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/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.support_copy
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' v' : V\np : Walk G u' v'\n⊢ support (Walk.copy p (_ : u' = u') (_ : v' = v')) = support p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u' v' : V\np : Walk G u v\nhu : u = u'\nhv : v = v'\n⊢ support (Walk.copy p hu hv) = support p",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' v' : V\np : Walk G u' v'\n⊢ support (Walk.copy p (_ : u' = u') (_ : v' = v')) = support p",
"tactic": "rfl"
}
] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/LinearAlgebra/CrossProduct.lean
|
dot_cross_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nv w : Fin 3 → R\n⊢ w ⬝ᵥ ↑(↑crossProduct v) w = 0",
"tactic": "rw [← cross_anticomm, Matrix.dotProduct_neg, dot_self_cross, neg_zero]"
}
] |
[
109,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
ofMul_multiset_prod
|
[
{
"state_after": "ι : Type ?u.997044\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid α\ns : Multiset α\n⊢ Multiset.prod s = Multiset.sum s",
"state_before": "ι : Type ?u.997044\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid α\ns : Multiset α\n⊢ ↑ofMul (Multiset.prod s) = Multiset.sum (Multiset.map (↑ofMul) s)",
"tactic": "simp [ofMul]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.997044\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid α\ns : Multiset α\n⊢ Multiset.prod s = Multiset.sum s",
"tactic": "rfl"
}
] |
[
2323,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2322,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
isUpperSet_iUnion₂
|
[] |
[
139,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.sum_single_index
|
[] |
[
116,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
ContinuousAt.pow
|
[] |
[
613,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.BinaryBicone.inlCokernelCofork_π
|
[] |
[
1628,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1628,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.prod_snd_mk
|
[
{
"state_after": "case fz\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\nα β : TypeVec (succ n✝)\na : α Fin2.fz\nb : β Fin2.fz\n⊢ prod.snd Fin2.fz (prod.mk Fin2.fz a b) = b\n\ncase fs\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝¹ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α β : TypeVec n✝} (a : α a✝) (b : β a✝), prod.snd a✝ (prod.mk a✝ a b) = b\nα β : TypeVec (succ n✝)\na : α (Fin2.fs a✝)\nb : β (Fin2.fs a✝)\n⊢ prod.snd (Fin2.fs a✝) (prod.mk (Fin2.fs a✝) a b) = b",
"state_before": "n : ℕ\nα β : TypeVec n\ni : Fin2 n\na : α i\nb : β i\n⊢ prod.snd i (prod.mk i a b) = b",
"tactic": "induction' i with _ _ _ i_ih"
},
{
"state_after": "case fs\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝¹ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α β : TypeVec n✝} (a : α a✝) (b : β a✝), prod.snd a✝ (prod.mk a✝ a b) = b\nα β : TypeVec (succ n✝)\na : α (Fin2.fs a✝)\nb : β (Fin2.fs a✝)\n⊢ prod.snd (Fin2.fs a✝) (prod.mk (Fin2.fs a✝) a b) = b",
"state_before": "case fz\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\nα β : TypeVec (succ n✝)\na : α Fin2.fz\nb : β Fin2.fz\n⊢ prod.snd Fin2.fz (prod.mk Fin2.fz a b) = b\n\ncase fs\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝¹ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α β : TypeVec n✝} (a : α a✝) (b : β a✝), prod.snd a✝ (prod.mk a✝ a b) = b\nα β : TypeVec (succ n✝)\na : α (Fin2.fs a✝)\nb : β (Fin2.fs a✝)\n⊢ prod.snd (Fin2.fs a✝) (prod.mk (Fin2.fs a✝) a b) = b",
"tactic": "simp_all [prod.snd, prod.mk]"
},
{
"state_after": "no goals",
"state_before": "case fs\nn : ℕ\nα✝ β✝ : TypeVec n\ni : Fin2 n\na✝¹ : α✝ i\nb✝ : β✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α β : TypeVec n✝} (a : α a✝) (b : β a✝), prod.snd a✝ (prod.mk a✝ a b) = b\nα β : TypeVec (succ n✝)\na : α (Fin2.fs a✝)\nb : β (Fin2.fs a✝)\n⊢ prod.snd (Fin2.fs a✝) (prod.mk (Fin2.fs a✝) a b) = b",
"tactic": "apply i_ih"
}
] |
[
556,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/CategoryTheory/Limits/Final.lean
|
CategoryTheory.Functor.final_of_initial_op
|
[] |
[
115,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_congr
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.77478\nγ : Type ?u.77481\ns✝ t s : Set α\ninst✝¹ inst✝ : Fintype ↑s\n⊢ toFinset s = toFinset s",
"state_before": "α : Type u_1\nβ : Type ?u.77478\nγ : Type ?u.77481\ns✝ t✝ s t : Set α\ninst✝¹ : Fintype ↑s\ninst✝ : Fintype ↑t\nh : s = t\n⊢ toFinset s = toFinset t",
"tactic": "subst h"
},
{
"state_after": "case h.e_3.h\nα : Type u_1\nβ : Type ?u.77478\nγ : Type ?u.77481\ns✝ t s : Set α\ninst✝¹ inst✝ : Fintype ↑s\n⊢ inst✝¹ = inst✝",
"state_before": "α : Type u_1\nβ : Type ?u.77478\nγ : Type ?u.77481\ns✝ t s : Set α\ninst✝¹ inst✝ : Fintype ↑s\n⊢ toFinset s = toFinset s",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_3.h\nα : Type u_1\nβ : Type ?u.77478\nγ : Type ?u.77481\ns✝ t s : Set α\ninst✝¹ inst✝ : Fintype ↑s\n⊢ inst✝¹ = inst✝",
"tactic": "exact Subsingleton.elim _ _"
}
] |
[
613,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Data/Int/GCD.lean
|
Int.gcd_assoc
|
[] |
[
268,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mk_compl_eq_mk_compl_infinite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Infinite α\ns t : Set α\nhs : (#↑s) < (#α)\nht : (#↑t) < (#α)\n⊢ (#↑(sᶜ)) = (#↑(tᶜ))",
"tactic": "rw [mk_compl_of_infinite s hs, mk_compl_of_infinite t ht]"
}
] |
[
1163,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_map_mul
|
[] |
[
199,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.image_eq_preimage
|
[] |
[
1019,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1018,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.val_le
|
[] |
[
62,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_nat_tsum
|
[
{
"state_after": "α : Type ?u.259758\nβ : Type ?u.259761\nγ : Type ?u.259764\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\n⊢ HasSum (fun i => f i) (∑' (n : ℕ), f n)",
"state_before": "α : Type ?u.259758\nβ : Type ?u.259761\nγ : Type ?u.259764\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\n⊢ Tendsto (fun n => ∑ i in Finset.range n, f i) atTop (𝓝 (∑' (n : ℕ), f n))",
"tactic": "rw [← hasSum_iff_tendsto_nat]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.259758\nβ : Type ?u.259761\nγ : Type ?u.259764\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\n⊢ HasSum (fun i => f i) (∑' (n : ℕ), f n)",
"tactic": "exact ENNReal.summable.hasSum"
}
] |
[
935,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
932,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.coe_coord_of_subsingleton_eq_one
|
[
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\n⊢ ↑(coord b i) = 1",
"tactic": "ext q"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "have hp : (range b).Subsingleton := by\n rw [← image_univ]\n apply Subsingleton.image\n apply subsingleton_of_subsingleton"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "haveI := AffineSubspace.subsingleton_of_subsingleton_span_eq_top hp b.tot"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "let s : Finset ι := {i}"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "have hi : i ∈ s := by simp"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\nhw : Finset.sum s (Function.const ι 1) = 1\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "have hw : s.sum (Function.const ι (1 : k)) = 1 := by simp"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\nhw : Finset.sum s (Function.const ι 1) = 1\nhq : q = ↑(Finset.affineCombination k s ↑b) (Function.const ι 1)\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\nhw : Finset.sum s (Function.const ι 1) = 1\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "have hq : q = s.affineCombination k b (Function.const ι (1 : k)) := by simp"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\nhw : Finset.sum s (Function.const ι 1) = 1\nhq : q = ↑(Finset.affineCombination k s ↑b) (Function.const ι 1)\n⊢ ↑(coord b i) q = OfNat.ofNat 1 q",
