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tasksource 14

[ { "state_after": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ ↑f x ∈ closure s", "state_before": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ x ∈ ↑(comap f (closure s))", "tactic": "rw [SetLike.mem_coe, mem_comap]" }, { "state_after": "no goals", "state_before": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ ↑f x ∈ closure s", "tactic": "exact subset_closure hx" } ]

[ 2926, 91 ]

[ 2925, 1 ]

[]

[ 142, 26 ]

[ 141, 1 ]

[ { "state_after": "case a\nα : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ erase s aᶜ ↔ a✝ ∈ insert a (sᶜ)", "state_before": "α : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na : α\n⊢ erase s aᶜ = insert a (sᶜ)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case a\nα : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ erase s aᶜ ↔ a✝ ∈ insert a (sᶜ)", "tactic": "simp only [or_iff_not_imp_left, mem_insert, not_and, mem_compl, mem_erase]" } ]

[ 224, 77 ]

[ 222, 1 ]

[]

[ 247, 29 ]

[ 245, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\ns : α\nh : ¬P s\n⊢ extend m s = ⊤", "tactic": "simp [extend, h]" } ]

[ 1325, 81 ]

[ 1325, 1 ]

[]

[ 113, 50 ]

[ 109, 1 ]

[ { "state_after": "a b c : Int\nh✝ : a ≤ b + c\nh : -b + a ≤ -b + (b + c)\n⊢ -b + a ≤ c", "state_before": "a b c : Int\nh : a ≤ b + c\n⊢ -b + a ≤ c", "tactic": "have h := Int.add_le_add_left h (-b)" }, { "state_after": "no goals", "state_before": "a b c : Int\nh✝ : a ≤ b + c\nh : -b + a ≤ -b + (b + c)\n⊢ -b + a ≤ c", "tactic": "rwa [Int.neg_add_cancel_left] at h" } ]

[ 980, 37 ]

[ 978, 11 ]

[]

[ 380, 32 ]

[ 379, 1 ]

[]

[ 337, 6 ]

[ 336, 1 ]

[]

[ 934, 6 ]

[ 933, 1 ]

[ { "state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective ((forget Profinite).map f)\n\ncase mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Function.Surjective ((forget Profinite).map f) → Epi f", "state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f ↔ Function.Surjective ((forget Profinite).map f)", "tactic": "constructor" }, { "state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → ∀ (b : (forget Profinite).obj Y), ∃ a, (forget Profinite).map f a = b", "state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective ((forget Profinite).map f)", "tactic": "dsimp [Function.Surjective]" }, { "state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ (∃ b, ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ b) → ¬Epi f", "state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → ∀ (b : (forget Profinite).obj Y), ∃ a, (forget Profinite).map f a = b", "tactic": "contrapose!" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False", "state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ (∃ b, ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ b) → ¬Epi f", "tactic": "rintro ⟨y, hy⟩ hf" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False", "tactic": "skip" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False", "tactic": "let C := Set.range f" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\n⊢ False", "tactic": "have hC : IsClosed C := (isCompact_range f.continuous).isClosed" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\n⊢ False", "tactic": "let U := Cᶜ" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ False", "tactic": "have hyU : y ∈ U := by\n refine' Set.mem_compl _\n rintro ⟨y', hy'⟩\n exact hy y' hy'" }, { "state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\n⊢ False", "tactic": "have hUy : U ∈ 𝓝 y := hC.compl_mem_nhds hyU" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False", "state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\n⊢ False", "tactic": "obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_clopen.mem_nhds_iff.mp hUy" }, { "state_after": "no goals", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False", "tactic": "classical\n let Z := of (ULift.{u} <| Fin 2)\n let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩\n let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩\n have H : h = g := by\n rw [← cancel_epi f]\n ext x\n apply ULift.ext\n dsimp [LocallyConstant.ofClopen]\n simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]\n rw [if_neg]\n refine' mt (fun α => hVU α) _\n simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]\n apply_fun fun e => (e y).down at H\n dsimp [LocallyConstant.ofClopen] at H\n rw [if_pos hyV] at H\n exact top_ne_bot H" }, { "state_after": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ ¬y ∈ C", "state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ y ∈ U", "tactic": "refine' Set.mem_compl _" }, { "state_after": "case intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\ny' : (forget Profinite).obj X\nhy' : (forget Profinite).map f y' = y\n⊢ False", "state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ ¬y ∈ C", "tactic": "rintro ⟨y', hy'⟩" }, { "state_after": "no goals", "state_before": "case intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\ny' : (forget Profinite).obj X\nhy' : (forget Profinite).map f y' = y\n⊢ False", "tactic": "exact hy y' hy'" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False", "tactic": "let Z := of (ULift.{u} <| Fin 2)" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\n⊢ False", "tactic": "let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\n⊢ False", "tactic": "let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : h = g\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ False", "tactic": "have H : h = g := by\n rw [← cancel_epi f]\n ext x\n apply ULift.ext\n dsimp [LocallyConstant.ofClopen]\n simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]\n rw [if_neg]\n refine' mt (fun α => hVU α) _\n simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : ((forget Profinite).map h y).down = ((forget Profinite).map g y).down\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : h = g\n⊢ False", "tactic": "apply_fun fun e => (e y).down at H" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = if y ∈ V then 0 else 1\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : ((forget Profinite).map h y).down = ((forget Profinite).map g y).down\n⊢ False", "tactic": "dsimp [LocallyConstant.ofClopen] at H" }, { "state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = 0\n⊢ False", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = if y ∈ V then 0 else 1\n⊢ False", "tactic": "rw [if_pos hyV] at H" }, { "state_after": "no goals", "state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = 0\n⊢ False", "tactic": "exact top_ne_bot H" }, { "state_after": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ f ≫ h = f ≫ g", "state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ h = g", "tactic": "rw [← cancel_epi f]" }, { "state_after": "case w\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ (forget Profinite).map (f ≫ h) x = (forget Profinite).map (f ≫ g) x", "state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ f ≫ h = f ≫ g", "tactic": "ext x" }, { "state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ h) x).down = ((forget Profinite).map (f ≫ g) x).down", "state_before": "case w\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ (forget Profinite).map (f ≫ h) x = (forget Profinite).map (f ≫ g) x", "tactic": "apply ULift.ext" }, { "state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ ContinuousMap.mk fun x => { down := 1 }) x).down =\n ((forget Profinite).map (f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down", "state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ h) x).down = ((forget Profinite).map (f ≫ g) x).down", "tactic": "dsimp [LocallyConstant.ofClopen]" }, { "state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ 1 = if (forget Profinite).map f x ∈ V then 0 else 1", "state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ ContinuousMap.mk fun x => { down := 1 }) x).down =\n ((forget Profinite).map (f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down", "tactic": "simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]" }, { "state_after": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ V", "state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ 1 = if (forget Profinite).map f x ∈ V then 0 else 1", "tactic": "rw [if_neg]" }, { "state_after": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ U", "state_before": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ V", "tactic": "refine' mt (fun α => hVU α) _" }, { "state_after": "no goals", "state_before": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ U", "tactic": "simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]" }, { "state_after": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi ((forget Profinite).map f) → Epi f", "state_before": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Function.Surjective ((forget Profinite).map f) → Epi f", "tactic": "rw [← CategoryTheory.epi_iff_surjective]" }, { "state_after": "no goals", "state_before": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi ((forget Profinite).map f) → Epi f", "tactic": "apply (forget Profinite).epi_of_epi_map" } ]

