file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.sOppSide_comm
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.95363\nP : Type u_3\nP' : Type ?u.95369\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ SOppSide s x y ↔ SOppSide s y x",
"tactic": "rw [SOppSide, SOppSide, wOppSide_comm, and_comm (b := x ∉ s)]"
}
] |
[
219,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.EventuallyEq.inter
|
[] |
[
1568,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1566,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean
|
Finset.mem_dfinsupp_iff
|
[
{
"state_after": "case refine'_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\n⊢ (∃ a,\n a ∈ pi s t ∧\n ↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n a =\n f) →\n support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\n\ncase refine'_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\n⊢ (support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i) →\n ∃ a,\n a ∈ pi s t ∧\n ↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n a =\n f",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\n⊢ f ∈ dfinsupp s t ↔ support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i",
"tactic": "refine' mem_map.trans ⟨_, _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\n⊢ support\n (↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n f) ⊆\n s ∧\n ∀ (i : ι),\n i ∈ s →\n ↑(↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n f)\n i ∈\n t i",
"state_before": "case refine'_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\n⊢ (∃ a,\n a ∈ pi s t ∧\n ↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n a =\n f) →\n support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i",
"tactic": "rintro ⟨f, hf, rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\n⊢ support (Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) ⊆ s ∧\n ∀ (i : ι), i ∈ s → ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i ∈ t i",
"state_before": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\n⊢ support\n (↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n f) ⊆\n s ∧\n ∀ (i : ι),\n i ∈ s →\n ↑(↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n f)\n i ∈\n t i",
"tactic": "rw [Function.Embedding.coeFn_mk]"
},
{
"state_after": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\ni : ι\nhi : i ∈ s\n⊢ ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i ∈ t i",
"state_before": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\n⊢ support (Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) ⊆ s ∧\n ∀ (i : ι), i ∈ s → ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i ∈ t i",
"tactic": "refine' ⟨support_mk_subset, fun i hi => _⟩"
},
{
"state_after": "case h.e'_4\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\ni : ι\nhi : i ∈ s\n⊢ ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i = f i hi",
"state_before": "case refine'_1.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\ni : ι\nhi : i ∈ s\n⊢ ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i ∈ t i",
"tactic": "convert mem_pi.1 hf i hi"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nf : (a : ι) → a ∈ s → (fun i => α i) a\nhf : f ∈ pi s t\ni : ι\nhi : i ∈ s\n⊢ ↑(Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s)) i = f i hi",
"tactic": "exact mk_of_mem hi"
},
{
"state_after": "case refine'_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\n⊢ (↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n fun i x => ↑f i) =\n f",
"state_before": "case refine'_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\n⊢ (support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i) →\n ∃ a,\n a ∈ pi s t ∧\n ↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n a =\n f",
"tactic": "refine' fun h => ⟨fun i _ => f i, mem_pi.2 h.2, _⟩"
},
{
"state_after": "case refine'_2.h\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\ni : ι\n⊢ ↑(↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n fun i x => ↑f i)\n i =\n ↑f i",
"state_before": "case refine'_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\n⊢ (↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n fun i x => ↑f i) =\n f",
"tactic": "ext i"
},
{
"state_after": "case refine'_2.h\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\ni : ι\n⊢ (if i ∈ s then ↑f i else 0) = ↑f i",
"state_before": "case refine'_2.h\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\ni : ι\n⊢ ↑(↑{ toFun := fun f => Dfinsupp.mk s fun i => f ↑i (_ : ↑i ∈ ↑s),\n inj' := (_ : Function.Injective (Dfinsupp.mk s ∘ fun f i => f ↑i (_ : ↑i ∈ ↑s))) }\n fun i x => ↑f i)\n i =\n ↑f i",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h\nι : Type u_1\nα : ι → Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (α i)\ns : Finset ι\nf : Π₀ (i : ι), α i\nt : (i : ι) → Finset (α i)\ninst✝ : (i : ι) → DecidableEq (α i)\nh : support f ⊆ s ∧ ∀ (i : ι), i ∈ s → ↑f i ∈ t i\ni : ι\n⊢ (if i ∈ s then ↑f i else 0) = ↑f i",
"tactic": "exact ite_eq_left_iff.2 fun hi => (not_mem_support_iff.1 fun H => hi <| h.1 H).symm"
}
] |
[
61,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
mem_uniform_prod
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.155123\nt₁ : UniformSpace α\nt₂ : UniformSpace β\na : Set (α × α)\nb : Set (β × β)\nha : a ∈ 𝓤 α\nhb : b ∈ 𝓤 β\n⊢ {p | (p.fst.fst, p.snd.fst) ∈ a ∧ (p.fst.snd, p.snd.snd) ∈ b} ∈\n comap (fun p => (p.fst.fst, p.snd.fst)) (𝓤 α) ⊓ comap (fun p => (p.fst.snd, p.snd.snd)) (𝓤 β)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.155123\nt₁ : UniformSpace α\nt₂ : UniformSpace β\na : Set (α × α)\nb : Set (β × β)\nha : a ∈ 𝓤 α\nhb : b ∈ 𝓤 β\n⊢ {p | (p.fst.fst, p.snd.fst) ∈ a ∧ (p.fst.snd, p.snd.snd) ∈ b} ∈ 𝓤 (α × β)",
"tactic": "rw [uniformity_prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.155123\nt₁ : UniformSpace α\nt₂ : UniformSpace β\na : Set (α × α)\nb : Set (β × β)\nha : a ∈ 𝓤 α\nhb : b ∈ 𝓤 β\n⊢ {p | (p.fst.fst, p.snd.fst) ∈ a ∧ (p.fst.snd, p.snd.snd) ∈ b} ∈\n comap (fun p => (p.fst.fst, p.snd.fst)) (𝓤 α) ⊓ comap (fun p => (p.fst.snd, p.snd.snd)) (𝓤 β)",
"tactic": "exact inter_mem_inf (preimage_mem_comap ha) (preimage_mem_comap hb)"
}
] |
[
1597,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1594,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.card_image₂_le
|
[] |
[
59,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.Equivalence_mk'_unit
|
[] |
[
132,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Data/Finset/NoncommProd.lean
|
Multiset.noncommProd_coe
|
[
{
"state_after": "F : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm : Set.Pairwise {x | x ∈ ↑l} Commute\n⊢ noncommFold (fun x x_1 => x * x_1) (↑l) comm 1 = List.prod l",
"state_before": "F : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm : Set.Pairwise {x | x ∈ ↑l} Commute\n⊢ noncommProd (↑l) comm = List.prod l",
"tactic": "rw [noncommProd]"
},
{
"state_after": "F : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm : Set.Pairwise {x | x ∈ ↑l} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l",
"state_before": "F : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm : Set.Pairwise {x | x ∈ ↑l} Commute\n⊢ noncommFold (fun x x_1 => x * x_1) (↑l) comm 1 = List.prod l",
"tactic": "simp only [noncommFold_coe]"
},
{
"state_after": "case nil\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\ncomm : Set.Pairwise {x | x ∈ ↑[]} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod []\n\ncase cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl)",
"state_before": "F : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm : Set.Pairwise {x | x ∈ ↑l} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l",
"tactic": "induction' l with hd tl hl"
},
{
"state_after": "no goals",
"state_before": "case nil\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\ncomm : Set.Pairwise {x | x ∈ ↑[]} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod []",
"tactic": "simp"
},
{
"state_after": "case cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\n⊢ Set.Pairwise {x | x ∈ ↑tl} Commute",
"state_before": "case cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\n⊢ List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl)",
"tactic": "rw [List.prod_cons, List.foldr, hl]"
},
{
"state_after": "case cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\nx : α\nhx : x ∈ {x | x ∈ ↑tl}\ny : α\nhy : y ∈ {x | x ∈ ↑tl}\n⊢ x ≠ y → Commute x y",
"state_before": "case cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\n⊢ Set.Pairwise {x | x ∈ ↑tl} Commute",
"tactic": "intro x hx y hy"
},
{
"state_after": "no goals",
"state_before": "case cons\nF : Type ?u.7239\nι : Type ?u.7242\nα : Type u_1\nβ : Type ?u.7248\nγ : Type ?u.7251\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nl : List α\ncomm✝ : Set.Pairwise {x | x ∈ ↑l} Commute\nhd : α\ntl : List α\nhl : Set.Pairwise {x | x ∈ ↑tl} Commute → List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x ∈ ↑(hd :: tl)} Commute\nx : α\nhx : x ∈ {x | x ∈ ↑tl}\ny : α\nhy : y ∈ {x | x ∈ ↑tl}\n⊢ x ≠ y → Commute x y",
"tactic": "exact comm (List.mem_cons_of_mem _ hx) (List.mem_cons_of_mem _ hy)"
}
] |
[
134,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Isometry.lean
|
QuadraticForm.Equivalent.refl
|
[] |
[
109,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.L1.SimpleFunc.integrable
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3383671\nι : Type ?u.3383674\nE : Type u_2\nF : Type ?u.3383680\n𝕜 : Type ?u.3383683\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ Lp.simpleFunc E 1 μ }\n⊢ Memℒp (↑(Lp.simpleFunc.toSimpleFunc f)) 1",
"state_before": "α : Type u_1\nβ : Type ?u.3383671\nι : Type ?u.3383674\nE : Type u_2\nF : Type ?u.3383680\n𝕜 : Type ?u.3383683\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ Lp.simpleFunc E 1 μ }\n⊢ Integrable ↑(Lp.simpleFunc.toSimpleFunc f)",
"tactic": "rw [← memℒp_one_iff_integrable]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3383671\nι : Type ?u.3383674\nE : Type u_2\nF : Type ?u.3383680\n𝕜 : Type ?u.3383683\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ Lp.simpleFunc E 1 μ }\n⊢ Memℒp (↑(Lp.simpleFunc.toSimpleFunc f)) 1",
"tactic": "exact Lp.simpleFunc.memℒp f"
}
] |
[
1087,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1085,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.prod_mono
|
[] |
[
1723,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1722,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
OrderIso.essSup_apply
