Datasets:

tasksource

/

leandojo

like 5

[ { "state_after": "no goals", "state_before": "l : Type ?u.82029\nm : Type ?u.82032\nn : Type u_1\no : Type ?u.82038\nm' : o → Type ?u.82043\nn' : o → Type ?u.82048\nR : Type ?u.82051\nS : Type ?u.82054\nα : Type v\nβ : Type w\nγ : Type ?u.82061\ninst✝² : DecidableEq n\ninst✝¹ : AddZeroClass α\ninst✝ : One α\nM : Matrix n n α\ni : n\n⊢ bit1 M i i = bit1 (M i i)", "tactic": "simp [bit1_apply]" } ]

[ 589, 20 ]

[ 588, 1 ]

[]

[ 447, 20 ]

[ 446, 1 ]

[ { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "tactic": "let k2nl :=\n lt_of_add_lt_add_right <|\n show 2 * n - k + k < n + k by\n rw [tsub_add_cancel_of_le]\n rw [two_mul]; exact add_lt_add_left kn n\n exact k2n" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "tactic": "have xle : xn a1 (2 * n - k) ≤ xn a1 n := le_of_lt <| strictMono_x a1 k2nl" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "tactic": "suffices xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k) by rw [this, Int.ofNat_sub xle]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k % xn a1 n = (xn a1 n - xn a1 (2 * n - k)) % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)", "tactic": "rw [← Nat.mod_eq_of_lt (Nat.sub_lt (x_pos a1 n) (x_pos a1 (2 * n - k)))]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k % xn a1 n = (xn a1 n - xn a1 (2 * n - k)) % xn a1 n", "tactic": "apply ModEq.add_right_cancel' (xn a1 (2 * n - k))" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]", "tactic": "rw [tsub_add_cancel_of_le xle]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\nt : xn a1 (2 * n - (2 * n - k)) + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "tactic": "have t := xn_modEq_x2n_sub_lem a1 k2nl.le" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\nt : xn a1 k + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\nt : xn a1 (2 * n - (2 * n - k)) + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "tactic": "rw [tsub_tsub_cancel_of_le k2n] at t" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\nt : xn a1 k + xn a1 (2 * n - k) ≡ 0 [MOD xn a1 n]\n⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n [MOD xn a1 n]", "tactic": "exact t.trans dvd_rfl.zero_modEq_nat" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ 2 * n < n + k\n\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ 2 * n - k + k < n + k", "tactic": "rw [tsub_add_cancel_of_le]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ n + n < n + k\n\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ 2 * n < n + k\n\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "tactic": "rw [two_mul]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ n + n < n + k\n\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "tactic": "exact add_lt_add_left kn n" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\n⊢ k ≤ 2 * n", "tactic": "exact k2n" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nk : ℕ\nkn : k > n\nk2n : k ≤ 2 * n\nk2nl : 2 * n - k < n :=\n lt_of_add_lt_add_right\n (let_fun this :=\n Eq.mpr (id (tsub_add_cancel_of_le k2n ▸ Eq.refl (2 * n - k + k < n + k)))\n (Eq.mpr (id (two_mul n ▸ Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n this)\nxle : xn a1 (2 * n - k) ≤ xn a1 n\nthis : xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)\n⊢ ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "tactic": "rw [this, Int.ofNat_sub xle]" }, { "state_after": "case refl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ xn a1 i % xn a1 n < xn a1 (n + 1) % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j = n\n⊢ xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n", "tactic": "cases jn" }, { "state_after": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 (n + 1) % xn a1 n)", "state_before": "case refl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ xn a1 i % xn a1 n < xn a1 (n + 1) % xn a1 n", "tactic": "apply Int.lt_of_ofNat_lt_ofNat" }, { "state_after": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 n) - ↑(xn a1 (n - 1))", "state_before": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 (n + 1) % xn a1 n)", "tactic": "rw [lem2 (n + 1) (Nat.lt_succ_self _) j2n,\n show 2 * n - (n + 1) = n - 1 by\n rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]]" }, { "state_after": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n", "state_before": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 n) - ↑(xn a1 (n - 1))", "tactic": "refine' lt_sub_left_of_add_lt (Int.ofNat_lt_ofNat_of_lt _)" }, { "state_after": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n\n\ncase refl.a.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nein : i = n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n", "state_before": "case refl.a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n", "tactic": "cases' lt_or_eq_of_le <| Nat.le_of_succ_le_succ ij with lin ein" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\n⊢ 2 * n - (n + 1) = n - 1", "tactic": "rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]" }, { "state_after": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "state_before": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n", "tactic": "rw [Nat.mod_eq_of_lt (strictMono_x _ lin)]" }, { "state_after": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "state_before": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "have ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n := by\n rw [← two_mul, mul_comm,\n show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)],\n xn_succ]\n exact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)" }, { "state_after": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "state_before": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "have npm : (n - 1).succ = n := Nat.succ_pred_eq_of_pos npos" }, { "state_after": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "state_before": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "have il : i ≤ n - 1 := by\n apply Nat.le_of_succ_le_succ\n rw [npm]\n exact lin" }, { "state_after": "case refl.a.inl.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nill : i < n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n\n\ncase refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "state_before": "case refl.a.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "cases' lt_or_eq_of_le il with ill ile" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) * 2 ≤ xn a1 (n - 1) * a + Pell.d a1 * yn a1 (n - 1)", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n", "tactic": "rw [← two_mul, mul_comm,\n show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)],\n xn_succ]" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 (n - 1) * 2 ≤ xn a1 (n - 1) * a + Pell.d a1 * yn a1 (n - 1)", "tactic": "exact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\n⊢ xn a1 n = xn a1 (n - 1 + 1)", "tactic": "rw [tsub_add_cancel_of_le (succ_le_of_lt npos)]" }, { "state_after": "case a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ succ i ≤ succ (n - 1)", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ i ≤ n - 1", "tactic": "apply Nat.le_of_succ_le_succ" }, { "state_after": "case a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ succ i ≤ n", "state_before": "case a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ succ i ≤ succ (n - 1)", "tactic": "rw [npm]" }, { "state_after": "no goals", "state_before": "case a\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\n⊢ succ i ≤ n", "tactic": "exact lin" }, { "state_after": "no goals", "state_before": "case refl.a.inl.inl\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nill : i < n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "exact lt_of_lt_of_le (Nat.add_lt_add_left (strictMono_x a1 ill) _) ll" }, { "state_after": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n", "state_before": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 i < xn a1 n", "tactic": "rw [ile]" }, { "state_after": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 (n - 1) ≠ xn a1 n", "state_before": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n", "tactic": "apply lt_of_le_of_ne ll" }, { "state_after": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ 2 * xn a1 (n - 1) ≠ xn a1 n", "state_before": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ xn a1 (n - 1) + xn a1 (n - 1) ≠ xn a1 n", "tactic": "rw [← two_mul]" }, { "state_after": "no goals", "state_before": "case refl.a.inl.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\n⊢ 2 * xn a1 (n - 1) ≠ xn a1 n", "tactic": "exact fun e =>\n ntriv <| by\n let ⟨a2, s1⟩ :=\n @eq_of_xn_modEq_lem2 _ a1 (n - 1)\n (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])\n have n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos\n rw [ile, a2, n1]; exact ⟨rfl, rfl, rfl, rfl⟩" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\n⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\n⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2", "tactic": "let ⟨a2, s1⟩ :=\n @eq_of_xn_modEq_lem2 _ a1 (n - 1)\n (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\n⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2", "tactic": "have n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n⊢ 2 = 2 ∧ 1 = 1 ∧ 1 - 1 = 0 ∧ 1 + 1 = 2", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n⊢ a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2", "tactic": "rw [ile, a2, n1]" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n⊢ 2 = 2 ∧ 1 = 1 ∧ 1 - 1 = 0 ∧ 1 + 1 = 2", "tactic": "exact ⟨rfl, rfl, rfl, rfl⟩" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n\nnpm : succ (n - 1) = n\nil : i ≤ n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\n⊢ 2 * xn a1 (n - 1) = xn a1 (n - 1 + 1)", "tactic": "rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)]" }, { "state_after": "case refl.a.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nein : i = n\n⊢ xn a1 (n - 1) < xn a1 n", "state_before": "case refl.a.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nein : i = n\n⊢ xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n", "tactic": "rw [ein, Nat.mod_self, add_zero]" }, { "state_after": "no goals", "state_before": "case refl.a.inr\na : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nein : i = n\n⊢ xn a1 (n - 1) < xn a1 n", "tactic": "exact strictMono_x _ (Nat.pred_lt npos.ne')" }, { "state_after": "a : ℕ\na1✝ : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : 2 + 1 ≤ 2 * 1\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1✝ k % xn a1✝ n) = ↑(xn a1✝ n) - ↑(xn a1✝ (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1✝ j % xn a1✝ n < xn a1✝ (j + 1) % xn a1✝ n\nh : i < j\nx✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n⊢ False", "state_before": "a : ℕ\na1✝ : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1✝ k % xn a1✝ n) = ↑(xn a1✝ n) - ↑(xn a1✝ (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1✝ j % xn a1✝ n < xn a1✝ (j + 1) % xn a1✝ n\nh : i < j\nx✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n⊢ False", "tactic": "rw [n1, j2] at j2n" }, { "state_after": "no goals", "state_before": "a : ℕ\na1✝ : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : 2 + 1 ≤ 2 * 1\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1✝ k % xn a1✝ n) = ↑(xn a1✝ n) - ↑(xn a1✝ (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1✝ j % xn a1✝ n < xn a1✝ (j + 1) % xn a1✝ n\nh : i < j\nx✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n⊢ False", "tactic": "exact absurd j2n (by decide)" }, { "state_after": "no goals", "state_before": "a : ℕ\na1✝ : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : 2 + 1 ≤ 2 * 1\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1✝ k % xn a1✝ n) = ↑(xn a1✝ n) - ↑(xn a1✝ (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1✝ j % xn a1✝ n < xn a1✝ (j + 1) % xn a1✝ n\nh : i < j\nx✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n⊢ ¬2 + 1 ≤ 2 * 1", "tactic": "decide" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n⊢ xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n⊢ xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n", "tactic": "rw [h]" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n⊢ xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n", "tactic": "exact s" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ ↑(xn a1 n) - ↑(xn a1 (2 * n - j)) < ↑(xn a1 n) - ↑(xn a1 (2 * n - (j + 1)))", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ ↑(xn a1 j % xn a1 n) < ↑(xn a1 (j + 1) % xn a1 n)", "tactic": "rw [lem2 j jn (le_of_lt j2n), lem2 (j + 1) (Nat.le_succ_of_le jn) j2n]" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ 2 * n - (j + 1) < 2 * n - j", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ ↑(xn a1 n) - ↑(xn a1 (2 * n - j)) < ↑(xn a1 n) - ↑(xn a1 (2 * n - (j + 1)))", "tactic": "refine' sub_lt_sub_left (Int.ofNat_lt_ofNat_of_lt <| strictMono_x _ _) _" }, { "state_after": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ pred (2 * n - j) < 2 * n - j", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ 2 * n - (j + 1) < 2 * n - j", "tactic": "rw [Nat.sub_succ]" }, { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * n\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ (k : ℕ), k > n → k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn : j > n\nlem1 : j ≠ n → xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n → xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n⊢ pred (2 * n - j) < 2 * n - j", "tactic": "exact Nat.pred_lt (_root_.ne_of_gt <| tsub_pos_of_lt j2n)" } ]

