file_path
stringlengths 11
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stringclasses 4
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---|---|---|---|---|---|---|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.head_congr
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ {s t : WSeq α}, s ~ʷ t → ∀ {o : Option α}, o ∈ head s → o ∈ head t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ {s t : WSeq α}, s ~ʷ t → head s ~ head t",
"tactic": "suffices ∀ {s t : WSeq α}, s ~ʷ t → ∀ {o}, o ∈ head s → o ∈ head t from fun s t h o =>\n ⟨this h, this h.symm⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\n⊢ o ∈ head t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ ∀ {s t : WSeq α}, s ~ʷ t → ∀ {o : Option α}, o ∈ head s → o ∈ head t",
"tactic": "intro s t h o ho"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\n⊢ o ∈ head t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\n⊢ o ∈ head t",
"tactic": "rcases @Computation.exists_of_mem_map _ _ _ _ (destruct s) ho with ⟨ds, dsm, dse⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\n⊢ Prod.fst <$> ds ∈ head t",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\n⊢ o ∈ head t",
"tactic": "rw [← dse]"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Prod.fst <$> ds ∈ head t",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\n⊢ Prod.fst <$> ds ∈ head t",
"tactic": "cases' destruct_congr h with l r"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndt : Option (α × WSeq α)\ndtm : dt ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) ds dt\n⊢ Prod.fst <$> ds ∈ head t",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Prod.fst <$> ds ∈ head t",
"tactic": "rcases l dsm with ⟨dt, dtm, dst⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.none.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\ndtm : none ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none none\n⊢ Prod.fst <$> none ∈ head t\n\ncase intro.intro.intro.intro.intro.none.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\nb : α × WSeq α\ndtm : some b ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none (some b)\n⊢ Prod.fst <$> none ∈ head t\n\ncase intro.intro.intro.intro.intro.some.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α × WSeq α\ndsm : some a ∈ destruct s\ndse : Prod.fst <$> some a = o\ndtm : none ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some a) none\n⊢ Prod.fst <$> some a ∈ head t\n\ncase intro.intro.intro.intro.intro.some.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α × WSeq α\ndsm : some a ∈ destruct s\ndse : Prod.fst <$> some a = o\nb : α × WSeq α\ndtm : some b ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some a) (some b)\n⊢ Prod.fst <$> some a ∈ head t",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nds : Option (α × WSeq α)\ndsm : ds ∈ destruct s\ndse : Prod.fst <$> ds = o\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndt : Option (α × WSeq α)\ndtm : dt ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) ds dt\n⊢ Prod.fst <$> ds ∈ head t",
"tactic": "cases' ds with a <;> cases' dt with b"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.none.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\ndtm : none ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none none\n⊢ Prod.fst <$> none ∈ head t",
"tactic": "apply Computation.mem_map _ dtm"
},
{
"state_after": "case intro.intro.intro.intro.intro.none.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\nfst✝ : α\nsnd✝ : WSeq α\ndtm : some (fst✝, snd✝) ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none (some (fst✝, snd✝))\n⊢ Prod.fst <$> none ∈ head t",
"state_before": "case intro.intro.intro.intro.intro.none.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\nb : α × WSeq α\ndtm : some b ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none (some b)\n⊢ Prod.fst <$> none ∈ head t",
"tactic": "cases b"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.none.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndsm : none ∈ destruct s\ndse : Prod.fst <$> none = o\nfst✝ : α\nsnd✝ : WSeq α\ndtm : some (fst✝, snd✝) ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) none (some (fst✝, snd✝))\n⊢ Prod.fst <$> none ∈ head t",
"tactic": "cases dst"
},
{
"state_after": "case intro.intro.intro.intro.intro.some.none.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndtm : none ∈ destruct t\nfst✝ : α\nsnd✝ : WSeq α\ndsm : some (fst✝, snd✝) ∈ destruct s\ndse : Prod.fst <$> some (fst✝, snd✝) = o\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (fst✝, snd✝)) none\n⊢ Prod.fst <$> some (fst✝, snd✝) ∈ head t",
"state_before": "case intro.intro.intro.intro.intro.some.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α × WSeq α\ndsm : some a ∈ destruct s\ndse : Prod.fst <$> some a = o\ndtm : none ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some a) none\n⊢ Prod.fst <$> some a ∈ head t",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.some.none.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\ndtm : none ∈ destruct t\nfst✝ : α\nsnd✝ : WSeq α\ndsm : some (fst✝, snd✝) ∈ destruct s\ndse : Prod.fst <$> some (fst✝, snd✝) = o\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (fst✝, snd✝)) none\n⊢ Prod.fst <$> some (fst✝, snd✝) ∈ head t",
"tactic": "cases dst"
},
{
"state_after": "case intro.intro.intro.intro.intro.some.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nb : α × WSeq α\ndtm : some b ∈ destruct t\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some b)\n⊢ Prod.fst <$> some (a, s') ∈ head t",
"state_before": "case intro.intro.intro.intro.intro.some.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α × WSeq α\ndsm : some a ∈ destruct s\ndse : Prod.fst <$> some a = o\nb : α × WSeq α\ndtm : some b ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some a) (some b)\n⊢ Prod.fst <$> some a ∈ head t",
"tactic": "cases' a with a s'"
},
{
"state_after": "case intro.intro.intro.intro.intro.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\nb : α\nt' : WSeq α\ndtm : some (b, t') ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Prod.fst <$> some (a, s') ∈ head t",
"state_before": "case intro.intro.intro.intro.intro.some.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nb : α × WSeq α\ndtm : some b ∈ destruct t\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some b)\n⊢ Prod.fst <$> some (a, s') ∈ head t",
"tactic": "cases' b with b t'"
},
{
"state_after": "case intro.intro.intro.intro.intro.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\nb : α\nt' : WSeq α\ndtm : some (b, t') ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Prod.fst <$> some (b, s') ∈ head t",
"state_before": "case intro.intro.intro.intro.intro.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\nb : α\nt' : WSeq α\ndtm : some (b, t') ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Prod.fst <$> some (a, s') ∈ head t",
"tactic": "rw [dst.left]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\no : Option α\nho : o ∈ head s\nl : ∀ {a : Option (α × WSeq α)}, a ∈ destruct s → ∃ b, b ∈ destruct t ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\nr : ∀ {b : Option (α × WSeq α)}, b ∈ destruct t → ∃ a, a ∈ destruct s ∧ BisimO (fun x x_1 => x ~ʷ x_1) a b\na : α\ns' : WSeq α\ndsm : some (a, s') ∈ destruct s\ndse : Prod.fst <$> some (a, s') = o\nb : α\nt' : WSeq α\ndtm : some (b, t') ∈ destruct t\ndst : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Prod.fst <$> some (b, s') ∈ head t",
"tactic": "exact @Computation.mem_map _ _ (@Functor.map _ _ (α × WSeq α) _ Prod.fst)\n (some (b, t')) (destruct t) dtm"
}
] |
[
1145,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1127,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.exists_not_mem_row
|
[
{
"state_after": "case intro\nμ : YoungDiagram\ni j : ℕ\nhj :\n ¬j ∈\n Finset.preimage μ.cells (Prod.mk i)\n (_ :\n ∀ (x : ℕ),\n x ∈ Prod.mk i ⁻¹' ↑μ.cells → ∀ (x_2 : ℕ), x_2 ∈ Prod.mk i ⁻¹' ↑μ.cells → (i, x) = (i, x_2) → x = x_2)\n⊢ ∃ j, ¬(i, j) ∈ μ",
"state_before": "μ : YoungDiagram\ni : ℕ\n⊢ ∃ j, ¬(i, j) ∈ μ",
"tactic": "obtain ⟨j, hj⟩ :=\n Infinite.exists_not_mem_finset\n (μ.cells.preimage (Prod.mk i) fun _ _ _ _ h => by\n cases h\n rfl)"
},
{
"state_after": "case intro\nμ : YoungDiagram\ni j : ℕ\nhj : ¬(i, j) ∈ μ.cells\n⊢ ∃ j, ¬(i, j) ∈ μ\n\nμ : YoungDiagram\ni x✝³ : ℕ\nx✝² : x✝³ ∈ Prod.mk i ⁻¹' ↑μ.cells\nx✝¹ : ℕ\nx✝ : x✝¹ ∈ Prod.mk i ⁻¹' ↑μ.cells\nh : (i, x✝³) = (i, x✝¹)\n⊢ x✝³ = x✝¹",
"state_before": "case intro\nμ : YoungDiagram\ni j : ℕ\nhj :\n ¬j ∈\n Finset.preimage μ.cells (Prod.mk i)\n (_ :\n ∀ (x : ℕ),\n x ∈ Prod.mk i ⁻¹' ↑μ.cells → ∀ (x_2 : ℕ), x_2 ∈ Prod.mk i ⁻¹' ↑μ.cells → (i, x) = (i, x_2) → x = x_2)\n⊢ ∃ j, ¬(i, j) ∈ μ",
"tactic": "rw [Finset.mem_preimage] at hj"
},
{
"state_after": "no goals",
"state_before": "case intro\nμ : YoungDiagram\ni j : ℕ\nhj : ¬(i, j) ∈ μ.cells\n⊢ ∃ j, ¬(i, j) ∈ μ\n\nμ : YoungDiagram\ni x✝³ : ℕ\nx✝² : x✝³ ∈ Prod.mk i ⁻¹' ↑μ.cells\nx✝¹ : ℕ\nx✝ : x✝¹ ∈ Prod.mk i ⁻¹' ↑μ.cells\nh : (i, x✝³) = (i, x✝¹)\n⊢ x✝³ = x✝¹",
"tactic": "exact ⟨j, hj⟩"
},
{
"state_after": "case refl\nμ : YoungDiagram\ni x✝² : ℕ\nx✝¹ x✝ : x✝² ∈ Prod.mk i ⁻¹' ↑μ.cells\n⊢ x✝² = x✝²",
"state_before": "μ : YoungDiagram\ni x✝³ : ℕ\nx✝² : x✝³ ∈ Prod.mk i ⁻¹' ↑μ.cells\nx✝¹ : ℕ\nx✝ : x✝¹ ∈ Prod.mk i ⁻¹' ↑μ.cells\nh : (i, x✝³) = (i, x✝¹)\n⊢ x✝³ = x✝¹",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nμ : YoungDiagram\ni x✝² : ℕ\nx✝¹ x✝ : x✝² ∈ Prod.mk i ⁻¹' ↑μ.cells\n⊢ x✝² = x✝²",
"tactic": "rfl"
}
] |
[
303,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
11
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.le_boundedBy
|
[
{
"state_after": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\n⊢ ∀ (a : Set α), (↑μ a ≤ ⨆ (_ : Set.Nonempty a), m a) ↔ ↑μ a ≤ m a",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\n⊢ μ ≤ boundedBy m ↔ ∀ (s : Set α), ↑μ s ≤ m s",
"tactic": "rw [boundedBy , le_ofFunction, forall_congr']"
},
{
"state_after": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\ns : Set α\n⊢ (↑μ s ≤ ⨆ (_ : Set.Nonempty s), m s) ↔ ↑μ s ≤ m s",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\n⊢ ∀ (a : Set α), (↑μ a ≤ ⨆ (_ : Set.Nonempty a), m a) ↔ ↑μ a ≤ m a",
"tactic": "intro s"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\nμ : OuterMeasure α\ns : Set α\n⊢ (↑μ s ≤ ⨆ (_ : Set.Nonempty s), m s) ↔ ↑μ s ≤ m s",
"tactic": "cases' s.eq_empty_or_nonempty with h h <;> simp [h, Set.not_nonempty_empty]"
}
] |
[
875,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
873,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nthLe_tail
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\ni : ℕ\nh : i < length (tail l)\n⊢ i + 1 < length l",
"tactic": "simpa [← lt_tsub_iff_right] using h"
},
{
"state_after": "case nil\nι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ni : ℕ\nh : i < length (tail [])\nh' : optParam (i + 1 < length []) (_ : i + 1 < length [])\n⊢ nthLe (tail []) i h = nthLe [] (i + 1) h'\n\ncase cons\nι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ni : ℕ\nhead✝ : α\ntail✝ : List α\nh : i < length (tail (head✝ :: tail✝))\nh' : optParam (i + 1 < length (head✝ :: tail✝)) (_ : i + 1 < length (head✝ :: tail✝))\n⊢ nthLe (tail (head✝ :: tail✝)) i h = nthLe (head✝ :: tail✝) (i + 1) h'",
"state_before": "ι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\ni : ℕ\nh : i < length (tail l)\nh' : optParam (i + 1 < length l) (_ : i + 1 < length l)\n⊢ nthLe (tail l) i h = nthLe l (i + 1) h'",
