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tasksource 14

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[ { "state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nn : σ →₀ ℕ\nhn : n ∈ support (↑C a)\nthis : n ∈ {0}\n⊢ (sum n fun x e => e) ≤ 0", "state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nn : σ →₀ ℕ\nhn : n ∈ support (↑C a)\n⊢ (sum n fun x e => e) ≤ 0", "tactic": "have := Finsupp.support_single_subset hn" }, { "state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nn : σ →₀ ℕ\nhn : n ∈ support (↑C a)\nthis : n = 0\n⊢ (sum n fun x e => e) ≤ 0", "state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nn : σ →₀ ℕ\nhn : n ∈ support (↑C a)\nthis : n ∈ {0}\n⊢ (sum n fun x e => e) ≤ 0", "tactic": "rw [Finset.mem_singleton] at this" }, { "state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nhn : 0 ∈ support (↑C a)\n⊢ (sum 0 fun x e => e) ≤ 0", "state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nn : σ →₀ ℕ\nhn : n ∈ support (↑C a)\nthis : n = 0\n⊢ (sum n fun x e => e) ≤ 0", "tactic": "subst this" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.373398\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\na : R\nhn : 0 ∈ support (↑C a)\n⊢ (sum 0 fun x e => e) ≤ 0", "tactic": "exact le_rfl" } ]

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[ { "state_after": "α : Type u_1\nβ : Type ?u.181361\nβ₂ : Type ?u.181364\nγ : Type ?u.181367\nι : Sort ?u.181370\nι' : Sort ?u.181373\nκ : ι → Sort ?u.181378\nκ' : ι' → Sort ?u.181383\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : ℕ → α\n⊢ (⨅ (i : ℕ) (_ : i > 0), f i) = ⨅ (b : ℕ) (_ : b ∈ {i | 0 < i}), f b", "state_before": "α : Type u_1\nβ : Type ?u.181361\nβ₂ : Type ?u.181364\nγ : Type ?u.181367\nι : Sort ?u.181370\nι' : Sort ?u.181373\nκ : ι → Sort ?u.181378\nκ' : ι' → Sort ?u.181383\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : ℕ → α\n⊢ (⨅ (i : ℕ) (_ : i > 0), f i) = ⨅ (i : ℕ), f (i + 1)", "tactic": "rw [← iInf_range, Nat.range_succ]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.181361\nβ₂ : Type ?u.181364\nγ : Type ?u.181367\nι : Sort ?u.181370\nι' : Sort ?u.181373\nκ : ι → Sort ?u.181378\nκ' : ι' → Sort ?u.181383\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : ℕ → α\n⊢ (⨅ (i : ℕ) (_ : i > 0), f i) = ⨅ (b : ℕ) (_ : b ∈ {i | 0 < i}), f b", "tactic": "simp" } ]

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[ { "state_after": "no goals", "state_before": "a b c : Prop\n⊢ a ∧ b ∧ c ↔ b ∧ a ∧ c", "tactic": "rw [← and_assoc, ← and_assoc, @and_comm a b]" } ]

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[ { "state_after": "case zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ∑ k in range zero, ↑(Nat.choose zero k) * bernoulli k = if zero = 1 then 1 else 0\n\ncase succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ n), ↑(Nat.choose (succ n) k) * bernoulli k = if succ n = 1 then 1 else 0", "state_before": "A : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range n, ↑(Nat.choose n k) * bernoulli k = if n = 1 then 1 else 0", "tactic": "cases' n with n n" }, { "state_after": "case succ.zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ∑ k in range (succ zero), ↑(Nat.choose (succ zero) k) * bernoulli k = if succ zero = 1 then 1 else 0\n\ncase succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ (succ n)), ↑(Nat.choose (succ (succ n)) k) * bernoulli k = if succ (succ n) = 1 then 1 else 0", "state_before": "case succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ n), ↑(Nat.choose (succ n) k) * bernoulli k = if succ n = 1 then 1 else 0", "tactic": "cases' n with n n" }, { "state_after": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "state_before": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ (succ n)), ↑(Nat.choose (succ (succ n)) k) * bernoulli k = if succ (succ n) = 1 then 1 else 0", "tactic": "suffices (∑ i in range n, ↑((n + 2).choose (i + 2)) * bernoulli (i + 2)) = n / 2 by\n simp only [this, sum_range_succ', cast_succ, bernoulli_one, bernoulli_zero, choose_one_right,\n mul_one, choose_zero_right, cast_zero, if_false, zero_add, succ_succ_ne_one]\n ring" }, { "state_after": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf : ∑ k in range (succ (succ n)), ↑(Nat.choose (succ (succ n)) k) * bernoulli' k = ↑(succ (succ n))\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "state_before": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "tactic": "have f := sum_bernoulli' n.succ.succ" }, { "state_after": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "state_before": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf : ∑ k in range (succ (succ n)), ↑(Nat.choose (succ (succ n)) k) * bernoulli' k = ↑(succ (succ n))\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "tactic": "simp_rw [sum_range_succ', bernoulli'_one, choose_one_right, cast_succ, ← eq_sub_iff_add_eq] at f" }, { "state_after": "case succ.succ.refine'_1\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) =\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1)\n\ncase succ.succ.refine'_2\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1) =\n ↑n / 2", "state_before": "case succ.succ\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2", "tactic": "refine' Eq.trans _ (Eq.trans f _)" }, { "state_after": "no goals", "state_before": "case zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ∑ k in range zero, ↑(Nat.choose zero k) * bernoulli k = if zero = 1 then 1 else 0", "tactic": "simp" }, { "state_after": "case succ.zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(Nat.choose (succ zero) 0) * bernoulli 0 = if succ zero = 1 then 1 else 0", "state_before": "case succ.zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ∑ k in range (succ zero), ↑(Nat.choose (succ zero) k) * bernoulli k = if succ zero = 1 then 1 else 0", "tactic": "rw [sum_range_one]" }, { "state_after": "no goals", "state_before": "case succ.zero\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(Nat.choose (succ zero) 0) * bernoulli 0 = if succ zero = 1 then 1 else 0", "tactic": "simp" }, { "state_after": "A : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2\n⊢ ↑n / 2 + (↑n + 1 + 1) * (-1 / 2) + 1 = 0", "state_before": "A : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2\n⊢ ∑ k in range (succ (succ n)), ↑(Nat.choose (succ (succ n)) k) * bernoulli k = if succ (succ n) = 1 then 1 else 0", "tactic": "simp only [this, sum_range_succ', cast_succ, bernoulli_one, bernoulli_zero, choose_one_right,\n mul_one, choose_zero_right, cast_zero, if_false, zero_add, succ_succ_ne_one]" }, { "state_after": "no goals", "state_before": "A : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) = ↑n / 2\n⊢ ↑n / 2 + (↑n + 1 + 1) * (-1 / 2) + 1 = 0", "tactic": "ring" }, { "state_after": "case succ.succ.refine'_1.e_f\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ (fun i => ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2)) = fun k =>\n ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1)", "state_before": "case succ.succ.refine'_1\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ∑ i in range n, ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2) =\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1)", "tactic": "congr" }, { "state_after": "case succ.succ.refine'_1.e_f.h\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\nx : ℕ\n⊢ ↑(Nat.choose (n + 2) (x + 2)) * bernoulli (x + 2) = ↑(Nat.choose (succ (succ n)) (x + 1 + 1)) * bernoulli' (x + 1 + 1)", "state_before": "case succ.succ.refine'_1.e_f\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ (fun i => ↑(Nat.choose (n + 2) (i + 2)) * bernoulli (i + 2)) = fun k =>\n ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1)", "tactic": "funext x" }, { "state_after": "no goals", "state_before": "case succ.succ.refine'_1.e_f.h\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\nx : ℕ\n⊢ ↑(Nat.choose (n + 2) (x + 2)) * bernoulli (x + 2) = ↑(Nat.choose (succ (succ n)) (x + 1 + 1)) * bernoulli' (x + 1 + 1)", "tactic": "rw [bernoulli_eq_bernoulli'_of_ne_one (succ_ne_zero x ∘ succ.inj)]" }, { "state_after": "case succ.succ.refine'_2\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ↑n + 1 + 1 - (↑0 + 1) - (↑n + 1 + 1) * 2⁻¹ = ↑n / 2", "state_before": "case succ.succ.refine'_2\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1) =\n ↑n / 2", "tactic": "simp only [one_div, mul_one, bernoulli'_zero, cast_one, choose_zero_right, add_sub_cancel,\n zero_add, choose_one_right, cast_succ, cast_add, cast_one, bernoulli'_one, one_div]" }, { "state_after": "no goals", "state_before": "case succ.succ.refine'_2\nA : Type ?u.636004\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nf :\n ∑ k in range n, ↑(Nat.choose (succ (succ n)) (k + 1 + 1)) * bernoulli' (k + 1 + 1) =\n ↑n + 1 + 1 - ↑(Nat.choose (succ (succ n)) 0) * bernoulli' 0 -\n ↑(Nat.choose (succ (succ n)) (0 + 1)) * bernoulli' (0 + 1)\n⊢ ↑n + 1 + 1 - (↑0 + 1) - (↑n + 1 + 1) * 2⁻¹ = ↑n / 2", "tactic": "ring" } ]

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[ { "state_after": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ (A ⬝ adjugate A) i j = (det A • 1) i j", "state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\n⊢ A ⬝ adjugate A = det A • 1", "tactic": "ext (i j)" }, { "state_after": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ ∑ j_1 : n, A i j_1 * adjugate A j_1 j = if i = j then det A else 0", "state_before": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ (A ⬝ adjugate A) i j = (det A • 1) i j", "tactic": "rw [mul_apply, Pi.smul_apply, Pi.smul_apply, one_apply, smul_eq_mul, mul_boole]" }, { "state_after": "no goals", "state_before": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ ∑ j_1 : n, A i j_1 * adjugate A j_1 j = if i = j then det A else 0", "tactic": "simp [mul_adjugate_apply, sum_cramer_apply, cramer_transpose_row_self, Pi.single_apply, eq_comm]" } ]

[ 295, 99 ]

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[ 192, 1 ]