[ 744, 70 ]

[ 670, 1 ]

[]

[ 321, 6 ]

[ 320, 1 ]

[ { "state_after": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\n⊢ x ∈ radical I ∨ y ∈ radical I", "state_before": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : (x * y) ^ m ∈ I\n⊢ x ∈ radical I ∨ y ∈ radical I", "tactic": "rw [mul_pow] at hxy" }, { "state_after": "case inl\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\nh : x ^ m ∈ I\n⊢ x ∈ radical I ∨ y ∈ radical I\n\ncase inr\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\nh : y ^ m ∈ radical I\n⊢ x ∈ radical I ∨ y ∈ radical I", "state_before": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\n⊢ x ∈ radical I ∨ y ∈ radical I", "tactic": "cases' hi.2 hxy with h h" }, { "state_after": "no goals", "state_before": "case inl\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\nh : x ^ m ∈ I\n⊢ x ∈ radical I ∨ y ∈ radical I", "tactic": "exact Or.inl ⟨m, h⟩" }, { "state_after": "no goals", "state_before": "case inr\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx✝ : x * y ∈ radical I\nm : ℕ\nhxy : x ^ m * y ^ m ∈ I\nh : y ^ m ∈ radical I\n⊢ x ∈ radical I ∨ y ∈ radical I", "tactic": "exact Or.inr (mem_radical_of_pow_mem h)" } ]

[ 1865, 47 ]

[ 1860, 1 ]

[ { "state_after": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nhf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "choose U hUx hU using hf" }, { "state_after": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X), N x ∈ upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "choose N hN using fun x => (hU x).bddAbove" }, { "state_after": "case hN\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X), N x ∈ upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X), N x ∈ upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y" }, { "state_after": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "case hN\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X), N x ∈ upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\n⊢ ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "exact fun x n hn y hy => by_contra fun hne => hn.lt.not_le <| hN x ⟨y, hne, hy⟩" }, { "state_after": "case hN\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "replace hN : ∀ (x), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y" }, { "state_after": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "state_before": "case hN\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n > N x → ∀ (y : X), y ∈ U x → f (n + 1) y = f n y\n⊢ ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "exact fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn" }, { "state_after": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\n⊢ ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = (fun x => f (N x + 1) x) p.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\n⊢ ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = F p.snd", "tactic": "refine ⟨fun x => f (N x + 1) x, fun x => ?_⟩" }, { "state_after": "case h\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\n⊢ ∀ (a : ℕ × X), a ∈ {x_1 | N x < x_1} ×ˢ U x → f a.fst a.snd = f (N a.snd + 1) a.snd", "state_before": "ι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\n⊢ ∀ᶠ (p : ℕ × X) in atTop ×ˢ 𝓝 x, f p.fst p.snd = (fun x => f (N x + 1) x) p.snd", "tactic": "filter_upwards [Filter.prod_mem_prod (eventually_gt_atTop (N x)) (hUx x)]" }, { "state_after": "case h.mk.intro\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\nn : ℕ\ny : X\nhn : N x < n\nhy : y ∈ U x\n⊢ f (n, y).fst (n, y).snd = f (N (n, y).snd + 1) (n, y).snd", "state_before": "case h\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\n⊢ ∀ (a : ℕ × X), a ∈ {x_1 | N x < x_1} ×ˢ U x → f a.fst a.snd = f (N a.snd + 1) a.snd", "tactic": "rintro ⟨n, y⟩ ⟨hn : N x < n, hy : y ∈ U x⟩" }, { "state_after": "no goals", "state_before": "case h.mk.intro\nι : Type ?u.9232\nι' : Type ?u.9235\nα : Type ?u.9238\nX : Type u_2\nY : Type ?u.9244\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf✝ g : ι → Set X\nπ : X → Sort u_1\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x ≠ f n x}) i ∩ U x)}\nN : X → ℕ\nhN : ∀ (x : X) (n : ℕ), n ≥ N x + 1 → ∀ (y : X), y ∈ U x → f n y = f (N x + 1) y\nx : X\nn : ℕ\ny : X\nhn : N x < n\nhy : y ∈ U x\n⊢ f (n, y).fst (n, y).snd = f (N (n, y).snd + 1) (n, y).snd", "tactic": "calc\n f n y = f (N x + 1) y := hN _ _ hn _ hy\n _ = f (max (N x + 1) (N y + 1)) y := (hN _ _ (le_max_left _ _) _ hy).symm\n _ = f (N y + 1) y := hN _ _ (le_max_right _ _) _ (mem_of_mem_nhds <| hUx y)" } ]

[ 173, 80 ]

[ 158, 1 ]