"tactic": "cases l"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ni : ℕ\nh : i < length (tail [])\nh' : optParam (i + 1 < length []) (_ : i + 1 < length [])\n⊢ nthLe (tail []) i h = nthLe [] (i + 1) h'",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.45408\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ni : ℕ\nhead✝ : α\ntail✝ : List α\nh : i < length (tail (head✝ :: tail✝))\nh' : optParam (i + 1 < length (head✝ :: tail✝)) (_ : i + 1 < length (head✝ :: tail✝))\n⊢ nthLe (tail (head✝ :: tail✝)) i h = nthLe (head✝ :: tail✝) (i + 1) h'",
"tactic": "rfl"
}
] |
[
951,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
945,
9
] |
Mathlib/Analysis/Calculus/Deriv/Star.lean
|
HasDerivAt.star
|
[] |
[
50,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
18
] |
Mathlib/Deprecated/Group.lean
|
Multiplicative.isMulHom
|
[] |
[
451,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean
|
CategoryTheory.Sieve.effectiveEpimorphic_family
|
[
{
"state_after": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π) → EffectiveEpiFamily X π\n\ncase mpr\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ EffectiveEpiFamily X π → Presieve.EffectiveEpimorphic (Presieve.ofArrows X π)",
"state_before": "C : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π) ↔ EffectiveEpiFamily X π",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X π)).arrows))\n⊢ EffectiveEpiFamily X π",
"state_before": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π) → EffectiveEpiFamily X π",
"tactic": "intro (h : Nonempty _)"
},
{
"state_after": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))\n⊢ EffectiveEpiFamily X π",
"state_before": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X π)).arrows))\n⊢ EffectiveEpiFamily X π",
"tactic": "rw [Sieve.generateFamily_eq] at h"
},
{
"state_after": "case mp.effectiveEpiFamily\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))\n⊢ Nonempty (EffectiveEpiFamilyStruct X π)",
"state_before": "case mp\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))\n⊢ EffectiveEpiFamily X π",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp.effectiveEpiFamily\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))\n⊢ Nonempty (EffectiveEpiFamilyStruct X π)",
"tactic": "apply Nonempty.map (effectiveEpiFamilyStructOfIsColimit _ _) h"
},
{
"state_after": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π)",
"state_before": "case mpr\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\n⊢ EffectiveEpiFamily X π → Presieve.EffectiveEpimorphic (Presieve.ofArrows X π)",
"tactic": "rintro ⟨h⟩"
},
{
"state_after": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X π)).arrows))",
"state_before": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Presieve.EffectiveEpimorphic (Presieve.ofArrows X π)",
"tactic": "show Nonempty _"
},
{
"state_after": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))",
"state_before": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X π)).arrows))",
"tactic": "rw [Sieve.generateFamily_eq]"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk\nC : Type u_3\ninst✝ : Category C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nh : Nonempty (EffectiveEpiFamilyStruct X π)\n⊢ Nonempty (IsColimit (Presieve.cocone (generateFamily X π).arrows))",
"tactic": "apply Nonempty.map (isColimitOfEffectiveEpiFamilyStruct _ _) h"
}
] |
[
434,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_mk
|
[] |
[
473,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.of_mem_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh this : a ∈ append s₁ s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"tactic": "have := h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\n⊢ a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh this : a ∈ append s₁ s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"tactic": "revert this"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\nss : Seq α\ne : append s₁ s₂ = ss\n⊢ a ∈ ss → a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\n⊢ a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂",
"tactic": "generalize e : append s₁ s₂ = ss"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh✝ : a ∈ append s₁ s₂\nss : Seq α\ne : append s₁ s₂ = ss\nh : a ∈ ss\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh : a ∈ append s₁ s₂\nss : Seq α\ne : append s₁ s₂ = ss\n⊢ a ∈ ss → a ∈ s₁ ∨ a ∈ s₂",
"tactic": "intro h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\n⊢ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = ss → a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₁ s₂ : Seq α\na : α\nh✝ : a ∈ append s₁ s₂\nss : Seq α\ne : append s₁ s₂ = ss\nh : a ∈ ss\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"tactic": "revert s₁"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\n⊢ ∀ (b : α) (s' : Seq α),\n (a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂) →\n ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = cons b s' → a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\n⊢ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = ss → a ∈ s₁ ∨ a ∈ s₂",
"tactic": "apply mem_rec_on h _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ a ∈ append s₁ s₂ → append s₁ s₂ = cons b s' → a ∈ s₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\n⊢ ∀ (b : α) (s' : Seq α),\n (a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂) →\n ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = cons b s' → a ∈ s₁ ∨ a ∈ s₂",
"tactic": "intro b s' o s₁"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ a ∈ append nil s₂ → append nil s₂ = cons b s' → a ∈ nil ∨ a ∈ s₂\n\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ ∀ (c : α) (t₁ : Seq α), a ∈ append (cons c t₁) s₂ → append (cons c t₁) s₂ = cons b s' → a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ a ∈ append s₁ s₂ → append s₁ s₂ = cons b s' → a ∈ s₁ ∨ a ∈ s₂",
"tactic": "apply s₁.recOn _ fun c t₁ => _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nm : a ∈ append nil s₂\ne✝ : append nil s₂ = cons b s'\n⊢ a ∈ nil ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ a ∈ append nil s₂ → append nil s₂ = cons b s' → a ∈ nil ∨ a ∈ s₂",
"tactic": "intro m _"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nm : a ∈ append nil s₂\ne✝ : append nil s₂ = cons b s'\n⊢ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nm : a ∈ append nil s₂\ne✝ : append nil s₂ = cons b s'\n⊢ a ∈ nil ∨ a ∈ s₂",
"tactic": "apply Or.inr"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nm : a ∈ append nil s₂\ne✝ : append nil s₂ = cons b s'\n⊢ a ∈ s₂",
"tactic": "simpa using m"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\n⊢ ∀ (c : α) (t₁ : Seq α), a ∈ append (cons c t₁) s₂ → append (cons c t₁) s₂ = cons b s' → a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "intro c t₁ m e"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "have this := congr_arg destruct e"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\ne' : a = c\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "cases' show a = c ∨ a ∈ append t₁ s₂ by simpa using m with e' m"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\n⊢ a = c ∨ a ∈ append t₁ s₂",
"tactic": "simpa using m"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\ne' : a = c\n⊢ c ∈ cons c t₁ ∨ c ∈ s₂",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\ne' : a = c\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "rw [e']"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\ne' : a = c\n⊢ c ∈ cons c t₁ ∨ c ∈ s₂",
"tactic": "exact Or.inl (mem_cons _ _)"
},
{
"state_after": "case inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "cases' show c = b ∧ append t₁ s₂ = s' by simpa with i1 i2"
},
{
"state_after": "case inr.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' s₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\ne' : a = b\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂\n\ncase inr.intro.inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' s₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\nIH : ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"state_before": "case inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "cases' o with e' IH"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' : Seq α\no : a = b ∨ ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\ns₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\n⊢ c = b ∧ append t₁ s₂ = s'",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' s₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\ne' : a = b\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "simp [i1, e']"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ : Seq α\na : α\nss : Seq α\nh : a ∈ ss\nb : α\ns' s₁ : Seq α\nc : α\nt₁ : Seq α\nm✝ : a ∈ append (cons c t₁) s₂\ne : append (cons c t₁) s₂ = cons b s'\nthis : destruct (append (cons c t₁) s₂) = destruct (cons b s')\nm : a ∈ append t₁ s₂\ni1 : c = b\ni2 : append t₁ s₂ = s'\nIH : ∀ {s₁ : Seq α}, a ∈ append s₁ s₂ → append s₁ s₂ = s' → a ∈ s₁ ∨ a ∈ s₂\n⊢ a ∈ cons c t₁ ∨ a ∈ s₂",
"tactic": "exact Or.imp_left (mem_cons_of_mem _) (IH m i2)"
}
] |
[
886,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.div_iInter_subset
|
[] |
[
792,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Init/IteSimp.lean
|
if_true_right_eq_or
|
[
{
"state_after": "no goals",
"state_before": "p : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then q else True) = (¬p ∨ q)",
"tactic": "by_cases p <;> simp [h]"
}
] |
[
25,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
24,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_prod_const
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.230579\nι : Sort u_3\nι' : Sort ?u.230585\nι₂ : Sort ?u.230588\nκ : ι → Sort ?u.230593\nκ₁ : ι → Sort ?u.230598\nκ₂ : ι → Sort ?u.230603\nκ' : ι' → Sort ?u.230608\ns : ι → Set α\nt : Set β\nx✝ : α × β\n⊢ x✝ ∈ (⋃ (i : ι), s i) ×ˢ t ↔ x✝ ∈ ⋃ (i : ι), s i ×ˢ t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.230579\nι : Sort u_3\nι' : Sort ?u.230585\nι₂ : Sort ?u.230588\nκ : ι → Sort ?u.230593\nκ₁ : ι → Sort ?u.230598\nκ₂ : ι → Sort ?u.230603\nκ' : ι' → Sort ?u.230608\ns : ι → Set α\nt : Set β\n⊢ (⋃ (i : ι), s i) ×ˢ t = ⋃ (i : ι), s i ×ˢ t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.230579\nι : Sort u_3\nι' : Sort ?u.230585\nι₂ : Sort ?u.230588\nκ : ι → Sort ?u.230593\nκ₁ : ι → Sort ?u.230598\nκ₂ : ι → Sort ?u.230603\nκ' : ι' → Sort ?u.230608\ns : ι → Set α\nt : Set β\nx✝ : α × β\n⊢ x✝ ∈ (⋃ (i : ι), s i) ×ˢ t ↔ x✝ ∈ ⋃ (i : ι), s i ×ˢ t",
"tactic": "simp"
}
] |
[
1783,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1781,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.snd
|
[] |
[
802,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_inter_of_mem'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18576\nγ : Type ?u.18579\nδ : Type ?u.18582\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\nh : t ∈ 𝓝[s] a\n⊢ 𝓝[s ∩ t] a = 𝓝[s] a",
"tactic": "rw [inter_comm, nhdsWithin_inter_of_mem h]"
}
] |
[
277,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiff.contDiffOn
|
[] |
[
1438,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1437,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.NonemptyCompacts.ext
|
[] |
[
243,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
11
] |
Mathlib/Data/Sym/Card.lean
|
Sym2.card_image_diag
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Set.InjOn Quotient.mk' ↑(Finset.diag s)",
"state_before": "α : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Finset.card (image Quotient.mk' (Finset.diag s)) = Finset.card s",
"tactic": "rw [card_image_of_injOn, diag_card]"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx : (x₀, x₁) ∈ ↑(Finset.diag s)\nx₂✝ : α × α\na✝ : x₂✝ ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' x₂✝\n⊢ (x₀, x₁) = x₂✝",