[ 413, 44 ]

[ 379, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))\n (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val))", "state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ (modify m k f).size = Bucket.size (modify m k f).buckets", "tactic": "dsimp [modify, cond]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))", "state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))\n (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val))", "tactic": "rw [Bucket.update_update]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)", "state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))", "tactic": "simp [h, Bucket.size]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))", "state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)", "tactic": "refine have ⟨_, _, h₁, _, eq⟩ := Bucket.exists_of_update ..; eq ▸ ?_" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))", "tactic": "simp [h, h₁, Bucket.size_eq]" } ]

[ 271, 31 ]

[ 265, 1 ]

[ { "state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) s", "state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x x_1 => x ≤ x_1) s", "tactic": "refine' ⟨_, Bounded.mono (Set.inter_subset_left s _)⟩" }, { "state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b})", "state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) s", "tactic": "rw [← @bounded_le_inter_lt _ s _ a]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b})", "tactic": "exact Bounded.mono fun x ⟨hx, hx'⟩ => ⟨hx, le_of_lt hx'⟩" } ]

[ 345, 59 ]

[ 341, 1 ]

[]

[ 2009, 6 ]

[ 2006, 1 ]

[ { "state_after": "x : PGame\n⊢ ((∀ (i : LeftMoves x), moveLeft x i ⧏ 0) ∧ ∀ (j : RightMoves 0), x ⧏ moveRight 0 j) ↔\n ∀ (i : LeftMoves x), moveLeft x i ⧏ 0", "state_before": "x : PGame\n⊢ x ≤ 0 ↔ ∀ (i : LeftMoves x), moveLeft x i ⧏ 0", "tactic": "rw [le_iff_forall_lf]" }, { "state_after": "no goals", "state_before": "x : PGame\n⊢ ((∀ (i : LeftMoves x), moveLeft x i ⧏ 0) ∧ ∀ (j : RightMoves 0), x ⧏ moveRight 0 j) ↔\n ∀ (i : LeftMoves x), moveLeft x i ⧏ 0", "tactic": "simp" } ]

[ 651, 7 ]

[ 649, 1 ]

[]

[ 568, 84 ]

[ 568, 1 ]

[]

[ 694, 73 ]

[ 691, 11 ]

[]

[ 775, 28 ]

[ 773, 1 ]

[]

[ 467, 54 ]

[ 466, 1 ]

