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tasksource 14

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[ 277, 49 ]

[ 275, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.723974\np : PosNum\n⊢ neg p + 1 = succ (neg p)", "tactic": "cases p <;> rfl" } ]

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[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.213751\nδ : Type ?u.213754\nι : Type ?u.213757\nR : Type ?u.213760\nR' : Type ?u.213763\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : AEMeasurable f\n⊢ ∀ (s : Set β), MeasurableSet s → ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.213751\nδ : Type ?u.213754\nι : Type ?u.213757\nR : Type ?u.213760\nR' : Type ?u.213763\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : AEMeasurable f\n⊢ ↑(map f μ) = OuterMeasure.trim (↑(OuterMeasure.map f) ↑μ)", "tactic": "rw [← trimmed, OuterMeasure.trim_eq_trim_iff]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.213751\nδ : Type ?u.213754\nι : Type ?u.213757\nR : Type ?u.213760\nR' : Type ?u.213763\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nf : α → β\nhf : AEMeasurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.213751\nδ : Type ?u.213754\nι : Type ?u.213757\nR : Type ?u.213760\nR' : Type ?u.213763\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : AEMeasurable f\n⊢ ∀ (s : Set β), MeasurableSet s → ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s", "tactic": "intro s hs" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.213751\nδ : Type ?u.213754\nι : Type ?u.213757\nR : Type ?u.213760\nR' : Type ?u.213763\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nf : α → β\nhf : AEMeasurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s", "tactic": "rw [map_apply_of_aemeasurable hf hs, OuterMeasure.map_apply]" } ]