"state_before": "α : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Set.InjOn Quotient.mk' ↑(Finset.diag s)",
"tactic": "rintro ⟨x₀, x₁⟩ hx _ _ h"
},
{
"state_after": "case mk.refl\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx a✝ : (x₀, x₁) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₀, x₁)\n⊢ (x₀, x₁) = (x₀, x₁)\n\ncase mk.swap\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx : (x₀, x₁) ∈ ↑(Finset.diag s)\na✝ : (x₁, x₀) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₁, x₀)\n⊢ (x₀, x₁) = (x₁, x₀)",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx : (x₀, x₁) ∈ ↑(Finset.diag s)\nx₂✝ : α × α\na✝ : x₂✝ ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' x₂✝\n⊢ (x₀, x₁) = x₂✝",
"tactic": "cases Quotient.eq'.1 h"
},
{
"state_after": "no goals",
"state_before": "case mk.refl\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx a✝ : (x₀, x₁) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₀, x₁)\n⊢ (x₀, x₁) = (x₀, x₁)",
"tactic": "rfl"
},
{
"state_after": "case mk.swap\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\na✝ : (x₁, x₀) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₁, x₀)\nhx : x₀ ∈ s ∧ x₀ = x₁\n⊢ (x₀, x₁) = (x₁, x₀)",
"state_before": "case mk.swap\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\nhx : (x₀, x₁) ∈ ↑(Finset.diag s)\na✝ : (x₁, x₀) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₁, x₀)\n⊢ (x₀, x₁) = (x₁, x₀)",
"tactic": "simp only [mem_coe, mem_diag] at hx"
},
{
"state_after": "no goals",
"state_before": "case mk.swap\nα : Type u_1\nβ : Type ?u.72186\ninst✝ : DecidableEq α\ns : Finset α\nx₀ x₁ : α\na✝ : (x₁, x₀) ∈ ↑(Finset.diag s)\nh : Quotient.mk' (x₀, x₁) = Quotient.mk' (x₁, x₀)\nhx : x₀ ∈ s ∧ x₀ = x₁\n⊢ (x₀, x₁) = (x₁, x₀)",
"tactic": "rw [hx.2]"
}
] |
[
141,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Algebra/Lie/Classical.lean
|
LieAlgebra.Orthogonal.jd_transform
|
[
{
"state_after": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ (PD l R)ᵀ ⬝ JD l R ⬝ PD l R = 2 • S l R",
"state_before": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\n⊢ (PD l R)ᵀ ⬝ JD l R ⬝ PD l R = 2 • S l R",
"tactic": "have h : (PD l R)ᵀ ⬝ JD l R = Matrix.fromBlocks 1 1 1 (-1) := by\n simp [PD, JD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_multiply]"
},
{
"state_after": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ fromBlocks (1 ⬝ 1 + 1 ⬝ 1) (1 ⬝ (-1) + 1 ⬝ 1) (1 ⬝ 1 + (-1) ⬝ 1) (1 ⬝ (-1) + (-1) ⬝ 1) =\n fromBlocks (2 • 1) (2 • 0) (2 • 0) (2 • -1)",
"state_before": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ (PD l R)ᵀ ⬝ JD l R ⬝ PD l R = 2 • S l R",
"tactic": "rw [h, PD, s_as_blocks, Matrix.fromBlocks_multiply, Matrix.fromBlocks_smul]"
},
{
"state_after": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ fromBlocks (1 ⬝ 1 + 1 ⬝ 1) (1 ⬝ (-1) + 1 ⬝ 1) (1 ⬝ 1 + (-1) ⬝ 1) (1 ⬝ (-1) + (-1) ⬝ 1) =\n fromBlocks (2 • 1) (2 • 0) (2 • 0) (2 • -1)",
"state_before": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ fromBlocks (1 ⬝ 1 + 1 ⬝ 1) (1 ⬝ (-1) + 1 ⬝ 1) (1 ⬝ 1 + (-1) ⬝ 1) (1 ⬝ (-1) + (-1) ⬝ 1) =\n fromBlocks (2 • 1) (2 • 0) (2 • 0) (2 • -1)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\nh : (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)\n⊢ fromBlocks (1 ⬝ 1 + 1 ⬝ 1) (1 ⬝ (-1) + 1 ⬝ 1) (1 ⬝ 1 + (-1) ⬝ 1) (1 ⬝ (-1) + (-1) ⬝ 1) =\n fromBlocks (2 • 1) (2 • 0) (2 • 0) (2 • -1)",
"tactic": "simp [two_smul]"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.110673\np : Type ?u.110676\nq : Type ?u.110679\nl : Type u_1\nR : Type u₂\ninst✝⁷ : DecidableEq n\ninst✝⁶ : DecidableEq p\ninst✝⁵ : DecidableEq q\ninst✝⁴ : DecidableEq l\ninst✝³ : CommRing R\ninst✝² : Fintype p\ninst✝¹ : Fintype q\ninst✝ : Fintype l\n⊢ (PD l R)ᵀ ⬝ JD l R = fromBlocks 1 1 1 (-1)",
"tactic": "simp [PD, JD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_multiply]"
}
] |
[
279,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
iSup_eq_iSup_finset'
|
[
{
"state_after": "F : Type ?u.419112\nα : Type u_1\nβ : Type ?u.419118\nγ : Type ?u.419121\nι : Type ?u.419124\nκ : Type ?u.419127\nι' : Sort u_2\ninst✝ : CompleteLattice α\ns : ι' → α\n⊢ (⨆ (x : PLift ι'), s (↑Equiv.plift x)) = ⨆ (i : PLift ι'), s i.down",
"state_before": "F : Type ?u.419112\nα : Type u_1\nβ : Type ?u.419118\nγ : Type ?u.419121\nι : Type ?u.419124\nκ : Type ?u.419127\nι' : Sort u_2\ninst✝ : CompleteLattice α\ns : ι' → α\n⊢ (⨆ (i : ι'), s i) = ⨆ (t : Finset (PLift ι')) (i : PLift ι') (_ : i ∈ t), s i.down",
"tactic": "rw [← iSup_eq_iSup_finset, ← Equiv.plift.surjective.iSup_comp]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.419112\nα : Type u_1\nβ : Type ?u.419118\nγ : Type ?u.419121\nι : Type ?u.419124\nκ : Type ?u.419127\nι' : Sort u_2\ninst✝ : CompleteLattice α\ns : ι' → α\n⊢ (⨆ (x : PLift ι'), s (↑Equiv.plift x)) = ⨆ (i : PLift ι'), s i.down",
"tactic": "rfl"
}
] |
[
1827,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1825,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_type
|
[] |
[
160,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_lt
|
[] |
[
288,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
right_sub_midpoint
|
[] |
[
157,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_two_pi_sub
|
[] |
[
1165,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_lt_zpow
|
[] |
[
355,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_of_range_eq_ker
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : IsNoetherian R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhf : Injective ↑f\nhg : Surjective ↑g\nh : LinearMap.range f = LinearMap.ker g\n⊢ ∀ (a : Submodule R N), map f (comap f a) = a ⊓ LinearMap.range f",
"tactic": "simp [Submodule.map_comap_eq, inf_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherian R M\ninst✝ : IsNoetherian R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhf : Injective ↑f\nhg : Surjective ↑g\nh : LinearMap.range f = LinearMap.ker g\n⊢ ∀ (a : Submodule R N), comap g (map g a) = a ⊔ LinearMap.range f",
"tactic": "simp [Submodule.comap_map_eq, h]"
}
] |
[
406,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/CategoryTheory/Sites/Pretopology.lean
|
CategoryTheory.Pretopology.toGrothendieck_bot
|
[] |
[
218,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.surjective
|
[] |
[
563,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
11
] |
Mathlib/NumberTheory/Multiplicity.lean
|
multiplicity.Nat.pow_sub_pow
|
[
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n\ncase inr\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : x ≤ y\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n",
"state_before": "R : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n",
"tactic": "obtain hyx | hyx := le_total y x"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity ↑p ↑(x ^ n - y ^ n) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n",
"tactic": "iterate 2 rw [← Int.coe_nat_multiplicity]"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity ↑p ↑(x ^ n - y ^ n) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"tactic": "rw [Int.ofNat_sub (Nat.pow_le_pow_of_le_left hyx n)]"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑(x - y)\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"tactic": "rw [← Int.coe_nat_dvd] at hxy hx"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑x - ↑y\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity (↑p) (↑x - ↑y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑(x - y)\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"tactic": "rw [Int.coe_nat_sub hyx] at *"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑x - ↑y\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑x ^ n - ↑y ^ n) = multiplicity (↑p) (↑x - ↑y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑x - ↑y\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑(x ^ n) - ↑(y ^ n)) = multiplicity (↑p) (↑x - ↑y) + multiplicity p n",
"tactic": "push_cast at *"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑x - ↑y\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity (↑p) (↑x ^ n - ↑y ^ n) = multiplicity (↑p) (↑x - ↑y) + multiplicity p n",
"tactic": "exact Int.pow_sub_pow hp hp1 hxy hx n"
},
{
"state_after": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity ↑p ↑(x ^ n - y ^ n) = multiplicity ↑p ↑(x - y) + multiplicity p n",
"state_before": "case inl\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : y ≤ x\n⊢ multiplicity ↑p ↑(x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n",
"tactic": "rw [← Int.coe_nat_multiplicity]"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type ?u.838285\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhyx : x ≤ y\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n",
"tactic": "simp only [Nat.sub_eq_zero_iff_le.mpr hyx,\n Nat.sub_eq_zero_iff_le.mpr (Nat.pow_le_pow_of_le_left hyx n), multiplicity.zero,\n PartENat.top_add]"
}
] |
[
241,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Topology/Basic.lean
|
isOpen_univ
|
[] |
[
122,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.494047\nE : Type ?u.494050\nF : Type ?u.494053\nG : Type ?u.494056\nE' : Type ?u.494059\nF' : Type ?u.494062\nG' : Type ?u.494065\nE'' : Type ?u.494068\nF'' : Type ?u.494071\nG'' : Type ?u.494074\nR : Type u_1\nR' : Type ?u.494080\n𝕜 : Type u_3\n𝕜' : Type ?u.494086\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝ : NormOneClass R\nf : α → R\ng : α → 𝕜\nh : IsBigOWith c l f g\n⊢ IsBigOWith (c ^ 0) l (fun x => f x ^ 0) fun x => g x ^ 0",
"tactic": "simpa using h.pow' 0"
}
] |
[
1623,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1620,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.mk_Iic_real
|
[] |
[
263,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BoundedOrderHom.ext
|
[] |
[
600,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.isIdeal
|
[] |
[
141,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
11
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.open_source'
|
[
{
"state_after": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ (localTrivAsLocalEquiv Z i).source ∈\n ⋃ (i : ι) (s : Set (B × F)) (_ : IsOpen s),\n {(localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' s}",
"state_before": "ι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ IsOpen (localTrivAsLocalEquiv Z i).source",
"tactic": "apply TopologicalSpace.GenerateOpen.basic"
},
{
"state_after": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ ∃ i_1 i_2,\n IsOpen i_2 ∧\n (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i_1).source ∩ ↑(localTrivAsLocalEquiv Z i_1) ⁻¹' i_2",
"state_before": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ (localTrivAsLocalEquiv Z i).source ∈\n ⋃ (i : ι) (s : Set (B × F)) (_ : IsOpen s),\n {(localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' s}",
"tactic": "simp only [exists_prop, mem_iUnion, mem_singleton_iff]"
},
{
"state_after": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ (localTrivAsLocalEquiv Z i).source =\n (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ",
"state_before": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ ∃ i_1 i_2,\n IsOpen i_2 ∧\n (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i_1).source ∩ ↑(localTrivAsLocalEquiv Z i_1) ⁻¹' i_2",
"tactic": "refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩"
},
{
"state_after": "case a.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\np : TotalSpace Z\n⊢ p ∈ (localTrivAsLocalEquiv Z i).source ↔\n p ∈ (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ",
"state_before": "case a\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\n⊢ (localTrivAsLocalEquiv Z i).source =\n (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ",
"tactic": "ext p"
},
{
"state_after": "no goals",