[ { "state_after": "no goals", "state_before": "M : Type ?u.368159\nN : Type ?u.368162\nG : Type ?u.368165\nR : Type u_1\nS : Type ?u.368171\nF : Type ?u.368174\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ+\ninst✝ : CommMonoid R\nζ : { x // x ∈ rootsOfUnity k R }\nm : ℕ\n⊢ ↑↑(ζ ^ m) = ↑↑ζ ^ m", "tactic": "rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]" } ]

[ 136, 50 ]

[ 134, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na b : α\nh : R b a\n⊢ destutter' R b (a :: l) = b :: destutter' R a l", "tactic": "rw [destutter', if_pos h]" } ]

[ 52, 28 ]

[ 51, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.12132\nα : Type ?u.12135\nM : Type u_1\nN : Type ?u.12141\nP : Type ?u.12144\nM₀ : Type ?u.12147\nG : Type ?u.12150\nR : Type ?u.12153\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na✝ a : M\nl : List M\n⊢ prod (a :: l) = foldr (fun x x_1 => x * x_1) 1 (a :: l)", "tactic": "rw [prod_cons, foldr_cons, prod_eq_foldr]" } ]

[ 75, 61 ]

[ 73, 1 ]

[]

[ 151, 22 ]

[ 149, 11 ]

[ { "state_after": "no goals", "state_before": "α : Type u_3\nα' : Type ?u.47893\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_2\nγ' : Type ?u.47905\nδ : Type u_1\nδ' : Type ?u.47911\nε : Type ?u.47914\nε' : Type ?u.47917\nζ : Type ?u.47920\nζ' : Type ?u.47923\nν : Type ?u.47926\nf f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\ng : γ → δ\nf' : β' → α → δ\ng' : β → β'\nh_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' (g' b) a\n⊢ image2 f' ((fun b => g' b) '' t) ((fun a => a) '' s) = image2 f' (g' '' t) s", "tactic": "rw [image_id']" } ]

[ 408, 70 ]

[ 405, 1 ]

[ { "state_after": "case mp\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : ↑f x = ↑f y\n⊢ y⁻¹ * x ∈ ker f\n\ncase mpr\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : y⁻¹ * x ∈ ker f\n⊢ ↑f x = ↑f y", "state_before": "G : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\n⊢ ↑f x = ↑f y ↔ y⁻¹ * x ∈ ker f", "tactic": "constructor <;> intro h" }, { "state_after": "no goals", "state_before": "case mp\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : ↑f x = ↑f y\n⊢ y⁻¹ * x ∈ ker f", "tactic": "rw [mem_ker, map_mul, h, ← map_mul, inv_mul_self, map_one]" }, { "state_after": "no goals", "state_before": "case mpr\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : y⁻¹ * x ∈ ker f\n⊢ ↑f x = ↑f y", "tactic": "rw [← one_mul x, ← mul_inv_self y, mul_assoc, map_mul, f.mem_ker.1 h, mul_one]" } ]

[ 2788, 83 ]

[ 2785, 1 ]

[ { "state_after": "no goals", "state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nc : Cone F\nX : C\nf : X ⟶ c.pt\n⊢ lift F (Cone.extend c f) = f ≫ lift F c", "tactic": "aesop_cat" } ]

[ 312, 67 ]

[ 311, 1 ]

[]

[ 635, 51 ]

[ 633, 1 ]

[ { "state_after": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (eqToHom (_ : X C i = X C i') ≫ d C i' j) = imageSubobject (d C i' j)", "state_before": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (d C i j) = imageSubobject (d C i' j)", "tactic": "rw [← eqToHom_comp_d C r r']" }, { "state_after": "no goals", "state_before": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (eqToHom (_ : X C i = X C i') ≫ d C i' j) = imageSubobject (d C i' j)", "tactic": "apply imageSubobject_iso_comp" } ]

[ 344, 32 ]

[ 341, 1 ]

[ { "state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ ∀ (x : E), x ∈ s → ∀ (y : E), y ∈ s → JoinedIn s x y", "state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ IsPathConnected s", "tactic": "refine' isPathConnected_iff.mpr ⟨hne, _⟩" }, { "state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\n⊢ JoinedIn s x y", "state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ ∀ (x : E), x ∈ s → ∀ (y : E), y ∈ s → JoinedIn s x y", "tactic": "intro x x_in y y_in" }, { "state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : [x-[ℝ]y] ⊆ s\n⊢ JoinedIn s x y", "state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\n⊢ JoinedIn s x y", "tactic": "have H := hconv.segment_subset x_in y_in" }, { "state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : ↑(lineMap x y) '' Icc 0 1 ⊆ s\n⊢ JoinedIn s x y", "state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : [x-[ℝ]y] ⊆ s\n⊢ JoinedIn s x y", "tactic": "rw [segment_eq_image_lineMap] at H" }, { "state_after": "no goals", "state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : ↑(lineMap x y) '' Icc 0 1 ⊆ s\n⊢ JoinedIn s x y", "tactic": "exact\n JoinedIn.ofLine AffineMap.lineMap_continuous.continuousOn (lineMap_apply_zero _ _)\n (lineMap_apply_one _ _) H" } ]

[ 350, 32 ]

