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tasksource 14

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[ { "state_after": "case intro\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\nu : Filter ι\ninst✝¹ : NeBot u\ninst✝ : IsCountablyGenerated u\nf : ι → α → E\ng : α → E\nhf : ∀ (n : ι), AEMeasurable (f n)\nh_tendsto : TendstoInMeasure μ f u g\nns : ℕ → ι\nhns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ AEMeasurable g", "state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\nu : Filter ι\ninst✝¹ : NeBot u\ninst✝ : IsCountablyGenerated u\nf : ι → α → E\ng : α → E\nhf : ∀ (n : ι), AEMeasurable (f n)\nh_tendsto : TendstoInMeasure μ f u g\n⊢ AEMeasurable g", "tactic": "obtain ⟨ns, hns⟩ := h_tendsto.exists_seq_tendsto_ae'" }, { "state_after": "no goals", "state_before": "case intro\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\nu : Filter ι\ninst✝¹ : NeBot u\ninst✝ : IsCountablyGenerated u\nf : ι → α → E\ng : α → E\nhf : ∀ (n : ι), AEMeasurable (f n)\nh_tendsto : TendstoInMeasure μ f u g\nns : ℕ → ι\nhns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ AEMeasurable g", "tactic": "exact aemeasurable_of_tendsto_metrizable_ae atTop (fun n => hf (ns n)) hns" } ]

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[ { "state_after": "no goals", "state_before": "M : Type u_1\nS T : Set M\ninst✝ : MulZeroClass M\n⊢ 0 ∈ centralizer S", "tactic": "simp [mem_centralizer_iff]" } ]

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[ { "state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nm : M\ns t : { x // x ∈ S }\n⊢ Localization.liftOn (Localization.mk r s)\n (fun r s =>\n liftOn (mk m t) (fun p => mk (SMul.smul r p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n (_ :\n ∀ {a c : R} {b d : { x // x ∈ S }},\n ↑(Localization.r S) (a, b) (c, d) →\n (fun r s =>\n liftOn (mk m t) (fun p => mk (r • p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n a b =\n (fun r s =>\n liftOn (mk m t) (fun p => mk (r • p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n c d) =\n mk (SMul.smul r m) (s * t)", "state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nm : M\ns t : { x // x ∈ S }\n⊢ Localization.mk r s • mk m t = mk (r • m) (s * t)", "tactic": "dsimp only [HSMul.hSMul, SMul.smul]" }, { "state_after": "no goals", "state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nm : M\ns t : { x // x ∈ S }\n⊢ Localization.liftOn (Localization.mk r s)\n (fun r s =>\n liftOn (mk m t) (fun p => mk (SMul.smul r p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n (_ :\n ∀ {a c : R} {b d : { x // x ∈ S }},\n ↑(Localization.r S) (a, b) (c, d) →\n (fun r s =>\n liftOn (mk m t) (fun p => mk (r • p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n a b =\n (fun r s =>\n liftOn (mk m t) (fun p => mk (r • p.fst) (s * p.snd))\n (_ :\n ∀ (p p' : M × { x // x ∈ S }),\n p ≈ p' → (fun p => mk (r • p.fst) (s * p.snd)) p = (fun p => mk (r • p.fst) (s * p.snd)) p'))\n c d) =\n mk (SMul.smul r m) (s * t)", "tactic": "rw [Localization.liftOn_mk, liftOn_mk]" } ]

[ 338, 41 ]

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[ { "state_after": "no goals", "state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.809158\ninst✝² : MeasurableSpace α\nK : Type ?u.809164\ninst✝¹ : Zero β\ninst✝ : Zero γ\ng : β → γ\nhg : g 0 = 0\nf : α →ₛ β\ns : Set α\nx : α\nhs : MeasurableSet s\n⊢ ↑(map g (restrict f s)) x = ↑(restrict (map g f) s) x", "tactic": "simp [hs, Set.indicator_comp_of_zero hg]" }, { "state_after": "no goals", "state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.809158\ninst✝² : MeasurableSpace α\nK : Type ?u.809164\ninst✝¹ : Zero β\ninst✝ : Zero γ\ng : β → γ\nhg : g 0 = 0\nf : α →ₛ β\ns : Set α\nx : α\nhs : ¬MeasurableSet s\n⊢ ↑(map g (restrict f s)) x = ↑(restrict (map g f) s) x", "tactic": "simp [restrict_of_not_measurable hs, hg]" } ]

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[ { "state_after": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ ↑t ⊆ f '' s → ∃ s', ↑s' ⊆ s ∧ image f s' = t\n\ncase mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ (∃ s', ↑s' ⊆ s ∧ image f s' = t) → ↑t ⊆ f '' s", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ ↑t ⊆ f '' s ↔ ∃ s', ↑s' ⊆ s ∧ image f s' = t", "tactic": "constructor" }, { "state_after": "case mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ (∃ s', ↑s' ⊆ s ∧ image f s' = t) → ↑t ⊆ f '' s\n\ncase mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ ↑t ⊆ f '' s → ∃ s', ↑s' ⊆ s ∧ image f s' = t", "state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ ↑t ⊆ f '' s → ∃ s', ↑s' ⊆ s ∧ image f s' = t\n\ncase mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ (∃ s', ↑s' ⊆ s ∧ image f s' = t) → ↑t ⊆ f '' s", "tactic": "swap" }, { "state_after": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\nh : ↑t ⊆ f '' s\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = t", "state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ ↑t ⊆ f '' s → ∃ s', ↑s' ⊆ s ∧ image f s' = t", "tactic": "intro h" }, { "state_after": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\nh : ↑t ⊆ f '' s\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = t", "state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\nh : ↑t ⊆ f '' s\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = t", "tactic": "letI : CanLift β s (f ∘ (↑)) fun y => y ∈ f '' s := ⟨fun y ⟨x, hxt, hy⟩ => ⟨⟨x, hxt⟩, hy⟩⟩" }, { "state_after": "case mp.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = image (f ∘ Subtype.val) t", "state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\nh : ↑t ⊆ f '' s\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = t", "tactic": "lift t to Finset s using h" }, { "state_after": "case mp.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\n⊢ image f (map (Embedding.subtype fun x => x ∈ s) t) = image (f ∘ Subtype.val) t", "state_before": "case mp.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\n⊢ ∃ s', ↑s' ⊆ s ∧ image f s' = image (f ∘ Subtype.val) t", "tactic": "refine' ⟨t.map (Embedding.subtype _), map_subtype_subset _, _⟩" }, { "state_after": "case mp.intro.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\ny : β\n⊢ y ∈ image f (map (Embedding.subtype fun x => x ∈ s) t) ↔ y ∈ image (f ∘ Subtype.val) t", "state_before": "case mp.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\n⊢ image f (map (Embedding.subtype fun x => x ∈ s) t) = image (f ∘ Subtype.val) t", "tactic": "ext y" }, { "state_after": "no goals", "state_before": "case mp.intro.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nthis : CanLift β (↑s) (f ∘ Subtype.val) fun y => y ∈ f '' s :=\n { prf := (_ : ∀ (y : β), y ∈ f '' s → ∃ y_1, (f ∘ Subtype.val) y_1 = y) }\nt : Finset ↑s\ny : β\n⊢ y ∈ image f (map (Embedding.subtype fun x => x ∈ s) t) ↔ y ∈ image (f ∘ Subtype.val) t", "tactic": "simp" }, { "state_after": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nt : Finset α\nht : ↑t ⊆ s\n⊢ ↑(image f t) ⊆ f '' s", "state_before": "case mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nt : Finset β\nf : α → β\n⊢ (∃ s', ↑s' ⊆ s ∧ image f s' = t) → ↑t ⊆ f '' s", "tactic": "rintro ⟨t, ht, rfl⟩" }, { "state_after": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nt : Finset α\nht : ↑t ⊆ s\n⊢ f '' ↑t ⊆ f '' s", "state_before": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nt : Finset α\nht : ↑t ⊆ s\n⊢ ↑(image f t) ⊆ f '' s", "tactic": "rw [coe_image]" }, { "state_after": "no goals", "state_before": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.140906\ninst✝ : DecidableEq β\ns : Set α\nf : α → β\nt : Finset α\nht : ↑t ⊆ s\n⊢ f '' ↑t ⊆ f '' s", "tactic": "exact Set.image_subset f ht" } ]

[ 755, 14 ]

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[ 1567, 1 ]

[ { "state_after": "case h.h\nα : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na✝ b c d a x : α\n⊢ ↑(symm (mulRight a)) x = ↑(mulRight a⁻¹) x", "state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na✝ b c d a : α\n⊢ symm (mulRight a) = mulRight a⁻¹", "tactic": "ext x" }, { "state_after": "no goals", "state_before": "case h.h\nα : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na✝ b c d a x : α\n⊢ ↑(symm (mulRight a)) x = ↑(mulRight a⁻¹) x", "tactic": "rfl" } ]

[ 109, 6 ]

[ 107, 1 ]

[]

[ 557, 6 ]

[ 556, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.8870\nγ : Type ?u.8873\nδ : Type ?u.8876\ninst✝ : CommMonoidWithZero α\np : α\nhp : Prime p\ns✝ : Multiset α\na : α\ns : Multiset α\nih : p ∣ Multiset.prod s → ∃ a, a ∈ s ∧ p ∣ a\nh : p ∣ Multiset.prod (a ::ₘ s)\n⊢ p ∣ a * Multiset.prod s", "tactic": "simpa using h" } ]

[ 41, 43 ]

[ 34, 1 ]

[ { "state_after": "case h1\nX : Type u_1\nx : X\n⊢ ↑(AddMonoidHom.comp toFreeAbelianGroup (singleAddHom x)) 1 =\n ↑(↑(AddMonoidHom.flip (smulAddHom ℤ (FreeAbelianGroup X))) (of x)) 1", "state_before": "X : Type u_1\nx : X\n⊢ AddMonoidHom.comp toFreeAbelianGroup (singleAddHom x) =\n ↑(AddMonoidHom.flip (smulAddHom ℤ (FreeAbelianGroup X))) (of x)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h1\nX : Type u_1\nx : X\n⊢ ↑(AddMonoidHom.comp toFreeAbelianGroup (singleAddHom x)) 1 =\n ↑(↑(AddMonoidHom.flip (smulAddHom ℤ (FreeAbelianGroup X))) (of x)) 1", "tactic": "simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,\n toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]" } ]

[ 57, 57 ]

[ 52, 1 ]