[ { "state_after": "case zero\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn : ℕ\nhn✝ : card s✝ ≤ n\ns : Finset α\nhn : card s ≤ zero\n⊢ ∃ t, s ⊆ t ∧ card t = zero\n\ncase succ\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "state_before": "α : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns : Finset α\nn : ℕ\nhn : card s ≤ n\n⊢ ∃ t, s ⊆ t ∧ card t = n", "tactic": "induction' n with n IH generalizing s" }, { "state_after": "no goals", "state_before": "case zero\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn : ℕ\nhn✝ : card s✝ ≤ n\ns : Finset α\nhn : card s ≤ zero\n⊢ ∃ t, s ⊆ t ∧ card t = zero", "tactic": "exact ⟨s, subset_refl _, Nat.eq_zero_of_le_zero hn⟩" }, { "state_after": "case succ.inl\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s = succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n\n\ncase succ.inr\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "state_before": "case succ\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "tactic": "cases' hn.eq_or_lt with hn' hn'" }, { "state_after": "case succ.inr.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "state_before": "case succ.inr\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "tactic": "obtain ⟨t, hs, ht⟩ := IH _ (Nat.le_of_lt_succ hn')" }, { "state_after": "case succ.inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\nx : α\nhx : ¬x ∈ t\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "state_before": "case succ.inr.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "tactic": "obtain ⟨x, hx⟩ := exists_not_mem_finset t" }, { "state_after": "case succ.inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\nx : α\nhx : ¬x ∈ t\n⊢ card (cons x t hx) = succ n", "state_before": "case succ.inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\nx : α\nhx : ¬x ∈ t\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "tactic": "refine' ⟨Finset.cons x t hx, hs.trans (Finset.subset_cons _), _⟩" }, { "state_after": "no goals", "state_before": "case succ.inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s < succ n\nt : Finset α\nhs : s ⊆ t\nht : card t = n\nx : α\nhx : ¬x ∈ t\n⊢ card (cons x t hx) = succ n", "tactic": "simp [hx, ht]" }, { "state_after": "no goals", "state_before": "case succ.inl\nα : Type u_1\nβ : Type ?u.80843\nγ : Type ?u.80846\ninst✝ : Infinite α\ns✝ : Finset α\nn✝ : ℕ\nhn✝ : card s✝ ≤ n✝\nn : ℕ\nIH : ∀ (s : Finset α), card s ≤ n → ∃ t, s ⊆ t ∧ card t = n\ns : Finset α\nhn : card s ≤ succ n\nhn' : card s = succ n\n⊢ ∃ t, s ⊆ t ∧ card t = succ n", "tactic": "exact ⟨s, subset_refl _, hn'⟩" } ]

[ 1125, 18 ]

[ 1116, 1 ]

[ { "state_after": "α : Type u\ninst✝ : CanonicallyOrderedMonoid α\na b c d : α\n⊢ a ≤ a * b", "state_before": "α : Type u\ninst✝ : CanonicallyOrderedMonoid α\na b c d : α\n⊢ a ≤ b * a", "tactic": "rw [mul_comm]" }, { "state_after": "no goals", "state_before": "α : Type u\ninst✝ : CanonicallyOrderedMonoid α\na b c d : α\n⊢ a ≤ a * b", "tactic": "exact le_self_mul" } ]

[ 164, 20 ]

[ 162, 1 ]

[]

[ 235, 56 ]

[ 232, 1 ]

[]

[ 939, 71 ]

[ 939, 16 ]

[]

[ 188, 6 ]

[ 187, 1 ]

[]

[ 158, 44 ]

[ 156, 1 ]

[]

[ 338, 18 ]

[ 337, 1 ]

[]

[ 1196, 46 ]

[ 1192, 1 ]

[]

[ 515, 95 ]

[ 512, 1 ]

[]

[ 256, 10 ]

[ 255, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.336860\nγ : Type ?u.336863\nδ : Type ?u.336866\nι : Type u_2\nR : Type ?u.336872\nR' : Type ?u.336875\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμ : ι → Measure α\ni : ι\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(μ i) s ≤ ↑↑(sum μ) s", "tactic": "simpa only [sum_apply μ hs] using ENNReal.le_tsum i" } ]

[ 2039, 54 ]

[ 2038, 1 ]

[]

[ 94, 61 ]

[ 92, 1 ]

[]

[ 1124, 55 ]

[ 1123, 1 ]

[ { "state_after": "α : Type u_1\nι : Type ?u.44160\nγ : Type ?u.44163\nA : Type ?u.44166\nB : Type ?u.44169\nC : Type ?u.44172\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.47302\nM : Type u_2\nM' : Type ?u.47308\nN : Type u_3\nP : Type ?u.47314\nG : Type ?u.47317\nH : Type ?u.47320\nR : Type ?u.47323\nS : Type ?u.47326\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 in f.support, if a = a_1 then b a_1 (↑f a_1) else 1) = if a ∈ f.support then b a (↑f a) else 1", "state_before": "α : Type u_1\nι : Type ?u.44160\nγ : Type ?u.44163\nA : Type ?u.44166\nB : Type ?u.44169\nC : Type ?u.44172\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.47302\nM : Type u_2\nM' : Type ?u.47308\nN : Type u_3\nP : Type ?u.47314\nG : Type ?u.47317\nH : Type ?u.47320\nR : Type ?u.47323\nS : Type ?u.47326\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (prod f fun x v => if a = x then b x v else 1) = if a ∈ f.support then b a (↑f a) else 1", "tactic": "dsimp [Finsupp.prod]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nι : Type ?u.44160\nγ : Type ?u.44163\nA : Type ?u.44166\nB : Type ?u.44169\nC : Type ?u.44172\ninst✝⁶ : AddCommMonoid A\ninst✝⁵ : AddCommMonoid B\ninst✝⁴ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.47302\nM : Type u_2\nM' : Type ?u.47308\nN : Type u_3\nP : Type ?u.47314\nG : Type ?u.47317\nH : Type ?u.47320\nR : Type ?u.47323\nS : Type ?u.47326\ninst✝³ : Zero M\ninst✝² : Zero M'\ninst✝¹ : CommMonoid N\ninst✝ : DecidableEq α\nf : α →₀ M\na : α\nb : α → M → N\n⊢ (∏ a_1 in f.support, if a = a_1 then b a_1 (↑f a_1) else 1) = if a ∈ f.support then b a (↑f a) else 1", "tactic": "rw [f.support.prod_ite_eq]" } ]

[ 111, 29 ]

[ 108, 1 ]

[]

[ 426, 36 ]

[ 423, 1 ]

[]

[ 1818, 6 ]

[ 1817, 1 ]

[]

[ 1046, 75 ]

[ 1045, 1 ]

[ { "state_after": "case mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nx✝ : FreeGroup α\nL : List (α × Bool)\n⊢ reduce (toWord (Quot.mk Red.Step L)) = toWord (Quot.mk Red.Step L)", "state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\n⊢ ∀ (x : FreeGroup α), reduce (toWord x) = toWord x", "tactic": "rintro ⟨L⟩" }, { "state_after": "no goals", "state_before": "case mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nx✝ : FreeGroup α\nL : List (α × Bool)\n⊢ reduce (toWord (Quot.mk Red.Step L)) = toWord (Quot.mk Red.Step L)", "tactic": "exact reduce.idem" } ]

[ 1315, 20 ]

[ 1313, 1 ]

[ { "state_after": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\n⊢ ∀ (a b : A[X]), minpoly A x ∣ a * b → minpoly A x ∣ a ∨ minpoly A x ∣ b", "state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\n⊢ Prime (minpoly A x)", "tactic": "refine' ⟨minpoly.ne_zero hx, not_isUnit A x, _⟩" }, { "state_after": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\n⊢ ∀ (a b : A[X]), minpoly A x ∣ a * b → minpoly A x ∣ a ∨ minpoly A x ∣ b", "tactic": "rintro p q ⟨d, h⟩" }, { "state_after": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\nthis : ↑(Polynomial.aeval x) (p * q) = 0\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "state_before": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "tactic": "have : Polynomial.aeval x (p * q) = 0 := by simp [h, aeval A x]" }, { "state_after": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\nthis : ↑(Polynomial.aeval x) p = 0 ∨ ↑(Polynomial.aeval x) q = 0\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "state_before": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\nthis : ↑(Polynomial.aeval x) (p * q) = 0\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "tactic": "replace : Polynomial.aeval x p = 0 ∨ Polynomial.aeval x q = 0 := by simpa" }, { "state_after": "no goals", "state_before": "case intro\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\nthis : ↑(Polynomial.aeval x) p = 0 ∨ ↑(Polynomial.aeval x) q = 0\n⊢ minpoly A x ∣ p ∨ minpoly A x ∣ q", "tactic": "exact Or.imp (dvd A x) (dvd A x) this" }, { "state_after": "no goals", "state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\n⊢ ↑(Polynomial.aeval x) (p * q) = 0", "tactic": "simp [h, aeval A x]" }, { "state_after": "no goals", "state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\np q d : A[X]\nh : p * q = minpoly A x * d\nthis : ↑(Polynomial.aeval x) (p * q) = 0\n⊢ ↑(Polynomial.aeval x) p = 0 ∨ ↑(Polynomial.aeval x) q = 0", "tactic": "simpa" } ]