"state_before": "case a.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.35703\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ : ι\nb : B\na : F\ni : ι\np : TotalSpace Z\n⊢ p ∈ (localTrivAsLocalEquiv Z i).source ↔\n p ∈ (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ",
"tactic": "simp only [localTrivAsLocalEquiv_apply, prod_mk_mem_set_prod_eq, mem_inter_iff, and_self_iff,\n mem_localTrivAsLocalEquiv_source, and_true, mem_univ, mem_preimage]"
}
] |
[
581,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/RingTheory/Localization/InvSubmonoid.lean
|
IsLocalization.finiteType_of_monoid_fg
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\nthis : Monoid.FG { x // x ∈ invSubmonoid M S }\n⊢ Algebra.FiniteType R S",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\n⊢ Algebra.FiniteType R S",
"tactic": "have := Monoid.fg_of_surjective _ (toInvSubmonoid_surjective M S)"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\nthis : Submonoid.FG (invSubmonoid M S)\n⊢ Algebra.FiniteType R S",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\nthis : Monoid.FG { x // x ∈ invSubmonoid M S }\n⊢ Algebra.FiniteType R S",
"tactic": "rw [Monoid.fg_iff_submonoid_fg] at this"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ Algebra.FiniteType R S",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\nthis : Submonoid.FG (invSubmonoid M S)\n⊢ Algebra.FiniteType R S",
"tactic": "rcases this with ⟨s, hs⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ Algebra.adjoin R ↑s = ⊤",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ Algebra.FiniteType R S",
"tactic": "refine' ⟨⟨s, _⟩⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ ⊤ ≤ Algebra.adjoin R ↑s",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ Algebra.adjoin R ↑s = ⊤",
"tactic": "rw [eq_top_iff]"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ Algebra.adjoin R ↑s",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\n⊢ ⊤ ≤ Algebra.adjoin R ↑s",
"tactic": "rintro x -"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R ↑s))",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ Algebra.adjoin R ↑s",
"tactic": "change x ∈ (Subalgebra.toSubmodule (Algebra.adjoin R _ : Subalgebra R S) : Set S)"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ ↑⊤",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R ↑s))",
"tactic": "rw [Algebra.adjoin_eq_span, hs, span_invSubmonoid]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.287534\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\ninst✝ : Monoid.FG { x // x ∈ M }\ns : Finset S\nhs : Submonoid.closure ↑s = invSubmonoid M S\nx : S\n⊢ x ∈ ↑⊤",
"tactic": "trivial"
}
] |
[
130,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/CategoryTheory/Over.lean
|
CategoryTheory.Over.over_right
|
[
{
"state_after": "no goals",
"state_before": "T : Type u₁\ninst✝ : Category T\nX : T\nU : Over X\n⊢ U.right = { as := PUnit.unit }",
"tactic": "simp only"
}
] |
[
70,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
Rat.toNNRat_one
|
[] |
[
344,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/Fintype/Powerset.lean
|
Fintype.card_set
|
[] |
[
77,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.add_right
|
[] |
[
207,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj.comp_eq
|
[] |
[
44,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
11
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.outer_Ioc
|
[
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) = ofReal (↑f b - ↑f a)",
"tactic": "refine'\n le_antisymm\n (by\n rw [← f.length_Ioc]\n apply outer_le_length)\n (le_iInf₂ fun s hs => ENNReal.le_of_forall_pos_le_add fun ε εpos h => _)"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "let δ := ε / 2"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "have δpos : 0 < (δ : ℝ≥0∞) := by simpa using εpos.ne'"
},
{
"state_after": "case intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "rcases ENNReal.exists_pos_sum_of_countable δpos.ne' ℕ with ⟨ε', ε'0, hε⟩"
},
{
"state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "case intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "obtain ⟨a', ha', aa'⟩ : ∃ a', f a' - f a < δ ∧ a < a' := by\n have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n exact (f.right_continuous a).mono Ioi_subset_Ici_self\n have B : f a - f a < δ := by rwa [sub_self, NNReal.coe_pos, ← ENNReal.coe_pos]\n exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists"
},
{
"state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\nthis : ∀ (i : ℕ), ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "have : ∀ i, ∃ p : ℝ × ℝ, s i ⊆ Ioo p.1 p.2 ∧\n (ofReal (f p.2 - f p.1) : ℝ≥0∞) < f.length (s i) + ε' i := by\n intro i\n have hl :=\n ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (ε'0 i).ne')\n conv at hl =>\n lhs\n rw [length]\n simp only [iInf_lt_iff, exists_prop] at hl\n rcases hl with ⟨p, q', spq, hq'⟩\n have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n exact (f.right_continuous q').mono Ioi_subset_Ici_self\n rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with ⟨q, hq, q'q⟩\n exact ⟨⟨p, q⟩, spq.trans (Ioc_subset_Ioo_right q'q), hq⟩"
},
{
"state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\nthis : ∀ (i : ℕ), ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "choose g hg using this"
},
{
"state_after": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "have I_subset : Icc a' b ⊆ ⋃ i, Ioo (g i).1 (g i).2 :=\n calc\n Icc a' b ⊆ Ioc a b := fun x hx => ⟨aa'.trans_le hx.1, hx.2⟩\n _ ⊆ ⋃ i, s i := hs\n _ ⊆ ⋃ i, Ioo (g i).1 (g i).2 := iUnion_mono fun i => (hg i).1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ ofReal (↑f b - ↑f a) ≤ (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "calc\n ofReal (f b - f a) = ofReal (f b - f a' + (f a' - f a)) := by rw [sub_add_sub_cancel]\n _ ≤ ofReal (f b - f a') + ofReal (f a' - f a) := ENNReal.ofReal_add_le\n _ ≤ (∑' i, ofReal (f (g i).2 - f (g i).1)) + ofReal δ :=\n (add_le_add (f.length_subadditive_Icc_Ioo I_subset) (ENNReal.ofReal_le_ofReal ha'.le))\n _ ≤ (∑' i, f.length (s i) + ε' i) + δ :=\n (add_le_add (ENNReal.tsum_le_tsum fun i => (hg i).2.le)\n (by simp only [ENNReal.ofReal_coe_nnreal, le_rfl]))\n _ = (∑' i, f.length (s i)) + (∑' i, (ε' i : ℝ≥0∞)) + δ := by rw [ENNReal.tsum_add]\n _ ≤ (∑' i, f.length (s i)) + δ + δ := (add_le_add (add_le_add le_rfl hε.le) le_rfl)\n _ = (∑' i : ℕ, f.length (s i)) + ε := by simp [add_assoc, ENNReal.add_halves]"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) ≤ length f (Ioc a b)",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) ≤ ofReal (↑f b - ↑f a)",
"tactic": "rw [← f.length_Ioc]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) ≤ length f (Ioc a b)",
"tactic": "apply outer_le_length"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\n⊢ 0 < ↑δ",
"tactic": "simpa using εpos.ne'"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\nA : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a\n⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'",
"tactic": "have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n exact (f.right_continuous a).mono Ioi_subset_Ici_self"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\nA : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a\nB : ↑f a - ↑f a < ↑δ\n⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\nA : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a\n⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'",
"tactic": "have B : f a - f a < δ := by rwa [sub_self, NNReal.coe_pos, ← ENNReal.coe_pos]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\nA : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a\nB : ↑f a - ↑f a < ↑δ\n⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'",
"tactic": "exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ContinuousWithinAt (fun r => ↑f r) (Ioi a) a",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a",
"tactic": "refine' ContinuousWithinAt.sub _ continuousWithinAt_const"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\n⊢ ContinuousWithinAt (fun r => ↑f r) (Ioi a) a",
"tactic": "exact (f.right_continuous a).mono Ioi_subset_Ici_self"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\nA : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a\n⊢ ↑f a - ↑f a < ↑δ",
"tactic": "rwa [sub_self, NNReal.coe_pos, ← ENNReal.coe_pos]"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\n⊢ ∀ (i : ℕ), ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "intro i"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : length f (s i) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "have hl :=\n ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (ε'0 i).ne')"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : (⨅ (a : ℝ) (b : ℝ) (_ : s i ⊆ Ioc a b), ofReal (↑f b - ↑f a)) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : length f (s i) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "conv at hl =>\n lhs\n rw [length]"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : ∃ i_1 i_2, s i ⊆ Ioc i_1 i_2 ∧ ofReal (↑f i_2 - ↑f i_1) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : (⨅ (a : ℝ) (b : ℝ) (_ : s i ⊆ Ioc a b), ofReal (↑f b - ↑f a)) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "simp only [iInf_lt_iff, exists_prop] at hl"
},
{
"state_after": "case intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\nhl : ∃ i_1 i_2, s i ⊆ Ioc i_1 i_2 ∧ ofReal (↑f i_2 - ↑f i_1) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "rcases hl with ⟨p, q', spq, hq'⟩"
},
{
"state_after": "case intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\nthis : ContinuousWithinAt (fun r => ofReal (↑f r - ↑f p)) (Ioi q') q'\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "case intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n exact (f.right_continuous q').mono Ioi_subset_Ici_self"
},
{
"state_after": "case intro.intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\nthis : ContinuousWithinAt (fun r => ofReal (↑f r - ↑f p)) (Ioi q') q'\nq : ℝ\nhq : ofReal (↑f q - ↑f p) < length f (s i) + ↑(ε' i)\nq'q : q' < q\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"state_before": "case intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\nthis : ContinuousWithinAt (fun r => ofReal (↑f r - ↑f p)) (Ioi q') q'\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with ⟨q, hq, q'q⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nf : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\nthis : ContinuousWithinAt (fun r => ofReal (↑f r - ↑f p)) (Ioi q') q'\nq : ℝ\nhq : ofReal (↑f q - ↑f p) < length f (s i) + ↑(ε' i)\nq'q : q' < q\n⊢ ∃ p, s i ⊆ Ioo p.fst p.snd ∧ ofReal (↑f p.snd - ↑f p.fst) < length f (s i) + ↑(ε' i)",
"tactic": "exact ⟨⟨p, q⟩, spq.trans (Ioc_subset_Ioo_right q'q), hq⟩"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ContinuousWithinAt (fun r => ↑f r - ↑f p) (Ioi q') q'",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ContinuousWithinAt (fun r => ofReal (↑f r - ↑f p)) (Ioi q') q'",
"tactic": "apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt"
},
{
"state_after": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ContinuousWithinAt (fun r => ↑f r) (Ioi q') q'",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ContinuousWithinAt (fun r => ↑f r - ↑f p) (Ioi q') q'",
"tactic": "refine' ContinuousWithinAt.sub _ continuousWithinAt_const"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ni : ℕ\np q' : ℝ\nspq : s i ⊆ Ioc p q'\nhq' : ofReal (↑f q' - ↑f p) < length f (s i) + ↑(ε' i)\n⊢ ContinuousWithinAt (fun r => ↑f r) (Ioi q') q'",
"tactic": "exact (f.right_continuous q').mono Ioi_subset_Ici_self"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ ofReal (↑f b - ↑f a) = ofReal (↑f b - ↑f a' + (↑f a' - ↑f a))",