[ 342, 11 ]

[ { "state_after": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ (∃ y, ↑(Algebra.linearMap R A) y = x) ↔ x ∈ Set.range ↑(algebraMap R A)", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ x ∈ span R ↑{1} ↔ x ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin R ∅)", "tactic": "erw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, one_eq_range,\n LinearMap.mem_range, Algebra.mem_bot]" }, { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ (∃ y, ↑(Algebra.linearMap R A) y = x) ↔ x ∈ Set.range ↑(algebraMap R A)", "tactic": "rfl" }, { "state_after": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s) * ↑Subalgebra.toSubmodule (Algebra.adjoin R {a}))", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R (insert a s)))", "tactic": "rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]" }, { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s) * ↑Subalgebra.toSubmodule (Algebra.adjoin R {a}))", "tactic": "exact\n FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)\n (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s)" } ]

[ 244, 8 ]

[ 230, 1 ]

[]

[ 1413, 6 ]

[ 1412, 1 ]

[ { "state_after": "case h\nβ : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)", "state_before": "β : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\n⊢ sigmaComparison G f ≫ G.map (Sigma.desc g) = Sigma.desc fun j => G.map (g j)", "tactic": "ext j" }, { "state_after": "no goals", "state_before": "case h\nβ : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)", "tactic": "simp only [Discrete.functor_obj, ι_comp_sigmaComparison_assoc, ← G.map_comp, colimit.ι_desc,\n Cofan.mk_pt, Cofan.mk_ι_app]" } ]

[ 279, 33 ]

[ 274, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nα' : Type ?u.2396048\nβ : Type u_2\nβ' : Type ?u.2396054\nγ : Type ?u.2396057\nE : Type u_3\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nf : α × β → E\nh1f : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ Integrable f ↔ (∀ᵐ (x : α) ∂μ, Integrable fun y => f (x, y)) ∧ Integrable fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν", "tactic": "simp [Integrable, h1f, hasFiniteIntegral_prod_iff', h1f.norm.integral_prod_right',\n h1f.prod_mk_left]" } ]

[ 280, 22 ]

[ 276, 1 ]

[ { "state_after": "no goals", "state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl : List α\n⊢ mapM' f l = mapM f l", "tactic": "simp [go, mapM]" }, { "state_after": "no goals", "state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl : List α\nacc : List β\n⊢ mapM.loop f [] acc = do\n let __do_lift ← mapM' f []\n pure (reverse acc ++ __do_lift)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl✝ : List α\na : α\nl : List α\nacc : List β\n⊢ mapM.loop f (a :: l) acc = do\n let __do_lift ← mapM' f (a :: l)\n pure (reverse acc ++ __do_lift)", "tactic": "simp [go l]" } ]

[ 216, 34 ]

[ 212, 1 ]

[]

[ 743, 52 ]

[ 743, 9 ]

[]

[ 255, 19 ]

[ 254, 1 ]

[ { "state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t → s = t\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ s = t → f ⁻¹' s = f ⁻¹' t", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t ↔ s = t", "tactic": "constructor" }, { "state_after": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t s : Set α\nf : β → α\nhs ht : s ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' s", "state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ s = t → f ⁻¹' s = f ⁻¹' t", "tactic": "rintro rfl" }, { "state_after": "no goals", "state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t s : Set α\nf : β → α\nhs ht : s ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' s", "tactic": "rfl" }, { "state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s = t", "state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t → s = t", "tactic": "intro h" }, { "state_after": "case mp.h₁\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s ⊆ t\n\ncase mp.h₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ t ⊆ s", "state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s = t", "tactic": "apply Subset.antisymm" }, { "state_after": "no goals", "state_before": "case mp.h₁\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s ⊆ t", "tactic": "rw [← preimage_subset_preimage_iff hs, h]" }, { "state_after": "no goals", "state_before": "case mp.h₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ t ⊆ s", "tactic": "rw [← preimage_subset_preimage_iff ht, h]" } ]

[ 836, 18 ]

[ 829, 1 ]

[ { "state_after": "case intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ ∃ v, minDegree G = degree G v", "state_before": "ι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\n⊢ ∃ v, minDegree G = degree G v", "tactic": "obtain ⟨t, ht : _ = _⟩ := min_of_nonempty (univ_nonempty.image fun v => G.degree v)" }, { "state_after": "case intro.intro.intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ ∃ v, minDegree G = degree G v", "state_before": "case intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ ∃ v, minDegree G = degree G v", "tactic": "obtain ⟨v, _, rfl⟩ := mem_image.mp (mem_of_min ht)" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ ∃ v, minDegree G = degree G v", "tactic": "refine' ⟨v, by simp [minDegree, ht]⟩" }, { "state_after": "no goals", "state_before": "ι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ minDegree G = degree G v", "tactic": "simp [minDegree, ht]" } ]

[ 1518, 39 ]

[ 1514, 1 ]

[]

[ 213, 77 ]

[ 209, 1 ]

[]

[ 166, 31 ]

