤ b₂\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "wlog h₂ : a₂ ≤ b₂ generalizing a₂ b₂"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHcg : ContinuousOn g (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₂ : a₂ ≤ b₂\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHcg : ContinuousOn g (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "calc\n (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =\n ∫ x in Icc a₁ b₁, ∫ y in Icc a₂ b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) := by\n simp only [intervalIntegral.integral_of_le, h₁, h₂,\n set_integral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]\n _ = ∫ x in Icc a₁ b₁ ×ˢ Icc a₂ b₂, f' x (1, 0) + g' x (0, 1) := (set_integral_prod _ Hi).symm\n _ = (((∫ x in a₁..b₁, g (x, b₂)) - ∫ x in a₁..b₁, g (x, a₂)) + ∫ y in a₂..b₂, f (b₁, y)) -\n ∫ y in a₂..b₂, f (a₁, y) := by\n rw [Icc_prod_Icc] at *\n apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n (a₁, a₂) (b₁, b₂) ⟨h₁, h₂⟩ s <;> assumption"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min b₁ a₁) (max b₁ a₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min b₁ a₁) (max b₁ a₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nthis :\n ∀ (a₁ b₁ : ℝ),\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n a₁ ≤ b₁ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) +\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\nh₁ : ¬a₁ ≤ b₁\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "specialize this b₁ a₁"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min b₁ a₁) (max b₁ a₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min b₁ a₁) (max b₁ a₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[b₁, a₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (-∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((-∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - -∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "simp only [intervalIntegral.integral_symm b₁ a₁]"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ -((((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) =\n (((-∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - -∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ (-∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((-∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - -∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₁))).trans _"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : ¬a₁ ≤ b₁\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₁ ≤ a₁ →\n (∫ (x : ℝ) in b₁..a₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)\n⊢ -((((∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - ∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (a₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) =\n (((-∫ (x : ℝ) in b₁..a₁, g (x, b₂)) - -∫ (x : ℝ) in b₁..a₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "abel"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min b₂ a₂) (max b₂ a₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min b₂ a₂) (max b₂ a₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nthis :\n ∀ (a₂ b₂ : ℝ),\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n a₂ ≤ b₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) +\n ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)\nh₂ : ¬a₂ ≤ b₂\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "specialize this b₂ a₂"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min b₂ a₂) (max b₂ a₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min b₂ a₂) (max b₂ a₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[b₂, a₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (-∫ (x : ℝ) in a₁..b₁, ∫ (x_1 : ℝ) in b₂..a₂, ↑(f' (x, x_1)) (1, 0) + ↑(g' (x, x_1)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (x : ℝ) in b₂..a₂, f (b₁, x)) -\n -∫ (x : ℝ) in b₂..a₂, f (a₁, x)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ -((((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (x : ℝ) in b₂..a₂, f (b₁, x)) -\n -∫ (x : ℝ) in b₂..a₂, f (a₁, x)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ (-∫ (x : ℝ) in a₁..b₁, ∫ (x_1 : ℝ) in b₂..a₂, ↑(f' (x, x_1)) (1, 0) + ↑(g' (x, x_1)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (x : ℝ) in b₂..a₂, f (b₁, x)) -\n -∫ (x : ℝ) in b₂..a₂, f (a₁, x)",