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\n\ncase refine'_2\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\n\ncase refine'_3\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) fun x => ↑g (f x)\n\ncase refine'_4\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) fun x => ↑g (f x)",
"state_before": "α : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ ↑g (essSup f μ) = essSup (fun x => ↑g (f x)) μ",
"tactic": "refine' OrderIso.limsup_apply g _ _ _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\n\ncase refine'_2\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\n\ncase refine'_3\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) fun x => ↑g (f x)\n\ncase refine'_4\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) fun x => ↑g (f x)",
"tactic": "all_goals isBoundedDefault"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type u_3\nm✝ : MeasurableSpace α\nμ✝ ν : MeasureTheory.Measure α\ninst✝¹ : CompleteLattice β\nm : MeasurableSpace α\nγ : Type u_2\ninst✝ : CompleteLattice γ\nf : α → β\nμ : MeasureTheory.Measure α\ng : β ≃o γ\n⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) fun x => ↑g (f x)",
"tactic": "isBoundedDefault"
}
] |
[
189,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mul_eq_max
|
[
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ a ≤ a * b",
"tactic": "simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans hb) a"
},
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ b ≤ a * b",
"tactic": "simpa only [one_mul] using mul_le_mul_right' (one_le_aleph0.trans ha) b"
}
] |
[
556,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.spanSingleton_mul_coeIdeal_eq_coeIdeal
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1518060\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1518267\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nI J : Ideal R₁\nz : K\n⊢ spanSingleton R₁⁰ z * ↑I = ↑J ↔ Ideal.span {(sec R₁⁰ z).fst} * I = Ideal.span {↑(sec R₁⁰ z).snd} * J",
"tactic": "erw [← mk'_mul_coeIdeal_eq_coeIdeal K (IsLocalization.sec R₁⁰ z).2.prop,\n IsLocalization.mk'_sec K z]"
}
] |
[
1461,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1455,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_self_eq_norm_sq
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2297537\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\n⊢ ↑re (inner x x) = ‖x‖ ^ 2",
"tactic": "rw [pow_two, inner_self_eq_norm_mul_norm]"
}
] |
[
1014,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.filter_filterMap
|
[] |
[
2159,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2157,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Balanced.zoom
|
[
{
"state_after": "case refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nhp : Path.Balanced c₀ n₀ path black 0\n⊢ ∃ c n, Balanced nil c n ∧ Path.Balanced c₀ n₀ path c n",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nhp : Path.Balanced c₀ n₀ path black 0\ne : zoom cut nil path = (t', path')\n⊢ ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nhp : Path.Balanced c₀ n₀ path black 0\n⊢ ∃ c n, Balanced nil c n ∧ Path.Balanced c₀ n₀ path c n",
"tactic": "exact ⟨_, _, .nil, hp⟩"
},
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut x✝ (left red path v✝ y✝)\n | Ordering.gt => zoom cut y✝ (right red x✝ v✝ path)\n | Ordering.eq => (node red x✝ v✝ y✝, path)) =\n (t', path') →\n ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\n⊢ zoom cut (node red x✝ v✝ y✝) path = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "unfold zoom"
},
{
"state_after": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut x✝¹ (left red path v✝ y✝) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n\n\ncase h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut y✝ (right red x✝¹ v✝ path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n\n\ncase h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node red x✝¹ v✝ y✝, path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut x✝ (left red path v✝ y✝)\n | Ordering.gt => zoom cut y✝ (right red x✝ v✝ path)\n | Ordering.eq => (node red x✝ v✝ y✝, path)) =\n (t', path') →\n ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut x✝¹ (left red path v✝ y✝) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "exact ha.zoom (.redL hb hp)"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut y✝ (right red x✝¹ v✝ path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "exact hb.zoom (.redR ha hp)"
},
{
"state_after": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node red x✝¹ v✝ y✝, path) = (t', path')\n⊢ ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node red x✝¹ v✝ y✝, path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "intro e"
},
{
"state_after": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ c n, Balanced (node red x✝¹ v✝ y✝) c n ∧ Path.Balanced c₀ n₀ path c n",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node red x✝¹ v✝ y✝, path) = (t', path')\n⊢ ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn✝ n : Nat\nx✝¹ y✝ : RBNode α✝\nv✝ : α✝\nha : Balanced x✝¹ black n\nhb : Balanced y✝ black n\nhp : Path.Balanced c₀ n₀ path red n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ c n, Balanced (node red x✝¹ v✝ y✝) c n ∧ Path.Balanced c₀ n₀ path c n",
"tactic": "exact ⟨_, _, .red ha hb, hp⟩"
},
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut x✝ (left black path v✝ y✝)\n | Ordering.gt => zoom cut y✝ (right black x✝ v✝ path)\n | Ordering.eq => (node black x✝ v✝ y✝, path)) =\n (t', path') →\n ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\n⊢ zoom cut (node black x✝ v✝ y✝) path = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "unfold zoom"
},
{
"state_after": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut x✝¹ (left black path v✝ y✝) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n\n\ncase h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut y✝ (right black x✝¹ v✝ path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n\n\ncase h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node black x✝¹ v✝ y✝, path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut x✝ (left black path v✝ y✝)\n | Ordering.gt => zoom cut y✝ (right black x✝ v✝ path)\n | Ordering.eq => (node black x✝ v✝ y✝, path)) =\n (t', path') →\n ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut x✝¹ (left black path v✝ y✝) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "exact ha.zoom (.blackL hb hp)"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut y✝ (right black x✝¹ v✝ path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "exact hb.zoom (.blackR ha hp)"
},
{
"state_after": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node black x✝¹ v✝ y✝, path) = (t', path')\n⊢ ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node black x✝¹ v✝ y✝, path) = (t', path') → ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "intro e"
},
{
"state_after": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ c n, Balanced (node black x✝¹ v✝ y✝) c n ∧ Path.Balanced c₀ n₀ path c n",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\ne : (node black x✝¹ v✝ y✝, path) = (t', path')\n⊢ ∃ c n, Balanced t' c n ∧ Path.Balanced c₀ n₀ path' c n",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case h_3.refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nc₀ : RBColor\nn₀ : Nat\nc : RBColor\nn : Nat\nx✝¹ : RBNode α✝\nc₁✝ : RBColor\nn✝ : Nat\ny✝ : RBNode α✝\nc₂✝ : RBColor\nv✝ : α✝\nha : Balanced x✝¹ c₁✝ n✝\nhb : Balanced y✝ c₂✝ n✝\nhp : Path.Balanced c₀ n₀ path black (n✝ + 1)\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ ∃ c n, Balanced (node black x✝¹ v✝ y✝) c n ∧ Path.Balanced c₀ n₀ path c n",
"tactic": "exact ⟨_, _, .black ha hb, hp⟩"
}
] |
[
135,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
123,
11
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.comap_codRestrict
|
[
{
"state_after": "case intro.mk.intro.refl\nR : Type u_1\nR₁ : Type ?u.1019468\nR₂ : Type u_3\nR₃ : Type ?u.1019474\nR₄ : Type ?u.1019477\nS : Type ?u.1019480\nK : Type ?u.1019483\nK₂ : Type ?u.1019486\nM : Type u_2\nM' : Type ?u.1019492\nM₁ : Type ?u.1019495\nM₂ : Type u_4\nM₃ : Type ?u.1019501\nM₄ : Type ?u.1019504\nN : Type ?u.1019507\nN₂ : Type ?u.1019510\nι : Type ?u.1019513\nV : Type ?u.1019516\nV₂ : Type ?u.1019519\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring R₃\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\np : Submodule R M\nf : M₂ →ₛₗ[σ₂₁] M\nhf : ∀ (c : M₂), ↑f c ∈ p\np' : Submodule R { x // x ∈ p }\nx : M₂\nproperty✝ : f.toAddHom.1 x ∈ p\nh : { val := f.toAddHom.1 x, property := property✝ } ∈ ↑p'\n⊢ x ∈ comap (codRestrict p f hf) p'",
"state_before": "R : Type u_1\nR₁ : Type ?u.1019468\nR₂ : Type u_3\nR₃ : Type ?u.1019474\nR₄ : Type ?u.1019477\nS : Type ?u.1019480\nK : Type ?u.1019483\nK₂ : Type ?u.1019486\nM : Type u_2\nM' : Type ?u.1019492\nM₁ : Type ?u.1019495\nM₂ : Type u_4\nM₃ : Type ?u.1019501\nM₄ : Type ?u.1019504\nN : Type ?u.1019507\nN₂ : Type ?u.1019510\nι : Type ?u.1019513\nV : Type ?u.1019516\nV₂ : Type ?u.1019519\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring R₃\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\np : Submodule R M\nf : M₂ →ₛₗ[σ₂₁] M\nhf : ∀ (c : M₂), ↑f c ∈ p\np' : Submodule R { x // x ∈ p }\nx : M₂\n⊢ x ∈ comap f (map (Submodule.subtype p) p') → x ∈ comap (codRestrict p f hf) p'",
"tactic": "rintro ⟨⟨_, _⟩, h, ⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.refl\nR : Type u_1\nR₁ : Type ?u.1019468\nR₂ : Type u_3\nR₃ : Type ?u.1019474\nR₄ : Type ?u.1019477\nS : Type ?u.1019480\nK : Type ?u.1019483\nK₂ : Type ?u.1019486\nM : Type u_2\nM' : Type ?u.1019492\nM₁ : Type ?u.1019495\nM₂ : Type u_4\nM₃ : Type ?u.1019501\nM₄ : Type ?u.1019504\nN : Type ?u.1019507\nN₂ : Type ?u.1019510\nι : Type ?u.1019513\nV : Type ?u.1019516\nV₂ : Type ?u.1019519\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring R₃\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\np : Submodule R M\nf : M₂ →ₛₗ[σ₂₁] M\nhf : ∀ (c : M₂), ↑f c ∈ p\np' : Submodule R { x // x ∈ p }\nx : M₂\nproperty✝ : f.toAddHom.1 x ∈ p\nh : { val := f.toAddHom.1 x, property := property✝ } ∈ ↑p'\n⊢ x ∈ comap (codRestrict p f hf) p'",
"tactic": "exact h"
}
] |
[
1188,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1186,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.uniformContinuous_infNndist_pt
|
[] |
[
674,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
673,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.swap_swap
|
[] |
[
163,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso_hom_desc
