Datasets:

tasksource

/

leandojo

like 5

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[ 184, 28 ]

[ 183, 1 ]

[ { "state_after": "case refine_1\nι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝¹ y : α\nf : Perm α\nx x✝ : α\nh : ↑f x✝ ∈ ↑(support (cycleOf f x))\n⊢ SameCycle f x x✝ ∧ x ∈ support f\n\ncase refine_2\nι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝¹ y : α\nf : Perm α\nx x✝ : α\nh : x✝ ∈ ↑(support (cycleOf f x))\n⊢ SameCycle f x (↑f x✝) ∧ x ∈ support f", "state_before": "ι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\n⊢ ∀ (a : α), ↑f a ∈ ↑(support (cycleOf f x)) ↔ a ∈ ↑(support (cycleOf f x))", "tactic": "refine fun _ ↦ ⟨fun h ↦ mem_support_cycleOf_iff.2 ?_, fun h ↦ mem_support_cycleOf_iff.2 ?_⟩" }, { "state_after": "no goals", "state_before": "case refine_1\nι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝¹ y : α\nf : Perm α\nx x✝ : α\nh : ↑f x✝ ∈ ↑(support (cycleOf f x))\n⊢ SameCycle f x x✝ ∧ x ∈ support f", "tactic": "exact ⟨sameCycle_apply_right.1 (mem_support_cycleOf_iff.1 h).1,\n(mem_support_cycleOf_iff.1 h).2⟩" }, { "state_after": "no goals", "state_before": "case refine_2\nι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝¹ y : α\nf : Perm α\nx x✝ : α\nh : x✝ ∈ ↑(support (cycleOf f x))\n⊢ SameCycle f x (↑f x✝) ∧ x ∈ support f", "tactic": "exact ⟨sameCycle_apply_right.2 (mem_support_cycleOf_iff.1 h).1,\n(mem_support_cycleOf_iff.1 h).2⟩" }, { "state_after": "ι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx a : α\nha : SameCycle f x a ∧ x ∈ support f\nb : α\nhb : SameCycle f x b ∧ x ∈ support f\n⊢ SameCycle f a b", "state_before": "ι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx a : α\nha : a ∈ ↑(support (cycleOf f x))\nb : α\nhb : b ∈ ↑(support (cycleOf f x))\n⊢ SameCycle f a b", "tactic": "rw [mem_coe, mem_support_cycleOf_iff] at ha hb" }, { "state_after": "no goals", "state_before": "ι : Type ?u.2583510\nα : Type u_1\nβ : Type ?u.2583516\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx a : α\nha : SameCycle f x a ∧ x ∈ support f\nb : α\nhb : SameCycle f x b ∧ x ∈ support f\n⊢ SameCycle f a b", "tactic": "exact ha.1.symm.trans hb.1" } ]

[ 1193, 36 ]

[ 1183, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.430806\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.430815\nι : Type ?u.430818\nκ : Type ?u.430821\ninst✝ : CompleteLattice β\na : α\ns : α → β\n⊢ (⨅ (x : α) (_ : x ∈ {a}), s x) = s a", "tactic": "simp" } ]

[ 1935, 92 ]

[ 1935, 1 ]

[]

[ 752, 6 ]

[ 750, 1 ]

[ { "state_after": "no goals", "state_before": "α : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\n⊢ ite b x y = ite c u v", "tactic": "cases Decidable.em c with\n| inl h => rw [if_pos h]; subst b; rw [if_pos h]; exact h₂ h\n| inr h => rw [if_neg h]; subst b; rw [if_neg h]; exact h₃ h" }, { "state_after": "case inl\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\n⊢ ite b x y = u", "state_before": "case inl\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\n⊢ ite b x y = ite c u v", "tactic": "rw [if_pos h]" }, { "state_after": "case inl\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\ns : Decidable c\n⊢ ite c x y = u", "state_before": "case inl\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\n⊢ ite b x y = u", "tactic": "subst b" }, { "state_after": "case inl\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\ns : Decidable c\n⊢ x = u", "state_before": "case inl\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\ns : Decidable c\n⊢ ite c x y = u", "tactic": "rw [if_pos h]" }, { "state_after": "no goals", "state_before": "case inl\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : c\ns : Decidable c\n⊢ x = u", "tactic": "exact h₂ h" }, { "state_after": "case inr\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\n⊢ ite b x y = v", "state_before": "case inr\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\n⊢ ite b x y = ite c u v", "tactic": "rw [if_neg h]" }, { "state_after": "case inr\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\ns : Decidable c\n⊢ ite c x y = v", "state_before": "case inr\nα : Sort u_1\nb c : Prop\nx y u v : α\ns : Decidable b\ninst✝ : Decidable c\nh₁ : b = c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\n⊢ ite b x y = v", "tactic": "subst b" }, { "state_after": "case inr\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\ns : Decidable c\n⊢ y = v", "state_before": "case inr\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\ns : Decidable c\n⊢ ite c x y = v", "tactic": "rw [if_neg h]" }, { "state_after": "no goals", "state_before": "case inr\nα : Sort u_1\nc : Prop\nx y u v : α\ninst✝ : Decidable c\nh₂ : c → x = u\nh₃ : ¬c → y = v\nh : ¬c\ns : Decidable c\n⊢ y = v", "tactic": "exact h₃ h" } ]

[ 60, 63 ]

[ 56, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.89441\nα : Type u_1\nβ : Type ?u.89447\ninst✝ : LinearOrderedSemifield α\na✝ b c d e : α\nm n : ℤ\na : α\n⊢ a / 2 + a / 2 = a", "tactic": "rw [div_add_div_same, ← two_mul, mul_div_cancel_left a two_ne_zero]" } ]

[ 498, 70 ]

[ 497, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.218847\nγ : Type ?u.218850\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nh : ¬a ∈ s\n⊢ s ⊆ insert a t ↔ s ⊆ t", "tactic": "rw [subset_insert_iff, erase_eq_of_not_mem h]" } ]

[ 2003, 48 ]

[ 2002, 1 ]

[]

[ 92, 81 ]

[ 91, 1 ]

[]

[ 341, 85 ]

[ 340, 1 ]

[]

[ 420, 31 ]

[ 419, 1 ]

[ { "state_after": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsFiniteMeasure μ\nhms : MeasurableSet s\nhci : ↑↑μ (s ∩ t) ≠ 0\n⊢ 0 < (↑↑μ s)⁻¹ * ↑↑μ (s ∩ t)", "state_before": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsFiniteMeasure μ\nhms : MeasurableSet s\nhci : ↑↑μ (s ∩ t) ≠ 0\n⊢ 0 < ↑↑(μ[|s]) t", "tactic": "rw [cond_apply _ hms]" }, { "state_after": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsFiniteMeasure μ\nhms : MeasurableSet s\nhci : ↑↑μ (s ∩ t) ≠ 0\n⊢ (↑↑μ s)⁻¹ ≠ 0", "state_before": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsFiniteMeasure μ\nhms : MeasurableSet s\nhci : ↑↑μ (s ∩ t) ≠ 0\n⊢ 0 < (↑↑μ s)⁻¹ * ↑↑μ (s ∩ t)", "tactic": "refine' ENNReal.mul_pos _ hci" }, { "state_after": "no goals", "state_before": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsFiniteMeasure μ\nhms : MeasurableSet s\nhci : ↑↑μ (s ∩ t) ≠ 0\n⊢ (↑↑μ s)⁻¹ ≠ 0", "tactic": "exact ENNReal.inv_ne_zero.mpr (measure_ne_top _ _)" } ]

[ 124, 53 ]

[ 120, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na : α\n⊢ SurjOn f (Ioi a ∪ {a}) (Ioi (f a) ∪ {f a})", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na : α\n⊢ SurjOn f (Ici a) (Ici (f a))", "tactic": "rw [← Ioi_union_left, ← Ioi_union_left]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na : α\n⊢ SurjOn f (Ioi a ∪ {a}) (Ioi (f a) ∪ {f a})", "tactic": "exact\n (surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union\n (@image_singleton _ _ f a ▸ surjOn_image _ _)" } ]

[ 84, 52 ]

[ 79, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.34618\ninst✝ : OrderedRing α\na b c d : α\nh : b ≤ a\nhc : c ≤ 0\n⊢ a * c ≤ b * c", "tactic": "simpa only [mul_neg, neg_le_neg_iff] using mul_le_mul_of_nonneg_right h (neg_nonneg.2 hc)" } ]

[ 356, 92 ]

[ 355, 1 ]

[ { "state_after": "no goals", "state_before": "n✝ m✝ m n : ℕ\ni : Fin m\n⊢ ↑(↑(castAdd n) i) < m", "tactic": "simp" } ]

[ 1142, 7 ]

[ 1141, 1 ]

[]

[ 337, 11 ]

[ 336, 1 ]