"tactic": "rw [sub_add_sub_cancel]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ ofReal ↑δ ≤ ↑δ",
"tactic": "simp only [ENNReal.ofReal_coe_nnreal, le_rfl]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ (∑' (i : ℕ), length f (s i) + ↑(ε' i)) + ↑δ = ((∑' (i : ℕ), length f (s i)) + ∑' (i : ℕ), ↑(ε' i)) + ↑δ",
"tactic": "rw [ENNReal.tsum_add]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\ns : ℕ → Set ℝ\nhs : Ioc a b ⊆ ⋃ (i : ℕ), s i\nε : ℝ≥0\nεpos : 0 < ε\nh : (∑' (i : ℕ), length f (s i)) < ⊤\nδ : ℝ≥0 := ε / 2\nδpos : 0 < ↑δ\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑δ\na' : ℝ\nha' : ↑f a' - ↑f a < ↑δ\naa' : a < a'\ng : ℕ → ℝ × ℝ\nhg : ∀ (i : ℕ), s i ⊆ Ioo (g i).fst (g i).snd ∧ ofReal (↑f (g i).snd - ↑f (g i).fst) < length f (s i) + ↑(ε' i)\nI_subset : Icc a' b ⊆ ⋃ (i : ℕ), Ioo (g i).fst (g i).snd\n⊢ (∑' (i : ℕ), length f (s i)) + ↑δ + ↑δ = (∑' (i : ℕ), length f (s i)) + ↑ε",
"tactic": "simp [add_assoc, ENNReal.add_halves]"
}
] |
[
447,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
dist_le_of_le_geometric_two_of_tendsto₀
|
[] |
[
417,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Std/Logic.lean
|
iff_true_right
|
[] |
[
65,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
65,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean
|
dslope_of_ne
|
[] |
[
46,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
two_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocSemiring α\nn : α\n⊢ 1 * n + 1 * n = n + n",
"tactic": "rw [one_mul]"
}
] |
[
181,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Topology/Algebra/MulAction.lean
|
ContinuousAt.smul
|
[] |
[
112,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.as_sum_range'
|
[] |
[
425,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_univ
|
[] |
[
352,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion
|
[
{
"state_after": "case hb\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsTopologicalBasis (range fun i => (b i).carrier)\n\ncase hb'\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"state_before": "ι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), ↑(b i)",
"tactic": "apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1"
},
{
"state_after": "case h.e'_3\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ (range fun i => (b i).carrier) = SetLike.coe '' range b",
"state_before": "case hb\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ IsTopologicalBasis (range fun i => (b i).carrier)",
"tactic": "convert (config := {transparency := .default}) hb"
},
{
"state_after": "case h.e'_3.h\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU x✝ : Set α\n⊢ (x✝ ∈ range fun i => (b i).carrier) ↔ x✝ ∈ SetLike.coe '' range b",
"state_before": "case h.e'_3\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ (range fun i => (b i).carrier) = SetLike.coe '' range b",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU x✝ : Set α\n⊢ (x✝ ∈ range fun i => (b i).carrier) ↔ x✝ ∈ SetLike.coe '' range b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case hb'\nι✝ : Type ?u.35576\nα : Type u_2\nβ : Type ?u.35582\nγ : Type ?u.35585\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"tactic": "exact hb'"
}
] |
[
337,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMinOn.fderivWithin_eq_zero
|
[
{
"state_after": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑0 y = 0",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑(fderivWithin ℝ f s a) y = 0",
"tactic": "rw [fderivWithin_zero_of_not_differentiableWithinAt hf]"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y ∈ posTangentConeAt s a\nhy' : -y ∈ posTangentConeAt s a\nhf : ¬DifferentiableWithinAt ℝ f s a\n⊢ ↑0 y = 0",
"tactic": "rfl"
}
] |
[
185,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
abs_pow
|
[] |
[
707,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_sSup_of_directedOn
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : R\nthis : Nonempty ↑S\n⊢ x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s",
"tactic": "haveI : Nonempty S := Sne.to_subtype"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x ≤ x_1) S\nx : R\nthis : Nonempty ↑S\n⊢ x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s",
"tactic": "simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,\n exists_prop]"
}
] |
[
1110,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1106,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.Nontrivial.image
|
[] |
[
1257,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1254,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph0_le_aleph
|
[
{
"state_after": "o : Ordinal\n⊢ ω ≤ ω + o",
"state_before": "o : Ordinal\n⊢ ℵ₀ ≤ aleph o",
"tactic": "rw [aleph, aleph0_le_aleph']"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\n⊢ ω ≤ ω + o",
"tactic": "apply Ordinal.le_add_right"
}
] |
[
294,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.graph_fst_eq_zero_snd
|
[] |
[
828,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Sized.node3L
|
[] |
[
376,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.prod_divisorsAntidiagonal'
|
[
{
"state_after": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ x in divisorsAntidiagonal n,\n f (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).fst (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).snd =\n ∏ i in divisors n, f (n / i) i",
"state_before": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ i in divisorsAntidiagonal n, f i.fst i.snd = ∏ i in divisors n, f (n / i) i",
"tactic": "rw [← map_swap_divisorsAntidiagonal, Finset.prod_map]"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ x in divisorsAntidiagonal n,\n f (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).fst (↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ)) x).snd =\n ∏ i in divisors n, f (n / i) i",
"tactic": "exact prod_divisorsAntidiagonal fun i j => f j i"
}
] |
[
480,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMaxOn.inter
|
[] |
[
125,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.normed_neg
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type ?u.1158619\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace ℝ X\ninst✝¹ : HasContDiffBump E\nc : E\nf✝ : ContDiffBump c\nx✝ : E\nn : ℕ∞\ninst✝ : MeasurableSpace E\nμ : MeasureTheory.Measure E\nf : ContDiffBump 0\nx : E\n⊢ ContDiffBump.normed f μ (-x) = ContDiffBump.normed f μ x",
"tactic": "simp_rw [f.normed_def, f.neg]"
}
] |
[
509,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/RingTheory/JacobsonIdeal.lean
|
Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot
|
[
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\n⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥",
"tactic": "have hf : Function.Surjective (Ideal.Quotient.mk I) := Submodule.Quotient.mk_surjective I"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical I = jacobson I → radical ⊥ = jacobson ⊥\n\ncase mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical ⊥ = jacobson ⊥ → radical I = jacobson I",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical I = jacobson I ↔ radical ⊥ = jacobson ⊥",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\n⊢ radical ⊥ = jacobson ⊥",
"state_before": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical I = jacobson I → radical ⊥ = jacobson ⊥",
"tactic": "intro h"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\nthis : map (Quotient.mk I) (radical I) = map (Quotient.mk I) (jacobson I)\n⊢ radical ⊥ = jacobson ⊥",
"state_before": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\n⊢ radical ⊥ = jacobson ⊥",
"tactic": "have := congr_arg (map (Ideal.Quotient.mk I)) h"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\nthis : radical (map (Quotient.mk I) I) = jacobson (map (Quotient.mk I) I)\n⊢ radical ⊥ = jacobson ⊥",
"state_before": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\nthis : map (Quotient.mk I) (radical I) = map (Quotient.mk I) (jacobson I)\n⊢ radical ⊥ = jacobson ⊥",
"tactic": "rw [map_radical_of_surjective hf (le_of_eq mk_ker),\n map_jacobson_of_surjective hf (le_of_eq mk_ker)] at this"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical I = jacobson I\nthis : radical (map (Quotient.mk I) I) = jacobson (map (Quotient.mk I) I)\n⊢ radical ⊥ = jacobson ⊥",
"tactic": "simpa using this"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\n⊢ radical I = jacobson I",
"state_before": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\n⊢ radical ⊥ = jacobson ⊥ → radical I = jacobson I",
"tactic": "intro h"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\nthis : comap (Quotient.mk I) (radical ⊥) = comap (Quotient.mk I) (jacobson ⊥)\n⊢ radical I = jacobson I",
"state_before": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\n⊢ radical I = jacobson I",
"tactic": "have := congr_arg (comap (Ideal.Quotient.mk I)) h"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\nthis : radical (RingHom.ker (Quotient.mk I)) = jacobson (RingHom.ker (Quotient.mk I))\n⊢ radical I = jacobson I",
"state_before": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\nthis : comap (Quotient.mk I) (radical ⊥) = comap (Quotient.mk I) (jacobson ⊥)\n⊢ radical I = jacobson I",
"tactic": "rw [comap_radical, comap_jacobson_of_surjective hf,\n ← RingHom.ker_eq_comap_bot (Ideal.Quotient.mk I)] at this"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nhf : Function.Surjective ↑(Quotient.mk I)\nh : radical ⊥ = jacobson ⊥\nthis : radical (RingHom.ker (Quotient.mk I)) = jacobson (RingHom.ker (Quotient.mk I))\n⊢ radical I = jacobson I",
"tactic": "simpa using this"
}
] |
[
305,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.AdjoinSimple.isIntegral_gen
|
[
{
"state_after": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\n⊢ IsIntegral F (gen F α) ↔ IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (gen F α))",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\n⊢ IsIntegral F (gen F α) ↔ IsIntegral F α",
"tactic": "conv_rhs => rw [← AdjoinSimple.algebraMap_gen F α]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\n⊢ IsIntegral F (gen F α) ↔ IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (gen F α))",
"tactic": "rw [isIntegral_algebraMap_iff (algebraMap F⟮α⟯ E).injective]"
}
] |
[
502,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.surjective
|
[
{
"state_after": "case inl\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhE : Subsingleton E\n⊢ Surjective f\n\ncase inr\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ Surjective f",
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"tactic": "cases' hc with hE hc"
},
{
"state_after": "case inl\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhE : Subsingleton E\nthis : Subsingleton F\n⊢ Surjective f",
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"tactic": "haveI : Subsingleton F := (Equiv.subsingleton_congr f'.toEquiv).1 hE"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhE : Subsingleton E\nthis : Subsingleton F\n⊢ Surjective f",
"tactic": "exact surjective_to_subsingleton _"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"state_before": "case inr\n𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ Surjective f",
