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tasksource 28

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x (bit0 p) = bit0 (eval₂ f x p)", "tactic": "rw [bit0, eval₂_add, bit0]" } ]

[ 104, 94 ]

[ 104, 1 ]

[ { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : n = 0\n⊢ i = j", "tactic": "simp_all" }, { "state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ xn a1 j % xn a1 n < xn a1 i % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ False", "tactic": "refine' _root_.ne_of_gt _ h" }, { "state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ 0 < xn a1 i % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ xn a1 j % xn a1 n < xn a1 i % xn a1 n", "tactic": "rw [jn, Nat.mod_self]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\n⊢ 0 < xn a1 i % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ 0 < xn a1 i % xn a1 n", "tactic": "have x0 : 0 < xn a1 0 % xn a1 n := by\n rw [Nat.mod_eq_of_lt (strictMono_x a1 (Nat.pos_of_ne_zero npos))]\n exact Nat.succ_pos _" }, { "state_after": "case zero\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\nij : zero ≤ j\nh : xn a1 zero ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ zero = 0 ∧ j = 2)\nij' : zero < j\n⊢ 0 < xn a1 zero % xn a1 n\n\ncase succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < j\n⊢ 0 < xn a1 (succ i) % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\n⊢ 0 < xn a1 i % xn a1 n", "tactic": "cases' i with i" }, { "state_after": "case succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < j\n⊢ 0 < xn a1 (succ i) % xn a1 n", "state_before": "case zero\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\nij : zero ≤ j\nh : xn a1 zero ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ zero = 0 ∧ j = 2)\nij' : zero < j\n⊢ 0 < xn a1 zero % xn a1 n\n\ncase succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < j\n⊢ 0 < xn a1 (succ i) % xn a1 n", "tactic": "exact x0" }, { "state_after": "case succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < n\n⊢ 0 < xn a1 (succ i) % xn a1 n", "state_before": "case succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < j\n⊢ 0 < xn a1 (succ i) % xn a1 n", "tactic": "rw [jn] at ij'" }, { "state_after": "no goals", "state_before": "case succ\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < n\n⊢ 0 < xn a1 (succ i) % xn a1 n", "tactic": "exact\n x0.trans\n (eq_of_xn_modEq_lem3 _ (Nat.pos_of_ne_zero npos) (Nat.succ_pos _) (le_trans ij j2n)\n (_root_.ne_of_lt ij') fun ⟨_, n1, _, i2⟩ => by\n rw [n1, i2] at ij'; exact absurd ij' (by decide))" }, { "state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ 0 < xn a1 0", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ 0 < xn a1 0 % xn a1 n", "tactic": "rw [Nat.mod_eq_of_lt (strictMono_x a1 (Nat.pos_of_ne_zero npos))]" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : ¬n = 0\nij' : i < j\njn : j = n\n⊢ 0 < xn a1 0", "tactic": "exact Nat.succ_pos _" }, { "state_after": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ succ i = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 = 0\ni2 : succ i = 2\n⊢ False", "state_before": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : succ i < n\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ succ i = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 = 0\ni2 : succ i = 2\n⊢ False", "tactic": "rw [n1, i2] at ij'" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ succ i = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 = 0\ni2 : succ i = 2\n⊢ False", "tactic": "exact absurd ij' (by decide)" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : succ i ≤ j\nh : xn a1 (succ i) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ succ i = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ succ i = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 = 0\ni2 : succ i = 2\n⊢ ¬2 < 1", "tactic": "decide" } ]

[ 767, 97 ]

[ 747, 1 ]

[]

[ 2742, 30 ]

[ 2741, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na : ZNum\n⊢ ↑(0 + a) = ↑0 + ↑a", "tactic": "cases a <;> exact (_root_.zero_add _).symm" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nb : ZNum\n⊢ ↑(b + 0) = ↑b + ↑0", "tactic": "cases b <;> exact (_root_.add_zero _).symm" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(pos a + neg b) = ↑(pos a) + ↑(neg b)", "tactic": "simpa only [sub_eq_add_neg] using PosNum.cast_sub' (α := α) _ _" }, { "state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑↑b + ↑(-↑a) = ↑(-↑a + ↑b)", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑b + -↑a = -↑a + ↑b", "tactic": "rw [← PosNum.cast_to_int a, ← PosNum.cast_to_int b, ← Int.cast_neg, ← Int.cast_add (-a)]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑↑b + ↑(-↑a) = ↑(-↑a + ↑b)", "tactic": "simp [add_comm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ -↑(a + b) = -↑a + -↑b", "tactic": "rw [PosNum.cast_add, neg_eq_iff_eq_neg, neg_add_rev, neg_neg, neg_neg,\n ← PosNum.cast_to_int a, ← PosNum.cast_to_int b, ← Int.cast_add, ← Int.cast_add, add_comm]" } ]

[ 1336, 100 ]

[ 1323, 1 ]

[ { "state_after": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (cancel m₁ m₂)", "state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (cancel m₁ m₂)", "tactic": "simp" }, { "state_after": "case h\nctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (List.reverse [] ++ m₁, List.reverse [] ++ m₂)", "state_before": "ctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (cancel m₁ m₂)", "tactic": "apply denote_eq_cancelAux" }, { "state_after": "no goals", "state_before": "case h\nctx : Context\nm₁ m₂ : Poly\nh : denote_eq ctx (m₁, m₂)\n⊢ denote_eq ctx (List.reverse [] ++ m₁, List.reverse [] ++ m₂)", "tactic": "simp [h]" } ]

[ 416, 44 ]

[ 415, 1 ]

[]

[ 57, 61 ]

[ 56, 1 ]

[]

[ 244, 6 ]

[ 243, 1 ]

[]

[ 2243, 29 ]

[ 2242, 1 ]

[]

[ 338, 21 ]

[ 336, 1 ]

[ { "state_after": "no goals", "state_before": "V : Type u_1\nP : Type ?u.38982\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv₁ v₂ v₃ v : V\n⊢ ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃", "tactic": "simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v" } ]

[ 112, 61 ]

[ 111, 1 ]

[]

[ 790, 6 ]

[ 789, 1 ]

[ { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "obtain ⟨U, hU_symm, hU_comp, hB⟩ :\n ∃ U : ℕ → Set (X × X),\n (∀ n, SymmetricRel (U n)) ∧\n (∀ ⦃m n⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ (𝓤 X).HasAntitoneBasis U := by\n rcases UniformSpace.has_seq_basis X with ⟨V, hB, hV_symm⟩\n rcases hB.subbasis_with_rel fun m =>\n hB.tendsto_smallSets.eventually\n (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with\n ⟨φ, -, hφ_comp, hφB⟩\n exact ⟨V ∘ φ, fun n => hV_symm _, hφ_comp, hφB⟩" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "letI := UniformSpace.separationSetoid X" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "set d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "have hd_symm : ∀ x y, d x y = d y x := by\n intro x y\n simp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "have hr : (1 / 2 : ℝ≥0) ∈ Ioo (0 : ℝ≥0) 1 := ⟨half_pos one_pos, NNReal.half_lt_self one_ne_zero⟩" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "letI I := PseudoMetricSpace.ofPreNNDist d (fun x => hd₀.2 (Setoid.refl _)) hd_symm" }, { "state_after": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "have hdist_le : ∀ x y, dist x y ≤ d x y := PseudoMetricSpace.dist_ofPreNNDist_le _ _ _" }, { "state_after": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\n⊢ ∀ (i : ℕ), True → ∃ i', True ∧ U i' ⊆ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ i}\n\ncase intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\n⊢ ∀ (i' : ℕ), True → ∃ i, True ∧ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ i} ⊆ U i'", "state_before": "case intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\n⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹", "tactic": "refine' ⟨I, uniformSpace_eq <| (uniformity_basis_dist_pow hr.1 hr.2).ext hB.toHasBasis _ _⟩" }, { "state_after": "case intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nV : ℕ → Set (X × X)\nhB : HasAntitoneBasis (𝓤 X) V\nhV_symm : ∀ (n : ℕ), SymmetricRel (V n)\n⊢ ∃ U, (∀ (n : ℕ), SymmetricRel (U n)) ∧ (∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ HasAntitoneBasis (𝓤 X) U", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\n⊢ ∃ U, (∀ (n : ℕ), SymmetricRel (U n)) ∧ (∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ HasAntitoneBasis (𝓤 X) U", "tactic": "rcases UniformSpace.has_seq_basis X with ⟨V, hB, hV_symm⟩" }, { "state_after": "case intro.intro.intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nV : ℕ → Set (X × X)\nhB : HasAntitoneBasis (𝓤 X) V\nhV_symm : ∀ (n : ℕ), SymmetricRel (V n)\nφ : ℕ → ℕ\nhφ_comp : ∀ ⦃m n : ℕ⦄, m < n → ((fun x x_1 => x ○ x_1) (V (φ n))^[2]) (V (φ n)) ⊆ V (φ m)\nhφB : HasAntitoneBasis (𝓤 X) (V ∘ φ)\n⊢ ∃ U, (∀ (n : ℕ), SymmetricRel (U n)) ∧ (∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ HasAntitoneBasis (𝓤 X) U", "state_before": "case intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nV : ℕ → Set (X × X)\nhB : HasAntitoneBasis (𝓤 X) V\nhV_symm : ∀ (n : ℕ), SymmetricRel (V n)\n⊢ ∃ U, (∀ (n : ℕ), SymmetricRel (U n)) ∧ (∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ HasAntitoneBasis (𝓤 X) U", "tactic": "rcases hB.subbasis_with_rel fun m =>\n hB.tendsto_smallSets.eventually\n (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with\n ⟨φ, -, hφ_comp, hφB⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nV : ℕ → Set (X × X)\nhB : HasAntitoneBasis (𝓤 X) V\nhV_symm : ∀ (n : ℕ), SymmetricRel (V n)\nφ : ℕ → ℕ\nhφ_comp : ∀ ⦃m n : ℕ⦄, m < n → ((fun x x_1 => x ○ x_1) (V (φ n))^[2]) (V (φ n)) ⊆ V (φ m)\nhφB : HasAntitoneBasis (𝓤 X) (V ∘ φ)\n⊢ ∃ U, (∀ (n : ℕ), SymmetricRel (U n)) ∧ (∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ HasAntitoneBasis (𝓤 X) U", "tactic": "exact ⟨V ∘ φ, fun n => hV_symm _, hφ_comp, hφB⟩" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ d x y = 0 ↔ x ≈ y", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\n⊢ ∀ {x y : X}, d x y = 0 ↔ x ≈ y", "tactic": "intro x y" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ d x y = 0 ↔ ∀ (i : ℕ), True → (x, y) ∈ U i", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ d x y = 0 ↔ x ≈ y", "tactic": "refine' Iff.trans _ hB.mem_separationRel.symm" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ (if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0) = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ d x y = 0 ↔ ∀ (i : ℕ), True → (x, y) ∈ U i", "tactic": "simp only [true_imp_iff]" }, { "state_after": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ Nat.find h = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i\n\ncase inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : ¬∃ n, ¬(x, y) ∈ U n\n⊢ 0 = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n⊢ (if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0) = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "tactic": "split_ifs with h" }, { "state_after": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh✝ : ∃ n, ¬(x, y) ∈ U n\nh : ¬∀ (x_1 : ℕ), (x, y) ∈ U x_1\n⊢ (1 / 2) ^ Nat.find h✝ = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "state_before": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ Nat.find h = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "tactic": "rw [← not_forall] at h" }, { "state_after": "no goals", "state_before": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh✝ : ∃ n, ¬(x, y) ∈ U n\nh : ¬∀ (x_1 : ℕ), (x, y) ∈ U x_1\n⊢ (1 / 2) ^ Nat.find h✝ = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "tactic": "simp [h, pow_eq_zero_iff']" }, { "state_after": "no goals", "state_before": "case inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : ¬∃ n, ¬(x, y) ∈ U n\n⊢ 0 = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i", "tactic": "simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nx y : X\n⊢ d x y = d y x", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\n⊢ ∀ (x y : X), d x y = d y x", "tactic": "intro x y" }, { "state_after": "no goals", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nx y : X\n⊢ d x y = d y x", "tactic": "simp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\n⊢ (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\n⊢ ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n", "tactic": "intro x y n" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\n⊢ ((1 / 2) ^ n ≤ if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0) ↔ ¬(x, y) ∈ U n", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\n⊢ (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n", "tactic": "dsimp only []" }, { "state_after": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ (1 / 2) ^ Nat.find h ↔ ¬(x, y) ∈ U n\n\ncase inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ¬∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ 0 ↔ ¬(x, y) ∈ U n", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\n⊢ ((1 / 2) ^ n ≤ if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0) ↔ ¬(x, y) ∈ U n", "tactic": "split_ifs with h" }, { "state_after": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (∃ m, m ≤ n ∧ ¬(x, y) ∈ U m) ↔ ¬(x, y) ∈ U n", "state_before": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ (1 / 2) ^ Nat.find h ↔ ¬(x, y) ∈ U n", "tactic": "rw [(strictAnti_pow hr.1 hr.2).le_iff_le, Nat.find_le_iff]" }, { "state_after": "no goals", "state_before": "case inl\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∃ n, ¬(x, y) ∈ U n\n⊢ (∃ m, m ≤ n ∧ ¬(x, y) ∈ U m) ↔ ¬(x, y) ∈ U n", "tactic": "exact ⟨fun ⟨m, hmn, hm⟩ hn => hm (hB.antitone hmn hn), fun h => ⟨n, le_rfl, h⟩⟩" }, { "state_after": "case inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∀ (n : ℕ), (x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ 0 ↔ ¬(x, y) ∈ U n", "state_before": "case inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ¬∃ n, ¬(x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ 0 ↔ ¬(x, y) ∈ U n", "tactic": "push_neg at h" }, { "state_after": "no goals", "state_before": "case inr\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nx y : X\nn : ℕ\nh : ∀ (n : ℕ), (x, y) ∈ U n\n⊢ (1 / 2) ^ n ≤ 0 ↔ ¬(x, y) ∈ U n", "tactic": "simp only [h, not_true, (pow_pos hr.1 _).not_le]" }, { "state_after": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\n⊢ ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y", "tactic": "refine' PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x₁ x₂ x₃ x₄ => _" }, { "state_after": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\n\ncase neg\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ¬∃ n, ¬(x₁, x₄) ∈ U n\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "state_before": "X✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "tactic": "by_cases H : ∃ n, (x₁, x₄) ∉ U n" }, { "state_after": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ (1 / 2) ^ Nat.find H ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "state_before": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "tactic": "refine' (dif_pos H).trans_le _" }, { "state_after": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ (1 / 2) ^ (Nat.find H + 1) ≤ max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "state_before": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ (1 / 2) ^ Nat.find H ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "tactic": "rw [← NNReal.div_le_iff' two_ne_zero, ← mul_one_div (_ ^ _), ← pow_succ']" }, { "state_after": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ ¬((x₁, x₂) ∈ U (Nat.find H + 1) ∧ (x₂, x₃) ∈ U (Nat.find H + 1) ∧ (x₃, x₄) ∈ U (Nat.find H + 1))", "state_before": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ (1 / 2) ^ (Nat.find H + 1) ≤ max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "tactic": "simp only [le_max_iff, hle_d, ← not_and_or]" }, { "state_after": "case pos.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\nh₁₂ : (x₁, x₂) ∈ U (Nat.find H + 1)\nh₂₃ : (x₂, x₃) ∈ U (Nat.find H + 1)\nh₃₄ : (x₃, x₄) ∈ U (Nat.find H + 1)\n⊢ False", "state_before": "case pos\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\n⊢ ¬((x₁, x₂) ∈ U (Nat.find H + 1) ∧ (x₂, x₃) ∈ U (Nat.find H + 1) ∧ (x₃, x₄) ∈ U (Nat.find H + 1))", "tactic": "rintro ⟨h₁₂, h₂₃, h₃₄⟩" }, { "state_after": "case pos.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\nh₁₂ : (x₁, x₂) ∈ U (Nat.find H + 1)\nh₂₃ : (x₂, x₃) ∈ U (Nat.find H + 1)\nh₃₄ : (x₃, x₄) ∈ U (Nat.find H + 1)\n⊢ (x₁, x₄) ∈ U (Nat.find H + 1) ○ (U (Nat.find H + 1) ○ U (Nat.find H + 1))", "state_before": "case pos.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\nh₁₂ : (x₁, x₂) ∈ U (Nat.find H + 1)\nh₂₃ : (x₂, x₃) ∈ U (Nat.find H + 1)\nh₃₄ : (x₃, x₄) ∈ U (Nat.find H + 1)\n⊢ False", "tactic": "refine' Nat.find_spec H (hU_comp (lt_add_one <| Nat.find H) _)" }, { "state_after": "no goals", "state_before": "case pos.intro.intro\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ∃ n, ¬(x₁, x₄) ∈ U n\nh₁₂ : (x₁, x₂) ∈ U (Nat.find H + 1)\nh₂₃ : (x₂, x₃) ∈ U (Nat.find H + 1)\nh₃₄ : (x₃, x₄) ∈ U (Nat.find H + 1)\n⊢ (x₁, x₄) ∈ U (Nat.find H + 1) ○ (U (Nat.find H + 1) ○ U (Nat.find H + 1))", "tactic": "exact ⟨x₂, h₁₂, x₃, h₂₃, h₃₄⟩" }, { "state_after": "no goals", "state_before": "case neg\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nx₁ x₂ x₃ x₄ : X\nH : ¬∃ n, ¬(x₁, x₄) ∈ U n\n⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))", "tactic": "exact (dif_neg H).trans_le (zero_le _)" }, { "state_after": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nhn : True\nx : X × X\nhx : x ∈ U n\n⊢ ↑(d x.fst x.snd) < (fun a => ↑a) (1 / 2) ^ n", "state_before": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\n⊢ ∀ (i : ℕ), True → ∃ i', True ∧ U i' ⊆ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ i}", "tactic": "refine' fun n hn => ⟨n, hn, fun x hx => (hdist_le _ _).trans_lt _⟩" }, { "state_after": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nhn : True\nx : X × X\nhx : x ∈ U n\n⊢ ↑(d x.fst x.snd) < ↑(1 / 2) ^ n", "state_before": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nhn : True\nx : X × X\nhx : x ∈ U n\n⊢ ↑(d x.fst x.snd) < (fun a => ↑a) (1 / 2) ^ n", "tactic": "change _ < (toReal _) ^ _" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_1\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nhn : True\nx : X × X\nhx : x ∈ U n\n⊢ ↑(d x.fst x.snd) < ↑(1 / 2) ^ n", "tactic": "rwa [← NNReal.coe_pow, NNReal.coe_lt_coe, ← not_le, hle_d, Classical.not_not, Prod.mk.eta]" }, { "state_after": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : x ∈ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ (n + 1)}\n⊢ x ∈ U n", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\n⊢ ∀ (i' : ℕ), True → ∃ i, True ∧ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ i} ⊆ U i'", "tactic": "refine' fun n _ => ⟨n + 1, trivial, fun x hx => _⟩" }, { "state_after": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : dist x.fst x.snd < (fun a => ↑a) (1 / 2) ^ (n + 1)\n⊢ x ∈ U n", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : x ∈ {p | dist p.fst p.snd < (fun a => ↑a) (1 / 2) ^ (n + 1)}\n⊢ x ∈ U n", "tactic": "rw [mem_setOf_eq] at hx" }, { "state_after": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ dist x.fst x.snd", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : dist x.fst x.snd < (fun a => ↑a) (1 / 2) ^ (n + 1)\n⊢ x ∈ U n", "tactic": "contrapose! hx" }, { "state_after": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ ↑(d x.fst x.snd) / 2", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ dist x.fst x.snd", "tactic": "refine' le_trans _ ((div_le_iff' (zero_lt_two' ℝ)).2 (hd_le x.1 x.2))" }, { "state_after": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ ↑(d x.fst x.snd) / 2", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ ↑(d x.fst x.snd) / 2", "tactic": "change (toReal _) ^ _ ≤ _" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.refine'_2\nX✝ : Type ?u.116434\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsCountablyGenerated (𝓤 X)\nU : ℕ → Set (X × X)\nhU_symm : ∀ (n : ℕ), SymmetricRel (U n)\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : HasAntitoneBasis (𝓤 X) U\nthis : Setoid X := separationSetoid X\nd : X → X → ℝ≥0 := fun x y => if h : ∃ n, ¬(x, y) ∈ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y\nhd_symm : ∀ (x y : X), d x y = d y x\nhr : 1 / 2 ∈ Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm\nhdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)\nhle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ ¬(x, y) ∈ U n\nhd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y\nn : ℕ\nx✝ : True\nx : X × X\nhx : ¬x ∈ U n\n⊢ ↑(1 / 2) ^ (n + 1) ≤ ↑(d x.fst x.snd) / 2", "tactic": "rwa [← NNReal.coe_two, ← NNReal.coe_div, ← NNReal.coe_pow, NNReal.coe_le_coe, pow_succ',\n mul_one_div, NNReal.div_le_iff two_ne_zero, div_mul_cancel _ (two_ne_zero' ℝ≥0), hle_d,\n Prod.mk.eta]" } ]

[ 259, 19 ]

[ 189, 11 ]

[]

[ 340, 21 ]

[ 338, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ng : β → γ\nhg : UniformInducing g\nf : α → β\nhf : UniformInducing f\n⊢ comap (fun x => ((g ∘ f) x.fst, (g ∘ f) x.snd)) (𝓤 γ) =\n comap ((fun x => (g x.fst, g x.snd)) ∘ fun x => (f x.fst, f x.snd)) (𝓤 γ)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ng : β → γ\nhg : UniformInducing g\nf : α → β\nhf : UniformInducing f\n⊢ comap (fun x => ((g ∘ f) x.fst, (g ∘ f) x.snd)) (𝓤 γ) = 𝓤 α", "tactic": "rw [← hf.1, ← hg.1, comap_comap]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ng : β → γ\nhg : UniformInducing g\nf : α → β\nhf : UniformInducing f\n⊢ comap (fun x => ((g ∘ f) x.fst, (g ∘ f) x.snd)) (𝓤 γ) =\n comap ((fun x => (g x.fst, g x.snd)) ∘ fun x => (f x.fst, f x.snd)) (𝓤 γ)", "tactic": "rfl" } ]

[ 73, 45 ]

[ 71, 1 ]

[]

[ 356, 6 ]

[ 355, 1 ]

[ { "state_after": "l : Type ?u.127855\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA B : Matrix n n α\ninst✝ : Invertible A\nthis : Invertible (det A) := detInvertibleOfInvertible A\n⊢ ⅟A = A⁻¹", "state_before": "l : Type ?u.127855\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA B : Matrix n n α\ninst✝ : Invertible A\n⊢ ⅟A = A⁻¹", "tactic": "letI := detInvertibleOfInvertible A" }, { "state_after": "no goals", "state_before": "l : Type ?u.127855\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA B : Matrix n n α\ninst✝ : Invertible A\nthis : Invertible (det A) := detInvertibleOfInvertible A\n⊢ ⅟A = A⁻¹", "tactic": "rw [inv_def, Ring.inverse_invertible, invOf_eq]" } ]

[ 272, 50 ]

[ 270, 1 ]

[]

[ 922, 71 ]

[ 921, 1 ]

[]

[ 87, 56 ]

[ 86, 1 ]

[]

[ 421, 50 ]

[ 420, 1 ]

[]

[ 196, 66 ]

[ 194, 1 ]

[]

[ 318, 74 ]

[ 317, 1 ]

[]

[ 142, 44 ]

[ 140, 1 ]

[]

[ 671, 44 ]

[ 669, 1 ]

[ { "state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl l' : List α\nh : l ~r l'\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ next l (nthLe l k hk) hx = next l' (nthLe l k hk) (_ : nthLe l k hk ∈ l')", "state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl l' : List α\nh : l ~r l'\nhn : Nodup l\nx : α\nhx : x ∈ l\n⊢ next l x hx = next l' x (_ : x ∈ l')", "tactic": "obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx" }, { "state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ next l (nthLe l k hk) hx = next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk ∈ rotate l n)", "state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl l' : List α\nh : l ~r l'\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ next l (nthLe l k hk) hx = next l' (nthLe l k hk) (_ : nthLe l k hk ∈ l')", "tactic": "obtain ⟨n, rfl⟩ := id h" }, { "state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk ∈ rotate l n)", "state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ next l (nthLe l k hk) hx = next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk ∈ rotate l n)", "tactic": "rw [next_nthLe _ hn]" }, { "state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n)\n (nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)))\n (_ :\n nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)) ∈\n rotate l n)", "state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk ∈ rotate l n)", "tactic": "simp_rw [← nthLe_rotate' _ n k]" }, { "state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe (rotate l n) ((length l - n % length l + (k + 1) % length l) % length l)\n (_ : (length l - n % length l + (k + 1) % length l) % length l < length (rotate l n)) =\n nthLe (rotate l n) (((length l - n % length l + k) % length l + 1) % length (rotate l n))\n (_ : ((length l - n % length l + k) % length l + 1) % length (rotate l n) < length (rotate l n))", "state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n)\n (nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)))\n (_ :\n nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)) ∈\n rotate l n)", "tactic": "rw [next_nthLe _ (h.nodup_iff.mp hn), ← nthLe_rotate' _ n]" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhn : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nn : ℕ\nh : l ~r rotate l n\n⊢ nthLe (rotate l n) ((length l - n % length l + (k + 1) % length l) % length l)\n (_ : (length l - n % length l + (k + 1) % length l) % length l < length (rotate l n)) =\n nthLe (rotate l n) (((length l - n % length l + k) % length l + 1) % length (rotate l n))\n (_ : ((length l - n % length l + k) % length l + 1) % length (rotate l n) < length (rotate l n))", "tactic": "simp [add_assoc]" } ]

[ 434, 19 ]

[ 427, 1 ]

[ { "state_after": "no goals", "state_before": "n m : ℕ\nh : n = m\ni : Fin n\n⊢ ↑(↑(cast h) i) = ↑i", "tactic": "simp" } ]

[ 1070, 73 ]

[ 1070, 1 ]