[ { "state_after": "case intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ π ρ : Perm α\nhc1 : IsConj σ π\nhc2 : IsConj τ ρ\nhd1 : Disjoint σ τ\nhd2 : Disjoint π ρ\nval✝ : Fintype α\n⊢ IsConj (σ * τ) (π * ρ)", "state_before": "ι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ π ρ : Perm α\nhc1 : IsConj σ π\nhc2 : IsConj τ ρ\nhd1 : Disjoint σ τ\nhd2 : Disjoint π ρ\n⊢ IsConj (σ * τ) (π * ρ)", "tactic": "cases nonempty_fintype α" }, { "state_after": "case intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ ρ : Perm α\nhc2 : IsConj τ ρ\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) ρ\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * ρ)", "state_before": "case intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ π ρ : Perm α\nhc1 : IsConj σ π\nhc2 : IsConj τ ρ\nhd1 : Disjoint σ τ\nhd2 : Disjoint π ρ\nval✝ : Fintype α\n⊢ IsConj (σ * τ) (π * ρ)", "tactic": "obtain ⟨f, rfl⟩ := isConj_iff.1 hc1" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ ρ : Perm α\nhc2 : IsConj τ ρ\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) ρ\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * ρ)", "tactic": "obtain ⟨g, rfl⟩ := isConj_iff.1 hc2" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "have hd1' := coe_inj.2 hd1.support_mul" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹) ∪ support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "have hd2' := coe_inj.2 hd2.support_mul" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹) ∪ support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "rw [coe_union] at *" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "have hd1'' := disjoint_coe.2 (disjoint_iff_disjoint_support.1 hd1)" }, { "state_after": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "have hd2'' := disjoint_coe.2 (disjoint_iff_disjoint_support.1 hd2)" }, { "state_after": "case intro.intro.intro.refine'_1\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\n⊢ { x // x ∈ ↑(support (σ * τ)) } ≃ { x // x ∈ ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) }\n\ncase intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\n⊢ ∀ (x : α) (hx : x ∈ ↑(support (σ * τ))),\n ↑(↑?intro.intro.intro.refine'_1 { val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹)) ↑(↑?intro.intro.intro.refine'_1 { val := x, property := hx })", "state_before": "case intro.intro.intro\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\n⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))", "tactic": "refine' isConj_of_support_equiv _ _" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_1.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\na : α\n⊢ a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))", "tactic": "simp only [Set.mem_image, toEmbedding_apply, exists_eq_right, support_conj, coe_map,\n apply_eq_iff_eq]" }, { "state_after": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\n⊢ ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := x, property := hx })", "state_before": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\n⊢ ∀ (x : α) (hx : x ∈ ↑(support (σ * τ))),\n ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans\n (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := x, property := hx })", "tactic": "intro x hx" }, { "state_after": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "state_before": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\n⊢ ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans\n ((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))\n (subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans\n ((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))\n { val := x, property := hx })", "tactic": "simp only [trans_apply, symm_trans_apply, Equiv.Set.ofEq_apply, Equiv.Set.ofEq_symm_apply,\n Equiv.sumCongr_apply]" }, { "state_after": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx✝ : x ∈ ↑(support (σ * τ))\nhx : x ∈ ↑(support σ) ∨ x ∈ ↑(support τ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx✝ } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "state_before": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "rw [hd1', Set.mem_union] at hx" }, { "state_after": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : x ∈ ↑(support σ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))\n\ncase intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : x ∈ ↑(support τ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "state_before": "case intro.intro.intro.refine'_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx✝ : x ∈ ↑(support (σ * τ))\nhx : x ∈ ↑(support σ) ∨ x ∈ ↑(support τ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx✝ } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "cases' hx with hxσ hxτ" }, { "state_after": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "state_before": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : x ∈ ↑(support σ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "rw [mem_coe, mem_support] at hxσ" }, { "state_after": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inl\n {\n val :=\n ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3053395 }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inl\n {\n val :=\n ↑{ val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3053769 })))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)", "state_before": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "rw [Set.union_apply_left hd1''.le_bot _, Set.union_apply_left hd1''.le_bot _]" }, { "state_after": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)", "state_before": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inl\n {\n val :=\n ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3053395 }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inl\n {\n val :=\n ↑{ val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3053769 })))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)", "tactic": "simp only [subtypeEquiv_apply, Perm.coe_mul, Sum.map_inl, comp_apply,\n Set.union_symm_apply_left, Subtype.coe_mk, apply_eq_iff_eq]" }, { "state_after": "case intro.intro.intro.refine'_2.inl.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\nh : ↑(g * τ * g⁻¹) (↑f x) = ↑f x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))\n\ncase intro.intro.intro.refine'_2.inl.refine_1\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ¬↑(f * σ * f⁻¹) (↑f x) = ↑f x", "state_before": "case intro.intro.intro.refine'_2.inl\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))", "tactic": "have h := (hd2 (f x)).resolve_left ?_" }, { "state_after": "case intro.intro.intro.refine'_2.inl.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\nh : ↑g (↑τ (↑g⁻¹ (↑f x))) = ↑f x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))", "state_before": "case intro.intro.intro.refine'_2.inl.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\nh : ↑(g * τ * g⁻¹) (↑f x) = ↑f x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))", "tactic": "rw [mul_apply, mul_apply] at h" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_2.inl.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\nh : ↑g (↑τ (↑g⁻¹ (↑f x))) = ↑f x\n⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))", "tactic": "rw [h, inv_apply_self, (hd1 x).resolve_left hxσ]" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_2.inl.refine_1\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ¬↑(f * σ * f⁻¹) (↑f x) = ↑f x", "tactic": "rwa [mul_apply, mul_apply, inv_apply_self, apply_eq_iff_eq]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)", "tactic": "rwa [Subtype.coe_mk, Perm.mul_apply, (hd1 x).resolve_left hxσ, mem_coe,\n apply_mem_support, mem_support]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxσ : ↑σ x ≠ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support σ)", "tactic": "rwa [Subtype.coe_mk, mem_coe, mem_support]" }, { "state_after": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "state_before": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : x ∈ ↑(support τ)\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "rw [mem_coe, ← apply_mem_support, mem_support] at hxτ" }, { "state_after": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inr\n {\n val :=\n ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3057896 }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inr\n {\n val :=\n ↑{ val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3058005 })))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)", "state_before": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))\n { val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))", "tactic": "rw [Set.union_apply_right hd1''.le_bot _, Set.union_apply_right hd1''.le_bot _]" }, { "state_after": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)", "state_before": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inr\n {\n val :=\n ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3057896 }))) =\n ↑(f * σ * f⁻¹ * (g * τ * g⁻¹))\n ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm\n (Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))\n (↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))\n (Sum.inr\n {\n val :=\n ↑{ val := x,\n property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },\n property := ?m.3058005 })))\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)\n\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)", "tactic": "simp only [subtypeEquiv_apply, Perm.coe_mul, Sum.map_inr, comp_apply,\n Set.union_symm_apply_right, Subtype.coe_mk, apply_eq_iff_eq]" }, { "state_after": "case intro.intro.intro.refine'_2.inr.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\nh : ↑(f * σ * f⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))\n\ncase intro.intro.intro.refine'_2.inr.refine_1\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ¬↑(g * τ * g⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)", "state_before": "case intro.intro.intro.refine'_2.inr\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))", "tactic": "have h := (hd2 (g (τ x))).resolve_right ?_" }, { "state_after": "case intro.intro.intro.refine'_2.inr.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\nh : ↑f (↑σ (↑f⁻¹ (↑g (↑τ x)))) = ↑g (↑τ x)\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))", "state_before": "case intro.intro.intro.refine'_2.inr.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\nh : ↑(f * σ * f⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))", "tactic": "rw [mul_apply, mul_apply] at h" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_2.inr.refine_2\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\nh : ↑f (↑σ (↑f⁻¹ (↑g (↑τ x)))) = ↑g (↑τ x)\n⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))", "tactic": "rw [inv_apply_self, h, (hd1 (τ x)).resolve_right hxτ]" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_2.inr.refine_1\nι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ¬↑(g * τ * g⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)", "tactic": "rwa [mul_apply, mul_apply, inv_apply_self, apply_eq_iff_eq]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := ↑(σ * τ) x,\n property :=\n (_ :\n ↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈\n ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)", "tactic": "rwa [Subtype.coe_mk, Perm.mul_apply, (hd1 (τ x)).resolve_right hxτ,\n mem_coe, mem_support]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.3042712\nα✝ : Type ?u.3042715\nβ : Type ?u.3042718\ninst✝² : DecidableEq α✝\ninst✝¹ : Fintype α✝\nσ✝ τ✝ : Perm α✝\nα : Type u_1\ninst✝ : Finite α\nσ τ : Perm α\nhd1 : Disjoint σ τ\nval✝ : Fintype α\nf : Perm α\nhc1 : IsConj σ (f * σ * f⁻¹)\ng : Perm α\nhc2 : IsConj τ (g * τ * g⁻¹)\nhd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)\nhd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)\nhd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))\nhd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)\nhd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))\nx : α\nhx : x ∈ ↑(support (σ * τ))\nhxτ : ↑τ (↑τ x) ≠ ↑τ x\n⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈\n ↑(support τ)", "tactic": "rwa [Subtype.coe_mk, mem_coe, ← apply_mem_support, mem_support]" } ]

[ 1815, 74 ]

[ 1771, 1 ]

[ { "state_after": "case inl\nG₀ : Type u_1\nα : Type u_2\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero α\ninst✝⁴ : MulActionWithZero G₀ α\ninst✝³ : MeasurableSpace G₀\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSMul G₀ α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet (0 • s)\n\ncase inr\nG₀ : Type u_1\nα : Type u_2\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero α\ninst✝⁴ : MulActionWithZero G₀ α\ninst✝³ : MeasurableSpace G₀\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSMul G₀ α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet s\na : G₀\nha : a ≠ 0\n⊢ MeasurableSet (a • s)", "state_before": "G₀ : Type u_1\nα : Type u_2\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero α\ninst✝⁴ : MulActionWithZero G₀ α\ninst✝³ : MeasurableSpace G₀\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSMul G₀ α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet s\na : G₀\n⊢ MeasurableSet (a • s)", "tactic": "rcases eq_or_ne a 0 with (rfl | ha)" }, { "state_after": "no goals", "state_before": "case inl\nG₀ : Type u_1\nα : Type u_2\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero α\ninst✝⁴ : MulActionWithZero G₀ α\ninst✝³ : MeasurableSpace G₀\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSMul G₀ α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet (0 • s)\n\ncase inr\nG₀ : Type u_1\nα : Type u_2\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero α\ninst✝⁴ : MulActionWithZero G₀ α\ninst✝³ : MeasurableSpace G₀\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSMul G₀ α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet s\na : G₀\nha : a ≠ 0\n⊢ MeasurableSet (a • s)", "tactic": "exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]" } ]

[ 47, 85 ]

[ 42, 1 ]

[]

[ 524, 31 ]

[ 522, 1 ]

[ { "state_after": "⊢ Subtype.val '' range (IccExtend arcsin.proof_2 ↑(OrderIso.symm sinOrderIso)) = Icc (-(π / 2)) (π / 2)", "state_before": "⊢ range arcsin = Icc (-(π / 2)) (π / 2)", "tactic": "rw [arcsin, range_comp Subtype.val]" }, { "state_after": "no goals", "state_before": "⊢ Subtype.val '' range (IccExtend arcsin.proof_2 ↑(OrderIso.symm sinOrderIso)) = Icc (-(π / 2)) (π / 2)", "tactic": "simp [Icc]" } ]

[ 48, 13 ]

[ 46, 1 ]

[ { "state_after": "ι : Type ?u.334826\nι' : Type ?u.334829\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.334838\ninst✝ : SemilatticeInf α\na : α\nf : β → ↑(Iio a)\nl : Filter β\n⊢ Tendsto (Subtype.val ∘ f) l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot", "state_before": "ι : Type ?u.334826\nι' : Type ?u.334829\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.334838\ninst✝ : SemilatticeInf α\na : α\nf : β → ↑(Iio a)\nl : Filter β\n⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot", "tactic": "rw [atBot_Iio_eq, tendsto_comap_iff]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.334826\nι' : Type ?u.334829\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.334838\ninst✝ : SemilatticeInf α\na : α\nf : β → ↑(Iio a)\nl : Filter β\n⊢ Tendsto (Subtype.val ∘ f) l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot", "tactic": "rfl" } ]

[ 1602, 44 ]

[ 1600, 1 ]