"tactic": "apply forall_of_forall_mem_closedBall (fun y : F => ∃ a, f a = y) (f 0) _"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"tactic": "have hc' : (0 : ℝ) < N⁻¹ - c := by rw [sub_pos]; exact hc"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"tactic": "let p : ℝ → Prop := fun R => closedBall (f 0) R ⊆ Set.range f"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhp : ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"tactic": "have hp : ∀ᶠ r : ℝ in atTop, p ((N⁻¹ - c) * r) := by\n have hr : ∀ᶠ r : ℝ in atTop, 0 ≤ r := eventually_ge_atTop 0\n refine' hr.mono fun r hr => Subset.trans _ (image_subset_range f (closedBall 0 r))\n refine' hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse hr _\n exact subset_univ _"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhp : ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)\n⊢ ∀ (x : ℝ), p x → ∀ (y : F), y ∈ closedBall (f 0) x → (fun y => ∃ a, f a = y) y",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhp : ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)\n⊢ ∃ᶠ (R : ℝ) in atTop, ∀ (y : F), y ∈ closedBall (f 0) R → (fun y => ∃ a, f a = y) y",
"tactic": "refine' ((tendsto_id.const_mul_atTop hc').frequently hp.frequently).mono _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhp : ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)\n⊢ ∀ (x : ℝ), p x → ∀ (y : F), y ∈ closedBall (f 0) x → (fun y => ∃ a, f a = y) y",
"tactic": "exact fun R h y hy => h hy"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ ↑c < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c",
"tactic": "rw [sub_pos]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\n⊢ ↑c < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹",
"tactic": "exact hc"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\n⊢ ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\n⊢ ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)",
"tactic": "have hr : ∀ᶠ r : ℝ in atTop, 0 ≤ r := eventually_ge_atTop 0"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr✝ : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall (f 0) ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r) ⊆ f '' closedBall 0 r",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\n⊢ ∀ᶠ (r : ℝ) in atTop, p ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r)",
"tactic": "refine' hr.mono fun r hr => Subset.trans _ (image_subset_range f (closedBall 0 r))"
},
{
"state_after": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr✝ : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall 0 r ⊆ univ",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr✝ : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall (f 0) ((↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c) * r) ⊆ f '' closedBall 0 r",
"tactic": "refine' hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse hr _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type ?u.379684\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nG' : Type ?u.379787\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nε : ℝ\ninst✝¹ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc : c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nhc' : 0 < ↑‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - ↑c\np : ℝ → Prop := fun R => closedBall (f 0) R ⊆ range f\nhr✝ : ∀ᶠ (r : ℝ) in atTop, 0 ≤ r\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall 0 r ⊆ univ",
"tactic": "exact subset_univ _"
}
] |
[
401,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
11
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_sqrt_two
|
[
{
"state_after": "no goals",
"state_before": "⊢ Irrational (Real.sqrt 2)",
"tactic": "simpa using Nat.prime_two.irrational_sqrt"
}
] |
[
111,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
AEMeasurable.max
|
[] |
[
798,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
8
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_empty_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.16420\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t✝ s₁ s₂ t₁ t₂ u : Set E\nx y : E\nt : Set E\n⊢ convexJoin 𝕜 ∅ t = ∅",
"tactic": "simp [convexJoin]"
}
] |
[
62,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfp_mul_one
|
[
{
"state_after": "a : Ordinal\nha : 0 < a\n⊢ (fun a_1 => sup fun n => ((fun x => a * x)^[n]) a_1) 1 = sup fun n => a ^ ↑n\n\na : Ordinal\nha : 0 < a\n⊢ 0 < a",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ nfp (fun x => a * x) 1 = a ^ ω",
"tactic": "rw [← sup_iterate_eq_nfp, ← sup_opow_nat]"
},
{
"state_after": "a : Ordinal\nha : 0 < a\n⊢ (sup fun n => ((fun x => a * x)^[n]) 1) = sup fun n => a ^ ↑n",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ (fun a_1 => sup fun n => ((fun x => a * x)^[n]) a_1) 1 = sup fun n => a ^ ↑n",
"tactic": "dsimp"
},
{
"state_after": "case e_f\na : Ordinal\nha : 0 < a\n⊢ (fun n => ((fun x => a * x)^[n]) 1) = fun n => a ^ ↑n",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ (sup fun n => ((fun x => a * x)^[n]) 1) = sup fun n => a ^ ↑n",
"tactic": "congr"
},
{
"state_after": "case e_f.h\na : Ordinal\nha : 0 < a\nn : ℕ\n⊢ ((fun x => a * x)^[n]) 1 = a ^ ↑n",
"state_before": "case e_f\na : Ordinal\nha : 0 < a\n⊢ (fun n => ((fun x => a * x)^[n]) 1) = fun n => a ^ ↑n",
"tactic": "funext n"
},
{
"state_after": "case e_f.h.zero\na : Ordinal\nha : 0 < a\n⊢ ((fun x => a * x)^[Nat.zero]) 1 = a ^ ↑Nat.zero\n\ncase e_f.h.succ\na : Ordinal\nha : 0 < a\nn : ℕ\nhn : ((fun x => a * x)^[n]) 1 = a ^ ↑n\n⊢ ((fun x => a * x)^[Nat.succ n]) 1 = a ^ ↑(Nat.succ n)",
"state_before": "case e_f.h\na : Ordinal\nha : 0 < a\nn : ℕ\n⊢ ((fun x => a * x)^[n]) 1 = a ^ ↑n",
"tactic": "induction' n with n hn"
},
{
"state_after": "case e_f.h.succ\na : Ordinal\nha : 0 < a\nn : ℕ\nhn : ((fun x => a * x)^[n]) 1 = a ^ ↑n\n⊢ ((fun x => a * x)^[Nat.succ n]) 1 = a ^ ↑(1 + n)",
"state_before": "case e_f.h.succ\na : Ordinal\nha : 0 < a\nn : ℕ\nhn : ((fun x => a * x)^[n]) 1 = a ^ ↑n\n⊢ ((fun x => a * x)^[Nat.succ n]) 1 = a ^ ↑(Nat.succ n)",
"tactic": "nth_rw 2 [Nat.succ_eq_one_add]"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.succ\na : Ordinal\nha : 0 < a\nn : ℕ\nhn : ((fun x => a * x)^[n]) 1 = a ^ ↑n\n⊢ ((fun x => a * x)^[Nat.succ n]) 1 = a ^ ↑(1 + n)",
"tactic": "rw [Nat.cast_add, Nat.cast_one, opow_add, opow_one, iterate_succ_apply', hn]"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.zero\na : Ordinal\nha : 0 < a\n⊢ ((fun x => a * x)^[Nat.zero]) 1 = a ^ ↑Nat.zero",
"tactic": "rw [Nat.cast_zero, opow_zero, iterate_zero_apply]"
},
{
"state_after": "no goals",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ 0 < a",
"tactic": "exact ha"
}
] |
[
625,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedLinearMap.continuous
|
[] |
[
178,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
lt_mul_of_one_lt_of_lt_of_nonneg
|
[] |
[
910,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.rotateL
|
[
{
"state_after": "case nil\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) nil o₂\nH1 : ¬size l + size nil ≤ 1\nH2 : delta * size l < size nil\nH3 : 2 * size nil ≤ 9 * size l + 5 ∨ size nil ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x nil) o₂\n\ncase node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l < size (Ordnode.node rs rl rx rr)\nH3 : 2 * size (Ordnode.node rs rl rx rr) ≤ 9 * size l + 5 ∨ size (Ordnode.node rs rl rx rr) ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size l < size r\nH3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x r) o₂",
"tactic": "cases' r with rs rl rx rr"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * size (Ordnode.node rs rl rx rr) ≤ 9 * size l + 5 ∨ size (Ordnode.node rs rl rx rr) ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l < size (Ordnode.node rs rl rx rr)\nH3 : 2 * size (Ordnode.node rs rl rx rr) ≤ 9 * size l + 5 ∨ size (Ordnode.node rs rl rx rr) ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "rw [hr.2.size_eq, Nat.lt_succ_iff] at H2"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr + 1) ≤ 9 * size l + 5 ∨ size rl + size rr + 1 ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * size (Ordnode.node rs rl rx rr) ≤ 9 * size l + 5 ∨ size (Ordnode.node rs rl rx rr) ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "rw [hr.2.size_eq] at H3"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr + 1) ≤ 9 * size l + 5 ∨ size rl + size rr + 1 ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=\n H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by\n intro l0; rw [l0] at H3\n exact\n (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>\n (or_iff_left_of_imp <| by intro; linarith).1 H3"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros ; linarith"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "have hlp : size l > 0 → ¬size rl + size rr ≤ 1 := fun l0 hb =>\n absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by decide)"
},
{
"state_after": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\n⊢ Valid' o₁ (if size rl < ratio * size rr then Ordnode.node3L l x rl rx rr else Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\n⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂",
"tactic": "rw [rotateL]"
},
{
"state_after": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\n⊢ Valid' o₁ (Ordnode.node3L l x rl rx rr) o₂\n\ncase node.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\n⊢ Valid' o₁ (if size rl < ratio * size rr then Ordnode.node3L l x rl rx rr else Ordnode.node4L l x rl rx rr) o₂",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) nil o₂\nH1 : ¬size l + size nil ≤ 1\nH2 : delta * size l < size nil\nH3 : 2 * size nil ≤ 9 * size l + 5 ∨ size nil ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateL l x nil) o₂",
"tactic": "cases H2"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\n⊢ size rl + size rr ≤ 2",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\n⊢ size l = 0 → size rl + size rr ≤ 2",
"tactic": "intro l0"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\n⊢ size rl + size rr ≤ 2",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\n⊢ size rl + size rr ≤ 2",
"tactic": "rw [l0] at H3"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\n⊢ size rl + size rr ≤ 2",
"tactic": "exact\n (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\nh : 2 * (size rl + size rr) ≤ 9 * 0 + 3\n⊢ 0 < 2",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2\nl0 : size l = 0\nh : 2 * (size rl + size rr) ≤ 9 * 0 + 3\n⊢ 9 * 0 + 3 ≤ 2 * 2",
"tactic": "decide"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nl0 : 1 ≤ size l\na✝ : size rl + size rr ≤ 2\n⊢ 2 * (size rl + size rr) ≤ 9 * size l + 3",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nl0 : 1 ≤ size l\n⊢ size rl + size rr ≤ 2 → 2 * (size rl + size rr) ≤ 9 * size l + 3",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nl0 : 1 ≤ size l\na✝ : size rl + size rr ≤ 2\n⊢ 2 * (size rl + size rr) ≤ 9 * size l + 3",
"tactic": "linarith"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\na✝² b✝ : ℕ\na✝¹ : 1 ≤ a✝²\na✝ : a✝² + b✝ ≤ 2\n⊢ b✝ ≤ 1",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\n⊢ ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\na✝² b✝ : ℕ\na✝¹ : 1 ≤ a✝²\na✝ : a✝² + b✝ ≤ 2\n⊢ b✝ ≤ 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nl0 : size l > 0\nhb : size rl + size rr ≤ 1\n⊢ ¬delta * Nat.succ 0 ≤ 1",
"tactic": "decide"
},
{
"state_after": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\n⊢ Valid' o₁ (Ordnode.node3L l x rl rx rr) o₂",
"state_before": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\n⊢ Valid' o₁ (Ordnode.node3L l x rl rx rr) o₂",
"tactic": "have rr0 : size rr > 0 :=\n (mul_lt_mul_left (by decide)).1 (lt_of_le_of_lt (Nat.zero_le _) h : ratio * 0 < _)"
},
{
"state_after": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"state_before": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\n⊢ Valid' o₁ (Ordnode.node3L l x rl rx rr) o₂",
"tactic": "suffices BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr) by\n exact hl.node3L hr.left hr.right this.1 this.2"
},
{