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[ { "state_after": "case mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ IsEquiv v v' → ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\n\ncase mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ (∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1) → IsEquiv v v'", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ IsEquiv v v' ↔ ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1", "tactic": "constructor" }, { "state_after": "case mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : IsEquiv v v'\nx : K\n⊢ ↑v x = 1 ↔ ↑v' x = 1", "state_before": "case mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ IsEquiv v v' → ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1", "tactic": "intro h x" }, { "state_after": "no goals", "state_before": "case mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : IsEquiv v v'\nx : K\n⊢ ↑v x = 1 ↔ ↑v' x = 1", "tactic": "simpa using @IsEquiv.val_eq _ _ _ _ _ _ v v' h x 1" }, { "state_after": "case mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\n⊢ IsEquiv v v'", "state_before": "case mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ (∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1) → IsEquiv v v'", "tactic": "intro h" }, { "state_after": "case mpr.h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\n⊢ ∀ {x : K}, ↑v x ≤ 1 ↔ ↑v' x ≤ 1", "state_before": "case mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\n⊢ IsEquiv v v'", "tactic": "apply isEquiv_of_val_le_one" }, { "state_after": "case mpr.h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v x ≤ 1 ↔ ↑v' x ≤ 1", "state_before": "case mpr.h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\n⊢ ∀ {x : K}, ↑v x ≤ 1 ↔ ↑v' x ≤ 1", "tactic": "intro x" }, { "state_after": "case mpr.h.mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v x ≤ 1 → ↑v' x ≤ 1\n\ncase mpr.h.mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v' x ≤ 1 → ↑v x ≤ 1", "state_before": "case mpr.h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v x ≤ 1 ↔ ↑v' x ≤ 1", "tactic": "constructor" }, { "state_after": "case mpr.h.mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\n⊢ ↑v' x ≤ 1", "state_before": "case mpr.h.mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v x ≤ 1 → ↑v' x ≤ 1", "tactic": "intro hx" }, { "state_after": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v' x ≤ 1\n\ncase mpr.h.mp.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x = 1\n⊢ ↑v' x ≤ 1", "state_before": "case mpr.h.mp\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\n⊢ ↑v' x ≤ 1", "tactic": "cases' lt_or_eq_of_le hx with hx' hx'" }, { "state_after": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v' x ≤ 1", "state_before": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v' x ≤ 1", "tactic": "have : v (1 + x) = 1 := by\n rw [← v.map_one]\n apply map_add_eq_of_lt_left\n simpa" }, { "state_after": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v' x ≤ 1", "state_before": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v' x ≤ 1", "tactic": "rw [h] at this" }, { "state_after": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v' (-1 + (1 + x)) ≤ 1", "state_before": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v' x ≤ 1", "tactic": "rw [show x = -1 + (1 + x) by simp]" }, { "state_after": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ max (↑v' (-1)) (↑v' (1 + x)) ≤ 1", "state_before": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v' (-1 + (1 + x)) ≤ 1", "tactic": "refine' le_trans (v'.map_add _ _) _" }, { "state_after": "no goals", "state_before": "case mpr.h.mp.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ max (↑v' (-1)) (↑v' (1 + x)) ≤ 1", "tactic": "simp [this]" }, { "state_after": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v (1 + x) = ↑v 1", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v (1 + x) = 1", "tactic": "rw [← v.map_one]" }, { "state_after": "case h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v x < ↑v 1", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v (1 + x) = ↑v 1", "tactic": "apply map_add_eq_of_lt_left" }, { "state_after": "no goals", "state_before": "case h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\n⊢ ↑v x < ↑v 1", "tactic": "simpa" }, { "state_after": "no goals", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x < 1\nthis : ↑v' (1 + x) = 1\n⊢ x = -1 + (1 + x)", "tactic": "simp" }, { "state_after": "case mpr.h.mp.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v' x = 1\n⊢ ↑v' x ≤ 1", "state_before": "case mpr.h.mp.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v x = 1\n⊢ ↑v' x ≤ 1", "tactic": "rw [h] at hx'" }, { "state_after": "no goals", "state_before": "case mpr.h.mp.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v x ≤ 1\nhx' : ↑v' x = 1\n⊢ ↑v' x ≤ 1", "tactic": "exact le_of_eq hx'" }, { "state_after": "case mpr.h.mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\n⊢ ↑v x ≤ 1", "state_before": "case mpr.h.mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\n⊢ ↑v' x ≤ 1 → ↑v x ≤ 1", "tactic": "intro hx" }, { "state_after": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v x ≤ 1\n\ncase mpr.h.mpr.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x = 1\n⊢ ↑v x ≤ 1", "state_before": "case mpr.h.mpr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\n⊢ ↑v x ≤ 1", "tactic": "cases' lt_or_eq_of_le hx with hx' hx'" }, { "state_after": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v x ≤ 1", "state_before": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v x ≤ 1", "tactic": "have : v' (1 + x) = 1 := by\n rw [← v'.map_one]\n apply map_add_eq_of_lt_left\n simpa" }, { "state_after": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v x ≤ 1", "state_before": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v' (1 + x) = 1\n⊢ ↑v x ≤ 1", "tactic": "rw [← h] at this" }, { "state_after": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v (-1 + (1 + x)) ≤ 1", "state_before": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v x ≤ 1", "tactic": "rw [show x = -1 + (1 + x) by simp]" }, { "state_after": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ max (↑v (-1)) (↑v (1 + x)) ≤ 1", "state_before": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ ↑v (-1 + (1 + x)) ≤ 1", "tactic": "refine' le_trans (v.map_add _ _) _" }, { "state_after": "no goals", "state_before": "case mpr.h.mpr.inl\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ max (↑v (-1)) (↑v (1 + x)) ≤ 1", "tactic": "simp [this]" }, { "state_after": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v' (1 + x) = ↑v' 1", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v' (1 + x) = 1", "tactic": "rw [← v'.map_one]" }, { "state_after": "case h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v' x < ↑v' 1", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v' (1 + x) = ↑v' 1", "tactic": "apply map_add_eq_of_lt_left" }, { "state_after": "no goals", "state_before": "case h\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\n⊢ ↑v' x < ↑v' 1", "tactic": "simpa" }, { "state_after": "no goals", "state_before": "K : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x < 1\nthis : ↑v (1 + x) = 1\n⊢ x = -1 + (1 + x)", "tactic": "simp" }, { "state_after": "case mpr.h.mpr.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v x = 1\n⊢ ↑v x ≤ 1", "state_before": "case mpr.h.mpr.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v' x = 1\n⊢ ↑v x ≤ 1", "tactic": "rw [← h] at hx'" }, { "state_after": "no goals", "state_before": "case mpr.h.mpr.inr\nK : Type u_3\nF : Type ?u.3253919\nR : Type ?u.3253922\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3253934\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1\nx : K\nhx : ↑v' x ≤ 1\nhx' : ↑v x = 1\n⊢ ↑v x ≤ 1", "tactic": "exact le_of_eq hx'" } ]

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[ { "state_after": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nj : J\n⊢ (IsLimit.conePointsIsoOfEquivalence (limit.isLimit F) (limit.isLimit G) e w).inv ≫ limit.π F j =\n limit.π G (e.functor.obj j) ≫ w.hom.app j", "state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nj : J\n⊢ (isoOfEquivalence e w).inv ≫ limit.π F j = limit.π G (e.functor.obj j) ≫ w.hom.app j", "tactic": "simp only [HasLimit.isoOfEquivalence, IsLimit.conePointsIsoOfEquivalence_hom]" }, { "state_after": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nj : J\n⊢ limit.lift F ((Cones.postcompose w.hom).obj (Cone.whisker e.functor (limit.cone G))) ≫ limit.π F j =\n limit.π G (e.functor.obj j) ≫ w.hom.app j", "state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nj : J\n⊢ (IsLimit.conePointsIsoOfEquivalence (limit.isLimit F) (limit.isLimit G) e w).inv ≫ limit.π F j =\n limit.π G (e.functor.obj j) ≫ w.hom.app j", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasLimit F\nG : K ⥤ C\ninst✝ : HasLimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nj : J\n⊢ limit.lift F ((Cones.postcompose w.hom).obj (Cone.whisker e.functor (limit.cone G))) ≫ limit.π F j =\n limit.π G (e.functor.obj j) ≫ w.hom.app j", "tactic": "simp" } ]