[ { "state_after": "case inl\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : Fintype.card α < r\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))\n\ncase inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : r ≤ Fintype.card α\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "state_before": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "tactic": "obtain hr' | hr' := lt_or_le (Fintype.card α) r" }, { "state_after": "case inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "state_before": "case inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : r ≤ Fintype.card α\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "tactic": "replace h𝒜 := card_mul_le_card_shadow_mul h𝒜" }, { "state_after": "case inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) r\n\ncase inr.b0\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ 0 < Nat.choose (Fintype.card α) r\n\ncase inr.d0\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ 0 < Nat.choose (Fintype.card α) (r - 1)", "state_before": "case inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "tactic": "rw [div_le_div_iff] <;> norm_cast" }, { "state_after": "case inl\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : Fintype.card α < r\n⊢ 0 ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "state_before": "case inl\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : Fintype.card α < r\n⊢ ↑(card 𝒜) / ↑(Nat.choose (Fintype.card α) r) ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "tactic": "rw [choose_eq_zero_of_lt hr', cast_zero, div_zero]" }, { "state_after": "no goals", "state_before": "case inl\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nh𝒜 : Set.Sized r ↑𝒜\nhr' : Fintype.card α < r\n⊢ 0 ≤ ↑(card ((∂ ) 𝒜)) / ↑(Nat.choose (Fintype.card α) (r - 1))", "tactic": "exact div_nonneg (cast_nonneg _) (cast_nonneg _)" }, { "state_after": "case inr.zero\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nhr : zero ≠ 0\nhr' : zero ≤ Fintype.card α\nh𝒜 : card 𝒜 * zero ≤ card ((∂ ) 𝒜) * (Fintype.card α - zero + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (zero - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) zero\n\ncase inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : succ r ≠ 0\nhr' : succ r ≤ Fintype.card α\nh𝒜 : card 𝒜 * succ r ≤ card ((∂ ) 𝒜) * (Fintype.card α - succ r + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (succ r - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (succ r)", "state_before": "case inr\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) r", "tactic": "cases' r with r" }, { "state_after": "case inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - (r + 1) + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1)", "state_before": "case inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : succ r ≠ 0\nhr' : succ r ≤ Fintype.card α\nh𝒜 : card 𝒜 * succ r ≤ card ((∂ ) 𝒜) * (Fintype.card α - succ r + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (succ r - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (succ r)", "tactic": "rw [Nat.succ_eq_add_one] at *" }, { "state_after": "case inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1)", "state_before": "case inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - (r + 1) + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1)", "tactic": "rw [tsub_add_eq_add_tsub hr', add_tsub_add_eq_tsub_right] at h𝒜" }, { "state_after": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) * (r + 1) ≤\n card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1) * (r + 1)", "state_before": "case inr.succ\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1)", "tactic": "apply le_of_mul_le_mul_right _ (pos_iff_ne_zero.2 hr)" }, { "state_after": "case h.e'_3\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) * (r + 1) = card 𝒜 * (r + 1) * Nat.choose (Fintype.card α) r\n\ncase h.e'_4\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1) * (r + 1) =\n card ((∂ ) 𝒜) * (Fintype.card α - r) * Nat.choose (Fintype.card α) r", "state_before": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) * (r + 1) ≤\n card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1) * (r + 1)", "tactic": "convert Nat.mul_le_mul_right ((Fintype.card α).choose r) h𝒜 using 1" }, { "state_after": "no goals", "state_before": "case inr.zero\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nhr : zero ≠ 0\nhr' : zero ≤ Fintype.card α\nh𝒜 : card 𝒜 * zero ≤ card ((∂ ) 𝒜) * (Fintype.card α - zero + 1)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (zero - 1) ≤ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) zero", "tactic": "exact (hr rfl).elim" }, { "state_after": "case h.e'_3\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ Nat.choose (Fintype.card α) r * (r + 1) = (r + 1) * Nat.choose (Fintype.card α) r ∨ 𝒜 = ∅", "state_before": "case h.e'_3\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card 𝒜 * Nat.choose (Fintype.card α) (r + 1 - 1) * (r + 1) = card 𝒜 * (r + 1) * Nat.choose (Fintype.card α) r", "tactic": "simp [mul_assoc, Nat.choose_succ_right_eq]" }, { "state_after": "no goals", "state_before": "case h.e'_3\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ Nat.choose (Fintype.card α) r * (r + 1) = (r + 1) * Nat.choose (Fintype.card α) r ∨ 𝒜 = ∅", "tactic": "exact Or.inl (mul_comm _ _)" }, { "state_after": "case h.e'_4\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ Nat.choose (Fintype.card α) r * (Fintype.card α - r) = (Fintype.card α - r) * Nat.choose (Fintype.card α) r ∨\n card ((∂ ) 𝒜) = 0", "state_before": "case h.e'_4\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ card ((∂ ) 𝒜) * Nat.choose (Fintype.card α) (r + 1) * (r + 1) =\n card ((∂ ) 𝒜) * (Fintype.card α - r) * Nat.choose (Fintype.card α) r", "tactic": "simp only [mul_assoc, choose_succ_right_eq, mul_eq_mul_left_iff]" }, { "state_after": "no goals", "state_before": "case h.e'_4\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : card 𝒜 * (r + 1) ≤ card ((∂ ) 𝒜) * (Fintype.card α - r)\n⊢ Nat.choose (Fintype.card α) r * (Fintype.card α - r) = (Fintype.card α - r) * Nat.choose (Fintype.card α) r ∨\n card ((∂ ) 𝒜) = 0", "tactic": "exact Or.inl (mul_comm _ _)" }, { "state_after": "no goals", "state_before": "case inr.b0\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ 0 < Nat.choose (Fintype.card α) r", "tactic": "exact Nat.choose_pos hr'" }, { "state_after": "no goals", "state_before": "case inr.d0\n𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : card 𝒜 * r ≤ card ((∂ ) 𝒜) * (Fintype.card α - r + 1)\n⊢ 0 < Nat.choose (Fintype.card α) (r - 1)", "tactic": "exact Nat.choose_pos (r.pred_le.trans hr')" } ]

[ 113, 47 ]

[ 94, 1 ]

[]

[ 1567, 6 ]

[ 1566, 1 ]

[]

[ 1299, 23 ]

[ 1298, 1 ]

[]

[ 478, 23 ]

[ 477, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.181670\nγ : Type ?u.181673\nι : Sort u_2\nι' : Sort ?u.181679\nι₂ : Sort ?u.181682\nκ : ι → Sort u_3\nκ₁ : ι → Sort ?u.181692\nκ₂ : ι → Sort ?u.181697\nκ' : ι' → Sort ?u.181702\ns : Set α\nt : (i : ι) → κ i → Set α\n⊢ (s ∪ ⋂ (i : ι) (j : κ i), t i j) = ⋂ (i : ι) (j : κ i), s ∪ t i j", "tactic": "simp_rw [union_distrib_iInter_left]" } ]

[ 1404, 92 ]

[ 1403, 1 ]

[]

[ 696, 65 ]

[ 693, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜₁ : Type u_3\n𝕜₂ : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜₁\ninst✝⁶ : SeminormedRing 𝕜₂\nσ₁₂ : 𝕜₁ →+* 𝕜₂\nM₁ : Type u_4\nM₂ : Type u_2\nM₃ : Type ?u.72218\ninst✝⁵ : SeminormedAddCommGroup M₁\ninst✝⁴ : TopologicalSpace M₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : NormedSpace 𝕜₁ M₁\ninst✝¹ : Module 𝕜₂ M₂\ninst✝ : ContinuousConstSMul 𝕜₂ M₂\nf : M₁ →ₛₗ[σ₁₂] M₂\nhf : IsCompactOperator ↑f\nS : Set M₁\nhS : Metric.Bounded S\n⊢ IsVonNBounded 𝕜₁ S", "tactic": "rwa [NormedSpace.isVonNBounded_iff, ← Metric.bounded_iff_isBounded]" } ]

[ 135, 77 ]

[ 131, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.212863\nγ : Type ?u.212866\nι : Sort u_3\nι' : Sort ?u.212872\nκ : ι → Sort ?u.212877\nκ' : ι' → Sort ?u.212882\ninst✝¹ : SupSet α\ninst✝ : SupSet β\nf : ι → α × β\n⊢ swap (iSup f) = ⨆ (i : ι), swap (f i)", "tactic": "simp_rw [iSup, swap_sSup, ←range_comp, Function.comp]" } ]

[ 1883, 56 ]

[ 1882, 1 ]

[ { "state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type ?u.131928\ninst✝ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Type u_2\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\n⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ (s : Finset ι), ∑ i in s, f i x') s x", "state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type ?u.131928\ninst✝ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Type u_2\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\n⊢ LowerSemicontinuousWithinAt (fun x' => ∑' (i : ι), f i x') s x", "tactic": "simp_rw [ENNReal.tsum_eq_iSup_sum]" }, { "state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type ?u.131928\ninst✝ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Type u_2\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\nb : Finset ι\n⊢ LowerSemicontinuousWithinAt (fun x' => ∑ i in b, f i x') s x", "state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type ?u.131928\ninst✝ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Type u_2\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\n⊢ LowerSemicontinuousWithinAt (fun x' => ⨆ (s : Finset ι), ∑ i in s, f i x') s x", "tactic": "refine lowerSemicontinuousWithinAt_iSup fun b => ?_" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type ?u.131928\ninst✝ : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Type u_2\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\nb : Finset ι\n⊢ LowerSemicontinuousWithinAt (fun x' => ∑ i in b, f i x') s x", "tactic": "exact lowerSemicontinuousWithinAt_sum fun i _hi => h i" } ]

[ 642, 57 ]

[ 637, 1 ]

[ { "state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ HasFDerivAt (f^[n]) (f' ^ n) x", "tactic": "refine' HasFDerivAtFilter.iterate hf _ hx n" }, { "state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : ContinuousAt f x\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "tactic": "have := hf.continuousAt" }, { "state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : Tendsto f (𝓝 x) (𝓝 (f x))\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : ContinuousAt f x\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "tactic": "unfold ContinuousAt at this" }, { "state_after": "case h.e'_5.h.e'_3\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : Tendsto f (𝓝 x) (𝓝 (f x))\n⊢ x = f x", "state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : Tendsto f (𝓝 x) (𝓝 (f x))\n⊢ Tendsto f (𝓝 x) (𝓝 x)", "tactic": "convert this" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.e'_3\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.140011\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.140106\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.140201\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : Tendsto f (𝓝 x) (𝓝 (f x))\n⊢ x = f x", "tactic": "exact hx.symm" } ]

[ 234, 16 ]

[ 227, 11 ]

[ { "state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\n⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0", "state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\n⊢ (∫ (x : α) in s, f x ∂Measure.dirac a) = if a ∈ s then f a else 0", "tactic": "rw [restrict_dirac]" }, { "state_after": "case inl\nα : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\nh✝ : a ∈ s\n⊢ (∫ (x : α), f x ∂Measure.dirac a) = f a\n\ncase inr\nα : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\nh✝ : ¬a ∈ s\n⊢ (∫ (x : α), f x ∂0) = 0", "state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\n⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0", "tactic": "split_ifs" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\nh✝ : a ∈ s\n⊢ (∫ (x : α), f x ∂Measure.dirac a) = f a", "tactic": "exact integral_dirac _ _" }, { "state_after": "no goals", "state_before": "case inr\nα : Type u_1\nE : Type u_2\nF : Type ?u.1496551\n𝕜 : Type ?u.1496554\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1499245\ninst✝⁴ : TopologicalSpace X\ninst✝³ : FirstCountableTopology X\nν : Measure α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\na : α\ns : Set α\ninst✝ : Decidable (a ∈ s)\nh✝ : ¬a ∈ s\n⊢ (∫ (x : α), f x ∂0) = 0", "tactic": "exact integral_zero_measure _" } ]

[ 1617, 34 ]

[ 1611, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Sort ?u.276164\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : Subgraph G\ns : Set V\n⊢ deleteVerts G' (G'.verts ∩ s) = deleteVerts G' s", "tactic": "ext <;> simp (config := { contextual := true }) [imp_false]" } ]

[ 1249, 62 ]

[ 1248, 1 ]

[]

[ 796, 91 ]

[ 796, 9 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_3\nR : Type u_1\nM : Type u_2\ninst✝¹⁵ : IsROrC 𝕜\ninst✝¹⁴ : NormedCommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : NormedAlgebra 𝕜 R\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module Rᵐᵒᵖ M\ninst✝⁹ : IsCentralScalar R M\ninst✝⁸ : Module 𝕜 M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalRing R\ninst✝⁴ : TopologicalAddGroup M\ninst✝³ : ContinuousSMul R M\ninst✝² : CompleteSpace R\ninst✝¹ : T2Space R\ninst✝ : T2Space M\nx : tsze R M\n⊢ snd (exp 𝕜 x) = exp 𝕜 (fst x) • snd x", "tactic": "rw [exp_def, snd_add, snd_inl, snd_inr, zero_add]" } ]

[ 158, 52 ]

[ 157, 1 ]

[]

[ 196, 36 ]

[ 194, 1 ]

[]

[ 136, 26 ]

[ 135, 1 ]

[]

[ 178, 48 ]

[ 176, 11 ]

[]

[ 230, 6 ]

[ 229, 1 ]