[ { "state_after": "θ : Angle\n⊢ toReal θ = toReal 0 ↔ θ = 0", "state_before": "θ : Angle\n⊢ toReal θ = 0 ↔ θ = 0", "tactic": "nth_rw 1 [← toReal_zero]" }, { "state_after": "no goals", "state_before": "θ : Angle\n⊢ toReal θ = toReal 0 ↔ θ = 0", "tactic": "exact toReal_inj" } ]

[ 594, 19 ]

[ 592, 1 ]

[]

[ 58, 18 ]

[ 57, 11 ]

[ { "state_after": "R : Type u_1\ninst✝ : CommRing R\nu v w x : Fin 3 → R\n⊢ vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 0 +\n vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 1 +\n vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 2 =\n (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)", "state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w x : Fin 3 → R\n⊢ ↑(↑crossProduct u) v ⬝ᵥ ↑(↑crossProduct w) x = u ⬝ᵥ w * v ⬝ᵥ x - u ⬝ᵥ x * v ⬝ᵥ w", "tactic": "simp_rw [cross_apply, vec3_dotProduct]" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\nu v w x : Fin 3 → R\n⊢ (u 1 * v 2 - u 2 * v 1) * (w 1 * x 2 - w 2 * x 1) + (u 2 * v 0 - u 0 * v 2) * (w 2 * x 0 - w 0 * x 2) +\n (u 0 * v 1 - u 1 * v 0) * (w 0 * x 1 - w 1 * x 0) =\n (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)", "state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w x : Fin 3 → R\n⊢ vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 0 +\n vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 1 +\n vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 *\n vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 2 =\n (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)", "tactic": "norm_num" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : CommRing R\nu v w x : Fin 3 → R\n⊢ (u 1 * v 2 - u 2 * v 1) * (w 1 * x 2 - w 2 * x 1) + (u 2 * v 0 - u 0 * v 2) * (w 2 * x 0 - w 0 * x 2) +\n (u 0 * v 1 - u 1 * v 0) * (w 0 * x 1 - w 1 * x 0) =\n (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)", "tactic": "ring" } ]

[ 132, 7 ]

[ 128, 1 ]

[]

[ 1530, 73 ]

[ 1528, 1 ]

[ { "state_after": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhs : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)!) s\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!) c\nhs : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)!) s\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "tactic": "replace hc := hasSum_coe.mpr hc" }, { "state_after": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhs : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)!) s\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "tactic": "replace hs := (hs.div_const ‖q‖).smul_const q" }, { "state_after": "case inl\nc s : ℝ\nhq : 0.re = 0\nhc : HasSum (fun a => ↑((-1) ^ a * ‖0‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => 0) (↑c + (s / ‖0‖) • 0)\n\ncase inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "tactic": "obtain rfl | hq0 := eq_or_ne q 0" }, { "state_after": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "state_before": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)", "tactic": "simp_rw [expSeries_apply_eq]" }, { "state_after": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "state_before": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "tactic": "have hq2 : q ^ 2 = -normSq q := sq_eq_neg_normSq.mpr hq" }, { "state_after": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "state_before": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "tactic": "have hqn := norm_ne_zero_iff.mpr hq0" }, { "state_after": "case inr.refine'_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun k => (↑(2 * k)!)⁻¹ • q ^ (2 * k)) ↑c\n\ncase inr.refine'_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun k => (↑(2 * k + 1)!)⁻¹ • q ^ (2 * k + 1)) ((s / ‖q‖) • q)", "state_before": "case inr\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑c + (s / ‖q‖) • q)", "tactic": "refine' HasSum.even_add_odd _ _" }, { "state_after": "case inl\nc s : ℝ\nhq : 0.re = 0\nhc : HasSum (fun a => ↑((-1) ^ a * ‖0‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\n⊢ HasSum (fun n => Pi.single 0 1 n) ↑c", "state_before": "case inl\nc s : ℝ\nhq : 0.re = 0\nhc : HasSum (fun a => ↑((-1) ^ a * ‖0‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\n⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => 0) (↑c + (s / ‖0‖) • 0)", "tactic": "simp_rw [expSeries_apply_zero, norm_zero, div_zero, zero_smul, add_zero]" }, { "state_after": "case inl\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ HasSum (fun n => Pi.single 0 1 n) ↑c", "state_before": "case inl\nc s : ℝ\nhq : 0.re = 0\nhc : HasSum (fun a => ↑((-1) ^ a * ‖0‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\n⊢ HasSum (fun n => Pi.single 0 1 n) ↑c", "tactic": "simp_rw [norm_zero] at hc" }, { "state_after": "case h.e'_5\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ (fun n => Pi.single 0 1 n) = fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)", "state_before": "case inl\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ HasSum (fun n => Pi.single 0 1 n) ↑c", "tactic": "convert hc using 1" }, { "state_after": "case h.e'_5.h.zero\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ Pi.single 0 1 Nat.zero = ↑((-1) ^ Nat.zero * 0 ^ (2 * Nat.zero) / ↑(2 * Nat.zero)!)\n\ncase h.e'_5.h.succ\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\nn : ℕ\n⊢ Pi.single 0 1 (Nat.succ n) = ↑((-1) ^ Nat.succ n * 0 ^ (2 * Nat.succ n) / ↑(2 * Nat.succ n)!)", "state_before": "case h.e'_5\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ (fun n => Pi.single 0 1 n) = fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)", "tactic": "ext (_ | n) : 1" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.zero\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\n⊢ Pi.single 0 1 Nat.zero = ↑((-1) ^ Nat.zero * 0 ^ (2 * Nat.zero) / ↑(2 * Nat.zero)!)", "tactic": "rw [pow_zero, Nat.zero_eq, MulZeroClass.mul_zero, pow_zero, Nat.factorial_zero, Nat.cast_one,\n div_one, one_mul, Pi.single_eq_same, coe_one]" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.succ\nc s : ℝ\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * ‖0‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖0‖) • 0) ((s / ‖0‖) • 0)\nhc : HasSum (fun a => ↑((-1) ^ a * 0 ^ (2 * a) / ↑(2 * a)!)) ↑c\nn : ℕ\n⊢ Pi.single 0 1 (Nat.succ n) = ↑((-1) ^ Nat.succ n * 0 ^ (2 * Nat.succ n) / ↑(2 * Nat.succ n)!)", "tactic": "rw [zero_pow (mul_pos two_pos (Nat.succ_pos _)), MulZeroClass.mul_zero, zero_div,\n Pi.single_eq_of_ne n.succ_ne_zero, coe_zero]" }, { "state_after": "case h.e'_5\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ (fun k => (↑(2 * k)!)⁻¹ • q ^ (2 * k)) = fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)", "state_before": "case inr.refine'_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun k => (↑(2 * k)!)⁻¹ • q ^ (2 * k)) ↑c", "tactic": "convert hc using 1" }, { "state_after": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\n⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)", "state_before": "case h.e'_5\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ (fun k => (↑(2 * k)!)⁻¹ • q ^ (2 * k)) = fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)", "tactic": "ext n : 1" }, { "state_after": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)", "state_before": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\n⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)", "tactic": "letI k : ℝ := ↑(2 * n)!" }, { "state_after": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • (-↑normSq q) ^ n) = k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n))\n\ncase h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / k)", "state_before": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)", "tactic": "calc\n k⁻¹ • q ^ (2 * n) = k⁻¹ • (-normSq q) ^ n := by rw [pow_mul, hq2]; norm_cast\n _ = k⁻¹ • ↑((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n)) := ?_\n _ = ↑((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n) / k) := ?_" }, { "state_after": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • (-↑(↑normSq q)) ^ n = ↑(k⁻¹ • (-↑normSq q) ^ n)", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • q ^ (2 * n) = ↑(k⁻¹ • (-↑normSq q) ^ n)", "tactic": "rw [pow_mul, hq2]" }, { "state_after": "no goals", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • (-↑(↑normSq q)) ^ n = ↑(k⁻¹ • (-↑normSq q) ^ n)", "tactic": "norm_cast" }, { "state_after": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • (-↑normSq q) ^ n) = k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n))", "state_before": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • (-↑normSq q) ^ n) = k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n))", "tactic": "congr 1" }, { "state_after": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • ((-1) ^ n * (‖q‖ * ‖q‖) ^ n)) = k⁻¹ • ↑((-1) ^ n * (‖q‖ * ‖q‖) ^ n)", "state_before": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • (-↑normSq q) ^ n) = k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n))", "tactic": "rw [neg_pow, normSq_eq_norm_mul_self, pow_mul, sq]" }, { "state_after": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ (↑(2 * n)!)⁻¹ • ((-1) ^ n * (↑‖q‖ * ↑‖q‖) ^ n) = (↑(2 * n)!)⁻¹ • ((-1) ^ n * (↑‖q‖ * ↑‖q‖) ^ n)", "state_before": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ • ((-1) ^ n * (‖q‖ * ‖q‖) ^ n)) = k⁻¹ • ↑((-1) ^ n * (‖q‖ * ‖q‖) ^ n)", "tactic": "push_cast" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ (↑(2 * n)!)⁻¹ • ((-1) ^ n * (↑‖q‖ * ↑‖q‖) ^ n) = (↑(2 * n)!)⁻¹ • ((-1) ^ n * (↑‖q‖ * ↑‖q‖) ^ n)", "tactic": "rfl" }, { "state_after": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑k⁻¹ * ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) * k⁻¹)", "state_before": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / k)", "tactic": "rw [← coe_mul_eq_smul, div_eq_mul_inv]" }, { "state_after": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ * (↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n))) = ↑(↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n) * k⁻¹)", "state_before": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑k⁻¹ * ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) * k⁻¹)", "tactic": "norm_cast" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ * (↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n))) = ↑(↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n) * k⁻¹)", "tactic": "ring_nf" }, { "state_after": "case h.e'_5\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ (fun k => (↑(2 * k + 1)!)⁻¹ • q ^ (2 * k + 1)) = fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q", "state_before": "case inr.refine'_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ HasSum (fun k => (↑(2 * k + 1)!)⁻¹ • q ^ (2 * k + 1)) ((s / ‖q‖) • q)", "tactic": "convert hs using 1" }, { "state_after": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\n⊢ (↑(2 * n + 1)!)⁻¹ • q ^ (2 * n + 1) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q", "state_before": "case h.e'_5\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\n⊢ (fun k => (↑(2 * k + 1)!)⁻¹ • q ^ (2 * k + 1)) = fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q", "tactic": "ext n : 1" }, { "state_after": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (↑(2 * n + 1)!)⁻¹ • q ^ (2 * n + 1) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q", "state_before": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\n⊢ (↑(2 * n + 1)!)⁻¹ • q ^ (2 * n + 1) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q", "tactic": "let k : ℝ := ↑(2 * n + 1)!" }, { "state_after": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • (↑((-↑normSq q) ^ n) * q) = k⁻¹ • ((-1) ^ n * ‖q‖ ^ (2 * n)) • q\n\ncase h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • ((-1) ^ n * ‖q‖ ^ (2 * n)) • q = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖) • q", "state_before": "case h.e'_5.h\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (↑(2 * n + 1)!)⁻¹ • q ^ (2 * n + 1) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q", "tactic": "calc\n k⁻¹ • q ^ (2 * n + 1) = k⁻¹ • ((-normSq q) ^ n * q) := by\n rw [pow_succ', pow_mul, hq2]\n norm_cast\n _ = k⁻¹ • ((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n)) • q := ?_\n _ = ((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖) • q := ?_" }, { "state_after": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • ((-↑(↑normSq q)) ^ n * q) = k⁻¹ • (↑((-↑normSq q) ^ n) * q)", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • q ^ (2 * n + 1) = k⁻¹ • (↑((-↑normSq q) ^ n) * q)", "tactic": "rw [pow_succ', pow_mul, hq2]" }, { "state_after": "no goals", "state_before": "q : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • ((-↑(↑normSq q)) ^ n * q) = k⁻¹ • (↑((-↑normSq q) ^ n) * q)", "tactic": "norm_cast" }, { "state_after": "case h.e'_5.h.calc_1.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ ↑((-↑normSq q) ^ n) * q = ((-1) ^ n * ‖q‖ ^ (2 * n)) • q", "state_before": "case h.e'_5.h.calc_1\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • (↑((-↑normSq q) ^ n) * q) = k⁻¹ • ((-1) ^ n * ‖q‖ ^ (2 * n)) • q", "tactic": "congr 1" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.calc_1.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ ↑((-↑normSq q) ^ n) * q = ((-1) ^ n * ‖q‖ ^ (2 * n)) • q", "tactic": "rw [neg_pow, normSq_eq_norm_mul_self, pow_mul, sq, ← coe_mul_eq_smul]" }, { "state_after": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n))) • q = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖) • q", "state_before": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • ((-1) ^ n * ‖q‖ ^ (2 * n)) • q = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖) • q", "tactic": "rw [smul_smul]" }, { "state_after": "case h.e'_5.h.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖", "state_before": "case h.e'_5.h.calc_2\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n))) • q = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖) • q", "tactic": "congr 1" }, { "state_after": "case h.e'_5.h.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (↑(2 * n + 1)!)⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * (↑(2 * n + 1)!)⁻¹)", "state_before": "case h.e'_5.h.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖", "tactic": "simp_rw [pow_succ', mul_div_assoc, div_div_cancel_left' hqn]" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nc s : ℝ\nhc : HasSum (fun a => ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c\nhs : HasSum (fun z => ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(↑normSq q)\nhqn : ‖q‖ ≠ 0\nn : ℕ\nk : ℝ := ↑(2 * n + 1)!\n⊢ (↑(2 * n + 1)!)⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * (↑(2 * n + 1)!)⁻¹)", "tactic": "ring" } ]