[ { "state_after": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nh : f ∈ []\nhl : List.Pairwise Disjoint []\n⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod []) x\n\ncase cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\n⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod (hd :: tl)) x", "state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh : f ∈ l\nhl : List.Pairwise Disjoint l\n⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod l) x", "tactic": "induction' l with hd tl IH" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nh : f ∈ []\nhl : List.Pairwise Disjoint []\n⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod []) x", "tactic": "simp at h" }, { "state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\n⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod (hd :: tl)) x", "tactic": "intro x hx" }, { "state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "tactic": "rw [List.pairwise_cons] at hl" }, { "state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f = hd ∨ f ∈ tl\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f ∈ hd :: tl\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "tactic": "rw [List.mem_cons] at h" }, { "state_after": "case cons.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh : f ∈ l\nhl✝ : List.Pairwise Disjoint l\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nx : α\nhx : x ∈ support f\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint f a') ∧ List.Pairwise Disjoint tl\n⊢ ↑f x = ↑(List.prod (f :: tl)) x\n\ncase cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nh : f = hd ∨ f ∈ tl\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "tactic": "rcases h with (rfl | h)" }, { "state_after": "no goals", "state_before": "case cons.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh : f ∈ l\nhl✝ : List.Pairwise Disjoint l\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nx : α\nhx : x ∈ support f\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint f a') ∧ List.Pairwise Disjoint tl\n⊢ ↑f x = ↑(List.prod (f :: tl)) x", "tactic": "rw [List.prod_cons, mul_apply,\n not_mem_support.mp ((disjoint_prod_right tl hl.left).mem_imp hx)]" }, { "state_after": "case cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ¬↑f x ∈ support hd", "state_before": "case cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ↑f x = ↑(List.prod (hd :: tl)) x", "tactic": "rw [List.prod_cons, mul_apply, ← IH h hl.right _ hx, eq_comm, ← not_mem_support]" }, { "state_after": "case cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ↑f x ∈ support f", "state_before": "case cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ¬↑f x ∈ support hd", "tactic": "refine' (hl.left _ h).symm.mem_imp _" }, { "state_after": "no goals", "state_before": "case cons.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : f ∈ l\nhl✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x\nhl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ support f\nh : f ∈ tl\n⊢ ↑f x ∈ support f", "tactic": "simpa using hx" } ]

[ 526, 21 ]

[ 513, 1 ]

[ { "state_after": "no goals", "state_before": "num den : Int\n⊢ num /. -den = -num /. den", "tactic": "rw [← neg_divInt_neg, Int.neg_neg]" } ]

[ 134, 99 ]

[ 134, 1 ]

[]

[ 193, 52 ]

[ 191, 1 ]

[ { "state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\nhpα : ¬p ∣ card α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\n⊢ Nonempty ↑(fixedPoints G α)", "state_before": "p : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\nhpα : ¬p ∣ card α\ninst✝ : Finite ↑(fixedPoints G α)\n⊢ Nonempty ↑(fixedPoints G α)", "tactic": "cases nonempty_fintype (fixedPoints G α)" }, { "state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\nhpα : ¬p ∣ card α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\n⊢ card ↑(fixedPoints G α) ≠ 0", "state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\nhpα : ¬p ∣ card α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\n⊢ Nonempty ↑(fixedPoints G α)", "tactic": "rw [← card_pos_iff, pos_iff_ne_zero]" }, { "state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\nhpα : card ↑(fixedPoints G α) = 0\n⊢ p ∣ card α", "state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\nhpα : ¬p ∣ card α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\n⊢ card ↑(fixedPoints G α) ≠ 0", "tactic": "contrapose! hpα" }, { "state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\nhpα : card ↑(fixedPoints G α) = 0\n⊢ card α ≡ card ↑(fixedPoints G α) [MOD p]", "state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\nhpα : card ↑(fixedPoints G α) = 0\n⊢ p ∣ card α", "tactic": "rw [← Nat.modEq_zero_iff_dvd, ← hpα]" }, { "state_after": "no goals", "state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_1\ninst✝² : MulAction G α\ninst✝¹ : Fintype α\ninst✝ : Finite ↑(fixedPoints G α)\nval✝ : Fintype ↑(fixedPoints G α)\nhpα : card ↑(fixedPoints G α) = 0\n⊢ card α ≡ card ↑(fixedPoints G α) [MOD p]", "tactic": "exact hG.card_modEq_card_fixedPoints α" } ]

[ 231, 46 ]

[ 223, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nx y : FreeSemigroup α\n⊢ length (x * y) = length x + length y", "tactic": "simp [length, ← add_assoc, add_right_comm, List.length, List.length_append]" } ]

[ 498, 78 ]

[ 497, 1 ]

[ { "state_after": "case inl\nx y : ℝ≥0∞\n⊢ (x * y) ^ 0 = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ 0 < 0 then ⊤ else x ^ 0 * y ^ 0\n\ncase inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "state_before": "x y : ℝ≥0∞\nz : ℝ\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "rcases eq_or_ne z 0 with (rfl | hz)" }, { "state_after": "case inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "state_before": "case inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "replace hz := hz.lt_or_lt" }, { "state_after": "case inr.inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nthis :\n ∀ (x y : ℝ≥0∞) (z : ℝ),\n z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z\nhxy : ¬x ≤ y\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z\n\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "state_before": "case inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "wlog hxy : x ≤ y" }, { "state_after": "case inl\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : 0 ≤ y\n⊢ (0 * y) ^ z = if (0 = 0 ∧ y = ⊤ ∨ 0 = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else 0 ^ z * y ^ z\n\ncase inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\nhx0 : x ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "rcases eq_or_ne x 0 with (rfl | hx0)" }, { "state_after": "case inr.inl\nx : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhx0 : x ≠ 0\nhxy : x ≤ 0\n⊢ (x * 0) ^ z = if (x = 0 ∧ 0 = ⊤ ∨ x = ⊤ ∧ 0 = 0) ∧ z < 0 then ⊤ else x ^ z * 0 ^ z\n\ncase inr.inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\nhx0 : x ≠ 0\nhy0 : y ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "state_before": "case inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\nhx0 : x ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "rcases eq_or_ne y 0 with (rfl | hy0)" }, { "state_after": "case inr.inr.top\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhy0 : y ≠ 0\nhxy : ⊤ ≤ y\nhx0 : ⊤ ≠ 0\n⊢ (⊤ * y) ^ z = if (⊤ = 0 ∧ y = ⊤ ∨ ⊤ = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else ⊤ ^ z * y ^ z\n\ncase inr.inr.coe\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhy0 : y ≠ 0\nx✝ : ℝ≥0\nhxy : ↑x✝ ≤ y\nhx0 : ↑x✝ ≠ 0\n⊢ (↑x✝ * y) ^ z = if (↑x✝ = 0 ∧ y = ⊤ ∨ ↑x✝ = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * y ^ z", "state_before": "case inr.inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : x ≤ y\nhx0 : x ≠ 0\nhy0 : y ≠ 0\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "induction x using ENNReal.recTopCoe" }, { "state_after": "case inr.inr.coe.top\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝ : ℝ≥0\nhx0 : ↑x✝ ≠ 0\nhy0 : ⊤ ≠ 0\nhxy : ↑x✝ ≤ ⊤\n⊢ (↑x✝ * ⊤) ^ z = if (↑x✝ = 0 ∧ ⊤ = ⊤ ∨ ↑x✝ = ⊤ ∧ ⊤ = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * ⊤ ^ z\n\ncase inr.inr.coe.coe\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝¹ : ℝ≥0\nhx0 : ↑x✝¹ ≠ 0\nx✝ : ℝ≥0\nhy0 : ↑x✝ ≠ 0\nhxy : ↑x✝¹ ≤ ↑x✝\n⊢ (↑x✝¹ * ↑x✝) ^ z = if (↑x✝¹ = 0 ∧ ↑x✝ = ⊤ ∨ ↑x✝¹ = ⊤ ∧ ↑x✝ = 0) ∧ z < 0 then ⊤ else ↑x✝¹ ^ z * ↑x✝ ^ z", "state_before": "case inr.inr.coe\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhy0 : y ≠ 0\nx✝ : ℝ≥0\nhxy : ↑x✝ ≤ y\nhx0 : ↑x✝ ≠ 0\n⊢ (↑x✝ * y) ^ z = if (↑x✝ = 0 ∧ y = ⊤ ∨ ↑x✝ = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * y ^ z", "tactic": "induction y using ENNReal.recTopCoe" }, { "state_after": "case inr.inr.coe.coe\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝¹ : ℝ≥0\nhx0 : ↑x✝¹ ≠ 0\nx✝ : ℝ≥0\nhy0 : ↑x✝ ≠ 0\nhxy : ↑x✝¹ ≤ ↑x✝\n⊢ (↑x✝¹ * ↑x✝) ^ z = ↑x✝¹ ^ z * ↑x✝ ^ z", "state_before": "case inr.inr.coe.coe\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝¹ : ℝ≥0\nhx0 : ↑x✝¹ ≠ 0\nx✝ : ℝ≥0\nhy0 : ↑x✝ ≠ 0\nhxy : ↑x✝¹ ≤ ↑x✝\n⊢ (↑x✝¹ * ↑x✝) ^ z = if (↑x✝¹ = 0 ∧ ↑x✝ = ⊤ ∨ ↑x✝¹ = ⊤ ∧ ↑x✝ = 0) ∧ z < 0 then ⊤ else ↑x✝¹ ^ z * ↑x✝ ^ z", "tactic": "simp only [*, false_and_iff, and_false_iff, false_or_iff, if_false]" }, { "state_after": "case inr.inr.coe.coe\nz : ℝ\nx✝¹ x✝ : ℝ≥0\nhz : z < 0 ∨ 0 < z\nhx0 : ¬x✝¹ = 0\nhy0 : ¬x✝ = 0\nhxy : x✝¹ ≤ x✝\n⊢ ↑(x✝¹ * x✝) ^ z = ↑x✝¹ ^ z * ↑x✝ ^ z", "state_before": "case inr.inr.coe.coe\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝¹ : ℝ≥0\nhx0 : ↑x✝¹ ≠ 0\nx✝ : ℝ≥0\nhy0 : ↑x✝ ≠ 0\nhxy : ↑x✝¹ ≤ ↑x✝\n⊢ (↑x✝¹ * ↑x✝) ^ z = ↑x✝¹ ^ z * ↑x✝ ^ z", "tactic": "norm_cast at *" }, { "state_after": "case inr.inr.coe.coe\nz : ℝ\nx✝¹ x✝ : ℝ≥0\nhz : z < 0 ∨ 0 < z\nhx0 : ¬x✝¹ = 0\nhy0 : ¬x✝ = 0\nhxy : x✝¹ ≤ x✝\n⊢ ↑(x✝¹ ^ z * x✝ ^ z) = ↑x✝¹ ^ z * ↑x✝ ^ z", "state_before": "case inr.inr.coe.coe\nz : ℝ\nx✝¹ x✝ : ℝ≥0\nhz : z < 0 ∨ 0 < z\nhx0 : ¬x✝¹ = 0\nhy0 : ¬x✝ = 0\nhxy : x✝¹ ≤ x✝\n⊢ ↑(x✝¹ * x✝) ^ z = ↑x✝¹ ^ z * ↑x✝ ^ z", "tactic": "rw [coe_rpow_of_ne_zero (mul_ne_zero hx0 hy0), NNReal.mul_rpow]" }, { "state_after": "no goals", "state_before": "case inr.inr.coe.coe\nz : ℝ\nx✝¹ x✝ : ℝ≥0\nhz : z < 0 ∨ 0 < z\nhx0 : ¬x✝¹ = 0\nhy0 : ¬x✝ = 0\nhxy : x✝¹ ≤ x✝\n⊢ ↑(x✝¹ ^ z * x✝ ^ z) = ↑x✝¹ ^ z * ↑x✝ ^ z", "tactic": "norm_cast" }, { "state_after": "no goals", "state_before": "case inl\nx y : ℝ≥0∞\n⊢ (x * y) ^ 0 = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ 0 < 0 then ⊤ else x ^ 0 * y ^ 0", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case inr.inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nthis :\n ∀ (x y : ℝ≥0∞) (z : ℝ),\n z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z\nhxy : ¬x ≤ y\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z", "tactic": "convert this y x z hz (le_of_not_le hxy) using 2 <;> simp only [mul_comm, and_comm, or_comm]" }, { "state_after": "no goals", "state_before": "case inl\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhxy : 0 ≤ y\n⊢ (0 * y) ^ z = if (0 = 0 ∧ y = ⊤ ∨ 0 = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else 0 ^ z * y ^ z", "tactic": "induction y using ENNReal.recTopCoe <;> cases' hz with hz hz <;> simp [*, hz.not_lt]" }, { "state_after": "no goals", "state_before": "case inr.inl\nx : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhx0 : x ≠ 0\nhxy : x ≤ 0\n⊢ (x * 0) ^ z = if (x = 0 ∧ 0 = ⊤ ∨ x = ⊤ ∧ 0 = 0) ∧ z < 0 then ⊤ else x ^ z * 0 ^ z", "tactic": "exact (hx0 (bot_unique hxy)).elim" }, { "state_after": "no goals", "state_before": "case inr.inr.top\ny : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nhy0 : y ≠ 0\nhxy : ⊤ ≤ y\nhx0 : ⊤ ≠ 0\n⊢ (⊤ * y) ^ z = if (⊤ = 0 ∧ y = ⊤ ∨ ⊤ = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else ⊤ ^ z * y ^ z", "tactic": "cases' hz with hz hz <;> simp [hz, top_unique hxy]" }, { "state_after": "case inr.inr.coe.top\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝ : ℝ≥0\nhx0 : ¬x✝ = 0\nhy0 : ⊤ ≠ 0\nhxy : ↑x✝ ≤ ⊤\n⊢ (↑x✝ * ⊤) ^ z = if (↑x✝ = 0 ∧ ⊤ = ⊤ ∨ ↑x✝ = ⊤ ∧ ⊤ = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * ⊤ ^ z", "state_before": "case inr.inr.coe.top\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝ : ℝ≥0\nhx0 : ↑x✝ ≠ 0\nhy0 : ⊤ ≠ 0\nhxy : ↑x✝ ≤ ⊤\n⊢ (↑x✝ * ⊤) ^ z = if (↑x✝ = 0 ∧ ⊤ = ⊤ ∨ ↑x✝ = ⊤ ∧ ⊤ = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * ⊤ ^ z", "tactic": "rw [ne_eq, coe_eq_zero] at hx0" }, { "state_after": "no goals", "state_before": "case inr.inr.coe.top\nz : ℝ\nhz : z < 0 ∨ 0 < z\nx✝ : ℝ≥0\nhx0 : ¬x✝ = 0\nhy0 : ⊤ ≠ 0\nhxy : ↑x✝ ≤ ⊤\n⊢ (↑x✝ * ⊤) ^ z = if (↑x✝ = 0 ∧ ⊤ = ⊤ ∨ ↑x✝ = ⊤ ∧ ⊤ = 0) ∧ z < 0 then ⊤ else ↑x✝ ^ z * ⊤ ^ z", "tactic": "cases' hz with hz hz <;> simp [*]" } ]