[ 164, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.11263\nδ : Type ?u.11266\na : α\ns t : Multiset α\nf g : α → Multiset β\n⊢ bind (s + t) f = bind s f + bind t f", "tactic": "simp [bind]" } ]

[ 119, 74 ]

[ 119, 1 ]

[]

[ 1879, 78 ]

[ 1879, 1 ]

[]

[ 1083, 99 ]

[ 1083, 1 ]

[]

[ 390, 48 ]

[ 388, 1 ]

[ { "state_after": "no goals", "state_before": "K : Type u\ninst✝¹ : Field K\nL : Type u_1\ninst✝ : Field L\nf : K →+* L\na : L\n⊢ eval f a 1 = 1", "tactic": "simp [eval]" } ]

[ 1503, 52 ]

[ 1503, 1 ]

[]

[ 932, 89 ]

[ 928, 1 ]

[ { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx✝ y : E\nhs : IsCompact s\nhδ : 0 ≤ δ\nx : E\n⊢ s / closedBall x δ = x⁻¹ • cthickening δ s", "tactic": "simp [div_eq_mul_inv, mul_comm, hs.mul_closedBall hδ]" } ]

[ 293, 56 ]

[ 291, 1 ]

[]

[ 62, 56 ]

[ 61, 1 ]

[]

[ 55, 14 ]

[ 54, 1 ]

[]

[ 1162, 60 ]

[ 1160, 1 ]

[]

[ 73, 60 ]

[ 73, 11 ]

[ { "state_after": "no goals", "state_before": "G : Type u_1\nP : Type u_2\ninst✝¹ : AddGroup G\ninst✝ : AddTorsor G P\nx y : P\nh : Injective bit0\n⊢ ↑(pointReflection x) y = y ↔ y = x", "tactic": "rw [pointReflection_apply, eq_comm, eq_vadd_iff_vsub_eq, ← neg_vsub_eq_vsub_rev,\n neg_eq_iff_add_eq_zero, ← bit0, ← bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq, eq_comm]" } ]

[ 471, 89 ]

[ 468, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\ninst✝ : CommRing R\nW : ModuleCat R\nX : ModuleCat R\nY : ModuleCat R\nZ : ModuleCat R\n⊢ tensorHom (associator W X Y).hom (𝟙 Z) ≫\n (associator W (tensorObj X Y) Z).hom ≫ tensorHom (𝟙 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom ≫ (associator W X (tensorObj Y Z)).hom", "tactic": "convert pentagon_aux R W X Y Z using 1" } ]

[ 135, 41 ]

[ 131, 1 ]

[]

[ 2018, 39 ]

[ 2016, 1 ]

[]

[ 166, 19 ]

[ 165, 1 ]

[]

[ 97, 81 ]

[ 97, 1 ]

[ { "state_after": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : ¬SameRay ℝ x y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "simp only [sameRay_iff_inv_norm_smul_eq, not_or, ← Ne.def] at h" }, { "state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "rcases h with ⟨hx, hy, hne⟩" }, { "state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "rw [← norm_pos_iff] at hx hy" }, { "state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "have hxy : 0 < ‖x‖ + ‖y‖ := add_pos hx hy" }, { "state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "have :=\n combo_mem_ball_of_ne (inv_norm_smul_mem_closed_unit_ball x)\n (inv_norm_smul_mem_closed_unit_ball y) hne (div_pos hx hxy) (div_pos hy hxy)\n (by rw [← add_div, div_self hxy.ne'])" }, { "state_after": "no goals", "state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖", "tactic": "rwa [mem_ball_zero_iff, div_eq_inv_mul, div_eq_inv_mul, mul_smul, mul_smul, smul_inv_smul₀ hx.ne',\n smul_inv_smul₀ hy.ne', ← smul_add, norm_smul, Real.norm_of_nonneg (inv_pos.2 hxy).le, ←\n div_eq_inv_mul, div_lt_one hxy] at this" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x‖ / (‖x‖ + ‖y‖) + ‖y‖ / (‖x‖ + ‖y‖) = 1", "tactic": "rw [← add_div, div_self hxy.ne']" } ]

[ 194, 44 ]

[ 183, 1 ]

[ { "state_after": "no goals", "state_before": "f g : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ↑f fun x => x - ↑n", "tactic": "simpa only [sub_eq_add_neg] using\n (f.commute_add_int n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv" } ]

[ 358, 98 ]

[ 356, 1 ]