[ 92, 11 ]

[ 45, 1 ]

[]

[ 136, 42 ]

[ 135, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nα : F ≅ G\nX : C\nZ : D\ng g' : F.obj X ⟶ Z\n⊢ α.inv.app X ≫ g = α.inv.app X ≫ g' ↔ g = g'", "tactic": "simp only [cancel_epi, refl]" } ]

[ 137, 83 ]

[ 136, 1 ]

[]

[ 76, 23 ]

[ 75, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ ↑↑count {i | ↑ε ≤ (ENNReal.some ∘ a) i} ≤ ↑c / ↑ε", "state_before": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ ↑↑count {i | ε ≤ a i} ≤ ↑c / ↑ε", "tactic": "rw [show (fun i => ε ≤ a i) = fun i => (ε : ℝ≥0∞) ≤ ((↑) ∘ a) i by\n funext i\n simp only [ENNReal.coe_le_coe, Function.comp]]" }, { "state_after": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ (∑' (i : α), (ENNReal.some ∘ a) i) ≤ ↑c", "state_before": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ ↑↑count {i | ↑ε ≤ (ENNReal.some ∘ a) i} ≤ ↑c / ↑ε", "tactic": "apply\n ENNReal.count_const_le_le_of_tsum_le (measurable_coe_nnreal_ennreal.comp a_mble) _\n (by exact_mod_cast ε_ne_zero) (@ENNReal.coe_ne_top ε)" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ (∑' (i : α), (ENNReal.some ∘ a) i) = ↑(∑' (i : α), a i)", "state_before": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ (∑' (i : α), (ENNReal.some ∘ a) i) ≤ ↑c", "tactic": "convert ENNReal.coe_le_coe.mpr tsum_le_c" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ (∑' (i : α), (ENNReal.some ∘ a) i) = ↑(∑' (i : α), a i)", "tactic": "erw [ENNReal.tsum_coe_eq a_summable.hasSum]" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\ni : α\n⊢ (ε ≤ a i) = (↑ε ≤ (ENNReal.some ∘ a) i)", "state_before": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ (fun i => ε ≤ a i) = fun i => ↑ε ≤ (ENNReal.some ∘ a) i", "tactic": "funext i" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\ni : α\n⊢ (ε ≤ a i) = (↑ε ≤ (ENNReal.some ∘ a) i)", "tactic": "simp only [ENNReal.coe_le_coe, Function.comp]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.1696412\nγ : Type ?u.1696415\nδ : Type ?u.1696418\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0\na_mble : Measurable a\na_summable : Summable a\nc : ℝ≥0\ntsum_le_c : (∑' (i : α), a i) ≤ c\nε : ℝ≥0\nε_ne_zero : ε ≠ 0\n⊢ ↑ε ≠ 0", "tactic": "exact_mod_cast ε_ne_zero" } ]

[ 1432, 46 ]

[ 1422, 1 ]

[]

[ 227, 25 ]

[ 225, 1 ]

[]

[ 892, 35 ]

[ 891, 1 ]

[ { "state_after": "no goals", "state_before": "n : ℕ\nb : Bool\n⊢ bit b n = 0 ↔ n = 0 ∧ b = false", "tactic": "cases b <;> simp [Nat.bit0_eq_zero, Nat.bit1_ne_zero]" } ]

[ 62, 56 ]

[ 61, 1 ]

[]

[ 229, 6 ]

[ 228, 1 ]

[]

[ 606, 64 ]

[ 603, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ range succ = {i | 0 < i}", "tactic": "ext (_ | i) <;> simp [succ_pos, succ_ne_zero, Set.mem_setOf]" } ]

[ 31, 63 ]

[ 30, 11 ]

[ { "state_after": "case hH\nG : Type u_1\nG' : Type ?u.612020\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.612029\ninst✝¹ : AddGroup A\nN : Type ?u.612035\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ H ≤ H ⊔ K\n\ncase hK\nG : Type u_1\nG' : Type ?u.612020\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.612029\ninst✝¹ : AddGroup A\nN : Type ?u.612035\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ K ≤ H ⊔ K", "state_before": "G : Type u_1\nG' : Type ?u.612020\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.612029\ninst✝¹ : AddGroup A\nN : Type ?u.612035\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ Codisjoint (subgroupOf H (H ⊔ K)) (subgroupOf K (H ⊔ K))", "tactic": "rw [codisjoint_iff, sup_subgroupOf_eq, subgroupOf_self]" }, { "state_after": "no goals", "state_before": "case hH\nG : Type u_1\nG' : Type ?u.612020\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.612029\ninst✝¹ : AddGroup A\nN : Type ?u.612035\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ H ≤ H ⊔ K\n\ncase hK\nG : Type u_1\nG' : Type ?u.612020\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.612029\ninst✝¹ : AddGroup A\nN : Type ?u.612035\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ K ≤ H ⊔ K", "tactic": "exacts [le_sup_left, le_sup_right]" } ]

[ 3174, 37 ]

[ 3171, 1 ]

[]

[ 1100, 37 ]

[ 1097, 1 ]

[]

[ 1604, 20 ]

[ 1603, 11 ]

[]

[ 753, 71 ]

[ 750, 1 ]

[]

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[ { "state_after": "α : Type u_1\nβ : Type ?u.308183\nγ : Type ?u.308186\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : i ∈ s\nv : δ i\n⊢ piecewise s (update f i v) (update g i v) = piecewise s (update f i v) g", "state_before": "α : Type u_1\nβ : Type ?u.308183\nγ : Type ?u.308186\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : i ∈ s\nv : δ i\n⊢ update (piecewise s f g) i v = piecewise s (update f i v) g", "tactic": "rw [update_piecewise]" }, { "state_after": "α : Type u_1\nβ : Type ?u.308183\nγ : Type ?u.308186\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : i ∈ s\nv : δ i\nj : α\nhj : ¬j ∈ s\n⊢ j ≠ i", "state_before": "α : Type u_1\nβ : Type ?u.308183\nγ : Type ?u.308186\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : i ∈ s\nv : δ i\n⊢ piecewise s (update f i v) (update g i v) = piecewise s (update f i v) g", "tactic": "refine' s.piecewise_congr (fun _ _ => rfl) fun j hj => update_noteq _ _ _" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.308183\nγ : Type ?u.308186\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : i ∈ s\nv : δ i\nj : α\nhj : ¬j ∈ s\n⊢ j ≠ i", "tactic": "exact fun h => hj (h.symm ▸ hi)" } ]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type ?u.211340\nβ : Type ?u.211343\nγ : Type ?u.211346\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\nh : IsLimit c\n⊢ a < b * c ↔ ∃ c', c' < c ∧ a < b * c'", "tactic": "simpa only [not_ball, not_le, bex_def] using not_congr (@mul_le_of_limit b c a h)" } ]