[ 501, 12 ]

[ 483, 1 ]

[ { "state_after": "C : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\nJ : Type w\ninst✝⁴ : SmallCategory J\nF : J ⥤ C\ninst✝³ : PreservesLimit F (forget C)\nB : C\nι : Type w\ninst✝² : Nonempty ι\nX : ι → C\nf : (j : ι) → X j ⟶ B\ninst✝¹ : HasWidePullback B X f\ninst✝ : PreservesLimit (wideCospan B X f) (forget C)\nx y : (forget C).obj (widePullback B X f)\nh : ∀ (j : ι), (forget C).map (π f j) x = (forget C).map (π f j) y\n⊢ (forget C).map (base f) x = (forget C).map (base f) y", "state_before": "C : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\nJ : Type w\ninst✝⁴ : SmallCategory J\nF : J ⥤ C\ninst✝³ : PreservesLimit F (forget C)\nB : C\nι : Type w\ninst✝² : Nonempty ι\nX : ι → C\nf : (j : ι) → X j ⟶ B\ninst✝¹ : HasWidePullback B X f\ninst✝ : PreservesLimit (wideCospan B X f) (forget C)\nx y : (forget C).obj (widePullback B X f)\nh : ∀ (j : ι), (forget C).map (π f j) x = (forget C).map (π f j) y\n⊢ x = y", "tactic": "apply Concrete.widePullback_ext _ _ _ _ h" }, { "state_after": "C : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\nJ : Type w\ninst✝⁴ : SmallCategory J\nF : J ⥤ C\ninst✝³ : PreservesLimit F (forget C)\nB : C\nι : Type w\ninst✝² : Nonempty ι\nX : ι → C\nf : (j : ι) → X j ⟶ B\ninst✝¹ : HasWidePullback B X f\ninst✝ : PreservesLimit (wideCospan B X f) (forget C)\nx y : (forget C).obj (widePullback B X f)\nh : ∀ (j : ι), (forget C).map (π f j) x = (forget C).map (π f j) y\ninhabited_h : Inhabited ι\n⊢ (forget C).map (base f) x = (forget C).map (base f) y", "state_before": "C : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\nJ : Type w\ninst✝⁴ : SmallCategory J\nF : J ⥤ C\ninst✝³ : PreservesLimit F (forget C)\nB : C\nι : Type w\ninst✝² : Nonempty ι\nX : ι → C\nf : (j : ι) → X j ⟶ B\ninst✝¹ : HasWidePullback B X f\ninst✝ : PreservesLimit (wideCospan B X f) (forget C)\nx y : (forget C).obj (widePullback B X f)\nh : ∀ (j : ι), (forget C).map (π f j) x = (forget C).map (π f j) y\n⊢ (forget C).map (base f) x = (forget C).map (base f) y", "tactic": "inhabit ι" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\nJ : Type w\ninst✝⁴ : SmallCategory J\nF : J ⥤ C\ninst✝³ : PreservesLimit F (forget C)\nB : C\nι : Type w\ninst✝² : Nonempty ι\nX : ι → C\nf : (j : ι) → X j ⟶ B\ninst✝¹ : HasWidePullback B X f\ninst✝ : PreservesLimit (wideCospan B X f) (forget C)\nx y : (forget C).obj (widePullback B X f)\nh : ∀ (j : ι), (forget C).map (π f j) x = (forget C).map (π f j) y\ninhabited_h : Inhabited ι\n⊢ (forget C).map (base f) x = (forget C).map (base f) y", "tactic": "simp only [← π_arrow f default, comp_apply, h]" } ]

[ 85, 49 ]

[ 79, 1 ]

[]

[ 987, 78 ]

[ 987, 1 ]

[]

[ 415, 29 ]

[ 413, 1 ]

[]

[ 107, 79 ]

[ 107, 1 ]

[]

[ 231, 53 ]

[ 229, 11 ]

[ { "state_after": "no goals", "state_before": "ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\n⊢ Dependent f ↔ ¬Independent f", "tactic": "rw [dependent_iff, independent_iff]" } ]

[ 101, 38 ]

[ 100, 1 ]

[ { "state_after": "α : Type ?u.455352\nβ : Type ?u.455355\nγ : Type ?u.455358\nδ : Type ?u.455361\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalAddGroup α\nf✝ g : β → α\na✝ a₁ a₂ : α\nM : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nf : ℕ → M\nk : ℕ\na : M\nh : HasSum (fun n => f (n + k)) a\n⊢ HasSum (f ∘ Subtype.val) a", "state_before": "α : Type ?u.455352\nβ : Type ?u.455355\nγ : Type ?u.455358\nδ : Type ?u.455361\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalAddGroup α\nf✝ g : β → α\na✝ a₁ a₂ : α\nM : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nf : ℕ → M\nk : ℕ\na : M\nh : HasSum (fun n => f (n + k)) a\n⊢ HasSum f (∑ i in range k, f i + a)", "tactic": "refine ((range k).hasSum f).add_compl ?_" }, { "state_after": "no goals", "state_before": "α : Type ?u.455352\nβ : Type ?u.455355\nγ : Type ?u.455358\nδ : Type ?u.455361\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalAddGroup α\nf✝ g : β → α\na✝ a₁ a₂ : α\nM : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nf : ℕ → M\nk : ℕ\na : M\nh : HasSum (fun n => f (n + k)) a\n⊢ HasSum (f ∘ Subtype.val) a", "tactic": "rwa [← (notMemRangeEquiv k).symm.hasSum_iff]" } ]

[ 961, 47 ]

[ 958, 1 ]

[]

[ 1175, 53 ]

[ 1173, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.2249\nf : α ↪ β\ns : Finset α\nb : β\n⊢ (∃ a, a ∈ s.val ∧ ↑f a = b) ↔ ∃ a, a ∈ s ∧ ↑f a = b", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.2249\nf : α ↪ β\ns : Finset α\nb : β\n⊢ (∃ a, a ∈ s.val ∧ ↑f a = b) ↔ ∃ a, a ∈ s ∧ ↑f a = b", "tactic": "simp only [exists_prop]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.2249\nf : α ↪ β\ns : Finset α\nb : β\n⊢ (∃ a, a ∈ s.val ∧ ↑f a = b) ↔ ∃ a, a ∈ s ∧ ↑f a = b", "tactic": "rfl" } ]

[ 71, 51 ]

[ 70, 1 ]

[]

[ 102, 6 ]

[ 99, 1 ]

[]

[ 141, 29 ]

[ 140, 1 ]