"state_after": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)\n\ncase node.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"state_before": "case node.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"tactic": "cases' Nat.eq_zero_or_pos (size l) with l0 l0"
},
{
"state_after": "case node.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"state_before": "case node.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"tactic": "replace H3 := H3p l0"
},
{
"state_after": "case node.inl.inr.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"state_before": "case node.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"tactic": "rcases hr.3.1.resolve_left (hlp l0) with ⟨_, hb₂⟩"
},
{
"state_after": "case node.inl.inr.intro.refine'_1\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size l ≤ delta * size rl\n\ncase node.inl.inr.intro.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size rl ≤ delta * size l\n\ncase node.inl.inr.intro.refine'_3\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size l + size rl + 1 ≤ delta * size rr\n\ncase node.inl.inr.intro.refine'_4\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size rr ≤ delta * (size l + size rl + 1)",
"state_before": "case node.inl.inr.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"tactic": "refine' ⟨Or.inr ⟨_, _⟩, Or.inr ⟨_, _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\n⊢ 0 < ratio",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nthis : BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)\n⊢ Valid' o₁ (Ordnode.node3L l x rl rx rr) o₂",
"tactic": "exact hl.node3L hr.left hr.right this.1 this.2"
},
{
"state_after": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\n⊢ BalancedSz (size l) (size rl) ∧ BalancedSz (size l + size rl + 1) (size rr)",
"tactic": "rw [l0]"
},
{
"state_after": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "replace H3 := H3_0 l0"
},
{
"state_after": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "have := hr.3.1"
},
{
"state_after": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)\n\ncase node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "cases' Nat.eq_zero_or_pos (size rl) with rl0 rl0"
},
{
"state_after": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "have rr1 : size rr = 1 := le_antisymm (ablem rl0 H3) rr0"
},
{
"state_after": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rr + size rl ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"state_before": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "rw [add_comm] at H3"
},
{
"state_after": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rr + size rl ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 1 ∧ BalancedSz (0 + 1 + 1) 1",
"state_before": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rr + size rl ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "rw [rr1, show size rl = 1 from le_antisymm (ablem rr0 H3) rl0]"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rr + size rl ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl > 0\nrr1 : size rr = 1\n⊢ BalancedSz 0 1 ∧ BalancedSz (0 + 1 + 1) 1",
"tactic": "decide"
},
{
"state_after": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz 0 (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 0 ∧ BalancedSz (0 + 0 + 1) (size rr)",
"state_before": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz (size rl) (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 (size rl) ∧ BalancedSz (0 + size rl + 1) (size rr)",
"tactic": "rw [rl0] at this ⊢"
},
{
"state_after": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz 0 (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 0 ∧ BalancedSz (0 + 0 + 1) 1",
"state_before": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz 0 (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 0 ∧ BalancedSz (0 + 0 + 1) (size rr)",
"tactic": "rw [le_antisymm (balancedSz_zero.1 this.symm) rr0]"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nthis : BalancedSz 0 (size rr)\nrl0 : size rl = 0\n⊢ BalancedSz 0 0 ∧ BalancedSz (0 + 0 + 1) 1",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.intro.refine'_1\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size l ≤ delta * size rl",
"tactic": "exact Valid'.rotateL_lemma₁ H2 hb₂"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.intro.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size rl ≤ delta * size l",
"tactic": "exact Nat.le_of_lt_succ (Valid'.rotateL_lemma₂ H3 h)"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.intro.refine'_3\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size l + size rl + 1 ≤ delta * size rr",
"tactic": "exact Valid'.rotateL_lemma₃ H2 h"
},
{
"state_after": "no goals",
"state_before": "case node.inl.inr.intro.refine'_4\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : size rl < ratio * size rr\nrr0 : size rr > 0\nl0 : size l > 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3\nleft✝ : size rl ≤ delta * size rr\nhb₂ : size rr ≤ delta * size rl\n⊢ size rr ≤ delta * (size l + size rl + 1)",
"tactic": "exact\n le_trans hb₂\n (Nat.mul_le_mul_left _ <| le_trans (Nat.le_add_left _ _) (Nat.le_add_right _ _))"
},
{
"state_after": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂\n\ncase node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "cases' Nat.eq_zero_or_pos (size rl) with rl0 rl0"
},
{
"state_after": "case node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "refine' hl.node4L hr.left hr.right rl0 _"
},
{
"state_after": "case node.inr.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr\n\ncase node.inr.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "cases' Nat.eq_zero_or_pos (size l) with l0 l0"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "exact\n Or.inr ⟨l0, not_lt.1 h, H2, Valid'.rotateL_lemma₄ (H3p l0), (hr.3.1.resolve_left (hlp l0)).1⟩"
},
{
"state_after": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ratio = 0 ∨ size rr = 0\nrl0 : size rl = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "rw [rl0, not_lt, le_zero_iff, Nat.mul_eq_zero] at h"
},
{
"state_after": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ratio = 0 ∨ size rr = 0\nrl0 : size rl = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "replace h := h.resolve_left (by decide)"
},
{
"state_after": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "erw [rl0, h, le_zero_iff, Nat.mul_eq_zero] at H2"
},
{
"state_after": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬0 + (0 + 0 + 1) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"state_before": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "rw [hr.2.size_eq, rl0, h, H2.resolve_left (by decide)] at H1"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬0 + (0 + 0 + 1) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ Valid' o₁ (Ordnode.node4L l x rl rx rr) o₂",
"tactic": "cases H1 (by decide)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ratio = 0 ∨ size rr = 0\nrl0 : size rl = 0\n⊢ ¬ratio = 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + (0 + 0 + 1) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ ¬delta = 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬0 + (0 + 0 + 1) ≤ 1\nH2 : delta = 0 ∨ size l = 0\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nrl0 : size rl = 0\nh : size rr = 0\n⊢ 0 + (0 + 0 + 1) ≤ 1",
"tactic": "decide"
},
{
"state_after": "case node.inr.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "replace H3 := H3_0 l0"
},
{
"state_after": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr\n\ncase node.inr.inr.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "cases' Nat.eq_zero_or_pos (size rr) with rr0 rr0"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inl.inr\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr > 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "exact Or.inl ⟨l0, le_antisymm (ablem rr0 <| by rwa [add_comm]) rl0, ablem rl0 H3⟩"
},
{
"state_after": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\nthis : BalancedSz (size rl) (size rr)\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "have := hr.3.1"
},
{
"state_after": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\nthis : BalancedSz (size rl) 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"state_before": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\nthis : BalancedSz (size rl) (size rr)\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "rw [rr0] at this"
},
{
"state_after": "no goals",
"state_before": "case node.inr.inr.inl.inl\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr = 0\nthis : BalancedSz (size rl) 0\n⊢ size l = 0 ∧ size rl = 1 ∧ size rr ≤ 1 ∨\n 0 < size l ∧\n ratio * size rr ≤ size rl ∧\n delta * size l ≤ size rl + size rr ∧ 3 * (size rl + size rr) ≤ 16 * size l + 9 ∧ size rl ≤ delta * size rr",
"tactic": "exact Or.inl ⟨l0, le_antisymm (balancedSz_zero.1 this) rl0, rr0.symm ▸ zero_le_one⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nhr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂\nH1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1\nH2 : delta * size l ≤ size rl + size rr\nH3_0 : size l = 0 → size rl + size rr ≤ 2\nH3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3\nablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1\nhlp : size l > 0 → ¬size rl + size rr ≤ 1\nh : ¬size rl < ratio * size rr\nrl0 : size rl > 0\nl0 : size l = 0\nH3 : size rl + size rr ≤ 2\nrr0 : size rr > 0\n⊢ size rr + size rl ≤ 2",
"tactic": "rwa [add_comm]"
}
] |
[
1292,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1235,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.comapₗ_apply
|
[
{
"state_after": "case hc\nα : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β✝\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\nμ : Measure β\nhs : MeasurableSet s\n⊢ Injective f ∧ ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\n\ncase hc\nα : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β✝\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\nμ : Measure β\nhs : MeasurableSet s\n⊢ Injective f ∧ ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β✝\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\nμ : Measure β\nhs : MeasurableSet s\n⊢ ↑↑(↑(comapₗ f) μ) s = ↑↑μ (f '' s)",
"tactic": "rw [comapₗ, dif_pos, liftLinear_apply _ hs, OuterMeasure.comap_apply]"
},
{
"state_after": "no goals",
"state_before": "case hc\nα : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β✝\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\nμ : Measure β\nhs : MeasurableSet s\n⊢ Injective f ∧ ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\n\ncase hc\nα : Type u_2\nβ✝ : Type ?u.221666\nγ : Type ?u.221669\nδ : Type ?u.221672\nι : Type ?u.221675\nR : Type ?u.221678\nR' : Type ?u.221681\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β✝\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)\nμ : Measure β\nhs : MeasurableSet s\n⊢ Injective f ∧ ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)",
"tactic": "exact ⟨hfi, hf⟩"
}
] |
[
1301,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree
|
[
{
"state_after": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nhx : x ≠ 0\ns : K[X]\nhs : s ≠ 0\n⊢ num x * s * denom x = num x * (s * denom x)",
"state_before": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nhx : x ≠ 0\ns : K[X]\nhs : s ≠ 0\n⊢ ↑(natDegree (num x * s)) - ↑(natDegree (s * denom x)) = intDegree x",
"tactic": "apply natDegree_sub_eq_of_prod_eq (mul_ne_zero (num_ne_zero hx) hs)\n (mul_ne_zero hs x.denom_ne_zero) (num_ne_zero hx) x.denom_ne_zero"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\nx : RatFunc K\nhx : x ≠ 0\ns : K[X]\nhs : s ≠ 0\n⊢ num x * s * denom x = num x * (s * denom x)",
"tactic": "rw [mul_assoc]"
}
] |
[