[ { "state_after": "f : Bool → Bool → Bool\nh : f false false = false\nk : ℕ\n⊢ ∀ (m n : ℕ), testBit (bitwise' f m n) k = f (testBit m k) (testBit n k)", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm n k : ℕ\n⊢ testBit (bitwise' f m n) k = f (testBit m k) (testBit n k)", "tactic": "revert m n" }, { "state_after": "case zero\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\na : Bool\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ testBit (bit (f a b) (bitwise' f m' n')) zero = f (testBit (bit a m') zero) (testBit (bit b n') zero)\n\ncase succ\nf : Bool → Bool → Bool\nh : f false false = false\nk : ℕ\nIH : ∀ (m n : ℕ), testBit (bitwise' f m n) k = f (testBit m k) (testBit n k)\nm n : ℕ\na : Bool\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ testBit (bit (f a b) (bitwise' f m' n')) (succ k) = f (testBit (bit a m') (succ k)) (testBit (bit b n') (succ k))", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nk : ℕ\n⊢ ∀ (m n : ℕ), testBit (bitwise' f m n) k = f (testBit m k) (testBit n k)", "tactic": "induction' k with k IH <;>\nintros m n <;>\napply bitCasesOn m <;> intros a m' <;>\napply bitCasesOn n <;> intros b n' <;>\nrw [bitwise'_bit h]" }, { "state_after": "no goals", "state_before": "case zero\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\na : Bool\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ testBit (bit (f a b) (bitwise' f m' n')) zero = f (testBit (bit a m') zero) (testBit (bit b n') zero)", "tactic": "simp [testBit_zero]" }, { "state_after": "no goals", "state_before": "case succ\nf : Bool → Bool → Bool\nh : f false false = false\nk : ℕ\nIH : ∀ (m n : ℕ), testBit (bitwise' f m n) k = f (testBit m k) (testBit n k)\nm n : ℕ\na : Bool\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ testBit (bit (f a b) (bitwise' f m' n')) (succ k) = f (testBit (bit a m') (succ k)) (testBit (bit b n') (succ k))", "tactic": "simp [testBit_succ, IH]" } ]

[ 500, 28 ]

[ 492, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ ∀ (a : Bool), xor (!a) a = true", "tactic": "decide" } ]

[ 274, 59 ]

[ 274, 1 ]

[ { "state_after": "ι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\n⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x", "state_before": "ι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nH : ∀ (x : X), ∃ U, U ∈ 𝓝 x ∧ ∃ g, ContinuousOn g U ∧ ∀ (y : X), y ∈ U → g y ∈ t y\n⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x", "tactic": "choose U hU g hgc hgt using H" }, { "state_after": "case intro\nι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\nf : PartitionOfUnity X X univ\nhf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)\n⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x", "state_before": "ι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\n⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x", "tactic": "obtain ⟨f, hf⟩ := PartitionOfUnity.exists_isSubordinate isClosed_univ (fun x => interior (U x))\n (fun x => isOpen_interior) fun x _ => mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩" }, { "state_after": "case intro\nι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\nf : PartitionOfUnity X X univ\nhf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)\nx i : X\nhi : ↑(PartitionOfUnity.toFun univ f i) x ≠ 0\n⊢ x ∈ U i", "state_before": "case intro\nι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\nf : PartitionOfUnity X X univ\nhf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)\n⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x", "tactic": "refine' ⟨⟨fun x => ∑ᶠ i, f i x • g i x,\n hf.continuous_finsum_smul (fun i => isOpen_interior) fun i => (hgc i).mono interior_subset⟩,\n fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩" }, { "state_after": "no goals", "state_before": "case intro\nι : Type ?u.6042\nX : Type u_2\nE : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : NormalSpace X\ninst✝³ : ParacompactSpace X\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nt : X → Set E\nht : ∀ (x : X), Convex ℝ (t x)\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\ng : X → X → E\nhgc : ∀ (x : X), ContinuousOn (g x) (U x)\nhgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y\nf : PartitionOfUnity X X univ\nhf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)\nx i : X\nhi : ↑(PartitionOfUnity.toFun univ f i) x ≠ 0\n⊢ x ∈ U i", "tactic": "exact interior_subset (hf _ <| subset_closure hi)" } ]

[ 63, 52 ]

[ 54, 1 ]

[]

[ 941, 46 ]

[ 940, 1 ]

[]

[ 112, 90 ]

[ 112, 1 ]

[]

[ 2012, 44 ]

[ 2011, 1 ]

[ { "state_after": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\n⊢ CompleteLattice.Independent fun i => range (ϕ i)", "state_before": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\n⊢ CompleteLattice.Independent fun i => range (ϕ i)", "tactic": "cases nonempty_fintype ι" }, { "state_after": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\n⊢ Disjoint ((fun i => range (ϕ i)) i) (⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j)", "state_before": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\n⊢ CompleteLattice.Independent fun i => range (ϕ i)", "tactic": "rintro i" }, { "state_after": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\n⊢ ((fun i => range (ϕ i)) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j) ≤ ⊥", "state_before": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\n⊢ Disjoint ((fun i => range (ϕ i)) i) (⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j)", "tactic": "rw [disjoint_iff_inf_le]" }, { "state_after": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ ↑((fun i => range (ϕ i)) i).toSubmonoid\nhxp : f ∈ ↑(⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j).toSubmonoid\n⊢ f ∈ ⊥", "state_before": "case intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\n⊢ ((fun i => range (ϕ i)) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j) ≤ ⊥", "tactic": "rintro f ⟨hxi, hxp⟩" }, { "state_after": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (j : ι) (_ : ¬j = i), range (ϕ j))\n⊢ f ∈ ⊥", "state_before": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ ↑((fun i => range (ϕ i)) i).toSubmonoid\nhxp : f ∈ ↑(⨆ (j : ι) (_ : j ≠ i), (fun i => range (ϕ i)) j).toSubmonoid\n⊢ f ∈ ⊥", "tactic": "dsimp at hxi hxp" }, { "state_after": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\nhxp : f ∈ ↑(range (noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm))\n⊢ f ∈ ⊥\n\ncase intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\n⊢ ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)", "state_before": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (j : ι) (_ : ¬j = i), range (ϕ j))\n⊢ f ∈ ⊥", "tactic": "rw [iSup_subtype', ← noncommPiCoprod_range] at hxp" }, { "state_after": "case intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\n⊢ ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)\n\ncase intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\nhxp : f ∈ ↑(range (noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm))\n⊢ f ∈ ⊥", "state_before": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\nhxp : f ∈ ↑(range (noncommPiCoprod (fun x => ϕ ↑x) ?intro.intro.hcomm))\n⊢ f ∈ ⊥\n\ncase intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\n⊢ ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)", "tactic": "rotate_left" }, { "state_after": "case intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\n⊢ f ∈ ⊥", "state_before": "case intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\nhxp :\n f ∈\n ↑(range\n (noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y))))\n⊢ f ∈ ⊥", "tactic": "cases' hxp with g hgf" }, { "state_after": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\n⊢ f ∈ ⊥", "state_before": "case intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\n⊢ f ∈ ⊥", "tactic": "cases' hxi with g' hg'f" }, { "state_after": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ f ∈ ⊥", "state_before": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\n⊢ f ∈ ⊥", "tactic": "have hxi : orderOf f ∣ Fintype.card (H i) := by\n rw [← hg'f]\n exact (orderOf_map_dvd _ _).trans orderOf_dvd_card_univ" }, { "state_after": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f ∈ ⊥", "state_before": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ f ∈ ⊥", "tactic": "have hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H j) := by\n rw [← hgf, ← Fintype.card_pi]\n exact (orderOf_map_dvd _ _).trans orderOf_dvd_card_univ" }, { "state_after": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f = 1", "state_before": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f ∈ ⊥", "tactic": "change f = 1" }, { "state_after": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ orderOf f ∣ 1", "state_before": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f = 1", "tactic": "rw [← pow_one f, ← orderOf_dvd_iff_pow_eq_one]" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ orderOf f ∣ 1", "state_before": "case intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ orderOf f ∣ 1", "tactic": "obtain ⟨c, hc⟩ := Nat.dvd_gcd hxp hxi" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = orderOf f * c", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ orderOf f ∣ 1", "tactic": "use c" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i))", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = orderOf f * c", "tactic": "rw [← hc]" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = 1", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ 1 = Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i))", "tactic": "symm" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ Nat.coprime (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i))", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = 1", "tactic": "rw [← Nat.coprime_iff_gcd_eq_one]" }, { "state_after": "case intro.intro.intro.intro.intro.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ ∀ (i_1 : { j // j ≠ i }), i_1 ∈ Finset.univ → Nat.coprime (Fintype.card (H ↑i_1)) (Fintype.card (H i))", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ Nat.coprime (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i))", "tactic": "apply Nat.coprime_prod_left" }, { "state_after": "case intro.intro.intro.intro.intro.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\nj : { j // j ≠ i }\na✝ : j ∈ Finset.univ\n⊢ Nat.coprime (Fintype.card (H ↑j)) (Fintype.card (H i))", "state_before": "case intro.intro.intro.intro.intro.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\n⊢ ∀ (i_1 : { j // j ≠ i }), i_1 ∈ Finset.univ → Nat.coprime (Fintype.card (H ↑i_1)) (Fintype.card (H i))", "tactic": "intro j _" }, { "state_after": "case intro.intro.intro.intro.intro.a.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\nj : { j // j ≠ i }\na✝ : j ∈ Finset.univ\n⊢ ↑j ≠ i", "state_before": "case intro.intro.intro.intro.intro.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\nj : { j // j ≠ i }\na✝ : j ∈ Finset.univ\n⊢ Nat.coprime (Fintype.card (H ↑j)) (Fintype.card (H i))", "tactic": "apply hcoprime" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.a.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : Nat.gcd (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)) (Fintype.card (H i)) = orderOf f * c\nj : { j // j ≠ i }\na✝ : j ∈ Finset.univ\n⊢ ↑j ≠ i", "tactic": "exact j.2" }, { "state_after": "case intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ni✝ j✝ : { j // ¬j = i }\nhj : i✝ ≠ j✝\n⊢ ∀ (x : H ↑i✝) (y : H ↑j✝), Commute (↑(ϕ ↑i✝) x) (↑(ϕ ↑j✝) y)", "state_before": "case intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\n⊢ ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)", "tactic": "intro _ _ hj" }, { "state_after": "case intro.intro.hcomm.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ni✝ j✝ : { j // ¬j = i }\nhj : i✝ ≠ j✝\n⊢ ↑i✝ ≠ ↑j✝", "state_before": "case intro.intro.hcomm\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ni✝ j✝ : { j // ¬j = i }\nhj : i✝ ≠ j✝\n⊢ ∀ (x : H ↑i✝) (y : H ↑j✝), Commute (↑(ϕ ↑i✝) x) (↑(ϕ ↑j✝) y)", "tactic": "apply hcomm" }, { "state_after": "no goals", "state_before": "case intro.intro.hcomm.a\nG : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxi : f ∈ Set.range ↑(ϕ i)\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ni✝ j✝ : { j // ¬j = i }\nhj : i✝ ≠ j✝\n⊢ ↑i✝ ≠ ↑j✝", "tactic": "exact hj ∘ Subtype.ext" }, { "state_after": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\n⊢ orderOf (↑(ϕ i) g') ∣ Fintype.card (H i)", "state_before": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\n⊢ orderOf f ∣ Fintype.card (H i)", "tactic": "rw [← hg'f]" }, { "state_after": "no goals", "state_before": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\n⊢ orderOf (↑(ϕ i) g') ∣ Fintype.card (H i)", "tactic": "exact (orderOf_map_dvd _ _).trans orderOf_dvd_card_univ" }, { "state_after": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ orderOf\n (↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g) ∣\n Fintype.card ((a : { j // j ≠ i }) → H ↑a)", "state_before": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)", "tactic": "rw [← hgf, ← Fintype.card_pi]" }, { "state_after": "no goals", "state_before": "G : Type u_3\ninst✝³ : Group G\nι : Type u_1\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_2\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), Commute (↑(ϕ i) x) (↑(ϕ j) y)\nf✝ g✝ : (i : ι) → H i\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : ∀ (i j : ι), i ≠ j → Nat.coprime (Fintype.card (H i)) (Fintype.card (H j))\nval✝ : Fintype ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ (x : { j // ¬j = i }), range (ϕ ↑x))\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf :\n ↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g =\n f\ng' : H i\nhg'f : ↑(ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ orderOf\n (↑(noncommPiCoprod (fun x => ϕ ↑x)\n (_ : ∀ (i_1 j : { j // ¬j = i }), i_1 ≠ j → ∀ (x : H ↑i_1) (y : H ↑j), Commute (↑(ϕ ↑i_1) x) (↑(ϕ ↑j) y)))\n g) ∣\n Fintype.card ((a : { j // j ≠ i }) → H ↑a)", "tactic": "exact (orderOf_map_dvd _ _).trans orderOf_dvd_card_univ" } ]

[ 274, 14 ]

[ 241, 1 ]

[]

[ 183, 52 ]

[ 181, 1 ]

[]

[ 703, 62 ]

[ 701, 11 ]

[ { "state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : ReflTransGen r a b\n⊢ b = a ∨ TransGen r a b\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : b = a ∨ TransGen r a b\n⊢ ReflTransGen r a b", "state_before": "α : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\n⊢ ReflTransGen r a b ↔ b = a ∨ TransGen r a b", "tactic": "refine' ⟨fun h ↦ _, fun h ↦ _⟩" }, { "state_after": "case refine'_1.refl\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na c d : α\n⊢ a = a ∨ TransGen r a a\n\ncase refine'_1.tail\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c✝ d c : α\nhac : ReflTransGen r a c\nhcb : r c b\n⊢ b = a ∨ TransGen r a b", "state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : ReflTransGen r a b\n⊢ b = a ∨ TransGen r a b", "tactic": "cases' h with c _ hac hcb" }, { "state_after": "no goals", "state_before": "case refine'_1.refl\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na c d : α\n⊢ a = a ∨ TransGen r a a", "tactic": "exact Or.inl rfl" }, { "state_after": "no goals", "state_before": "case refine'_1.tail\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c✝ d c : α\nhac : ReflTransGen r a c\nhcb : r c b\n⊢ b = a ∨ TransGen r a b", "tactic": "exact Or.inr (TransGen.tail' hac hcb)" }, { "state_after": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\nb c d : α\n⊢ ReflTransGen r b b\n\ncase refine'_2.inr\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : TransGen r a b\n⊢ ReflTransGen r a b", "state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : b = a ∨ TransGen r a b\n⊢ ReflTransGen r a b", "tactic": "rcases h with (rfl | h)" }, { "state_after": "no goals", "state_before": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\nb c d : α\n⊢ ReflTransGen r b b", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case refine'_2.inr\nα : Type u_1\nβ : Type ?u.24500\nγ : Type ?u.24503\nδ : Type ?u.24506\nr : α → α → Prop\na b c d : α\nh : TransGen r a b\n⊢ ReflTransGen r a b", "tactic": "exact h.to_reflTransGen" } ]