[ 249, 40 ]

[ 244, 1 ]

[ { "state_after": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n⊢ ⊥ = ⊤", "state_before": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : Module.rank K V = 0\n⊢ ⊥ = ⊤", "tactic": "haveI : FiniteDimensional _ _ := finiteDimensional_of_rank_eq_zero h" }, { "state_after": "case h\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n⊢ finrank K { x // x ∈ ⊥ } = finrank K V", "state_before": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n⊢ ⊥ = ⊤", "tactic": "apply eq_top_of_finrank_eq" }, { "state_after": "no goals", "state_before": "case h\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n⊢ finrank K { x // x ∈ ⊥ } = finrank K V", "tactic": "rw [finrank_bot, finrank_eq_zero_of_rank_eq_zero h]" } ]

[ 669, 54 ]

[ 666, 1 ]

[]

[ 3237, 6 ]

[ 3236, 1 ]

[]

[ 923, 10 ]

[ 922, 1 ]

[ { "state_after": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Finset S\nf : S → R\n⊢ ∑ i in s, trop (f i) = Multiset.sum (Multiset.map trop (Multiset.map f s.val))", "state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Finset S\nf : S → R\n⊢ trop (inf s f) = ∑ i in s, trop (f i)", "tactic": "convert Multiset.trop_inf (s.val.map f)" }, { "state_after": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Finset S\nf : S → R\n⊢ ∑ i in s, trop (f i) = Multiset.sum (Multiset.map (fun i => trop (f i)) s.val)", "state_before": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Finset S\nf : S → R\n⊢ ∑ i in s, trop (f i) = Multiset.sum (Multiset.map trop (Multiset.map f s.val))", "tactic": "simp only [Multiset.map_map, Function.comp_apply]" }, { "state_after": "no goals", "state_before": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Finset S\nf : S → R\n⊢ ∑ i in s, trop (f i) = Multiset.sum (Multiset.map (fun i => trop (f i)) s.val)", "tactic": "rfl" } ]

[ 101, 6 ]

[ 97, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : DivisionRing R\np q : R[X]\nh : p ≠ 0\n⊢ Monic (p * ↑C (leadingCoeff p)⁻¹)", "tactic": "rw [Monic, leadingCoeff_mul, leadingCoeff_C,\n mul_inv_cancel (show leadingCoeff p ≠ 0 from mt leadingCoeff_eq_zero.1 h)]" } ]

[ 129, 79 ]

[ 127, 1 ]

[]

[ 157, 41 ]

[ 156, 1 ]

[]

[ 254, 85 ]

[ 253, 1 ]

[ { "state_after": "α : Type u_1\ni : Nat\nl : List α\nh : length l ≤ i\nthis : take i l ++ drop i l = l\n⊢ take i l = l", "state_before": "α : Type u_1\ni : Nat\nl : List α\nh : length l ≤ i\n⊢ take i l = l", "tactic": "have := take_append_drop i l" }, { "state_after": "α : Type u_1\ni : Nat\nl : List α\nh : length l ≤ i\nthis : take i l = l\n⊢ take i l = l", "state_before": "α : Type u_1\ni : Nat\nl : List α\nh : length l ≤ i\nthis : take i l ++ drop i l = l\n⊢ take i l = l", "tactic": "rw [drop_length_le h, append_nil] at this" }, { "state_after": "no goals", "state_before": "α : Type u_1\ni : Nat\nl : List α\nh : length l ≤ i\nthis : take i l = l\n⊢ take i l = l", "tactic": "exact this" } ]

[ 145, 56 ]

[ 143, 1 ]

[]

[ 160, 26 ]

[ 159, 1 ]

[ { "state_after": "no goals", "state_before": "n : ℕ\nx r₁ r₂ : ℝ\n⊢ expNear n x r₁ - expNear n x r₂ = x ^ n / ↑(Nat.factorial n) * (r₁ - r₂)", "tactic": "simp [expNear, mul_sub]" } ]

[ 1764, 26 ]

[ 1762, 1 ]

[]

[ 170, 32 ]

[ 169, 1 ]

[]

[ 220, 26 ]

[ 219, 1 ]

[]

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[ { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)) →\n n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "tactic": "apply Nat.strongInductionOn n" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)) →\n n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)) →\n n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "tactic": "clear n" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)) →\n n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "tactic": "intro n ih" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ∧\n {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ⊆ {n} ∪ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ vars (xInTermsOfW p ℚ n) ∧ vars (xInTermsOfW p ℚ n) ⊆ range (n + 1)", "tactic": "rw [xInTermsOfW_eq, mul_comm, vars_C_mul _ (nonzero_of_invertible _),\n vars_sub_of_disjoint, vars_X, range_succ, insert_eq]" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ∧\n {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ⊆ {n} ∪ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "tactic": "on_goal 1 =>\n simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]\n intro i\n rw [mem_union, mem_union]\n apply Or.imp id" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ ¬n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "tactic": "on_goal 2 => rw [vars_X, disjoint_singleton_left]" }, { "state_after": "case intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni j : ℕ\nhj : j < n\nH : i < j + 1\n⊢ i ∈ range n\n\ncase hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n\n\ncase hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ ¬n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))", "tactic": "all_goals\n intro H\n replace H := vars_sum_subset _ _ H\n rw [mem_biUnion] at H\n rcases H with ⟨j, hj, H⟩\n rw [vars_C_mul] at H\n swap\n . apply pow_ne_zero\n exact_mod_cast hp.1.ne_zero\n rw [mem_range] at hj\n replace H := (ih j hj).2 (vars_pow _ _ H)\n rw [mem_range] at H" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "state_before": "case intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni j : ℕ\nhj : j < n\nH : i < j + 1\n⊢ i ∈ range n\n\ncase hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "tactic": ". rw [mem_range]\n linarith" }, { "state_after": "no goals", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "tactic": ". linarith" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) ⊆ {n} ∪ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ∧\n {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ⊆ {n} ∪ range n", "tactic": "simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∪ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) ⊆ {n} ∪ range n", "tactic": "intro i" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∨ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∨ i ∈ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∪ range n", "tactic": "rw [mem_union, mem_union]" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∨ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∨ i ∈ range n", "tactic": "apply Or.imp id" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) ⊆ {n} ∪ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ n ∈ {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ∧\n {n} ∪ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)) ⊆ {n} ∪ range n", "tactic": "simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∪ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) ⊆ {n} ∪ range n", "tactic": "intro i" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∨ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∨ i ∈ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∪ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∪ range n", "tactic": "rw [mem_union, mem_union]" }, { "state_after": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ range n", "state_before": "p : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni : ℕ\n⊢ i ∈ {n} ∨ i ∈ vars (∑ x in range n, ↑C (↑p ^ x) * xInTermsOfW p ℚ x ^ p ^ (n - x)) → i ∈ {n} ∨ i ∈ range n", "tactic": "apply Or.imp id" }, { "state_after": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ ¬n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "tactic": "rw [vars_X, disjoint_singleton_left]" }, { "state_after": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ ¬n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ Disjoint (vars (X n)) (vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)))", "tactic": "rw [vars_X, disjoint_singleton_left]" }, { "state_after": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))\n⊢ False", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\n⊢ ¬n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))", "tactic": "intro H" }, { "state_after": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : n ∈ Finset.biUnion (range n) fun i => vars (↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))\n⊢ False", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : n ∈ vars (∑ i in range n, ↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))\n⊢ False", "tactic": "replace H := vars_sum_subset _ _ H" }, { "state_after": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : ∃ a, a ∈ range n ∧ n ∈ vars (↑C (↑p ^ a) * xInTermsOfW p ℚ a ^ p ^ (n - a))\n⊢ False", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : n ∈ Finset.biUnion (range n) fun i => vars (↑C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i))\n⊢ False", "tactic": "rw [mem_biUnion] at H" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "state_before": "case hpq\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nH : ∃ a, a ∈ range n ∧ n ∈ vars (↑C (↑p ^ a) * xInTermsOfW p ℚ a ^ p ^ (n - a))\n⊢ False", "tactic": "rcases H with ⟨j, hj, H⟩" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False\n\ncase hpq.intro.intro.ha\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ^ j ≠ 0", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "tactic": "rw [vars_C_mul] at H" }, { "state_after": "case hpq.intro.intro.ha\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ^ j ≠ 0\n\ncase hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False\n\ncase hpq.intro.intro.ha\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ^ j ≠ 0", "tactic": "swap" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "state_before": "case hpq.intro.intro.ha\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ^ j ≠ 0\n\ncase hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "tactic": ". apply pow_ne_zero\n exact_mod_cast hp.1.ne_zero" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "tactic": "rw [mem_range] at hj" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n ∈ range (j + 1)\n⊢ False", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n ∈ vars (xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ False", "tactic": "replace H := (ih j hj).2 (vars_pow _ _ H)" }, { "state_after": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n ∈ range (j + 1)\n⊢ False", "tactic": "rw [mem_range] at H" }, { "state_after": "case hpq.intro.intro.ha.h\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ≠ 0", "state_before": "case hpq.intro.intro.ha\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ^ j ≠ 0", "tactic": "apply pow_ne_zero" }, { "state_after": "no goals", "state_before": "case hpq.intro.intro.ha.h\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j ∈ range n\nH : n ∈ vars (↑C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j))\n⊢ ↑p ≠ 0", "tactic": "exact_mod_cast hp.1.ne_zero" }, { "state_after": "case intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni j : ℕ\nhj : j < n\nH : i < j + 1\n⊢ i < n", "state_before": "case intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni j : ℕ\nhj : j < n\nH : i < j + 1\n⊢ i ∈ range n", "tactic": "rw [mem_range]" }, { "state_after": "no goals", "state_before": "case intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\ni j : ℕ\nhj : j < n\nH : i < j + 1\n⊢ i < n", "tactic": "linarith" }, { "state_after": "no goals", "state_before": "case hpq.intro.intro\np : ℕ\nR : Type ?u.289237\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (m : ℕ), m < n → m ∈ vars (xInTermsOfW p ℚ m) ∧ vars (xInTermsOfW p ℚ m) ⊆ range (m + 1)\nj : ℕ\nhj : j < n\nH : n < j + 1\n⊢ False", "tactic": "linarith" } ]

