Datasets:

tasksource

/

leandojo

like 5

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[ 469, 1 ]

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[ 258, 1 ]

[ { "state_after": "P : Type u_1\ninst✝ : Preorder P\nIF : PrimePair P\nsrc✝ : IsProper IF.I := I_isProper IF\n⊢ IsPFilter ↑IF.F", "state_before": "P : Type u_1\ninst✝ : Preorder P\nIF : PrimePair P\nsrc✝ : IsProper IF.I := I_isProper IF\n⊢ IsPFilter (↑IF.Iᶜ)", "tactic": "rw [IF.compl_I_eq_F]" }, { "state_after": "no goals", "state_before": "P : Type u_1\ninst✝ : Preorder P\nIF : PrimePair P\nsrc✝ : IsProper IF.I := I_isProper IF\n⊢ IsPFilter ↑IF.F", "tactic": "exact IF.F.isPFilter" } ]

[ 116, 29 ]

[ 112, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na : α\nb : β\nh : ↑f a = b\n⊢ Acc s b → Acc r a", "state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na : α\n⊢ Acc s (↑f a) → Acc r a", "tactic": "generalize h : f a = b" }, { "state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na : α\nb : β\nh : ↑f a = b\nac : Acc s b\n⊢ Acc r a", "state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na : α\nb : β\nh : ↑f a = b\n⊢ Acc s b → Acc r a", "tactic": "intro ac" }, { "state_after": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na✝ : α\nb : β\nh✝ : ↑f a✝ = b\nx✝ : β\nH : ∀ (y : β), s y x✝ → Acc s y\nIH : ∀ (y : β), s y x✝ → ∀ (a : α), ↑f a = y → Acc r a\na : α\nh : ↑f a = x✝\n⊢ Acc r a", "state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na : α\nb : β\nh : ↑f a = b\nac : Acc s b\n⊢ Acc r a", "tactic": "induction' ac with _ H IH generalizing a" }, { "state_after": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na✝ : α\nb : β\nh : ↑f a✝ = b\na : α\nH : ∀ (y : β), s y (↑f a) → Acc s y\nIH : ∀ (y : β), s y (↑f a) → ∀ (a : α), ↑f a = y → Acc r a\n⊢ Acc r a", "state_before": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na✝ : α\nb : β\nh✝ : ↑f a✝ = b\nx✝ : β\nH : ∀ (y : β), s y x✝ → Acc s y\nIH : ∀ (y : β), s y x✝ → ∀ (a : α), ↑f a = y → Acc r a\na : α\nh : ↑f a = x✝\n⊢ Acc r a", "tactic": "subst h" }, { "state_after": "no goals", "state_before": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.9632\nδ : Type ?u.9635\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nF : Type u_1\ninst✝ : RelHomClass F r s\nf : F\na✝ : α\nb : β\nh : ↑f a✝ = b\na : α\nH : ∀ (y : β), s y (↑f a) → Acc s y\nIH : ∀ (y : β), s y (↑f a) → ∀ (a : α), ↑f a = y → Acc r a\n⊢ Acc r a", "tactic": "exact ⟨_, fun a' h => IH (f a') (map_rel f h) _ rfl⟩" } ]

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[ { "state_after": "no goals", "state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q : K[X]\nhq : q ≠ 0\n⊢ RatFunc.mk p q = { toFractionRing := Localization.mk p { val := q, property := (_ : q ∈ K[X]⁰) } }", "tactic": "rw [mk_def_of_ne, Localization.mk_eq_mk'] <;> exact hq" } ]

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[ { "state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.822903\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx : E\nthis : IsCompl K Kᗮ\n⊢ ↑(orthogonalProjection K) x = ↑(Submodule.linearProjOfIsCompl K Kᗮ (_ : IsCompl K Kᗮ)) x", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.822903\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx : E\n⊢ ↑(orthogonalProjection K) x = ↑(Submodule.linearProjOfIsCompl K Kᗮ (_ : IsCompl K Kᗮ)) x", "tactic": "have : IsCompl K Kᗮ := Submodule.isCompl_orthogonal_of_completeSpace" }, { "state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.822903\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx : E\nthis : IsCompl K Kᗮ\n⊢ ↑(orthogonalProjection K)\n (↑(↑(Submodule.linearProjOfIsCompl K Kᗮ this) x) +\n ↑(↑(Submodule.linearProjOfIsCompl Kᗮ K (_ : IsCompl Kᗮ K)) x)) =\n ↑(Submodule.linearProjOfIsCompl K Kᗮ (_ : IsCompl K Kᗮ)) x", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.822903\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx : E\nthis : IsCompl K Kᗮ\n⊢ ↑(orthogonalProjection K) x = ↑(Submodule.linearProjOfIsCompl K Kᗮ (_ : IsCompl K Kᗮ)) x", "tactic": "conv_lhs => rw [← Submodule.linear_proj_add_linearProjOfIsCompl_eq_self this x]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.822903\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx : E\nthis : IsCompl K Kᗮ\n⊢ ↑(orthogonalProjection K)\n (↑(↑(Submodule.linearProjOfIsCompl K Kᗮ this) x) +\n ↑(↑(Submodule.linearProjOfIsCompl Kᗮ K (_ : IsCompl Kᗮ K)) x)) =\n ↑(Submodule.linearProjOfIsCompl K Kᗮ (_ : IsCompl K Kᗮ)) x", "tactic": "rw [map_add, orthogonalProjection_mem_subspace_eq_self,\n orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero (Submodule.coe_mem _), add_zero]" } ]

[ 850, 100 ]

[ 844, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.135719\nE : Type ?u.135722\nF : Type u_3\nG : Type ?u.135728\nE' : Type u_2\nF' : Type ?u.135734\nG' : Type ?u.135737\nE'' : Type ?u.135740\nF'' : Type ?u.135743\nG'' : Type ?u.135746\nR : Type ?u.135749\nR' : Type ?u.135752\n𝕜 : Type ?u.135755\n𝕜' : Type ?u.135758\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x => -f' x) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g", "state_before": "α : Type u_1\nβ : Type ?u.135719\nE : Type ?u.135722\nF : Type u_3\nG : Type ?u.135728\nE' : Type u_2\nF' : Type ?u.135734\nG' : Type ?u.135737\nE'' : Type ?u.135740\nF'' : Type ?u.135743\nG'' : Type ?u.135746\nR : Type ?u.135749\nR' : Type ?u.135752\n𝕜 : Type ?u.135755\n𝕜' : Type ?u.135758\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (fun x => -f' x) =o[l] g ↔ f' =o[l] g", "tactic": "simp only [IsLittleO_def]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.135719\nE : Type ?u.135722\nF : Type u_3\nG : Type ?u.135728\nE' : Type u_2\nF' : Type ?u.135734\nG' : Type ?u.135737\nE'' : Type ?u.135740\nF'' : Type ?u.135743\nG'' : Type ?u.135746\nR : Type ?u.135749\nR' : Type ?u.135752\n𝕜 : Type ?u.135755\n𝕜' : Type ?u.135758\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x => -f' x) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g", "tactic": "exact forall₂_congr fun _ _ => isBigOWith_neg_left" } ]