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[ { "state_after": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n⊢ iteratedFDerivWithin 𝕜 Nat.zero f univ = iteratedFDeriv 𝕜 Nat.zero f\n\ncase succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nIH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f\n⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f univ = iteratedFDeriv 𝕜 (Nat.succ n) f", "state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f", "tactic": "induction' n with n IH" }, { "state_after": "case zero.h.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝¹ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nx : E\nx✝ : Fin Nat.zero → E\n⊢ ↑(iteratedFDerivWithin 𝕜 Nat.zero f univ x) x✝ = ↑(iteratedFDeriv 𝕜 Nat.zero f x) x✝", "state_before": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n⊢ iteratedFDerivWithin 𝕜 Nat.zero f univ = iteratedFDeriv 𝕜 Nat.zero f", "tactic": "ext x" }, { "state_after": "no goals", "state_before": "case zero.h.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝¹ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nx : E\nx✝ : Fin Nat.zero → E\n⊢ ↑(iteratedFDerivWithin 𝕜 Nat.zero f univ x) x✝ = ↑(iteratedFDeriv 𝕜 Nat.zero f x) x✝", "tactic": "simp" }, { "state_after": "case succ.h.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nIH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f\nx : E\nm : Fin (Nat.succ n) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n) f univ x) m = ↑(iteratedFDeriv 𝕜 (Nat.succ n) f x) m", "state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nIH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f\n⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f univ = iteratedFDeriv 𝕜 (Nat.succ n) f", "tactic": "ext (x m)" }, { "state_after": "no goals", "state_before": "case succ.h.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nIH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f\nx : E\nm : Fin (Nat.succ n) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n) f univ x) m = ↑(iteratedFDeriv 𝕜 (Nat.succ n) f x) m", "tactic": "rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]" } ]

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[ { "state_after": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\n⊢ ↑(matrixDecomposition o) (𝟙 (⨁ fun a => s (f a))) ((fun b => f b) b)\n { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } { val := a, property := j_property } =\n OfNat.ofNat 1 { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) }\n { val := a, property := j_property }", "state_before": "C : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\ni : ι\n⊢ ↑(matrixDecomposition o) (𝟙 (⨁ fun a => s (f a))) i = 1", "tactic": "ext (⟨b, ⟨⟩⟩⟨a, j_property⟩)" }, { "state_after": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\n⊢ ↑(matrixDecomposition o) (𝟙 (⨁ fun a => s (f a))) ((fun b => f b) b)\n { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } { val := a, property := j_property✝ } =\n OfNat.ofNat 1 { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) }\n { val := a, property := j_property✝ }", "state_before": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\n⊢ ↑(matrixDecomposition o) (𝟙 (⨁ fun a => s (f a))) ((fun b => f b) b)\n { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } { val := a, property := j_property } =\n OfNat.ofNat 1 { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) }\n { val := a, property := j_property }", "tactic": "simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property" }, { "state_after": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n if { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ } then\n 𝟙 (s (f b))\n else 0", "state_before": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\n⊢ ↑(matrixDecomposition o) (𝟙 (⨁ fun a => s (f a))) ((fun b => f b) b)\n { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } { val := a, property := j_property✝ } =\n OfNat.ofNat 1 { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) }\n { val := a, property := j_property✝ }", "tactic": "simp only [Category.comp_id, Category.id_comp, Category.assoc, End.one_def, eqToHom_refl,\n Matrix.one_apply, HomOrthogonal.matrixDecomposition_apply, biproduct.components]" }, { "state_after": "case a.mk.refl.h.mk.inl\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ }\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 𝟙 (s (f b))\n\ncase a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬{ val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ }\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "state_before": "case a.mk.refl.h.mk\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n if { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ } then\n 𝟙 (s (f b))\n else 0", "tactic": "split_ifs with h" }, { "state_after": "case a.mk.refl.h.mk.inl.refl\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb : α\nj_property✝ : b ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f b = f b\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := b, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) b ≫ biproduct.π (fun b => s (f b)) b =\n 𝟙 (s (f b))", "state_before": "case a.mk.refl.h.mk.inl\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : { val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ }\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 𝟙 (s (f b))", "tactic": "cases h" }, { "state_after": "no goals", "state_before": "case a.mk.refl.h.mk.inl.refl\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb : α\nj_property✝ : b ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f b = f b\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := b, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) b ≫ biproduct.π (fun b => s (f b)) b =\n 𝟙 (s (f b))", "tactic": "simp" }, { "state_after": "case a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬b = a\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "state_before": "case a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬{ val := b, property := (_ : (fun b => f b) b = (fun b => f b) b) } = { val := a, property := j_property✝ }\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "tactic": "simp at h" }, { "state_after": "case a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬b = a\nthis : biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b = 0\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "state_before": "case a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬b = a\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "tactic": "have : biproduct.ι (fun a ↦ s (f a)) a ≫ biproduct.π (fun b ↦ s (f b)) b = 0 := by\n simpa using biproduct.ι_π_ne _ (Ne.symm h)" }, { "state_after": "no goals", "state_before": "case a.mk.refl.h.mk.inr\nC : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬b = a\nthis : biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b = 0\n⊢ eqToHom (_ : s (f b) = s (f ↑{ val := a, property := j_property✝ })) ≫\n biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b =\n 0", "tactic": "rw [this, comp_zero]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝³ : Category C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Fintype α\nf : α → ι\nb a : α\nj_property✝ : a ∈ (fun a => f a) ⁻¹' {(fun b => f b) b}\nj_property : f a = f b\nh : ¬b = a\n⊢ biproduct.ι (fun a => s (f a)) a ≫ biproduct.π (fun b => s (f b)) b = 0", "tactic": "simpa using biproduct.ι_π_ne _ (Ne.symm h)" } ]