[ 283, 13 ]

[ 257, 1 ]

[ { "state_after": "no goals", "state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.522014\ninst✝¹ : Semiring k\ninst✝ : MulOneClass G\nf : MonoidAlgebra k G\nr : k\nx a : G\n⊢ 1 * a = x ↔ a = x", "tactic": "rw [one_mul]" } ]

[ 589, 50 ]

[ 587, 1 ]

[ { "state_after": "case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ↑f ⁻¹' ball 0 ε ⊆ ball 0 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\n⊢ ∃ K, AntilipschitzWith K ↑f", "tactic": "rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩" }, { "state_after": "case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ∀ (x : E), ‖↑f x‖ < ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "state_before": "case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ↑f ⁻¹' ball 0 ε ⊆ ball 0 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "tactic": "simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε" }, { "state_after": "case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "state_before": "case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ∀ (x : E), ‖↑f x‖ < ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "tactic": "lift ε to ℝ≥0 using ε0.le" }, { "state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ K, AntilipschitzWith K ↑f", "state_before": "case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ↑f", "tactic": "rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩" }, { "state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ K, AntilipschitzWith K ↑f", "tactic": "refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩" }, { "state_after": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖\n\ncase neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "by_cases hx : f x = 0" }, { "state_after": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)" }, { "state_after": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "rw [← h.1] at hc" }, { "state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑σ₁₂ c ^ n • ↑f x‖ < ↑ε\nhle : ‖↑σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖↑σ₁₂ c‖ * ‖↑f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩" }, { "state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖↑f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑σ₁₂ c ^ n • ↑f x‖ < ↑ε\nhle : ‖↑σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖↑σ₁₂ c‖ * ‖↑f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle" }, { "state_after": "no goals", "state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖↑f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "calc\n ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by\n rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]\n _ ≤ ‖c ^ n‖⁻¹ * 1 := (mul_le_mul_of_nonneg_left (hε _ hlt).le (inv_nonneg.2 (norm_nonneg _)))\n _ ≤ ε⁻¹ * ‖c‖ * ‖f x‖ := by rwa [mul_one]" }, { "state_after": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhf : Filter.comap (↑f) (𝓝 (↑f x)) ≤ 𝓝 0\nhx : ↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "state_before": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "rw [← hx] at hf" }, { "state_after": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nhf : Filter.comap (↑f) (𝓝 (↑f 0)) ≤ 𝓝 0\nhx : ↑f 0 = 0\n⊢ ‖0‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f 0‖", "state_before": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhf : Filter.comap (↑f) (𝓝 (↑f x)) ≤ 𝓝 0\nhx : ↑f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f x‖", "tactic": "obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|\n ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)" }, { "state_after": "no goals", "state_before": "case pos\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nhf : Filter.comap (↑f) (𝓝 (↑f 0)) ≤ 𝓝 0\nhx : ↑f 0 = 0\n⊢ ‖0‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖↑f 0‖", "tactic": "exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖↑f x‖\n⊢ ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖", "tactic": "rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2566867\nE : Type u_3\nEₗ : Type ?u.2566873\nF : Type u_4\nFₗ : Type ?u.2566879\nG : Type ?u.2566882\nGₗ : Type ?u.2566885\n𝓕 : Type ?u.2566888\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : Filter.comap (↑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖↑f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖↑σ₁₂ c‖\nx : E\nhx : ¬↑f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖↑f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖↑f x‖\n⊢ ‖c ^ n‖⁻¹ * 1 ≤ ↑ε⁻¹ * ‖c‖ * ‖↑f x‖", "tactic": "rwa [mul_one]" } ]

[ 1452, 46 ]

[ 1432, 1 ]

[]

[ 1030, 6 ]

[ 1029, 1 ]

[]

[ 229, 25 ]

[ 226, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type ?u.37777\nR₂ : Type ?u.37780\nK : Type ?u.37783\nM✝ : Type ?u.37786\nM₂ : Type ?u.37789\nV : Type ?u.37792\nS : Type ?u.37795\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M✝\ninst✝⁴ : Module R M✝\nx : M✝\np p' : Submodule R M✝\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M✝\nM : Type u_1\ninst✝ : AddCommGroup M\ns : AddSubgroup M\n⊢ toAddSubgroup (span ℤ ↑s) = s", "tactic": "rw [span_int_eq_addSubgroup_closure, s.closure_eq]" } ]

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[ 206, 1 ]

[]

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[ { "state_after": "case nil\nα : Type u_1\nβ : Type u_2\nl₂ : List β\n⊢ length ([] ×ˢ l₂) = length [] * length l₂\n\ncase cons\nα : Type u_1\nβ : Type u_2\nl₂ : List β\nx : α\nl₁ : List α\nIH : length (l₁ ×ˢ l₂) = length l₁ * length l₂\n⊢ length ((x :: l₁) ×ˢ l₂) = length (x :: l₁) * length l₂", "state_before": "α : Type u_1\nβ : Type u_2\nl₁ : List α\nl₂ : List β\n⊢ length (l₁ ×ˢ l₂) = length l₁ * length l₂", "tactic": "induction' l₁ with x l₁ IH" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nβ : Type u_2\nl₂ : List β\n⊢ length ([] ×ˢ l₂) = length [] * length l₂", "tactic": "exact (zero_mul _).symm" }, { "state_after": "no goals", "state_before": "case cons\nα : Type u_1\nβ : Type u_2\nl₂ : List β\nx : α\nl₁ : List α\nIH : length (l₁ ×ˢ l₂) = length l₁ * length l₂\n⊢ length ((x :: l₁) ×ˢ l₂) = length (x :: l₁) * length l₂", "tactic": "simp only [length, product_cons, length_append, IH, right_distrib, one_mul, length_map,\n add_comm]" } ]