[ 524, 30 ]

[ 517, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.248450\ninst✝ : Fintype ι\na : ℝ\nn : ℕ\n⊢ ↑n = ↑↑volume (Ioo (a - ↑n) a)", "tactic": "simp" } ]

[ 166, 63 ]

[ 161, 1 ]

[]

[ 1368, 35 ]

[ 1367, 1 ]

[]

[ 653, 44 ]

[ 652, 1 ]

[ { "state_after": "case inl\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid\n\ncase inr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n✝ : A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid", "state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n✝ : A\nn : ℕ\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid", "tactic": "obtain rfl | hn := Decidable.eq_or_ne n 0" }, { "state_after": "case inl\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ 1 ≤ 1.toAddSubmonoid", "state_before": "case inl\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid", "tactic": "rw [pow_zero, pow_zero]" }, { "state_after": "no goals", "state_before": "case inl\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ 1 ≤ 1.toAddSubmonoid", "tactic": "exact le_one_toAddSubmonoid" }, { "state_after": "no goals", "state_before": "case inr\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n✝ : A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid", "tactic": "exact (pow_toAddSubmonoid M hn).ge" } ]

[ 432, 39 ]

[ 428, 1 ]

[ { "state_after": "case mp\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ\n\ncase mpr\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ\n⊢ totalVariation s ≪ μ", "state_before": "α : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\n⊢ totalVariation s ≪ μ ↔ (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ", "tactic": "constructor <;> intro h" }, { "state_after": "case mp.left\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).posPart ≪ μ\n\ncase mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).negPart ≪ μ", "state_before": "case mp\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ", "tactic": "constructor" }, { "state_after": "case mp.left\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0\n⊢ ↑↑(toJordanDecomposition s).posPart S = 0\n\ncase mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "state_before": "case mp.left\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).posPart ≪ μ\n\ncase mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).negPart ≪ μ", "tactic": "all_goals\n refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _\n have := h hS₂\n rw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this" }, { "state_after": "no goals", "state_before": "case mp.left\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0\n⊢ ↑↑(toJordanDecomposition s).posPart S = 0\n\ncase mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "tactic": "exacts [this.1, this.2]" }, { "state_after": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "state_before": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\n⊢ (toJordanDecomposition s).negPart ≪ μ", "tactic": "refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _" }, { "state_after": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(totalVariation s) S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "state_before": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "tactic": "have := h hS₂" }, { "state_after": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(toJordanDecomposition s).posPart S = 0 ∧ ↑↑(toJordanDecomposition s).negPart S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "state_before": "case mp.right\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : totalVariation s ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\nthis : ↑↑(totalVariation s) S = 0\n⊢ ↑↑(toJordanDecomposition s).negPart S = 0", "tactic": "rw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this" }, { "state_after": "case mpr\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\n⊢ ↑↑(totalVariation s) S = 0", "state_before": "case mpr\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ\n⊢ totalVariation s ≪ μ", "tactic": "refine' Measure.AbsolutelyContinuous.mk fun S _ hS₂ => _" }, { "state_after": "no goals", "state_before": "case mpr\nα : Type u_1\nβ : Type ?u.93781\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nh : (toJordanDecomposition s).posPart ≪ μ ∧ (toJordanDecomposition s).negPart ≪ μ\nS : Set α\nx✝ : MeasurableSet S\nhS₂ : ↑↑μ S = 0\n⊢ ↑↑(totalVariation s) S = 0", "tactic": "rw [totalVariation, Measure.add_apply, h.1 hS₂, h.2 hS₂, add_zero]" } ]

[ 548, 71 ]

[ 537, 1 ]

[]

[ 1531, 16 ]

[ 1529, 1 ]

[ { "state_after": "case top\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\n⊢ (a + b) * ⊤ = a * ⊤ + b * ⊤\n\ncase coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c", "state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b c : WithTop α\n⊢ (a + b) * c = a * c + b * c", "tactic": "induction' c using WithTop.recTopCoe with c" }, { "state_after": "no goals", "state_before": "case top\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\n⊢ (a + b) * ⊤ = a * ⊤ + b * ⊤", "tactic": "by_cases ha : a = 0 <;> simp [ha]" }, { "state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : c = 0\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : ¬c = 0\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c", "state_before": "case coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c", "tactic": "by_cases hc : c = 0" }, { "state_after": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : ¬c = 0\n⊢ (Option.bind (a + b) fun a => Option.some (a * c)) =\n (Option.bind a fun a => Option.some (a * c)) + Option.bind b fun a => Option.some (a * c)", "state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : ¬c = 0\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c", "tactic": "simp [mul_coe hc]" }, { "state_after": "case neg.none.none\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\n⊢ (Option.bind (none + none) fun a => Option.some (a * c)) =\n (Option.bind none fun a => Option.some (a * c)) + Option.bind none fun a => Option.some (a * c)\n\ncase neg.none.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝ : α\n⊢ (Option.bind (none + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind none fun a => Option.some (a * c)) + Option.bind (Option.some val✝) fun a => Option.some (a * c)\n\ncase neg.some.none\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝ : α\n⊢ (Option.bind (Option.some val✝ + none) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝) fun a => Option.some (a * c)) + Option.bind none fun a => Option.some (a * c)\n\ncase neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : ¬c = 0\n⊢ (Option.bind (a + b) fun a => Option.some (a * c)) =\n (Option.bind a fun a => Option.some (a * c)) + Option.bind b fun a => Option.some (a * c)", "tactic": "cases a <;> cases b" }, { "state_after": "no goals", "state_before": "case neg.none.none\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\n⊢ (Option.bind (none + none) fun a => Option.some (a * c)) =\n (Option.bind none fun a => Option.some (a * c)) + Option.bind none fun a => Option.some (a * c)\n\ncase neg.none.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝ : α\n⊢ (Option.bind (none + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind none fun a => Option.some (a * c)) + Option.bind (Option.some val✝) fun a => Option.some (a * c)\n\ncase neg.some.none\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝ : α\n⊢ (Option.bind (Option.some val✝ + none) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝) fun a => Option.some (a * c)) + Option.bind none fun a => Option.some (a * c)\n\ncase neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "tactic": "repeat' first | rfl |exact congr_arg some (add_mul _ _ _)" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\na b : WithTop α\nc : α\nhc : c = 0\n⊢ (a + b) * ↑c = a * ↑c + b * ↑c", "tactic": "simp [hc]" }, { "state_after": "no goals", "state_before": "case neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "tactic": "first | rfl |exact congr_arg some (add_mul _ _ _)" }, { "state_after": "case neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "state_before": "case neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case neg.some.some\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CanonicallyOrderedCommSemiring α\nc : α\nhc : ¬c = 0\nval✝¹ val✝ : α\n⊢ (Option.bind (Option.some val✝¹ + Option.some val✝) fun a => Option.some (a * c)) =\n (Option.bind (Option.some val✝¹) fun a => Option.some (a * c)) +\n Option.bind (Option.some val✝) fun a => Option.some (a * c)", "tactic": "exact congr_arg some (add_mul _ _ _)" } ]

[ 195, 62 ]

[ 188, 9 ]

[]

[ 166, 26 ]

[ 165, 1 ]

[ { "state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.275320\nι' : Sort ?u.275323\nι₂ : Sort ?u.275326\nκ : ι → Sort ?u.275331\nκ₁ : ι → Sort ?u.275336\nκ₂ : ι → Sort ?u.275341\nκ' : ι' → Sort ?u.275346\nf : α → β → γ\ns : Set α\nt : Set β\nx✝ : γ\n⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ seq (f '' s) t", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.275320\nι' : Sort ?u.275323\nι₂ : Sort ?u.275326\nκ : ι → Sort ?u.275331\nκ₁ : ι → Sort ?u.275336\nκ₂ : ι → Sort ?u.275341\nκ' : ι' → Sort ?u.275346\nf : α → β → γ\ns : Set α\nt : Set β\n⊢ image2 f s t = seq (f '' s) t", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.275320\nι' : Sort ?u.275323\nι₂ : Sort ?u.275326\nκ : ι → Sort ?u.275331\nκ₁ : ι → Sort ?u.275336\nκ₂ : ι → Sort ?u.275341\nκ' : ι' → Sort ?u.275346\nf : α → β → γ\ns : Set α\nt : Set β\nx✝ : γ\n⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ seq (f '' s) t", "tactic": "simp" } ]

[ 2010, 7 ]

[ 2008, 1 ]

[ { "state_after": "case intro\nd✝ d : ℤ\nhd : d ≤ 0\nx : ℤ√d\nu : (ℤ√d)ˣ\n⊢ norm x = norm (x * ↑u)", "state_before": "d✝ d : ℤ\nhd : d ≤ 0\nx y : ℤ√d\nh : Associated x y\n⊢ norm x = norm y", "tactic": "obtain ⟨u, rfl⟩ := h" }, { "state_after": "no goals", "state_before": "case intro\nd✝ d : ℤ\nhd : d ≤ 0\nx : ℤ√d\nu : (ℤ√d)ˣ\n⊢ norm x = norm (x * ↑u)", "tactic": "rw [norm_mul, (norm_eq_one_iff' hd _).mpr u.isUnit, mul_one]" } ]

[ 630, 63 ]

[ 627, 1 ]

[ { "state_after": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nB_decidableEq : DecidableEq α\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ B_decidableLE B_decidableEq B_decidableLT", "state_before": "ι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_le B_lt : α → α → Prop\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nB_decidableEq : DecidableEq α\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nh : toPartialOrder = toPartialOrder\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ B_decidableLE B_decidableEq B_decidableLT", "tactic": "cases h" }, { "state_after": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nB_decidableEq : DecidableEq α\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE B_decidableEq B_decidableLT", "state_before": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nB_decidableEq : DecidableEq α\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ B_decidableLE B_decidableEq B_decidableLT", "tactic": "obtain rfl : A_decidableLE = B_decidableLE := Subsingleton.elim _ _" }, { "state_after": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq B_decidableLT", "state_before": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nB_decidableEq : DecidableEq α\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE B_decidableEq B_decidableLT", "tactic": "obtain rfl : A_decidableEq = B_decidableEq := Subsingleton.elim _ _" }, { "state_after": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq B_decidableLT", "tactic": "obtain rfl : A_decidableLT = B_decidableLT := Subsingleton.elim _ _" }, { "state_after": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nthis : A_min = B_min\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "have : A_min = B_min := by\n funext a b\n exact (A_min_def _ _).trans (B_min_def _ _).symm" }, { "state_after": "case refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nthis : A_min = B_min\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "cases this" }, { "state_after": "case refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nthis : A_max = B_max\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "have : A_max = B_max := by\n funext a b\n exact (A_max_def _ _).trans (B_max_def _ _).symm" }, { "state_after": "case refl.refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nthis : A_max = B_max\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "cases this" }, { "state_after": "case refl.refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nthis : A_compare = B_compare\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "state_before": "case refl.refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "have : A_compare = B_compare := by\n funext a b\n exact (A_compare_canonical _ _).trans (B_compare_canonical _ _).symm" }, { "state_after": "no goals", "state_before": "case refl.refl.refl\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nthis : A_compare = B_compare\n⊢ mk le_total✝¹ A_decidableLE A_decidableEq A_decidableLT = mk le_total✝ A_decidableLE A_decidableEq A_decidableLT", "tactic": "congr" }, { "state_after": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\na b : α\n⊢ A_min a b = B_min a b", "state_before": "ι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\n⊢ A_min = B_min", "tactic": "funext a b" }, { "state_after": "no goals", "state_before": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min B_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\na b : α\n⊢ A_min a b = B_min a b", "tactic": "exact (A_min_def _ _).trans (B_min_def _ _).symm" }, { "state_after": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\na b : α\n⊢ A_max a b = B_max a b", "state_before": "ι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\n⊢ A_max = B_max", "tactic": "funext a b" }, { "state_after": "no goals", "state_before": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_max : α → α → α\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\na b : α\n⊢ A_max a b = B_max a b", "tactic": "exact (A_max_def _ _).trans (B_max_def _ _).symm" }, { "state_after": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\na b : α\n⊢ A_compare a b = B_compare a b", "state_before": "ι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\n⊢ A_compare = B_compare", "tactic": "funext a b" }, { "state_after": "no goals", "state_before": "case h.h\nι : Type ?u.13117\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.13128\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : LinearOrder α\nA_le A_lt : α → α → Prop\nle_refl✝¹ : ∀ (a : α), a ≤ a\nle_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝¹ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝¹ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nA_min A_max : α → α → α\nA_compare : α → α → Ordering\nle_total✝¹ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nA_decidableLE : DecidableRel fun x x_1 => x ≤ x_1\nA_decidableEq : DecidableEq α\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nA_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\nA_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_compare : α → α → Ordering\nle_refl✝ : ∀ (a : α), a ≤ a\nle_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c\nlt_iff_le_not_le✝ : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a\nle_antisymm✝ : ∀ (a b : α), a ≤ b → b ≤ a → a = b\nle_total✝ : ∀ (a b : α), a ≤ b ∨ b ≤ a\nB_compare_canonical : ∀ (a b : α), compare a b = compareOfLessAndEq a b\nB_min_def : ∀ (a b : α), min a b = if a ≤ b then a else b\nB_max_def : ∀ (a b : α), max a b = if a ≤ b then b else a\na b : α\n⊢ A_compare a b = B_compare a b", "tactic": "exact (A_compare_canonical _ _).trans (B_compare_canonical _ _).symm" } ]