[ { "state_after": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\n⊢ ∀ᶠ (t : R) in 𝓝 0, inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n⊢ ∀ᶠ (t : R) in 𝓝 0, inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "tactic": "nontriviality R" }, { "state_after": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\n⊢ ∃ ε, ε > 0 ∧ ∀ ⦃y : R⦄, dist y 0 < ε → inverse (↑x + y) = inverse (1 + ↑x⁻¹ * y) * ↑x⁻¹", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\n⊢ ∀ᶠ (t : R) in 𝓝 0, inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "tactic": "rw [Metric.eventually_nhds_iff]" }, { "state_after": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\nt : R\nht : dist t 0 < ‖↑x⁻¹‖⁻¹\n⊢ inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\n⊢ ∃ ε, ε > 0 ∧ ∀ ⦃y : R⦄, dist y 0 < ε → inverse (↑x + y) = inverse (1 + ↑x⁻¹ * y) * ↑x⁻¹", "tactic": "refine ⟨‖(↑x⁻¹ : R)‖⁻¹, by cancel_denoms, fun t ht ↦ ?_⟩" }, { "state_after": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\nt : R\nht : ‖t‖ < ‖↑x⁻¹‖⁻¹\n⊢ inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\nt : R\nht : dist t 0 < ‖↑x⁻¹‖⁻¹\n⊢ inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "tactic": "rw [dist_zero_right] at ht" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\nt : R\nht : ‖t‖ < ‖↑x⁻¹‖⁻¹\n⊢ inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "tactic": "rw [← x.add_val t ht, inverse_unit, Units.add, Units.copy_eq, mul_inv_rev, Units.val_mul,\n ← inverse_unit, Units.oneSub_val, sub_neg_eq_add]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\n✝ : Nontrivial R\n⊢ ‖↑x⁻¹‖⁻¹ > 0", "tactic": "cancel_denoms" } ]

[ 129, 54 ]

[ 122, 1 ]

[ { "state_after": "case h\nι : Type u_5\nι' : Type ?u.181637\nR : Type u_1\nR₂ : Type ?u.181643\nK : Type ?u.181646\nM : Type u_3\nM' : Type ?u.181652\nM'' : Type ?u.181655\nV : Type u\nV' : Type ?u.181660\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M →ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ x = ↑f₂ x", "state_before": "ι : Type u_5\nι' : Type ?u.181637\nR : Type u_1\nR₂ : Type ?u.181643\nK : Type ?u.181646\nM : Type u_3\nM' : Type ?u.181652\nM'' : Type ?u.181655\nV : Type u\nV' : Type ?u.181660\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M →ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\n⊢ f₁ = f₂", "tactic": "ext x" }, { "state_after": "case h\nι : Type u_5\nι' : Type ?u.181637\nR : Type u_1\nR₂ : Type ?u.181643\nK : Type ?u.181646\nM : Type u_3\nM' : Type ?u.181652\nM'' : Type ?u.181655\nV : Type u\nV' : Type ?u.181660\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M →ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a) = ↑f₂ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a)", "state_before": "case h\nι : Type u_5\nι' : Type ?u.181637\nR : Type u_1\nR₂ : Type ?u.181643\nK : Type ?u.181646\nM : Type u_3\nM' : Type ?u.181652\nM'' : Type ?u.181655\nV : Type u\nV' : Type ?u.181660\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M →ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ x = ↑f₂ x", "tactic": "rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]" }, { "state_after": "no goals", "state_before": "case h\nι : Type u_5\nι' : Type ?u.181637\nR : Type u_1\nR₂ : Type ?u.181643\nK : Type ?u.181646\nM : Type u_3\nM' : Type ?u.181652\nM'' : Type ?u.181655\nV : Type u\nV' : Type ?u.181660\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M →ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a) = ↑f₂ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a)", "tactic": "simp only [LinearMap.map_sum, LinearMap.map_smulₛₗ, h]" } ]

[ 281, 57 ]

[ 278, 1 ]

[ { "state_after": "no goals", "state_before": "n₁ n₂ d₁ d₂ : Int\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\n⊢ n₁ /. d₁ + n₂ /. d₂ = (n₁ * d₂ + n₂ * d₁) /. (d₁ * d₂)", "tactic": "rcases Int.eq_nat_or_neg d₁ with ⟨_, rfl | rfl⟩ <;>\nrcases Int.eq_nat_or_neg d₂ with ⟨_, rfl | rfl⟩ <;>\nsimp_all [Int.ofNat_eq_zero, Int.neg_eq_zero, divInt_neg', Int.mul_neg,\n Int.ofNat_mul_ofNat, Int.neg_add, Int.neg_mul, mkRat_add_mkRat]" } ]

[ 204, 68 ]

[ 199, 1 ]

[ { "state_after": "R : Type u\nS : Type v\ninst✝ : CommRing R\nI : Ideal R\nhi : IsLocal I\nx : R\nh : I ⊔ span {x} = ⊤\np : R\nhpi : p ∈ I\nq : R\nhq : q ∈ span {x}\nhpq : p + q = 1\nr : R\nhr : q = x * r\n⊢ -p ∈ I", "state_before": "R : Type u\nS : Type v\ninst✝ : CommRing R\nI : Ideal R\nhi : IsLocal I\nx : R\nh : I ⊔ span {x} = ⊤\np : R\nhpi : p ∈ I\nq : R\nhq : q ∈ span {x}\nhpq : p + q = 1\nr : R\nhr : q = x * r\n⊢ r * x - 1 ∈ I", "tactic": "rw [← hpq, mul_comm, ← hr, ← neg_sub, add_sub_cancel]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\ninst✝ : CommRing R\nI : Ideal R\nhi : IsLocal I\nx : R\nh : I ⊔ span {x} = ⊤\np : R\nhpi : p ∈ I\nq : R\nhq : q ∈ span {x}\nhpq : p + q = 1\nr : R\nhr : q = x * r\n⊢ -p ∈ I", "tactic": "exact I.neg_mem hpi" } ]

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[ 387, 1 ]

[]

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[ { "state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.4171\nπ : ι → Type ?u.4176\ninst✝ : TopologicalSpace α\ns✝ t✝ u v s t : Set α\nhs : IsPreconnected s\nht : IsPreconnected t\nx : α\nhxs : x ∈ s\nhxt : x ∈ t\n⊢ IsPreconnected (s ∪ t)", "state_before": "α : Type u\nβ : Type v\nι : Type ?u.4171\nπ : ι → Type ?u.4176\ninst✝ : TopologicalSpace α\ns✝ t✝ u v s t : Set α\nH : Set.Nonempty (s ∩ t)\nhs : IsPreconnected s\nht : IsPreconnected t\n⊢ IsPreconnected (s ∪ t)", "tactic": "rcases H with ⟨x, hxs, hxt⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.4171\nπ : ι → Type ?u.4176\ninst✝ : TopologicalSpace α\ns✝ t✝ u v s t : Set α\nhs : IsPreconnected s\nht : IsPreconnected t\nx : α\nhxs : x ∈ s\nhxt : x ∈ t\n⊢ IsPreconnected (s ∪ t)", "tactic": "exact hs.union x hxs hxt ht" } ]

[ 151, 30 ]

[ 148, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\nγ : Type u_1\ninst✝² : LocallyCompactSpace α\ninst✝¹ : AddCommMonoid β\ninst✝ : ContinuousAdd β\nf : γ → C(α, β)\ng : C(α, β)\nhf : HasSum f g\nx : α\nevₓ : C(α, β) →+ β := AddMonoidHom.comp (Pi.evalAddMonoidHom (fun a => β) x) coeFnAddMonoidHom\n⊢ HasSum (fun i => ↑(f i) x) (↑g x)", "state_before": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\nγ : Type u_1\ninst✝² : LocallyCompactSpace α\ninst✝¹ : AddCommMonoid β\ninst✝ : ContinuousAdd β\nf : γ → C(α, β)\ng : C(α, β)\nhf : HasSum f g\nx : α\n⊢ HasSum (fun i => ↑(f i) x) (↑g x)", "tactic": "let evₓ : AddMonoidHom C(α, β) β := (Pi.evalAddMonoidHom _ x).comp coeFnAddMonoidHom" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\nγ : Type u_1\ninst✝² : LocallyCompactSpace α\ninst✝¹ : AddCommMonoid β\ninst✝ : ContinuousAdd β\nf : γ → C(α, β)\ng : C(α, β)\nhf : HasSum f g\nx : α\nevₓ : C(α, β) →+ β := AddMonoidHom.comp (Pi.evalAddMonoidHom (fun a => β) x) coeFnAddMonoidHom\n⊢ HasSum (fun i => ↑(f i) x) (↑g x)", "tactic": "exact hf.map evₓ (ContinuousMap.continuous_eval_const' x)" } ]

[ 431, 60 ]

[ 427, 1 ]

[]

[ 1000, 6 ]

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[]

[ 114, 6 ]

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[]

[ 36, 6 ]

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[]

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[ { "state_after": "no goals", "state_before": "ι : Type u_2\nβ : ι → Type ?u.33701\nr : ι → ι → Prop\ns : {i : ι} → β i → β i → Prop\nα : Type u_1\ninst✝² : Preorder ι\ninst✝¹ : DecidableEq ι\ninst✝ : Preorder α\nf : ι → α\ni j : ι\nh₁ : i < j\nh₂ : f j < f i\n⊢ (fun x x_1 x_2 => x_1 < x_2) i (↑toLex (f ∘ ↑(Equiv.swap i j)) i) (↑toLex f i)", "tactic": "simpa only [Pi.toLex_apply, Function.comp_apply, Equiv.swap_apply_left] using h₂" } ]

[ 245, 86 ]

[ 242, 1 ]

[]

[ 1063, 29 ]

[ 1060, 1 ]

[ { "state_after": "𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ Nat.gcd m n * (addOrderOf ↑(p / ↑n) / Nat.gcd (addOrderOf ↑(p / ↑n)) m) = n\n\ncase h\n𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ IsOfFinAddOrder ↑(p / ↑n)", "state_before": "𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ Nat.gcd m n * addOrderOf ↑(↑m / ↑n * p) = n", "tactic": "rw [mul_comm_div, ← nsmul_eq_mul, coe_nsmul, addOrderOf_nsmul'']" }, { "state_after": "𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ Nat.gcd n m ∣ n", "state_before": "𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ Nat.gcd m n * (addOrderOf ↑(p / ↑n) / Nat.gcd (addOrderOf ↑(p / ↑n)) m) = n", "tactic": "rw [addOrderOf_period_div hn, Nat.gcd_comm, Nat.mul_div_cancel']" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ Nat.gcd n m ∣ n", "tactic": "exact n.gcd_dvd_left m" }, { "state_after": "case h\n𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ 0 < n", "state_before": "case h\n𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ IsOfFinAddOrder ↑(p / ↑n)", "tactic": "rw [← addOrderOf_pos_iff, addOrderOf_period_div hn]" }, { "state_after": "no goals", "state_before": "case h\n𝕜 : Type u_1\nB : Type ?u.183351\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : 𝕜\nhp : Fact (0 < p)\nn m : ℕ\nhn : 0 < n\n⊢ 0 < n", "tactic": "exact hn" } ]

[ 392, 13 ]

[ 386, 1 ]