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nr s t : Cocone F\nP : IsColimit s\nQ : IsColimit t\n⊢ ∀ (j : J), s.ι.app j ≫ (coconePointUniqueUpToIso P Q).hom ≫ desc Q r = r.ι.app j",
"tactic": "simp"
}
] |
[
677,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.pointwise_smul_le_pointwise_smul_iff₀
|
[] |
[
169,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.disjoint_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\n⊢ (∀ (a : β) (x : α), x ∈ s → ↑f x = a → ∀ (b : β) (x : α), x ∈ t → ↑f x = b → a ≠ b) ↔\n ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\n⊢ _root_.Disjoint (map f s) (map f t) ↔ _root_.Disjoint s t",
"tactic": "simp only [disjoint_iff_ne, mem_map, exists_prop, exists_imp, and_imp]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b) →\n ∀ (a : β) (x : α), x ∈ s → ↑f x = a → ∀ (b : β) (x : α), x ∈ t → ↑f x = b → a ≠ b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\n⊢ (∀ (a : β) (x : α), x ∈ s → ↑f x = a → ∀ (b : β) (x : α), x ∈ t → ↑f x = b → a ≠ b) ↔\n ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b",
"tactic": "refine' ⟨fun h a ha b hb hab => h _ _ ha rfl _ _ hb rfl <| congr_arg _ hab, _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\nh : ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\n⊢ ↑f a ≠ ↑f b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b) →\n ∀ (a : β) (x : α), x ∈ s → ↑f x = a → ∀ (b : β) (x : α), x ∈ t → ↑f x = b → a ≠ b",
"tactic": "rintro h _ a ha rfl _ b hb rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31419\nf✝ : α ↪ β\ns✝ s t : Finset α\nf : α ↪ β\nh : ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\n⊢ ↑f a ≠ ↑f b",
"tactic": "exact f.injective.ne (h _ ha _ hb)"
}
] |
[
198,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Order/Antichain.lean
|
isAntichain_and_least_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.11078\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\nt : Set α\na b : α\ninst✝ : Preorder α\n⊢ IsAntichain (fun x x_1 => x ≤ x_1) {a} ∧ IsLeast {a} a",
"state_before": "α : Type u_1\nβ : Type ?u.11078\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\ns t : Set α\na b : α\ninst✝ : Preorder α\n⊢ s = {a} → IsAntichain (fun x x_1 => x ≤ x_1) s ∧ IsLeast s a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11078\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\nt : Set α\na b : α\ninst✝ : Preorder α\n⊢ IsAntichain (fun x x_1 => x ≤ x_1) {a} ∧ IsLeast {a} a",
"tactic": "exact ⟨isAntichain_singleton _ _, isLeast_singleton⟩"
}
] |
[
227,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
FiniteDimensional.basisUnique.repr_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type u_1\ninst✝ : Unique ι\nh : finrank K V = 1\nv : V\ni : ι\nhv : v = 0\n⊢ ↑(↑(basisUnique ι h).repr v) i = 0",
"tactic": "rw [hv, LinearEquiv.map_zero, Finsupp.zero_apply]"
}
] |
[
274,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.add_imJ
|
[] |
[
893,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
9
] |
Mathlib/Data/W/Basic.lean
|
WType.toSigma_ofSigma
|
[] |
[
70,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_quot_le
|
[] |
[
1966,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1965,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.coe_map
|
[] |
[
1516,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1515,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.nat_le_order
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : ¬↑n ≤ order φ\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\n⊢ ↑n ≤ order φ",
"tactic": "by_contra H"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : order φ < ↑n\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : ¬↑n ≤ order φ\n⊢ False",
"tactic": "rw [not_le] at H"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : order φ < ↑n\nthis : (order φ).Dom\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : order φ < ↑n\n⊢ False",
"tactic": "have : (order φ).Dom := PartENat.dom_of_le_natCast H.le"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nthis : (order φ).Dom\nH : Part.get (order φ) this < n\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nH : order φ < ↑n\nthis : (order φ).Dom\n⊢ False",
"tactic": "rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ : PowerSeries R\nn : ℕ\nh : ∀ (i : ℕ), i < n → ↑(coeff R i) φ = 0\nthis : (order φ).Dom\nH : Part.get (order φ) this < n\n⊢ False",
"tactic": "exact coeff_order this (h _ H)"
}
] |
[
2296,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2292,
1
] |
Mathlib/Logic/Equiv/List.lean
|
Denumerable.raise_lower
|
[
{
"state_after": "α : Type ?u.45406\nβ : Type ?u.45409\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nm : ℕ\nl : List ℕ\nn : ℕ\nh : Sorted (fun x x_1 => x ≤ x_1) (n :: m :: l)\nthis : n ≤ m\n⊢ raise (lower (m :: l) n) n = m :: l",
"state_before": "α : Type ?u.45406\nβ : Type ?u.45409\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nm : ℕ\nl : List ℕ\nn : ℕ\nh : Sorted (fun x x_1 => x ≤ x_1) (n :: m :: l)\n⊢ raise (lower (m :: l) n) n = m :: l",
"tactic": "have : n ≤ m := List.rel_of_sorted_cons h _ (l.mem_cons_self _)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.45406\nβ : Type ?u.45409\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nm : ℕ\nl : List ℕ\nn : ℕ\nh : Sorted (fun x x_1 => x ≤ x_1) (n :: m :: l)\nthis : n ≤ m\n⊢ raise (lower (m :: l) n) n = m :: l",
"tactic": "simp [raise, lower, tsub_add_cancel_of_le this, raise_lower h.of_cons]"
}
] |
[
313,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.isImage_source_target_of_disjoint
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31535\nδ : Type ?u.31538\ne : LocalEquiv α β\ne'✝ : LocalEquiv β γ\ne' : LocalEquiv α β\nhs : Disjoint e.source e'.source\nht : Disjoint e.target e'.target\n⊢ ↑e '' (e.source ∩ e'.source) = e.target ∩ e'.target",
"tactic": "rw [hs.inter_eq, ht.inter_eq, image_empty]"
}
] |
[
482,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.div_mem_div
|
[] |
[
447,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.nnnorm_lt_iff_of_nonempty
|
[] |
[
247,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.algHom_ext
|
[
{
"state_after": "case H\nS : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type ?u.381212\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ ψ : Unitization R A →ₐ[S] B\nh : ∀ (a : A), ↑φ ↑a = ↑ψ ↑a\nh' : ∀ (r : R), ↑φ (↑(algebraMap R (Unitization R A)) r) = ↑ψ (↑(algebraMap R (Unitization R A)) r)\nx : Unitization R A\n⊢ ↑φ x = ↑ψ x",
"state_before": "S : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type ?u.381212\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ ψ : Unitization R A →ₐ[S] B\nh : ∀ (a : A), ↑φ ↑a = ↑ψ ↑a\nh' : ∀ (r : R), ↑φ (↑(algebraMap R (Unitization R A)) r) = ↑ψ (↑(algebraMap R (Unitization R A)) r)\n⊢ φ = ψ",
"tactic": "ext x"
},
{
"state_after": "case H.h\nS : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type ?u.381212\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ ψ : Unitization R A →ₐ[S] B\nh : ∀ (a : A), ↑φ ↑a = ↑ψ ↑a\nh' : ∀ (r : R), ↑φ (↑(algebraMap R (Unitization R A)) r) = ↑ψ (↑(algebraMap R (Unitization R A)) r)\nr✝ : R\na✝ : A\n⊢ ↑φ (inl r✝ + ↑a✝) = ↑ψ (inl r✝ + ↑a✝)",
"state_before": "case H\nS : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type ?u.381212\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ ψ : Unitization R A →ₐ[S] B\nh : ∀ (a : A), ↑φ ↑a = ↑ψ ↑a\nh' : ∀ (r : R), ↑φ (↑(algebraMap R (Unitization R A)) r) = ↑ψ (↑(algebraMap R (Unitization R A)) r)\nx : Unitization R A\n⊢ ↑φ x = ↑ψ x",
"tactic": "induction x using Unitization.ind"
},
{
"state_after": "no goals",
"state_before": "case H.h\nS : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMulAction S A\ninst✝² : IsScalarTower S R A\nC : Type ?u.381212\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ ψ : Unitization R A →ₐ[S] B\nh : ∀ (a : A), ↑φ ↑a = ↑ψ ↑a\nh' : ∀ (r : R), ↑φ (↑(algebraMap R (Unitization R A)) r) = ↑ψ (↑(algebraMap R (Unitization R A)) r)\nr✝ : R\na✝ : A\n⊢ ↑φ (inl r✝ + ↑a✝) = ↑ψ (inl r✝ + ↑a✝)",
"tactic": "simp only [map_add, ← algebraMap_eq_inl, h, h']"
}
] |
[
643,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finSuccEquiv'_symm_coe_below
|
[] |
[
184,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.infDist_le_infDist_add_dist
|
[
{
"state_after": "ι : Sort ?u.61351\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ ENNReal.toReal (infEdist x s) ≤ ENNReal.toReal (infEdist y s) + ENNReal.toReal (edist x y)",
"state_before": "ι : Sort ?u.61351\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infDist x s ≤ infDist y s + dist x y",
"tactic": "rw [infDist, infDist, dist_edist]"
},
{
"state_after": "ι : Sort ?u.61351\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infEdist y s = ⊤ → infEdist x s = ⊤",
"state_before": "ι : Sort ?u.61351\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ ENNReal.toReal (infEdist x s) ≤ ENNReal.toReal (infEdist y s) + ENNReal.toReal (edist x y)",
"tactic": "refine ENNReal.toReal_le_add' infEdist_le_infEdist_add_edist ?_ (flip absurd (edist_ne_top _ _))"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.61351\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infEdist y s = ⊤ → infEdist x s = ⊤",
"tactic": "simp only [infEdist_eq_top_iff, imp_self]"
}
] |
[
530,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\n⊢ f ≤ᵐ[μ] g",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\n⊢ f ≤ᵐ[μ] g",
"tactic": "have A :\n ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p) = 0 := by\n intro ε N p εpos\n let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p\n have s_meas : MeasurableSet s := by\n have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf\n have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const\n exact (A.inter B).inter (measurable_spanningSets μ p)\n have s_lt_top : μ s < ∞ :=\n (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top μ p)\n have A : (∫⁻ x in s, g x ∂μ) + ε * μ s ≤ (∫⁻ x in s, g x ∂μ) + 0 :=\n calc\n (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ _ in s, ε ∂μ := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm\n _ ≤ ∫⁻ x in s, f x ∂μ :=\n (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top\n have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by\n apply ne_of_lt\n calc\n (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=\n set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * μ s := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < ∞ := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]\n have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A\n simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\n⊢ f ≤ᵐ[μ] g",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\n⊢ f ≤ᵐ[μ] g",