[ { "state_after": "no goals", "state_before": "f : ℝ → ℝ\nf' a b : ℝ\nh : IsLocalMin f a\nhf : HasDerivAt f f' a\n⊢ f' = 0", "tactic": "simpa using FunLike.congr_fun (h.hasFDerivAt_eq_zero (hasDerivAt_iff_hasFDerivAt.1 hf)) 1" } ]

[ 231, 92 ]

[ 230, 1 ]

[]

[ 303, 69 ]

[ 301, 1 ]

[]

[ 212, 31 ]

[ 210, 11 ]

[]

[ 523, 6 ]

[ 521, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nX : Type ?u.20200\nι : Type ?u.20203\ninst✝ : PseudoMetricSpace α\nx y : α\nc : ℝ≥0\n⊢ edist x y < ↑c ↔ nndist x y < c", "tactic": "rw [edist_nndist, ENNReal.coe_lt_coe]" } ]

[ 333, 40 ]

[ 332, 1 ]

[]

[ 589, 43 ]

[ 587, 1 ]

[ { "state_after": "α : Type u_1\na : Array α\n⊢ (if h : size a ≤ 1 then a\n else\n reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "state_before": "α : Type u_1\na : Array α\n⊢ (reverse a).data = List.reverse a.data", "tactic": "simp only [reverse]" }, { "state_after": "case inl\nα : Type u_1\na : Array α\nh✝ : size a ≤ 1\n⊢ a.data = List.reverse a.data\n\ncase inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\n⊢ (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "state_before": "α : Type u_1\na : Array α\n⊢ (if h : size a ≤ 1 then a\n else\n reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "tactic": "split" }, { "state_after": "α : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n⊢ List.get?\n (if h : i < { val := j, isLt := hj }.val then\n let_fun this := (_ : { val := j, isLt := hj }.val - 1 - (i + 1) < { val := j, isLt := hj }.val - i);\n let as_1 := swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj };\n let_fun this := (_ : { val := j, isLt := hj }.val - 1 < size as_1);\n reverse.loop as_1 (i + 1) { val := { val := j, isLt := hj }.val - 1, isLt := this }\n else as).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "α : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n⊢ List.get? (reverse.loop as i { val := j, isLt := hj }).data k = List.get? (List.reverse a.data) k", "tactic": "rw [reverse.loop]" }, { "state_after": "α : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n⊢ List.get?\n (if h : i < j then\n reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt := (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }\n else as).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "α : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n⊢ List.get?\n (if h : i < { val := j, isLt := hj }.val then\n let_fun this := (_ : { val := j, isLt := hj }.val - 1 - (i + 1) < { val := j, isLt := hj }.val - i);\n let as_1 := swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj };\n let_fun this := (_ : { val := j, isLt := hj }.val - 1 < size as_1);\n reverse.loop as_1 (i + 1) { val := { val := j, isLt := hj }.val - 1, isLt := this }\n else as).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "dsimp" }, { "state_after": "case inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : i < j\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt :=\n (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k\n\ncase inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\n⊢ List.get? as.data k = List.get? (List.reverse a.data) k", "state_before": "α : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\n⊢ List.get?\n (if h : i < j then\n reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt := (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }\n else as).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "split <;> rename_i h₁" }, { "state_after": "case inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : i < j\nthis : j - 1 - (i + 1) < j - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt :=\n (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "case inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : i < j\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt :=\n (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "have := reverse.termination h₁" }, { "state_after": "case inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j + 1 - 1 - (i + 1) < j + 1 - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n { val := j + 1 - 1,\n isLt :=\n (_ :\n j + 1 - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "case inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : i < j\nthis : j - 1 - (i + 1) < j - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)\n { val := j - 1,\n isLt :=\n (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "match j with | j+1 => ?_" }, { "state_after": "case inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n { val := j,\n isLt :=\n (_ : j < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "case inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j + 1 - 1 - (i + 1) < j + 1 - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n { val := j + 1 - 1,\n isLt :=\n (_ :\n j + 1 - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "simp at *" }, { "state_after": "case inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n { val := j,\n isLt :=\n (_ : j < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "state_before": "case inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ List.get?\n (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)\n { val := j,\n isLt :=\n (_ : j < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data\n k =\n List.get? (List.reverse a.data) k", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case inl.h\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ i + 1 + j + 1 = size a", "tactic": "rwa [Nat.add_right_comm i]" }, { "state_after": "no goals", "state_before": "case inl.h₂\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) = size a", "tactic": "simp [size_swap, h₂]" }, { "state_after": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\n⊢ ∀ (k : Nat),\n List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "intro k" }, { "state_after": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ (if { val := j + 1, isLt := hj }.val = k then some as[{ val := i, isLt := (_ : i < size as) }.val]\n else\n if { val := i, isLt := (_ : i < size as) }.val = k then some as[{ val := j + 1, isLt := hj }.val] else as[k]?) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "rw [← getElem?_eq_data_get?, get?_swap]" }, { "state_after": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ (if j + 1 = k then List.get? a.data i\n else\n if i = k then List.get? a.data (j + 1)\n else if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ (if { val := j + 1, isLt := hj }.val = k then some as[{ val := i, isLt := (_ : i < size as) }.val]\n else\n if { val := i, isLt := (_ : i < size as) }.val = k then some as[{ val := j + 1, isLt := hj }.val] else as[k]?) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "simp [getElem?_eq_data_get?, getElem_eq_data_get, ← List.get?_eq_get, H, Nat.le_of_lt h₁]" }, { "state_after": "case inl.H.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : j + 1 = k\n⊢ List.get? a.data i = if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\n\ncase inl.H.inr\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\n⊢ (if i = k then List.get? a.data (j + 1)\n else if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inl.H\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\n⊢ (if j + 1 = k then List.get? a.data i\n else\n if i = k then List.get? a.data (j + 1)\n else if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "split <;> rename_i h₂" }, { "state_after": "case inl.H.inr.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : i = k\n⊢ List.get? a.data (j + 1) = if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\n\ncase inl.H.inr.inr\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : ¬i = k\n⊢ (if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inl.H.inr\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\n⊢ (if i = k then List.get? a.data (j + 1)\n else if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "split <;> rename_i h₃" }, { "state_after": "no goals", "state_before": "case inl.H.inr.inr\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : ¬i = k\n⊢ (if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =\n if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "simp only [Nat.succ_le, Nat.lt_iff_le_and_ne.trans (and_iff_left h₃),\n Nat.lt_succ.symm.trans (Nat.lt_iff_le_and_ne.trans (and_iff_left (Ne.symm h₂)))]" }, { "state_after": "case inl.H.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : j + 1 = k\n⊢ List.get? a.data i = List.get? (List.reverse a.data) (j + 1)", "state_before": "case inl.H.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : j + 1 = k\n⊢ List.get? a.data i = if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "simp [← h₂, Nat.not_le.2 (Nat.lt_succ_self _)]" }, { "state_after": "no goals", "state_before": "case inl.H.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : j + 1 = k\n⊢ List.get? a.data i = List.get? (List.reverse a.data) (j + 1)", "tactic": "exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm" }, { "state_after": "no goals", "state_before": "α : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : j + 1 = k\n⊢ j + 1 + i + 1 = i + (j + 1) + 1", "tactic": "simp_arith" }, { "state_after": "case inl.H.inr.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : i = k\n⊢ List.get? a.data (j + 1) = List.get? (List.reverse a.data) i", "state_before": "case inl.H.inr.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : i = k\n⊢ List.get? a.data (j + 1) = if i + 1 ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "simp [← h₃, Nat.not_le.2 (Nat.lt_succ_self _)]" }, { "state_after": "no goals", "state_before": "case inl.H.inr.inl\nα : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : i = k\n⊢ List.get? a.data (j + 1) = List.get? (List.reverse a.data) i", "tactic": "exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm" }, { "state_after": "no goals", "state_before": "α : Type u_1\na as : Array α\ni j✝ : Nat\nh₂✝ : size as = size a\nk✝ j : Nat\nhj : j + 1 < size as\nh : i + (j + 1) + 1 = size a\nH :\n ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : i < j + 1\nthis : j - (i + 1) < j + 1 - i\nk : Nat\nh₂ : ¬j + 1 = k\nh₃ : i = k\n⊢ i + (j + 1) + 1 = i + (j + 1) + 1", "tactic": "simp_arith" }, { "state_after": "case inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\n⊢ (if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k) = List.get? (List.reverse a.data) k", "state_before": "case inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\n⊢ List.get? as.data k = List.get? (List.reverse a.data) k", "tactic": "rw [H]" }, { "state_after": "case inr.inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂✝ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\nh₂ : i ≤ k ∧ k ≤ j\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k\n\ncase inr.inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂✝ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\nh₂ : ¬(i ≤ k ∧ k ≤ j)\n⊢ List.get? (List.reverse a.data) k = List.get? (List.reverse a.data) k", "state_before": "case inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\n⊢ (if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k) = List.get? (List.reverse a.data) k", "tactic": "split <;> rename_i h₂" }, { "state_after": "case inr.inl.refl\nα : Type u_1\na as : Array α\ni : Nat\nh₂✝ : size as = size a\nk : Nat\nhj : i < size as\nh : i + i + 1 = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ i then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : ¬i < i\nh₂ : i ≤ k ∧ k ≤ i\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "state_before": "case inr.inl\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂✝ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\nh₂ : i ≤ k ∧ k ≤ j\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "cases Nat.le_antisymm (Nat.not_lt.1 h₁) (Nat.le_trans h₂.1 h₂.2)" }, { "state_after": "case inr.inl.refl.refl\nα : Type u_1\na as : Array α\ni : Nat\nh₂✝ : size as = size a\nhj : i < size as\nh : i + i + 1 = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ i then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : ¬i < i\nh₂ : i ≤ i ∧ i ≤ i\n⊢ List.get? a.data i = List.get? (List.reverse a.data) i", "state_before": "case inr.inl.refl\nα : Type u_1\na as : Array α\ni : Nat\nh₂✝ : size as = size a\nk : Nat\nhj : i < size as\nh : i + i + 1 = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ i then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : ¬i < i\nh₂ : i ≤ k ∧ k ≤ i\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "cases Nat.le_antisymm h₂.1 h₂.2" }, { "state_after": "no goals", "state_before": "case inr.inl.refl.refl\nα : Type u_1\na as : Array α\ni : Nat\nh₂✝ : size as = size a\nhj : i < size as\nh : i + i + 1 = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ i then List.get? a.data k else List.get? (List.reverse a.data) k\nh₁ : ¬i < i\nh₂ : i ≤ i ∧ i ≤ i\n⊢ List.get? a.data i = List.get? (List.reverse a.data) i", "tactic": "exact (List.get?_reverse' _ _ h).symm" }, { "state_after": "no goals", "state_before": "case inr.inr\nα : Type u_1\na as : Array α\ni j : Nat\nhj : j < size as\nh : i + j + 1 = size a\nh₂✝ : size as = size a\nH : ∀ (k : Nat), List.get? as.data k = if i ≤ k ∧ k ≤ j then List.get? a.data k else List.get? (List.reverse a.data) k\nk : Nat\nh₁ : ¬i < j\nh₂ : ¬(i ≤ k ∧ k ≤ j)\n⊢ List.get? (List.reverse a.data) k = List.get? (List.reverse a.data) k", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\na : Array α\nh✝ : size a ≤ 1\n⊢ a.data = List.reverse a.data", "tactic": "match a with | ⟨[]⟩ | ⟨[_]⟩ => rfl" }, { "state_after": "no goals", "state_before": "α : Type u_1\na : Array α\nhead✝ : α\nh✝ : size { data := [head✝] } ≤ 1\n⊢ { data := [head✝] }.data = List.reverse { data := [head✝] }.data", "tactic": "rfl" }, { "state_after": "case inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\n⊢ (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "state_before": "case inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\n⊢ (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "tactic": "have := Nat.sub_add_cancel (Nat.le_of_not_le ‹_›)" }, { "state_after": "case inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\n⊢ List.get? a.data k = if 0 ≤ k ∧ k ≤ size a - 1 then List.get? a.data k else List.get? (List.reverse a.data) k", "state_before": "case inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\n⊢ (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =\n List.reverse a.data", "tactic": "refine List.ext <| go _ _ _ _ (by simp [this]) rfl fun k => ?_" }, { "state_after": "case inr.inl\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : 0 ≤ k ∧ k ≤ size a - 1\n⊢ List.get? a.data k = List.get? a.data k\n\ncase inr.inr\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "state_before": "case inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\n⊢ List.get? a.data k = if 0 ≤ k ∧ k ≤ size a - 1 then List.get? a.data k else List.get? (List.reverse a.data) k", "tactic": "split" }, { "state_after": "case inr.inr\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "state_before": "case inr.inl\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : 0 ≤ k ∧ k ≤ size a - 1\n⊢ List.get? a.data k = List.get? a.data k\n\ncase inr.inr\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "{rfl}" }, { "state_after": "case inr.inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "state_before": "case inr.inr\nα : Type u_1\na : Array α\nh✝¹ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh✝ : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "rename_i h" }, { "state_after": "case inr.inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : size a ≤ k\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "state_before": "case inr.inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : ¬(0 ≤ k ∧ k ≤ size a - 1)\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "simp [← show k < _ + 1 ↔ _ from Nat.lt_succ (n := a.size - 1), this] at h" }, { "state_after": "no goals", "state_before": "case inr.inr\nα : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\nk : Nat\nh : size a ≤ k\n⊢ List.get? a.data k = List.get? (List.reverse a.data) k", "tactic": "rw [List.get?_eq_none.2 ‹_›, List.get?_eq_none.2 (a.data.length_reverse ▸ ‹_›)]" }, { "state_after": "no goals", "state_before": "α : Type u_1\na : Array α\nh✝ : ¬size a ≤ 1\nthis : size a - 1 + 1 = size a\n⊢ 0 + (size a - 1) + 1 = size a", "tactic": "simp [this]" } ]