"tactic": "rw [Pi.one_apply, hq, b.coord_apply_combination_of_mem hi hw, Function.const_apply]"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ Set.Subsingleton (↑b '' univ)",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ Set.Subsingleton (range ↑b)",
"tactic": "rw [← image_univ]"
},
{
"state_after": "case hs\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ Set.Subsingleton univ",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ Set.Subsingleton (↑b '' univ)",
"tactic": "apply Subsingleton.image"
},
{
"state_after": "no goals",
"state_before": "case hs\nι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\n⊢ Set.Subsingleton univ",
"tactic": "apply subsingleton_of_subsingleton"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\n⊢ i ∈ s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\n⊢ Finset.sum s (Function.const ι 1) = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.129150\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Subsingleton ι\ni : ι\nq : P\nhp : Set.Subsingleton (range ↑b)\nthis : Subsingleton P\ns : Finset ι := {i}\nhi : i ∈ s\nhw : Finset.sum s (Function.const ι 1) = 1\n⊢ q = ↑(Finset.affineCombination k s ↑b) (Function.const ι 1)",
"tactic": "simp"
}
] |
[
254,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
ClosedEmbedding.measurable
|
[] |
[
829,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
Antitone.mem_lowerBounds_image
|
[] |
[
1314,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1313,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
Antitone.tendsto_alternating_series_of_tendsto_zero
|
[] |
[
645,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMax.hasFDerivAt_eq_zero
|
[] |
[
204,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.extend_symm_preimage_inter_range_eventuallyEq
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.164976\nM' : Type ?u.164979\nH' : Type ?u.164982\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhs : s ⊆ (extend f I).source\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)] ↑(extend f I) '' s",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.164976\nM' : Type ?u.164979\nH' : Type ?u.164982\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhs : s ⊆ f.source\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)] ↑(extend f I) '' s",
"tactic": "rw [← f.extend_source I] at hs"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.164976\nM' : Type ?u.164979\nH' : Type ?u.164982\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhs : s ⊆ (extend f I).source\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)]\n (extend f I).target ∩ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.164976\nM' : Type ?u.164979\nH' : Type ?u.164982\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhs : s ⊆ (extend f I).source\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)] ↑(extend f I) '' s",
"tactic": "rw [(f.extend I).image_eq_target_inter_inv_preimage hs]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.164976\nM' : Type ?u.164979\nH' : Type ?u.164982\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhs : s ⊆ (extend f I).source\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)]\n (extend f I).target ∩ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s",
"tactic": "exact f.extend_symm_preimage_inter_range_eventuallyEq_aux I hx"
}
] |
[
957,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/Data/Polynomial/Reverse.lean
|
Polynomial.coeff_zero_reverse
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf✝ f : R[X]\n⊢ coeff (reverse f) 0 = leadingCoeff f",
"tactic": "rw [coeff_reverse, revAt_le (zero_le f.natDegree), tsub_zero, leadingCoeff]"
}
] |
[
259,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.eq_inv_mul_iff_mul_eq
|
[] |
[
326,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
11
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval_eq
|
[] |
[
1129,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1127,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean
|
MeasureTheory.Content.outerMeasure_opens
|
[] |
[
258,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Topology/Order/LowerTopology.lean
|
LowerTopology.isOpen_iff_generate_Ici_compl
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3811\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : LowerTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s",
"state_before": "α : Type u_1\nβ : Type ?u.3811\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : LowerTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s",
"tactic": "rw [topology_eq α]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3811\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : LowerTopology α\ns : Set α\n⊢ IsOpen s ↔ GenerateOpen {t | ∃ a, Ici aᶜ = t} s",
"tactic": "rfl"
}
] |
[
172,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/Separation.lean
|
Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete
|
[
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\nι : Type u_1\np : ι → Prop\nt : ι → Set α\ns : Set α\ninst✝ : DiscreteTopology ↑s\nx : α\nhb : HasBasis (𝓝 x) p t\nhx : x ∈ s\ni : ι\nhi : p i\nhix : t i ∩ s ⊆ {x}\n⊢ ∃ i, p i ∧ t i ∩ s = {x}",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\nι : Type u_1\np : ι → Prop\nt : ι → Set α\ns : Set α\ninst✝ : DiscreteTopology ↑s\nx : α\nhb : HasBasis (𝓝 x) p t\nhx : x ∈ s\n⊢ ∃ i, p i ∧ t i ∩ s = {x}",
"tactic": "rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with\n ⟨i, hi, hix⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\nι : Type u_1\np : ι → Prop\nt : ι → Set α\ns : Set α\ninst✝ : DiscreteTopology ↑s\nx : α\nhb : HasBasis (𝓝 x) p t\nhx : x ∈ s\ni : ι\nhi : p i\nhix : t i ∩ s ⊆ {x}\n⊢ ∃ i, p i ∧ t i ∩ s = {x}",
"tactic": "exact ⟨i, hi, hix.antisymm <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩"
}
] |
[
831,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.frequently_or_distrib_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.164816\nι : Sort x\nf : Filter α\ninst✝ : NeBot f\np : α → Prop\nq : Prop\n⊢ (∃ᶠ (x : α) in f, p x ∨ q) ↔ (∃ᶠ (x : α) in f, p x) ∨ q",
"tactic": "simp"
}
] |
[
1350,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1349,
1
] |
src/lean/Init/Core.lean
|
recSubsingleton
|
[] |
[
902,
22
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
893,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.isClosed_iff_zeroLocus
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nZ : Set (PrimeSpectrum R)\n⊢ IsClosed Z ↔ ∃ s, Z = zeroLocus s",
"tactic": "rw [← isOpen_compl_iff, isOpen_iff, compl_compl]"
}
] |
[
422,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
AddMonoidAlgebra.modOf_apply_of_exists_add
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng g' : G\nh : ∃ d, g' = g + d\n⊢ ¬¬∃ g₂, g' = g + g₂",
"tactic": "rwa [Classical.not_not]"
}
] |
[
144,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.Perm.prodExtendRight_apply_eq
|
[] |
[
831,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_re
|
[] |
[
144,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
FundamentalGroupoid.map_eq
|
[] |
[
372,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Data/Set/Pointwise/BigOperators.lean
|
Set.finset_prod_singleton
|
[] |
[
174,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.mul_self_abs
|
[] |
[
969,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
968,
1
] |
Mathlib/Order/RelClasses.lean
|
IsStrictOrder.swap
|
[] |
[
104,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.mul_mul_invOf_self_cancel