"tactic": "obtain ⟨u, _, u_pos, u_lim⟩ :\n ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (nhds 0) :=\n exists_seq_strictAnti_tendsto (0 : ℝ≥0)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\n⊢ f ≤ᵐ[μ] g",
"state_before": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\n⊢ f ≤ᵐ[μ] g",
"tactic": "let s := fun n : ℕ => {x | g x + u n ≤ f x ∧ g x ≤ (n : ℝ≥0)} ∩ spanningSets μ n"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\n⊢ f ≤ᵐ[μ] g",
"state_before": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\n⊢ f ≤ᵐ[μ] g",
"tactic": "have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nB : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n\n⊢ f ≤ᵐ[μ] g",
"state_before": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\n⊢ f ≤ᵐ[μ] g",
"tactic": "have B : {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n := by\n intro x hx\n simp at hx\n have L1 : ∀ᶠ n in atTop, g x + u n ≤ f x := by\n have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=\n tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n simp at this\n exact eventually_le_of_tendsto_lt hx this\n have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=\n haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by\n simp only [ENNReal.coe_nat]\n exact ENNReal.tendsto_nat_nhds_top\n eventually_ge_of_tendsto_gt (hx.trans_le le_top) this\n apply Set.mem_iUnion.2\n exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nB : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n\n⊢ ↑↑μ ({x | (fun x => f x ≤ g x) x}ᶜ) ≤ 0",
"state_before": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nB : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n\n⊢ f ≤ᵐ[μ] g",
"tactic": "refine' le_antisymm _ bot_le"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nB : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n\n⊢ ↑↑μ ({x | (fun x => f x ≤ g x) x}ᶜ) ≤ 0",
"tactic": "calc\n μ ({x : α | (fun x : α => f x ≤ g x) x}ᶜ) ≤ μ (⋃ n, s n) := measure_mono B\n _ ≤ ∑' n, μ (s n) := (measure_iUnion_le _)\n _ = 0 := by simp only [μs, tsum_zero]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\n⊢ ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "intro ε N p εpos"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "have s_meas : MeasurableSet s := by\n have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf\n have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const\n exact (A.inter B).inter (measurable_spanningSets μ p)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "have s_lt_top : μ s < ∞ :=\n (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top μ p)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "have A : (∫⁻ x in s, g x ∂μ) + ε * μ s ≤ (∫⁻ x in s, g x ∂μ) + 0 :=\n calc\n (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ _ in s, ε ∂μ := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm\n _ ≤ ∫⁻ x in s, f x ∂μ :=\n (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\nB : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by\n apply ne_of_lt\n calc\n (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=\n set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * μ s := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < ∞ := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\nB : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤\nthis : ↑ε * ↑↑μ s ≤ 0\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\nB : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\nB : (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤\nthis : ↑ε * ↑↑μ s ≤ 0\n⊢ ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0",
"tactic": "simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\nA : MeasurableSet {x | g x + ↑ε ≤ f x}\n⊢ MeasurableSet s",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\n⊢ MeasurableSet s",
"tactic": "have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\nA : MeasurableSet {x | g x + ↑ε ≤ f x}\nB : MeasurableSet {x | g x ≤ ↑N}\n⊢ MeasurableSet s",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\nA : MeasurableSet {x | g x + ↑ε ≤ f x}\n⊢ MeasurableSet s",
"tactic": "have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\nA : MeasurableSet {x | g x + ↑ε ≤ f x}\nB : MeasurableSet {x | g x ≤ ↑N}\n⊢ MeasurableSet s",
"tactic": "exact (A.inter B).inter (measurable_spanningSets μ p)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s = (∫⁻ (x : α) in s, g x ∂μ) + ∫⁻ (x : α) in s, ↑ε ∂μ",
"tactic": "simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0",
"tactic": "rw [add_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\n⊢ (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ",
"tactic": "exact h s s_meas s_lt_top"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ (∫⁻ (x : α) in s, g x ∂μ) < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ (∫⁻ (x : α) in s, g x ∂μ) ≠ ⊤",
"tactic": "apply ne_of_lt"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ (∫⁻ (x : α) in s, g x ∂μ) < ⊤",
"tactic": "calc\n (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ _ in s, N ∂μ :=\n set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * μ s := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < ∞ := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ (∫⁻ (x : α) in s, ↑N ∂μ) = ↑N * ↑↑μ s",
"tactic": "simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np✝ : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nε N : ℝ≥0\np : ℕ\nεpos : 0 < ε\ns : Set α := {x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p\ns_meas : MeasurableSet s\ns_lt_top : ↑↑μ s < ⊤\nA : (∫⁻ (x : α) in s, g x ∂μ) + ↑ε * ↑↑μ s ≤ (∫⁻ (x : α) in s, g x ∂μ) + 0\n⊢ ↑N * ↑↑μ s < ⊤",
"tactic": "simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : x ∈ {x | f x ≤ g x}ᶜ\n⊢ x ∈ ⋃ (n : ℕ), s n",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\n⊢ {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\n⊢ x ∈ ⋃ (n : ℕ), s n",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : x ∈ {x | f x ≤ g x}ᶜ\n⊢ x ∈ ⋃ (n : ℕ), s n",
"tactic": "simp at hx"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\n⊢ x ∈ ⋃ (n : ℕ), s n",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\n⊢ x ∈ ⋃ (n : ℕ), s n",
"tactic": "have L1 : ∀ᶠ n in atTop, g x + u n ≤ f x := by\n have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=\n tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n simp at this\n exact eventually_le_of_tendsto_lt hx this"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\nL2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n\n⊢ x ∈ ⋃ (n : ℕ), s n",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\n⊢ x ∈ ⋃ (n : ℕ), s n",
"tactic": "have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=\n haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by\n simp only [ENNReal.coe_nat]\n exact ENNReal.tendsto_nat_nhds_top\n eventually_ge_of_tendsto_gt (hx.trans_le le_top) this"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\nL2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n\n⊢ ∃ i, x ∈ s i",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\nL2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n\n⊢ x ∈ ⋃ (n : ℕ), s n",
"tactic": "apply Set.mem_iUnion.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\nL2 : ∀ᶠ (n : ℕ) in atTop, g x ≤ ↑↑n\n⊢ ∃ i, x ∈ s i",
"tactic": "exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nthis : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x + ↑0))\n⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\n⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x",
"tactic": "have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=\n tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nthis : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x))\n⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nthis : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x + ↑0))\n⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x",
"tactic": "simp at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nthis : Tendsto (fun n => g x + ↑(u n)) atTop (𝓝 (g x))\n⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x",
"tactic": "exact eventually_le_of_tendsto_lt hx this"
},
{
"state_after": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\n⊢ Tendsto (fun n => ↑n) atTop (𝓝 ⊤)",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\n⊢ Tendsto (fun n => ↑↑n) atTop (𝓝 ⊤)",
"tactic": "simp only [ENNReal.coe_nat]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nx : α\nhx : g x < f x\nL1 : ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x\n⊢ Tendsto (fun n => ↑n) atTop (𝓝 ⊤)",
"tactic": "exact ENNReal.tendsto_nat_nhds_top"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.48454\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\np : ℝ≥0∞\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → ↑↑μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ → ℝ≥0\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\ns : ℕ → Set α := fun n => {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n\nμs : ∀ (n : ℕ), ↑↑μ (s n) = 0\nB : {x | f x ≤ g x}ᶜ ⊆ ⋃ (n : ℕ), s n\n⊢ (∑' (n : ℕ), ↑↑μ (s n)) = 0",
"tactic": "simp only [μs, tsum_zero]"
}
] |
[
225,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_monomial_mul'
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m (p * ↑(monomial s) r) = if s ≤ m then coeff (m - s) p * r else 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m (↑(monomial s) r * p) = if s ≤ m then r * coeff (m - s) p else 0",
"tactic": "rw [mul_comm, mul_comm r]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\n⊢ coeff m (p * ↑(monomial s) r) = if s ≤ m then coeff (m - s) p * r else 0",
"tactic": "exact coeff_mul_monomial' _ _ _ _"
}
] |
[
790,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
786,
1
] |
Mathlib/Algebra/GradedMulAction.lean
|
SetLike.coe_GSmul
|
[] |
[
142,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.swapCore_self
|
[
{
"state_after": "α : Sort u_1\ninst✝ : DecidableEq α\nr a : α\n⊢ (if r = a then a else if r = a then a else r) = r",
"state_before": "α : Sort u_1\ninst✝ : DecidableEq α\nr a : α\n⊢ swapCore a a r = r",
"tactic": "unfold swapCore"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\ninst✝ : DecidableEq α\nr a : α\n⊢ (if r = a then a else if r = a then a else r) = r",
"tactic": "split_ifs <;> simp [*]"
}
] |
[
1517,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1515,
1
] |
Mathlib/AlgebraicGeometry/RingedSpace.lean
|
AlgebraicGeometry.RingedSpace.mem_basicOpen