[ { "state_after": "case a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ lift (#ι) ≤ ?b\n\ncase a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ ?b ≤ lift (Module.rank R M)\n\ncase b\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ Cardinal", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ lift (#ι) ≤ lift (Module.rank R M)", "tactic": "apply le_trans" }, { "state_after": "no goals", "state_before": "case a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ lift (#ι) ≤ ?b", "tactic": "exact Cardinal.lift_mk_le'.mpr ⟨(Equiv.ofInjective _ hv.injective).toEmbedding⟩" }, { "state_after": "case a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι", "state_before": "case a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ lift (#↑(range v)) ≤ lift (Module.rank R M)", "tactic": "simp only [Cardinal.lift_le, Module.rank_def]" }, { "state_after": "case a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ ?a.b✝\n\ncase a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ ?a.b✝ ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι\n\ncase a.b\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ Cardinal", "state_before": "case a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι", "tactic": "apply le_trans" }, { "state_after": "case a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ ?a.b✝ ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι\n\ncase a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ ?a.b✝\n\ncase a.b\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ Cardinal", "state_before": "case a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ ?a.b✝\n\ncase a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ ?a.b✝ ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι\n\ncase a.b\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ Cardinal", "tactic": "swap" }, { "state_after": "no goals", "state_before": "case a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ ?a.b✝ ≤ ⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι", "tactic": "exact le_ciSup (Cardinal.bddAbove_range.{v, v} _) ⟨range v, hv.coe_range⟩" }, { "state_after": "no goals", "state_before": "case a.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.100046\nR : Type u\ninst✝⁷ : Ring R\nM : Type v\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nM' : Type v'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nM₁ : Type v\ninst✝² : AddCommGroup M₁\ninst✝¹ : Module R M₁\ninst✝ : Nontrivial R\nι : Type w\nv : ι → M\nhv : LinearIndependent R v\n⊢ (#↑(range v)) ≤ (#↑↑{ val := range v, property := (_ : LinearIndependent R Subtype.val) })", "tactic": "exact le_rfl" } ]

[ 256, 19 ]

[ 247, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.450946\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\ni : Fin (length (enumFrom n l))\n⊢ ↑i < length l", "tactic": "simpa [length_enumFrom] using i.2" }, { "state_after": "ι : Type ?u.450946\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\ni : Fin (length (enumFrom n l))\nhi : optParam (↑i < length l) (_ : ↑i < length l)\n⊢ get? (enumFrom n l) ↑i = some (n + ↑i, get l { val := ↑i, isLt := hi })", "state_before": "ι : Type ?u.450946\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\ni : Fin (length (enumFrom n l))\nhi : optParam (↑i < length l) (_ : ↑i < length l)\n⊢ get (enumFrom n l) i = (n + ↑i, get l { val := ↑i, isLt := hi })", "tactic": "rw [← Option.some_inj, ← get?_eq_get]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.450946\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\ni : Fin (length (enumFrom n l))\nhi : optParam (↑i < length l) (_ : ↑i < length l)\n⊢ get? (enumFrom n l) ↑i = some (n + ↑i, get l { val := ↑i, isLt := hi })", "tactic": "simp [enumFrom_get?, get?_eq_get hi]" } ]

[ 3947, 39 ]

[ 3943, 1 ]

[ { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\nn_le_m : n ≤ m\nterminated_at_n : TerminatedAt g n\n⊢ continuants g m = continuants g n", "tactic": "simp only [nth_cont_eq_succ_nth_cont_aux,\n continuantsAux_stable_of_terminated (Nat.pred_le_iff.mp n_le_m) terminated_at_n]" } ]

[ 75, 85 ]

[ 72, 1 ]

[]

[ 279, 61 ]

[ 279, 9 ]

[ { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ ↑(thickenedIndicator δ₁_pos E) x ≤ ↑(thickenedIndicator δ₂_pos E) x", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\n⊢ ↑(thickenedIndicator δ₁_pos E) ≤ ↑(thickenedIndicator δ₂_pos E)", "tactic": "intro x" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ thickenedIndicatorAux δ₁ E x ≤ thickenedIndicatorAux δ₂ E x", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ ↑(thickenedIndicator δ₁_pos E) x ≤ ↑(thickenedIndicator δ₂_pos E) x", "tactic": "apply (toNNReal_le_toNNReal thickenedIndicatorAux_lt_top.ne thickenedIndicatorAux_lt_top.ne).mpr" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ thickenedIndicatorAux δ₁ E x ≤ thickenedIndicatorAux δ₂ E x", "tactic": "apply thickenedIndicatorAux_mono hle" } ]

[ 224, 39 ]

[ 220, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.90830\nα : Type u_1\nβ : Type ?u.90836\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a ≤ a \\ b ⊔ b", "tactic": "rw [sup_comm, ← sdiff_le_iff]" } ]

[ 537, 73 ]

[ 537, 1 ]

[ { "state_after": "case inl\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : IsEmpty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))\n\ncase inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "cases isEmpty_or_nonempty ι" }, { "state_after": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "state_before": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "let y := (orthogonalProjection (⨆ i, U i).topologicalClosure x : E)" }, { "state_after": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "state_before": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "have proj_x : ∀ i, orthogonalProjection (U i) x = orthogonalProjection (U i) y := fun i =>\n (orthogonalProjection_orthogonalProjection_of_le\n ((le_iSup U i).trans (iSup U).le_topologicalClosure) _).symm" }, { "state_after": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "suffices ∀ ε > 0, ∃ I, ∀ i ≥ I, ‖(orthogonalProjection (U i) y : E) - y‖ < ε by\n simpa only [proj_x, NormedAddCommGroup.tendsto_atTop] using this" }, { "state_after": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "intro ε hε" }, { "state_after": "case inr.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : dist y a < ε\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "obtain ⟨a, ha, hay⟩ : ∃ a ∈ ⨆ i, U i, dist y a < ε := by\n have y_mem : y ∈ (⨆ i, U i).topologicalClosure := Submodule.coe_mem _\n rw [← SetLike.mem_coe, Submodule.topologicalClosure_coe, Metric.mem_closure_iff] at y_mem\n exact y_mem ε hε" }, { "state_after": "case inr.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : dist y a < ε\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "rw [dist_eq_norm] at hay" }, { "state_after": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "obtain ⟨I, hI⟩ : ∃ I, a ∈ U I := by rwa [Submodule.mem_iSup_of_directed _ hU.directed_le] at ha" }, { "state_after": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "state_before": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\n⊢ ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "refine' ⟨I, fun i (hi : I ≤ i) => _⟩" }, { "state_after": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) < ε", "state_before": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε", "tactic": "rw [norm_sub_rev, orthogonalProjection_minimal]" }, { "state_after": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) ≤ ‖y - a‖", "state_before": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) < ε", "tactic": "refine' lt_of_le_of_lt _ hay" }, { "state_after": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) ≤ ‖y - ↑{ val := a, property := (_ : a ∈ U i) }‖", "state_before": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) ≤ ‖y - a‖", "tactic": "change _ ≤ ‖y - (⟨a, hU hi hI⟩ : U i)‖" }, { "state_after": "no goals", "state_before": "case inr.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\nI : ι\nhI : a ∈ U I\ni : ι\nhi : I ≤ i\n⊢ (⨅ (x : { x // x ∈ U i }), ‖y - ↑x‖) ≤ ‖y - ↑{ val := a, property := (_ : a ∈ U i) }‖", "tactic": "exact ciInf_le ⟨0, Set.forall_range_iff.mpr fun _ => norm_nonneg _⟩ _" }, { "state_after": "case inl\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : IsEmpty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) ⊥\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "state_before": "case inl\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : IsEmpty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "rw [filter_eq_bot_of_isEmpty (atTop : Filter ι)]" }, { "state_after": "no goals", "state_before": "case inl\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : IsEmpty ι\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) ⊥\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "exact tendsto_bot" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nthis : ∀ (ε : ℝ), ε > 0 → ∃ I, ∀ (i : ι), i ≥ I → ‖↑(↑(orthogonalProjection (U i)) y) - y‖ < ε\n⊢ Tendsto (fun i => ↑(↑(orthogonalProjection (U i)) x)) atTop\n (𝓝 ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x))", "tactic": "simpa only [proj_x, NormedAddCommGroup.tendsto_atTop] using this" }, { "state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\ny_mem : y ∈ Submodule.topologicalClosure (⨆ (i : ι), U i)\n⊢ ∃ a, (a ∈ ⨆ (i : ι), U i) ∧ dist y a < ε", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\n⊢ ∃ a, (a ∈ ⨆ (i : ι), U i) ∧ dist y a < ε", "tactic": "have y_mem : y ∈ (⨆ i, U i).topologicalClosure := Submodule.coe_mem _" }, { "state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\ny_mem : ∀ (ε : ℝ), ε > 0 → ∃ b, b ∈ ↑(⨆ (i : ι), U i) ∧ dist y b < ε\n⊢ ∃ a, (a ∈ ⨆ (i : ι), U i) ∧ dist y a < ε", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\ny_mem : y ∈ Submodule.topologicalClosure (⨆ (i : ι), U i)\n⊢ ∃ a, (a ∈ ⨆ (i : ι), U i) ∧ dist y a < ε", "tactic": "rw [← SetLike.mem_coe, Submodule.topologicalClosure_coe, Metric.mem_closure_iff] at y_mem" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\ny_mem : ∀ (ε : ℝ), ε > 0 → ∃ b, b ∈ ↑(⨆ (i : ι), U i) ∧ dist y b < ε\n⊢ ∃ a, (a ∈ ⨆ (i : ι), U i) ∧ dist y a < ε", "tactic": "exact y_mem ε hε" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.850669\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace { x // x ∈ U i }\nhU : Monotone U\nx : E\nh✝ : Nonempty ι\ny : E := ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)\nproj_x : ∀ (i : ι), ↑(orthogonalProjection (U i)) x = ↑(orthogonalProjection (U i)) y\nε : ℝ\nhε : ε > 0\na : E\nha : a ∈ ⨆ (i : ι), U i\nhay : ‖y - a‖ < ε\n⊢ ∃ I, a ∈ U I", "tactic": "rwa [Submodule.mem_iSup_of_directed _ hU.directed_le] at ha" } ]

[ 908, 72 ]

[ 884, 1 ]

[ { "state_after": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.708718\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.711370\nM' : Type ?u.711373\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n⊢ eval₂ (algebraMap R R[X]) p p' • ↑(eval p) (↑(map R[X] (lsingle R 0)) q) =\n eval₂ C p p' • ↑(eval p) (↑(map R[X] (lsingle R 0)) q)", "state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.708718\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.711370\nM' : Type ?u.711373\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n⊢ ↑(comp p) (p' • q) = Polynomial.comp p' p • ↑(comp p) q", "tactic": "rw [comp_apply, map_smul, eval_smul, Polynomial.comp, Polynomial.eval_map, comp_apply]" }, { "state_after": "no goals", "state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.708718\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type ?u.711370\nM' : Type ?u.711373\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n⊢ eval₂ (algebraMap R R[X]) p p' • ↑(eval p) (↑(map R[X] (lsingle R 0)) q) =\n eval₂ C p p' • ↑(eval p) (↑(map R[X] (lsingle R 0)) q)", "tactic": "rfl" } ]