[ 151, 25 ]

[ 138, 1 ]

[ { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f (insert x s ∩ u)\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "rcases h.contDiffOn' (ENat.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f (insert x s ∩ u)\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "set t := insert x s ∩ u" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "have A : t =ᶠ[𝓝[≠] x] s := by\n simp only [set_eventuallyEq_iff_inf_principal, ← nhdsWithin_inter']\n rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,\n diff_eq_compl_inter]\n exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "have B : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t :=\n iteratedFDerivWithin_eventually_congr_set' _ A.symm _" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\nC : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=\n hu.differentiableOn_iteratedFDerivWithin (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x\n ⟨mem_insert _ _, xu⟩" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\nC : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) s x\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\nC : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "rw [differentiableWithinAt_congr_set' _ A] at C" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\nA : t =ᶠ[𝓝[{x}ᶜ] x] s\nB : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t\nC : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) s x\n⊢ DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x", "tactic": "exact C.congr_of_eventuallyEq (B.filter_mono inf_le_left) B.self_of_nhds" }, { "state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ 𝓝[{x}ᶜ ∩ (insert x s ∩ u)] x = 𝓝[{x}ᶜ ∩ s] x", "state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ t =ᶠ[𝓝[{x}ᶜ] x] s", "tactic": "simp only [set_eventuallyEq_iff_inf_principal, ← nhdsWithin_inter']" }, { "state_after": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ x ∈ {x}\n\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ u ∈ 𝓝[{x}ᶜ ∩ insert x s] x", "state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ 𝓝[{x}ᶜ ∩ (insert x s ∩ u)] x = 𝓝[{x}ᶜ ∩ s] x", "tactic": "rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,\n diff_eq_compl_inter]" }, { "state_after": "no goals", "state_before": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ x ∈ {x}\n\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x ∈ u\nt : Set E := insert x s ∩ u\nhu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t\n⊢ u ∈ 𝓝[{x}ᶜ ∩ insert x s] x", "tactic": "exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]" } ]

[ 1097, 75 ]

[ 1081, 1 ]

[ { "state_after": "no goals", "state_before": "f g : CircleDeg1Lift\nx : ℝ\nn : ℤ\n⊢ ↑translate (↑Multiplicative.ofAdd x) ^ n = ↑translate (↑Multiplicative.ofAdd (↑n * x))", "tactic": "simp only [← zsmul_eq_mul, ofAdd_zsmul, MonoidHom.map_zpow]" } ]

[ 310, 62 ]

[ 308, 1 ]

[]

[ 665, 41 ]

[ 661, 1 ]

[]

[ 50, 26 ]

[ 48, 1 ]

[]

[ 192, 34 ]

[ 191, 1 ]

[]

[ 37, 25 ]

[ 36, 1 ]