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[ { "state_after": "c : Char\n⊢ UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 1 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 2 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 3 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 4", "state_before": "c : Char\n⊢ csize c = 1 ∨ csize c = 2 ∨ csize c = 3 ∨ csize c = 4", "tactic": "simp only [csize, Char.utf8Size]" }, { "state_after": "no goals", "state_before": "c : Char\n⊢ UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 1 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 2 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 3 ∨\n UInt32.toNat\n (if c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1 then UInt32.ofNatCore 1 Char.utf8Size.proof_2\n else\n if c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3 then UInt32.ofNatCore 2 Char.utf8Size.proof_4\n else\n if c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5 then UInt32.ofNatCore 3 Char.utf8Size.proof_6\n else UInt32.ofNatCore 4 Char.utf8Size.proof_7) =\n 4", "tactic": "repeat (first | split | (solve | simp))" }, { "state_after": "no goals", "state_before": "case inr.inr.inr\nc : Char\nh✝² : ¬c.val ≤ UInt32.ofNatCore 127 Char.utf8Size.proof_1\nh✝¹ : ¬c.val ≤ UInt32.ofNatCore 2047 Char.utf8Size.proof_3\nh✝ : ¬c.val ≤ UInt32.ofNatCore 65535 Char.utf8Size.proof_5\n⊢ UInt32.toNat (UInt32.ofNatCore 4 Char.utf8Size.proof_7) = 1 ∨\n UInt32.toNat (UInt32.ofNatCore 4 Char.utf8Size.proof_7) = 2 ∨\n UInt32.toNat (UInt32.ofNatCore 4 Char.utf8Size.proof_7) = 3 ∨\n UInt32.toNat (UInt32.ofNatCore 4 Char.utf8Size.proof_7) = 4", "tactic": "simp" } ]

[ 15, 42 ]

[ 11, 9 ]

[]

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[]

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[ { "state_after": "R✝ : Type u\na✝ b✝ : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\na b : AddMonoidAlgebra R ℕ\n⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }", "state_before": "R✝ : Type u\na✝ b✝ : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\na b : AddMonoidAlgebra R ℕ\n⊢ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }", "tactic": "rw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg]" }, { "state_after": "no goals", "state_before": "R✝ : Type u\na✝ b✝ : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\na b : AddMonoidAlgebra R ℕ\n⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }", "tactic": "rfl" } ]

[ 179, 6 ]

[ 177, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.259014\nγ : Type ?u.259017\nσ : Type u_2\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nc : α → Bool\nf g : α →. σ\nhc : Computable c\nhf : Partrec f\nhg : Partrec g\ncf : Code\nef : eval cf = fun n => Part.bind ↑(decode n) fun a => Part.map encode (f a)\ncg : Code\neg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode (g a)\na : α\n⊢ (Part.bind (eval (bif c a then cf else cg) (encode a)) fun b => ↑(decode (a, b).snd)) = bif c a then f a else g a", "tactic": "cases c a <;> simp [ef, eg, encodek]" } ]

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[ 119, 1 ]

[]

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[ 62, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.320195\nγ : Type ?u.320198\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns : Finset α\na : α\n⊢ a ∈ filter q (filter p s) ↔ a ∈ filter (fun a => p a ∧ q a) s", "tactic": "simp only [mem_filter, and_assoc, Bool.decide_and, Bool.decide_coe, Bool.and_eq_true]" } ]

[ 2678, 90 ]

[ 2676, 1 ]

[ { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.270732\nβ : Type ?u.270735\ninst✝⁴ : LinearOrderedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nh_conv : Convex 𝕜 s\np q : 𝕜\nhp : 0 ≤ p\nhq : 0 ≤ q\nv₁ : E\nh₁ : v₁ ∈ s\nv₂ : E\nh₂ : v₂ ∈ s\n⊢ (fun x x_1 => x + x_1) ((fun x => p • x) v₁) ((fun x => q • x) v₂) ∈ (p + q) • s", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.270732\nβ : Type ?u.270735\ninst✝⁴ : LinearOrderedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nh_conv : Convex 𝕜 s\np q : 𝕜\nhp : 0 ≤ p\nhq : 0 ≤ q\n⊢ p • s + q • s ⊆ (p + q) • s", "tactic": "rintro _ ⟨_, _, ⟨v₁, h₁, rfl⟩, ⟨v₂, h₂, rfl⟩, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.270732\nβ : Type ?u.270735\ninst✝⁴ : LinearOrderedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nh_conv : Convex 𝕜 s\np q : 𝕜\nhp : 0 ≤ p\nhq : 0 ≤ q\nv₁ : E\nh₁ : v₁ ∈ s\nv₂ : E\nh₂ : v₂ ∈ s\n⊢ (fun x x_1 => x + x_1) ((fun x => p • x) v₁) ((fun x => q • x) v₂) ∈ (p + q) • s", "tactic": "exact h_conv.exists_mem_add_smul_eq h₁ h₂ hp hq" } ]

[ 553, 50 ]

[ 550, 1 ]

[]

[ 416, 21 ]

[ 415, 1 ]

[]

[ 390, 40 ]

[ 389, 1 ]

[]

[ 576, 18 ]

[ 575, 1 ]

[ { "state_after": "no goals", "state_before": "p q : ℝ\nh : 1 < p\nH : q = p / (p - 1)\n⊢ 1 / p + 1 / q = 1", "tactic": "field_simp [H, ne_of_gt (lt_trans zero_lt_one h)]" } ]

[ 123, 92 ]

[ 122, 1 ]

[]

[ 56, 65 ]

[ 56, 9 ]

[]

[ 489, 50 ]

[ 486, 1 ]

[ { "state_after": "m n : ℕ\n⊢ max m n ≤ max m n * max m n", "state_before": "m n : ℕ\n⊢ m + n ≤ max m n ^ 2 + min m n", "tactic": "rw [sq, ← min_add_max, add_comm, add_le_add_iff_right]" }, { "state_after": "no goals", "state_before": "m n : ℕ\n⊢ max m n ≤ max m n * max m n", "tactic": "exact le_mul_self _" } ]

[ 175, 24 ]

[ 172, 1 ]

[ { "state_after": "no goals", "state_before": "α✝ : Type ?u.16686\ninst✝ : TopologicalSpace α✝\nα : Type u_1\nm : α → EReal\nf : Filter α\n⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Iio ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x", "tactic": "simp only [true_implies, mem_Iio]" } ]

[ 173, 81 ]

[ 171, 1 ]

[]

[ 553, 78 ]

[ 551, 1 ]