1636,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1631,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Presieve.functorPullback_id
|
[] |
[
186,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.mem_carrier
|
[] |
[
153,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_mul
|
[
{
"state_after": "no goals",
"state_before": "x✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\n⊢ x ^ (y * z) = (x ^ y) ^ z",
"tactic": "rw [← Complex.ofReal_inj, Complex.ofReal_cpow (rpow_nonneg_of_nonneg hx _),\n Complex.ofReal_cpow hx, Complex.ofReal_mul, Complex.cpow_mul, Complex.ofReal_cpow hx] <;>\n simp only [(Complex.ofReal_mul _ _).symm, (Complex.ofReal_log hx).symm, Complex.ofReal_im,\n neg_lt_zero, pi_pos, le_of_lt pi_pos]"
}
] |
[
324,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.dirac_bind
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.40352\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(bind m dirac) s = ↑↑m s",
"state_before": "α : Type u_1\nβ : Type ?u.40352\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measure α\n⊢ bind m dirac = m",
"tactic": "ext1 s hs"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.40352\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nm : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(bind m dirac) s = ↑↑m s",
"tactic": "simp only [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs,\n Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter]"
}
] |
[
208,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.neg
|
[] |
[
284,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMin.sub
|
[] |
[
422,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
8
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.eraseLead_coeff_natDegree
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ coeff (eraseLead f) (natDegree f) = 0",
"tactic": "simp [eraseLead_coeff]"
}
] |
[
54,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Order/Lattice.lean
|
sup_inf_left
|
[] |
[
761,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
760,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
|
CategoryTheory.Limits.WidePullback.eq_lift_of_comp_eq
|
[
{
"state_after": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ g = lift f fs w",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\n⊢ (∀ (j : J), g ≫ π arrows j = fs j) → g ≫ base arrows = f → g = lift f fs w",
"tactic": "intro h1 h2"
},
{
"state_after": "case x\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ ∀ (j : WidePullbackShape J),\n g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app j = (WidePullbackShape.mkCone f fs w).π.app j",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ g = lift f fs w",
"tactic": "apply\n (limit.isLimit (WidePullbackShape.wideCospan B objs arrows)).uniq\n (WidePullbackShape.mkCone f fs <| w)"
},
{
"state_after": "case x.none\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app none =\n (WidePullbackShape.mkCone f fs w).π.app none\n\ncase x.some\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\nval✝ : J\n⊢ g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app (some val✝) =\n (WidePullbackShape.mkCone f fs w).π.app (some val✝)",
"state_before": "case x\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ ∀ (j : WidePullbackShape J),\n g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app j = (WidePullbackShape.mkCone f fs w).π.app j",
"tactic": "rintro (_ | _)"
},
{
"state_after": "no goals",
"state_before": "case x.none\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\n⊢ g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app none =\n (WidePullbackShape.mkCone f fs w).π.app none",
"tactic": "apply h2"
},
{
"state_after": "no goals",
"state_before": "case x.some\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g ≫ π arrows j = fs j\nh2 : g ≫ base arrows = f\nval✝ : J\n⊢ g ≫ (limit.cone (WidePullbackShape.wideCospan B objs arrows)).π.app (some val✝) =\n (WidePullbackShape.mkCone f fs w).π.app (some val✝)",
"tactic": "apply h1"
}
] |
[
361,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/RingTheory/Ideal/AssociatedPrime.lean
|
associatedPrimes.nonempty
|
[
{
"state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 0\n⊢ Set.Nonempty (associatedPrimes R M)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\n⊢ Set.Nonempty (associatedPrimes R M)",
"tactic": "obtain ⟨x, hx⟩ := exists_ne (0 : M)"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 0\nP : Ideal R\nhP : IsAssociatedPrime P M\nright✝ : Submodule.annihilator (Submodule.span R {x}) ≤ P\n⊢ Set.Nonempty (associatedPrimes R M)",
"state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 0\n⊢ Set.Nonempty (associatedPrimes R M)",
"tactic": "obtain ⟨P, hP, _⟩ := exists_le_isAssociatedPrime_of_isNoetherianRing R x hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.65376\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsNoetherianRing R\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 0\nP : Ideal R\nhP : IsAssociatedPrime P M\nright✝ : Submodule.annihilator (Submodule.span R {x}) ≤ P\n⊢ Set.Nonempty (associatedPrimes R M)",
"tactic": "exact ⟨P, hP⟩"
}
] |
[
133,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.fiber_eq
|
[] |
[
117,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
mul_singleton_mem_nhds
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\n⊢ MulOpposite.op a • s ∈ 𝓝 (b * a)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\n⊢ s * {a} ∈ 𝓝 (b * a)",
"tactic": "simp only [← iUnion_op_smul_set, mem_singleton_iff, iUnion_iUnion_eq_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\n⊢ MulOpposite.op a • s ∈ 𝓝 (b * a)",
"tactic": "exact smul_mem_nhds _ h"
}
] |
[
1318,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1316,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.eq_empty_of_forall_not_mem
|
[] |
[
579,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.mem_of_formPerm_apply_ne
|
[] |
[
87,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PNat.factorMultiset_le_iff'
|
[
{
"state_after": "m : ℕ+\nv : PrimeMultiset\nh : factorMultiset m ≤ factorMultiset (PrimeMultiset.prod v) ↔ m ∣ PrimeMultiset.prod v := factorMultiset_le_iff\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"state_before": "m : ℕ+\nv : PrimeMultiset\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"tactic": "let h := @factorMultiset_le_iff m v.prod"
},
{
"state_after": "m : ℕ+\nv : PrimeMultiset\nh : factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"state_before": "m : ℕ+\nv : PrimeMultiset\nh : factorMultiset m ≤ factorMultiset (PrimeMultiset.prod v) ↔ m ∣ PrimeMultiset.prod v := factorMultiset_le_iff\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"tactic": "rw [v.factorMultiset_prod] at h"
},
{
"state_after": "no goals",
"state_before": "m : ℕ+\nv : PrimeMultiset\nh : factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v\n⊢ factorMultiset m ≤ v ↔ m ∣ PrimeMultiset.prod v",
"tactic": "exact h"
}
] |
[
344,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
|
Complex.hasStrictDerivAt_tan
|
[
{
"state_after": "case h.e'_7\nx : ℂ\nh : cos x ≠ 0\n⊢ ↑1 / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2",
"state_before": "x : ℂ\nh : cos x ≠ 0\n⊢ HasStrictDerivAt tan (↑1 / cos x ^ 2) x",
"tactic": "convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1"
},
{
"state_after": "case h.e'_7\nx : ℂ\nh : ¬cos x = 0\n⊢ (sin x ^ 2 + cos x ^ 2) / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2",
"state_before": "case h.e'_7\nx : ℂ\nh : cos x ≠ 0\n⊢ ↑1 / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2",
"tactic": "rw_mod_cast [← sin_sq_add_cos_sq x]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\nx : ℂ\nh : ¬cos x = 0\n⊢ (sin x ^ 2 + cos x ^ 2) / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2",
"tactic": "ring"
}
] |
[
33,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_le_abs_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : 0 ≤ a\nhab : a ≤ b\n⊢ abs a ≤ abs b",
"tactic": "rwa [abs_of_nonneg ha, abs_of_nonneg (ha.trans hab)]"
}
] |
[
127,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_csSup_iff'
|
[] |
[
1048,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.IsTopologicalBasis.isOpen_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ IsOpen s ↔ ∀ (a : α), a ∈ s → ∃ t, t ∈ b ∧ a ∈ t ∧ t ⊆ s",
"tactic": "simp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]"
}
] |
[
160,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/CategoryTheory/Category/TwoP.lean
|
TwoP.coe_of
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.29828\nG : Type ?u.29831\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ (a * b)⁻¹ = a⁻¹ * b⁻¹",
"tactic": "simp"
}
] |
[
495,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
|
FormalMultilinearSeries.removeZero_coeff_zero
|
[] |
[
101,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.ghostFun_nat_cast
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.681371\nT : Type ?u.681374\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.681389\nβ : Type ?u.681392\nx y : 𝕎 R\ni : ℕ\n⊢ WittVector.ghostFun (Nat.unaryCast i) = ↑i",
"tactic": "induction i <;>\n simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.coe_nat]"
}
] |
[
201,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
9
] |
Mathlib/RingTheory/EuclideanDomain.lean
|
gcd_ne_zero_of_left
|
[] |
[
40,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Control/Monad/Basic.lean
|
map_eq_bind_pure_comp
|
[] |
[
54,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Topology/Sober.lean
|
IsGenericPoint.isIrreducible
|
[] |
[
79,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
11
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
fderivWithin.log
|
[] |
[
196,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.unifIntegrable_subsingleton
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\n⊢ UnifIntegrable f p μ",
"tactic": "intro ε hε"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\nhι : Nonempty ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\n\ncase neg\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\nhι : ¬Nonempty ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "by_cases hι : Nonempty ι"
},
{
"state_after": "case pos.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\nhι : Nonempty ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "cases' hι with i"
},
{
"state_after": "case pos.intro.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case pos.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨δ, hδpos, hδ⟩ := (hf i).snorm_indicator_le μ hp_one hp_top hε"
},
{
"state_after": "case pos.intro.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nj : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\n⊢ snorm (Set.indicator s (f j)) p μ ≤ ENNReal.ofReal ε",
"state_before": "case pos.intro.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "refine' ⟨δ, hδpos, fun j s hs hμs => _⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\ni : ι\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε\nj : ι\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≤ ENNReal.ofReal δ\n⊢ snorm (Set.indicator s (f j)) p μ ≤ ENNReal.ofReal ε",
"tactic": "convert hδ s hs hμs"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf✝ : α → β\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nf : ι → α → β\nhf : ∀ (i : ι), Memℒp (f i) p\nε : ℝ\nhε : 0 < ε\nhι : ¬Nonempty ι\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩"
}
] |
[
421,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_mem_prod
|
[] |
[
65,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
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