[ { "state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "tactic": "rcases(F a).prop with ⟨f, hf, ha⟩" }, { "state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\ng : PartialIso α β\nhg : g ∈ our_ideal\nhb : (↑(G b), b) ∈ ↑g\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "tactic": "rcases(G b).prop with ⟨g, hg, hb⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\ng : PartialIso α β\nhg : g ∈ our_ideal\nhb : (↑(G b), b) ∈ ↑g\nm : PartialIso α β\nleft✝ : m ∈ ↑our_ideal\nfm : f ≤ m\ngm : g ≤ m\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\ng : PartialIso α β\nhg : g ∈ our_ideal\nhb : (↑(G b), b) ∈ ↑g\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "tactic": "rcases our_ideal.directed _ hf _ hg with ⟨m, _, fm, gm⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Encodable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Encodable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nto_cofinal : α ⊕ β → Cofinal (PartialIso α β) := fun p => Sum.recOn p (definedAtLeft β) (definedAtRight α)\nour_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal\nF : (a : α) → { b // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun a => funOfIdeal a our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inl a) ∧ x ∈ idealOfCofinals default to_cofinal)\nG : (b : β) → { a // ∃ f, f ∈ our_ideal ∧ (a, b) ∈ ↑f } :=\n fun b => invOfIdeal b our_ideal (_ : ∃ x, x ∈ to_cofinal (Sum.inr b) ∧ x ∈ idealOfCofinals default to_cofinal)\na : α\nb : β\nf : PartialIso α β\nhf : f ∈ our_ideal\nha : (a, ↑(F a)) ∈ ↑f\ng : PartialIso α β\nhg : g ∈ our_ideal\nhb : (↑(G b), b) ∈ ↑g\nm : PartialIso α β\nleft✝ : m ∈ ↑our_ideal\nfm : f ≤ m\ngm : g ≤ m\n⊢ cmp a ((fun b => ↑(G b)) b) = cmp ((fun a => ↑(F a)) a) b", "tactic": "exact m.prop (a, _) (fm ha) (_, b) (gm hb)" } ]

[ 235, 50 ]

[ 223, 1 ]

[ { "state_after": "ι : Type u_1\nM : Type ?u.58340\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₁) =\n Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₂)", "state_before": "ι : Type u_1\nM : Type ?u.58340\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ compl π₁ = compl π₂", "tactic": "dsimp only [compl]" }, { "state_after": "case e_p\nι : Type u_1\nM : Type ?u.58340\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ (fun π' => Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₁) = fun π' =>\n Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₂", "state_before": "ι : Type u_1\nM : Type ?u.58340\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₁) =\n Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₂)", "tactic": "congr 1" }, { "state_after": "no goals", "state_before": "case e_p\nι : Type u_1\nM : Type ?u.58340\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ (fun π' => Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₁) = fun π' =>\n Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π₂", "tactic": "rw [h]" } ]

[ 378, 9 ]

[ 375, 1 ]

[]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type ?u.133965\nβ : Type ?u.133968\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ (⋃ (n : ℕ), Ioc a ↑n) = Ioi a \\ {⊤}", "tactic": "simp only [← Ioi_inter_Iic, ← inter_iUnion, iUnion_Iic_coe_nat, diff_eq]" } ]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.223921\nγ : Type ?u.223924\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nh : ¬a ∈ s\n⊢ ¬a ∈ s \\ t", "tactic": "simp [h]" } ]

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[]

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[ { "state_after": "ι : Sort ?u.98897\nα✝ : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α✝\nδ✝ ε : ℝ\ns t : Set α✝\nx✝ : α✝\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nx y : α\nδ : ℝ\nE : Set α\nh : y ∈ E\nh' : dist x y ≤ δ\n⊢ edist x y ≤ ENNReal.ofReal δ", "state_before": "ι : Sort ?u.98897\nα✝ : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α✝\nδ✝ ε : ℝ\ns t : Set α✝\nx✝ : α✝\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nx y : α\nδ : ℝ\nE : Set α\nh : y ∈ E\nh' : dist x y ≤ δ\n⊢ x ∈ cthickening δ E", "tactic": "apply mem_cthickening_of_edist_le x y δ E h" }, { "state_after": "ι : Sort ?u.98897\nα✝ : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α✝\nδ✝ ε : ℝ\ns t : Set α✝\nx✝ : α✝\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nx y : α\nδ : ℝ\nE : Set α\nh : y ∈ E\nh' : dist x y ≤ δ\n⊢ ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ", "state_before": "ι : Sort ?u.98897\nα✝ : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α✝\nδ✝ ε : ℝ\ns t : Set α✝\nx✝ : α✝\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nx y : α\nδ : ℝ\nE : Set α\nh : y ∈ E\nh' : dist x y ≤ δ\n⊢ edist x y ≤ ENNReal.ofReal δ", "tactic": "rw [edist_dist]" }, { "state_after": "no goals", "state_before": "ι : Sort ?u.98897\nα✝ : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α✝\nδ✝ ε : ℝ\ns t : Set α✝\nx✝ : α✝\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nx y : α\nδ : ℝ\nE : Set α\nh : y ∈ E\nh' : dist x y ≤ δ\n⊢ ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ", "tactic": "exact ENNReal.ofReal_le_ofReal h'" } ]

[ 1041, 36 ]