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[ { "state_after": "F : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)", "state_before": "F : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\n⊢ count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)", "tactic": "letI := Classical.decEq α" }, { "state_after": "case refine'_1\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ count b (Finset.sup ∅ f) = Finset.sup ∅ fun a => count b (f a)\n\ncase refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ ∀ ⦃a : α⦄ {s : Finset α},\n ¬a ∈ s →\n (count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)) →\n count b (Finset.sup (insert a s) f) = Finset.sup (insert a s) fun a => count b (f a)", "state_before": "F : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)", "tactic": "refine' s.induction _ _" }, { "state_after": "no goals", "state_before": "case refine'_1\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ count b (Finset.sup ∅ f) = Finset.sup ∅ fun a => count b (f a)", "tactic": "exact count_zero _" }, { "state_after": "case refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns✝ : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\ni : α\ns : Finset α\na✝ : ¬i ∈ s\nih : count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)\n⊢ count b (Finset.sup (insert i s) f) = Finset.sup (insert i s) fun a => count b (f a)", "state_before": "case refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\n⊢ ∀ ⦃a : α⦄ {s : Finset α},\n ¬a ∈ s →\n (count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)) →\n count b (Finset.sup (insert a s) f) = Finset.sup (insert a s) fun a => count b (f a)", "tactic": "intro i s _ ih" }, { "state_after": "case refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns✝ : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\ni : α\ns : Finset α\na✝ : ¬i ∈ s\nih : count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)\n⊢ max (count b (f i)) (Finset.sup s fun a => count b (f a)) = count b (f i) ⊔ Finset.sup s fun a => count b (f a)", "state_before": "case refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns✝ : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\ni : α\ns : Finset α\na✝ : ¬i ∈ s\nih : count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)\n⊢ count b (Finset.sup (insert i s) f) = Finset.sup (insert i s) fun a => count b (f a)", "tactic": "rw [Finset.sup_insert, sup_eq_union, count_union, Finset.sup_insert, ih]" }, { "state_after": "no goals", "state_before": "case refine'_2\nF : Type ?u.410886\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.410895\nι : Type ?u.410898\nκ : Type ?u.410901\ninst✝ : DecidableEq β\ns✝ : Finset α\nf : α → Multiset β\nb : β\nthis : DecidableEq α := Classical.decEq α\ni : α\ns : Finset α\na✝ : ¬i ∈ s\nih : count b (Finset.sup s f) = Finset.sup s fun a => count b (f a)\n⊢ max (count b (f i)) (Finset.sup s fun a => count b (f a)) = count b (f i) ⊔ Finset.sup s fun a => count b (f a)", "tactic": "rfl" } ]

[ 1752, 8 ]

[ 1745, 1 ]

[ { "state_after": "case refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ ∀ (X : C ⥤ Karoubi D),\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =\n (𝟭 (C ⥤ Karoubi D)).obj X\n\ncase refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ ∀ (X Y : C ⥤ Karoubi D) (f : X ⟶ Y),\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map f =\n eqToHom (_ : ?m.30774.obj X = ?m.30775.obj X) ≫\n (𝟭 (C ⥤ Karoubi D)).map f ≫\n eqToHom\n (_ :\n (𝟭 (C ⥤ Karoubi D)).obj Y =\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj Y)", "state_before": "C : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) = 𝟭 (C ⥤ Karoubi D)", "tactic": "refine' Functor.ext _ _" }, { "state_after": "case refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (𝟭 (C ⥤ Karoubi D)).obj F", "state_before": "case refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ ∀ (X : C ⥤ Karoubi D),\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =\n (𝟭 (C ⥤ Karoubi D)).obj X", "tactic": "intro F" }, { "state_after": "case refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ ∀ (X : C),\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj X\n\ncase refine'_1.refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ ∀ (X Y : C) (f : X ⟶ Y),\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).map f =\n eqToHom (_ : ?m.31081.obj X = ?m.31082.obj X) ≫\n ((𝟭 (C ⥤ Karoubi D)).obj F).map f ≫\n eqToHom\n (_ :\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj Y =\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj Y)", "state_before": "case refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (𝟭 (C ⥤ Karoubi D)).obj F", "tactic": "refine' Functor.ext _ _" }, { "state_after": "case refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj X", "state_before": "case refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ ∀ (X : C),\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj X", "tactic": "intro X" }, { "state_after": "case refine'_1.refine'_1.h_p\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p ≫\n eqToHom ?refine'_1.refine'_1.h_X =\n eqToHom ?refine'_1.refine'_1.h_X ≫ (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).p\n\ncase refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).X", "state_before": "case refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj X", "tactic": "ext" }, { "state_after": "case refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).X", "state_before": "case refine'_1.refine'_1.h_p\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p ≫\n eqToHom ?refine'_1.refine'_1.h_X =\n eqToHom ?refine'_1.refine'_1.h_X ≫ (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).p\n\ncase refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).X", "tactic": ". simp" }, { "state_after": "no goals", "state_before": "case refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).X", "tactic": ". simp" }, { "state_after": "no goals", "state_before": "case refine'_1.refine'_1.h_p\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p ≫\n eqToHom ?refine'_1.refine'_1.h_X =\n eqToHom ?refine'_1.refine'_1.h_X ≫ (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).p", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\nX : C\n⊢ (((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((𝟭 (C ⥤ Karoubi D)).obj F).obj X).X", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case refine'_1.refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C ⥤ Karoubi D\n⊢ ∀ (X Y : C) (f : X ⟶ Y),\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).map f =\n eqToHom\n (_ :\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj X) ≫\n ((𝟭 (C ⥤ Karoubi D)).obj F).map f ≫\n eqToHom\n (_ :\n ((𝟭 (C ⥤ Karoubi D)).obj F).obj Y =\n ((functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj Y)", "tactic": "aesop_cat" }, { "state_after": "case refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF G : C ⥤ Karoubi D\nφ : F ⟶ G\n⊢ (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map φ =\n eqToHom\n (_ :\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (𝟭 (C ⥤ Karoubi D)).obj F) ≫\n (𝟭 (C ⥤ Karoubi D)).map φ ≫\n eqToHom\n (_ :\n (𝟭 (C ⥤ Karoubi D)).obj G =\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj G)", "state_before": "case refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ ∀ (X Y : C ⥤ Karoubi D) (f : X ⟶ Y),\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map f =\n eqToHom\n (_ :\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =\n (𝟭 (C ⥤ Karoubi D)).obj X) ≫\n (𝟭 (C ⥤ Karoubi D)).map f ≫\n eqToHom\n (_ :\n (𝟭 (C ⥤ Karoubi D)).obj Y =\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj Y)", "tactic": "intro F G φ" }, { "state_after": "no goals", "state_before": "case refine'_2\nC : Type u_1\nD : Type u_2\nE : Type ?u.30040\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF G : C ⥤ Karoubi D\nφ : F ⟶ G\n⊢ (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map φ =\n eqToHom\n (_ :\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (𝟭 (C ⥤ Karoubi D)).obj F) ≫\n (𝟭 (C ⥤ Karoubi D)).map φ ≫\n eqToHom\n (_ :\n (𝟭 (C ⥤ Karoubi D)).obj G =\n (functorExtension₁ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj G)", "tactic": "aesop_cat" } ]

[ 128, 14 ]

[ 116, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.637\nM₀ : Type ?u.640\nG₀ : Type u_1\nM₀' : Type ?u.646\nG₀' : Type ?u.649\nF : Type ?u.652\nF' : Type ?u.655\ninst✝ : GroupWithZero G₀\na x y x' y' : G₀\nha : a = 0\n⊢ SemiconjBy a⁻¹ x y ↔ SemiconjBy a y x", "tactic": "simp only [ha, inv_zero, SemiconjBy.zero_left]" } ]

[ 40, 54 ]

[ 38, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40788\nι : Sort ?u.40791\nπ : α → Type ?u.40796\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : SurjOn f₁ s t\nH : EqOn f₁ f₂ s\n⊢ SurjOn f₂ s t", "tactic": "rwa [SurjOn, ← H.image_eq]" } ]

[ 785, 29 ]

[ 784, 1 ]

[]

[ 1275, 6 ]

[ 1274, 1 ]

[]

[ 1039, 27 ]

[ 1038, 1 ]

[]

[ 76, 53 ]

[ 75, 1 ]

[]

[ 291, 6 ]

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[]

[ 539, 6 ]

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[ { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\n⊢ PathConnectedSpace X → IsPathConnected univ\n\ncase mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\n⊢ IsPathConnected univ → PathConnectedSpace X", "state_before": "X : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\n⊢ PathConnectedSpace X ↔ IsPathConnected univ", "tactic": "constructor" }, { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\n⊢ IsPathConnected univ", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\n⊢ PathConnectedSpace X → IsPathConnected univ", "tactic": "intro h" }, { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\n⊢ IsPathConnected univ", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\n⊢ IsPathConnected univ", "tactic": "haveI := @PathConnectedSpace.Nonempty X _ _" }, { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\n⊢ IsPathConnected univ", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\n⊢ IsPathConnected univ", "tactic": "inhabit X" }, { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\n⊢ ∀ {y : X}, y ∈ univ → JoinedIn univ default y", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\n⊢ IsPathConnected univ", "tactic": "refine' ⟨default, mem_univ _, _⟩" }, { "state_after": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\ny : X\n_hy : y ∈ univ\n⊢ JoinedIn univ default y", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\n⊢ ∀ {y : X}, y ∈ univ → JoinedIn univ default y", "tactic": "intros y _hy" }, { "state_after": "no goals", "state_before": "case mp\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.689194\nF : Set X\nh : PathConnectedSpace X\nthis : Nonempty X\ninhabited_h : Inhabited X\ny : X\n_hy : y ∈ univ\n⊢ JoinedIn univ default y", "tactic": "simpa using PathConnectedSpace.Joined default y" }, { "state_after": "case mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : IsPathConnected univ\n⊢ PathConnectedSpace X", "state_before": "case mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\n⊢ IsPathConnected univ → PathConnectedSpace X", "tactic": "intro h" }, { "state_after": "case mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : IsPathConnected univ\nh' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y\n⊢ PathConnectedSpace X", "state_before": "case mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : IsPathConnected univ\n⊢ PathConnectedSpace X", "tactic": "have h' := h.joinedIn" }, { "state_after": "case mpr.intro\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.689194\nF : Set X\nh' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y\nx : X\nh : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y\n⊢ PathConnectedSpace X", "state_before": "case mpr\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.689194\nF : Set X\nh : IsPathConnected univ\nh' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y\n⊢ PathConnectedSpace X", "tactic": "cases' h with x h" }, { "state_after": "no goals", "state_before": "case mpr.intro\nX : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.689194\nF : Set X\nh' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y\nx : X\nh : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y\n⊢ PathConnectedSpace X", "tactic": "exact ⟨⟨x⟩, by simpa using h'⟩" }, { "state_after": "no goals", "state_before": "X : Type u_1\nY : Type ?u.689179\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.689194\nF : Set X\nh' : ∀ (x : X), x ∈ univ → ∀ (y : X), y ∈ univ → JoinedIn univ x y\nx : X\nh : x ∈ univ ∧ ∀ {y : X}, y ∈ univ → JoinedIn univ x y\n⊢ ∀ (x y : X), Joined x y", "tactic": "simpa using h'" } ]