[ { "state_after": "case intro\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\n⊢ Tendsto (fun s => ∑' (b : { x // ¬x ∈ s }), (fun i => ↑(f i)) ↑b) atTop (𝓝 0)", "state_before": "α✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0∞\nhf : (∑' (x : α), f x) ≠ ⊤\n⊢ Tendsto (fun s => ∑' (b : { x // ¬x ∈ s }), f ↑b) atTop (𝓝 0)", "tactic": "lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hf" }, { "state_after": "case h.e'_3.h\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\nx✝ : Finset α\n⊢ (∑' (b : { x // ¬x ∈ x✝ }), (fun i => ↑(f i)) ↑b) = ↑(∑' (b : { x // ¬x ∈ x✝ }), f ↑b)", "state_before": "case intro\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\n⊢ Tendsto (fun s => ∑' (b : { x // ¬x ∈ s }), (fun i => ↑(f i)) ↑b) atTop (𝓝 0)", "tactic": "convert ENNReal.tendsto_coe.2 (NNReal.tendsto_tsum_compl_atTop_zero f)" }, { "state_after": "case h.e'_3.h\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\nx✝ : Finset α\n⊢ Summable fun b => f ↑b", "state_before": "case h.e'_3.h\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\nx✝ : Finset α\n⊢ (∑' (b : { x // ¬x ∈ x✝ }), (fun i => ↑(f i)) ↑b) = ↑(∑' (b : { x // ¬x ∈ x✝ }), f ↑b)", "tactic": "rw [ENNReal.coe_tsum]" }, { "state_after": "no goals", "state_before": "case h.e'_3.h\nα✝ : Type ?u.262957\nβ : Type ?u.262960\nγ : Type ?u.262963\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α✝ → ℝ≥0∞\nα : Type u_1\nf : α → ℝ≥0\nhf : (∑' (x : α), (fun i => ↑(f i)) x) ≠ ⊤\nx✝ : Finset α\n⊢ Summable fun b => f ↑b", "tactic": "exact NNReal.summable_comp_injective (tsum_coe_ne_top_iff_summable.1 hf) Subtype.coe_injective" } ]

[ 970, 97 ]

[ 965, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\n⊢ ∃ t, t ⊆ range s ∧ Set.Finite t ∧ range s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ a}", "state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\n⊢ TotallyBounded (range s)", "tactic": "refine' totallyBounded_iff_subset.2 fun a ha => _" }, { "state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ ∃ t, t ⊆ range s ∧ Set.Finite t ∧ range s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ a}", "state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\n⊢ ∃ t, t ⊆ range s ∧ Set.Finite t ∧ range s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ a}", "tactic": "cases' cauchySeq_iff.1 hs a ha with n hn" }, { "state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ range s ⊆ ⋃ (y : α) (_ : y ∈ s '' {k | k ≤ n}), {x | (x, y) ∈ a}", "state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ ∃ t, t ⊆ range s ∧ Set.Finite t ∧ range s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ a}", "tactic": "refine' ⟨s '' { k | k ≤ n }, image_subset_range _ _, (finite_le_nat _).image _, _⟩" }, { "state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ ∀ (y : ℕ), s y ∈ ⋃ (y : ℕ) (_ : y ∈ {k | k ≤ n}), {x | (x, s y) ∈ a}", "state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ range s ⊆ ⋃ (y : α) (_ : y ∈ s '' {k | k ≤ n}), {x | (x, y) ∈ a}", "tactic": "rw [range_subset_iff, biUnion_image]" }, { "state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\n⊢ s m ∈ ⋃ (y : ℕ) (_ : y ∈ {k | k ≤ n}), {x | (x, s y) ∈ a}", "state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\n⊢ ∀ (y : ℕ), s y ∈ ⋃ (y : ℕ) (_ : y ∈ {k | k ≤ n}), {x | (x, s y) ∈ a}", "tactic": "intro m" }, { "state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}", "state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\n⊢ s m ∈ ⋃ (y : ℕ) (_ : y ∈ {k | k ≤ n}), {x | (x, s y) ∈ a}", "tactic": "rw [mem_iUnion₂]" }, { "state_after": "case intro.inl\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\nhm : m ≤ n\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}\n\ncase intro.inr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\nhm : n ≤ m\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}", "state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}", "tactic": "cases' le_total m n with hm hm" }, { "state_after": "no goals", "state_before": "case intro.inl\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\nhm : m ≤ n\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}\n\ncase intro.inr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\nn : ℕ\nhn : ∀ (k : ℕ), k ≥ n → ∀ (l : ℕ), l ≥ n → (s k, s l) ∈ a\nm : ℕ\nhm : n ≤ m\n⊢ ∃ i j, s m ∈ {x | (x, s i) ∈ a}", "tactic": "exacts [⟨m, hm, refl_mem_uniformity ha⟩, ⟨n, le_refl n, hn m hm n le_rfl⟩]" } ]

[ 618, 77 ]

[ 609, 1 ]

[]

[ 1561, 6 ]

[ 1560, 1 ]

[]

[ 1465, 23 ]

[ 1464, 1 ]

[]

[ 605, 6 ]

[ 602, 1 ]

[ { "state_after": "Q : Type ?u.78946\ninst✝ : Quandle Q\nn : ℕ\na b : ZMod n\n⊢ dihedralAct n a (dihedralAct n a b) = b", "state_before": "Q : Type ?u.78946\ninst✝ : Quandle Q\nn : ℕ\na : ZMod n\n⊢ Function.Involutive (dihedralAct n a)", "tactic": "intro b" }, { "state_after": "Q : Type ?u.78946\ninst✝ : Quandle Q\nn : ℕ\na b : ZMod n\n⊢ 2 * a - (2 * a - b) = b", "state_before": "Q : Type ?u.78946\ninst✝ : Quandle Q\nn : ℕ\na b : ZMod n\n⊢ dihedralAct n a (dihedralAct n a b) = b", "tactic": "dsimp only [dihedralAct]" }, { "state_after": "no goals", "state_before": "Q : Type ?u.78946\ninst✝ : Quandle Q\nn : ℕ\na b : ZMod n\n⊢ 2 * a - (2 * a - b) = b", "tactic": "simp" } ]

[ 494, 7 ]

[ 491, 1 ]

[]

[ 282, 6 ]

[ 281, 1 ]

[ { "state_after": "case intro\na b c d : ℤ\nhb : b ≠ 0\nh : b * a = c * d\nk : ℤ\nhk : c = b * k\n⊢ a = c / b * d", "state_before": "a b c d : ℤ\nhb : b ≠ 0\nhbc : b ∣ c\nh : b * a = c * d\n⊢ a = c / b * d", "tactic": "cases' hbc with k hk" }, { "state_after": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * k * d\n⊢ a = b * k / b * d", "state_before": "case intro\na b c d : ℤ\nhb : b ≠ 0\nh : b * a = c * d\nk : ℤ\nhk : c = b * k\n⊢ a = c / b * d", "tactic": "subst hk" }, { "state_after": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * k * d\n⊢ a = k * d", "state_before": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * k * d\n⊢ a = b * k / b * d", "tactic": "rw [Int.mul_ediv_cancel_left _ hb]" }, { "state_after": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * (k * d)\n⊢ a = k * d", "state_before": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * k * d\n⊢ a = k * d", "tactic": "rw [mul_assoc] at h" }, { "state_after": "no goals", "state_before": "case intro\na b d : ℤ\nhb : b ≠ 0\nk : ℤ\nh : b * a = b * (k * d)\n⊢ a = k * d", "tactic": "apply mul_left_cancel₀ hb h" } ]