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[]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "R : Type ?u.1640706\nM✝ : Type ?u.1640709\ninst✝²⁰ : Semiring R\ninst✝¹⁹ : AddCommMonoid M✝\ninst✝¹⁸ : Module R M✝\nR₁ : Type ?u.1640745\nM₁ : Type ?u.1640748\ninst✝¹⁷ : Ring R₁\ninst✝¹⁶ : AddCommGroup M₁\ninst✝¹⁵ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁴ : CommSemiring R₂\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R₂ M₂\nR₃ : Type ?u.1641547\nM₃ : Type ?u.1641550\ninst✝¹¹ : CommRing R₃\ninst✝¹⁰ : AddCommGroup M₃\ninst✝⁹ : Module R₃ M₃\nV : Type ?u.1642138\nK : Type ?u.1642141\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nB✝ : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_4\no : Type u_3\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : DecidableEq n\nb : Basis n R₂ M₂\nM₂' : Type u_5\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R₂ M₂'\nc : Basis o R₂ M₂'\ninst✝ : DecidableEq o\nB : BilinForm R₂ M₂\nM : Matrix o n R₂\nN : Matrix n o R₂\n⊢ M ⬝ ↑(toMatrix b) B ⬝ N = ↑(toMatrix c) (comp B (↑(Matrix.toLin c b) Mᵀ) (↑(Matrix.toLin c b) N))", "tactic": "simp only [B.toMatrix_comp b c, toMatrix_toLin, transpose_transpose]" } ]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\n⊢ ∃ x x_1 y x_2 _hxy, infsep s = dist x y", "tactic": "classical\n cases nonempty_fintype s\n simp_rw [hs.infsep_of_fintype]\n rcases@Finset.exists_mem_eq_inf' _ _ _ s.offDiag.toFinset (by simpa) (uncurry dist) with\n ⟨w, hxy, hed⟩\n simp_rw [mem_toFinset] at hxy\n exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x x_1 y x_2 _hxy, infsep s = dist x y", "state_before": "α : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\n⊢ ∃ x x_1 y x_2 _hxy, infsep s = dist x y", "tactic": "cases nonempty_fintype s" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x x_1 y x_2 _hxy, infsep s = dist x y", "tactic": "simp_rw [hs.infsep_of_fintype]" }, { "state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhxy : w ∈ toFinset (offDiag s)\nhed : Finset.inf' (toFinset (offDiag s)) ?m.80020 (uncurry dist) = uncurry dist w\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "tactic": "rcases@Finset.exists_mem_eq_inf' _ _ _ s.offDiag.toFinset (by simpa) (uncurry dist) with\n ⟨w, hxy, hed⟩" }, { "state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhed : Finset.inf' (toFinset (offDiag s)) ?m.80020 (uncurry dist) = uncurry dist w\nhxy : w ∈ offDiag s\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhxy : w ∈ toFinset (offDiag s)\nhed : Finset.inf' (toFinset (offDiag s)) ?m.80020 (uncurry dist) = uncurry dist w\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "tactic": "simp_rw [mem_toFinset] at hxy" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhed : Finset.inf' (toFinset (offDiag s)) ?m.80020 (uncurry dist) = uncurry dist w\nhxy : w ∈ offDiag s\n⊢ ∃ x h y h h, Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist) = dist x y", "tactic": "exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.77742\ninst✝¹ : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ Finset.Nonempty (toFinset (offDiag s))", "tactic": "simpa" } ]

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[]

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[]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\na b c d : α\ninst✝² : CommSemigroup α\ninst✝¹ : Zero α\ninst✝ : Preorder α\n⊢ PosMulStrictMono α ↔ MulPosStrictMono α", "tactic": "simp only [PosMulStrictMono, MulPosStrictMono, mul_comm]" } ]

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[]

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[]

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[ { "state_after": "α : Type ?u.176030\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.176039\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_3\nι : Type ?u.176054\nf✝ : κ → β\ng✝ g : β → γ\nf : κ → β\nhg : DenseRange g\nhf : DenseRange f\ncg : Continuous g\n⊢ Dense (g '' range f)", "state_before": "α : Type ?u.176030\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.176039\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_3\nι : Type ?u.176054\nf✝ : κ → β\ng✝ g : β → γ\nf : κ → β\nhg : DenseRange g\nhf : DenseRange f\ncg : Continuous g\n⊢ DenseRange (g ∘ f)", "tactic": "rw [DenseRange, range_comp]" }, { "state_after": "no goals", "state_before": "α : Type ?u.176030\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.176039\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_3\nι : Type ?u.176054\nf✝ : κ → β\ng✝ g : β → γ\nf : κ → β\nhg : DenseRange g\nhf : DenseRange f\ncg : Continuous g\n⊢ Dense (g '' range f)", "tactic": "exact hg.dense_image cg hf" } ]

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[]