[ 352, 6 ]

[ 349, 1 ]

[]

[ 966, 38 ]

[ 963, 1 ]

[ { "state_after": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j", "state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\n⊢ lp.single p i (-a) = -lp.single p i a", "tactic": "refine' ext (funext (fun (j : α) => _))" }, { "state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\nhi : j = i\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j\n\ncase neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j", "state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j", "tactic": "by_cases hi : j = i" }, { "state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nj : α\na : E j\n⊢ ↑(lp.single p j (-a)) j = ↑(-lp.single p j a) j", "state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\nhi : j = i\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j", "tactic": "subst hi" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nj : α\na : E j\n⊢ ↑(lp.single p j (-a)) j = ↑(-lp.single p j a) j", "tactic": "simp [lp.single_apply_self]" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type ?u.866464\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (-a)) j = ↑(-lp.single p i a) j", "tactic": "simp [lp.single_apply_ne p i _ hi]" } ]

[ 1021, 39 ]

[ 1016, 11 ]

[]

[ 355, 40 ]

[ 352, 1 ]

[]

[ 481, 6 ]

[ 480, 1 ]

[]

[ 1728, 36 ]

[ 1726, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 1\n⊢ ∃ a, s = {a}", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\n⊢ ncard s = 1 ↔ ∃ a, s = {a}", "tactic": "refine' ⟨fun h ↦ _, by rintro ⟨a, rfl⟩; rw [ncard_singleton]⟩" }, { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 1\nhft : Fintype ↑s\n⊢ ∃ a, s = {a}", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 1\n⊢ ∃ a, s = {a}", "tactic": "have hft := (Finite_of_ncard_ne_zero (ne_zero_of_eq_one h)).fintype" }, { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\n⊢ ∃ a, s = {a}", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 1\nhft : Fintype ↑s\n⊢ ∃ a, s = {a}", "tactic": "simp_rw [ncard_eq_toFinset_card', @Finset.card_eq_one _ (toFinset s)] at h" }, { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : toFinset s = {a}\n⊢ s = {a}", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\n⊢ ∃ a, s = {a}", "tactic": "refine' h.imp fun a ha ↦ _" }, { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : toFinset s = {a}\n⊢ ∀ (x : α), x ∈ s ↔ x = a", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : toFinset s = {a}\n⊢ s = {a}", "tactic": "simp_rw [Set.ext_iff, mem_singleton_iff]" }, { "state_after": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : ∀ (a_1 : α), a_1 ∈ s ↔ a_1 = a\n⊢ ∀ (x : α), x ∈ s ↔ x = a", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : toFinset s = {a}\n⊢ ∀ (x : α), x ∈ s ↔ x = a", "tactic": "simp only [Finset.ext_iff, mem_toFinset, Finset.mem_singleton] at ha" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na✝ b x y : α\nf : α → β\nhft : Fintype ↑s\nh : ∃ a, toFinset s = {a}\na : α\nha : ∀ (a_1 : α), a_1 ∈ s ↔ a_1 = a\n⊢ ∀ (x : α), x ∈ s ↔ x = a", "tactic": "exact ha" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.138753\nt : Set α\na✝ b x y : α\nf : α → β\na : α\n⊢ ncard {a} = 1", "state_before": "α : Type u_1\nβ : Type ?u.138753\ns t : Set α\na b x y : α\nf : α → β\n⊢ (∃ a, s = {a}) → ncard s = 1", "tactic": "rintro ⟨a, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.138753\nt : Set α\na✝ b x y : α\nf : α → β\na : α\n⊢ ncard {a} = 1", "tactic": "rw [ncard_singleton]" } ]

[ 644, 11 ]

[ 637, 9 ]

[ { "state_after": "no goals", "state_before": "r : ℝ≥0\np : ℝ\nhp : 0 ≤ p\n⊢ r ≤ toNNReal p ↔ ↑r ≤ p", "tactic": "rw [← NNReal.coe_le_coe, Real.coe_toNNReal p hp]" } ]

[ 678, 51 ]

[ 677, 1 ]

[ { "state_after": "case ofFractionRing\nK : Type u\ninst✝¹ : CommRing K\nR : Type u_1\ninst✝ : SMul R (FractionRing K[X])\nc : R\ntoFractionRing✝ : FractionRing K[X]\n⊢ (c • { toFractionRing := toFractionRing✝ }).toFractionRing = c • { toFractionRing := toFractionRing✝ }.toFractionRing", "state_before": "K : Type u\ninst✝¹ : CommRing K\nR : Type u_1\ninst✝ : SMul R (FractionRing K[X])\nc : R\np : RatFunc K\n⊢ (c • p).toFractionRing = c • p.toFractionRing", "tactic": "cases p" }, { "state_after": "no goals", "state_before": "case ofFractionRing\nK : Type u\ninst✝¹ : CommRing K\nR : Type u_1\ninst✝ : SMul R (FractionRing K[X])\nc : R\ntoFractionRing✝ : FractionRing K[X]\n⊢ (c • { toFractionRing := toFractionRing✝ }).toFractionRing = c • { toFractionRing := toFractionRing✝ }.toFractionRing", "tactic": "rw [← ofFractionRing_smul]" } ]

[ 455, 29 ]

[ 452, 1 ]

[]

[ 506, 19 ]

[ 505, 1 ]

[]

[ 116, 87 ]

[ 114, 1 ]

[]

[ 221, 58 ]

[ 220, 1 ]

[ { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ p %ₘ q = p", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\n⊢ p %ₘ q = p", "tactic": "have : ¬degree q ≤ degree p := not_le_of_gt h" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ p %ₘ q = p", "tactic": "unfold modByMonic divModByMonicAux" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).snd\n else p) =\n p", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).snd\n else p) =\n p", "tactic": "rw [dif_pos hq, if_neg (mt And.left this)]" } ]

[ 217, 91 ]

[ 214, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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⊢ f'' =o[⊤] g'' → ∀ (x : α), f'' x = 0", "state_before": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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⊢ f'' =o[⊤] g'' ↔ ∀ (x : α), f'' x = 0", "tactic": "refine' ⟨_, fun h => (isLittleO_zero g'' ⊤).congr (fun x => (h x).symm) fun x => rfl⟩" }, { "state_after": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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 → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖) → ∀ (x : α), f'' x = 0", "state_before": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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⊢ f'' =o[⊤] g'' → ∀ (x : α), f'' x = 0", "tactic": "simp only [isLittleO_iff, eventually_top]" }, { "state_after": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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\nh : ∀ ⦃c : ℝ⦄, 0 < c → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖\nx : α\n⊢ ‖f'' x‖ ≤ 0", "state_before": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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 → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖) → ∀ (x : α), f'' x = 0", "tactic": "refine' fun h x => norm_le_zero_iff.1 _" }, { "state_after": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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\nh : ∀ ⦃c : ℝ⦄, 0 < c → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖\nx : α\nthis : Tendsto (fun c => c * ‖g'' x‖) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ‖f'' x‖ ≤ 0", "state_before": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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\nh : ∀ ⦃c : ℝ⦄, 0 < c → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖\nx : α\n⊢ ‖f'' x‖ ≤ 0", "tactic": "have : Tendsto (fun c : ℝ => c * ‖g'' x‖) (𝓝[>] 0) (𝓝 0) :=\n ((continuous_id.mul continuous_const).tendsto' _ _ (MulZeroClass.zero_mul _)).mono_left\n inf_le_left" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.285927\nE : Type ?u.285930\nF : Type ?u.285933\nG : Type ?u.285936\nE' : Type ?u.285939\nF' : Type ?u.285942\nG' : Type ?u.285945\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.285954\nR : Type ?u.285957\nR' : Type ?u.285960\n𝕜 : Type ?u.285963\n𝕜' : Type ?u.285966\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\nh : ∀ ⦃c : ℝ⦄, 0 < c → ∀ (x : α), ‖f'' x‖ ≤ c * ‖g'' x‖\nx : α\nthis : Tendsto (fun c => c * ‖g'' x‖) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ‖f'' x‖ ≤ 0", "tactic": "exact\n le_of_tendsto_of_tendsto tendsto_const_nhds this\n (eventually_nhdsWithin_iff.2 <| eventually_of_forall fun c hc => h hc x)" } ]

[ 1291, 79 ]

[ 1282, 1 ]

[ { "state_after": "G : Type u\ninst✝ : Monoid G\n⊢ ExponentExists G ↔ (if h : ExponentExists G then Nat.find h else 0) ≠ 0", "state_before": "G : Type u\ninst✝ : Monoid G\n⊢ ExponentExists G ↔ exponent G ≠ 0", "tactic": "rw [exponent]" }, { "state_after": "case inl\nG : Type u\ninst✝ : Monoid G\nh : ExponentExists G\n⊢ ExponentExists G ↔ Nat.find h ≠ 0\n\ncase inr\nG : Type u\ninst✝ : Monoid G\nh : ¬ExponentExists G\n⊢ ExponentExists G ↔ 0 ≠ 0", "state_before": "G : Type u\ninst✝ : Monoid G\n⊢ ExponentExists G ↔ (if h : ExponentExists G then Nat.find h else 0) ≠ 0", "tactic": "split_ifs with h" }, { "state_after": "no goals", "state_before": "case inl\nG : Type u\ninst✝ : Monoid G\nh : ExponentExists G\n⊢ ExponentExists G ↔ Nat.find h ≠ 0", "tactic": "simp [h, @not_lt_zero' ℕ]" }, { "state_after": "no goals", "state_before": "case inr\nG : Type u\ninst✝ : Monoid G\nh : ¬ExponentExists G\n⊢ ExponentExists G ↔ 0 ≠ 0", "tactic": "tauto" } ]

[ 86, 10 ]

[ 81, 1 ]

[]

[ 2442, 71 ]

[ 2440, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ns : Set α\nx✝ : MeasurableSet s\nt : Set α\nht : t = ∅\n⊢ ↑⊤ (t ∩ s) + ↑⊤ (t \\ s) ≤ ↑⊤ t", "tactic": "simp [ht]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ns : Set α\nx✝ : MeasurableSet s\nt : Set α\nht : Set.Nonempty t\n⊢ ↑⊤ (t ∩ s) + ↑⊤ (t \\ s) ≤ ↑⊤ t", "tactic": "simp only [ht, top_apply, le_top]" } ]

[ 1106, 42 ]

[ 1102, 1 ]