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.16996\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m n α\nB : Matrix n n α\ninst✝ : Invertible B\n⊢ A ⬝ B ⬝ ⅟B = A",
"tactic": "rw [Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.mul_one]"
}
] |
[
97,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
11
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.dual_map₂
|
[] |
[
220,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_top_iff
|
[
{
"state_after": "case mp\nR : Type u_1\nM : Type u_2\nP : Type ?u.20594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid P\ninst✝¹ : Module R M\ninst✝ : Module R P\nh : IsNoetherian R { x // x ∈ ⊤ }\n⊢ IsNoetherian R M\n\ncase mpr\nR : Type u_1\nM : Type u_2\nP : Type ?u.20594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid P\ninst✝¹ : Module R M\ninst✝ : Module R P\nh : IsNoetherian R M\n⊢ IsNoetherian R { x // x ∈ ⊤ }",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.20594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid P\ninst✝¹ : Module R M\ninst✝ : Module R P\n⊢ IsNoetherian R { x // x ∈ ⊤ } ↔ IsNoetherian R M",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u_1\nM : Type u_2\nP : Type ?u.20594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid P\ninst✝¹ : Module R M\ninst✝ : Module R P\nh : IsNoetherian R { x // x ∈ ⊤ }\n⊢ IsNoetherian R M",
"tactic": "exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl)"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\nM : Type u_2\nP : Type ?u.20594\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid P\ninst✝¹ : Module R M\ninst✝ : Module R P\nh : IsNoetherian R M\n⊢ IsNoetherian R { x // x ∈ ⊤ }",
"tactic": "exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl).symm"
}
] |
[
142,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSet.of_subtype_image
|
[] |
[
552,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.lt_def
|
[] |
[
636,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.mem_neLocus
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf✝ g✝ f g : Π₀ (a : α), N a\na : α\n⊢ a ∈ neLocus f g ↔ ↑f a ≠ ↑g a",
"tactic": "simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,\n and_iff_right_iff_imp] using Ne.ne_or_ne _"
}
] |
[
46,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.eq_one_of_isUnit
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\n⊢ p = 1",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhm : Monic p\nhpu : IsUnit p\n⊢ p = 1",
"tactic": "nontriviality R"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\n⊢ p = 1",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\n⊢ p = 1",
"tactic": "obtain ⟨q, h⟩ := hpu.exists_right_inv"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\nthis : natDegree (p * q) = natDegree p + natDegree q\n⊢ p = 1",
"state_before": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\n⊢ p = 1",
"tactic": "have := hm.natDegree_mul' (right_ne_zero_of_mul_eq_one h)"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\nthis : natDegree p = 0 ∧ natDegree q = 0\n⊢ p = 1",
"state_before": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\nthis : natDegree (p * q) = natDegree p + natDegree q\n⊢ p = 1",
"tactic": "rw [h, natDegree_one, eq_comm, add_eq_zero_iff] at this"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhm : Monic p\nhpu : IsUnit p\n✝ : Nontrivial R\nq : R[X]\nh : p * q = 1\nthis : natDegree p = 0 ∧ natDegree q = 0\n⊢ p = 1",
"tactic": "exact hm.natDegree_eq_zero_iff_eq_one.mp this.1"
}
] |
[
252,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.bot_lie
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅⊥, N⁆ ≤ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅⊥, N⁆ = ⊥",
"tactic": "suffices ⁅(⊥ : LieIdeal R L), N⁆ ≤ ⊥ by exact le_bot_iff.mp this"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅⊥, N⁆ ≤ ⊥",
"tactic": "rw [lieIdeal_oper_eq_span, lieSpan_le]"
},
{
"state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nhx : x ∈ ⊥\nn : { x // x ∈ N }\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ m ∈ ↑⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑⊥",
"tactic": "rintro m ⟨⟨x, hx⟩, n, hn⟩"
},
{
"state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nhx : x ∈ ⊥\nn : { x // x ∈ N }\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx }, ↑n⁆ ∈ ↑⊥",
"state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nhx : x ∈ ⊥\nn : { x // x ∈ N }\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ m ∈ ↑⊥",
"tactic": "rw [← hn]"
},
{
"state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nn : { x // x ∈ N }\nhx : x ∈ ⊥\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx }, ↑n⁆ ∈ ↑⊥",
"state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nhx : x ∈ ⊥\nn : { x // x ∈ N }\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx }, ↑n⁆ ∈ ↑⊥",
"tactic": "change x ∈ (⊥ : LieIdeal R L) at hx"
},
{
"state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nn : { x // x ∈ N }\nhx✝ : x ∈ ⊥\nhx : x = 0\nhn : ⁅↑{ val := x, property := hx✝ }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx✝ }, ↑n⁆ ∈ ↑⊥",
"state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nn : { x // x ∈ N }\nhx : x ∈ ⊥\nhn : ⁅↑{ val := x, property := hx }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx }, ↑n⁆ ∈ ↑⊥",
"tactic": "rw [mem_bot] at hx"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\nx : L\nn : { x // x ∈ N }\nhx✝ : x ∈ ⊥\nhx : x = 0\nhn : ⁅↑{ val := x, property := hx✝ }, ↑n⁆ = m\n⊢ ⁅↑{ val := x, property := hx✝ }, ↑n⁆ ∈ ↑⊥",
"tactic": "simp [hx]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : 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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nthis : ⁅⊥, N⁆ ≤ ⊥\n⊢ ⁅⊥, N⁆ = ⊥",
"tactic": "exact le_bot_iff.mp this"
}
] |
[
145,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.num_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nh : num x = 0\n⊢ x = 0",
"tactic": "rw [← num_div_denom x, h, RingHom.map_zero, zero_div]"
}
] |
[
1253,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1252,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasLimit.isoOfEquivalence_hom_π
|
[
{
"state_after": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ IsLimit.lift (limit.isLimit G)\n ((Cones.equivalenceOfReindexing (Equivalence.symm e)\n ((isoWhiskerLeft e.inverse w).symm ≪≫ Equivalence.invFunIdAssoc e G)).functor.obj\n (limit.cone F)) ≫\n limit.π G k =\n limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map ((Equivalence.counit e).app k)",
"state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ (isoOfEquivalence e w).hom ≫ limit.π G k =\n limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map ((Equivalence.counit e).app k)",
"tactic": "simp only [HasLimit.isoOfEquivalence, IsLimit.conePointsIsoOfEquivalence_hom]"
},
{
"state_after": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ limit.lift G\n ((Cones.postcompose (whiskerLeft e.inverse w.inv ≫ (Equivalence.invFunIdAssoc e G).hom)).obj\n (Cone.whisker e.inverse (limit.cone F))) ≫\n limit.π G k =\n limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map ((Equivalence.counit e).app k)",
"state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ IsLimit.lift (limit.isLimit G)\n ((Cones.equivalenceOfReindexing (Equivalence.symm e)\n ((isoWhiskerLeft e.inverse w).symm ≪≫ Equivalence.invFunIdAssoc e G)).functor.obj\n (limit.cone F)) ≫\n limit.π G k =\n limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map ((Equivalence.counit e).app k)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ limit.lift G\n ((Cones.postcompose (whiskerLeft e.inverse w.inv ≫ (Equivalence.invFunIdAssoc e G).hom)).obj\n (Cone.whisker e.inverse (limit.cone F))) ≫\n limit.π G k =\n limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map ((Equivalence.counit e).app k)",
"tactic": "simp"
}
] |
[
392,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Data/PFun.lean