[ 30, 30 ]

[ 24, 1 ]

[ { "state_after": "a : Ordinal\n⊢ a ^ 0 * a = a", "state_before": "a : Ordinal\n⊢ a ^ 1 = a", "tactic": "rw [← succ_zero, opow_succ]" }, { "state_after": "no goals", "state_before": "a : Ordinal\n⊢ a ^ 0 * a = a", "tactic": "simp only [opow_zero, one_mul]" } ]

[ 81, 62 ]

[ 80, 1 ]

[]

[ 584, 6 ]

[ 583, 1 ]

[ { "state_after": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny✝ y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nhJ : J ≤ I\ny : F\nhy : Tendsto (integralSum f vol) (toFilteriUnion l I (Prepartition.single I J hJ)) (𝓝 y)\n⊢ HasIntegral J l f vol y", "state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nhJ : J ≤ I\n⊢ ∃ y, HasIntegral J l f vol y ∧ Tendsto (integralSum f vol) (toFilteriUnion l I (Prepartition.single I J hJ)) (𝓝 y)", "tactic": "refine (cauchy_map_iff_exists_tendsto.1\n (h.cauchy_map_integralSum_toFilteriUnion (.single I J hJ))).imp fun y hy ↦ ⟨?_, hy⟩" }, { "state_after": "no goals", "state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny✝ y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nhJ : J ≤ I\ny : F\nhy : Tendsto (integralSum f vol) (toFilteriUnion l I (Prepartition.single I J hJ)) (𝓝 y)\n⊢ HasIntegral J l f vol y", "tactic": "convert hy.comp (l.tendsto_embedBox_toFilteriUnion_top hJ)" } ]

[ 534, 61 ]

[ 529, 1 ]

[]

[ 1572, 79 ]

[ 1569, 1 ]

[ { "state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ 𝟙 X = 0 ↔ f = 0", "state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ IsZero X ↔ f = 0", "tactic": "rw [iff_id_eq_zero]" }, { "state_after": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ 𝟙 X = 0 → f = 0\n\ncase mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ f = 0 → 𝟙 X = 0", "state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ 𝟙 X = 0 ↔ f = 0", "tactic": "constructor" }, { "state_after": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : 𝟙 X = 0\n⊢ f = 0", "state_before": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ 𝟙 X = 0 → f = 0", "tactic": "intro h" }, { "state_after": "no goals", "state_before": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : 𝟙 X = 0\n⊢ f = 0", "tactic": "rw [← Category.id_comp f, h, zero_comp]" }, { "state_after": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : f = 0\n⊢ 𝟙 X = 0", "state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ f = 0 → 𝟙 X = 0", "tactic": "intro h" }, { "state_after": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : f = 0\n⊢ f ≫ retraction f = 0", "state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : f = 0\n⊢ 𝟙 X = 0", "tactic": "rw [← IsSplitMono.id f]" }, { "state_after": "no goals", "state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : f = 0\n⊢ f ≫ retraction f = 0", "tactic": "simp [h]" } ]

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[ { "state_after": "no goals", "state_before": "α : Type ?u.44179\nβ : Type ?u.44182\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nm : ℕ\nl : List ℕ\nn : ℕ\n⊢ lower (raise (m :: l) n) n = m :: l", "tactic": "rw [raise, lower, add_tsub_cancel_right, lower_raise l]" } ]

[ 306, 76 ]

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[ { "state_after": "α : Type u_1\nα' : Type ?u.34144\nβ : Type u_3\nβ' : Type ?u.34150\nγ : Type u_2\nγ' : Type ?u.34156\nδ : Type ?u.34159\nδ' : Type ?u.34162\nε : Type ?u.34165\nε' : Type ?u.34168\nζ : Type ?u.34171\nζ' : Type ?u.34174\nν : Type ?u.34177\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t", "state_before": "α : Type u_1\nα' : Type ?u.34144\nβ : Type u_3\nβ' : Type ?u.34150\nγ : Type u_2\nγ' : Type ?u.34156\nδ : Type ?u.34159\nδ' : Type ?u.34162\nε : Type ?u.34165\nε' : Type ?u.34168\nζ : Type ?u.34171\nζ' : Type ?u.34174\nν : Type ?u.34177\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ ↑(image₂ f (s ∪ s') t) = ↑(image₂ f s t ∪ image₂ f s' t)", "tactic": "push_cast" }, { "state_after": "no goals", "state_before": "α : Type u_1\nα' : Type ?u.34144\nβ : Type u_3\nβ' : Type ?u.34150\nγ : Type u_2\nγ' : Type ?u.34156\nδ : Type ?u.34159\nδ' : Type ?u.34162\nε : Type ?u.34165\nε' : Type ?u.34168\nζ : Type ?u.34171\nζ' : Type ?u.34174\nν : Type ?u.34177\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t", "tactic": "exact image2_union_left" } ]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "x : ℂ\ny : ℝ\n⊢ ↑abs (x ^ ↑y) = ↑abs x ^ y", "tactic": "rcases eq_or_ne x 0 with (rfl | hx) <;> [rcases eq_or_ne y 0 with (rfl | hy); skip] <;>\n simp [*, abs_cpow_of_ne_zero]" } ]

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[ { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_3\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf✝ : E × F → G\nf : E →L[𝕜] F →L[𝕜] G\nx : E\ny : F\n⊢ ‖f‖ * ‖x‖ * ‖y‖ ≤ max ‖f‖ 1 * ‖x‖ * ‖y‖", "tactic": "apply_rules [mul_le_mul_of_nonneg_right, norm_nonneg, le_max_left]" } ]

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[ { "state_after": "α : Type u_2\nα' : Type ?u.4457703\nβ : Type u_1\nβ' : Type ?u.4457709\nγ : Type ?u.4457712\nE : Type ?u.4457715\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ (∫⁻ (x : β), ↑↑μ (Prod.mk x ⁻¹' (Prod.swap ⁻¹' s)) ∂ν) = ∫⁻ (y : β), ↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂ν", "state_before": "α : Type u_2\nα' : Type ?u.4457703\nβ : Type u_1\nβ' : Type ?u.4457709\nγ : Type ?u.4457712\nE : Type ?u.4457715\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ ↑↑(Measure.prod μ ν) s = ∫⁻ (y : β), ↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂ν", "tactic": "rw [← prod_swap, map_apply measurable_swap hs, prod_apply (measurable_swap hs)]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nα' : Type ?u.4457703\nβ : Type u_1\nβ' : Type ?u.4457709\nγ : Type ?u.4457712\nE : Type ?u.4457715\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ (∫⁻ (x : β), ↑↑μ (Prod.mk x ⁻¹' (Prod.swap ⁻¹' s)) ∂ν) = ∫⁻ (y : β), ↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂ν", "tactic": "rfl" } ]