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[ { "state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n⊢ IsLieAbelian { x // x ∈ ↑I } ↔ ∀ (y : L) (x : { x // x ∈ I }) (x_1 : { x // x ∈ I }), ⁅↑x, ↑x_1⁆ = y → y = 0", "state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n⊢ IsLieAbelian { x // x ∈ ↑I } ↔ ⁅I, I⁆ = ⊥", "tactic": "simp only [_root_.eq_bot_iff, lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le,\n LieSubmodule.bot_coe, Set.subset_singleton_iff, Set.mem_setOf_eq, exists_imp]" }, { "state_after": "no goals", "state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n⊢ IsLieAbelian { x // x ∈ ↑I } ↔ ∀ (y : L) (x : { x // x ∈ I }) (x_1 : { x // x ∈ I }), ⁅↑x, ↑x_1⁆ = y → y = 0", "tactic": "refine'\n ⟨fun h z x y hz =>\n hz.symm.trans\n (((I : LieSubalgebra R L).coe_bracket x y).symm.trans\n ((coe_zero_iff_zero _ _).mpr (by apply h.trivial))),\n fun h => ⟨fun x y => ((I : LieSubalgebra R L).coe_zero_iff_zero _).mp (h _ x y rfl)⟩⟩" }, { "state_after": "no goals", "state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nh : IsLieAbelian { x // x ∈ ↑I }\nz : L\nx : { x // x ∈ I }\ny : { x // x ∈ I }\nhz : ⁅↑x, ↑y⁆ = z\n⊢ ⁅x, y⁆ = 0", "tactic": "apply h.trivial" } ]

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[]

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[ { "state_after": "ι : Type ?u.1334907\nι' : Type ?u.1334910\n𝕜 : Type ?u.1334913\ninst✝⁹ : IsROrC 𝕜\nE : Type ?u.1334919\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.1334939\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_2\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nv : OrthonormalBasis (Fin 2) ℝ F\nf : F ≃ₗᵢ[ℝ] F'\n⊢ LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.trans (LinearIsometryEquiv.symm v.repr) f) =\n LinearIsometryEquiv.trans (LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.symm v.repr)) f", "state_before": "ι : Type ?u.1334907\nι' : Type ?u.1334910\n𝕜 : Type ?u.1334913\ninst✝⁹ : IsROrC 𝕜\nE : Type ?u.1334919\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.1334939\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_2\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nv : OrthonormalBasis (Fin 2) ℝ F\nf : F ≃ₗᵢ[ℝ] F'\n⊢ isometryOfOrthonormal (OrthonormalBasis.map v f) = LinearIsometryEquiv.trans (isometryOfOrthonormal v) f", "tactic": "simp [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_assoc, OrthonormalBasis.map]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.1334907\nι' : Type ?u.1334910\n𝕜 : Type ?u.1334913\ninst✝⁹ : IsROrC 𝕜\nE : Type ?u.1334919\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.1334939\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type u_2\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nv : OrthonormalBasis (Fin 2) ℝ F\nf : F ≃ₗᵢ[ℝ] F'\n⊢ LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.trans (LinearIsometryEquiv.symm v.repr) f) =\n LinearIsometryEquiv.trans (LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.symm v.repr)) f", "tactic": "rw [LinearIsometryEquiv.trans_assoc]" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.45708\nγ : Type ?u.45711\nι : Sort ?u.45714\nκ : ι → Sort ?u.45719\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\n⊢ ↑s = univ ↔ s = ⊥", "tactic": "simp [SetLike.ext'_iff]" } ]

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[ { "state_after": "case h.e'_7\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nc : 𝕜 → 𝕜\nh : E → 𝕜\nc' : 𝕜\nz : E\nS : Set E\nhc : HasDerivWithinAt c c' s x\nhx : c x ≠ 0\n⊢ -c' / c x ^ 2 = -(c x ^ 2)⁻¹ * c'", "state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nc : 𝕜 → 𝕜\nh : E → 𝕜\nc' : 𝕜\nz : E\nS : Set E\nhc : HasDerivWithinAt c c' s x\nhx : c x ≠ 0\n⊢ HasDerivWithinAt (fun y => (c y)⁻¹) (-c' / c x ^ 2) s x", "tactic": "convert (hasDerivAt_inv hx).comp_hasDerivWithinAt x hc using 1" }, { "state_after": "no goals", "state_before": "case h.e'_7\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nc : 𝕜 → 𝕜\nh : E → 𝕜\nc' : 𝕜\nz : E\nS : Set E\nhc : HasDerivWithinAt c c' s x\nhx : c x ≠ 0\n⊢ -c' / c x ^ 2 = -(c x ^ 2)⁻¹ * c'", "tactic": "field_simp" } ]

[ 138, 13 ]

[ 135, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.60876\ninst✝ : StrictOrderedSemiring α\na b c d : α\n⊢ a < c → b < d → 0 ≤ a → 0 ≤ b → a * b < c * d", "tactic": "classical\nexact Decidable.mul_lt_mul''" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.60876\ninst✝ : StrictOrderedSemiring α\na b c d : α\n⊢ a < c → b < d → 0 ≤ a → 0 ≤ b → a * b < c * d", "tactic": "exact Decidable.mul_lt_mul''" } ]