|
PFun.mem_preimage
|
[] |
[
435,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sup_sdiff_cancel'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.92580\nα : Type u_1\nβ : Type ?u.92586\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\nhab : a ≤ b\nhbc : b ≤ c\n⊢ b ⊔ c \\ a = c",
"tactic": "rw [sup_sdiff_eq_sup hab, sup_of_le_right hbc]"
}
] |
[
577,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
1
] |
src/lean/Init/Prelude.lean
|
Nat.lt_of_le_of_ne
|
[] |
[
1673,
51
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
1670,
11
] |
Mathlib/Order/Compare.lean
|
eq_iff_eq_of_cmp_eq_cmp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ✝ : Type ?u.20888\ninst✝¹ : LinearOrder α\nx y : α\nβ : Type u_2\ninst✝ : LinearOrder β\nx' y' : β\nh : cmp x y = cmp x' y'\n⊢ x = y ↔ x' = y'",
"tactic": "rw [le_antisymm_iff, le_antisymm_iff, le_iff_le_of_cmp_eq_cmp h,\n le_iff_le_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 h)]"
}
] |
[
267,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.HasFiniteIntegral.congr'
|
[] |
[
156,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.631576\nR₂ : Type u_3\nR₃ : Type ?u.631582\nR₄ : Type ?u.631585\nS : Type ?u.631588\nK : Type ?u.631591\nK₂ : Type ?u.631594\nM : Type u_1\nM' : Type ?u.631600\nM₁ : Type ?u.631603\nM₂ : Type u_4\nM₃ : Type ?u.631609\nM₄ : Type ?u.631612\nN : Type ?u.631615\nN₂ : Type ?u.631618\nι : Type ?u.631621\nV : Type ?u.631624\nV₂ : Type ?u.631627\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring R₂\ninst✝¹¹ : Semiring R₃\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : AddCommMonoid M₃\ninst✝⁷ : AddCommMonoid M'\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹ : RingHomInvPair σ₂₁ σ₁₂\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\nF : Type ?u.632164\nsc : SemilinearMapClass F σ₁₂ M M₂\n⊢ ∀ (x : M), x ∈ comap 0 q ↔ x ∈ ⊤",
"tactic": "simp"
}
] |
[
871,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
870,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.measure_toMeasurable_inter_of_cover
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\n⊢ ↑↑μ\n ((if h : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t then Exists.choose h\n else\n if h' : ∃ t_1 x, MeasurableSet t_1 ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t_1 ∩ u) = ↑↑μ (t ∩ u) then\n Exists.choose h'\n else Exists.choose (_ : ∃ t_1, t ⊆ t_1 ∧ MeasurableSet t_1 ∧ ↑↑μ t_1 = ↑↑μ t)) ∩\n s) =\n ↑↑μ (t ∩ s)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\n⊢ ↑↑μ (toMeasurable μ t ∩ s) = ↑↑μ (t ∩ s)",
"tactic": "rw [toMeasurable]"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ ↑↑μ (Exists.choose (_ : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t) ∩ s) = ↑↑μ (t ∩ s)\n\ncase inr\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ¬∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ ↑↑μ (Exists.choose A ∩ s) = ↑↑μ (t ∩ s)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\n⊢ ↑↑μ\n ((if h : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t then Exists.choose h\n else\n if h' : ∃ t_1 x, MeasurableSet t_1 ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t_1 ∩ u) = ↑↑μ (t ∩ u) then\n Exists.choose h'\n else Exists.choose (_ : ∃ t_1, t ⊆ t_1 ∧ MeasurableSet t_1 ∧ ↑↑μ t_1 = ↑↑μ t)) ∩\n s) =\n ↑↑μ (t ∩ s)",
"tactic": "split_ifs with ht"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"tactic": "let w n := toMeasurable μ (t ∩ v n)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"tactic": "have hw : ∀ n, μ (w n) < ∞ := by\n intro n\n simp_rw [measure_toMeasurable]\n exact (h'v n).lt_top"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"tactic": "set t' := ⋃ n, toMeasurable μ (t ∩ disjointed w n) with ht'"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\n⊢ ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)",
"tactic": "have tt' : t ⊆ t' :=\n calc\n t ⊆ ⋃ n, t ∩ disjointed w n := by\n rw [← inter_iUnion, iUnion_disjointed, inter_iUnion]\n intro x hx\n rcases mem_iUnion.1 (hv hx) with ⟨n, hn⟩\n refine' mem_iUnion.2 ⟨n, _⟩\n have : x ∈ t ∩ v n := ⟨hx, hn⟩\n exact ⟨hx, subset_toMeasurable μ _ this⟩\n _ ⊆ ⋃ n, toMeasurable μ (t ∩ disjointed w n) :=\n iUnion_mono fun n => subset_toMeasurable _ _"
},
{
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{
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"tactic": "refine MeasurableSet.disjointed (fun n => ?_) n"
},
{
"state_after": "no goals",
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"tactic": "exact measurableSet_toMeasurable _ _"
},
{
"state_after": "case hn\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ Pairwise (Disjoint on fun n => disjointed w n)\n\ncase h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ ∀ (i : ℕ), MeasurableSet (disjointed w i)",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ (∑' (n : ℕ), ↑↑(restrict μ (t ∩ u)) (disjointed w n)) = ↑↑(restrict μ (t ∩ u)) (⋃ (n : ℕ), disjointed w n)",
"tactic": "rw [measure_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case hn\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ Pairwise (Disjoint on fun n => disjointed w n)",
"tactic": "exact disjoint_disjointed _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\ni : ℕ\n⊢ MeasurableSet (disjointed w i)",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ ∀ (i : ℕ), MeasurableSet (disjointed w i)",
"tactic": "intro i"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\ni n : ℕ\n⊢ MeasurableSet (w n)",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\ni : ℕ\n⊢ MeasurableSet (disjointed w i)",
"tactic": "refine MeasurableSet.disjointed (fun n => ?_) i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\ni n : ℕ\n⊢ MeasurableSet (w n)",
"tactic": "exact measurableSet_toMeasurable _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nw : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), ↑↑μ (w n) < ⊤\nt' : Set α := ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\nht' : t' = ⋃ (n : ℕ), toMeasurable μ (t ∩ disjointed w n)\ntt' : t ⊆ t'\nu : Set α\nhu : MeasurableSet u\n⊢ ↑↑(restrict μ (t ∩ u)) univ = ↑↑μ (t ∩ u)",
"tactic": "rw [restrict_apply MeasurableSet.univ, univ_inter]"
},
{
"state_after": "case inl.H\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ Exists.choose (_ : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t) ∩ s =ᶠ[ae μ] t ∩ s",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ ↑↑μ (Exists.choose (_ : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t) ∩ s) = ↑↑μ (t ∩ s)",
"tactic": "apply measure_congr"
},
{
"state_after": "no goals",
"state_before": "case inl.H\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ Exists.choose (_ : ∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t) ∩ s =ᶠ[ae μ] t ∩ s",
"tactic": "exact ae_eq_set_inter ht.choose_spec.snd.2 (ae_eq_refl _)"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.746309\nγ : Type ?u.746312\nδ : Type ?u.746315\nι : Type ?u.746318\nR : Type ?u.746321\nR' : Type ?u.746324\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ (n : ℕ), v n\nh'v : ∀ (n : ℕ), ↑↑μ (t ∩ v n) ≠ ⊤\nA : ∃ t' x, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t' ∩ u) = ↑↑μ (t ∩ u)\nht : ¬∃ t_1 x, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t\n⊢ ↑↑μ (Exists.choose A ∩ s) = ↑↑μ (t ∩ s)",
"tactic": "exact A.choose_spec.snd.2 s hs"
}
] |
[
3699,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3637,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_toFormula
|
[