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[ { "state_after": "no goals", "state_before": "α : Type ?u.929509\na : ZNum\n⊢ ↑(0 * a) = ↑0 * ↑a", "tactic": "cases a <;> exact (MulZeroClass.zero_mul _).symm" }, { "state_after": "no goals", "state_before": "α : Type ?u.929509\nb : ZNum\n⊢ ↑(b * 0) = ↑b * ↑0", "tactic": "cases b <;> exact (MulZeroClass.mul_zero _).symm" }, { "state_after": "no goals", "state_before": "α : Type ?u.929509\na b : PosNum\n⊢ -↑(a * b) = ↑a * -↑b", "tactic": "rw [PosNum.cast_mul, neg_mul_eq_mul_neg]" }, { "state_after": "no goals", "state_before": "α : Type ?u.929509\na b : PosNum\n⊢ -↑(a * b) = -↑a * ↑b", "tactic": "rw [PosNum.cast_mul, neg_mul_eq_neg_mul]" }, { "state_after": "no goals", "state_before": "α : Type ?u.929509\na b : PosNum\n⊢ ↑(a * b) = -↑a * -↑b", "tactic": "rw [PosNum.cast_mul, neg_mul_neg]" } ]

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[]

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[ { "state_after": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ biproduct.ι (fun j => f j ⊗ X) j✝ ≫ (rightDistributor X f).inv =\n biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ (biproduct.ι f j ⊗ 𝟙 X)", "state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\n⊢ (rightDistributor X f).inv = ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ (biproduct.ι f j ⊗ 𝟙 X)", "tactic": "ext" }, { "state_after": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ (biproduct.ι (fun j => f j ⊗ X) j✝ ≫ biproduct.desc fun j => biproduct.ι f j ⊗ 𝟙 X) =\n biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ (biproduct.ι f j ⊗ 𝟙 X)", "state_before": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ biproduct.ι (fun j => f j ⊗ X) j✝ ≫ (rightDistributor X f).inv =\n biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ (biproduct.ι f j ⊗ 𝟙 X)", "tactic": "dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]" }, { "state_after": "no goals", "state_before": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ (biproduct.ι (fun j => f j ⊗ X) j✝ ≫ biproduct.desc fun j => biproduct.ι f j ⊗ 𝟙 X) =\n biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ (biproduct.ι f j ⊗ 𝟙 X)", "tactic": "simp only [biproduct.ι_desc, Preadditive.comp_sum, ne_eq, biproduct.ι_π_assoc, dite_comp,\n zero_comp, Finset.sum_dite_eq, Finset.mem_univ, eqToHom_refl, Category.id_comp, ite_true]" } ]

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[ { "state_after": "no goals", "state_before": "m✝ n : ℕ\ni : Fin (n + 1)\nm : Fin n\nh : i ≤ ↑Fin.castSucc m\n⊢ ↑(finSuccEquiv' i) (Fin.succ m) = some m", "tactic": "rw [← Fin.succAbove_above _ _ h, finSuccEquiv'_succAbove]" } ]

[ 158, 60 ]

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[ { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E × F → G\nh : IsBoundedBilinearMap 𝕜 f\nC : ℝ\nCpos : C > 0\nhC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖\nx✝ : E × F\nx : E\ny : F\n⊢ ‖f (x, y)‖ ≤ ?m.295592 h C Cpos hC * ‖‖(x, y).fst‖ * ‖(x, y).snd‖‖", "tactic": "simpa [mul_assoc] using hC x y" } ]

[ 371, 80 ]

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[ { "state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Balanced C\nX✝ Y✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\nfmono : Mono f\nfepi : Epi f\n⊢ IsIso f.unop.op", "state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Balanced C\nX✝ Y✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\nfmono : Mono f\nfepi : Epi f\n⊢ IsIso f", "tactic": "rw [← Quiver.Hom.op_unop f]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Balanced C\nX✝ Y✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\nfmono : Mono f\nfepi : Epi f\n⊢ IsIso f.unop.op", "tactic": "exact isIso_of_op _" } ]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "X : Type u_1\nY : Type ?u.60142\nZ : Type ?u.60145\nα : Type ?u.60148\nι : Type ?u.60151\nπ : ι → Type ?u.60156\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\n⊢ Filter.map mk (𝓝 x) = 𝓝 (mk x)", "tactic": "rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]" } ]

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[ { "state_after": "ι : Type ?u.112982\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ c a b : α\n⊢ card (Icc a b) - 1 - 1 = card (Icc a b) - 2", "state_before": "ι : Type ?u.112982\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ c a b : α\n⊢ card (Ioo a b) = card (Icc a b) - 2", "tactic": "rw [card_Ioo_eq_card_Ico_sub_one, card_Ico_eq_card_Icc_sub_one]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.112982\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ c a b : α\n⊢ card (Icc a b) - 1 - 1 = card (Icc a b) - 2", "tactic": "rfl" } ]

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[ { "state_after": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\n⊢ (⨆ (i : ι) (_ : p i), S i) ≤\n LinearMap.range\n (LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p))\n\ncase a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\n⊢ LinearMap.range\n (LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p)) ≤\n ⨆ (i : ι) (_ : p i), S i", "state_before": "ι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\n⊢ (⨆ (i : ι) (_ : p i), S i) =\n LinearMap.range\n (LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p))", "tactic": "apply le_antisymm" }, { "state_after": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\ni : ι\nhi : p i\ny : N\nhy : y ∈ S i\n⊢ ↑(LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p))\n (single i { val := y, property := hy }) =\n y", "state_before": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\n⊢ (⨆ (i : ι) (_ : p i), S i) ≤\n LinearMap.range\n (LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p))", "tactic": "refine' iSup₂_le fun i hi y hy => ⟨Dfinsupp.single i ⟨y, hy⟩, _⟩" }, { "state_after": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\ni : ι\nhi : p i\ny : N\nhy : y ∈ S i\n⊢ ↑(↑(lsum ℕ) fun i => Submodule.subtype (S i)) (single i { val := y, property := hy }) = y", "state_before": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\ni : ι\nhi : p i\ny : N\nhy : y ∈ S i\n⊢ ↑(LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p))\n (single i { val := y, property := hy }) =\n y", "tactic": "rw [LinearMap.comp_apply, filterLinearMap_apply, filter_single_pos _ _ hi]" }, { "state_after": "no goals", "state_before": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\ni : ι\nhi : p i\ny : N\nhy : y ∈ S i\n⊢ ↑(↑(lsum ℕ) fun i => Submodule.subtype (S i)) (single i { val := y, property := hy }) = y", "tactic": "simp only [lsum_apply_apply, sumAddHom_single, LinearMap.toAddMonoidHom_coe, coeSubtype]" }, { "state_after": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\n⊢ ↑(LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p)) v ∈\n ⨆ (i : ι) (_ : p i), S i", "state_before": "case a\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\n⊢ LinearMap.range\n (LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p)) ≤\n ⨆ (i : ι) (_ : p i), S i", "tactic": "rintro x ⟨v, rfl⟩" }, { "state_after": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (i : ι) (_ : p i), S i", "state_before": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\n⊢ ↑(LinearMap.comp (↑(lsum ℕ) fun i => Submodule.subtype (S i)) (filterLinearMap R (fun i => { x // x ∈ S i }) p)) v ∈\n ⨆ (i : ι) (_ : p i), S i", "tactic": "refine' dfinsupp_sumAddHom_mem _ _ _ fun i _ => _" }, { "state_after": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i", "state_before": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (i : ι) (_ : p i), S i", "tactic": "refine' mem_iSup_of_mem i _" }, { "state_after": "case pos\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\nhp : p i\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i\n\ncase neg\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\nhp : ¬p i\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i", "state_before": "case a.intro\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i", "tactic": "by_cases hp : p i" }, { "state_after": "no goals", "state_before": "case pos\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\nhp : p i\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i", "tactic": "simp [hp]" }, { "state_after": "no goals", "state_before": "case neg\nι : Type u_1\nR : Type u_2\nS✝ : Type ?u.366938\nM : ι → Type ?u.366943\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\nv : Π₀ (i : ι), { x // x ∈ S i }\ni : ι\nx✝ : ↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i ≠ 0\nhp : ¬p i\n⊢ ↑((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (S i)) i)) i)\n (↑(↑(filterLinearMap R (fun i => { x // x ∈ S i }) p) v) i) ∈\n ⨆ (_ : p i), S i", "tactic": "simp [hp]" } ]