[ 265, 26 ]

[ 230, 9 ]

[]

[ 698, 78 ]

[ 697, 1 ]

[]

[ 85, 18 ]

[ 84, 8 ]

[ { "state_after": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\nq : G ⧸ H\n⊢ ↑(toEquiv (_ : (Set.range fun q => f q) ∈ leftTransversals ↑H)) q =\n { val := f q, property := (_ : ∃ y, (fun q => f q) y = f q) }", "state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\nq : G ⧸ H\n⊢ ↑(↑(toEquiv (_ : (Set.range fun q => f q) ∈ leftTransversals ↑H)) q) = f q", "tactic": "refine' (Subtype.ext_iff.mp _).trans (Subtype.coe_mk (f q) ⟨q, rfl⟩)" }, { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\nq : G ⧸ H\n⊢ ↑(toEquiv (_ : (Set.range fun q => f q) ∈ leftTransversals ↑H)) q =\n { val := f q, property := (_ : ∃ y, (fun q => f q) y = f q) }", "tactic": "exact (toEquiv (range_mem_leftTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm" } ]

[ 359, 93 ]

[ 356, 1 ]

[]

[ 620, 54 ]

[ 617, 1 ]

[]

[ 1365, 47 ]

[ 1364, 1 ]

[ { "state_after": "case h.e'_2\nα : Type u_1\nβ : Type ?u.363947\nγ : Type ?u.363950\ninst✝ : DecidableEq α\na b : α\nn : ℕ\n⊢ count a (replicate n b) = List.count a (List.replicate n b)", "state_before": "α : Type u_1\nβ : Type ?u.363947\nγ : Type ?u.363950\ninst✝ : DecidableEq α\na b : α\nn : ℕ\n⊢ count a (replicate n b) = if a = b then n else 0", "tactic": "convert List.count_replicate a b n" }, { "state_after": "no goals", "state_before": "case h.e'_2\nα : Type u_1\nβ : Type ?u.363947\nγ : Type ?u.363950\ninst✝ : DecidableEq α\na b : α\nn : ℕ\n⊢ count a (replicate n b) = List.count a (List.replicate n b)", "tactic": "rw [←coe_count, coe_replicate]" } ]

[ 2440, 33 ]

[ 2438, 1 ]

[]

[ 72, 4 ]

[ 70, 1 ]

[ { "state_after": "no goals", "state_before": "G : Type u_1\nG' : Type ?u.665\nF : Type ?u.668\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\n⊢ ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁", "tactic": "rw [commutatorElement_def, mul_inv_eq_one, mul_inv_eq_iff_eq_mul]" } ]

[ 33, 68 ]

[ 32, 1 ]

[]

[ 121, 33 ]

[ 119, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.160671\nγ : Type ?u.160674\ninst✝ : DecidableEq α\ns s₁ s₂ t t₁ t₂ u v : Finset α\na b : α\n⊢ _root_.Disjoint s (t ∪ u) ↔ _root_.Disjoint s t ∧ _root_.Disjoint s u", "tactic": "simp only [disjoint_right, mem_union, or_imp, forall_and]" } ]

[ 1508, 60 ]

[ 1507, 1 ]

[]

[ 308, 25 ]

[ 307, 1 ]

[]

[ 1134, 10 ]

[ 1133, 1 ]

[]

[ 242, 16 ]

[ 241, 1 ]

[]

[ 117, 82 ]

[ 116, 1 ]

[ { "state_after": "case mp\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ 1 ≤ 1 / I → I ≤ 1\n\ncase mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ I ≤ 1 → 1 ≤ 1 / I", "state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ 1 ≤ 1 / I ↔ I ≤ 1", "tactic": "constructor" }, { "state_after": "case mp\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\nhI : 1 ≤ 1 / I\n⊢ I ≤ 1\n\ncase mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\nhI : I ≤ 1\n⊢ 1 ≤ 1 / I", "state_before": "case mp\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ 1 ≤ 1 / I → I ≤ 1\n\ncase mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ I ≤ 1 → 1 ≤ 1 / I", "tactic": "all_goals intro hI" }, { "state_after": "case mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\nhI : I ≤ 1\n⊢ 1 ≤ 1 / I", "state_before": "case mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\n⊢ I ≤ 1 → 1 ≤ 1 / I", "tactic": "intro hI" }, { "state_after": "no goals", "state_before": "case mp\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\nhI : 1 ≤ 1 / I\n⊢ I ≤ 1", "tactic": "rwa [le_div_iff_mul_le, one_mul] at hI" }, { "state_after": "no goals", "state_before": "case mpr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nI : Submodule R A\nhI : I ≤ 1\n⊢ 1 ≤ 1 / I", "tactic": "rwa [le_div_iff_mul_le, one_mul]" } ]

[ 712, 37 ]

[ 709, 1 ]

[]

[ 360, 6 ]

[ 359, 1 ]

[ { "state_after": "case inl\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nab : a ≤ b\n⊢ max a b - min a b = abs (a - b)\n\ncase inr\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nba : b ≤ a\n⊢ max a b - min a b = abs (a - b)", "state_before": "α : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\n⊢ max a b - min a b = abs (a - b)", "tactic": "cases' le_total a b with ab ba" }, { "state_after": "case inl\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nab : a ≤ b\n⊢ a - b ≤ 0", "state_before": "case inl\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nab : a ≤ b\n⊢ max a b - min a b = abs (a - b)", "tactic": "rw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub]" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nab : a ≤ b\n⊢ a - b ≤ 0", "tactic": "rwa [sub_nonpos]" }, { "state_after": "case inr\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nba : b ≤ a\n⊢ 0 ≤ a - b", "state_before": "case inr\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nba : b ≤ a\n⊢ max a b - min a b = abs (a - b)", "tactic": "rw [max_eq_left ba, min_eq_right ba, abs_of_nonneg]" }, { "state_after": "no goals", "state_before": "case inr\nα : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b : α\nba : b ≤ a\n⊢ 0 ≤ a - b", "tactic": "rwa [sub_nonneg]" } ]

[ 216, 21 ]

[ 211, 1 ]

[]

[ 2070, 31 ]

[ 2069, 1 ]

[ { "state_after": "a b c : Nat\nh : a + b < c\n⊢ succ (a + b) ≤ c", "state_before": "a b c : Nat\nh : a + b < c\n⊢ succ a + b ≤ c", "tactic": "simp [Nat.succ_add]" }, { "state_after": "no goals", "state_before": "a b c : Nat\nh : a + b < c\n⊢ succ (a + b) ≤ c", "tactic": "exact h" } ]

[ 681, 24 ]

[ 679, 1 ]

[ { "state_after": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x | EMetric.infEdist x s ≤ ENNReal.ofReal δ}⁻¹ = {x | EMetric.infEdist x⁻¹ s ≤ ENNReal.ofReal δ}", "state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ (cthickening δ s)⁻¹ = cthickening δ s⁻¹", "tactic": "simp_rw [cthickening, ← infEdist_inv]" }, { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x | EMetric.infEdist x s ≤ ENNReal.ofReal δ}⁻¹ = {x | EMetric.infEdist x⁻¹ s ≤ ENNReal.ofReal δ}", "tactic": "rfl" } ]

[ 89, 6 ]

[ 87, 1 ]