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula falsum) v ↔ Realize falsum (v ∘ Sum.inl) (v ∘ Sum.inr)\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (equal t₁✝ t₂✝)) v ↔ Realize (equal t₁✝ t₂✝) (v ∘ Sum.inl) (v ∘ Sum.inr)\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (rel R✝ ts✝)) v ↔ Realize (rel R✝ ts✝) (v ∘ Sum.inl) (v ∘ Sum.inr)\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ (v : α ⊕ Fin n✝ → M), Formula.Realize (toFormula f₁✝) v ↔ Realize f₁✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nih2 : ∀ (v : α ⊕ Fin n✝ → M), Formula.Realize (toFormula f₂✝) v ↔ Realize f₂✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (f₁✝ ⟹ f₂✝)) v ↔ Realize (f₁✝ ⟹ f₂✝) (v ∘ Sum.inl) (v ∘ Sum.inr)\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (∀'f✝)) v ↔ Realize (∀'f✝) (v ∘ Sum.inl) (v ∘ Sum.inr)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nφ : BoundedFormula L α n\nv : α ⊕ Fin n → M\n⊢ Formula.Realize (toFormula φ) v ↔ Realize φ (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 a8 a9 a0"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula falsum) v ↔ Realize falsum (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (equal t₁✝ t₂✝)) v ↔ Realize (equal t₁✝ t₂✝) (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "simp [BoundedFormula.Realize]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (rel R✝ ts✝)) v ↔ Realize (rel R✝ ts✝) (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "simp [BoundedFormula.Realize]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ (v : α ⊕ Fin n✝ → M), Formula.Realize (toFormula f₁✝) v ↔ Realize f₁✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nih2 : ∀ (v : α ⊕ Fin n✝ → M), Formula.Realize (toFormula f₂✝) v ↔ Realize f₂✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (f₁✝ ⟹ f₂✝)) v ↔ Realize (f₁✝ ⟹ f₂✝) (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,\n realize_imp]"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ (∀ (a : M),\n Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a)) ↔\n ∀ (a : M), Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ Formula.Realize (toFormula (∀'f✝)) v ↔ Realize (∀'f✝) (v ∘ Sum.inl) (v ∘ Sum.inr)",
"tactic": "rw [toFormula, Formula.Realize, realize_all, realize_all]"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\n⊢ (∀ (a : M),\n Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a)) ↔\n ∀ (a : M), Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"tactic": "refine' forall_congr' fun a => _"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a) ∘ Sum.inl)\n (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a) ∘ Sum.inr)\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"tactic": "have h := ih3 (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a))"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a) ∘ Sum.inl)\n (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a) ∘ Sum.inr)\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"tactic": "simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ Realize (toFormula f✝)\n (Sum.elim v (snoc default a ∘ ↑(castAdd 0)) ∘\n Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm))\n (snoc default a ∘ ↑(natAdd 1)) =\n Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) default",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ Realize (relabel (Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm)) (toFormula f✝)) v\n (snoc default a) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)",
"tactic": "rw [← h, realize_relabel, Formula.Realize, iff_iff_eq]"
},
{
"state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ (Realize (toFormula f✝)\n (fun x =>\n Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) x))\n fun x => snoc default a (↑(natAdd 1) x)) =\n Realize (toFormula f✝) (Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a)) default",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ Realize (toFormula f✝)\n (Sum.elim v (snoc default a ∘ ↑(castAdd 0)) ∘\n Sum.elim (Sum.inl ∘ Sum.inl) (Sum.map Sum.inr id ∘ ↑finSumFinEquiv.symm))\n (snoc default a ∘ ↑(natAdd 1)) =\n Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) default",
"tactic": "simp only [Function.comp]"
},
{
"state_after": "case all.e__v.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : α ⊕ Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) x) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) x\n\ncase all.e__xs.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin 0\n⊢ snoc default a (↑(natAdd 1) x) = default x",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\n⊢ (Realize (toFormula f✝)\n (fun x =>\n Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) x))\n fun x => snoc default a (↑(natAdd 1) x)) =\n Realize (toFormula f✝) (Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a)) default",
"tactic": "congr with x"
},
{
"state_after": "case all.e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nval✝ : α\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) (Sum.inl val✝)) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inl val✝)\n\ncase all.e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) (Sum.inr x)) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr x)",
"state_before": "case all.e__v.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : α ⊕ Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) x) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) x",
"tactic": "cases' x with _ x"
},
{
"state_after": "no goals",
"state_before": "case all.e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nval✝ : α\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) (Sum.inl val✝)) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inl val✝)",
"tactic": "simp"
},
{
"state_after": "case all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x))\n (Sum.inr (last n✝))) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (last n✝))\n\ncase all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝),\n Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x))\n (Sum.inr (↑castSucc i))) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (↑castSucc i))",
"state_before": "case all.e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x)) (Sum.inr x)) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr x)",
"tactic": "refine' Fin.lastCases _ _ x"
},
{
"state_after": "case all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ snoc default a (↑(castAdd 0) (id 0)) = snoc (fun x => v (Sum.inr x)) a (last n✝)",
"state_before": "case all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x))\n (Sum.inr (last n✝))) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (last n✝))",
"tactic": "rw [Sum.elim_inr, Sum.elim_inr,\n finSumFinEquiv_symm_last, Sum.map_inr, Sum.elim_inr]"
},
{
"state_after": "no goals",
"state_before": "case all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ snoc default a (↑(castAdd 0) (id 0)) = snoc (fun x => v (Sum.inr x)) a (last n✝)",
"tactic": "simp [Fin.snoc]"
},
{
"state_after": "case all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (↑(castAdd 1) i)",
"state_before": "case all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝),\n Sum.elim v (fun x => snoc default a (↑(castAdd 0) x))\n (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (↑finSumFinEquiv.symm x))\n (Sum.inr (↑castSucc i))) =\n Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (↑castSucc i))",
"tactic": "simp only [castSucc, Function.comp_apply, Sum.elim_inr,\n finSumFinEquiv_symm_apply_castAdd, Sum.map_inl, Sum.elim_inl]"
},
{
"state_after": "case all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (↑castSucc i)",
"state_before": "case all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (↑(castAdd 1) i)",
"tactic": "rw [← castSucc]"
},
{
"state_after": "no goals",