[ 647, 10 ]

[ 621, 1 ]

[]

[ 140, 99 ]

[ 139, 1 ]

[]

[ 3115, 69 ]

[ 3114, 1 ]

[ { "state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\nthis : IsIso f\n⊢ f = g", "state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\n⊢ f = g", "tactic": "haveI := hI.isIso_from f" }, { "state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\nthis✝ : IsIso f\nthis : IsIso g\n⊢ f = g", "state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\nthis : IsIso f\n⊢ f = g", "tactic": "haveI := hI.isIso_from g" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\nthis✝ : IsIso f\nthis : IsIso g\n⊢ f = g", "tactic": "exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))" } ]

[ 198, 54 ]

[ 195, 1 ]

[]

[ 548, 83 ]

[ 548, 1 ]

[]

[ 277, 28 ]

[ 275, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nδ : Type w\nx y : α\ns t : Stream' α\nh : x :: s = y :: t\n⊢ x = y", "tactic": "rw [← nth_zero_cons x s, h, nth_zero_cons]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nδ : Type w\nx y : α\ns t : Stream' α\nh : x :: s = y :: t\nn : ℕ\n⊢ nth s n = nth t n", "tactic": "rw [← nth_succ_cons n _ x, h, nth_succ_cons]" } ]

[ 101, 74 ]

[ 99, 1 ]

[]

[ 178, 6 ]

[ 176, 1 ]

[]

[ 1642, 70 ]

[ 1640, 1 ]

[]

[ 559, 25 ]

[ 555, 1 ]

[]

[ 92, 50 ]

[ 91, 11 ]

[]

[ 1033, 40 ]

[ 1032, 1 ]

[]

[ 105, 39 ]

[ 103, 1 ]

[]

[ 571, 77 ]

[ 571, 1 ]

[]

[ 1043, 18 ]

[ 1039, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 - g1 ≈ f2 - g2", "tactic": "simpa only [sub_eq_add_neg] using add_equiv_add hf (neg_equiv_neg hg)" } ]

[ 483, 98 ]

[ 482, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\n⊢ ∀ (b : β), b ∈ t → ∃ a ha, b = f a ha", "tactic": "intro b hb" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\n⊢ ∃ a ha, b = f a ha", "tactic": "set f' : s → t := fun x ↦ ⟨f x.1 x.2, hf _ _⟩" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\n⊢ ∃ a ha, b = f a ha", "tactic": "have finj : f'.Injective := by\n rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy\n simp only [Subtype.mk.injEq] at hxy ⊢\n apply hinj _ _ hx hy hxy" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\n⊢ ∃ a ha, b = f a ha", "tactic": "have hft := ht.fintype" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\n⊢ ∃ a ha, b = f a ha", "tactic": "have hft' := Fintype.ofInjective f' finj" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\n⊢ ∃ a ha, b = f a ha", "tactic": "simp_rw [ncard_eq_toFinset_card] at hst" }, { "state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∃ a ha, b = f a ha", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\n⊢ ∃ a ha, b = f a ha", "tactic": "set f'' : ∀ a, a ∈ s.toFinset → β := fun a h ↦ f a (by simpa using h)" }, { "state_after": "case h.e'_2.h.h.e'_1.a\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\nx✝ : α\n⊢ x✝ ∈ s ↔ x✝ ∈ toFinset s\n\ncase convert_1\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∀ (a : α) (ha : a ∈ toFinset s), f'' a ha ∈ toFinset t\n\ncase convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ toFinset s) (ha₂ : a₂ ∈ toFinset s), f'' a₁ ha₁ = f'' a₂ ha₂ → a₁ = a₂\n\ncase convert_3\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ Finset.card (toFinset t) ≤ Finset.card (toFinset s)", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∃ a ha, b = f a ha", "tactic": "convert @Finset.surj_on_of_inj_on_of_card_le _ _ _ t.toFinset f'' _ _ _ (by simpa) (by simpa)" }, { "state_after": "no goals", "state_before": "case convert_3\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ Finset.card (toFinset t) ≤ Finset.card (toFinset s)", "tactic": "rwa [←ncard_eq_toFinset_card', ←ncard_eq_toFinset_card']" }, { "state_after": "case mk.mk\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x✝ y✝ : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : f' { val := x, property := hx } = f' { val := y, property := hy }\n⊢ { val := x, property := hx } = { val := y, property := hy }", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\n⊢ Function.Injective f'", "tactic": "rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy" }, { "state_after": "case mk.mk\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x✝ y✝ : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : f x (_ : ↑{ val := x, property := hx } ∈ s) = f y (_ : ↑{ val := y, property := hy } ∈ s)\n⊢ x = y", "state_before": "case mk.mk\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x✝ y✝ : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : f' { val := x, property := hx } = f' { val := y, property := hy }\n⊢ { val := x, property := hx } = { val := y, property := hy }", "tactic": "simp only [Subtype.mk.injEq] at hxy ⊢" }, { "state_after": "no goals", "state_before": "case mk.mk\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x✝ y✝ : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : f x (_ : ↑{ val := x, property := hx } ∈ s) = f y (_ : ↑{ val := y, property := hy } ∈ s)\n⊢ x = y", "tactic": "apply hinj _ _ hx hy hxy" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na✝ b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\na : α\nh : a ∈ toFinset s\n⊢ a ∈ s", "tactic": "simpa using h" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ β", "tactic": "simpa" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ b ∈ toFinset t", "tactic": "simpa" }, { "state_after": "no goals", "state_before": "case h.e'_2.h.h.e'_1.a\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\nx✝ : α\n⊢ x✝ ∈ s ↔ x✝ ∈ toFinset s", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case convert_1\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∀ (a : α) (ha : a ∈ toFinset s), f'' a ha ∈ toFinset t", "tactic": "simp [hf]" }, { "state_after": "case convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\na₁ a₂ : α\nha₁ : a₁ ∈ toFinset s\nha₂ : a₂ ∈ toFinset s\nh : f'' a₁ ha₁ = f'' a₂ ha₂\n⊢ a₁ = a₂", "state_before": "case convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\n⊢ ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ toFinset s) (ha₂ : a₂ ∈ toFinset s), f'' a₁ ha₁ = f'' a₂ ha₂ → a₁ = a₂", "tactic": "intros a₁ a₂ ha₁ ha₂ h" }, { "state_after": "case convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\na₁ a₂ : α\nha₁✝ : a₁ ∈ toFinset s\nha₁ : a₁ ∈ s\nha₂✝ : a₂ ∈ toFinset s\nha₂ : a₂ ∈ s\nh : f'' a₁ ha₁✝ = f'' a₂ ha₂✝\n⊢ a₁ = a₂", "state_before": "case convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\na₁ a₂ : α\nha₁ : a₁ ∈ toFinset s\nha₂ : a₂ ∈ toFinset s\nh : f'' a₁ ha₁ = f'' a₂ ha₂\n⊢ a₁ = a₂", "tactic": "rw [mem_toFinset] at ha₁ ha₂" }, { "state_after": "no goals", "state_before": "case convert_2\nα : Type u_2\nβ : Type u_1\ns t✝ : Set α\na b✝ x y : α\nf✝ : α → β\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : ncard t ≤ ncard s\nht : autoParam (Set.Finite t) _auto✝\nb : β\nhb : b ∈ t\nf' : ↑s → ↑t := fun x => { val := f ↑x (_ : ↑x ∈ s), property := (_ : f ↑x (_ : ↑x ∈ s) ∈ t) }\nfinj : Function.Injective f'\nhft : Fintype ↑t\nhft' : Fintype ↑s\nf'' : (a : α) → a ∈ toFinset s → β := fun a h => f a (_ : a ∈ s)\na₁ a₂ : α\nha₁✝ : a₁ ∈ toFinset s\nha₁ : a₁ ∈ s\nha₂✝ : a₂ ∈ toFinset s\nha₂ : a₂ ∈ s\nh : f'' a₁ ha₁✝ = f'' a₂ ha₂✝\n⊢ a₁ = a₂", "tactic": "exact hinj _ _ ha₁ ha₂ h" } ]

[ 431, 59 ]

[ 411, 1 ]

[ { "state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ Group.IsNilpotent G ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "tactic": "rw [nilpotent_iff_finite_ascending_central_series]" }, { "state_after": "case mp\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) → ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥\n\ncase mpr\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥) → ∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤", "state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "tactic": "constructor" }, { "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "state_before": "case mp\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) → ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "tactic": "rintro ⟨n, H, hH, hn⟩" }, { "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ (fun m => H (n - m)) n = ⊥", "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥", "tactic": "refine ⟨n, fun m => H (n - m), is_decending_rev_series_of_is_ascending G hn hH, ?_⟩" }, { "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ H (n - n) = ⊥", "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ (fun m => H (n - m)) n = ⊥", "tactic": "dsimp only" }, { "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ H 0 = ⊥", "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ H (n - n) = ⊥", "tactic": "rw [tsub_self]" }, { "state_after": "no goals", "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ H 0 = ⊥", "tactic": "exact hH.1" }, { "state_after": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ ∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤", "state_before": "case mpr\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ (∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥) → ∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤", "tactic": "rintro ⟨n, H, hH, hn⟩" }, { "state_after": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ (fun m => H (n - m)) n = ⊤", "state_before": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ ∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤", "tactic": "refine ⟨n, fun m => H (n - m), is_ascending_rev_series_of_is_descending G hn hH, ?_⟩" }, { "state_after": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ H (n - n) = ⊤", "state_before": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ (fun m => H (n - m)) n = ⊤", "tactic": "dsimp only" }, { "state_after": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ H 0 = ⊤", "state_before": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ H (n - n) = ⊤", "tactic": "rw [tsub_self]" }, { "state_after": "no goals", "state_before": "case mpr.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\n⊢ H 0 = ⊤", "tactic": "exact hH.1" } ]

[ 278, 15 ]

[ 265, 1 ]

[ { "state_after": "case inl\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b : ℝ\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x ∈ closedBall z 0\nhy : y ∈ closedBall z 0\n⊢ a • x + b • y ∈ ball z 0\n\ncase inr\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ∈ closedBall z r\nhy : y ∈ closedBall z r\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r ≠ 0\n⊢ a • x + b • y ∈ ball z r", "state_before": "𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ∈ closedBall z r\nhy : y ∈ closedBall z r\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ ball z r", "tactic": "rcases eq_or_ne r 0 with (rfl | hr)" }, { "state_after": "case inl\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b : ℝ\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x = z\nhy : y = z\n⊢ a • x + b • y ∈ ball z 0", "state_before": "case inl\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b : ℝ\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x ∈ closedBall z 0\nhy : y ∈ closedBall z 0\n⊢ a • x + b • y ∈ ball z 0", "tactic": "rw [closedBall_zero, mem_singleton_iff] at hx hy" }, { "state_after": "no goals", "state_before": "case inl\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b : ℝ\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x = z\nhy : y = z\n⊢ a • x + b • y ∈ ball z 0", "tactic": "exact (hne (hx.trans hy.symm)).elim" }, { "state_after": "case inr\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ∈ closedBall z r\nhy : y ∈ closedBall z r\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r ≠ 0\n⊢ a • x + b • y ∈ interior (closedBall z r)", "state_before": "case inr\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ∈ closedBall z r\nhy : y ∈ closedBall z r\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r ≠ 0\n⊢ a • x + b • y ∈ ball z r", "tactic": "simp only [← interior_closedBall _ hr] at hx hy⊢" }, { "state_after": "no goals", "state_before": "case inr\n𝕜 : Type ?u.41739\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ∈ closedBall z r\nhy : y ∈ closedBall z r\nhne : x ≠ y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r ≠ 0\n⊢ a • x + b • y ∈ interior (closedBall z r)", "tactic": "exact strictConvex_closedBall ℝ z r hx hy hne ha hb hab" } ]

[ 163, 60 ]

[ 157, 1 ]

[ { "state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84669\nι : Sort ?u.84672\nπ : α → Type ?u.84677\nδ : α → Sort ?u.84682\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₁ (piecewise s f₁ f₂) (s₁ ∩ s)\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84669\nι : Sort ?u.84672\nπ : α → Type ?u.84677\nδ : α → Sort ?u.84682\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₂ (piecewise s f₁ f₂) (s₂ ∩ sᶜ)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.84669\nι : Sort ?u.84672\nπ : α → Type ?u.84677\nδ : α → Sort ?u.84682\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ MapsTo (piecewise s f₁ f₂) (Set.ite s s₁ s₂) (Set.ite t t₁ t₂)", "tactic": "refine' (h₁.congr _).union_union (h₂.congr _)" }, { "state_after": "no goals", "state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84669\nι : Sort ?u.84672\nπ : α → Type ?u.84677\nδ : α → Sort ?u.84682\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₁ (piecewise s f₁ f₂) (s₁ ∩ s)\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84669\nι : Sort ?u.84672\nπ : α → Type ?u.84677\nδ : α → Sort ?u.84682\ns✝ : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s✝)\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ EqOn f₂ (piecewise s f₁ f₂) (s₂ ∩ sᶜ)", "tactic": "exacts [(piecewise_eqOn s f₁ f₂).symm.mono (inter_subset_right _ _),\n (piecewise_eqOn_compl s f₁ f₂).symm.mono (inter_subset_right _ _)]" } ]