|
[
{
"state_after": "case mp\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\n⊢ ↑x ∈ basicOpen X f → IsUnit (↑(germ X.presheaf x) f)\n\ncase mpr\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\n⊢ IsUnit (↑(germ X.presheaf x) f) → ↑x ∈ basicOpen X f",
"state_before": "X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\n⊢ ↑x ∈ basicOpen X f ↔ IsUnit (↑(germ X.presheaf x) f)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx✝ x : { x // x ∈ U }\nhx : x ∈ {x | IsUnit (↑(germ X.presheaf x) f)}\na : ↑x = ↑x✝\n⊢ IsUnit (↑(germ X.presheaf x✝) f)",
"state_before": "case mp\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\n⊢ ↑x ∈ basicOpen X f → IsUnit (↑(germ X.presheaf x) f)",
"tactic": "rintro ⟨x, hx, a⟩"
},
{
"state_after": "case mp.intro.intro.refl\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : x ∈ {x | IsUnit (↑(germ X.presheaf x) f)}\na : ↑x = ↑x\n⊢ IsUnit (↑(germ X.presheaf x) f)",
"state_before": "case mp.intro.intro\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx✝ x : { x // x ∈ U }\nhx : x ∈ {x | IsUnit (↑(germ X.presheaf x) f)}\na : ↑x = ↑x✝\n⊢ IsUnit (↑(germ X.presheaf x✝) f)",
"tactic": "cases Subtype.eq a"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refl\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nhx : x ∈ {x | IsUnit (↑(germ X.presheaf x) f)}\na : ↑x = ↑x\n⊢ IsUnit (↑(germ X.presheaf x) f)",
"tactic": "exact hx"
},
{
"state_after": "case mpr\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nh : IsUnit (↑(germ X.presheaf x) f)\n⊢ ↑x ∈ basicOpen X f",
"state_before": "case mpr\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\n⊢ IsUnit (↑(germ X.presheaf x) f) → ↑x ∈ basicOpen X f",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj U.op)\nx : { x // x ∈ U }\nh : IsUnit (↑(germ X.presheaf x) f)\n⊢ ↑x ∈ basicOpen X f",
"tactic": "exact ⟨x, h, rfl⟩"
}
] |
[
171,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.map_id
|
[] |
[
414,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/RingTheory/LaurentSeries.lean
|
LaurentSeries.coeff_coe_powerSeries
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : Semiring R\nx : PowerSeries R\nn : ℕ\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) x) ↑n = ↑(PowerSeries.coeff R n) x",
"tactic": "rw [ofPowerSeries_apply_coeff]"
}
] |
[
59,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
IsCompact.measure_lt_top
|
[] |
[
3940,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3938,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.smul_modByMonic
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\n⊢ c • p %ₘ q = c • (p %ₘ q)\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : ¬Monic q\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"tactic": "by_cases hq : q.Monic"
},
{
"state_after": "case pos.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\nhR : Subsingleton R\n⊢ c • p %ₘ q = c • (p %ₘ q)\n\ncase pos.inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\nhR : Nontrivial R\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"state_before": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"tactic": "cases' subsingleton_or_nontrivial R with hR hR"
},
{
"state_after": "no goals",
"state_before": "case pos.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\nhR : Subsingleton R\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"tactic": "simp only [eq_iff_true_of_subsingleton]"
},
{
"state_after": "no goals",
"state_before": "case pos.inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\nhR : Nontrivial R\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"tactic": "exact\n(div_modByMonic_unique (c • (p /ₘ q)) (c • (p %ₘ q)) hq\n ⟨by rw [mul_smul_comm, ← smul_add, modByMonic_add_div p hq],\n (degree_smul_le _ _).trans_lt (degree_modByMonic_lt _ hq)⟩).2"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : Monic q\nhR : Nontrivial R\n⊢ c • (p %ₘ q) + q * c • (p /ₘ q) = c • p",
"tactic": "rw [mul_smul_comm, ← smul_add, modByMonic_add_div p hq]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np✝ q : R[X]\ninst✝ : Semiring S\nc : R\np : R[X]\nhq : ¬Monic q\n⊢ c • p %ₘ q = c • (p %ₘ q)",
"tactic": "simp_rw [modByMonic_eq_of_not_monic _ hq]"
}
] |
[
98,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
|
MeasureTheory.SimpleFunc.approxOn_zero
|
[] |
[
139,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/Cover.lean
|
covby_congr_left
|
[] |
[
318,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Std/Logic.lean
|
false_iff_true
|
[] |
[
96,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
96,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.gcd_eq_right_iff_dvd
|
[
{
"state_after": "m n : ℕ+\n⊢ m ∣ n ↔ gcd m n = m",
"state_before": "m n : ℕ+\n⊢ m ∣ n ↔ gcd n m = m",
"tactic": "rw [gcd_comm]"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ+\n⊢ m ∣ n ↔ gcd m n = m",
"tactic": "apply gcd_eq_left_iff_dvd"
}
] |
[
204,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.coe_eq_pure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43617\nγ : Type ?u.43620\nδ : Type ?u.43623\nι : Sort ?u.43626\nκ : ι → Sort ?u.43631\ninst✝ : Preorder α\ns : NonemptyInterval α\na : α\n⊢ ↑s = Interval.pure a ↔ s = pure a",
"tactic": "rw [← Interval.coe_inj, coe_pure_interval]"
}
] |
[
645,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Data/List/Basic.lean
|
List.sizeOf_lt_sizeOf_of_mem
|
[
{
"state_after": "case cons.head\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\n⊢ sizeOf x < 1 + sizeOf x + sizeOf t\n\ncase cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ sizeOf x < 1 + sizeOf h + sizeOf t",
"state_before": "ι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\n⊢ sizeOf x < sizeOf l",
"tactic": "induction' l with h t ih <;> cases hx <;> rw [cons.sizeOf_spec]"
},
{
"state_after": "no goals",
"state_before": "case cons.head\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\n⊢ sizeOf x < 1 + sizeOf x + sizeOf t",
"tactic": "exact lt_add_of_lt_of_nonneg (lt_one_add _) (Nat.zero_le _)"
},
{
"state_after": "case cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ 0 < 1 + sizeOf h",
"state_before": "case cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ sizeOf x < 1 + sizeOf h + sizeOf t",
"tactic": "refine lt_add_of_pos_of_le ?_ (le_of_lt (ih ‹_›))"
},
{
"state_after": "case cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ 0 < sizeOf h + 1",
"state_before": "case cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ 0 < 1 + sizeOf h",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "case cons.tail\nι : Type ?u.320638\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nx : α\nl : List α\nhx : x ∈ l\nh : α\nt : List α\nih : x ∈ t → sizeOf x < sizeOf t\na✝ : Mem x t\n⊢ 0 < sizeOf h + 1",
"tactic": "exact succ_pos _"
}
] |
[
3084,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3079,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.one_sub_lt_exp_minus_of_pos
|
[
{
"state_after": "case inl\ny : ℝ\nh : 0 < y\nh' : 1 ≤ y\n⊢ 1 - y < exp (-y)\n\ncase inr\ny : ℝ\nh : 0 < y\nh' : y < 1\n⊢ 1 - y < exp (-y)",
"state_before": "y : ℝ\nh : 0 < y\n⊢ 1 - y < exp (-y)",
"tactic": "cases' le_or_lt 1 y with h' h'"
},
{
"state_after": "case inr\ny : ℝ\nh : 0 < y\nh' : y < 1\n⊢ exp y < 1 / (1 - y)\n\ny : ℝ\nh : 0 < y\nh' : y < 1\n⊢ 0 < 1 - y",
"state_before": "case inr\ny : ℝ\nh : 0 < y\nh' : y < 1\n⊢ 1 - y < exp (-y)",
"tactic": "rw [exp_neg, lt_inv _ y.exp_pos, inv_eq_one_div]"
},
{
"state_after": "no goals",
"state_before": "case inl\ny : ℝ\nh : 0 < y\nh' : 1 ≤ y\n⊢ 1 - y < exp (-y)",
"tactic": "linarith [(-y).exp_pos]"
},
{
"state_after": "no goals",
"state_before": "case inr\ny : ℝ\nh : 0 < y\nh' : y < 1\n⊢ exp y < 1 / (1 - y)",
"tactic": "exact exp_bound_div_one_sub_of_interval' h h'"
},
{
"state_after": "no goals",
"state_before": "y : ℝ\nh : 0 < y\nh' : y < 1\n⊢ 0 < 1 - y",
"tactic": "linarith"
}
] |
[
1967,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1962,
1
] |
Mathlib/Computability/Language.lean
|
Language.mem_pow
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero ↔ ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l\n\ncase succ\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ x ∈ l ^ Nat.succ n ↔ ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"state_before": "α : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nn : ℕ\n⊢ x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "induction' n with n ihn generalizing x"
},
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero ↔ ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero ↔ ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "simp only [mem_one, pow_zero, length_eq_zero]"
},
{
"state_after": "case zero.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero → ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l\n\ncase zero.mpr\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ (∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l) → x ∈ l ^ Nat.zero",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero ↔ ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "constructor"
},
{
"state_after": "case zero.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\n⊢ ∃ S, [] = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"state_before": "case zero.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ x ∈ l ^ Nat.zero → ∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case zero.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\n⊢ ∃ S, [] = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "exact ⟨[], rfl, rfl, fun _ h ↦ by contradiction⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx x✝ : List α\nh : x✝ ∈ []\n⊢ x✝ ∈ l",
"tactic": "contradiction"
},
{
"state_after": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\n⊢ join w✝ ∈ l ^ Nat.zero",
"state_before": "case zero.mpr\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ x : List α\n⊢ (∃ S, x = join S ∧ length S = Nat.zero ∧ ∀ (y : List α), y ∈ S → y ∈ l) → x ∈ l ^ Nat.zero",
"tactic": "rintro ⟨_, rfl, h₀, _⟩"
},
{
"state_after": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\n⊢ ∀ (l : List α), l ∈ w✝ → l = []",
"state_before": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\n⊢ join w✝ ∈ l ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ w✝\n⊢ l✝ = []",
"state_before": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\n⊢ ∀ (l : List α), l ∈ w✝ → l = []",
"tactic": "intros _ h₁"
},
{
"state_after": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : w✝ = []\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ w✝\n⊢ l✝ = []",