[ 352, 16 ]

[ 337, 1 ]

[]

[ 90, 40 ]

[ 88, 1 ]

[ { "state_after": "case intro\nm : ℕ\nh : 2 ≤ 2 + m\n⊢ cycleType (finRotate (2 + m)) = {2 + m}", "state_before": "n : ℕ\nh : 2 ≤ n\n⊢ cycleType (finRotate n) = {n}", "tactic": "obtain ⟨m, rfl⟩ := exists_add_of_le h" }, { "state_after": "no goals", "state_before": "case intro\nm : ℕ\nh : 2 ≤ 2 + m\n⊢ cycleType (finRotate (2 + m)) = {2 + m}", "tactic": "rw [add_comm, cycleType_finRotate]" } ]

[ 150, 37 ]

[ 148, 1 ]

[]

[ 368, 39 ]

[ 366, 1 ]

[]

[ 794, 86 ]

[ 793, 1 ]

[]

[ 193, 40 ]

[ 192, 1 ]

[ { "state_after": "c : Char\ncs : List Char\nb e : Pos\n⊢ go₁ cs (0 + c) (b + c) (e + c) = go₁ cs 0 b e", "state_before": "c : Char\ncs : List Char\nb e : Pos\n⊢ go₁ (c :: cs) 0 (b + c) (e + c) = go₁ cs 0 b e", "tactic": "simp [go₁, Pos.ext_iff, Nat.ne_of_lt add_csize_pos]" }, { "state_after": "no goals", "state_before": "c : Char\ncs : List Char\nb e : Pos\n⊢ go₁ cs (0 + c) (b + c) (e + c) = go₁ cs 0 b e", "tactic": "apply go₁_add_right_cancel" } ]

[ 420, 29 ]

[ 417, 1 ]

[]

[ 441, 30 ]

[ 440, 11 ]

[ { "state_after": "no goals", "state_before": "R : Type ?u.146799\nm i : ℕ\nhi : i ∈ range (m + 1)\n⊢ i ≤ 2 * m + 1", "tactic": "linarith [mem_range.1 hi]" }, { "state_after": "no goals", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 * ∑ i in range (m + 1), choose (2 * m + 1) i =\n ∑ i in range (m + 1), choose (2 * m + 1) i + ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i)", "tactic": "rw [two_mul, this]" }, { "state_after": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 * m + 1 + 1 - (m + 1) = m + 1", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ ∑ i in Ico 0 (m + 1), choose (2 * m + 1) i +\n ∑ j in Ico (2 * m + 1 + 1 - (m + 1)) (2 * m + 1 + 1 - 0), choose (2 * m + 1) j =\n ∑ i in Ico 0 (m + 1), choose (2 * m + 1) i + ∑ i in Ico (m + 1) (2 * m + 2), choose (2 * m + 1) i", "tactic": "congr" }, { "state_after": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 + 1 - (m + 1) = m + 1", "state_before": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 * m + 1 + 1 - (m + 1) = m + 1", "tactic": "have A : m + 1 ≤ 2 * m + 1 := by linarith" }, { "state_after": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 - (m + 1) + 1 = m + 1", "state_before": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 + 1 - (m + 1) = m + 1", "tactic": "rw [add_comm, add_tsub_assoc_of_le A, ← add_comm]" }, { "state_after": "case e_a.e_s.e_a.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 - (m + 1) = m", "state_before": "case e_a.e_s.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 - (m + 1) + 1 = m + 1", "tactic": "congr" }, { "state_after": "case e_a.e_s.e_a.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 = m + (m + 1)", "state_before": "case e_a.e_s.e_a.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 - (m + 1) = m", "tactic": "rw [tsub_eq_iff_eq_add_of_le A]" }, { "state_after": "no goals", "state_before": "case e_a.e_s.e_a.e_a\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\nA : m + 1 ≤ 2 * m + 1\n⊢ 2 * m + 1 = m + (m + 1)", "tactic": "ring" }, { "state_after": "no goals", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ m + 1 ≤ 2 * m + 1", "tactic": "linarith" }, { "state_after": "no goals", "state_before": "case h\nR : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ m + 1 ≤ 2 * m + 1 + 1", "tactic": "linarith" }, { "state_after": "no goals", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ m + 1 ≤ 2 * m + 2", "tactic": "linarith" }, { "state_after": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 * (2 ^ 2) ^ m = 2 * 4 ^ m", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 ^ (2 * m + 1) = 2 * 4 ^ m", "tactic": "rw [pow_succ, pow_mul, mul_comm]" }, { "state_after": "no goals", "state_before": "R : Type ?u.146799\nm : ℕ\nthis : ∑ i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) = ∑ i in range (m + 1), choose (2 * m + 1) i\n⊢ 2 * (2 ^ 2) ^ m = 2 * 4 ^ m", "tactic": "rfl" } ]

[ 119, 64 ]

[ 98, 1 ]

[ { "state_after": "case h_nil\nα : Type u_1\nβ : Type ?u.19015\nn : ℕ\ninst✝ : DecidableEq α\na : α\n⊢ Finset.card (filter (fun i => a = Vector.get Vector.nil i) univ) = List.count a (Vector.toList Vector.nil)\n\ncase h_cons\nα : Type u_1\nβ : Type ?u.19015\nn✝ : ℕ\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx : α\nxs : Vector α n\nhxs : Finset.card (filter (fun i => a = Vector.get xs i) univ) = List.count a (Vector.toList xs)\n⊢ Finset.card (filter (fun i => a = Vector.get (x ::ᵥ xs) i) univ) = List.count a (Vector.toList (x ::ᵥ xs))", "state_before": "α : Type u_1\nβ : Type ?u.19015\nn : ℕ\ninst✝ : DecidableEq α\na : α\nv : Vector α n\n⊢ Finset.card (filter (fun i => a = Vector.get v i) univ) = List.count a (Vector.toList v)", "tactic": "induction' v using Vector.inductionOn with n x xs hxs" }, { "state_after": "no goals", "state_before": "case h_nil\nα : Type u_1\nβ : Type ?u.19015\nn : ℕ\ninst✝ : DecidableEq α\na : α\n⊢ Finset.card (filter (fun i => a = Vector.get Vector.nil i) univ) = List.count a (Vector.toList Vector.nil)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case h_cons\nα : Type u_1\nβ : Type ?u.19015\nn✝ : ℕ\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx : α\nxs : Vector α n\nhxs : Finset.card (filter (fun i => a = Vector.get xs i) univ) = List.count a (Vector.toList xs)\n⊢ Finset.card (filter (fun i => a = Vector.get (x ::ᵥ xs) i) univ) = List.count a (Vector.toList (x ::ᵥ xs))", "tactic": "simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp,\n Vector.get_cons_succ, hxs, List.count_cons', add_comm (ite (a = x) 1 0)]" } ]