[ { "state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i", "tactic": "have : ∀ m, (cyclotomic (p ^ (m + 1)) R = ∑ i in Finset.range p, (X ^ p ^ m) ^ i) ↔\n ((∑ i in Finset.range p, (X ^ p ^ m) ^ i) *\n ∏ x : ℕ in Finset.range (m + 1), cyclotomic (p ^ x) R) = X ^ p ^ (m + 1) - 1 := by\n intro m\n have := eq_cyclotomic_iff (R := R) (P := ∑ i in range p, (X ^ p ^ m) ^ i)\n (pow_pos hp.pos (m + 1))\n rw [eq_comm] at this\n rw [this, Nat.prod_properDivisors_prime_pow hp]" }, { "state_after": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i\n\ncase succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ cyclotomic (p ^ (Nat.succ n_n + 1)) R = ∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i", "tactic": "induction' n with n_n n_ih" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) * ∏ i in Nat.properDivisors (p ^ (Nat.succ n_n + 1)), cyclotomic i R =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "state_before": "case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ cyclotomic (p ^ (Nat.succ n_n + 1)) R = ∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i", "tactic": "rw [((eq_cyclotomic_iff (pow_pos hp.pos (n_n.succ + 1)) _).mpr _).symm]" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) * ∏ i in Nat.properDivisors (p ^ (Nat.succ n_n + 1)), cyclotomic i R =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "tactic": "rw [Nat.prod_properDivisors_prime_pow hp, Finset.prod_range_succ, n_ih]" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : (∑ i in range p, (X ^ p ^ n_n) ^ i) * ∏ x in range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "tactic": "rw [this] at n_ih" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : (∑ i in range p, (X ^ p ^ n_n) ^ i) * ∏ x in range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1", "tactic": "rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\n⊢ ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "tactic": "intro m" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n ∑ i in range p, (X ^ p ^ m) ^ i = cyclotomic (p ^ (m + 1)) R ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "tactic": "have := eq_cyclotomic_iff (R := R) (P := ∑ i in range p, (X ^ p ^ m) ^ i)\n (pow_pos hp.pos (m + 1))" }, { "state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n ∑ i in range p, (X ^ p ^ m) ^ i = cyclotomic (p ^ (m + 1)) R ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "tactic": "rw [eq_comm] at this" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1", "tactic": "rw [this, Nat.prod_properDivisors_prime_pow hp]" }, { "state_after": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis✝ :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nthis : Fact (Nat.Prime p)\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i", "state_before": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i", "tactic": "haveI := Fact.mk hp" }, { "state_after": "no goals", "state_before": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis✝ :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nthis : Fact (Nat.Prime p)\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i", "tactic": "simp [cyclotomic_prime]" } ]

[ 576, 95 ]

[ 561, 1 ]

[ { "state_after": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ a✝ ∈ image f s ↔ a✝ ∈ image g s", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\n⊢ image f s = image g s", "tactic": "ext" }, { "state_after": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ (∃ x x_1, f x = a✝) ↔ ∃ x x_1, g x = a✝", "state_before": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ a✝ ∈ image f s ↔ a✝ ∈ image g s", "tactic": "simp_rw [mem_image, ← bex_def]" }, { "state_after": "no goals", "state_before": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ (∃ x x_1, f x = a✝) ↔ ∃ x x_1, g x = a✝", "tactic": "exact bex_congr fun x hx => by rw [h hx]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\nx : α\nhx : x ∈ s\n⊢ f x = a✝ ↔ g x = a✝", "tactic": "rw [h hx]" } ]

[ 361, 43 ]

[ 358, 1 ]

[]

[ 668, 32 ]

[ 667, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.75384\nι : Sort ?u.75387\nι' : Sort ?u.75390\nf✝ : ι → α\ns t : Set α\nf : α → β\np : Set β → Prop\n⊢ (∃ s x, p s) ↔ ∃ s, p (f '' s)", "tactic": "simp" } ]

[ 813, 68 ]

[ 812, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.816393\nβ : Type ?u.816396\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ\nhx : 0 < x\n⊢ (ENNReal.ofReal x)⁻¹ = ENNReal.ofReal x⁻¹", "tactic": "rw [ENNReal.ofReal, ENNReal.ofReal, ← @coe_inv (Real.toNNReal x) (by simp [hx]), coe_eq_coe,\n ← Real.toNNReal_inv]" }, { "state_after": "no goals", "state_before": "α : Type ?u.816393\nβ : Type ?u.816396\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ\nhx : 0 < x\n⊢ Real.toNNReal x ≠ 0", "tactic": "simp [hx]" } ]

[ 2187, 25 ]

[ 2185, 1 ]

[]

[ 93, 10 ]

[ 85, 1 ]

[]

[ 458, 63 ]

[ 457, 11 ]

[]

[ 2123, 31 ]

[ 2122, 1 ]

[]

[ 112, 6 ]

[ 111, 1 ]

[]

[ 1131, 66 ]

[ 1126, 1 ]

[]

[ 164, 6 ]

[ 163, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.576510\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nx : R\ny : { x // x ∈ M }\nz₁ z₂ : S\n⊢ mk' S x y + z₁ = z₂ ↔ ↑(algebraMap R S) x + z₁ * ↑(algebraMap R S) ↑y = z₂ * ↑(algebraMap R S) ↑y", "tactic": "rw [← mk'_spec S x y, ← IsUnit.mul_left_inj (IsLocalization.map_units S y), right_distrib]" } ]

[ 296, 93 ]

[ 294, 1 ]

[]

[ 438, 41 ]

[ 436, 1 ]

[]

[ 75, 22 ]

[ 74, 1 ]