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[ { "state_after": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : degree f = ⊥\n⊢ f = 0", "state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\n⊢ degree f = ⊥ ↔ f = 0", "tactic": "refine' ⟨fun h => _, fun h => by rw [h, degree_zero]⟩" }, { "state_after": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\n⊢ f = 0", "state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : degree f = ⊥\n⊢ f = 0", "tactic": "rw [degree, Finset.max_eq_sup_withBot] at h" }, { "state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\nn : ℤ\n⊢ ↑f n = ↑0 n", "state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\n⊢ f = 0", "tactic": "ext n" }, { "state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\nn : ℤ\nf0 : ¬↑f n = ↑0 n\n⊢ False", "state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\nn : ℤ\n⊢ ↑f n = ↑0 n", "tactic": "refine' not_not.mp fun f0 => _" }, { "state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nn : ℤ\nf0 : ¬↑f n = ↑0 n\nh : ∀ (s : ℤ), ¬↑f s = 0 → False\n⊢ False", "state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : Finset.sup f.support WithBot.some = ⊥\nn : ℤ\nf0 : ¬↑f n = ↑0 n\n⊢ False", "tactic": "simp_rw [Finset.sup_eq_bot_iff, Finsupp.mem_support_iff, Ne.def, WithBot.coe_ne_bot] at h" }, { "state_after": "no goals", "state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nn : ℤ\nf0 : ¬↑f n = ↑0 n\nh : ∀ (s : ℤ), ¬↑f s = 0 → False\n⊢ False", "tactic": "exact h n f0" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : f = 0\n⊢ degree f = ⊥", "tactic": "rw [h, degree_zero]" } ]

[ 495, 15 ]

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[ { "state_after": "case refine'_1\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ biUnion π πi\nJ : Box ι\nhJ : J ∈ π\n⊢ restrict π' J ≤ πi J\n\ncase refine'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\n⊢ (π' ≤ π ∧ ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J) → π' ≤ biUnion π πi", "state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\n⊢ π' ≤ biUnion π πi ↔ π' ≤ π ∧ ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J", "tactic": "refine' ⟨fun H => ⟨H.trans (π.biUnion_le πi), fun J hJ => _⟩, _⟩" }, { "state_after": "case refine'_1\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ biUnion π πi\nJ : Box ι\nhJ : J ∈ π\n⊢ restrict π' J ≤ restrict (biUnion π πi) J", "state_before": "case refine'_1\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ biUnion π πi\nJ : Box ι\nhJ : J ∈ π\n⊢ restrict π' J ≤ πi J", "tactic": "rw [← π.restrict_biUnion πi hJ]" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ biUnion π πi\nJ : Box ι\nhJ : J ∈ π\n⊢ restrict π' J ≤ restrict (biUnion π πi) J", "tactic": "exact restrict_mono H" }, { "state_after": "case refine'_2.intro\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "state_before": "case refine'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\n⊢ (π' ≤ π ∧ ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J) → π' ≤ biUnion π πi", "tactic": "rintro ⟨H, Hi⟩ J' hJ'" }, { "state_after": "case refine'_2.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "state_before": "case refine'_2.intro\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "tactic": "rcases H hJ' with ⟨J, hJ, hle⟩" }, { "state_after": "case refine'_2.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\nthis : J' ∈ restrict π' J\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "state_before": "case refine'_2.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "tactic": "have : J' ∈ π'.restrict J :=\n π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right <| WithBot.coe_le_coe.2 hle).symm⟩" }, { "state_after": "case refine'_2.intro.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\nthis : J' ∈ restrict π' J\nJi : Box ι\nhJi : Ji ∈ πi J\nhlei : J' ≤ Ji\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "state_before": "case refine'_2.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\nthis : J' ∈ restrict π' J\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "tactic": "rcases Hi J hJ this with ⟨Ji, hJi, hlei⟩" }, { "state_after": "no goals", "state_before": "case refine'_2.intro.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\nH : π' ≤ π\nHi : ∀ (J : Box ι), J ∈ π → restrict π' J ≤ πi J\nJ' : Box ι\nhJ' : J' ∈ π'\nJ : Box ι\nhJ : J ∈ π\nhle : J' ≤ J\nthis : J' ∈ restrict π' J\nJi : Box ι\nhJi : Ji ∈ πi J\nhlei : J' ≤ Ji\n⊢ ∃ I', I' ∈ biUnion π πi ∧ J' ≤ I'", "tactic": "exact ⟨Ji, π.mem_biUnion.2 ⟨J, hJ, hJi⟩, hlei⟩" } ]

[ 565, 51 ]

[ 555, 1 ]

[]

[ 2766, 99 ]

[ 2766, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type ?u.111481\nA : Type ?u.111484\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℝ\nn : ℕ\n⊢ eval (cos θ) (T ℝ n) = cos (↑n * θ)", "tactic": "exact_mod_cast T_complex_cos θ n" } ]

[ 119, 95 ]

[ 119, 1 ]

[]

[ 75, 28 ]

[ 73, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.308464\ninst✝ : Preorder α\na b c : ℕ\nH3 : 2 * (b + c) ≤ 9 * a + 3\nh : b < 2 * c\n⊢ b < 3 * a + 1", "tactic": "linarith" } ]

[ 1224, 33 ]

[ 1223, 1 ]

[]

[ 4052, 6 ]

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[ { "state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g 1 = 1", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g (p ^ 0) = eval₂ f g p ^ 0", "tactic": "rw [pow_zero, pow_zero]" }, { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g 1 = 1", "tactic": "exact eval₂_one _ _" }, { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : ℕ\n⊢ eval₂ f g (p ^ (n + 1)) = eval₂ f g p ^ (n + 1)", "tactic": "rw [pow_add, pow_one, pow_add, pow_one, eval₂_mul, eval₂_pow]" } ]

[ 1015, 78 ]

[ 1011, 1 ]

[]

[ 350, 59 ]

[ 347, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.6644\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t u v : Set α\n⊢ AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u", "tactic": "simp [union_eq_iUnion, and_comm]" } ]

[ 109, 35 ]

[ 108, 1 ]

[]

[ 4495, 30 ]

[ 4493, 1 ]

[]

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[]

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[]