[ 1449, 71 ]

[ 1443, 1 ]

[]

[ 275, 6 ]

[ 274, 1 ]

[ { "state_after": "⊢ StrictMono (ord ∘ aleph) ∧ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b}", "state_before": "⊢ ord ∘ aleph = enumOrd {b | ord (card b) = b ∧ ω ≤ b}", "tactic": "rw [← eq_enumOrd _ ord_card_unbounded']" }, { "state_after": "⊢ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b}", "state_before": "⊢ StrictMono (ord ∘ aleph) ∧ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b}", "tactic": "use aleph_isNormal.strictMono" }, { "state_after": "⊢ (∀ (a : Ordinal), (ord ∘ aleph) a ∈ {b | ord (card b) = b ∧ ω ≤ b}) ∧\n ∀ (b : Ordinal), b ∈ {b | ord (card b) = b ∧ ω ≤ b} → ∃ a, (ord ∘ aleph) a = b", "state_before": "⊢ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b}", "tactic": "rw [range_eq_iff]" }, { "state_after": "case refine'_1\na : Ordinal\n⊢ ord (card ((ord ∘ aleph) a)) = (ord ∘ aleph) a\n\ncase refine'_2\na : Ordinal\n⊢ ω ≤ (ord ∘ aleph) a", "state_before": "⊢ (∀ (a : Ordinal), (ord ∘ aleph) a ∈ {b | ord (card b) = b ∧ ω ≤ b}) ∧\n ∀ (b : Ordinal), b ∈ {b | ord (card b) = b ∧ ω ≤ b} → ∃ a, (ord ∘ aleph) a = b", "tactic": "refine' ⟨fun a => ⟨_, _⟩, fun b hb => eq_aleph_of_eq_card_ord hb.1 hb.2⟩" }, { "state_after": "no goals", "state_before": "case refine'_1\na : Ordinal\n⊢ ord (card ((ord ∘ aleph) a)) = (ord ∘ aleph) a", "tactic": "rw [Function.comp_apply, card_ord]" }, { "state_after": "case refine'_2\na : Ordinal\n⊢ ℵ₀ ≤ aleph a", "state_before": "case refine'_2\na : Ordinal\n⊢ ω ≤ (ord ∘ aleph) a", "tactic": "rw [← ord_aleph0, Function.comp_apply, ord_le_ord]" }, { "state_after": "no goals", "state_before": "case refine'_2\na : Ordinal\n⊢ ℵ₀ ≤ aleph a", "tactic": "exact aleph0_le_aleph _" } ]

[ 401, 28 ]

[ 393, 1 ]

[]

[ 947, 41 ]

[ 945, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type ?u.196613\nδ : Type ?u.196616\nε : Type ?u.196619\nζ : Type ?u.196622\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.196633\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\na : (i : ι) → π i\n⊢ 𝓝 a = Filter.pi fun i => 𝓝 (a i)", "tactic": "simp only [nhds_iInf, nhds_induced, Filter.pi]" } ]

[ 1218, 49 ]

[ 1217, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.138257\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : LipschitzWith K f\ng : β → α\nhg : Function.RightInverse g f\nx y : β\n⊢ edist x y ≤ ↑K * edist (g x) (g y)", "tactic": "simpa only [hg _] using hf (g x) (g y)" } ]

[ 256, 82 ]

[ 254, 1 ]

[]

[ 758, 18 ]

[ 757, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ signAux (f * g) = signAux f * signAux g⁻¹", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ signAux (f * g) = signAux f * signAux g", "tactic": "rw [← signAux_inv g]" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ (∏ x in finPairsLT n, if ↑(f * g) x.fst ≤ ↑(f * g) x.snd then -1 else 1) =\n (∏ x in finPairsLT n, if ↑f x.fst ≤ ↑f x.snd then -1 else 1) *\n ∏ x in finPairsLT n, if ↑g⁻¹ x.fst ≤ ↑g⁻¹ x.snd then -1 else 1", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ signAux (f * g) = signAux f * signAux g⁻¹", "tactic": "unfold signAux" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ (∏ x in finPairsLT n, if ↑(f * g) x.fst ≤ ↑(f * g) x.snd then -1 else 1) =\n ∏ x in finPairsLT n, (if ↑f x.fst ≤ ↑f x.snd then -1 else 1) * if ↑g⁻¹ x.fst ≤ ↑g⁻¹ x.snd then -1 else 1", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ (∏ x in finPairsLT n, if ↑(f * g) x.fst ≤ ↑(f * g) x.snd then -1 else 1) =\n (∏ x in finPairsLT n, if ↑f x.fst ≤ ↑f x.snd then -1 else 1) *\n ∏ x in finPairsLT n, if ↑g⁻¹ x.fst ≤ ↑g⁻¹ x.snd then -1 else 1", "tactic": "rw [← prod_mul_distrib]" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ ∀ (a : (_ : Fin n) × Fin n) (ha : a ∈ finPairsLT n),\n (if ↑(f * g) a.fst ≤ ↑(f * g) a.snd then -1 else 1) =\n (if ↑f ((fun a x => signBijAux g a) a ha).fst ≤ ↑f ((fun a x => signBijAux g a) a ha).snd then -1 else 1) *\n if ↑g⁻¹ ((fun a x => signBijAux g a) a ha).fst ≤ ↑g⁻¹ ((fun a x => signBijAux g a) a ha).snd then -1 else 1", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ (∏ x in finPairsLT n, if ↑(f * g) x.fst ≤ ↑(f * g) x.snd then -1 else 1) =\n ∏ x in finPairsLT n, (if ↑f x.fst ≤ ↑f x.snd then -1 else 1) * if ↑g⁻¹ x.fst ≤ ↑g⁻¹ x.snd then -1 else 1", "tactic": "refine'\n prod_bij (fun a _ => signBijAux g a) signBijAux_mem _ signBijAux_inj\n (by simpa using signBijAux_surj)" }, { "state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑(f * g) { fst := a, snd := b }.fst ≤ ↑(f * g) { fst := a, snd := b }.snd then -1 else 1) =\n (if\n ↑f ((fun a x => signBijAux g a) { fst := a, snd := b } hab).fst ≤\n ↑f ((fun a x => signBijAux g a) { fst := a, snd := b } hab).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ ((fun a x => signBijAux g a) { fst := a, snd := b } hab).fst ≤\n ↑g⁻¹ ((fun a x => signBijAux g a) { fst := a, snd := b } hab).snd then\n -1\n else 1", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ ∀ (a : (_ : Fin n) × Fin n) (ha : a ∈ finPairsLT n),\n (if ↑(f * g) a.fst ≤ ↑(f * g) a.snd then -1 else 1) =\n (if ↑f ((fun a x => signBijAux g a) a ha).fst ≤ ↑f ((fun a x => signBijAux g a) a ha).snd then -1 else 1) *\n if ↑g⁻¹ ((fun a x => signBijAux g a) a ha).fst ≤ ↑g⁻¹ ((fun a x => signBijAux g a) a ha).snd then -1 else 1", "tactic": "rintro ⟨a, b⟩ hab" }, { "state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑(f * g) a ≤ ↑(f * g) b then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑(f * g) { fst := a, snd := b }.fst ≤ ↑(f * g) { fst := a, snd := b }.snd then -1 else 1) =\n (if\n ↑f ((fun a x => signBijAux g a) { fst := a, snd := b } hab).fst ≤\n ↑f ((fun a x => signBijAux g a) { fst := a, snd := b } hab).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ ((fun a x => signBijAux g a) { fst := a, snd := b } hab).fst ≤\n ↑g⁻¹ ((fun a x => signBijAux g a) { fst := a, snd := b } hab).snd then\n -1\n else 1", "tactic": "dsimp only [signBijAux]" }, { "state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑(f * g) a ≤ ↑(f * g) b then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "tactic": "rw [mul_apply, mul_apply]" }, { "state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b } ∈ finPairsLT n\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "tactic": "rw [mem_finPairsLT] at hab" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1\n\ncase neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "tactic": "by_cases h : g b < g a" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\n⊢ ∀ (b : (_ : Fin n) × Fin n), b ∈ finPairsLT n → ∃ a ha, b = (fun a x => signBijAux g a) a ha", "tactic": "simpa using signBijAux_surj" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g a, snd := ↑g b }.fst ≤ ↑f { fst := ↑g a, snd := ↑g b }.snd then -1 else 1) *\n if ↑g⁻¹ { fst := ↑g a, snd := ↑g b }.fst ≤ ↑g⁻¹ { fst := ↑g a, snd := ↑g b }.snd then -1 else 1", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "tactic": "rw [dif_pos h]" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n if a ≤ b then -if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1 else if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g a, snd := ↑g b }.fst ≤ ↑f { fst := ↑g a, snd := ↑g b }.snd then -1 else 1) *\n if ↑g⁻¹ { fst := ↑g a, snd := ↑g b }.fst ≤ ↑g⁻¹ { fst := ↑g a, snd := ↑g b }.snd then -1 else 1", "tactic": "simp only [not_le_of_gt hab, mul_one, mul_ite, mul_neg, Perm.inv_apply_self, if_false]" }, { "state_after": "case pos.inl.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ↑f (↑g a) ≤ ↑f (↑g b)\nh₂ : a ≤ b\n⊢ -1 = - -1\n\ncase pos.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ¬↑f (↑g a) ≤ ↑f (↑g b)\nh₃ : a ≤ b\n⊢ 1 = -1", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n if a ≤ b then -if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1 else if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1", "tactic": "split_ifs with h₁ h₂ h₃ <;> dsimp at *" }, { "state_after": "case pos.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ¬↑f (↑g a) ≤ ↑f (↑g b)\nh₃ : a ≤ b\n⊢ 1 = -1", "state_before": "case pos.inl.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ↑f (↑g a) ≤ ↑f (↑g b)\nh₂ : a ≤ b\n⊢ -1 = - -1\n\ncase pos.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ¬↑f (↑g a) ≤ ↑f (↑g b)\nh₃ : a ≤ b\n⊢ 1 = -1", "tactic": ". exact absurd hab (not_lt_of_ge h₂)" }, { "state_after": "no goals", "state_before": "case pos.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ¬↑f (↑g a) ≤ ↑f (↑g b)\nh₃ : a ≤ b\n⊢ 1 = -1", "tactic": ". exact absurd hab (not_lt_of_ge h₃)" }, { "state_after": "no goals", "state_before": "case pos.inl.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ↑f (↑g a) ≤ ↑f (↑g b)\nh₂ : a ≤ b\n⊢ -1 = - -1", "tactic": "exact absurd hab (not_lt_of_ge h₂)" }, { "state_after": "no goals", "state_before": "case pos.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : b < a\nh : ↑g b < ↑g a\nh₁ : ¬↑f (↑g a) ≤ ↑f (↑g b)\nh₃ : a ≤ b\n⊢ 1 = -1", "tactic": "exact absurd hab (not_lt_of_ge h₃)" }, { "state_after": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "state_before": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑f (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1) *\n if\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).fst ≤\n ↑g⁻¹ (if x : ↑g b < ↑g a then { fst := ↑g a, snd := ↑g b } else { fst := ↑g b, snd := ↑g a }).snd then\n -1\n else 1", "tactic": "rw [dif_neg h, inv_apply_self, inv_apply_self, if_pos hab.le]" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1\n\ncase neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ¬↑f (↑g b) ≤ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "state_before": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "tactic": "by_cases h₁ : f (g b) ≤ f (g a)" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\nthis : ↑f (↑g b) ≠ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "tactic": "have : f (g b) ≠ f (g a) := by\n rw [Ne.def, f.injective.eq_iff, g.injective.eq_iff]\n exact ne_of_lt hab" }, { "state_after": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\nthis : ↑f (↑g b) ≠ ↑f (↑g a)\n⊢ 1 = -1 * -1", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\nthis : ↑f (↑g b) ≠ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "tactic": "rw [if_pos h₁, if_neg (h₁.lt_of_ne this).not_le]" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\nthis : ↑f (↑g b) ≠ ↑f (↑g a)\n⊢ 1 = -1 * -1", "tactic": "rfl" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\n⊢ ¬b = a", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\n⊢ ↑f (↑g b) ≠ ↑f (↑g a)", "tactic": "rw [Ne.def, f.injective.eq_iff, g.injective.eq_iff]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ↑f (↑g b) ≤ ↑f (↑g a)\n⊢ ¬b = a", "tactic": "exact ne_of_lt hab" }, { "state_after": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ¬↑f (↑g b) ≤ ↑f (↑g a)\n⊢ -1 = 1 * -1", "state_before": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ¬↑f (↑g b) ≤ ↑f (↑g a)\n⊢ (if ↑f (↑g a) ≤ ↑f (↑g b) then -1 else 1) =\n (if ↑f { fst := ↑g b, snd := ↑g a }.fst ≤ ↑f { fst := ↑g b, snd := ↑g a }.snd then -1 else 1) * -1", "tactic": "rw [if_neg h₁, if_pos (lt_of_not_ge h₁).le]" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : { fst := a, snd := b }.snd < { fst := a, snd := b }.fst\nh : ¬↑g b < ↑g a\nh₁ : ¬↑f (↑g b) ≤ ↑f (↑g a)\n⊢ -1 = 1 * -1", "tactic": "rfl" } ]