"state_before": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : length w✝ = Nat.zero\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ w✝\n⊢ l✝ = []",
"tactic": "rw [length_eq_zero] at h₀"
},
{
"state_after": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : w✝ = []\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ []\n⊢ l✝ = []",
"state_before": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : w✝ = []\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ w✝\n⊢ l✝ = []",
"tactic": "rw [h₀] at h₁"
},
{
"state_after": "no goals",
"state_before": "case zero.mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝¹ m : Language α\na b x✝ : List α\nl : Language α\nx : List α\nw✝ : List (List α)\nh₀ : w✝ = []\nright✝ : ∀ (y : List α), y ∈ w✝ → y ∈ l\nl✝ : List α\nh₁ : l✝ ∈ []\n⊢ l✝ = []",
"tactic": "contradiction"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x) ↔\n ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ x ∈ l ^ Nat.succ n ↔ ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "simp only [pow_succ, mem_mul, ihn]"
},
{
"state_after": "case succ.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x) →\n ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l\n\ncase succ.mpr\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l) →\n ∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x) ↔\n ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "constructor"
},
{
"state_after": "case succ.mp.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nha : a ∈ l\nS : List (List α)\nhS : ∀ (y : List α), y ∈ S → y ∈ l\nihn : ∀ {x : List α}, x ∈ l ^ length S ↔ ∃ S_1, x = join S_1 ∧ length S_1 = length S ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ S_1, a ++ join S = join S_1 ∧ length S_1 = Nat.succ (length S) ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l",
"state_before": "case succ.mp\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x) →\n ∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l",
"tactic": "rintro ⟨a, b, ha, ⟨S, rfl, rfl, hS⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case succ.mp.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nha : a ∈ l\nS : List (List α)\nhS : ∀ (y : List α), y ∈ S → y ∈ l\nihn : ∀ {x : List α}, x ∈ l ^ length S ↔ ∃ S_1, x = join S_1 ∧ length S_1 = length S ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ S_1, a ++ join S = join S_1 ∧ length S_1 = Nat.succ (length S) ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l",
"tactic": "exact ⟨a :: S, rfl, rfl, forall_mem_cons.2 ⟨ha, hS⟩⟩"
},
{
"state_after": "case succ.mpr.intro.cons.intro.intro.refl\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nS : List (List α)\nhS : ∀ (y : List α), y ∈ a :: S → y ∈ l\nihn :\n ∀ {x : List α},\n x ∈ l ^ Nat.add (length S) 0 ↔\n ∃ S_1, x = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ a_1 b,\n a_1 ∈ l ∧\n (∃ S_1, b = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l) ∧\n a_1 ++ b = join (a :: S)",
"state_before": "case succ.mpr\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na b x✝¹ : List α\nl : Language α\nx✝ : List α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l\nx : List α\n⊢ (∃ S, x = join S ∧ length S = Nat.succ n ∧ ∀ (y : List α), y ∈ S → y ∈ l) →\n ∃ a b, a ∈ l ∧ (∃ S, b = join S ∧ length S = n ∧ ∀ (y : List α), y ∈ S → y ∈ l) ∧ a ++ b = x",
"tactic": "rintro ⟨_ | ⟨a, S⟩, rfl, hn, hS⟩ <;> cases hn"
},
{
"state_after": "case succ.mpr.intro.cons.intro.intro.refl\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nS : List (List α)\nhS : a ∈ l ∧ ∀ (x : List α), x ∈ S → x ∈ l\nihn :\n ∀ {x : List α},\n x ∈ l ^ Nat.add (length S) 0 ↔\n ∃ S_1, x = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ a_1 b,\n a_1 ∈ l ∧\n (∃ S_1, b = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l) ∧\n a_1 ++ b = join (a :: S)",
"state_before": "case succ.mpr.intro.cons.intro.intro.refl\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nS : List (List α)\nhS : ∀ (y : List α), y ∈ a :: S → y ∈ l\nihn :\n ∀ {x : List α},\n x ∈ l ^ Nat.add (length S) 0 ↔\n ∃ S_1, x = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ a_1 b,\n a_1 ∈ l ∧\n (∃ S_1, b = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l) ∧\n a_1 ++ b = join (a :: S)",
"tactic": "rw [forall_mem_cons] at hS"
},
{
"state_after": "no goals",
"state_before": "case succ.mpr.intro.cons.intro.intro.refl\nα : Type u_1\nβ : Type ?u.95661\nγ : Type ?u.95664\nl✝ m : Language α\na✝ b x✝ : List α\nl : Language α\nx a : List α\nS : List (List α)\nhS : a ∈ l ∧ ∀ (x : List α), x ∈ S → x ∈ l\nihn :\n ∀ {x : List α},\n x ∈ l ^ Nat.add (length S) 0 ↔\n ∃ S_1, x = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l\n⊢ ∃ a_1 b,\n a_1 ∈ l ∧\n (∃ S_1, b = join S_1 ∧ length S_1 = Nat.add (length S) 0 ∧ ∀ (y : List α), y ∈ S_1 → y ∈ l) ∧\n a_1 ++ b = join (a :: S)",
"tactic": "exact ⟨a, _, hS.1, ⟨S, rfl, rfl, hS.2⟩, rfl⟩"
}
] |
[
260,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ioo.one_sub_mem
|
[
{
"state_after": "α : Type ?u.62843\ninst✝¹ : StrictOrderedSemiring α\nβ : Type u_1\ninst✝ : OrderedRing β\nt : β\nht : 0 < t ∧ t < 1\n⊢ 0 < 1 - t ∧ 1 - t < 1",
"state_before": "α : Type ?u.62843\ninst✝¹ : StrictOrderedSemiring α\nβ : Type u_1\ninst✝ : OrderedRing β\nt : β\nht : t ∈ Ioo 0 1\n⊢ 1 - t ∈ Ioo 0 1",
"tactic": "rw [mem_Ioo] at *"
},
{
"state_after": "α : Type ?u.62843\ninst✝¹ : StrictOrderedSemiring α\nβ : Type u_1\ninst✝ : OrderedRing β\nt : β\nht : 0 < t ∧ t < 1\n⊢ 1 - t < 1",
"state_before": "α : Type ?u.62843\ninst✝¹ : StrictOrderedSemiring α\nβ : Type u_1\ninst✝ : OrderedRing β\nt : β\nht : 0 < t ∧ t < 1\n⊢ 0 < 1 - t ∧ 1 - t < 1",
"tactic": "refine' ⟨sub_pos.2 ht.2, _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.62843\ninst✝¹ : StrictOrderedSemiring α\nβ : Type u_1\ninst✝ : OrderedRing β\nt : β\nht : 0 < t ∧ t < 1\n⊢ 1 - t < 1",
"tactic": "exact lt_of_le_of_ne ((sub_le_self_iff 1).2 ht.1.le) (mt sub_eq_self.mp ht.1.ne')"
}
] |
[
376,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isGLB_lt_iff
|
[] |
[
1119,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1118,
1
] |
Mathlib/Topology/Instances/Int.lean
|
Int.dist_eq
|
[] |
[
31,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_succ'
|
[] |
[
56,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.curryLeft_norm
|
[] |
[
1436,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1434,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.compose
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.22803\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nl1 : α → β\nu1 : β → α\nl2 : β → γ\nu2 : γ → β\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : α\nb : γ\n⊢ (l2 ∘ l1) a ≤ b ↔ a ≤ (u1 ∘ u2) b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.22803\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nl1 : α → β\nu1 : β → α\nl2 : β → γ\nu2 : γ → β\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\n⊢ GaloisConnection (l2 ∘ l1) (u1 ∘ u2)",
"tactic": "intro a b"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.22803\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nl1 : α → β\nu1 : β → α\nl2 : β → γ\nu2 : γ → β\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : α\nb : γ\n⊢ l2 (l1 a) ≤ b ↔ a ≤ u1 (u2 b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.22803\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nl1 : α → β\nu1 : β → α\nl2 : β → γ\nu2 : γ → β\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : α\nb : γ\n⊢ (l2 ∘ l1) a ≤ b ↔ a ≤ (u1 ∘ u2) b",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.22803\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nl1 : α → β\nu1 : β → α\nl2 : β → γ\nu2 : γ → β\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : α\nb : γ\n⊢ l2 (l1 a) ≤ b ↔ a ≤ u1 (u2 b)",
"tactic": "rw [gc2, gc1]"
}
] |
[
335,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
11
] |
Mathlib/Order/Basic.lean
|
le_of_forall_lt'
|
[] |
[
545,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.snorm_indicator_le_of_bound
|
[
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε\n\ncase neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : ¬M ≤ 0\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "by_cases hM : M ≤ 0"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : ¬M ≤ 0\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "rw [not_le] at hM"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨(ε / M) ^ p.toReal, Real.rpow_pos_of_pos (div_pos hε hM) _, fun s hs hμ => _⟩"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : p = 0\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε\n\ncase neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "rw [snorm_indicator_eq_snorm_restrict hs]"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε",
"tactic": "have haebdd : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ M := by\n filter_upwards\n exact fun x => (hf x).le"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑(Measure.restrict μ s) Set.univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal M ≤ ENNReal.ofReal ε",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε",
"tactic": "refine' le_trans (snorm_le_of_ae_bound haebdd) _"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑μ s ^ (ENNReal.toReal p)⁻¹ ≤ ENNReal.ofReal ε / ENNReal.ofReal M\n\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ENNReal.ofReal M ≠ 0",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑(Measure.restrict μ s) Set.univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal M ≤ ENNReal.ofReal ε",
"tactic": "rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter,\n ← ENNReal.le_div_iff_mul_le (Or.inl _) (Or.inl ENNReal.ofReal_ne_top)]"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\n⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨1, zero_lt_one, fun s _ _ => _⟩"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ snorm (Set.indicator s 0) p μ ≤ ENNReal.ofReal ε\n\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ f = 0",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "rw [(_ : f = 0)]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ snorm (Set.indicator s 0) p μ ≤ ENNReal.ofReal ε",
"tactic": "simp [hε.le]"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\nx : α\n⊢ f x = OfNat.ofNat 0 x",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\n⊢ f = 0",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\nx : α\n⊢ ‖f x‖ ≤ 0",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\nx : α\n⊢ f x = OfNat.ofNat 0 x",