[ { "state_after": "case zero\nα : Type ?u.13242\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm : ℕ\n⊢ a ^ (m + Nat.zero) = a ^ m * a ^ Nat.zero\n\ncase succ\nα : Type ?u.13242\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm n : ℕ\nih : a ^ (m + n) = a ^ m * a ^ n\n⊢ a ^ (m + Nat.succ n) = a ^ m * a ^ Nat.succ n", "state_before": "α : Type ?u.13242\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm n : ℕ\n⊢ a ^ (m + n) = a ^ m * a ^ n", "tactic": "induction' n with n ih" }, { "state_after": "no goals", "state_before": "case zero\nα : Type ?u.13242\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm : ℕ\n⊢ a ^ (m + Nat.zero) = a ^ m * a ^ Nat.zero", "tactic": "rw [Nat.add_zero, pow_zero, mul_one]" }, { "state_after": "no goals", "state_before": "case succ\nα : Type ?u.13242\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nm n : ℕ\nih : a ^ (m + n) = a ^ m * a ^ n\n⊢ a ^ (m + Nat.succ n) = a ^ m * a ^ Nat.succ n", "tactic": "rw [pow_succ', ← mul_assoc, ← ih, ← pow_succ', Nat.add_assoc]" } ]

[ 122, 66 ]

[ 119, 1 ]

[]

[ 842, 78 ]

[ 839, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nh : a ∈ s\n⊢ s ∩ insert a t = insert a (s ∩ t)", "tactic": "rw [insert_inter_distrib, insert_eq_of_mem h]" } ]

[ 2003, 48 ]

[ 2002, 1 ]

[]

[ 282, 6 ]

[ 280, 1 ]

[ { "state_after": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nhf : LinearIndependent 𝕜 f\nval✝ : Fintype ι\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f, LinearIndependent 𝕜 g", "state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nhf : LinearIndependent 𝕜 f\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f, LinearIndependent 𝕜 g", "tactic": "cases nonempty_fintype ι" }, { "state_after": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nhf : LinearIndependent 𝕜 f\nval✝ : Fintype ι\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f, LinearIndependent 𝕜 g", "tactic": "simp only [Fintype.linearIndependent_iff'] at hf⊢" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "rcases LinearMap.exists_antilipschitzWith _ hf with ⟨K, K0, hK⟩" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 (∑ i : ι, ‖f i - f i‖))\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "have : Tendsto (fun g : ι → E => ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 <| ∑ i, ‖f i - f i‖) :=\n tendsto_finset_sum _ fun i _ =>\n Tendsto.norm <| ((continuous_apply i).tendsto _).sub tendsto_const_nhds" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 (∑ i : ι, ‖f i - f i‖))\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "simp only [sub_self, norm_zero, Finset.sum_const_zero] at this" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\n⊢ ∀ᶠ (g : ι → E) in 𝓝 f,\n LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "refine' (this.eventually (gt_mem_nhds <| inv_pos.2 K0)).mono fun g hg => _" }, { "state_after": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\n\ncase intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\n⊢ LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "replace hg : (∑ i, ‖g i - f i‖₊) < K⁻¹" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\n⊢ Function.Injective ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i))", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\n⊢ LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) = ⊥", "tactic": "rw [LinearMap.ker_eq_bot]" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\n⊢ dist\n ((↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)))\n v)\n ((↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)))\n u) ≤\n ↑(∑ i : ι, ‖g i - f i‖₊) * dist v u", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\n⊢ Function.Injective ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i))", "tactic": "refine' (hK.add_sub_lipschitzWith (LipschitzWith.of_dist_le_mul fun v u => _) hg).injective" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\n⊢ ‖∑ x : ι, (v x - u x) • (g x - f x)‖ ≤ ∑ x : ι, ‖g x - f x‖ * ‖v - u‖", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\n⊢ dist\n ((↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)))\n v)\n ((↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)))\n u) ≤\n ↑(∑ i : ι, ‖g i - f i‖₊) * dist v u", "tactic": "simp only [dist_eq_norm, LinearMap.lsum_apply, Pi.sub_apply, LinearMap.sum_apply,\n LinearMap.comp_apply, LinearMap.proj_apply, LinearMap.smulRight_apply, LinearMap.id_apply, ←\n Finset.sum_sub_distrib, ← smul_sub, ← sub_smul, NNReal.coe_sum, coe_nnnorm, Finset.sum_mul]" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖(v i - u i) • (g i - f i)‖ ≤ ‖g i - f i‖ * ‖v - u‖", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\n⊢ ‖∑ x : ι, (v x - u x) • (g x - f x)‖ ≤ ∑ x : ι, ‖g x - f x‖ * ‖v - u‖", "tactic": "refine' norm_sum_le_of_le _ fun i _ => _" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖g i - f i‖ * ‖v i - u i‖ ≤ ‖g i - f i‖ * ‖v - u‖", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖(v i - u i) • (g i - f i)‖ ≤ ‖g i - f i‖ * ‖v - u‖", "tactic": "rw [norm_smul, mul_comm]" }, { "state_after": "case intro.intro.intro.h\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖v i - u i‖ ≤ ‖v - u‖", "state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖g i - f i‖ * ‖v i - u i‖ ≤ ‖g i - f i‖ * ‖v - u‖", "tactic": "gcongr" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.h\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹\nv u : ι → 𝕜\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ ‖v i - u i‖ ≤ ‖v - u‖", "tactic": "exact norm_le_pi_norm (v - u) i" }, { "state_after": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ↑(∑ i : ι, ‖g i - f i‖₊) < ↑K⁻¹", "state_before": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ∑ i : ι, ‖g i - f i‖₊ < K⁻¹", "tactic": "rw [← NNReal.coe_lt_coe]" }, { "state_after": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ∑ x : ι, ‖g x - f x‖ < (↑K)⁻¹", "state_before": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ↑(∑ i : ι, ‖g i - f i‖₊) < ↑K⁻¹", "tactic": "push_cast" }, { "state_after": "no goals", "state_before": "case hg\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E\nval✝ : Fintype ι\nhf : LinearMap.ker (↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i)) = ⊥\nK : ℝ≥0\nK0 : K > 0\nhK : AntilipschitzWith K ↑(↑(LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)\ng : ι → E\nhg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹\n⊢ ∑ x : ι, ‖g x - f x‖ < (↑K)⁻¹", "tactic": "exact hg" } ]

[ 257, 34 ]

[ 235, 11 ]

[]

[ 198, 28 ]

[ 197, 11 ]

[ { "state_after": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.43778\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nh : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s", "state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.43778\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nh : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s\n⊢ ContDiffOn 𝕜 n f s", "tactic": "apply contDiffOn_of_differentiableOn" }, { "state_after": "no goals", "state_before": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.43778\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nh : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s", "tactic": "simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_differentiableOn_iff]" } ]

[ 151, 97 ]

[ 147, 1 ]

[ { "state_after": "m n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n find p", "state_before": "m n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\n⊢ { val := Nat.find h, isLt := (_ : Nat.find h < n) } ∈ find p", "tactic": "let ⟨i, hin, hi⟩ := h" }, { "state_after": "case none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nhf : find p = none\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n none\n\ncase some\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n some f", "state_before": "m n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n find p", "tactic": "cases' hf : find p with f" }, { "state_after": "case none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nhf : ∀ (i : Fin n), ¬p i\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n none", "state_before": "case none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nhf : find p = none\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n none", "tactic": "rw [find_eq_none_iff] at hf" }, { "state_after": "no goals", "state_before": "case none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nhf : ∀ (i : Fin n), ¬p i\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n none", "tactic": "exact (hf ⟨i, hin⟩ hi).elim" }, { "state_after": "case some.refine'_1\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ f ≤\n { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) }\n\ncase some.refine'_2\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ≤\n f", "state_before": "case some\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ∈\n some f", "tactic": "refine' Option.some_inj.2 (le_antisymm _ _)" }, { "state_after": "no goals", "state_before": "case some.refine'_1\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ f ≤\n { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) }", "tactic": "exact find_min' hf (Nat.find_spec h).snd" }, { "state_after": "no goals", "state_before": "case some.refine'_2\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ { val := Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }),\n isLt := (_ : Nat.find (_ : ∃ i hin, p { val := i, isLt := hin }) < n) } ≤\n f", "tactic": "exact Nat.find_min' _ ⟨f.2, by convert find_spec p hf⟩" }, { "state_after": "no goals", "state_before": "m n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\nh : ∃ i hin, p { val := i, isLt := hin }\ni : ℕ\nhin : i < n\nhi : p { val := i, isLt := hin }\nf : Fin n\nhf : find p = some f\n⊢ p { val := ↑f, isLt := (_ : ↑f < n) }", "tactic": "convert find_spec p hf" } ]

[ 927, 61 ]

[ 918, 1 ]

[ { "state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\n⊢ ∃ f_lim x, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n\ncase intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "tactic": "obtain ⟨f_lim, h_f_lim_meas, h_lim⟩ :\n ∃ (f_lim : α → E) (_ : StronglyMeasurable f_lim),\n ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (nhds (f_lim x))" }, { "state_after": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\n⊢ ∃ f_lim x, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n\ncase intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "tactic": "exact\n exists_stronglyMeasurable_limit_of_tendsto_ae (fun n => (hf n).1)\n (ae_tendsto_of_cauchy_snorm (fun n => (hf n).1) hp hB h_cau)" }, { "state_after": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "state_before": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "tactic": "have h_tendsto' : atTop.Tendsto (fun n => snorm (f n - f_lim) p μ) (𝓝 0) :=\n cauchy_tendsto_of_tendsto (fun m => (hf m).1) f_lim hB h_cau h_lim" }, { "state_after": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)\nh_ℒp_lim : Memℒp f_lim p\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "state_before": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "tactic": "have h_ℒp_lim : Memℒp f_lim p μ :=\n memℒp_of_cauchy_tendsto hp hf f_lim h_f_lim_meas.aestronglyMeasurable h_tendsto'" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u_2\nE : Type u_1\nF : Type ?u.8227274\nG : Type ?u.8227277\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf : ∀ (n : ℕ), Memℒp (f n) p\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nf_lim : α → E\nh_f_lim_meas : StronglyMeasurable f_lim\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nh_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)\nh_ℒp_lim : Memℒp f_lim p\n⊢ ∃ f_lim, Memℒp f_lim p ∧ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)", "tactic": "exact ⟨f_lim, h_ℒp_lim, h_tendsto'⟩" } ]

[ 1546, 38 ]