[ { "state_after": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ (inv h)\n\ncase x\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1", "state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ h⁻¹", "tactic": "rw [← SingleObj.inv_as_inv]" }, { "state_after": "no goals", "state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ (inv h)\n\ncase x\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1", "tactic": "rfl" } ]

[ 754, 35 ]

[ 753, 1 ]

[]

[ 2315, 6 ]

[ 2314, 1 ]

[ { "state_after": "case h\nx : EReal\n⊢ x ∈ range Real.toEReal ↔ x ∈ Ioo ⊥ ⊤", "state_before": "⊢ range Real.toEReal = Ioo ⊥ ⊤", "tactic": "ext x" }, { "state_after": "no goals", "state_before": "case h\nx : EReal\n⊢ x ∈ range Real.toEReal ↔ x ∈ Ioo ⊥ ⊤", "tactic": "induction x using EReal.rec <;> simp" } ]

[ 351, 39 ]

[ 349, 1 ]

[]

[ 195, 6 ]

[ 194, 1 ]

[]

[ 1126, 23 ]

[ 1125, 1 ]

[ { "state_after": "case pos\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : a = 0\n⊢ a ^ 2 * b / a = a * b\n\ncase neg\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : ¬a = 0\n⊢ a ^ 2 * b / a = a * b", "state_before": "G₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\n⊢ a ^ 2 * b / a = a * b", "tactic": "by_cases ha : a = 0" }, { "state_after": "no goals", "state_before": "case neg\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : ¬a = 0\n⊢ a ^ 2 * b / a = a * b", "tactic": "rw [sq, mul_assoc, mul_div_cancel_left _ ha]" }, { "state_after": "no goals", "state_before": "case pos\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : a = 0\n⊢ a ^ 2 * b / a = a * b", "tactic": "simp [ha]" } ]

[ 198, 47 ]

[ 195, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.53359\nβ : Type ?u.53362\nG : Type u_1\ninst✝ : Group G\na b c d : G\n⊢ a * b = 1 ↔ a⁻¹ = b", "tactic": "rw [mul_eq_one_iff_eq_inv, inv_eq_iff_eq_inv]" } ]

[ 675, 51 ]

[ 674, 1 ]

[]

[ 417, 6 ]

[ 416, 1 ]

[ { "state_after": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} → 𝓝 a = pure a\n\ncase mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝 a = pure a → IsOpen {a}", "state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} ↔ 𝓝 a = pure a", "tactic": "constructor" }, { "state_after": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a = pure a", "state_before": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} → 𝓝 a = pure a", "tactic": "intro h" }, { "state_after": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a ≤ pure a", "state_before": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a = pure a", "tactic": "apply le_antisymm _ (pure_le_nhds a)" }, { "state_after": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ {a} ∈ 𝓝 a", "state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a ≤ pure a", "tactic": "rw [le_pure_iff]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ {a} ∈ 𝓝 a", "tactic": "exact h.mem_nhds (mem_singleton a)" }, { "state_after": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : 𝓝 a = pure a\n⊢ IsOpen {a}", "state_before": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝 a = pure a → IsOpen {a}", "tactic": "intro h" }, { "state_after": "no goals", "state_before": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : 𝓝 a = pure a\n⊢ IsOpen {a}", "tactic": "simp [isOpen_iff_nhds, h]" } ]

[ 1257, 30 ]

[ 1250, 1 ]

[]

[ 633, 31 ]

[ 627, 1 ]

[ { "state_after": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = (fun s => s.fst + s.snd) (s₁, s₂) + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.fst + s.snd) a' = s", "state_before": "α : Type u_1\nr✝ r : α → α → Prop\n⊢ Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s => s.fst + s.snd", "tactic": "rintro ⟨s₁, s₂⟩ s ⟨t, a, hr, he⟩" }, { "state_after": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s", "state_before": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = (fun s => s.fst + s.snd) (s₁, s₂) + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.fst + s.snd) a' = s", "tactic": "dsimp at he ⊢" }, { "state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s", "state_before": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s", "tactic": "obtain ⟨ha, hb⟩ := add_singleton_eq_iff.1 he" }, { "state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α", "state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s", "tactic": "rw [hb]" }, { "state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ ∨ a ∈ s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α", "state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α", "tactic": "rw [add_assoc, mem_add] at ha" }, { "state_after": "case mk.intro.intro.intro.intro.inl\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\ncase mk.intro.intro.intro.intro.inr\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α", "state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ ∨ a ∈ s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α", "tactic": "obtain h | h := ha" }, { "state_after": "case mk.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ erase s₁ a + t + {a} = s₁ + t\n\ncase mk.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ (erase s₁ a + t, s₂).fst + (erase s₁ a + t, s₂).snd = erase (s₁ + s₂ + t) a", "state_before": "case mk.intro.intro.intro.intro.inl\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a", "tactic": "refine' ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, _⟩, _⟩" }, { "state_after": "no goals", "state_before": "case mk.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ erase s₁ a + t + {a} = s₁ + t", "tactic": "rw [add_comm, ← add_assoc, singleton_add, cons_erase h]" }, { "state_after": "no goals", "state_before": "case mk.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ (erase s₁ a + t, s₂).fst + (erase s₁ a + t, s₂).snd = erase (s₁ + s₂ + t) a", "tactic": "rw [add_assoc s₁, erase_add_left_pos _ h, add_right_comm, add_assoc]" }, { "state_after": "case mk.intro.intro.intro.intro.inr.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ erase (s₂ + t) a + {a} = s₂ + t\n\ncase mk.intro.intro.intro.intro.inr.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ (s₁, erase (s₂ + t) a).fst + (s₁, erase (s₂ + t) a).snd = erase (s₁ + s₂ + t) a", "state_before": "case mk.intro.intro.intro.intro.inr\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a", "tactic": "refine' ⟨(s₁, (s₂ + t).erase a), GameAdd.snd ⟨t, a, hr, _⟩, _⟩" }, { "state_after": "no goals", "state_before": "case mk.intro.intro.intro.intro.inr.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ erase (s₂ + t) a + {a} = s₂ + t", "tactic": "rw [add_comm, singleton_add, cons_erase h]" }, { "state_after": "no goals", "state_before": "case mk.intro.intro.intro.intro.inr.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ (s₁, erase (s₂ + t) a).fst + (s₁, erase (s₂ + t) a).snd = erase (s₁ + s₂ + t) a", "tactic": "rw [add_assoc, erase_add_right_pos _ h]" } ]