[ 92, 79 ]

[ 87, 1 ]

[]

[ 211, 35 ]

[ 210, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68690\nι : Type ?u.68693\ninst✝³ : Countable ι\nf g : α → β\nm : MeasurableSpace α\ninst✝² : One β\ninst✝¹ : TopologicalSpace β\ninst✝ : MetrizableSpace β\nhf : StronglyMeasurable f\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ MeasurableSet (mulSupport f)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68690\nι : Type ?u.68693\ninst✝³ : Countable ι\nf g : α → β\nm : MeasurableSpace α\ninst✝² : One β\ninst✝¹ : TopologicalSpace β\ninst✝ : MetrizableSpace β\nhf : StronglyMeasurable f\n⊢ MeasurableSet (mulSupport f)", "tactic": "borelize β" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68690\nι : Type ?u.68693\ninst✝³ : Countable ι\nf g : α → β\nm : MeasurableSpace α\ninst✝² : One β\ninst✝¹ : TopologicalSpace β\ninst✝ : MetrizableSpace β\nhf : StronglyMeasurable f\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ MeasurableSet (mulSupport f)", "tactic": "exact measurableSet_mulSupport hf.measurable" } ]

[ 379, 47 ]

[ 376, 8 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.137611\ninst✝ : LinearOrderedSemiring α\na b c d : α\n⊢ 1 < bit1 a ↔ 0 < a", "tactic": "rw [bit1, lt_add_iff_pos_left, bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2 : α))]" } ]

[ 915, 96 ]

[ 914, 1 ]

[ { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝ : DivisionMonoid G\na b c d : G\nhbc : Commute b c\nhbd : Commute b⁻¹ d\nhcd : Commute c⁻¹ d\n⊢ a / b / (c / d) = a / c / (b / d)", "tactic": "simp_rw [div_eq_mul_inv, mul_inv_rev, inv_inv, hbd.symm.eq, hcd.symm.eq,\n hbc.inv_inv.mul_mul_mul_comm]" } ]

[ 344, 34 ]

[ 341, 11 ]

[]

[ 712, 56 ]

[ 711, 1 ]

[]

[ 2201, 42 ]

[ 2199, 1 ]

[]

[ 1394, 92 ]

[ 1380, 1 ]

[]

[ 172, 15 ]

[ 171, 8 ]

[]

[ 1470, 35 ]

[ 1468, 1 ]

[]

[ 837, 34 ]

[ 835, 1 ]

[ { "state_after": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (Fin.last s₁.length)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (Fin.last s₁.length)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (Fin.last s₁.length))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (Fin.last s₁.length))))\n\ncase refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ ∀ (i : Fin s₁.length),\n Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (↑Fin.castSucc i)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (↑Fin.castSucc i)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (↑Fin.castSucc i))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (↑Fin.castSucc i))))", "state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ Iso (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc i), series (snoc s₁ x₁ hsat₁) (Fin.succ i))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e i)), series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e i)))", "tactic": "refine' Fin.lastCases _ _ i" }, { "state_after": "no goals", "state_before": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (Fin.last s₁.length)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (Fin.last s₁.length)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (Fin.last s₁.length))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (Fin.last s₁.length))))", "tactic": "simpa [top] using htop" }, { "state_after": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni✝ : Fin (snoc s₁ x₁ hsat₁).length\ni : Fin s₁.length\n⊢ Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (↑Fin.castSucc i)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (↑Fin.castSucc i)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (↑Fin.castSucc i))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (↑Fin.castSucc i))))", "state_before": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni : Fin (snoc s₁ x₁ hsat₁).length\n⊢ ∀ (i : Fin s₁.length),\n Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (↑Fin.castSucc i)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (↑Fin.castSucc i)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (↑Fin.castSucc i))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (↑Fin.castSucc i))))", "tactic": "intro i" }, { "state_after": "no goals", "state_before": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (top s₁) x₁\nhsat₂ : IsMaximal (top s₂) x₂\nhequiv : Equivalent s₁ s₂\nhtop : Iso (top s₁, x₁) (top s₂, x₂)\ne : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=\n Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm\ni✝ : Fin (snoc s₁ x₁ hsat₁).length\ni : Fin s₁.length\n⊢ Iso\n (series (snoc s₁ x₁ hsat₁) (↑Fin.castSucc (↑Fin.castSucc i)),\n series (snoc s₁ x₁ hsat₁) (Fin.succ (↑Fin.castSucc i)))\n (series (snoc s₂ x₂ hsat₂) (↑Fin.castSucc (↑e (↑Fin.castSucc i))),\n series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (↑Fin.castSucc i))))", "tactic": "simpa [Fin.succ_castSucc] using hequiv.choose_spec i" } ]

[ 676, 60 ]

[ 663, 11 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227504\nι : Type ?u.227507\ninst✝⁴ : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderClosedTopology β\ninst✝ : PseudoMetrizableSpace β\nf g : α → β\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nthis✝¹ : MeasurableSpace (β × β) := borel (β × β)\nthis✝ : BorelSpace (β × β)\n⊢ MeasurableSet {a | f a ≤ g a}", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227504\nι : Type ?u.227507\ninst✝⁴ : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderClosedTopology β\ninst✝ : PseudoMetrizableSpace β\nf g : α → β\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {a | f a ≤ g a}", "tactic": "borelize (β × β)" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227504\nι : Type ?u.227507\ninst✝⁴ : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderClosedTopology β\ninst✝ : PseudoMetrizableSpace β\nf g : α → β\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nthis✝¹ : MeasurableSpace (β × β) := borel (β × β)\nthis✝ : BorelSpace (β × β)\n⊢ MeasurableSet {a | f a ≤ g a}", "tactic": "exact (hf.prod_mk hg).measurable isClosed_le_prod.measurableSet" } ]

[ 884, 66 ]

[ 880, 1 ]