[ { "state_after": "case h.e'_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ f = fun x => ⨆ (n : ℕ), ↑(eapprox f n) x\n\nα : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (n : ℕ), P ↑(eapprox f n)", "state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ P f", "tactic": "convert h_iSup (fun n => (eapprox f n).measurable) (monotone_eapprox f) _ using 1" }, { "state_after": "case h.e'_1.h\nα : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\nx : α\n⊢ f x = ⨆ (n : ℕ), ↑(eapprox f n) x", "state_before": "case h.e'_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ f = fun x => ⨆ (n : ℕ), ↑(eapprox f n) x", "tactic": "ext1 x" }, { "state_after": "no goals", "state_before": "case h.e'_1.h\nα : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\nx : α\n⊢ f x = ⨆ (n : ℕ), ↑(eapprox f n) x", "tactic": "rw [iSup_eapprox_apply f hf]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nP : (α → ℝ≥0∞) → Prop\nh_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g)\nh_iSup :\n ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ (n : ℕ), f n x\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (n : ℕ), P ↑(eapprox f n)", "tactic": "exact fun n =>\n SimpleFunc.induction (fun c s hs => h_ind c hs)\n (fun f g hfg hf hg => h_add hfg f.measurable g.measurable hf hg) (eapprox f n)" } ]

[ 1351, 87 ]

[ 1337, 1 ]

[]

[ 2680, 65 ]

[ 2678, 1 ]

[ { "state_after": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhm : testBit m i = true\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\n⊢ 0 < m\n\ncase f\nn✝ m✝ i✝ : ℕ\nhn✝ : testBit n✝ i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm i : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit m i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j\n⊢ bit b n < m", "state_before": "n m i : ℕ\nhn : testBit n i = false\nhm : testBit m i = true\nhnm : ∀ (j : ℕ), i < j → testBit n j = testBit m j\n⊢ n < m", "tactic": "induction' n using Nat.binaryRec with b n hn' generalizing i m" }, { "state_after": "case f.z\nn✝ m✝ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m j\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit 0 i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit 0 j\n⊢ bit b n < 0\n\ncase f.f\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\n⊢ bit b n < bit b' m", "state_before": "case f\nn✝ m✝ i✝ : ℕ\nhn✝ : testBit n✝ i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm i : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit m i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j\n⊢ bit b n < m", "tactic": "induction' m using Nat.binaryRec with b' m hm' generalizing i" }, { "state_after": "case pos\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : i = 0\n⊢ bit b n < bit b' m\n\ncase neg\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬i = 0\n⊢ bit b n < bit b' m", "state_before": "case f.f\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\n⊢ bit b n < bit b' m", "tactic": "by_cases hi : i = 0" }, { "state_after": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\nhm : m ≤ 0\n⊢ testBit m i ≠ true", "state_before": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhm : testBit m i = true\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\n⊢ 0 < m", "tactic": "contrapose! hm" }, { "state_after": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\nhm : m = 0\n⊢ testBit m i ≠ true", "state_before": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\nhm : m ≤ 0\n⊢ testBit m i ≠ true", "tactic": "rw [le_zero_iff] at hm" }, { "state_after": "no goals", "state_before": "case z\nn m✝ i✝ : ℕ\nhn✝ : testBit n i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j\nm i : ℕ\nhn : testBit 0 i = false\nhnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j\nhm : m = 0\n⊢ testBit m i ≠ true", "tactic": "simp [hm]" }, { "state_after": "no goals", "state_before": "case f.z\nn✝ m✝ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m j\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit 0 i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit 0 j\n⊢ bit b n < 0", "tactic": "exact False.elim (Bool.ff_ne_tt ((zero_testBit i).symm.trans hm))" }, { "state_after": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhn : testBit (bit b n) 0 = false\nhm : testBit (bit b' m) 0 = true\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\n⊢ bit b n < bit b' m", "state_before": "case pos\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : i = 0\n⊢ bit b n < bit b' m", "tactic": "subst hi" }, { "state_after": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\n⊢ bit b n < bit b' m", "state_before": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhn : testBit (bit b n) 0 = false\nhm : testBit (bit b' m) 0 = true\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\n⊢ bit b n < bit b' m", "tactic": "simp only [testBit_zero] at hn hm" }, { "state_after": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\nthis : n = m\n⊢ bit b n < bit b' m", "state_before": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\n⊢ bit b n < bit b' m", "tactic": "have : n = m :=\n eq_of_testBit_eq fun i => by convert hnm (i + 1) (Nat.zero_lt_succ _) using 1\n <;> rw [testBit_succ]" }, { "state_after": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\nthis : n = m\n⊢ 2 * m < 2 * m + 1", "state_before": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\nthis : n = m\n⊢ bit b n < bit b' m", "tactic": "rw [hn, hm, this, bit_false, bit_true, bit0_val, bit1_val]" }, { "state_after": "no goals", "state_before": "case pos\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\nthis : n = m\n⊢ 2 * m < 2 * m + 1", "tactic": "exact lt_add_one _" }, { "state_after": "no goals", "state_before": "n✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\nhnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j\nhn : b = false\nhm : b' = true\ni : ℕ\n⊢ testBit n i = testBit m i", "tactic": "convert hnm (i + 1) (Nat.zero_lt_succ _) using 1\n<;> rw [testBit_succ]" }, { "state_after": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhn : testBit (bit b n) (succ i') = false\nhm : testBit (bit b' m) (succ i') = true\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\n⊢ bit b n < bit b' m", "state_before": "case neg\nn✝ m✝¹ i✝¹ : ℕ\nhn✝¹ : testBit n✝ i✝¹ = false\nhm✝¹ : testBit m✝¹ i✝¹ = true\nhnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i✝ : ℕ\nhn✝ : testBit (bit b n) i✝ = false\nhm✝ : testBit m✝ i✝ = true\nhnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni : ℕ\nhn : testBit (bit b n) i = false\nhm : testBit (bit b' m) i = true\nhnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬i = 0\n⊢ bit b n < bit b' m", "tactic": "obtain ⟨i', rfl⟩ := exists_eq_succ_of_ne_zero hi" }, { "state_after": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\nhn : testBit n i' = false\nhm : testBit m i' = true\n⊢ bit b n < bit b' m", "state_before": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhn : testBit (bit b n) (succ i') = false\nhm : testBit (bit b' m) (succ i') = true\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\n⊢ bit b n < bit b' m", "tactic": "simp only [testBit_succ] at hn hm" }, { "state_after": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\nhn : testBit n i' = false\nhm : testBit m i' = true\nthis : n < m\n⊢ bit b n < bit b' m", "state_before": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\nhn : testBit n i' = false\nhm : testBit m i' = true\n⊢ bit b n < bit b' m", "tactic": "have :=\n hn' _ hn hm fun j hj => by convert hnm j.succ (succ_lt_succ hj) using 1 <;> rw [testBit_succ]" }, { "state_after": "no goals", "state_before": "case neg.intro\nn✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\nhn : testBit n i' = false\nhm : testBit m i' = true\nthis : n < m\n⊢ bit b n < bit b' m", "tactic": "cases b <;> cases b'\n<;> simp only [bit_false, bit_true, bit0_val n, bit1_val n, bit0_val m, bit1_val m]\n<;> linarith only [this]" }, { "state_after": "no goals", "state_before": "n✝ m✝¹ i✝ : ℕ\nhn✝¹ : testBit n✝ i✝ = false\nhm✝¹ : testBit m✝¹ i✝ = true\nhnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j\nb : Bool\nn : ℕ\nhn' :\n ∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m\nm✝ i : ℕ\nhn✝ : testBit (bit b n) i = false\nhm✝ : testBit m✝ i = true\nhnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j\nb' : Bool\nm : ℕ\nhm' :\n ∀ (i : ℕ),\n testBit (bit b n) i = false →\n testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m\ni' : ℕ\nhnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j\nhi : ¬succ i' = 0\nhn : testBit n i' = false\nhm : testBit m i' = true\nj : ℕ\nhj : i' < j\n⊢ testBit n j = testBit m j", "tactic": "convert hnm j.succ (succ_lt_succ hj) using 1 <;> rw [testBit_succ]" } ]

[ 138, 29 ]

[ 116, 1 ]

[]

[ 833, 41 ]

[ 832, 1 ]

[ { "state_after": "M : Type u_2\nN : Type ?u.21456\nA : Type ?u.21459\ninst✝¹ : Mul M\ns : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\nι : Sort u_1\np : ι → Subsemigroup M\nm : M\n⊢ (∀ (b : Subsemigroup M), (∀ (i : ι), p i ≤ b) → closure {m} ≤ b) ↔\n ∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N", "state_before": "M : Type u_2\nN : Type ?u.21456\nA : Type ?u.21459\ninst✝¹ : Mul M\ns : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\nι : Sort u_1\np : ι → Subsemigroup M\nm : M\n⊢ (m ∈ ⨆ (i : ι), p i) ↔ ∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N", "tactic": "rw [← closure_singleton_le_iff_mem, le_iSup_iff]" }, { "state_after": "no goals", "state_before": "M : Type u_2\nN : Type ?u.21456\nA : Type ?u.21459\ninst✝¹ : Mul M\ns : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\nι : Sort u_1\np : ι → Subsemigroup M\nm : M\n⊢ (∀ (b : Subsemigroup M), (∀ (i : ι), p i ≤ b) → closure {m} ≤ b) ↔\n ∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N", "tactic": "simp only [closure_singleton_le_iff_mem]" } ]

[ 454, 43 ]

[ 451, 1 ]

[ { "state_after": "no goals", "state_before": "J : Type w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nP Q X Y : C\ninst✝ : HasBinaryBiproduct X Y\n⊢ (isoProd X Y).inv = lift prod.fst prod.snd", "tactic": "apply biprod.hom_ext <;> simp [Iso.inv_comp_eq]" } ]

[ 1448, 50 ]

[ 1446, 1 ]

[]

[ 527, 19 ]

[ 526, 1 ]

[]

[ 1531, 25 ]

[ 1529, 1 ]

[]

[ 278, 57 ]

[ 277, 1 ]

[]

[ 357, 16 ]

[ 356, 1 ]

[ { "state_after": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.83491\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\ns : Set α\nx : α\nhx : x ∈ s\n⊢ (fun x => (f x, g x)) x ∈ (f '' s) ×ˢ (g '' s)", "state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.83491\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\ns : Set α\n⊢ (fun x => (f x, g x)) '' s ⊆ (f '' s) ×ˢ (g '' s)", "tactic": "rintro _ ⟨x, hx, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.83491\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\ns : Set α\nx : α\nhx : x ∈ s\n⊢ (fun x => (f x, g x)) x ∈ (f '' s) ×ˢ (g '' s)", "tactic": "exact mk_mem_prod (mem_image_of_mem f hx) (mem_image_of_mem g hx)" } ]

[ 335, 68 ]

[ 332, 1 ]

[]

[ 314, 70 ]

[ 312, 1 ]