[ 127, 46 ]

[ 112, 1 ]

[]

[ 137, 15 ]

[ 136, 1 ]

[]

[ 673, 39 ]

[ 672, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nb : β\nf✝ g : α → β\ninst✝¹ : NonAssocSemiring β\ninst✝ : DecidableEq α\ns : Finset α\nf : α → β\na : α\n⊢ (∑ x in s, f x * if a = x then 1 else 0) = if a ∈ s then f a else 0", "tactic": "simp" } ]

[ 77, 71 ]

[ 76, 1 ]

[]

[ 149, 24 ]

[ 148, 1 ]

[]

[ 165, 37 ]

[ 164, 1 ]

[]

[ 255, 10 ]

[ 254, 1 ]

[]

[ 59, 6 ]

[ 58, 1 ]

[]

[ 41, 45 ]

[ 40, 1 ]

[ { "state_after": "case nil\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\nhl : ∀ (f : α → M), f ∈ [] → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod [])\n\ncase cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : ∀ (f_1 : α → M), f_1 ∈ f :: l → StronglyMeasurable f_1\n⊢ StronglyMeasurable (List.prod (f :: l))", "state_before": "α : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod l)", "tactic": "induction' l with f l ihl" }, { "state_after": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (List.prod (f :: l))", "state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : ∀ (f_1 : α → M), f_1 ∈ f :: l → StronglyMeasurable f_1\n⊢ StronglyMeasurable (List.prod (f :: l))", "tactic": "rw [List.forall_mem_cons] at hl" }, { "state_after": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (f * List.prod l)", "state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (List.prod (f :: l))", "tactic": "rw [List.prod_cons]" }, { "state_after": "no goals", "state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (f * List.prod l)", "tactic": "exact hl.1.mul (ihl hl.2)" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\nhl : ∀ (f : α → M), f ∈ [] → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod [])", "tactic": "exact stronglyMeasurable_one" } ]

[ 548, 28 ]

[ 543, 1 ]

[]

[ 393, 27 ]

[ 392, 1 ]

[]

[ 116, 46 ]

[ 115, 1 ]

[]

[ 69, 47 ]

[ 68, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type ?u.1103\nι : Type ?u.1106\nκ : Type ?u.1109\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nx : α × β\n⊢ x ∈ interedges r s t ↔ x.fst ∈ s ∧ x.snd ∈ t ∧ r x.fst x.snd", "tactic": "rw [interedges, mem_filter, Finset.mem_product, and_assoc]" } ]

[ 61, 61 ]

[ 60, 1 ]

[]

[ 1047, 33 ]

[ 1046, 1 ]

[ { "state_after": "no goals", "state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\n⊢ 0 ≤ 2", "tactic": "decide" }, { "state_after": "no goals", "state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\n⊢ abs 2 ≠ 0", "tactic": "decide" }, { "state_after": "no goals", "state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\n⊢ 2 ≠ 0", "tactic": "decide" }, { "state_after": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑k + ↑2", "state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑(k + 2)", "tactic": "rw [Nat.cast_add]" }, { "state_after": "no goals", "state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑k + ↑2", "tactic": "exact (le_add_iff_nonneg_left 2).2 (NonNeg.mk k)" }, { "state_after": "no goals", "state_before": "a✝ b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\na : ℕ\nx✝ : -[a+1] < 2\nh₁ : 0 ≤ -[a+1]\n⊢ -[a+1] = 0 ∨ -[a+1] = 1", "tactic": "cases h₁" } ]

[ 330, 33 ]

[ 318, 1 ]

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[ 68, 10 ]

[ 67, 1 ]

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[ 1107, 48 ]

[ 1105, 1 ]