[ { "state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ x ∈ asSubtype f ⁻¹' s ↔ x ∈ Subtype.val ⁻¹' preimage f s", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\n⊢ asSubtype f ⁻¹' s = Subtype.val ⁻¹' preimage f s", "tactic": "ext x" }, { "state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ fn f ↑x (_ : ↑x ∈ Dom f) ∈ s ↔ ∃ y, y ∈ s ∧ y ∈ f ↑x", "state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ x ∈ asSubtype f ⁻¹' s ↔ x ∈ Subtype.val ⁻¹' preimage f s", "tactic": "simp only [Set.mem_preimage, Set.mem_setOf_eq, PFun.asSubtype, PFun.mem_preimage]" }, { "state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ fn f ↑x (_ : ↑x ∈ Dom f) ∈ s ↔ ∃ y, y ∈ s ∧ y ∈ f ↑x", "state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ fn f ↑x (_ : ↑x ∈ Dom f) ∈ s ↔ ∃ y, y ∈ s ∧ y ∈ f ↑x", "tactic": "show f.fn x.val _ ∈ s ↔ ∃ y ∈ s, y ∈ f x.val" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.51072\nδ : Type ?u.51075\nε : Type ?u.51078\nι : Type ?u.51081\nf✝ f : α →. β\ns : Set β\nx : ↑(Dom f)\n⊢ fn f ↑x (_ : ↑x ∈ Dom f) ∈ s ↔ ∃ y, y ∈ s ∧ y ∈ f ↑x", "tactic": "exact\n Iff.intro (fun h => ⟨_, h, Part.get_mem _⟩) fun ⟨y, ys, fxy⟩ =>\n have : f.fn x.val x.property ∈ f x.val := Part.get_mem _\n Part.mem_unique fxy this ▸ ys" } ]

[ 534, 36 ]

[ 526, 1 ]

[]

[ 173, 19 ]

[ 172, 1 ]

[]

[ 327, 35 ]

[ 326, 1 ]

[ { "state_after": "case refine'_1\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\na : Ordinal\nha : a < Ordinal.lift (ord c)\n⊢ a < ord (lift c)\n\ncase refine'_2\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\n⊢ ord (lift c) ≤ Ordinal.lift (ord c)", "state_before": "α : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\n⊢ Ordinal.lift (ord c) = ord (lift c)", "tactic": "refine' le_antisymm (le_of_forall_lt fun a ha => _) _" }, { "state_after": "case refine'_1.intro.intro\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\na : Ordinal\nright✝ : a < ord c\nha : Ordinal.lift a < Ordinal.lift (ord c)\n⊢ Ordinal.lift a < ord (lift c)", "state_before": "case refine'_1\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\na : Ordinal\nha : a < Ordinal.lift (ord c)\n⊢ a < ord (lift c)", "tactic": "rcases Ordinal.lt_lift_iff.1 ha with ⟨a, rfl, _⟩" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\na : Ordinal\nright✝ : a < ord c\nha : Ordinal.lift a < Ordinal.lift (ord c)\n⊢ Ordinal.lift a < ord (lift c)", "tactic": "rwa [lt_ord, ← lift_card, lift_lt, ← lt_ord, ← Ordinal.lift_lt]" }, { "state_after": "no goals", "state_before": "case refine'_2\nα : Type ?u.229284\nβ : Type ?u.229287\nγ : Type ?u.229290\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\n⊢ ord (lift c) ≤ Ordinal.lift (ord c)", "tactic": "rw [ord_le, ← lift_card, card_ord]" } ]

[ 1444, 39 ]

[ 1440, 1 ]

[ { "state_after": "case refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1\n\ncase refine'_2\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ (∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1) → s * t = 1", "state_before": "F : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "refine' ⟨fun h => _, _⟩" }, { "state_after": "case refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "state_before": "case refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "have hst : (s * t).Nonempty := h.symm.subst one_nonempty" }, { "state_after": "case refine'_1.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "state_before": "case refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "obtain ⟨a, ha⟩ := hst.of_image2_left" }, { "state_after": "case refine'_1.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "state_before": "case refine'_1.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "obtain ⟨b, hb⟩ := hst.of_image2_right" }, { "state_after": "case refine'_1.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "state_before": "case refine'_1.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "have H : ∀ {a b}, a ∈ s → b ∈ t → a * b = (1 : α) := fun {a b} ha hb =>\n h.subset <| mem_image2_of_mem ha hb" }, { "state_after": "case refine'_1.intro.intro.refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\nx : α\nhx : x ∈ s\n⊢ x = a\n\ncase refine'_1.intro.intro.refine'_2\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\nx : α\nhx : x ∈ t\n⊢ x = b", "state_before": "case refine'_1.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\n⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "refine' ⟨a, b, _, _, H ha hb⟩ <;> refine' eq_singleton_iff_unique_mem.2 ⟨‹_›, fun x hx => _⟩" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro.refine'_1\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\nx : α\nhx : x ∈ s\n⊢ x = a", "tactic": "exact (eq_inv_of_mul_eq_one_left <| H hx hb).trans (inv_eq_of_mul_eq_one_left <| H ha hb)" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro.refine'_2\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\nh : s * t = 1\nhst : Set.Nonempty (s * t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nH : ∀ {a b : α}, a ∈ s → b ∈ t → a * b = 1\nx : α\nhx : x ∈ t\n⊢ x = b", "tactic": "exact (eq_inv_of_mul_eq_one_right <| H ha hx).trans (inv_eq_of_mul_eq_one_right <| H ha hb)" }, { "state_after": "case refine'_2.intro.intro.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\nb c : α\nh : b * c = 1\n⊢ {b} * {c} = 1", "state_before": "case refine'_2\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ (∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1) → s * t = 1", "tactic": "rintro ⟨b, c, rfl, rfl, h⟩" }, { "state_after": "no goals", "state_before": "case refine'_2.intro.intro.intro.intro\nF : Type ?u.80713\nα : Type u_1\nβ : Type ?u.80719\nγ : Type ?u.80722\ninst✝ : DivisionMonoid α\nb c : α\nh : b * c = 1\n⊢ {b} * {c} = 1", "tactic": "rw [singleton_mul_singleton, h, singleton_one]" } ]

[ 1050, 51 ]

[ 1039, 11 ]

[ { "state_after": "no goals", "state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n⊢ logb b x < y ↔ b ^ y < x", "tactic": "rw [← rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx]" } ]

[ 277, 101 ]

[ 276, 1 ]

[ { "state_after": "case a.h\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\nt : ↑I\n⊢ ↑(eval F 0) t = ↑p₀ t", "state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\n⊢ eval F 0 = p₀", "tactic": "ext t" }, { "state_after": "no goals", "state_before": "case a.h\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\nt : ↑I\n⊢ ↑(eval F 0) t = ↑p₀ t", "tactic": "simp [eval]" } ]

[ 89, 14 ]

[ 87, 1 ]

[]

[ 95, 10 ]

[ 94, 1 ]

[]

[ 4472, 17 ]

[ 4471, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.70104\nα : Type u_1\nβ : Type ?u.70110\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c✝ d a b c : α\n⊢ a ⇨ b ⇨ c = b ⇨ a ⇨ c", "tactic": "simp_rw [himp_himp, inf_comm]" } ]

[ 397, 95 ]

[ 397, 1 ]

[]

[ 2350, 70 ]

[ 2349, 1 ]

[]

[ 2485, 43 ]

[ 2484, 1 ]

[]

[ 2371, 38 ]

[ 2370, 1 ]

[]

[ 134, 54 ]

[ 132, 1 ]

[]

[ 173, 44 ]

[ 171, 1 ]