"state_before": "case all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin (n✝ + 1)\n⊢ ∀ (i : Fin n✝), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (↑castSucc i)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case all.e__xs.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.638594\nP : Type ?u.638597\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nv✝ : α ⊕ Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), Formula.Realize (toFormula f✝) v ↔ Realize f✝ (v ∘ Sum.inl) (v ∘ Sum.inr)\nv : α ⊕ Fin n✝ → M\na : M\nh :\n Formula.Realize (toFormula f✝) (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔\n Realize f✝ (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)\nx : Fin 0\n⊢ snoc default a (↑(natAdd 1) x) = default x",
"tactic": "exact Fin.elim0 x"
}
] |
[
942,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
1
] |
Mathlib/Data/Polynomial/Derivative.lean
|
Polynomial.derivative_one
|
[] |
[
142,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Associates.le_singleton_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.817526\nA : Type u_1\nK : Type ?u.817532\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : A\nn : ℕ\nI : Ideal A\n⊢ Associates.mk I ^ n ≤ Associates.mk (span {x}) ↔ x ∈ I ^ n",
"tactic": "simp_rw [← Associates.dvd_eq_le, ← Associates.mk_pow, Associates.mk_dvd_mk,\n Ideal.dvd_span_singleton]"
}
] |
[
778,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
closure_smul₀
|
[
{
"state_after": "case inl\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\n⊢ closure (0 • s) = 0 • closure s\n\ncase inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc✝ : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\nc : G₀\ns : Set E\nhc : c ≠ 0\n⊢ closure (c • s) = c • closure s",
"state_before": "M : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc✝ : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\nc : G₀\ns : Set E\n⊢ closure (c • s) = c • closure s",
"tactic": "rcases eq_or_ne c 0 with (rfl | hc)"
},
{
"state_after": "case inl.inl\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\n⊢ closure (0 • ∅) = 0 • closure ∅\n\ncase inl.inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\nhs : Set.Nonempty s\n⊢ closure (0 • s) = 0 • closure s",
"state_before": "case inl\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\n⊢ closure (0 • s) = 0 • closure s",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hs)"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\n⊢ closure (0 • ∅) = 0 • closure ∅",
"tactic": "simp"
},
{
"state_after": "case inl.inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\nhs : Set.Nonempty s\n⊢ closure 0 = 0",
"state_before": "case inl.inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\nhs : Set.Nonempty s\n⊢ closure (0 • s) = 0 • closure s",
"tactic": "rw [zero_smul_set hs, zero_smul_set hs.closure]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\nhs : Set.Nonempty s\n⊢ closure 0 = 0",
"tactic": "exact closure_singleton"
},
{
"state_after": "no goals",
"state_before": "case inr\nM : Type ?u.97023\nα : Type ?u.97026\nβ : Type ?u.97029\nG₀ : Type u_2\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : GroupWithZero G₀\ninst✝⁷ : MulAction G₀ α\ninst✝⁶ : ContinuousConstSMul G₀ α\ninst✝⁵ : TopologicalSpace β\nf : β → α\nb : β\nc✝ : G₀\ns✝ : Set β\nE : Type u_1\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\nc : G₀\ns : Set E\nhc : c ≠ 0\n⊢ closure (c • s) = c • closure s",
"tactic": "exact ((Homeomorph.smulOfNeZero c hc).image_closure s).symm"
}
] |
[
348,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_zero
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : p = 0\n⊢ snorm 0 p μ = 0\n\ncase neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\n⊢ snorm 0 p μ = 0",
"state_before": "α : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ snorm 0 p μ = 0",
"tactic": "by_cases h0 : p = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm 0 p μ = 0\n\ncase neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm 0 p μ = 0",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\n⊢ snorm 0 p μ = 0",
"tactic": "by_cases h_top : p = ∞"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : p ≠ 0\nh_top : ¬p = ⊤\n⊢ snorm 0 p μ = 0",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm 0 p μ = 0",
"tactic": "rw [← Ne.def] at h0"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : p ≠ 0\nh_top : ¬p = ⊤\n⊢ snorm 0 p μ = 0",
"tactic": "simp [snorm_eq_snorm' h0 h_top, ENNReal.toReal_pos h0 h_top]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : p = 0\n⊢ snorm 0 p μ = 0",
"tactic": "simp [h0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.922031\nF : Type u_2\nG : Type ?u.922037\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm 0 p μ = 0",
"tactic": "simp only [h_top, snorm_exponent_top, snormEssSup_zero]"
}
] |
[
210,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_coe_ne_top_iff_summable_coe
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.306594\nγ : Type ?u.306597\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf : α → ℝ≥0\n⊢ (∑' (a : α), ↑(f a)) ≠ ⊤ ↔ Summable fun a => f a",
"state_before": "α : Type u_1\nβ : Type ?u.306594\nγ : Type ?u.306597\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf : α → ℝ≥0\n⊢ (∑' (a : α), ↑(f a)) ≠ ⊤ ↔ Summable fun a => ↑(f a)",
"tactic": "rw [NNReal.summable_coe]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.306594\nγ : Type ?u.306597\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf : α → ℝ≥0\n⊢ (∑' (a : α), ↑(f a)) ≠ ⊤ ↔ Summable fun a => f a",
"tactic": "exact tsum_coe_ne_top_iff_summable"
}
] |
[
1075,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1072,
1
] |
Mathlib/CategoryTheory/Limits/Lattice.lean
|
CategoryTheory.Limits.CompleteLattice.coprod_eq_sup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nx y : α\n⊢ colimit (pair x y) = Finset.sup Finset.univ (pair x y).toPrefunctor.obj",
"tactic": "rw [finite_colimit_eq_finset_univ_sup (pair x y)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nx y : α\n⊢ x ⊔ (y ⊔ ⊥) = x ⊔ y",
"tactic": "rw [sup_bot_eq]"
}
] |
[
151,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Order/MinMax.lean
|
lt_max_of_lt_left
|
[] |
[
92,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Topology/Order/Basic.lean
|
isOpen_Ioo
|
[] |
[
306,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.le_iff_mem_closure
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx y : PrimeSpectrum R\n⊢ x ≤ y ↔ y ∈ closure {x}",
"tactic": "rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,\n vanishingIdeal_singleton, SetLike.coe_subset_coe]"
}
] |
[
932,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.AEStronglyMeasurable.integral_prod_right'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nα' : Type ?u.2386685\nβ : Type u_1\nβ' : Type ?u.2386691\nγ : Type ?u.2386694\nE : Type u_2\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace α'\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : MeasurableSpace β'\ninst✝⁴ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝³ : NormedAddCommGroup E\ninst✝² : SigmaFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α × β → E\nhf : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ (fun x => ∫ (y : β), f (x, y) ∂ν) =ᶠ[ae μ] fun x => ∫ (y : β), AEStronglyMeasurable.mk f hf (x, y) ∂ν",
"tactic": "filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx"
}
] |
[
204,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Icc.mul_le_left
|
[] |
[
136,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_union_right
|
[] |
[
1354,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1353,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_toIcoMod
|
[
{
"state_after": "θ ψ : ℝ\n⊢ ∃ k, toIcoMod two_pi_pos ψ θ - θ = 2 * π * ↑k",
"state_before": "θ ψ : ℝ\n⊢ ↑(toIcoMod two_pi_pos ψ θ) = ↑θ",
"tactic": "rw [angle_eq_iff_two_pi_dvd_sub]"
},
{
"state_after": "θ ψ : ℝ\n⊢ toIcoMod two_pi_pos ψ θ - θ = 2 * π * ↑(-toIcoDiv two_pi_pos ψ θ)",
"state_before": "θ ψ : ℝ\n⊢ ∃ k, toIcoMod two_pi_pos ψ θ - θ = 2 * π * ↑k",
"tactic": "refine' ⟨-toIcoDiv two_pi_pos ψ θ, _⟩"
},
{
"state_after": "no goals",