[ 1531, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\nf g : α →. β\ninst✝² : (x : α) → Decidable (f x).Dom\ninst✝¹ : (x : α) → Decidable (g x).Dom\ns t : Finset α\nb : β\ninst✝ : DecidableEq α\n⊢ pimage f (s ∩ t) ⊆ pimage f s ∩ pimage f t", "tactic": "simp only [← coe_subset, coe_pimage, coe_inter, PFun.image_inter]" } ]

[ 124, 68 ]

[ 123, 1 ]

[]

[ 788, 59 ]

[ 787, 1 ]

[ { "state_after": "no goals", "state_before": "α✝ : Type u_1\no : Option α✝\n⊢ ¬isSome o = true ↔ o = none", "tactic": "cases o <;> simp" } ]

[ 71, 19 ]

[ 70, 1 ]

[ { "state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.50331\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : (Filter.lift (𝓤 α) fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ Filter.lift (𝓤 α) (g ∘ preimage Prod.swap) ≤ f", "state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.50331\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : (Filter.lift (𝓤 α) fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ Filter.lift (map Prod.swap (𝓤 α)) g ≤ f", "tactic": "rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]" }, { "state_after": "no goals", "state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.50331\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : (Filter.lift (𝓤 α) fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ Filter.lift (𝓤 α) (g ∘ preimage Prod.swap) ≤ f", "tactic": "exact h" } ]

[ 554, 75 ]

[ 549, 1 ]

[]

[ 817, 43 ]

[ 815, 1 ]

[]

[ 1625, 36 ]

[ 1617, 1 ]

[]

[ 536, 71 ]

[ 536, 1 ]

[]

[ 35, 69 ]

[ 34, 1 ]

[]

[ 287, 30 ]

[ 286, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.12395\nm : Type ?u.12398\nn : Type ?u.12401\nR : Type ?u.12404\ninst✝ : Star α\n⊢ ∀ (v : Fin 0 → α), (circulant v)ᴴ = circulant (star fun i => v (-i))", "tactic": "simp [Injective]" } ]

[ 99, 44 ]

[ 96, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na✝ b✝ a'✝ b'✝ : PreEnvelGroup R\nha : PreEnvelGroupRel' R a✝ a'✝\nhb : PreEnvelGroupRel' R b✝ b'✝\n⊢ mapAux f (PreEnvelGroup.mul a✝ b✝) = mapAux f (PreEnvelGroup.mul a'✝ b'✝)", "tactic": "simp [toEnvelGroup.mapAux, well_def f ha, well_def f hb]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na✝ a'✝ : PreEnvelGroup R\nha : PreEnvelGroupRel' R a✝ a'✝\n⊢ mapAux f (PreEnvelGroup.inv a✝) = mapAux f (PreEnvelGroup.inv a'✝)", "tactic": "simp [toEnvelGroup.mapAux, well_def f ha]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na b c : PreEnvelGroup R\n⊢ mapAux f (PreEnvelGroup.mul (PreEnvelGroup.mul a b) c) = mapAux f (PreEnvelGroup.mul a (PreEnvelGroup.mul b c))", "tactic": "apply mul_assoc" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ mapAux f (PreEnvelGroup.mul unit a) = mapAux f a", "tactic": "simp [toEnvelGroup.mapAux]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ mapAux f (PreEnvelGroup.mul a unit) = mapAux f a", "tactic": "simp [toEnvelGroup.mapAux]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ mapAux f (PreEnvelGroup.mul (PreEnvelGroup.inv a) a) = mapAux f unit", "tactic": "simp [toEnvelGroup.mapAux]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\nx y : R\n⊢ mapAux f (PreEnvelGroup.mul (PreEnvelGroup.mul (incl x) (incl y)) (PreEnvelGroup.inv (incl x))) =\n mapAux f (incl (x ◃ y))", "tactic": "simp [toEnvelGroup.mapAux]" } ]

[ 739, 56 ]

[ 726, 1 ]

[ { "state_after": "f : StieltjesFunction\ns : Set ℝ\n⊢ ↑(OuterMeasure.trim (StieltjesFunction.outer f)) s ≤ ↑(StieltjesFunction.outer f) s", "state_before": "f : StieltjesFunction\n⊢ OuterMeasure.trim (StieltjesFunction.outer f) = StieltjesFunction.outer f", "tactic": "refine' le_antisymm (fun s => _) (OuterMeasure.le_trim _)" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ ↑(StieltjesFunction.outer f) s", "state_before": "f : StieltjesFunction\ns : Set ℝ\n⊢ ↑(OuterMeasure.trim (StieltjesFunction.outer f)) s ≤ ↑(StieltjesFunction.outer f) s", "tactic": "rw [OuterMeasure.trim_eq_iInf]" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ (∑' (i : ℕ), length f (t i)) + ↑ε", "state_before": "f : StieltjesFunction\ns : Set ℝ\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ ↑(StieltjesFunction.outer f) s", "tactic": "refine' le_iInf fun t => le_iInf fun ht => ENNReal.le_of_forall_pos_le_add fun ε ε0 h => _" }, { "state_after": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ (∑' (i : ℕ), length f (t i)) + ↑ε", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ (∑' (i : ℕ), length f (t i)) + ↑ε", "tactic": "rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 ε0).ne' ℕ with ⟨ε', ε'0, hε⟩" }, { "state_after": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n (∑' (i : ℕ), length f (t i)) + ∑' (i : ℕ), ↑(ε' i)", "state_before": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤ (∑' (i : ℕ), length f (t i)) + ↑ε", "tactic": "refine' le_trans _ (add_le_add_left (le_of_lt hε) _)" }, { "state_after": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n (∑' (i : ℕ), length f (t i)) + ∑' (i : ℕ), ↑(ε' i)", "tactic": "rw [← ENNReal.tsum_add]" }, { "state_after": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ofReal ↑(ε' i)\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "choose g hg using\n show ∀ i, ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal (ε' i) by\n intro i\n have hl :=\n ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'\n conv at hl =>\n lhs\n rw [length]\n simp only [iInf_lt_iff] at hl\n rcases hl with ⟨a, b, h₁, h₂⟩\n rw [← f.outer_Ioc] at h₂\n exact ⟨_, h₁, measurableSet_Ioc, le_of_lt <| by simpa using h₂⟩" }, { "state_after": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ofReal ↑(ε' i)\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "simp only [ofReal_coe_nnreal] at hg" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (_ : s ⊆ iUnion g) (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g)) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "case intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (t : Set ℝ) (_ : s ⊆ t) (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "apply iInf_le_of_le (iUnion g) _" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g)) ≤ ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (_ : s ⊆ iUnion g) (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g)) ≤\n ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "apply iInf_le_of_le (ht.trans <| iUnion_mono fun i => (hg i).1) _" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ ↑(StieltjesFunction.outer f) (iUnion g) ≤ ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ (⨅ (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g)) ≤ ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "apply iInf_le_of_le (MeasurableSet.iUnion fun i => (hg i).2.1) _" }, { "state_after": "no goals", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ng : ℕ → Set ℝ\nhg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i)\n⊢ ↑(StieltjesFunction.outer f) (iUnion g) ≤ ∑' (a : ℕ), length f (t a) + ↑(ε' a)", "tactic": "exact le_trans (f.outer.iUnion _) (ENNReal.tsum_le_tsum fun i => (hg i).2.2)" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\n⊢ ∀ (i : ℕ), ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "intro i" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : length f (t i) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "have hl :=\n ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : (⨅ (a : ℝ) (b : ℝ) (_ : t i ⊆ Ioc a b), ofReal (↑f b - ↑f a)) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : length f (t i) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "conv at hl =>\n lhs\n rw [length]" }, { "state_after": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : ∃ i_1 i_2 i_3, ofReal (↑f i_2 - ↑f i_1) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : (⨅ (a : ℝ) (b : ℝ) (_ : t i ⊆ Ioc a b), ofReal (↑f b - ↑f a)) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "simp only [iInf_lt_iff] at hl" }, { "state_after": "case intro.intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\na b : ℝ\nh₁ : t i ⊆ Ioc a b\nh₂ : ofReal (↑f b - ↑f a) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\nhl : ∃ i_1 i_2 i_3, ofReal (↑f i_2 - ↑f i_1) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "rcases hl with ⟨a, b, h₁, h₂⟩" }, { "state_after": "case intro.intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\na b : ℝ\nh₁ : t i ⊆ Ioc a b\nh₂ : ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "state_before": "case intro.intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\na b : ℝ\nh₁ : t i ⊆ Ioc a b\nh₂ : ofReal (↑f b - ↑f a) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "rw [← f.outer_Ioc] at h₂" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nf : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\na b : ℝ\nh₁ : t i ⊆ Ioc a b\nh₂ : ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ↑(ε' i)\n⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i)", "tactic": "exact ⟨_, h₁, measurableSet_Ioc, le_of_lt <| by simpa using h₂⟩" }, { "state_after": "no goals", "state_before": "f : StieltjesFunction\ns : Set ℝ\nt : ℕ → Set ℝ\nht : s ⊆ ⋃ (i : ℕ), t i\nε : ℝ≥0\nε0 : 0 < ε\nh : (∑' (i : ℕ), length f (t i)) < ⊤\nε' : ℕ → ℝ≥0\nε'0 : ∀ (i : ℕ), 0 < ε' i\nhε : (∑' (i : ℕ), ↑(ε' i)) < ↑ε\ni : ℕ\na b : ℝ\nh₁ : t i ⊆ Ioc a b\nh₂ : ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ↑(ε' i)\n⊢ ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ofReal ↑(ε' i)", "tactic": "simpa using h₂" } ]

[ 493, 79 ]

[ 470, 1 ]

[ { "state_after": "no goals", "state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.56747\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : Fintype V\ninst✝¹ : AddCommMonoid α\ninst✝ : One α\n⊢ trace (adjMatrix α G) = 0", "tactic": "simp [Matrix.trace]" } ]

[ 243, 22 ]

[ 242, 1 ]

[]

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[ 514, 1 ]

[ { "state_after": "ι : Type ?u.54429\nι' : Type ?u.54432\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.54441\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b, b ≥ a ∧ p (u b)) ↔\n ∀ (U : Set β), U ∈ F → ∀ (N : α), ∃ n, n ≥ N ∧ u n ∈ U", "state_before": "ι : Type ?u.54429\nι' : Type ?u.54432\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.54441\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ NeBot (F ⊓ map u atTop) ↔ ∀ (U : Set β), U ∈ F → ∀ (N : α), ∃ n, n ≥ N ∧ u n ∈ U", "tactic": "simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.54429\nι' : Type ?u.54432\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.54441\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b, b ≥ a ∧ p (u b)) ↔\n ∀ (U : Set β), U ∈ F → ∀ (N : α), ∃ n, n ≥ N ∧ u n ∈ U", "tactic": "rfl" } ]