"tactic": "rw [Pi.zero_apply, ← norm_le_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : M ≤ 0\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑μ s ≤ ENNReal.ofReal 1\nx : α\n⊢ ‖f x‖ ≤ 0",
"tactic": "exact (lt_of_lt_of_le (hf x) hM).le"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : p = 0\n⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε",
"tactic": "simp [hp]"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ ∀ (a : α), ‖f a‖ ≤ M",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M",
"tactic": "filter_upwards"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\n⊢ ∀ (a : α), ‖f a‖ ≤ M",
"tactic": "exact fun x => (hf x).le"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑μ s ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑μ s ^ (ENNReal.toReal p)⁻¹ ≤ ENNReal.ofReal ε / ENNReal.ofReal M",
"tactic": "rw [← one_div, ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hp hp_top)]"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p) ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ↑↑μ s ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p",
"tactic": "refine' le_trans hμ _"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p) ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p",
"tactic": "rw [← ENNReal.ofReal_rpow_of_pos (div_pos hε hM),\n ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp hp_top), ENNReal.ofReal_div_of_pos hM]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.121470\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_top : p ≠ ⊤\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhf : ∀ (x : α), ‖f x‖ < M\nhM : 0 < M\ns : Set α\nhs : MeasurableSet s\nhμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)\nhp : ¬p = 0\nhaebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M\n⊢ ENNReal.ofReal M ≠ 0",
"tactic": "simpa only [ENNReal.ofReal_eq_zero, not_le, Ne.def]"
}
] |
[
340,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean
|
TopCat.Presheaf.pushforward_map_app'
|
[] |
[
377,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
DoubleQuot.quotQuotEquivComm_mk_mk
|
[] |
[
654,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.iUnion_Icc_coe_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.135385\nβ : Type ?u.135388\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ (⋃ (n : ℕ), Icc a ↑n) = Ici a \\ {⊤}",
"tactic": "simp only [← Ici_inter_Iic, ← inter_iUnion, iUnion_Iic_coe_nat, diff_eq]"
}
] |
[
884,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
883,
1
] |
Mathlib/Logic/Basic.lean
|
exists_apply_eq
|
[] |
[
778,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
778,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.two_sinh
|
[] |
[
593,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Data/Set/Semiring.lean
|
SetSemiring.add_def
|
[] |
[
120,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_regular
|
[] |
[
550,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderOnWith.mono
|
[] |
[
153,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
11
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
MonoidHom.map_zpowers
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nA : Type ?u.30840\ninst✝¹ : AddGroup A\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\n⊢ Subgroup.map f (Subgroup.zpowers x) = Subgroup.zpowers (↑f x)",
"tactic": "rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]"
}
] |
[
157,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLinAlgEquiv'_toMatrixAlgEquiv'
|
[] |
[
470,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.inf_adj
|
[] |
[
355,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Data/List/Zip.lean
|
List.unzip_zip_left
|
[
{
"state_after": "α : Type u\nβ : Type u_1\nγ : Type ?u.86201\nδ : Type ?u.86204\nε : Type ?u.86207\nl₁ : List α\nh : length l₁ ≤ length []\n⊢ (unzip (zip [] [])).fst = []",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.86201\nδ : Type ?u.86204\nε : Type ?u.86207\nl₁ : List α\nh : length l₁ ≤ length []\n⊢ (unzip (zip l₁ [])).fst = l₁",
"tactic": "rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_le_zero h)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.86201\nδ : Type ?u.86204\nε : Type ?u.86207\nl₁ : List α\nh : length l₁ ≤ length []\n⊢ (unzip (zip [] [])).fst = []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.86201\nδ : Type ?u.86204\nε : Type ?u.86207\na : α\nl₁ : List α\nb : β\nl₂ : List β\nh : length (a :: l₁) ≤ length (b :: l₂)\n⊢ (unzip (zip (a :: l₁) (b :: l₂))).fst = a :: l₁",
"tactic": "simp only [zip_cons_cons, unzip_cons, unzip_zip_left (le_of_succ_le_succ h)]"
}
] |
[
238,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Topology/Separation.lean
|
point_disjoint_finset_opens_of_t2
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\nγ : Type ?u.129204\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space α\nx : α\ns : Finset α\nh : ¬x ∈ s\n⊢ SeparatedNhds {x} ↑s",
"tactic": "exact_mod_cast finset_disjoint_finset_opens_of_t2 {x} s (Finset.disjoint_singleton_left.mpr h)"
}
] |
[
1256,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1254,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Algebra.id_f
|
[] |
[
111,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.Disjoint.mem_imp
|
[] |
[
510,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/CategoryTheory/Limits/FullSubcategory.lean
|
CategoryTheory.Limits.hasLimitsOfShape_of_closed_under_limits
|
[] |
[
120,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Submonoid.pointwise_smul_le_pointwise_smul_iff
|
[] |
[
294,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousOn_fst
|
[] |
[
1265,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1264,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.one_kroneckerTMul_one
|
[] |
[
548,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.cons_head_tail
|
[] |
[
165,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.rev_inj
|
[] |
[
472,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.vec_one_mul
|
[
{
"state_after": "case h\nl : Type ?u.899745\nm : Type u_1\nn : Type u_2\no : Type ?u.899754\nm' : o → Type ?u.899759\nn' : o → Type ?u.899764\nR : Type ?u.899767\nS : Type ?u.899770\nα : Type v\nβ : Type w\nγ : Type ?u.899777\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx✝ : n\n⊢ vecMul 1 A x✝ = ∑ i : m, A i x✝",
"state_before": "l : Type ?u.899745\nm : Type u_1\nn : Type u_2\no : Type ?u.899754\nm' : o → Type ?u.899759\nn' : o → Type ?u.899764\nR : Type ?u.899767\nS : Type ?u.899770\nα : Type v\nβ : Type w\nγ : Type ?u.899777\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\n⊢ vecMul 1 A = fun j => ∑ i : m, A i j",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.899745\nm : Type u_1\nn : Type u_2\no : Type ?u.899754\nm' : o → Type ?u.899759\nn' : o → Type ?u.899764\nR : Type ?u.899767\nS : Type ?u.899770\nα : Type v\nβ : Type w\nγ : Type ?u.899777\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx✝ : n\n⊢ vecMul 1 A x✝ = ∑ i : m, A i x✝",
"tactic": "simp [vecMul, dotProduct]"
}
] |
[
1873,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1872,
1
] |
Mathlib/Data/List/FinRange.lean
|
List.nodup_ofFn_ofInjective
|
[
{
"state_after": "α✝ : Type u\nα : Type u_1\nn : ℕ\nf : Fin n → α\nhf : Function.Injective f\n⊢ Nodup (pmap (fun i hi => f { val := i, isLt := hi }) (range n) (_ : ∀ (x : ℕ), x ∈ range n → x < n))",
"state_before": "α✝ : Type u\nα : Type u_1\nn : ℕ\nf : Fin n → α\nhf : Function.Injective f\n⊢ Nodup (ofFn f)",
"tactic": "rw [ofFn_eq_pmap]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nα : Type u_1\nn : ℕ\nf : Fin n → α\nhf : Function.Injective f\n⊢ Nodup (pmap (fun i hi => f { val := i, isLt := hi }) (range n) (_ : ∀ (x : ℕ), x ∈ range n → x < n))",
"tactic": "exact (nodup_range n).pmap fun _ _ _ _ H => Fin.veq_of_eq <| hf H"
}
] |
[
60,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.monomial_mul_monomial
|
[
{
"state_after": "case h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\n⊢ ↑(coeff R k) (↑(monomial R m) a * ↑(monomial R n) b) = ↑(coeff R k) (↑(monomial R (m + n)) (a * b))",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\n⊢ ↑(monomial R m) a * ↑(monomial R n) b = ↑(monomial R (m + n)) (a * b)",
"tactic": "ext k"
},
{
"state_after": "case h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\n⊢ (if n ≤ k then (if k - n = m then a else 0) * b else 0) = if k = m + n then a * b else 0",
"state_before": "case h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\n⊢ ↑(coeff R k) (↑(monomial R m) a * ↑(monomial R n) b) = ↑(coeff R k) (↑(monomial R (m + n)) (a * b))",
"tactic": "simp only [coeff_mul_monomial, coeff_monomial]"
},
{
"state_after": "case h.inl.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : k - n = m\nh₃ : ¬k = m + n\n⊢ a * b = 0\n\ncase h.inl.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬k - n = m\nh₃ : k = m + n\n⊢ 0 * b = a * b\n\ncase h.inl.inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬k - n = m\nh₃ : ¬k = m + n\n⊢ 0 * b = 0\n\ncase h.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : ¬n ≤ k\nh₂ : k = m + n\n⊢ 0 = a * b",
"state_before": "case h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\n⊢ (if n ≤ k then (if k - n = m then a else 0) * b else 0) = if k = m + n then a * b else 0",
"tactic": "split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try rfl"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : ¬n ≤ k\nh₂ : ¬k = m + n\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case h.inl.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : k - n = m\nh₃ : ¬k = k\n⊢ a * b = 0",
"state_before": "case h.inl.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : k - n = m\nh₃ : ¬k = m + n\n⊢ a * b = 0",
"tactic": "rw [← h₂, tsub_add_cancel_of_le h₁] at h₃"
},
{
"state_after": "no goals",
"state_before": "case h.inl.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : k - n = m\nh₃ : ¬k = k\n⊢ a * b = 0",
"tactic": "exact (h₃ rfl).elim"
},
{
"state_after": "case h.inl.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬m = m\nh₃ : k = m + n\n⊢ 0 * b = a * b",
"state_before": "case h.inl.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬k - n = m\nh₃ : k = m + n\n⊢ 0 * b = a * b",
"tactic": "rw [h₃, add_tsub_cancel_right] at h₂"
},
{
"state_after": "no goals",
"state_before": "case h.inl.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬m = m\nh₃ : k = m + n\n⊢ 0 * b = a * b",
"tactic": "exact (h₂ rfl).elim"
},
{
"state_after": "no goals",
"state_before": "case h.inl.inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : n ≤ k\nh₂ : ¬k - n = m\nh₃ : ¬k = m + n\n⊢ 0 * b = 0",