"state_before": "θ ψ : ℝ\n⊢ toIcoMod two_pi_pos ψ θ - θ = 2 * π * ↑(-toIcoDiv two_pi_pos ψ θ)",
"tactic": "rw [toIcoMod_sub_self, zsmul_eq_mul, mul_comm]"
}
] |
[
511,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_of_tower
|
[
{
"state_after": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : SMul R S\ninst✝² : Module S M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R S M\nh : WellFounded fun x x_1 => x > x_1\n⊢ WellFounded fun x x_1 => x > x_1",
"state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : SMul R S\ninst✝² : Module S M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R S M\nh : IsNoetherian R M\n⊢ IsNoetherian S M",
"tactic": "rw [isNoetherian_iff_wellFounded] at h⊢"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : SMul R S\ninst✝² : Module S M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R S M\nh : WellFounded fun x x_1 => x > x_1\n⊢ WellFounded fun x x_1 => x > x_1",
"tactic": "refine' (Submodule.restrictScalarsEmbedding R S M).dual.wellFounded h"
}
] |
[
533,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
StrictMonoOn.dual
|
[] |
[
250,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
11
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.codomain_trivial_iff_range_trivial
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.86859\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.86868\nx✝² : NonAssocSemiring α\nx✝¹ : NonAssocSemiring β\nf : α →+* β\nx✝ y : α\nh : ↑f 1 = 0\nx : α\n⊢ ↑f x = 0",
"tactic": "rw [← mul_one x, map_mul, h, mul_zero]"
}
] |
[
605,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
LinearMap.leftInverse_splittingOfFunOnFintypeSurjective
|
[] |
[
1262,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1260,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.div_lt
|
[] |
[
919,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.inf_adj
|
[] |
[
266,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_mul_of_div_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.67078\nβ : Type ?u.67081\nG : Type u_1\ninst✝ : CommGroup G\na b c d : G\nh : a / b = c\n⊢ a = b * c",
"tactic": "simp [h.symm]"
}
] |
[
920,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
920,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
neg_abs_le_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ -abs a ≤ -a",
"tactic": "simpa using neg_abs_le_self (-a)"
}
] |
[
168,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.inv_eq_iff_eq
|
[] |
[
115,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.ofTop_apply
|
[] |
[
2103,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2102,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_congr_ae
|
[] |
[
82,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
Monotone.map_limsSup_of_continuousAt
|
[] |
[
386,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.coeSort_toFinset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs : Set.Finite s\nht : Set.Finite t\nh : Set.Finite s\n⊢ { x // x ∈ Finite.toFinset h } = ↑s",
"tactic": "rw [← Finset.coe_sort_coe _, h.coe_toFinset]"
}
] |
[
179,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_gt_gt
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})\n\ncase hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 ≤ᵐ[Measure.restrict μ {x | R < f x}] fun a => f a - R\n\ncase hfi\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ IntegrableOn (fun a => f a - R) {x | R < f x}",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ ENNReal.toReal (↑↑μ {x | R < f x}) * R < ∫ (x : α) in {x | R < f x}, f x ∂μ",
"tactic": "rw [← sub_pos, ← smul_eq_mul, ← set_integral_const, ← integral_sub hfint this,\n set_integral_pos_iff_support_of_nonneg_ae]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\n⊢ (∫⁻ (a : α) in {x | R < f x}, ↑‖(fun x => R) a‖₊ ∂μ) ≤ ∫⁻ (a : α) in {x | R < f x}, ↑‖f a‖₊ ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\n⊢ IntegrableOn (fun x => R) {x | R < f x}",
"tactic": "refine' ⟨aestronglyMeasurable_const, lt_of_le_of_lt _ hfint.2⟩"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\n⊢ Measurable fun a => (fun x => R) a\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nx : α\nhx : x ∈ {x | R < f x}\n⊢ ↑‖(fun x => R) x‖₊ ≤ ↑‖f x‖₊",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\n⊢ (∫⁻ (a : α) in {x | R < f x}, ↑‖(fun x => R) a‖₊ ∂μ) ≤ ∫⁻ (a : α) in {x | R < f x}, ↑‖f a‖₊ ∂μ",
"tactic": "refine'\n set_lintegral_mono (Measurable.nnnorm _).coe_nnreal_ennreal hfm.nnnorm.coe_nnreal_ennreal\n fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\n⊢ Measurable fun a => (fun x => R) a",
"tactic": "exact measurable_const"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nx : α\nhx : x ∈ {x | R < f x}\n⊢ { val := R, property := hR } ≤ { val := f x, property := (_ : 0 ≤ f x) }",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nx : α\nhx : x ∈ {x | R < f x}\n⊢ ↑‖(fun x => R) x‖₊ ≤ ↑‖f x‖₊",
"tactic": "simp only [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg hR,\n Real.nnnorm_of_nonneg (hR.trans <| le_of_lt hx), Subtype.mk_le_mk]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nx : α\nhx : x ∈ {x | R < f x}\n⊢ { val := R, property := hR } ≤ { val := f x, property := (_ : 0 ≤ f x) }",
"tactic": "exact le_of_lt hx"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : 0 < ↑↑μ {x | R < f x}\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})",
"tactic": "rw [← zero_lt_iff] at hμ"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : 0 < ↑↑μ {x | R < f x}\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ {x | R < f x} ⊆ support fun a => f a - R",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : 0 < ↑↑μ {x | R < f x}\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})",
"tactic": "rwa [Set.inter_eq_self_of_subset_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : 0 < ↑↑μ {x | R < f x}\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ {x | R < f x} ⊆ support fun a => f a - R",
"tactic": "exact fun x hx => Ne.symm (ne_of_lt <| sub_pos.2 hx)"
},
{
"state_after": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | R < f x}, OfNat.ofNat 0 x ≤ (fun a => f a - R) x",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ 0 ≤ᵐ[Measure.restrict μ {x | R < f x}] fun a => f a - R",
"tactic": "change ∀ᵐ x ∂μ.restrict _, _"
},
{
"state_after": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | R < f x} → OfNat.ofNat 0 x ≤ (fun a => f a - R) x\n\ncase hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ MeasurableSet {x | OfNat.ofNat 0 x ≤ (fun a => f a - R) x}",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | R < f x}, OfNat.ofNat 0 x ≤ (fun a => f a - R) x",
"tactic": "rw [ae_restrict_iff]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | R < f x} → OfNat.ofNat 0 x ≤ (fun a => f a - R) x",
"tactic": "exact eventually_of_forall fun x hx => sub_nonneg.2 <| le_of_lt hx"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ MeasurableSet {x | OfNat.ofNat 0 x ≤ (fun a => f a - R) x}",
"tactic": "exact measurableSet_le measurable_zero (hfm.sub measurable_const)"
},
{
"state_after": "no goals",
"state_before": "case hfi\nα : Type u_1\nβ : Type ?u.264063\nE : Type ?u.264066\nF : Type ?u.264069\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nR : ℝ\nf : α → ℝ\nhR : 0 ≤ R\nhfm : Measurable f\nhfint : IntegrableOn f {x | R < f x}\nhμ : ↑↑μ {x | R < f x} ≠ 0\nthis : IntegrableOn (fun x => R) {x | R < f x}\n⊢ IntegrableOn (fun a => f a - R) {x | R < f x}",
"tactic": "exact Integrable.sub hfint this"
}
] |
[
590,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSheaf_of_yoneda
|
[] |
[
759,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Std/Data/HashMap/WF.lean
|
Std.HashMap.Imp.Bucket.size_eq
|
[] |
[
34,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
33,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Cotrident.IsColimit.hom_ext