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[ { "state_after": "X : Type u_3\nY : Type ?u.7897\nZ : Type ?u.7900\nα✝ : Type ?u.7903\ninst✝ : TopologicalSpace X\nα : Type u_1\nβ : Type u_2\nf : X → α\ng : α → β\nh : IsLocallyConstant (g ∘ f)\ninj : Function.Injective g\ns : Set α\n⊢ IsOpen ((fun x => g (f x)) ⁻¹' (g '' s))", "state_before": "X : Type u_3\nY : Type ?u.7897\nZ : Type ?u.7900\nα✝ : Type ?u.7903\ninst✝ : TopologicalSpace X\nα : Type u_1\nβ : Type u_2\nf : X → α\ng : α → β\nh : IsLocallyConstant (g ∘ f)\ninj : Function.Injective g\ns : Set α\n⊢ IsOpen (f ⁻¹' s)", "tactic": "rw [← preimage_image_eq s inj, preimage_preimage]" }, { "state_after": "no goals", "state_before": "X : Type u_3\nY : Type ?u.7897\nZ : Type ?u.7900\nα✝ : Type ?u.7903\ninst✝ : TopologicalSpace X\nα : Type u_1\nβ : Type u_2\nf : X → α\ng : α → β\nh : IsLocallyConstant (g ∘ f)\ninj : Function.Injective g\ns : Set α\n⊢ IsOpen ((fun x => g (f x)) ⁻¹' (g '' s))", "tactic": "exact h (g '' s)" } ]

[ 222, 19 ]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : PredOrder α\na b : α\nha : ¬IsMin a\n⊢ Ioo (pred a) b = Ico a b", "tactic": "rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]" } ]

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[ 679, 1 ]

[]

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[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nas : List α\n⊢ as ++ nil = as", "tactic": "induction as with\n| nil => rfl\n| cons a as ih =>\n simp_all [HAppend.hAppend, Append.append, List.append]" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\n⊢ nil ++ nil = nil", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\nih : as ++ nil = as\n⊢ a :: as ++ nil = a :: as", "tactic": "simp_all [HAppend.hAppend, Append.append, List.append]" } ]

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[ 93, 9 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\n⊢ toList 1 x = []", "tactic": "simp [toList, cycleOf_one]" } ]

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[]

[ 284, 6 ]

[ 282, 1 ]

[ { "state_after": "case inl\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : a < b\n⊢ a :: as = b :: bs\n\ncase inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\n⊢ a :: as = b :: bs", "state_before": "α : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\n⊢ a :: as = b :: bs", "tactic": "by_cases hab : a < b" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : a < b\n⊢ a :: as = b :: bs", "tactic": "exact False.elim <| h₂ (List.lt.head _ _ hab)" }, { "state_after": "case inr.inl\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : b < a\n⊢ a :: as = b :: bs\n\ncase inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\n⊢ a :: as = b :: bs", "state_before": "case inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\n⊢ a :: as = b :: bs", "tactic": "by_cases hba : b < a" }, { "state_after": "no goals", "state_before": "case inr.inl\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : b < a\n⊢ a :: as = b :: bs", "tactic": "exact False.elim <| h₁ (List.lt.head _ _ hba)" }, { "state_after": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\n⊢ a :: as = b :: bs", "state_before": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\n⊢ a :: as = b :: bs", "tactic": "have h₁ : as ≤ bs := fun h => h₁ (List.lt.tail hba hab h)" }, { "state_after": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\n⊢ a :: as = b :: bs", "state_before": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\n⊢ a :: as = b :: bs", "tactic": "have h₂ : bs ≤ as := fun h => h₂ (List.lt.tail hab hba h)" }, { "state_after": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\nih : as = bs\n⊢ a :: as = b :: bs", "state_before": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\n⊢ a :: as = b :: bs", "tactic": "have ih : as = bs := le_antisymm h₁ h₂" }, { "state_after": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\nih : as = bs\nthis : a = b\n⊢ a :: as = b :: bs", "state_before": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\nih : as = bs\n⊢ a :: as = b :: bs", "tactic": "have : a = b := s.antisymm hab hba" }, { "state_after": "no goals", "state_before": "case inr.inr\nα : Type u_1\ninst✝ : LT α\ns : Antisymm fun x x_1 => ¬x < x_1\nas✝ bs✝ : List α\na : α\nas : List α\nb : α\nbs : List α\nh₁✝ : a :: as ≤ b :: bs\nh₂✝ : b :: bs ≤ a :: as\nhab : ¬a < b\nhba : ¬b < a\nh₁ : as ≤ bs\nh₂ : bs ≤ as\nih : as = bs\nthis : a = b\n⊢ a :: as = b :: bs", "tactic": "simp [this, ih]" } ]

[ 182, 24 ]

[ 168, 1 ]

[]

[ 132, 33 ]

[ 131, 1 ]