"tactic": "exact MulZeroClass.zero_mul b"
},
{
"state_after": "case h.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : ¬n ≤ m + n\nh₂ : k = m + n\n⊢ 0 = a * b",
"state_before": "case h.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : ¬n ≤ k\nh₂ : k = m + n\n⊢ 0 = a * b",
"tactic": "rw [h₂] at h₁"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\na b : R\nk : σ →₀ ℕ\nh₁ : ¬n ≤ m + n\nh₂ : k = m + n\n⊢ 0 = a * b",
"tactic": "exact (h₁ <| le_add_left le_rfl).elim"
}
] |
[
355,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/List/Duplicate.lean
|
List.Duplicate.mono_sublist
|
[
{
"state_after": "case slnil\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nhx : x ∈+ []\n⊢ x ∈+ []\n\ncase cons\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\na✝ : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ l₁\n⊢ x ∈+ y :: l₂\n\ncase cons₂\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ y :: l₁\n⊢ x ∈+ y :: l₂",
"state_before": "α : Type u_1\nl : List α\nx : α\nl' : List α\nhx : x ∈+ l\nh : l <+ l'\n⊢ x ∈+ l'",
"tactic": "induction' h with l₁ l₂ y _ IH l₁ l₂ y h IH"
},
{
"state_after": "no goals",
"state_before": "case slnil\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nhx : x ∈+ []\n⊢ x ∈+ []",
"tactic": "exact hx"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\na✝ : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ l₁\n⊢ x ∈+ y :: l₂",
"tactic": "exact (IH hx).duplicate_cons _"
},
{
"state_after": "case cons₂\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : y = x ∧ x ∈ l₁ ∨ x ∈+ l₁\n⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂",
"state_before": "case cons₂\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ y :: l₁\n⊢ x ∈+ y :: l₂",
"tactic": "rw [duplicate_cons_iff] at hx⊢"
},
{
"state_after": "case cons₂.inl.intro\nα : Type u_1\nl l' l₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nhx✝ : y ∈+ l\nIH : y ∈+ l₁ → y ∈+ l₂\nhx : y ∈ l₁\n⊢ y = y ∧ y ∈ l₂ ∨ y ∈+ l₂\n\ncase cons₂.inr\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ l₁\n⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂",
"state_before": "case cons₂\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : y = x ∧ x ∈ l₁ ∨ x ∈+ l₁\n⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂",
"tactic": "rcases hx with (⟨rfl, hx⟩ | hx)"
},
{
"state_after": "no goals",
"state_before": "case cons₂.inl.intro\nα : Type u_1\nl l' l₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nhx✝ : y ∈+ l\nIH : y ∈+ l₁ → y ∈+ l₂\nhx : y ∈ l₁\n⊢ y = y ∧ y ∈ l₂ ∨ y ∈+ l₂",
"tactic": "simp [h.subset hx]"
},
{
"state_after": "no goals",
"state_before": "case cons₂.inr\nα : Type u_1\nl : List α\nx : α\nl' : List α\nhx✝ : x ∈+ l\nl₁ l₂ : List α\ny : α\nh : l₁ <+ l₂\nIH : x ∈+ l₁ → x ∈+ l₂\nhx : x ∈+ l₁\n⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂",
"tactic": "simp [IH hx]"
}
] |
[
117,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_div'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30365\nG : Type ?u.30368\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ (a / b)⁻¹ = a⁻¹ / b⁻¹",
"tactic": "simp"
}
] |
[
500,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.closure_le
|
[] |
[
287,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.def'
|
[] |
[
171,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_out
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.23296\nβ : Type ?u.23299\nγ : Type ?u.23302\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ type (Quotient.out o).r = o",
"tactic": "rw [Ordinal.type, WellOrder.eta, Quotient.out_eq]"
}
] |
[
205,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.add_mem
|
[] |
[
323,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
11
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Nat.cast_min
|
[] |
[
174,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Topology/Constructions.lean
|
embedding_sigmaMk
|
[] |
[
1498,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1497,
1
] |
Mathlib/Data/List/Basic.lean
|
List.filter_true
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.380943\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\n⊢ filter (fun x => true) l = l",
"tactic": "induction l <;> simp [*, filter]"
}
] |
[
3591,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3590,
1
] |
Mathlib/MeasureTheory/Group/MeasurableEquiv.lean
|
measurableEmbedding_const_smul
|
[] |
[
61,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_atTop_of_geom_le
|
[] |
[
159,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.uniformIntegrable_of
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\n⊢ UniformIntegrable f p μ",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ UniformIntegrable f p μ",
"tactic": "set g : ι → α → β := fun i => (hf i).choose"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\n⊢ UniformIntegrable f p μ",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\n⊢ UniformIntegrable f p μ",
"tactic": "have hgmeas : ∀ i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hf i).1"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\n⊢ UniformIntegrable f p μ",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\n⊢ UniformIntegrable f p μ",
"tactic": "have hgeq : ∀ i, g i =ᵐ[μ] f i := fun i => (Exists.choose_spec <| hf i).2.symm"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\n⊢ UniformIntegrable f p μ",
"tactic": "refine' (uniformIntegrable_of' hp hp' hgmeas fun ε hε => _).ae_eq hgeq"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨C, hC⟩ := h ε hε"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\n⊢ Set.indicator {x | C ≤ ‖g i x‖₊} (g i) =ᵐ[μ] Set.indicator {x | C ≤ ‖f i x‖₊} (f i)",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨C, fun i => le_trans (le_of_eq <| snorm_congr_ae _) (hC i)⟩"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\n⊢ Set.indicator {x | C ≤ ‖g i x‖₊} (g i) =ᵐ[μ] Set.indicator {x | C ≤ ‖f i x‖₊} (f i)",
"tactic": "filter_upwards [(Exists.choose_spec <| hf i).2] with x hx"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x\n\ncase neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x",
"tactic": "by_cases hfx : x ∈ { x | C ≤ ‖f i x‖₊ }"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x",
"tactic": "rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]"
},
{
"state_after": "no goals",
"state_before": "case pos.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}",
"tactic": "rwa [Set.mem_setOf, hx] at hfx"
},
{
"state_after": "case neg.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ ¬x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ Set.indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}\n (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =\n Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x",
"tactic": "rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ng : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)\nhgmeas : ∀ (i : ι), StronglyMeasurable (g i)\nhgeq : ∀ (i : ι), g i =ᵐ[μ] f i\nε : ℝ\nhε : 0 < ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\nx : α\nhx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x\nhfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}\n⊢ ¬x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}",
"tactic": "rwa [Set.mem_setOf, hx] at hfx"
}
] |
[
855,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Combinatorics/Quiver/Basic.lean
|
Prefunctor.comp_id
|
[] |
[
121,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.sublists'_nil
|
[] |
[
36,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.exists_rename_eq_of_vars_subset_range
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) p = ↑(RingHom.id (MvPolynomial σ R)) p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ ↑(rename f) (↑(aeval fun i => Option.elim' 0 X (partialInv f i)) p) = p",
"tactic": "show (rename f).toRingHom.comp _ p = RingHom.id _ p"
},
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ RingHom.comp (RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) C =\n RingHom.comp (RingHom.id (MvPolynomial σ R)) C\n\ncase refine'_2\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ ∀ (i : σ),\n i ∈ vars p →\n i ∈ vars p →\n ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n ↑(RingHom.id (MvPolynomial σ R)) (X i)\n\ncase refine'_3\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ p = p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) p = ↑(RingHom.id (MvPolynomial σ R)) p",
"tactic": "refine' hom_congr_vars _ _ _"
},
{
"state_after": "case refine'_1.a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\nx✝ : R\n⊢ ↑(RingHom.comp (RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) C) x✝ =\n ↑(RingHom.comp (RingHom.id (MvPolynomial σ R)) C) x✝",
"state_before": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ RingHom.comp (RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) C =\n RingHom.comp (RingHom.id (MvPolynomial σ R)) C",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\nx✝ : R\n⊢ ↑(RingHom.comp (RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) C) x✝ =\n ↑(RingHom.comp (RingHom.id (MvPolynomial σ R)) C) x✝",
"tactic": "simp [algebraMap_eq]"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\ni : σ\nhip a✝ : i ∈ vars p\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n ↑(RingHom.id (MvPolynomial σ R)) (X i)",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ ∀ (i : σ),\n i ∈ vars p →\n i ∈ vars p →\n ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n ↑(RingHom.id (MvPolynomial σ R)) (X i)",
"tactic": "intro i hip _"
},
{
"state_after": "case refine'_2.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\ni : τ\nhip a✝ : f i ∈ vars p\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X (f i)) =\n ↑(RingHom.id (MvPolynomial σ R)) (X (f i))",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\ni : σ\nhip a✝ : i ∈ vars p\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =\n ↑(RingHom.id (MvPolynomial σ R)) (X i)",
"tactic": "rcases hf hip with ⟨i, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\ni : τ\nhip a✝ : f i ∈ vars p\n⊢ ↑(RingHom.comp ↑(rename f) ↑(aeval fun i => Option.elim' 0 X (partialInv f i))) (X (f i)) =\n ↑(RingHom.id (MvPolynomial σ R)) (X (f i))",
"tactic": "simp [partialInv_left hfi]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : τ → σ\nhfi : Injective f\nhf : ↑(vars p) ⊆ range f\n⊢ p = p",
"tactic": "rfl"
}
] |
[
883,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
872,
1
] |
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