[ 1145, 35 ]

[ 1134, 1 ]

[ { "state_after": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ abs (sin θ) = abs (sin ψ)", "state_before": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ abs (sin θ) = abs (sin ψ)", "tactic": "simp_rw [two_zsmul, ← two_nsmul] at h" }, { "state_after": "no goals", "state_before": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ abs (sin θ) = abs (sin ψ)", "tactic": "exact abs_sin_eq_of_two_nsmul_eq h" } ]

[ 490, 37 ]

[ 487, 1 ]

[]

[ 162, 36 ]

[ 161, 1 ]

[]

[ 150, 50 ]

[ 148, 1 ]

[]

[ 254, 34 ]

[ 253, 1 ]

[]

[ 382, 80 ]

[ 382, 1 ]

[]

[ 496, 41 ]

[ 494, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ b⁻¹ * a ≤ c ↔ a ≤ b * c", "tactic": "rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]" } ]

[ 124, 52 ]

[ 123, 1 ]

[ { "state_after": "a b : Int\n⊢ (if a ≤ b then b else a) = if b ≤ a then a else b", "state_before": "a b : Int\n⊢ max a b = max b a", "tactic": "simp only [Int.max_def]" }, { "state_after": "case pos\na b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\n⊢ b = a\n\ncase neg\na b : Int\nh₁ : ¬a ≤ b\nh₂ : ¬b ≤ a\n⊢ a = b", "state_before": "a b : Int\n⊢ (if a ≤ b then b else a) = if b ≤ a then a else b", "tactic": "by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]" }, { "state_after": "no goals", "state_before": "case pos\na b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\n⊢ b = a", "tactic": "exact Int.le_antisymm h₂ h₁" }, { "state_after": "no goals", "state_before": "case neg\na b : Int\nh₁ : ¬a ≤ b\nh₂ : ¬b ≤ a\n⊢ a = b", "tactic": "cases not_or_intro h₁ h₂ <| Int.le_total .." } ]

[ 712, 48 ]

[ 708, 11 ]

[]

[ 97, 49 ]

[ 95, 1 ]

[ { "state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ InnerProductGeometry.angle (p1 -ᵥ p2) (p2 -ᵥ p1) = π", "state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ ∠ p1 (midpoint ℝ p1 p2) p2 = π", "tactic": "simp [angle, hp1p2, -zero_lt_one]" }, { "state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ InnerProductGeometry.angle (p1 -ᵥ p2) (-(p1 -ᵥ p2)) = π", "state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ InnerProductGeometry.angle (p1 -ᵥ p2) (p2 -ᵥ p1) = π", "tactic": "rw [← neg_vsub_eq_vsub_rev p1 p2]" }, { "state_after": "case hx\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ p1 -ᵥ p2 ≠ 0", "state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ InnerProductGeometry.angle (p1 -ᵥ p2) (-(p1 -ᵥ p2)) = π", "tactic": "apply angle_self_neg_of_nonzero" }, { "state_after": "no goals", "state_before": "case hx\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\nhp1p2 : p1 ≠ p2\n⊢ p1 -ᵥ p2 ≠ 0", "tactic": "simpa only [ne_eq, vsub_eq_zero_iff_eq]" } ]

[ 257, 42 ]

[ 253, 1 ]

[]

[ 61, 70 ]

[ 60, 1 ]

[]

[ 27, 6 ]

[ 26, 1 ]

[ { "state_after": "case mk\nval✝ : ℕ\nproperty✝ : 0 < val✝\n⊢ ↑(↑factorizationEquiv { val := val✝, property := property✝ }) = factorization ↑{ val := val✝, property := property✝ }", "state_before": "n : ℕ+\n⊢ ↑(↑factorizationEquiv n) = factorization ↑n", "tactic": "cases n" }, { "state_after": "no goals", "state_before": "case mk\nval✝ : ℕ\nproperty✝ : 0 < val✝\n⊢ ↑(↑factorizationEquiv { val := val✝, property := property✝ }) = factorization ↑{ val := val✝, property := property✝ }", "tactic": "rfl" } ]

[ 335, 6 ]

[ 333, 1 ]

[]

[ 79, 6 ]

[ 78, 1 ]

[]

[ 230, 91 ]

[ 229, 1 ]

[]

[ 1076, 6 ]

[ 1075, 1 ]

[ { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\n⊢ Indep m1 m2", "tactic": "intro t1 t2 ht1 ht2" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "tactic": "let μ_inter := μ.restrict t2" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "tactic": "let ν := μ t2 • μ" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "tactic": "have h_univ : μ_inter Set.univ = ν Set.univ := by\n rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "tactic": "haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t2 μ ⟨measure_lt_top μ t2⟩" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑(Measure.restrict μ t2) t1 = ↑↑μ t2 * ↑↑μ t1", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2", "tactic": "rw [mul_comm, ← Measure.restrict_apply (h1 t1 ht1)]" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ ↑↑μ_inter t = ↑↑ν t", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑(Measure.restrict μ t2) t1 = ↑↑μ t2 * ↑↑μ t1", "tactic": "refine' ext_on_measurableSpace_of_generate_finite m p1 (fun t ht => _) h1 hpm1 hp1 h_univ ht1" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1✝ : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\nht1 : MeasurableSet t\n⊢ ↑↑μ_inter t = ↑↑ν t", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ ↑↑μ_inter t = ↑↑ν t", "tactic": "have ht1 : MeasurableSet[m] t := by\n refine' h1 _ _\n rw [hpm1]\n exact measurableSet_generateFrom ht" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1✝ : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\nht1 : MeasurableSet t\n⊢ ↑↑μ (t ∩ t2) = ↑↑μ t * ↑↑μ t2", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1✝ : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\nht1 : MeasurableSet t\n⊢ ↑↑μ_inter t = ↑↑ν t", "tactic": "rw [Measure.restrict_apply ht1, Measure.smul_apply, smul_eq_mul, mul_comm]" }, { "state_after": "no goals", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1✝ : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\nht1 : MeasurableSet t\n⊢ ↑↑μ (t ∩ t2) = ↑↑μ t * ↑↑μ t2", "tactic": "exact IndepSets.indep_aux h2 hp2 hpm2 hyp ht ht2" }, { "state_after": "no goals", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\n⊢ ↑↑μ_inter Set.univ = ↑↑ν Set.univ", "tactic": "rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ MeasurableSet t", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ MeasurableSet t", "tactic": "refine' h1 _ _" }, { "state_after": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ MeasurableSet t", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ MeasurableSet t", "tactic": "rw [hpm1]" }, { "state_after": "no goals", "state_before": "Ω : Type u_1\nι : Type ?u.1736711\nm1 m2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ {s | MeasurableSet s}\nht2 : t2 ∈ {s | MeasurableSet s}\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t2\nν : MeasureTheory.Measure Ω := ↑↑μ t2 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p1\n⊢ MeasurableSet t", "tactic": "exact measurableSet_generateFrom ht" } ]

[ 378, 51 ]

[ 361, 1 ]

[]

[ 1293, 6 ]

[ 1291, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.54011\nδ : Type ?u.54014\nε : Type ?u.54017\nι : Type ?u.54020\nf✝ f : α →. β\nx✝¹ : α\nx✝ : β\n⊢ x✝ ∈ comp f (PFun.id α) x✝¹ ↔ x✝ ∈ f x✝¹", "tactic": "simp" } ]

[ 593, 25 ]

[ 592, 1 ]

[ { "state_after": "G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nH₁ : Subgroup G\ninst✝¹ inst✝ : Normal H₁\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₁) (center (G ⧸ H₁))", "state_before": "G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nH₁ H₂ : Subgroup G\ninst✝¹ : Normal H₁\ninst✝ : Normal H₂\nh : H₁ = H₂\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₂) (center (G ⧸ H₂))", "tactic": "subst h" }, { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝³ : Group G\nH : Subgroup G\ninst✝² : Normal H\nH₁ : Subgroup G\ninst✝¹ inst✝ : Normal H₁\n⊢ comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₁) (center (G ⧸ H₁))", "tactic": "rfl" } ]

[ 593, 91 ]

[ 592, 9 ]

[ { "state_after": "case refine'_1\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s\n\ncase refine'_2\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ s ≤ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t)", "state_before": "ι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) = ↑↑μ s", "tactic": "refine' le_antisymm _ _" }, { "state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\ns' : Set α\nhsub : s ⊆ s'\nhs'm : MeasurableSet s'\nhs' : ↑↑μ s' = ↑↑μ s\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s", "state_before": "case refine'_1\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s", "tactic": "rcases exists_measurable_superset μ s with ⟨s', hsub, hs'm, hs'⟩" }, { "state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\ns' : Set α\nhsub : s ⊆ s'\nhs' : ↑↑μ s' = ↑↑μ s\nhs'm : NullMeasurableSet s'\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s", "state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\ns' : Set α\nhsub : s ⊆ s'\nhs'm : MeasurableSet s'\nhs' : ↑↑μ s' = ↑↑μ s\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s", "tactic": "replace hs'm : NullMeasurableSet s' μ := hs'm.nullMeasurableSet" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\ns' : Set α\nhsub : s ⊆ s'\nhs' : ↑↑μ s' = ↑↑μ s\nhs'm : NullMeasurableSet s'\n⊢ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t) ≤ ↑↑μ s", "tactic": "calc\n μ (s ∩ t) + μ (s \\ t) ≤ μ (s' ∩ t) + μ (s' \\ t) :=\n add_le_add (measure_mono <| inter_subset_inter_left _ hsub)\n (measure_mono <| diff_subset_diff_left hsub)\n _ = μ (s' ∩ t ∪ s' \\ t) :=\n (measure_union₀_aux (hs'm.inter ht) (hs'm.diff ht) <|\n (@disjoint_inf_sdiff _ s' t _).aedisjoint).symm\n _ = μ s' := (congr_arg μ (inter_union_diff _ _))\n _ = μ s := hs'" }, { "state_after": "no goals", "state_before": "case refine'_2\nι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ s ≤ ↑↑μ (s ∩ t) + ↑↑μ (s \\ t)", "tactic": "calc\n μ s = μ (s ∩ t ∪ s \\ t) := by rw [inter_union_diff]\n _ ≤ μ (s ∩ t) + μ (s \\ t) := measure_union_le _ _" }, { "state_after": "no goals", "state_before": "ι : Type ?u.14279\nα : Type u_1\nβ : Type ?u.14285\nγ : Type ?u.14288\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ s = ↑↑μ (s ∩ t ∪ s \\ t)", "tactic": "rw [inter_union_diff]" } ]