[ 423, 10 ]

[ 398, 1 ]

[]

[ 266, 26 ]

[ 265, 1 ]

[ { "state_after": "no goals", "state_before": "M : Type u_3\nι : Type u_1\nα : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf : ι → α → M\ns : Finset ι\nhf : ∀ (i : ι), i ∈ s → AEMeasurable (f i)\n⊢ AEMeasurable fun a => ∏ i in s, f i a", "tactic": "simpa only [← Finset.prod_apply] using s.aemeasurable_prod' hf" } ]

[ 957, 65 ]

[ 955, 1 ]

[ { "state_after": "case intro.mk.intro\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\nw✝ : M ⧸ p\nx : M\nhxq : Quot.mk Setoid.r x ∈ ↑q\n⊢ ↑(liftQ p f h) (Quot.mk Setoid.r x) ∈ map f (comap (mkQ p) q)", "state_before": "R : Type u_1\nM : Type u_3\nr : R\nx y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\n⊢ map (liftQ p f h) q ≤ map f (comap (mkQ p) q)", "tactic": "rintro _ ⟨⟨x⟩, hxq, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro.mk.intro\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\nw✝ : M ⧸ p\nx : M\nhxq : Quot.mk Setoid.r x ∈ ↑q\n⊢ ↑(liftQ p f h) (Quot.mk Setoid.r x) ∈ map f (comap (mkQ p) q)", "tactic": "exact ⟨x, hxq, rfl⟩" }, { "state_after": "case intro.intro\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\nx : M\nhxq : x ∈ ↑(comap (mkQ p) q)\n⊢ ↑f x ∈ map (liftQ p f h) q", "state_before": "R : Type u_1\nM : Type u_3\nr : R\nx y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\n⊢ map f (comap (mkQ p) q) ≤ map (liftQ p f h) q", "tactic": "rintro _ ⟨x, hxq, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq✝ : Submodule R₂ M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ ker f\nq : Submodule R (M ⧸ p)\nx : M\nhxq : x ∈ ↑(comap (mkQ p) q)\n⊢ ↑f x ∈ map (liftQ p f h) q", "tactic": "exact ⟨Quotient.mk x, hxq, rfl⟩" } ]

[ 469, 65 ]

[ 466, 1 ]

[ { "state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\nr : R\n✝ : Nontrivial R\n⊢ leadingCoeff (X ^ n + ↑C r) = 1", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\nr : R\n⊢ leadingCoeff (X ^ n + ↑C r) = 1", "tactic": "nontriviality R" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\nr : R\n✝ : Nontrivial R\n⊢ leadingCoeff (X ^ n + ↑C r) = 1", "tactic": "rw [leadingCoeff, natDegree_X_pow_add_C, coeff_add, coeff_X_pow_self, coeff_C,\n if_neg (pos_iff_ne_zero.mp hn), add_zero]" } ]

[ 1428, 46 ]

[ 1424, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\ninst✝ : IsProbabilityMeasure μ\nhf : PreErgodic f\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ ↑↑μ s = 0 ∨ ↑↑μ s = 1", "tactic": "simpa [hs] using hf.measure_self_or_compl_eq_zero hs hs'" } ]

[ 75, 59 ]

[ 73, 1 ]

[ { "state_after": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.327240\ninst✝²³ : Semiring R₁\ninst✝²² : Semiring R₂\ninst✝²¹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝²⁰ : TopologicalSpace M₁\ninst✝¹⁹ : AddCommMonoid M₁\nM'₁ : Type ?u.327327\ninst✝¹⁸ : TopologicalSpace M'₁\ninst✝¹⁷ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝¹⁶ : TopologicalSpace M₂\ninst✝¹⁵ : AddCommMonoid M₂\nM₃ : Type ?u.327345\ninst✝¹⁴ : TopologicalSpace M₃\ninst✝¹³ : AddCommMonoid M₃\nM₄ : Type ?u.327354\ninst✝¹² : TopologicalSpace M₄\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R₁ M₁\ninst✝⁹ : Module R₁ M'₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : RingHomSurjective σ₁₂\ninst✝⁵ : TopologicalSpace R₁\ninst✝⁴ : TopologicalSpace R₂\ninst✝³ : ContinuousSMul R₁ M₁\ninst✝² : ContinuousAdd M₁\ninst✝¹ : ContinuousSMul R₂ M₂\ninst✝ : ContinuousAdd M₂\nf : M₁ →SL[σ₁₂] M₂\nhf' : DenseRange ↑f\ns : Submodule R₁ M₁\nhs✝ : Submodule.topologicalClosure s = ⊤\nhs : ↑(Submodule.topologicalClosure s) = ↑⊤\n⊢ ↑(Submodule.topologicalClosure (Submodule.map (↑f) s)) = ↑⊤", "state_before": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.327240\ninst✝²³ : Semiring R₁\ninst✝²² : Semiring R₂\ninst✝²¹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝²⁰ : TopologicalSpace M₁\ninst✝¹⁹ : AddCommMonoid M₁\nM'₁ : Type ?u.327327\ninst✝¹⁸ : TopologicalSpace M'₁\ninst✝¹⁷ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝¹⁶ : TopologicalSpace M₂\ninst✝¹⁵ : AddCommMonoid M₂\nM₃ : Type ?u.327345\ninst✝¹⁴ : TopologicalSpace M₃\ninst✝¹³ : AddCommMonoid M₃\nM₄ : Type ?u.327354\ninst✝¹² : TopologicalSpace M₄\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R₁ M₁\ninst✝⁹ : Module R₁ M'₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : RingHomSurjective σ₁₂\ninst✝⁵ : TopologicalSpace R₁\ninst✝⁴ : TopologicalSpace R₂\ninst✝³ : ContinuousSMul R₁ M₁\ninst✝² : ContinuousAdd M₁\ninst✝¹ : ContinuousSMul R₂ M₂\ninst✝ : ContinuousAdd M₂\nf : M₁ →SL[σ₁₂] M₂\nhf' : DenseRange ↑f\ns : Submodule R₁ M₁\nhs : Submodule.topologicalClosure s = ⊤\n⊢ Submodule.topologicalClosure (Submodule.map (↑f) s) = ⊤", "tactic": "rw [SetLike.ext'_iff] at hs⊢" }, { "state_after": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.327240\ninst✝²³ : Semiring R₁\ninst✝²² : Semiring R₂\ninst✝²¹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝²⁰ : TopologicalSpace M₁\ninst✝¹⁹ : AddCommMonoid M₁\nM'₁ : Type ?u.327327\ninst✝¹⁸ : TopologicalSpace M'₁\ninst✝¹⁷ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝¹⁶ : TopologicalSpace M₂\ninst✝¹⁵ : AddCommMonoid M₂\nM₃ : Type ?u.327345\ninst✝¹⁴ : TopologicalSpace M₃\ninst✝¹³ : AddCommMonoid M₃\nM₄ : Type ?u.327354\ninst✝¹² : TopologicalSpace M₄\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R₁ M₁\ninst✝⁹ : Module R₁ M'₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : RingHomSurjective σ₁₂\ninst✝⁵ : TopologicalSpace R₁\ninst✝⁴ : TopologicalSpace R₂\ninst✝³ : ContinuousSMul R₁ M₁\ninst✝² : ContinuousAdd M₁\ninst✝¹ : ContinuousSMul R₂ M₂\ninst✝ : ContinuousAdd M₂\nf : M₁ →SL[σ₁₂] M₂\nhf' : DenseRange ↑f\ns : Submodule R₁ M₁\nhs✝ : Submodule.topologicalClosure s = ⊤\nhs : Dense ↑s\n⊢ Dense ↑(Submodule.map (↑f) s)", "state_before": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.327240\ninst✝²³ : Semiring R₁\ninst✝²² : Semiring R₂\ninst✝²¹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝²⁰ : TopologicalSpace M₁\ninst✝¹⁹ : AddCommMonoid M₁\nM'₁ : Type ?u.327327\ninst✝¹⁸ : TopologicalSpace M'₁\ninst✝¹⁷ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝¹⁶ : TopologicalSpace M₂\ninst✝¹⁵ : AddCommMonoid M₂\nM₃ : Type ?u.327345\ninst✝¹⁴ : TopologicalSpace M₃\ninst✝¹³ : AddCommMonoid M₃\nM₄ : Type ?u.327354\ninst✝¹² : TopologicalSpace M₄\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R₁ M₁\ninst✝⁹ : Module R₁ M'₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : RingHomSurjective σ₁₂\ninst✝⁵ : TopologicalSpace R₁\ninst✝⁴ : TopologicalSpace R₂\ninst✝³ : ContinuousSMul R₁ M₁\ninst✝² : ContinuousAdd M₁\ninst✝¹ : ContinuousSMul R₂ M₂\ninst✝ : ContinuousAdd M₂\nf : M₁ →SL[σ₁₂] M₂\nhf' : DenseRange ↑f\ns : Submodule R₁ M₁\nhs✝ : Submodule.topologicalClosure s = ⊤\nhs : ↑(Submodule.topologicalClosure s) = ↑⊤\n⊢ ↑(Submodule.topologicalClosure (Submodule.map (↑f) s)) = ↑⊤", "tactic": "simp only [Submodule.topologicalClosure_coe, Submodule.top_coe, ← dense_iff_closure_eq] at hs⊢" }, { "state_after": "no goals", "state_before": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.327240\ninst✝²³ : Semiring R₁\ninst✝²² : Semiring R₂\ninst✝²¹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_3\ninst✝²⁰ : TopologicalSpace M₁\ninst✝¹⁹ : AddCommMonoid M₁\nM'₁ : Type ?u.327327\ninst✝¹⁸ : TopologicalSpace M'₁\ninst✝¹⁷ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝¹⁶ : TopologicalSpace M₂\ninst✝¹⁵ : AddCommMonoid M₂\nM₃ : Type ?u.327345\ninst✝¹⁴ : TopologicalSpace M₃\ninst✝¹³ : AddCommMonoid M₃\nM₄ : Type ?u.327354\ninst✝¹² : TopologicalSpace M₄\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R₁ M₁\ninst✝⁹ : Module R₁ M'₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : RingHomSurjective σ₁₂\ninst✝⁵ : TopologicalSpace R₁\ninst✝⁴ : TopologicalSpace R₂\ninst✝³ : ContinuousSMul R₁ M₁\ninst✝² : ContinuousAdd M₁\ninst✝¹ : ContinuousSMul R₂ M₂\ninst✝ : ContinuousAdd M₂\nf : M₁ →SL[σ₁₂] M₂\nhf' : DenseRange ↑f\ns : Submodule R₁ M₁\nhs✝ : Submodule.topologicalClosure s = ⊤\nhs : Dense ↑s\n⊢ Dense ↑(Submodule.map (↑f) s)", "tactic": "exact hf'.dense_image f.continuous hs" } ]

[ 568, 40 ]

[ 561, 1 ]

[]

[ 850, 61 ]

[ 849, 1 ]

[]

[ 245, 56 ]

[ 244, 1 ]

[ { "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn : ℕ\n⊢ (↑derivative^[Nat.zero]) (X ^ n) = ↑(Nat.descFactorial n Nat.zero) * X ^ (n - Nat.zero)\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\nih : (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) * X ^ (n - k)\n⊢ (↑derivative^[Nat.succ k]) (X ^ n) = ↑(Nat.descFactorial n (Nat.succ k)) * X ^ (n - Nat.succ k)", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\n⊢ (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) * X ^ (n - k)", "tactic": "induction' k with k ih" }, { "state_after": "no goals", "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn : ℕ\n⊢ (↑derivative^[Nat.zero]) (X ^ n) = ↑(Nat.descFactorial n Nat.zero) * X ^ (n - Nat.zero)", "tactic": "erw [Function.iterate_zero_apply, tsub_zero, Nat.descFactorial_zero, Nat.cast_one, one_mul]" }, { "state_after": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\nih : (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) * X ^ (n - k)\n⊢ ↑(Nat.descFactorial n k) * (↑(n - k) * X ^ (n - (k + 1))) = ↑(n - k) * ↑(Nat.descFactorial n k) * X ^ (n - (k + 1))", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\nih : (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) * X ^ (n - k)\n⊢ (↑derivative^[Nat.succ k]) (X ^ n) = ↑(Nat.descFactorial n (Nat.succ k)) * X ^ (n - Nat.succ k)", "tactic": "rw [Function.iterate_succ_apply', ih, derivative_nat_cast_mul, derivative_X_pow, C_eq_nat_cast,\n Nat.succ_eq_add_one, Nat.descFactorial_succ, Nat.sub_sub, Nat.cast_mul]" }, { "state_after": "no goals", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\nih : (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) * X ^ (n - k)\n⊢ ↑(Nat.descFactorial n k) * (↑(n - k) * X ^ (n - (k + 1))) = ↑(n - k) * ↑(Nat.descFactorial n k) * X ^ (n - (k + 1))", "tactic": "simp [mul_comm, mul_assoc, mul_left_comm]" } ]

[ 508, 46 ]

[ 502, 1 ]

[ { "state_after": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv : V\n⊢ direction s₁ = direction (map (↑(constVAdd k P v)) s₁)", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nh : s₁ ∥ s₂\n⊢ direction s₁ = direction s₂", "tactic": "rcases h with ⟨v, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv : V\n⊢ direction s₁ = direction (map (↑(constVAdd k P v)) s₁)", "tactic": "simp" } ]

[ 1750, 7 ]

[ 1747, 1 ]