[ { "state_after": "ι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ (∀ (a : α), a ∈ m → 0 ≤ a) → 0 ≤ prod m", "state_before": "ι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\nh : ∀ (a : α), a ∈ m → 0 ≤ a\n⊢ 0 ≤ prod m", "tactic": "revert h" }, { "state_after": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ (∀ (a : α), a ∈ 0 → 0 ≤ a) → 0 ≤ prod 0\n\ncase refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ ∀ ⦃a : α⦄ {s : Multiset α},\n ((∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s) → (∀ (a_2 : α), a_2 ∈ a ::ₘ s → 0 ≤ a_2) → 0 ≤ prod (a ::ₘ s)", "state_before": "ι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ (∀ (a : α), a ∈ m → 0 ≤ a) → 0 ≤ prod m", "tactic": "refine' m.induction_on _ _" }, { "state_after": "case refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\na : α\ns : Multiset α\nhs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s\nih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1\n⊢ 0 ≤ prod (a ::ₘ s)", "state_before": "case refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ ∀ ⦃a : α⦄ {s : Multiset α},\n ((∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s) → (∀ (a_2 : α), a_2 ∈ a ::ₘ s → 0 ≤ a_2) → 0 ≤ prod (a ::ₘ s)", "tactic": "intro a s hs ih" }, { "state_after": "case refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\na : α\ns : Multiset α\nhs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s\nih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1\n⊢ 0 ≤ a * prod s", "state_before": "case refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\na : α\ns : Multiset α\nhs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s\nih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1\n⊢ 0 ≤ prod (a ::ₘ s)", "tactic": "rw [prod_cons]" }, { "state_after": "no goals", "state_before": "case refine'_2\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\na : α\ns : Multiset α\nhs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s\nih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1\n⊢ 0 ≤ a * prod s", "tactic": "exact mul_nonneg (ih _ <| mem_cons_self _ _) (hs fun a ha => ih _ <| mem_cons_of_mem ha)" }, { "state_after": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ 0 ≤ prod 0", "state_before": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ (∀ (a : α), a ∈ 0 → 0 ≤ a) → 0 ≤ prod 0", "tactic": "rintro -" }, { "state_after": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ 0 ≤ 1", "state_before": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ 0 ≤ prod 0", "tactic": "rw [prod_zero]" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type ?u.134844\nα : Type u_1\nβ : Type ?u.134850\nγ : Type ?u.134853\ninst✝ : OrderedCommSemiring α\nm : Multiset α\n⊢ 0 ≤ 1", "tactic": "exact zero_le_one" } ]

[ 441, 91 ]

[ 432, 1 ]

[]

[ 1889, 28 ]

[ 1888, 11 ]

[ { "state_after": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhK : IsPGroup p { x // x ∈ K }\nhHK : H ≤ K\na b : { x // x ∈ H }\nh : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b\n⊢ ↑(↑(Subgroup.inclusion hHK) a) = ↑(↑(Subgroup.inclusion hHK) b)", "state_before": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhK : IsPGroup p { x // x ∈ K }\nhHK : H ≤ K\na b : { x // x ∈ H }\nh : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b\n⊢ ↑a = ↑b", "tactic": "change ((Subgroup.inclusion hHK) a : G) = (Subgroup.inclusion hHK) b" }, { "state_after": "no goals", "state_before": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhK : IsPGroup p { x // x ∈ K }\nhHK : H ≤ K\na b : { x // x ∈ H }\nh : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b\n⊢ ↑(↑(Subgroup.inclusion hHK) a) = ↑(↑(Subgroup.inclusion hHK) b)", "tactic": "apply Subtype.ext_iff.mp h" } ]

[ 273, 34 ]

[ 269, 1 ]

[ { "state_after": "no goals", "state_before": "q : ℚ\nm : ℤ\nn : ℕ\nx : ℝ\n⊢ Irrational (↑n * x) ↔ n ≠ 0 ∧ Irrational x", "tactic": "rw [← cast_coe_nat, irrational_rat_mul_iff, Nat.cast_ne_zero]" } ]

[ 624, 64 ]

[ 623, 1 ]

[ { "state_after": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\n⊢ NormalSpace X", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\n⊢ NormalSpace X", "tactic": "have : ∀ s t : Set X, IsClosed s → IsClosed t →\n (∀ x ∈ s, ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := by\n \n intro s t hs _ H\n choose u v hu hv hxu htv huv using SetCoe.forall'.1 H\n rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with\n ⟨u', hu'o, hcov', hu'fin, hsub⟩\n refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,\n hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩\n rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]\n refine' fun i x hxt hxu ↦\n absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)\n exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩" }, { "state_after": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\n⊢ NormalSpace X", "tactic": "refine' ⟨fun s t hs ht hst ↦ this s t hs ht fun x hx ↦ _⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\nv u : Set X\nhv : IsOpen v\nhu : IsOpen u\nhtv : t ⊆ v\nhxu : {x} ⊆ u\nhuv : Disjoint v u\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "rcases this t {x} ht isClosed_singleton fun y hy ↦ (by\n simp_rw [singleton_subset_iff]\n exact t2_separation (hst.symm.ne_of_mem hy hx))\n with ⟨v, u, hv, hu, htv, hxu, huv⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\nv u : Set X\nhv : IsOpen v\nhu : IsOpen u\nhtv : t ⊆ v\nhxu : {x} ⊆ u\nhuv : Disjoint v u\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "exact ⟨u, v, hu, hv, singleton_subset_iff.1 hxu, htv, huv.symm⟩" }, { "state_after": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\n⊢ ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "intro s t hs _ H" }, { "state_after": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "choose u v hu hv hxu htv huv using SetCoe.forall'.1 H" }, { "state_after": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with\n ⟨u', hu'o, hcov', hu'fin, hsub⟩" }, { "state_after": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ t ⊆ closure (⋃ (i : ↑s), u' i)ᶜ", "state_before": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v", "tactic": "refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,\n hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩" }, { "state_after": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ ∀ (i : ↑s), t ⊆ closure (u' i)ᶜ", "state_before": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ t ⊆ closure (⋃ (i : ↑s), u' i)ᶜ", "tactic": "rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]" }, { "state_after": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu✝ : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\ni : ↑s\nx : X\nhxt : x ∈ t\nhxu : x ∈ closure (u' i)\n⊢ u' i ⊆ v iᶜ", "state_before": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\n⊢ ∀ (i : ↑s), t ⊆ closure (u' i)ᶜ", "tactic": "refine' fun i x hxt hxu ↦\n absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro\nι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\ns t : Set X\nhs : IsClosed s\na✝ : IsClosed t\nH : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v\nu v : ↑s → Set X\nhu : ∀ (x : ↑s), IsOpen (u x)\nhv : ∀ (x : ↑s), IsOpen (v x)\nhxu✝ : ∀ (x : ↑s), ↑x ∈ u x\nhtv : ∀ (x : ↑s), t ⊆ v x\nhuv : ∀ (x : ↑s), Disjoint (u x) (v x)\nu' : ↑s → Set X\nhu'o : ∀ (i : ↑s), IsOpen (u' i)\nhcov' : s ⊆ ⋃ (i : ↑s), u' i\nhu'fin : LocallyFinite u'\nhsub : ∀ (i : ↑s), u' i ⊆ u i\ni : ↑s\nx : X\nhxt : x ∈ t\nhxu : x ∈ closure (u' i)\n⊢ u' i ⊆ v iᶜ", "tactic": "exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩" }, { "state_after": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\ny : X\nhy : y ∈ t\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ x ∈ v ∧ Disjoint u v", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\ny : X\nhy : y ∈ t\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ {x} ⊆ v ∧ Disjoint u v", "tactic": "simp_rw [singleton_subset_iff]" }, { "state_after": "no goals", "state_before": "ι : Type u\nX : Type v\nY : Type w\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : ParacompactSpace X\nthis :\n ∀ (s t : Set X),\n IsClosed s →\n IsClosed t →\n (∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →\n ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nx : X\nhx : x ∈ s\ny : X\nhy : y ∈ t\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ x ∈ v ∧ Disjoint u v", "tactic": "exact t2_separation (hst.symm.ne_of_mem hy hx)" } ]

[ 323, 66 ]

[ 299, 1 ]

[ { "state_after": "case inl\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn p : R\nk : ℕ\nh : Nat.minFac 0 ^ ↑(Nat.factorization 0) (Nat.minFac 0) = 0\nhn : 0 ≠ 1\n⊢ IsPrimePow 0\n\ncase inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ IsPrimePow n", "state_before": "R : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\n⊢ IsPrimePow n", "tactic": "rcases eq_or_ne n 0 with (rfl | hn')" }, { "state_after": "case inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ 0 < ↑(Nat.factorization n) (Nat.minFac n)", "state_before": "case inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ IsPrimePow n", "tactic": "refine' ⟨_, _, (Nat.minFac_prime hn).prime, _, h⟩" }, { "state_after": "case inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ Nat.minFac n ∣ n", "state_before": "case inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ 0 < ↑(Nat.factorization n) (Nat.minFac n)", "tactic": "rw [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.factor_iff_mem_factorization,\n Nat.mem_factors_iff_dvd hn' (Nat.minFac_prime hn)]" }, { "state_after": "no goals", "state_before": "case inr\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n : ℕ\nh : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n\nhn : n ≠ 1\nhn' : n ≠ 0\n⊢ Nat.minFac n ∣ n", "tactic": "apply Nat.minFac_dvd" }, { "state_after": "no goals", "state_before": "case inl\nR : Type ?u.706\ninst✝ : CommMonoidWithZero R\nn p : R\nk : ℕ\nh : Nat.minFac 0 ^ ↑(Nat.factorization 0) (Nat.minFac 0) = 0\nhn : 0 ≠ 1\n⊢ IsPrimePow 0", "tactic": "simp_all" } ]

[ 37, 23 ]

[ 30, 1 ]

[]

[ 336, 100 ]

[ 333, 1 ]

[ { "state_after": "case zero\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\n⊢ smulRight 1 c ^ Nat.zero = smulRight 1 (c ^ Nat.zero)\n\ncase succ\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\nn : ℕ\nihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)\n⊢ smulRight 1 c ^ Nat.succ n = smulRight 1 (c ^ Nat.succ n)", "state_before": "R : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\nn : ℕ\n⊢ smulRight 1 c ^ n = smulRight 1 (c ^ n)", "tactic": "induction' n with n ihn" }, { "state_after": "case zero.h\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\n⊢ ↑(smulRight 1 c ^ Nat.zero) 1 = ↑(smulRight 1 (c ^ Nat.zero)) 1", "state_before": "case zero\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\n⊢ smulRight 1 c ^ Nat.zero = smulRight 1 (c ^ Nat.zero)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case zero.h\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\n⊢ ↑(smulRight 1 c ^ Nat.zero) 1 = ↑(smulRight 1 (c ^ Nat.zero)) 1", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case succ\nR : Type u_1\ninst✝¹⁵ : Ring R\nR₂ : Type ?u.888280\ninst✝¹⁴ : Ring R₂\nR₃ : Type ?u.888286\ninst✝¹³ : Ring R₃\nM : Type ?u.888292\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : AddCommGroup M\nM₂ : Type ?u.888301\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommGroup M₂\nM₃ : Type ?u.888310\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommGroup M₃\nM₄ : Type ?u.888319\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalRing R\nc : R\nn : ℕ\nihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)\n⊢ smulRight 1 c ^ Nat.succ n = smulRight 1 (c ^ Nat.succ n)", "tactic": "rw [pow_succ, ihn, mul_def, smulRight_comp, smul_eq_mul, pow_succ']" } ]