[ 451, 81 ]

[ 449, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nP : Ideal R\nq : R[X]\nc : (R ⧸ P)[X]\nhq : map (mk P) q = c * X ^ n\nhn0 : 0 < n\n⊢ eval 0 q ∈ P", "tactic": "rw [← coeff_zero_eq_eval_zero, ← eq_zero_iff_mem, ← coeff_map, hq,\ncoeff_zero_eq_eval_zero, coeff_zero_eq_eval_zero,\n eval_mul, eval_pow, eval_X, zero_pow hn0, MulZeroClass.mul_zero]" } ]

[ 73, 69 ]

[ 68, 1 ]

[]

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[ 338, 1 ]

[]

[ 648, 10 ]

[ 647, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : n = 0\nh : X ^ n - ↑C a = 0\n⊢ ↑Multiset.card (nthRoots n a) ≤ n", "tactic": "simp [Nat.zero_le, nthRoots, roots, h, dif_pos rfl, empty_eq_zero, Multiset.card_zero]" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : n = 0\nh : ¬X ^ n - ↑C a = 0\n⊢ degree (↑C (1 - a)) ≤ ↑0", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : n = 0\nh : ¬X ^ n - ↑C a = 0\n⊢ degree (X ^ n - ↑C a) ≤ ↑n", "tactic": "rw [hn, pow_zero, ← C_1, ← RingHom.map_sub]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : n = 0\nh : ¬X ^ n - ↑C a = 0\n⊢ degree (↑C (1 - a)) ≤ ↑0", "tactic": "exact degree_C_le" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ ↑n", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑Multiset.card (nthRoots n a) ≤ n", "tactic": "rw [← WithBot.coe_le_coe]" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ ↑n", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ ↑n", "tactic": "simp only [← Nat.cast_withBot]" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ degree (X ^ n - ↑C a)", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ ↑n", "tactic": "rw [← degree_X_pow_sub_C (Nat.pos_of_ne_zero hn) a]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn✝ : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nn : ℕ\na : R\nhn : ¬n = 0\n⊢ ↑(↑Multiset.card (nthRoots n a)) ≤ degree (X ^ n - ↑C a)", "tactic": "exact card_roots (X_pow_sub_C_ne_zero (Nat.pos_of_ne_zero hn) a)" } ]

[ 803, 69 ]

[ 789, 1 ]

[]

[ 1498, 34 ]

[ 1497, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : StrictOrderedSemiring α\ninst✝ : Archimedean α\nx : α\nn : ℕ\nh : x ≤ n • 1\n⊢ x ≤ ↑n", "tactic": "rwa [← nsmul_one]" } ]

[ 117, 86 ]

[ 115, 1 ]

[]

[ 60, 14 ]

[ 59, 1 ]

[ { "state_after": "case intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type ?u.86642\ninst✝² : Semiring α\ninst✝¹ : Semiring β\nn : α\ninst✝ : RingHomClass F α β\nf : F\nm : α\n⊢ Odd (↑f (2 * m + 1))", "state_before": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type ?u.86642\ninst✝² : Semiring α\ninst✝¹ : Semiring β\nm n : α\ninst✝ : RingHomClass F α β\nf : F\n⊢ Odd m → Odd (↑f m)", "tactic": "rintro ⟨m, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type ?u.86642\ninst✝² : Semiring α\ninst✝¹ : Semiring β\nn : α\ninst✝ : RingHomClass F α β\nf : F\nm : α\n⊢ Odd (↑f (2 * m + 1))", "tactic": "exact ⟨f m, by simp [two_mul]⟩" }, { "state_after": "no goals", "state_before": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type ?u.86642\ninst✝² : Semiring α\ninst✝¹ : Semiring β\nn : α\ninst✝ : RingHomClass F α β\nf : F\nm : α\n⊢ ↑f (2 * m + 1) = 2 * ↑f m + 1", "tactic": "simp [two_mul]" } ]

[ 359, 33 ]

[ 357, 1 ]

[]

[ 737, 38 ]

[ 736, 1 ]

[ { "state_after": "α : Type u_1\nf : SignType → α\n⊢ ↑(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_before": "α : Type u_1\nf : SignType → α\n⊢ Set.range f = {f zero, f neg, f pos}", "tactic": "classical rw [← Fintype.coe_image_univ, univ_eq]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : SignType → α\n⊢ ↑(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "tactic": "classical simp [Finset.coe_insert]" }, { "state_after": "α : Type u_1\nf : SignType → α\n⊢ ↑(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_before": "α : Type u_1\nf : SignType → α\n⊢ Set.range f = {f zero, f neg, f pos}", "tactic": "rw [← Fintype.coe_image_univ, univ_eq]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : SignType → α\n⊢ ↑(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "tactic": "simp [Finset.coe_insert]" } ]

[ 291, 37 ]

[ 289, 1 ]

[]

[ 241, 27 ]

[ 238, 1 ]

[]

[ 498, 47 ]

[ 495, 1 ]

[]

[ 650, 43 ]

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[ { "state_after": "no goals", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nh : ⊥ = ⊤\n⊢ False", "tactic": "cases hr.symm.trans (Subalgebra.bot_eq_top_iff_finrank_eq_one.1 h)" }, { "state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\n⊢ ∀ (a : Subalgebra F E), a = ⊥ ∨ a = ⊤", "tactic": "intro S" }, { "state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis : FiniteDimensional F E\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "haveI : FiniteDimensional F E := finiteDimensional_of_finrank_eq_succ hr" }, { "state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝ : FiniteDimensional F E\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis : FiniteDimensional F E\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "haveI : FiniteDimensional F S :=\n FiniteDimensional.finiteDimensional_submodule (Subalgebra.toSubmodule S)" }, { "state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝¹ : FiniteDimensional F E\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : finrank F { x // x ∈ S } ≤ 2\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝ : FiniteDimensional F E\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "have : finrank F S ≤ 2 := hr ▸ S.toSubmodule.finrank_le" }, { "state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nthis✝ : finrank F { x // x ∈ S } ≤ 2\nthis : 0 < finrank F { x // x ∈ S }\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝¹ : FiniteDimensional F E\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : finrank F { x // x ∈ S } ≤ 2\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "have : 0 < finrank F S := finrank_pos_iff.mpr inferInstance" }, { "state_after": "case «1»\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 1\nthis✝ : 1 ≤ 2\nthis : 0 < 1\n⊢ S = ⊥ ∨ S = ⊤\n\ncase «2»\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 2\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊥ ∨ S = ⊤", "state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nthis✝ : finrank F { x // x ∈ S } ≤ 2\nthis : 0 < finrank F { x // x ∈ S }\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "interval_cases h : finrank F { x // x ∈ S }" }, { "state_after": "case «1».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 1\nthis✝ : 1 ≤ 2\nthis : 0 < 1\n⊢ S = ⊥", "state_before": "case «1»\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 1\nthis✝ : 1 ≤ 2\nthis : 0 < 1\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "left" }, { "state_after": "no goals", "state_before": "case «1».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 1\nthis✝ : 1 ≤ 2\nthis : 0 < 1\n⊢ S = ⊥", "tactic": "exact Subalgebra.eq_bot_of_finrank_one h" }, { "state_after": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 2\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊤", "state_before": "case «2»\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 2\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊥ ∨ S = ⊤", "tactic": "right" }, { "state_after": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = finrank F E\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊤", "state_before": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = 2\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊤", "tactic": "rw [← hr] at h" }, { "state_after": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = finrank F E\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ ↑toSubmodule S = ⊤", "state_before": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = finrank F E\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ S = ⊤", "tactic": "rw [← Algebra.toSubmodule_eq_top]" }, { "state_after": "no goals", "state_before": "case «2».h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis✝² : FiniteDimensional F E\nthis✝¹ : FiniteDimensional F { x // x ∈ S }\nh : finrank F { x // x ∈ S } = finrank F E\nthis✝ : 2 ≤ 2\nthis : 0 < 2\n⊢ ↑toSubmodule S = ⊤", "tactic": "exact Submodule.eq_top_of_finrank_eq h" } ]

[ 1457, 49 ]

[ 1439, 1 ]

[]

[ 49, 6 ]

[ 47, 1 ]

[ { "state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\n⊢ lift (#ι) ≤ lift (#κ)", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\n⊢ lift (#ι) ≤ lift (#κ)", "tactic": "let Φ := fun k : κ => (b.repr (v k)).support" }, { "state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\nw₁ : (#ι) ≤ (#↑(range Φ))\n⊢ lift (#ι) ≤ lift (#κ)", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\n⊢ lift (#ι) ≤ lift (#κ)", "tactic": "have w₁ : (#ι) ≤ (#Set.range Φ) := by\n apply Cardinal.le_range_of_union_finset_eq_top\n exact union_support_maximal_linearIndependent_eq_range_basis b v i m" }, { "state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\nw₁ : (#ι) ≤ (#↑(range Φ))\nw₂ : lift (#↑(range Φ)) ≤ lift (#κ)\n⊢ lift (#ι) ≤ lift (#κ)", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\nw₁ : (#ι) ≤ (#↑(range Φ))\n⊢ lift (#ι) ≤ lift (#κ)", "tactic": "have w₂ : Cardinal.lift.{w'} (#Set.range Φ) ≤ Cardinal.lift.{w} (#κ) := Cardinal.mk_range_le_lift" }, { "state_after": "no goals", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\nw₁ : (#ι) ≤ (#↑(range Φ))\nw₂ : lift (#↑(range Φ)) ≤ lift (#κ)\n⊢ lift (#ι) ≤ lift (#κ)", "tactic": "exact (Cardinal.lift_le.mpr w₁).trans w₂" }, { "state_after": "case w\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\n⊢ (⋃ (a : κ), ↑(Φ a)) = ⊤", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\n⊢ (#ι) ≤ (#↑(range Φ))", "tactic": "apply Cardinal.le_range_of_union_finset_eq_top" }, { "state_after": "no goals", "state_before": "case w\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.208466\nR : Type u\ninst✝⁸ : Ring R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type v'\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nM₁ : Type v\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\nι : Type w\nb : Basis ι R M\ninst✝ : Infinite ι\nκ : Type w'\nv : κ → M\ni : LinearIndependent R v\nm : LinearIndependent.Maximal i\nΦ : κ → Finset ι := fun k => (↑b.repr (v k)).support\n⊢ (⋃ (a : κ), ↑(Φ a)) = ⊤", "tactic": "exact union_support_maximal_linearIndependent_eq_range_basis b v i m" } ]

[ 438, 43 ]

[ 430, 1 ]

[]

[ 1128, 6 ]

[ 1126, 1 ]