[ 320, 56 ]

[ 303, 1 ]

[]

[ 60, 61 ]

[ 59, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ns t : Set α\nx : α × α\na : α\n⊢ ∀ (x : α × α), x ∈ offDiag univ ↔ x ∈ diagonal αᶜ", "tactic": "simp" } ]

[ 583, 17 ]

[ 582, 1 ]

[]

[ 83, 30 ]

[ 80, 1 ]

[ { "state_after": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "state_before": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\n⊢ RingHom.IsIntegral\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))", "tactic": "intro x" }, { "state_after": "case intro.mk.mk\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nhu : u ∈ Algebra.algebraMapSubmonoid S M\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := hu }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := hu }).fst\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "state_before": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "tactic": "obtain ⟨⟨s, ⟨u, hu⟩⟩, hx⟩ := surj (Algebra.algebraMapSubmonoid S M) x" }, { "state_after": "case intro.mk.mk.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "state_before": "case intro.mk.mk\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nhu : u ∈ Algebra.algebraMapSubmonoid S M\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := hu }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := hu }).fst\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "tactic": "obtain ⟨v, hv⟩ := hu" }, { "state_after": "case intro.mk.mk.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "state_before": "case intro.mk.mk.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "tactic": "obtain ⟨v', hv'⟩ := isUnit_iff_exists_inv'.1 (map_units Rₘ ⟨v, hv.1⟩)" }, { "state_after": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1\n\ncase intro.mk.mk.intro.intro.refine'_2\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ IsIntegral Rₘ (x * ↑(algebraMap S Sₘ) u)", "state_before": "case intro.mk.mk.intro.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ RingHom.IsIntegralElem\n (IsLocalization.map Sₘ (algebraMap R S)\n (_ : M ≤ Submonoid.comap (algebraMap R S) (Algebra.algebraMapSubmonoid S M)))\n x", "tactic": "refine'\n @isIntegral_of_isIntegral_mul_unit Rₘ _ _ _ (localizationAlgebra M S) x (algebraMap S Sₘ u) v' _\n _" }, { "state_after": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : ↑(algebraMap Rₘ Sₘ) (v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) }) = ↑(algebraMap Rₘ Sₘ) 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "state_before": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "tactic": "replace hv' := congr_arg (@algebraMap Rₘ Sₘ _ _ (localizationAlgebra M S)) hv'" }, { "state_after": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' :\n ↑(algebraMap Rₘ Sₘ) v' *\n ↑(RingHom.comp (algebraMap Rₘ Sₘ) (algebraMap R Rₘ)) ↑{ val := v, property := (_ : v ∈ ↑M) } =\n 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "state_before": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : ↑(algebraMap Rₘ Sₘ) (v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) }) = ↑(algebraMap Rₘ Sₘ) 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "tactic": "rw [RingHom.map_mul, RingHom.map_one, ← RingHom.comp_apply _ (algebraMap R Rₘ)] at hv'" }, { "state_after": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' :\n ↑(algebraMap Rₘ Sₘ) v' * ↑(RingHom.comp (algebraMap S Sₘ) (algebraMap R S)) ↑{ val := v, property := (_ : v ∈ ↑M) } =\n 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "state_before": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' :\n ↑(algebraMap Rₘ Sₘ) v' *\n ↑(RingHom.comp (algebraMap Rₘ Sₘ) (algebraMap R Rₘ)) ↑{ val := v, property := (_ : v ∈ ↑M) } =\n 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "tactic": "erw [IsLocalization.map_comp\n (show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map)] at hv'" }, { "state_after": "no goals", "state_before": "case intro.mk.mk.intro.intro.refine'_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' :\n ↑(algebraMap Rₘ Sₘ) v' * ↑(RingHom.comp (algebraMap S Sₘ) (algebraMap R S)) ↑{ val := v, property := (_ : v ∈ ↑M) } =\n 1\n⊢ ↑(algebraMap Rₘ Sₘ) v' * ↑(algebraMap S Sₘ) u = 1", "tactic": "exact hv.2 ▸ hv'" }, { "state_after": "case intro.mk.mk.intro.intro.refine'_2.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\np : R[X]\nhp : Monic p ∧ eval₂ (algebraMap R S) s p = 0\n⊢ IsIntegral Rₘ (x * ↑(algebraMap S Sₘ) u)", "state_before": "case intro.mk.mk.intro.intro.refine'_2\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\n⊢ IsIntegral Rₘ (x * ↑(algebraMap S Sₘ) u)", "tactic": "obtain ⟨p, hp⟩ := H s" }, { "state_after": "no goals", "state_before": "case intro.mk.mk.intro.intro.refine'_2.intro\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.284769\ninst✝⁶ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\ninst✝¹ : Algebra S Sₘ\ninst✝ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\nH : Algebra.IsIntegral R S\nx : Sₘ\ns u : S\nv : R\nhv : v ∈ ↑M ∧ ↑(algebraMap R S) v = u\nhx :\n x * ↑(algebraMap S Sₘ) ↑(s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).snd =\n ↑(algebraMap S Sₘ) (s, { val := u, property := (_ : ∃ a, a ∈ ↑M ∧ ↑(algebraMap R S) a = u) }).fst\nv' : Rₘ\nhv' : v' * ↑(algebraMap R Rₘ) ↑{ val := v, property := (_ : v ∈ ↑M) } = 1\np : R[X]\nhp : Monic p ∧ eval₂ (algebraMap R S) s p = 0\n⊢ IsIntegral Rₘ (x * ↑(algebraMap S Sₘ) u)", "tactic": "exact hx.symm ▸ is_integral_localization_at_leadingCoeff p hp.2 (hp.1.symm ▸ M.one_mem)" } ]

[ 249, 92 ]

[ 231, 1 ]

[ { "state_after": "x y : ℝ\nS : Set ℝ\nh₁ : Set.Nonempty S\nh₂ : BddAbove S\n⊢ IsLUB S (choose (_ : ∃ x, IsLUB S x))", "state_before": "x y : ℝ\nS : Set ℝ\nh₁ : Set.Nonempty S\nh₂ : BddAbove S\n⊢ IsLUB S (sSup S)", "tactic": "simp only [sSup_def, dif_pos (And.intro h₁ h₂)]" }, { "state_after": "no goals", "state_before": "x y : ℝ\nS : Set ℝ\nh₁ : Set.Nonempty S\nh₂ : BddAbove S\n⊢ IsLUB S (choose (_ : ∃ x, IsLUB S x))", "tactic": "apply Classical.choose_spec" } ]

[ 744, 30 ]

[ 741, 11 ]

[]

[ 254, 6 ]

[ 253, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.1261185\nγ : Type ?u.1261188\nδ : Type ?u.1261191\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite μ\nC : ℝ≥0∞\nhC : C < ⊤\nf : α → E\nhf_meas : AEStronglyMeasurable f μ\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ) ≤ C\n⊢ SigmaFinite (Measure.trim μ (_ : m ≤ m))", "tactic": "rwa [@trim_eq_self _ m]" } ]

[ 1194, 98 ]

[ 1190, 1 ]

[ { "state_after": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nc : Cocone F\nj✝ : J\n⊢ ι (F ⋙ G) j✝ ≫ post F G ≫ G.map (desc F c) = ι (F ⋙ G) j✝ ≫ desc (F ⋙ G) (G.mapCocone c)", "state_before": "J : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nc : Cocone F\n⊢ post F G ≫ G.map (desc F c) = desc (F ⋙ G) (G.mapCocone c)", "tactic": "ext" }, { "state_after": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nc : Cocone F\nj✝ : J\n⊢ G.map (c.ι.app j✝) = (G.mapCocone c).ι.app j✝", "state_before": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nc : Cocone F\nj✝ : J\n⊢ ι (F ⋙ G) j✝ ≫ post F G ≫ G.map (desc F c) = ι (F ⋙ G) j✝ ≫ desc (F ⋙ G) (G.mapCocone c)", "tactic": "rw [← assoc, colimit.ι_post, ← G.map_comp, colimit.ι_desc, colimit.ι_desc]" }, { "state_after": "no goals", "state_before": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nc : Cocone F\nj✝ : J\n⊢ G.map (c.ι.app j✝) = (G.mapCocone c).ι.app j✝", "tactic": "rfl" } ]

[ 1039, 6 ]

[ 1035, 1 ]

[]

[ 301, 14 ]

[ 300, 1 ]

[]

[ 610, 20 ]

[ 609, 1 ]