[ 1437, 72 ]

[ 1432, 1 ]

[ { "state_after": "no goals", "state_before": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ns t : Set Ω\ninst✝ : IsProbabilityMeasure μ\n⊢ μ[|Set.univ] = μ", "tactic": "simp [cond, measure_univ, Measure.restrict_univ]" } ]

[ 101, 51 ]

[ 100, 1 ]

[ { "state_after": "no goals", "state_before": "p a b✝ : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\ncpm : coprime p m\nb : ℕ\ndvd : m ∣ b\n⊢ padicValNat p (b / m) = padicValNat p b", "tactic": "rw [padicValNat.div_of_dvd dvd, eq_zero_of_not_dvd (hp.out.coprime_iff_not_dvd.mp cpm),\n Nat.sub_zero]" } ]

[ 442, 18 ]

[ 439, 11 ]

[]

[ 546, 83 ]

[ 544, 1 ]

[]

[ 1465, 25 ]

[ 1463, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.197269\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ico a b \\ Iio c = Ico (max a c) b", "tactic": "rw [diff_eq, compl_Iio, Ico_inter_Ici, sup_eq_max]" } ]

[ 1816, 53 ]

[ 1815, 1 ]

[]

[ 539, 66 ]

[ 539, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.74361\nγ : Type ?u.74364\nδ : Type ?u.74367\nm : MeasurableSpace α\nμ ν : Measure α\n⊢ (∫⁻ (x : α), 0 ∂μ) = 0", "tactic": "simp" } ]

[ 171, 57 ]

[ 171, 1 ]

[ { "state_after": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\n⊢ adjugate (A ^ Nat.zero) = adjugate A ^ Nat.zero\n\ncase succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nk : ℕ\nIH : adjugate (A ^ k) = adjugate A ^ k\n⊢ adjugate (A ^ Nat.succ k) = adjugate A ^ Nat.succ k", "state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nk : ℕ\n⊢ adjugate (A ^ k) = adjugate A ^ k", "tactic": "induction' k with k IH" }, { "state_after": "no goals", "state_before": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\n⊢ adjugate (A ^ Nat.zero) = adjugate A ^ Nat.zero", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nk : ℕ\nIH : adjugate (A ^ k) = adjugate A ^ k\n⊢ adjugate (A ^ Nat.succ k) = adjugate A ^ Nat.succ k", "tactic": "rw [pow_succ', mul_eq_mul, adjugate_mul_distrib, IH, ← mul_eq_mul, pow_succ]" } ]

[ 514, 81 ]

[ 511, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_2\nR₁ : Type u_1\nR₂ : Type u_5\nR₃ : Type ?u.540765\nM : Type ?u.540768\nM₁ : Type u_3\nM₂ : Type u_4\nMₗ₁ : Type ?u.540777\nMₗ₁' : Type ?u.540780\nMₗ₂ : Type ?u.540783\nMₗ₂' : Type ?u.540786\nK : Type ?u.540789\nK₁ : Type ?u.540792\nK₂ : Type ?u.540795\nV : Type ?u.540798\nV₁ : Type ?u.540801\nV₂ : Type ?u.540804\nn : Type ?u.540807\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nI₁ : R₁ →+* R\nI₂ : R₂ →+* R\nI₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R\n⊢ SeparatingLeft (flip B) ↔ SeparatingRight B", "tactic": "rw [← flip_separatingRight, flip_flip]" } ]

[ 692, 91 ]

[ 691, 1 ]

[]

[ 726, 28 ]

[ 724, 1 ]

[]

[ 999, 30 ]

[ 997, 1 ]

[]

[ 40, 47 ]

[ 39, 1 ]

[]

[ 768, 6 ]

[ 767, 1 ]

[ { "state_after": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\n⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀", "state_before": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\n⊢ 0 < ↑(Content.outerMeasure (haarContent K₀)) ↑K₀", "tactic": "refine' zero_lt_one.trans_le _" }, { "state_after": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\n⊢ 1 ≤\n ⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }", "state_before": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\n⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀", "tactic": "rw [Content.outerMeasure_eq_iInf]" }, { "state_after": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\nU : Set G\nhU : IsOpen U\nhK₀ : ↑K₀ ⊆ U\n⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts", "state_before": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\n⊢ 1 ≤\n ⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }", "tactic": "refine' le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans _ <| le_iSup₂ K₀.toCompacts hK₀" }, { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : T2Space G\nK₀ : PositiveCompacts G\nU : Set G\nhU : IsOpen U\nhK₀ : ↑K₀ ⊆ U\n⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts", "tactic": "exact haarContent_self.ge" } ]

[ 586, 28 ]

[ 581, 1 ]

[ { "state_after": "c : Cardinal\n⊢ (∀ (o' : Ordinal) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c", "state_before": "c : Cardinal\n⊢ aleph' ω ≤ c ↔ ℵ₀ ≤ c", "tactic": "simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le]" }, { "state_after": "no goals", "state_before": "c : Cardinal\n⊢ (∀ (o' : Ordinal) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c", "tactic": "exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat])" }, { "state_after": "no goals", "state_before": "c : Cardinal\nn : ℕ\n⊢ (∀ (x : Ordinal), x = ↑n → aleph' x ≤ c) ↔ ↑n ≤ c", "tactic": "simp only [forall_eq, aleph'_nat]" } ]

[ 234, 90 ]

[ 231, 1 ]

[ { "state_after": "no goals", "state_before": "R✝ : Type ?u.43695\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑n = 0 ↔ addOrderOf 1 ∣ n", "tactic": "rw [← Nat.smul_one_eq_coe, addOrderOf_dvd_iff_nsmul_eq_zero]" } ]

[ 130, 77 ]

[ 129, 1 ]

[]

[ 234, 40 ]

[ 233, 1 ]

[ { "state_after": "case h.e'_2.h.a\nα : Type ?u.349869\nβ : Type ?u.349872\nγ : Type ?u.349875\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\na✝ : Ordinal\n⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a", "state_before": "α : Type ?u.349869\nβ : Type ?u.349872\nγ : Type ?u.349875\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\n⊢ blsub o f ≤ a ↔ ∀ (i : Ordinal) (h : i < o), f i h < a", "tactic": "convert bsup_le_iff.{_, v} (f := fun a ha => succ (f a ha)) (a := a) using 2" }, { "state_after": "no goals", "state_before": "case h.e'_2.h.a\nα : Type ?u.349869\nβ : Type ?u.349872\nγ : Type ?u.349875\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\na✝ : Ordinal\n⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a", "tactic": "simp_rw [succ_le_iff]" } ]

[ 1808, 24 ]

[ 1805, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.329647\nδ : Type ?u.329650\nι : Type ?u.329653\nR : Type ?u.329656\nR' : Type ?u.329659\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : MeasurableSpace α\nf : α → β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f (dirac a)) s = ↑↑(dirac (f a)) s", "tactic": "simp [hs, map_apply hf hs, hf hs, indicator_apply]" } ]

[ 2005, 72 ]

[ 2004, 1 ]

[ { "state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ ↑(formPerm l) (nthLe l k hk) = nthLe l k hk ↔ length l ≤ 1", "state_before": "α : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l) x = x ↔ length l ≤ 1", "tactic": "obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx" }, { "state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ (k + 1) % length l = k ↔ length l ≤ 1", "state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ ↑(formPerm l) (nthLe l k hk) = nthLe l k hk ↔ length l ≤ 1", "tactic": "rw [formPerm_apply_nthLe _ hl, hl.nthLe_inj_iff]" }, { "state_after": "case intro.intro.zero\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nhn : length l = Nat.zero\n⊢ (k + 1) % Nat.zero = k ↔ Nat.zero ≤ 1\n\ncase intro.intro.succ\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn✝ : ℕ\nhn : length l = Nat.succ n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ (k + 1) % length l = k ↔ length l ≤ 1", "tactic": "cases hn : l.length" }, { "state_after": "no goals", "state_before": "case intro.intro.zero\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nhn : length l = Nat.zero\n⊢ (k + 1) % Nat.zero = k ↔ Nat.zero ≤ 1", "tactic": "exact absurd k.zero_le (hk.trans_le hn.le).not_le" }, { "state_after": "case intro.intro.succ\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "state_before": "case intro.intro.succ\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn✝ : ℕ\nhn : length l = Nat.succ n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "tactic": "rw [hn] at hk" }, { "state_after": "case intro.intro.succ.inl\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\nhk' : k = n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1\n\ncase intro.intro.succ.inr\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\nhk' : k < n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "state_before": "case intro.intro.succ\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "tactic": "cases' (Nat.le_of_lt_succ hk).eq_or_lt with hk' hk'" }, { "state_after": "no goals", "state_before": "case intro.intro.succ.inl\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\nhk' : k = n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "tactic": "simp [← hk', Nat.succ_le_succ_iff, eq_comm]" }, { "state_after": "no goals", "state_before": "case intro.intro.succ.inr\nα : Type u_1\nβ : Type ?u.813892\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk✝ : k < length l\nhx : nthLe l k hk✝ ∈ l\nn✝ : ℕ\nhk : k < Nat.succ n✝\nhn : length l = Nat.succ n✝\nhk' : k < n✝\n⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1", "tactic": "simpa [Nat.mod_eq_of_lt (Nat.succ_lt_succ hk'), Nat.succ_lt_succ_iff] using\n k.zero_le.trans_lt hk'" } ]

[ 394, 31 ]

[ 383, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf g : α → ℝ\n⊢ ((fun x => exp (f x)) =o[l] fun x => exp (g x)) ↔ Tendsto (fun x => g x - f x) l atTop", "tactic": "simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,\n imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub]" } ]

[ 386, 55 ]

[ 383, 1 ]

[ { "state_after": "no goals", "state_before": "E : Type ?u.1055778\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ns : Set ℝ\nf f' : ℝ → ℝ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : ℝ), x ∈ s → HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ (∫ (x : ℝ) in f '' s, g x) = ∫ (x : ℝ) in s, abs (f' x) • g (f x)", "tactic": "simpa only [det_one_smulRight] using\n integral_image_eq_integral_abs_det_fderiv_smul volume hs\n (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g" } ]

[ 1256, 54 ]

[ 1251, 1 ]