"tactic": "refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nh₁ : a₁ ≤ b₁\nh₂ : ¬a₂ ≤ b₂\nthis :\n ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt f (f' x) x) →\n (∀ (x : ℝ × ℝ),\n x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \\ s → HasFDerivAt g (g' x) x) →\n IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →\n b₂ ≤ a₂ →\n (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)\n⊢ -((((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -\n ∫ (y : ℝ) in b₂..a₂, f (a₁, y)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (x : ℝ) in b₂..a₂, f (b₁, x)) -\n -∫ (x : ℝ) in b₂..a₂, f (a₁, x)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHcg : ContinuousOn g (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\n⊢ (∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)) =\n ∫ (x : ℝ) in Set.Icc a₁ b₁, ∫ (y : ℝ) in Set.Icc a₂ b₂, ↑(f' (x, y)) (1, 0) + ↑(g' (x, y)) (0, 1)",
"tactic": "simp only [intervalIntegral.integral_of_le, h₁, h₂,\n set_integral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc (a₁, a₂) (b₁, b₂))\nHcg : ContinuousOn g (Set.Icc (a₁, a₂) (b₁, b₂))\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc (a₁, a₂) (b₁, b₂))\n⊢ (∫ (x : ℝ × ℝ) in Set.Icc (a₁, a₂) (b₁, b₂), ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHcg : ContinuousOn g (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂)\n⊢ (∫ (x : ℝ × ℝ) in Set.Icc a₁ b₁ ×ˢ Set.Icc a₂ b₂, ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "rw [Icc_prod_Icc] at *"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\ns : Set (ℝ × ℝ)\nhs : Set.Countable s\na₁ b₁ : ℝ\nh₁ : a₁ ≤ b₁\na₂ b₂ : ℝ\nh₂ : a₂ ≤ b₂\nHcf : ContinuousOn f (Set.Icc (a₁, a₂) (b₁, b₂))\nHcg : ContinuousOn g (Set.Icc (a₁, a₂) (b₁, b₂))\nHdf : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt f (f' x) x\nHdg : ∀ (x : ℝ × ℝ), x ∈ Set.Ioo a₁ b₁ ×ˢ Set.Ioo a₂ b₂ \\ s → HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) (Set.Icc (a₁, a₂) (b₁, b₂))\n⊢ (∫ (x : ℝ × ℝ) in Set.Icc (a₁, a₂) (b₁, b₂), ↑(f' x) (1, 0) + ↑(g' x) (0, 1)) =\n (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -\n ∫ (y : ℝ) in a₂..b₂, f (a₁, y)",
"tactic": "apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n (a₁, a₂) (b₁, b₂) ⟨h₁, h₂⟩ s <;> assumption"
}
] |
[
527,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iio_union_Ici
|
[] |
[
1225,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1224,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
BddBelow.bddAbove_image2_of_bddAbove
|
[
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β → γ\ns : Set α\nt : Set β\na✝ : α\nb✝ : β\nh₀ : ∀ (b : β), Antitone (swap f b)\nh₁ : ∀ (a : α), Monotone (f a)\na : α\nha : a ∈ lowerBounds s\nb : β\nhb : b ∈ upperBounds t\n⊢ BddAbove (Set.image2 f s t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β → γ\ns : Set α\nt : Set β\na : α\nb : β\nh₀ : ∀ (b : β), Antitone (swap f b)\nh₁ : ∀ (a : α), Monotone (f a)\n⊢ BddBelow s → BddAbove t → BddAbove (Set.image2 f s t)",
"tactic": "rintro ⟨a, ha⟩ ⟨b, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β → γ\ns : Set α\nt : Set β\na✝ : α\nb✝ : β\nh₀ : ∀ (b : β), Antitone (swap f b)\nh₁ : ∀ (a : α), Monotone (f a)\na : α\nha : a ∈ lowerBounds s\nb : β\nhb : b ∈ upperBounds t\n⊢ BddAbove (Set.image2 f s t)",
"tactic": "exact ⟨f a b, mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds h₀ h₁ ha hb⟩"
}
] |
[
1530,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1527,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.embDomain_notin_range
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : β\nh : ¬a ∈ Set.range ↑f\n⊢ ↑(embDomain f v) a = 0",
"tactic": "classical\n refine' dif_neg (mt (fun h => _) h)\n rcases Finset.mem_map.1 h with ⟨a, _h, rfl⟩\n exact Set.mem_range_self a"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : β\nh✝ : ¬a ∈ Set.range ↑f\nh : a ∈ map f v.support\n⊢ a ∈ Set.range ↑f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : β\nh : ¬a ∈ Set.range ↑f\n⊢ ↑(embDomain f v) a = 0",
"tactic": "refine' dif_neg (mt (fun h => _) h)"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : α\n_h : a ∈ v.support\nh✝ : ¬↑f a ∈ Set.range ↑f\nh : ↑f a ∈ map f v.support\n⊢ ↑f a ∈ Set.range ↑f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : β\nh✝ : ¬a ∈ Set.range ↑f\nh : a ∈ map f v.support\n⊢ a ∈ Set.range ↑f",
"tactic": "rcases Finset.mem_map.1 h with ⟨a, _h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.218890\nι : Type ?u.218893\nM : Type u_3\nM' : Type ?u.218899\nN : Type ?u.218902\nP : Type ?u.218905\nG : Type ?u.218908\nH : Type ?u.218911\nR : Type ?u.218914\nS : Type ?u.218917\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nv : α →₀ M\na : α\n_h : a ∈ v.support\nh✝ : ¬↑f a ∈ Set.range ↑f\nh : ↑f a ∈ map f v.support\n⊢ ↑f a ∈ Set.range ↑f",
"tactic": "exact Set.mem_range_self a"
}
] |
[
868,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_bit1'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.39154\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nn : ℕ\n⊢ a ^ bit1 n = (a * a) ^ n * a",
"tactic": "rw [bit1, pow_succ', pow_bit0']"
}
] |
[
217,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.