[ { "state_after": "case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ x ∈ map (AlgEquiv.toLinearMap h) (I / J) ↔ x ∈ map (AlgEquiv.toLinearMap h) I / map (AlgEquiv.toLinearMap h) J", "state_before": "ι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\n⊢ map (AlgEquiv.toLinearMap h) (I / J) = map (AlgEquiv.toLinearMap h) I / map (AlgEquiv.toLinearMap h) J", "tactic": "ext x" }, { "state_after": "case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x) ↔\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y", "state_before": "case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ x ∈ map (AlgEquiv.toLinearMap h) (I / J) ↔ x ∈ map (AlgEquiv.toLinearMap h) I / map (AlgEquiv.toLinearMap h) J", "tactic": "simp only [mem_map, mem_div_iff_forall_mul_mem]" }, { "state_after": "case h.mp\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x) →\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\n\ncase h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y) →\n ∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x", "state_before": "case h\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x) ↔\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y", "tactic": "constructor" }, { "state_after": "case h.mp.intro.intro.intro.intro\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : A\nhx : ∀ (y : A), y ∈ J → x * y ∈ I\ny : A\nhy : y ∈ J\n⊢ ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = ↑(AlgEquiv.toLinearMap h) x * ↑(AlgEquiv.toLinearMap h) y", "state_before": "case h.mp\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x) →\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y", "tactic": "rintro ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩" }, { "state_after": "no goals", "state_before": "case h.mp.intro.intro.intro.intro\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : A\nhx : ∀ (y : A), y ∈ J → x * y ∈ I\ny : A\nhy : y ∈ J\n⊢ ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = ↑(AlgEquiv.toLinearMap h) x * ↑(AlgEquiv.toLinearMap h) y", "tactic": "exact ⟨x * y, hx _ hy, h.map_mul x y⟩" }, { "state_after": "case h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\n⊢ ∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x", "state_before": "case h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\n⊢ (∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y) →\n ∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x", "tactic": "rintro hx" }, { "state_after": "case h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\n⊢ ↑(AlgEquiv.symm h) x * z ∈ I", "state_before": "case h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\n⊢ ∃ y, (∀ (y_1 : A), y_1 ∈ J → y * y_1 ∈ I) ∧ ↑(AlgEquiv.toLinearMap h) y = x", "tactic": "refine' ⟨h.symm x, fun z hz => _, h.apply_symm_apply x⟩" }, { "state_after": "case h.mpr.intro.intro\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑(AlgEquiv.symm h) x * z ∈ I", "state_before": "case h.mpr\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\n⊢ ↑(AlgEquiv.symm h) x * z ∈ I", "tactic": "obtain ⟨xz, xz_mem, hxz⟩ := hx (h z) ⟨z, hz, rfl⟩" }, { "state_after": "case h.e'_4\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑(AlgEquiv.symm h) x * z = xz", "state_before": "case h.mpr.intro.intro\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑(AlgEquiv.symm h) x * z ∈ I", "tactic": "convert xz_mem" }, { "state_after": "case h.e'_4.a\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑h (↑(AlgEquiv.symm h) x * z) = ↑h xz", "state_before": "case h.e'_4\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑(AlgEquiv.symm h) x * z = xz", "tactic": "apply h.injective" }, { "state_after": "no goals", "state_before": "case h.e'_4.a\nι : Sort uι\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nM N : Submodule R A\nm n : A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : B\nhx :\n ∀ (y : B),\n (∃ y_1, y_1 ∈ J ∧ ↑(AlgEquiv.toLinearMap h) y_1 = y) → ∃ y_1, y_1 ∈ I ∧ ↑(AlgEquiv.toLinearMap h) y_1 = x * y\nz : A\nhz : z ∈ J\nxz : A\nxz_mem : xz ∈ I\nhxz : ↑(AlgEquiv.toLinearMap h) xz = x * ↑h z\n⊢ ↑h (↑(AlgEquiv.symm h) x * z) = ↑h xz", "tactic": "erw [h.map_mul, h.apply_symm_apply, hxz]" } ]

[ 741, 45 ]

[ 729, 11 ]

[ { "state_after": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ (leftAdjointUniq adj1 adj1).hom.app x✝ = (𝟙 F).app x✝", "state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\n⊢ (leftAdjointUniq adj1 adj1).hom = 𝟙 F", "tactic": "ext" }, { "state_after": "case w.h.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ ((leftAdjointUniq adj1 adj1).hom.app x✝).op = ((𝟙 F).app x✝).op", "state_before": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ (leftAdjointUniq adj1 adj1).hom.app x✝ = (𝟙 F).app x✝", "tactic": "apply Quiver.Hom.op_inj" }, { "state_after": "case w.h.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝).op = coyoneda.map ((𝟙 F).app x✝).op", "state_before": "case w.h.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ ((leftAdjointUniq adj1 adj1).hom.app x✝).op = ((𝟙 F).app x✝).op", "tactic": "apply coyoneda.map_injective" }, { "state_after": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝¹).op).app x✝ a✝ = (coyoneda.map ((𝟙 F).app x✝¹).op).app x✝ a✝", "state_before": "case w.h.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝ : C\n⊢ coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝).op = coyoneda.map ((𝟙 F).app x✝).op", "tactic": "ext" }, { "state_after": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝¹).op).app x✝ a✝ = (coyoneda.map ((𝟙 F).app x✝¹).op).app x✝ a✝", "state_before": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝¹).op).app x✝ a✝ = (coyoneda.map ((𝟙 F).app x✝¹).op).app x✝ a✝", "tactic": "funext" }, { "state_after": "no goals", "state_before": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x✝¹).op).app x✝ a✝ = (coyoneda.map ((𝟙 F).app x✝¹).op).app x✝ a✝", "tactic": "simp [leftAdjointsCoyonedaEquiv, leftAdjointUniq]" } ]

[ 213, 52 ]

[ 206, 1 ]

[ { "state_after": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\n⊢ ContinuousWithinAt (fun a => List.prod (List.map (fun i => f i a) l)) t x", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\n⊢ ContinuousOn (fun a => List.prod (List.map (fun i => f i a) l)) t", "tactic": "intro x hx" }, { "state_after": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\n⊢ ContinuousAt (restrict t fun a => List.prod (List.map (fun i => f i a) l)) { val := x, property := hx }", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\n⊢ ContinuousWithinAt (fun a => List.prod (List.map (fun i => f i a) l)) t x", "tactic": "rw [continuousWithinAt_iff_continuousAt_restrict _ hx]" }, { "state_after": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\n⊢ ContinuousAt (restrict t fun a => List.prod (List.map (fun i => f i a) l)) { val := x, property := hx }", "tactic": "refine' tendsto_list_prod _ fun i hi => _" }, { "state_after": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\nh : ContinuousWithinAt (f i) t x\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ (i : ι), i ∈ l → ContinuousOn (f i) t\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "tactic": "specialize h i hi x hx" }, { "state_after": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\nh : ContinuousAt (restrict t (f i)) { val := x, property := hx }\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\nh : ContinuousWithinAt (f i) t x\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "tactic": "rw [continuousWithinAt_iff_continuousAt_restrict _ hx] at h" }, { "state_after": "no goals", "state_before": "ι : Type u_1\nα : Type ?u.175300\nX : Type u_2\nM : Type u_3\nN : Type ?u.175309\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nx : X\nhx : x ∈ t\ni : ι\nhi : i ∈ l\nh : ContinuousAt (restrict t (f i)) { val := x, property := hx }\n⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property := hx }) (𝓝 (f i ↑{ val := x, property := hx }))", "tactic": "exact h" } ]

[ 555, 10 ]

[ 547, 1 ]

[]

[ 161, 49 ]

[ 159, 1 ]

[ { "state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (↑(algebraMap R A) r)", "state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\n⊢ ↑f x ∈ adjoin R (↑f '' (s ∪ {1}))", "tactic": "refine'\n @adjoin_induction R A _ _ _ _ (fun a => f a ∈ adjoin R (f '' (s ∪ {1}))) x h\n (fun a ha => subset_adjoin ⟨a, ⟨Set.subset_union_left _ _ ha, rfl⟩⟩) (fun r => _)\n (fun y z hy hz => by simpa [hy, hz] using Subalgebra.add_mem _ hy hz) fun y z hy hz => by\n simpa [hy, hz, hf y z] using Subalgebra.mul_mem _ hy hz" }, { "state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (↑(algebraMap R A) r)", "state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (↑(algebraMap R A) r)", "tactic": "have : f 1 ∈ adjoin R (f '' (s ∪ {1})) :=\n subset_adjoin ⟨1, ⟨Set.subset_union_right _ _ <| Set.mem_singleton 1, rfl⟩⟩" }, { "state_after": "case h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ ↑f (↑(algebraMap R A) r) = r • ↑f 1", "state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (↑(algebraMap R A) r)", "tactic": "convert Subalgebra.smul_mem (adjoin R (f '' (s ∪ {1}))) this r" }, { "state_after": "case h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ ↑f (r • 1) = r • ↑f 1", "state_before": "case h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ ↑f (↑(algebraMap R A) r) = r • ↑f 1", "tactic": "rw [algebraMap_eq_smul_one]" }, { "state_after": "no goals", "state_before": "case h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\nr : R\nthis : ↑f 1 ∈ adjoin R (↑f '' (s ∪ {1}))\n⊢ ↑f (r • 1) = r • ↑f 1", "tactic": "exact f.map_smul _ _" }, { "state_after": "no goals", "state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\ny z : A\nhy : (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) y\nhz : (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) z\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (y + z)", "tactic": "simpa [hy, hz] using Subalgebra.add_mem _ hy hz" }, { "state_after": "no goals", "state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns✝ t s : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), ↑f (a₁ * a₂) = ↑f a₁ * ↑f a₂\nh : x ∈ adjoin R s\ny z : A\nhy : (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) y\nhz : (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) z\n⊢ (fun a => ↑f a ∈ adjoin R (↑f '' (s ∪ {1}))) (y * z)", "tactic": "simpa [hy, hz, hf y z] using Subalgebra.mul_mem _ hy hz" } ]

[ 249, 23 ]

[ 238, 1 ]

[]

[ 671, 45 ]

[ 669, 1 ]

[]

[ 203, 51 ]

[ 201, 1 ]

[]

[ 1260, 28 ]

[ 1259, 1 ]

[]

[ 890, 27 ]

[ 889, 1 ]

[ { "state_after": "no goals", "state_before": "E : Type ?u.101139\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\n⊢ dist z (↑(starRingEnd ℂ) z) = 2 * Abs.abs z.im", "tactic": "rw [dist_comm, dist_conj_self]" } ]

[ 139, 99 ]

[ 139, 1 ]

[ { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\n⊢ ↑↑μ univ = ⊤", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\n⊢ ↑↑μ univ = ⊤", "tactic": "obtain ⟨K, hK, Kclosed, Kint⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ (1 : G) ∈ interior K := by\n rcases local_isCompact_isClosed_nhds_of_group (isOpen_univ.mem_nhds (mem_univ (1 : G))) with\n ⟨K, hK⟩\n exact ⟨K, hK.1, hK.2.1, hK.2.2.2⟩" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\n⊢ ↑↑μ univ = ⊤", "tactic": "have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, Kint⟩" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\nA : ∀ (L : Set G), IsCompact L → ∃ g, Disjoint L (g • K)\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\n⊢ ↑↑μ univ = ⊤", "tactic": "have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>\n exists_disjoint_smul_of_isCompact hL hK" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\nA : ∀ (L : Set G), IsCompact L → ∃ g, Disjoint L (g • K)\n⊢ ↑↑μ univ = ⊤", "tactic": "choose! g hg using A" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\n⊢ ↑↑μ univ = ⊤", "tactic": "set L : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (L n)) atTop (𝓝 (⊤ * ↑↑μ K))\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ univ = ⊤", "tactic": "have N : Tendsto (fun n => μ (L n)) atTop (𝓝 (∞ * μ K)) := by\n simp_rw [M]\n apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero)\n exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _)" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (((fun T => T ∪ g T • K)^[n]) K)) atTop (𝓝 ⊤)\n⊢ ↑↑μ univ = ⊤", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (L n)) atTop (𝓝 (⊤ * ↑↑μ K))\n⊢ ↑↑μ univ = ⊤", "tactic": "simp only [ENNReal.top_mul', K_pos.ne', if_false] at N" }, { "state_after": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (((fun T => T ∪ g T • K)^[n]) K)) atTop (𝓝 ⊤)\n⊢ ⊤ ≤ ↑↑μ univ", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (((fun T => T ∪ g T • K)^[n]) K)) atTop (𝓝 ⊤)\n⊢ ↑↑μ univ = ⊤", "tactic": "apply top_le_iff.1" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\nN : Tendsto (fun n => ↑↑μ (((fun T => T ∪ g T • K)^[n]) K)) atTop (𝓝 ⊤)\n⊢ ⊤ ≤ ↑↑μ univ", "tactic": "exact le_of_tendsto' N fun n => measure_mono (subset_univ _)" }, { "state_after": "case intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K ∧ IsClosed K ∧ K ⊆ univ ∧ 1 ∈ interior K\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ 1 ∈ interior K", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ 1 ∈ interior K", "tactic": "rcases local_isCompact_isClosed_nhds_of_group (isOpen_univ.mem_nhds (mem_univ (1 : G))) with\n ⟨K, hK⟩" }, { "state_after": "no goals", "state_before": "case intro\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K ∧ IsClosed K ∧ K ⊆ univ ∧ 1 ∈ interior K\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ 1 ∈ interior K", "tactic": "exact ⟨K, hK.1, hK.2.1, hK.2.2.2⟩" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\n⊢ IsCompact (L n)", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\n⊢ ∀ (n : ℕ), IsCompact (L n)", "tactic": "intro n" }, { "state_after": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\n⊢ IsCompact (L Nat.zero)\n\ncase succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\nIH : IsCompact (L n)\n⊢ IsCompact (L (Nat.succ n))", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\n⊢ IsCompact (L n)", "tactic": "induction' n with n IH" }, { "state_after": "no goals", "state_before": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\n⊢ IsCompact (L Nat.zero)", "tactic": "exact hK" }, { "state_after": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\nIH : IsCompact (L n)\n⊢ IsCompact (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K)", "state_before": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\nIH : IsCompact (L n)\n⊢ IsCompact (L (Nat.succ n))", "tactic": "simp_rw [iterate_succ']" }, { "state_after": "no goals", "state_before": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nn : ℕ\nIH : IsCompact (L n)\n⊢ IsCompact (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K)", "tactic": "apply IsCompact.union IH (hK.smul (g (L n)))" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\n⊢ IsClosed (L n)", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\n⊢ ∀ (n : ℕ), IsClosed (L n)", "tactic": "intro n" }, { "state_after": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\n⊢ IsClosed (L Nat.zero)\n\ncase succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\nIH : IsClosed (L n)\n⊢ IsClosed (L (Nat.succ n))", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\n⊢ IsClosed (L n)", "tactic": "induction' n with n IH" }, { "state_after": "no goals", "state_before": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\n⊢ IsClosed (L Nat.zero)", "tactic": "exact Kclosed" }, { "state_after": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\nIH : IsClosed (L n)\n⊢ IsClosed (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K)", "state_before": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\nIH : IsClosed (L n)\n⊢ IsClosed (L (Nat.succ n))", "tactic": "simp_rw [iterate_succ']" }, { "state_after": "no goals", "state_before": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nn : ℕ\nIH : IsClosed (L n)\n⊢ IsClosed (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K)", "tactic": "apply IsClosed.union IH (Kclosed.smul (g (L n)))" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\n⊢ ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\n⊢ ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K", "tactic": "intro n" }, { "state_after": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\n⊢ ↑↑μ (L Nat.zero) = ↑(Nat.zero + 1) * ↑↑μ K\n\ncase succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (L (Nat.succ n)) = ↑(Nat.succ n + 1) * ↑↑μ K", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\n⊢ ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K", "tactic": "induction' n with n IH" }, { "state_after": "no goals", "state_before": "case zero\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\n⊢ ↑↑μ (L Nat.zero) = ↑(Nat.zero + 1) * ↑↑μ K", "tactic": "simp only [one_mul, Nat.cast_one, iterate_zero, id.def, Nat.zero_eq, Nat.zero_add]" }, { "state_after": "no goals", "state_before": "case succ\n𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (L (Nat.succ n)) = ↑(Nat.succ n + 1) * ↑↑μ K", "tactic": "calc\n μ (L (n + 1)) = μ (L n) + μ (g (L n) • K) := by\n simp_rw [iterate_succ']\n exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet\n _ = (n + 1 + 1 : ℕ) * μ K := by\n simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul]" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K) =\n ↑↑μ (((fun T => T ∪ g T • K)^[n]) K) + ↑↑μ (g (((fun T => T ∪ g T • K)^[n]) K) • K)", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (L (n + 1)) = ↑↑μ (L n) + ↑↑μ (g (L n) • K)", "tactic": "simp_rw [iterate_succ']" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K) =\n ↑↑μ (((fun T => T ∪ g T • K)^[n]) K) + ↑↑μ (g (((fun T => T ∪ g T • K)^[n]) K) • K)", "tactic": "exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nn : ℕ\nIH : ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ ↑↑μ (L n) + ↑↑μ (g (L n) • K) = ↑(n + 1 + 1) * ↑↑μ K", "tactic": "simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul]" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ Tendsto (fun n => ↑(n + 1) * ↑↑μ K) atTop (𝓝 (⊤ * ↑↑μ K))", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ Tendsto (fun n => ↑↑μ (L n)) atTop (𝓝 (⊤ * ↑↑μ K))", "tactic": "simp_rw [M]" }, { "state_after": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ Tendsto (fun x => ↑(x + 1)) atTop (𝓝 ⊤)", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ Tendsto (fun n => ↑(n + 1) * ↑↑μ K) atTop (𝓝 (⊤ * ↑↑μ K))", "tactic": "apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero)" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.542119\nG : Type u_1\nH : Type ?u.542125\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : BorelSpace G\nμ✝ : Measure G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : IsMulLeftInvariant μ✝\ninst✝³ : LocallyCompactSpace G\ninst✝² : NoncompactSpace G\nμ : Measure G\ninst✝¹ : IsOpenPosMeasure μ\ninst✝ : IsMulLeftInvariant μ\nK : Set G\nhK : IsCompact K\nKclosed : IsClosed K\nKint : 1 ∈ interior K\nK_pos : 0 < ↑↑μ K\ng : Set G → G\nhg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)\nL : ℕ → Set G := fun n => ((fun T => T ∪ g T • K)^[n]) K\nLcompact : ∀ (n : ℕ), IsCompact (L n)\nLclosed : ∀ (n : ℕ), IsClosed (L n)\nM : ∀ (n : ℕ), ↑↑μ (L n) = ↑(n + 1) * ↑↑μ K\n⊢ Tendsto (fun x => ↑(x + 1)) atTop (𝓝 ⊤)", "tactic": "exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _)" } ]

[ 655, 63 ]

[ 612, 1 ]

[]

[ 320, 56 ]

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[]

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[]

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[]

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[ { "state_after": "case mp\na b r q : Int\nh : 0 < b\n⊢ a / b = q ∧ a % b = r → r + b * q = a ∧ 0 ≤ r ∧ r < b\n\ncase mpr\na b r q : Int\nh : 0 < b\n⊢ r + b * q = a ∧ 0 ≤ r ∧ r < b → a / b = q ∧ a % b = r", "state_before": "a b r q : Int\nh : 0 < b\n⊢ a / b = q ∧ a % b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < b", "tactic": "constructor" }, { "state_after": "case mp\na b r q : Int\nh : 0 < b\n⊢ a % b + b * (a / b) = a ∧ 0 ≤ a % b ∧ a % b < b", "state_before": "case mp\na b r q : Int\nh : 0 < b\n⊢ a / b = q ∧ a % b = r → r + b * q = a ∧ 0 ≤ r ∧ r < b", "tactic": "intro ⟨rfl, rfl⟩" }, { "state_after": "no goals", "state_before": "case mp\na b r q : Int\nh : 0 < b\n⊢ a % b + b * (a / b) = a ∧ 0 ≤ a % b ∧ a % b < b", "tactic": "exact ⟨emod_add_ediv a b, emod_nonneg _ (Int.ne_of_gt h), emod_lt_of_pos _ h⟩" }, { "state_after": "case mpr\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) / b = q ∧ (r + b * q) % b = r", "state_before": "case mpr\na b r q : Int\nh : 0 < b\n⊢ r + b * q = a ∧ 0 ≤ r ∧ r < b → a / b = q ∧ a % b = r", "tactic": "intro ⟨rfl, hz, hb⟩" }, { "state_after": "case mpr.left\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) / b = q\n\ncase mpr.right\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) % b = r", "state_before": "case mpr\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) / b = q ∧ (r + b * q) % b = r", "tactic": "constructor" }, { "state_after": "case mpr.left\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ 0 + q = q", "state_before": "case mpr.left\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) / b = q", "tactic": "rw [Int.add_mul_ediv_left r q (Int.ne_of_gt h), ediv_eq_zero_of_lt hz hb]" }, { "state_after": "no goals", "state_before": "case mpr.left\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ 0 + q = q", "tactic": "simp [Int.zero_add]" }, { "state_after": "no goals", "state_before": "case mpr.right\na b r q : Int\nh : 0 < b\nhz : 0 ≤ r\nhb : r < b\n⊢ (r + b * q) % b = r", "tactic": "rw [add_mul_emod_self_left, emod_eq_of_lt hz hb]" } ]

[ 510, 55 ]

[ 501, 11 ]

[]

[ 614, 6 ]

[ 613, 1 ]

[ { "state_after": "no goals", "state_before": "α✝ : Type u\nβ : Type v\nι : Type ?u.47137\nπ : ι → Type ?u.47142\ninst✝³ : TopologicalSpace α✝\ninst✝² : TopologicalSpace β\ns t : Set α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\n⊢ cocompact α = cofinite", "tactic": "simp only [cocompact, hasBasis_cofinite.eq_biInf, isCompact_iff_finite]" } ]

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[]

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[]

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[]

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