|
[] |
[
309,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.mul_lt_of_lt_ediv
|
[] |
[
430,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
11
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.map_powers
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_3\nA : Type ?u.225218\nB : Type ?u.225221\ninst✝² : Monoid M\nN : Type u_1\nF : Type u_2\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass F M N\nf : F\nm : M\n⊢ map f (powers m) = powers (↑f m)",
"tactic": "simp only [powers_eq_closure, map_mclosure f, Set.image_singleton]"
}
] |
[
516,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.derivedSeries_le_lowerCentralSeries
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ derivedSeries R L Nat.zero ≤ lowerCentralSeries R L L Nat.zero\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\n⊢ derivedSeries R L (Nat.succ k) ≤ lowerCentralSeries R L L (Nat.succ k)",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"tactic": "induction' k with k h"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ derivedSeries R L Nat.zero ≤ lowerCentralSeries R L L Nat.zero",
"tactic": "rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\nh' : derivedSeries R L k ≤ ⊤\n⊢ derivedSeries R L (Nat.succ k) ≤ lowerCentralSeries R L L (Nat.succ k)",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\n⊢ derivedSeries R L (Nat.succ k) ≤ lowerCentralSeries R L L (Nat.succ k)",
"tactic": "have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\nh' : derivedSeries R L k ≤ ⊤\n⊢ ⁅derivedSeriesOfIdeal R L k ⊤, derivedSeriesOfIdeal R L k ⊤⁆ ≤ ⁅⊤, lowerCentralSeries R L L k⁆",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\nh' : derivedSeries R L k ≤ ⊤\n⊢ derivedSeries R L (Nat.succ k) ≤ lowerCentralSeries R L L (Nat.succ k)",
"tactic": "rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\nh' : derivedSeries R L k ≤ ⊤\n⊢ ⁅derivedSeriesOfIdeal R L k ⊤, derivedSeriesOfIdeal R L k ⊤⁆ ≤ ⁅⊤, lowerCentralSeries R L L k⁆",
"tactic": "exact LieSubmodule.mono_lie _ _ _ _ h' h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nh : derivedSeries R L k ≤ lowerCentralSeries R L L k\n⊢ derivedSeries R L k ≤ ⊤",
"tactic": "simp only [le_top]"
}
] |
[
181,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.encode_inr
|
[] |
[
295,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_submatrix_sum_swap_sum_swap
|
[
{
"state_after": "no goals",
"state_before": "l✝ : Type ?u.20447\nm✝ : Type ?u.20450\nn✝ : Type ?u.20453\no✝ : Type ?u.20456\np : Type ?u.20459\nq : Type ?u.20462\nm' : o✝ → Type ?u.20467\nn' : o✝ → Type ?u.20472\np' : o✝ → Type ?u.20477\nR : Type ?u.20480\nS : Type ?u.20483\nα✝ : Type ?u.20486\nβ : Type ?u.20489\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type u_5\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\n⊢ submatrix (fromBlocks A B C D) Sum.swap Sum.swap = fromBlocks D C B A",
"tactic": "simp"
}
] |
[
181,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
AffineBasis.exists_affineBasis_of_finiteDimensional
|
[
{
"state_after": "case intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Set P\nb : AffineBasis (↑s) k P\nhb : ↑b = Subtype.val\n⊢ Nonempty (AffineBasis ι k P)",
"state_before": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\n⊢ Nonempty (AffineBasis ι k P)",
"tactic": "obtain ⟨s, b, hb⟩ := AffineBasis.exists_affineBasis k V P"
},
{
"state_after": "case intro.intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Finset P\nb : AffineBasis (↑↑s) k P\nhb : ↑b = Subtype.val\n⊢ Nonempty (AffineBasis ι k P)",
"state_before": "case intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Set P\nb : AffineBasis (↑s) k P\nhb : ↑b = Subtype.val\n⊢ Nonempty (AffineBasis ι k P)",
"tactic": "lift s to Finset P using b.finite_set"
},
{
"state_after": "case intro.intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Finset P\nb : AffineBasis (↑↑s) k P\nhb : ↑b = Subtype.val\n⊢ Fintype.card ↑↑s = Fintype.card ι",
"state_before": "case intro.intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Finset P\nb : AffineBasis (↑↑s) k P\nhb : ↑b = Subtype.val\n⊢ Nonempty (AffineBasis ι k P)",
"tactic": "refine' ⟨b.reindex <| Fintype.equivOfCardEq _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : DivisionRing k\ninst✝² : Module k V\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional k V\nh : Fintype.card ι = FiniteDimensional.finrank k V + 1\ns : Finset P\nb : AffineBasis (↑↑s) k P\nhb : ↑b = Subtype.val\n⊢ Fintype.card ↑↑s = Fintype.card ι",
"tactic": "rw [h, ← b.card_eq_finrank_add_one]"
}
] |
[
793,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.image_eq_preimage
|
[] |
[
567,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
11
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Monovary.sum_smul_comp_perm_le_sum_smul
|
[] |
[
249,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean
|
Projectivization.mk'_eq_mk
|
[] |
[
68,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.IsTheta.trans_eventuallyEq
|
[] |
[
132,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuous_if'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.376081\nδ : Type ?u.376084\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : α → Prop\nf g : α → β\ninst✝ : (a : α) → Decidable (p a)\nhpf : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto f (𝓝[{x | p x}] a) (𝓝 (if p a then f a else g a))\nhpg : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 (if p a then f a else g a))\nhf : ContinuousOn f {x | p x}\nhg : ContinuousOn g {x | ¬p x}\n⊢ ContinuousOn (fun a => if p a then f a else g a) univ",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.376081\nδ : Type ?u.376084\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : α → Prop\nf g : α → β\ninst✝ : (a : α) → Decidable (p a)\nhpf : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto f (𝓝[{x | p x}] a) (𝓝 (if p a then f a else g a))\nhpg : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 (if p a then f a else g a))\nhf : ContinuousOn f {x | p x}\nhg : ContinuousOn g {x | ¬p x}\n⊢ Continuous fun a => if p a then f a else g a",
"tactic": "rw [continuous_iff_continuousOn_univ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.376081\nδ : Type ?u.376084\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : α → Prop\nf g : α → β\ninst✝ : (a : α) → Decidable (p a)\nhpf : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto f (𝓝[{x | p x}] a) (𝓝 (if p a then f a else g a))\nhpg : ∀ (a : α), a ∈ frontier {x | p x} → Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 (if p a then f a else g a))\nhf : ContinuousOn f {x | p x}\nhg : ContinuousOn g {x | ¬p x}\n⊢ ContinuousOn (fun a => if p a then f a else g a) univ",
"tactic": "apply ContinuousOn.if' <;> simp [*] <;> assumption"
}
] |
[
1174,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean
|
Polynomial.natDegree_mul_C_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q r f : R[X]\na : R\n⊢ natDegree f + natDegree (↑C a) = natDegree f + 0",
"tactic": "rw [natDegree_C a]"
}
] |
[
105,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.tendsto_iterate_efixedPoint'
|
[] |
[
213,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.powers_eq_closure
|
[
{
"state_after": "case h\nM : Type u_1\nA : Type ?u.128599\nB : Type ?u.128602\ninst✝ : Monoid M\nn x✝ : M\n⊢ x✝ ∈ powers n ↔ x✝ ∈ closure {n}",
"state_before": "M : Type u_1\nA : Type ?u.128599\nB : Type ?u.128602\ninst✝ : Monoid M\nn : M\n⊢ powers n = closure {n}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nM : Type u_1\nA : Type ?u.128599\nB : Type ?u.128602\ninst✝ : Monoid M\nn x✝ : M\n⊢ x✝ ∈ powers n ↔ x✝ ∈ closure {n}",
"tactic": "exact mem_closure_singleton.symm"
}
] |
[
446,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
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