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[ { "state_after": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ WithBot.recBotCoe True p ↑(sup' s H f)", "state_before": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ p (sup' s H f)", "tactic": "show @WithBot.recBotCoe α (fun _ => Prop) True p ↑(s.sup' H f)" }, { "state_after": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ WithBot.recBotCoe True p (sup s (WithBot.some ∘ f))", "state_before": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ WithBot.recBotCoe True p ↑(sup' s H f)", "tactic": "rw [coe_sup']" }, { "state_after": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ ∀ (a₁ : WithBot α),\n WithBot.recBotCoe True p a₁ → ∀ (a₂ : WithBot α), WithBot.recBotCoe True p a₂ → WithBot.recBotCoe True p (a₁ ⊔ a₂)", "state_before": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ WithBot.recBotCoe True p (sup s (WithBot.some ∘ f))", "tactic": "refine' sup_induction trivial _ hs" }, { "state_after": "case none\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\nh₁ : WithBot.recBotCoe True p none\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p (none ⊔ a₂)\n\ncase some\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\na₁ : α\nh₁ : WithBot.recBotCoe True p (some a₁)\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p (some a₁ ⊔ a₂)", "state_before": "F : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\n⊢ ∀ (a₁ : WithBot α),\n WithBot.recBotCoe True p a₁ → ∀ (a₂ : WithBot α), WithBot.recBotCoe True p a₂ → WithBot.recBotCoe True p (a₁ ⊔ a₂)", "tactic": "rintro (_ | a₁) h₁ a₂ h₂" }, { "state_after": "case none\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\nh₁ : WithBot.recBotCoe True p none\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p a₂", "state_before": "case none\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\nh₁ : WithBot.recBotCoe True p none\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p (none ⊔ a₂)", "tactic": "rw [WithBot.none_eq_bot, bot_sup_eq]" }, { "state_after": "no goals", "state_before": "case none\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\nh₁ : WithBot.recBotCoe True p none\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p a₂", "tactic": "exact h₂" }, { "state_after": "no goals", "state_before": "case some\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\na₁ : α\nh₁ : WithBot.recBotCoe True p (some a₁)\na₂ : WithBot α\nh₂ : WithBot.recBotCoe True p a₂\n⊢ WithBot.recBotCoe True p (some a₁ ⊔ a₂)", "tactic": "cases a₂ using WithBot.recBotCoe with\n| bot => exact h₁\n| coe a₂ => exact hp a₁ h₁ a₂ h₂" }, { "state_after": "no goals", "state_before": "case some.bot\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\na₁ : α\nh₁ : WithBot.recBotCoe True p (some a₁)\nh₂ : WithBot.recBotCoe True p ⊥\n⊢ WithBot.recBotCoe True p (some a₁ ⊔ ⊥)", "tactic": "exact h₁" }, { "state_after": "no goals", "state_before": "case some.coe\nF : Type ?u.260693\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.260702\nι : Type ?u.260705\nκ : Type ?u.260708\ninst✝ : SemilatticeSup α\ns : Finset β\nH : Finset.Nonempty s\nf : β → α\np : α → Prop\nhp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)\nhs : ∀ (b : β), b ∈ s → p (f b)\na₁ : α\nh₁ : WithBot.recBotCoe True p (some a₁)\na₂ : α\nh₂ : WithBot.recBotCoe True p ↑a₂\n⊢ WithBot.recBotCoe True p (some a₁ ⊔ ↑a₂)", "tactic": "exact hp a₁ h₁ a₂ h₂" } ]

[ 867, 37 ]

[ 857, 1 ]

[]

[ 199, 17 ]

[ 198, 1 ]

[]

[ 69, 34 ]

[ 68, 11 ]

[]

[ 168, 78 ]

[ 166, 1 ]

[ { "state_after": "case inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nterminated_at_n : TerminatedAt g n\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n\n\ncase inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "cases' Decidable.em (g.TerminatedAt n) with terminated_at_n not_terminated_at_n" }, { "state_after": "case inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nterminated_at_n : TerminatedAt g n\nthis : squashGCF g n = g\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "state_before": "case inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nterminated_at_n : TerminatedAt g n\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "have : squashGCF g n = g := squashGCF_eq_self_of_terminated terminated_at_n" }, { "state_after": "no goals", "state_before": "case inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nterminated_at_n : TerminatedAt g n\nthis : squashGCF g n = g\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "simp only [this, convergents_stable_of_terminated n.le_succ terminated_at_n]" }, { "state_after": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\n⊢ ∃ gp_n, Stream'.Seq.get? g.s n = some gp_n\n\ncase inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "state_before": "case inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "obtain ⟨⟨a, b⟩, s_nth_eq⟩ : ∃ gp_n, g.s.get? n = some gp_n" }, { "state_after": "case inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\n⊢ ∃ gp_n, Stream'.Seq.get? g.s n = some gp_n\n\ncase inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "exact Option.ne_none_iff_exists'.mp not_terminated_at_n" }, { "state_after": "case inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\nb_ne_zero : b ≠ 0\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "state_before": "case inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "have b_ne_zero : b ≠ 0 := nth_part_denom_ne_zero (part_denom_eq_s_b s_nth_eq)" }, { "state_after": "case inr.intro.mk.zero\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ convergents g (Nat.zero + 1) = convergents (squashGCF g Nat.zero) Nat.zero\n\ncase inr.intro.mk.succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case inr.intro.mk\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) n = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g n\na b : K\ns_nth_eq : Stream'.Seq.get? g.s n = some { a := a, b := b }\nb_ne_zero : b ≠ 0\n⊢ convergents g (n + 1) = convergents (squashGCF g n) n", "tactic": "cases' n with n'" }, { "state_after": "case inr.intro.mk.succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case inr.intro.mk.zero\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ convergents g (Nat.zero + 1) = convergents (squashGCF g Nat.zero) Nat.zero\n\ncase inr.intro.mk.succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "case zero =>\n suffices (b * g.h + a) / b = g.h + a / b by\n simpa [squashGCF, s_nth_eq, convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq zeroth_continuant_aux_eq_one_zero\n first_continuant_aux_eq_h_one]\n calc\n (b * g.h + a) / b = b * g.h / b + a / b := by ring\n _ = g.h + a / b := by rw [mul_div_cancel_left _ b_ne_zero]" }, { "state_after": "no goals", "state_before": "case inr.intro.mk.succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "case succ =>\n obtain ⟨⟨pa, pb⟩, s_n'th_eq⟩ : ∃ gp_n', g.s.get? n' = some gp_n' :=\n g.s.ge_stable n'.le_succ s_nth_eq\n let g' := squashGCF g (n' + 1)\n set pred_conts := g.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq\n set ppred_conts := g.continuantsAux n' with n'th_conts_aux_eq\n let pA := pred_conts.a\n let pB := pred_conts.b\n let ppA := ppred_conts.a\n let ppB := ppred_conts.b\n set pred_conts' := g'.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq'\n set ppred_conts' := g'.continuantsAux n' with n'th_conts_aux_eq'\n let pA' := pred_conts'.a\n let pB' := pred_conts'.b\n let ppA' := ppred_conts'.a\n let ppB' := ppred_conts'.b\n have : g'.convergents (n' + 1) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') := by\n have : g'.s.get? n' = some ⟨pa, pb + a / b⟩ :=\n squashSeq_nth_of_not_terminated s_n'th_eq s_nth_eq\n rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux this n'th_conts_aux_eq'.symm succ_n'th_conts_aux_eq'.symm]\n rw [this]\n have : g.convergents (n' + 2) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) := by\n have : g.continuantsAux (n' + 2) = ⟨pb * pA + pa * ppA, pb * pB + pa * ppB⟩ :=\n continuantsAux_recurrence s_n'th_eq n'th_conts_aux_eq.symm succ_n'th_conts_aux_eq.symm\n rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq succ_n'th_conts_aux_eq.symm this]\n rw [this]\n suffices\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) by\n obtain ⟨eq1, eq2, eq3, eq4⟩ : pA' = pA ∧ pB' = pB ∧ ppA' = ppA ∧ ppB' = ppB := by\n simp [*, (continuantsAux_eq_continuantsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,\n (continuantsAux_eq_continuantsAux_squashGCF_of_le n'.le_succ).symm]\n symm\n simpa only [eq1, eq2, eq3, eq4, mul_div_cancel _ b_ne_zero]\n field_simp\n congr 1 <;> ring" }, { "state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ (b * g.h + a) / b = g.h + a / b", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ convergents g (Nat.zero + 1) = convergents (squashGCF g Nat.zero) Nat.zero", "tactic": "suffices (b * g.h + a) / b = g.h + a / b by\n simpa [squashGCF, s_nth_eq, convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq zeroth_continuant_aux_eq_one_zero\n first_continuant_aux_eq_h_one]" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ (b * g.h + a) / b = g.h + a / b", "tactic": "calc\n (b * g.h + a) / b = b * g.h / b + a / b := by ring\n _ = g.h + a / b := by rw [mul_div_cancel_left _ b_ne_zero]" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\nthis : (b * g.h + a) / b = g.h + a / b\n⊢ convergents g (Nat.zero + 1) = convergents (squashGCF g Nat.zero) Nat.zero", "tactic": "simpa [squashGCF, s_nth_eq, convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq zeroth_continuant_aux_eq_one_zero\n first_continuant_aux_eq_h_one]" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ (b * g.h + a) / b = b * g.h / b + a / b", "tactic": "ring" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) Nat.zero = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g Nat.zero\ns_nth_eq : Stream'.Seq.get? g.s Nat.zero = some { a := a, b := b }\n⊢ b * g.h / b + a / b = g.h + a / b", "tactic": "rw [mul_div_cancel_left _ b_ne_zero]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "obtain ⟨⟨pa, pb⟩, s_n'th_eq⟩ : ∃ gp_n', g.s.get? n' = some gp_n' :=\n g.s.ge_stable n'.le_succ s_nth_eq" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let g' := squashGCF g (n' + 1)" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "set pred_conts := g.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "set ppred_conts := g.continuantsAux n' with n'th_conts_aux_eq" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let pA := pred_conts.a" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let pB := pred_conts.b" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let ppA := ppred_conts.a" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let ppB := ppred_conts.b" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "set pred_conts' := g'.continuantsAux (n' + 1) with succ_n'th_conts_aux_eq'" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "set ppred_conts' := g'.continuantsAux n' with n'th_conts_aux_eq'" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let pA' := pred_conts'.a" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let pB' := pred_conts'.b" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let ppA' := ppred_conts'.a" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "let ppB' := ppred_conts'.b" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "have : g'.convergents (n' + 1) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') := by\n have : g'.s.get? n' = some ⟨pa, pb + a / b⟩ :=\n squashSeq_nth_of_not_terminated s_n'th_eq s_nth_eq\n rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux this n'th_conts_aux_eq'.symm succ_n'th_conts_aux_eq'.symm]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\n⊢ convergents g (Nat.succ n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\n⊢ convergents g (Nat.succ n' + 1) = convergents (squashGCF g (Nat.succ n')) (Nat.succ n')", "tactic": "rw [this]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ convergents g (Nat.succ n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\n⊢ convergents g (Nat.succ n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "have : g.convergents (n' + 2) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) := by\n have : g.continuantsAux (n' + 2) = ⟨pb * pA + pa * ppA, pb * pB + pa * ppB⟩ :=\n continuantsAux_recurrence s_n'th_eq n'th_conts_aux_eq.symm succ_n'th_conts_aux_eq.symm\n rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq succ_n'th_conts_aux_eq.symm this]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ convergents g (Nat.succ n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "rw [this]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "suffices\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) by\n obtain ⟨eq1, eq2, eq3, eq4⟩ : pA' = pA ∧ pB' = pB ∧ ppA' = ppA ∧ ppB' = ppB := by\n simp [*, (continuantsAux_eq_continuantsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,\n (continuantsAux_eq_continuantsAux_squashGCF_of_le n'.le_succ).symm]\n symm\n simpa only [eq1, eq2, eq3, eq4, mul_div_cancel _ b_ne_zero]" }, { "state_after": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ ((pb * b + a) * (continuantsAux g (n' + 1)).a + pa * (continuantsAux g n').a * b) /\n ((pb * b + a) * (continuantsAux g (n' + 1)).b + pa * (continuantsAux g n').b * b) =\n (b * (pb * (continuantsAux g (n' + 1)).a + pa * (continuantsAux g n').a) + a * (continuantsAux g (n' + 1)).a) /\n (b * (pb * (continuantsAux g (n' + 1)).b + pa * (continuantsAux g n').b) + a * (continuantsAux g (n' + 1)).b)", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "tactic": "field_simp" }, { "state_after": "no goals", "state_before": "case intro.mk\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ ((pb * b + a) * (continuantsAux g (n' + 1)).a + pa * (continuantsAux g n').a * b) /\n ((pb * b + a) * (continuantsAux g (n' + 1)).b + pa * (continuantsAux g n').b * b) =\n (b * (pb * (continuantsAux g (n' + 1)).a + pa * (continuantsAux g n').a) + a * (continuantsAux g (n' + 1)).a) /\n (b * (pb * (continuantsAux g (n' + 1)).b + pa * (continuantsAux g n').b) + a * (continuantsAux g (n' + 1)).b)", "tactic": "congr 1 <;> ring" }, { "state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : Stream'.Seq.get? g'.s n' = some { a := pa, b := pb + a / b }\n⊢ convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\n⊢ convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "have : g'.s.get? n' = some ⟨pa, pb + a / b⟩ :=\n squashSeq_nth_of_not_terminated s_n'th_eq s_nth_eq" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : Stream'.Seq.get? g'.s n' = some { a := pa, b := pb + a / b }\n⊢ convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux this n'th_conts_aux_eq'.symm succ_n'th_conts_aux_eq'.symm]" }, { "state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : continuantsAux g (n' + 2) = { a := pb * pA + pa * ppA, b := pb * pB + pa * ppB }\n⊢ convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\n⊢ convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "tactic": "have : g.continuantsAux (n' + 2) = ⟨pb * pA + pa * ppA, pb * pB + pa * ppB⟩ :=\n continuantsAux_recurrence s_n'th_eq n'th_conts_aux_eq.symm succ_n'th_conts_aux_eq.symm" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis : continuantsAux g (n' + 2) = { a := pb * pA + pa * ppA, b := pb * pB + pa * ppB }\n⊢ convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "tactic": "rw [convergent_eq_conts_a_div_conts_b,\n continuants_recurrenceAux s_nth_eq succ_n'th_conts_aux_eq.symm this]" }, { "state_after": "case intro.intro.intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\neq1 : pA' = pA\neq2 : pB' = pB\neq3 : ppA' = ppA\neq4 : ppB' = ppB\n⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "obtain ⟨eq1, eq2, eq3, eq4⟩ : pA' = pA ∧ pB' = pB ∧ ppA' = ppA ∧ ppB' = ppB := by\n simp [*, (continuantsAux_eq_continuantsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,\n (continuantsAux_eq_continuantsAux_squashGCF_of_le n'.le_succ).symm]" }, { "state_after": "case intro.intro.intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\neq1 : pA' = pA\neq2 : pB' = pB\neq3 : ppA' = ppA\neq4 : ppB' = ppB\n⊢ ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "state_before": "case intro.intro.intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\neq1 : pA' = pA\neq2 : pB' = pB\neq3 : ppA' = ppA\neq4 : ppB' = ppB\n⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =\n ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')", "tactic": "symm" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\neq1 : pA' = pA\neq2 : pB' = pB\neq3 : ppA' = ppA\neq4 : ppB' = ppB\n⊢ ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)", "tactic": "simpa only [eq1, eq2, eq3, eq4, mul_div_cancel _ b_ne_zero]" }, { "state_after": "no goals", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_part_denom_ne_zero : ∀ {b : K}, Stream'.Seq.get? (partialDenominators g) (Nat.succ n') = some b → b ≠ 0\nnot_terminated_at_n : ¬TerminatedAt g (Nat.succ n')\ns_nth_eq : Stream'.Seq.get? g.s (Nat.succ n') = some { a := a, b := b }\npa pb : K\ns_n'th_eq : Stream'.Seq.get? g.s n' = some { a := pa, b := pb }\ng' : GeneralizedContinuedFraction K := squashGCF g (n' + 1)\npred_conts : Pair K := continuantsAux g (n' + 1)\nsucc_n'th_conts_aux_eq : pred_conts = continuantsAux g (n' + 1)\nppred_conts : Pair K := continuantsAux g n'\nn'th_conts_aux_eq : ppred_conts = continuantsAux g n'\npA : K := pred_conts.a\npB : K := pred_conts.b\nppA : K := ppred_conts.a\nppB : K := ppred_conts.b\npred_conts' : Pair K := continuantsAux g' (n' + 1)\nsucc_n'th_conts_aux_eq' : pred_conts' = continuantsAux g' (n' + 1)\nppred_conts' : Pair K := continuantsAux g' n'\nn'th_conts_aux_eq' : ppred_conts' = continuantsAux g' n'\npA' : K := pred_conts'.a\npB' : K := pred_conts'.b\nppA' : K := ppred_conts'.a\nppB' : K := ppred_conts'.b\nthis✝¹ : convergents g' (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')\nthis✝ : convergents g (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\nthis :\n ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =\n (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)\n⊢ pA' = pA ∧ pB' = pB ∧ ppA' = ppA ∧ ppB' = ppB", "tactic": "simp [*, (continuantsAux_eq_continuantsAux_squashGCF_of_le <| le_refl <| n' + 1).symm,\n (continuantsAux_eq_continuantsAux_squashGCF_of_le n'.le_succ).symm]" } ]

[ 332, 23 ]

[ 268, 1 ]

[]

[ 708, 13 ]

[ 707, 1 ]

[]

[ 1880, 65 ]

[ 1879, 1 ]

[ { "state_after": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\n⊢ (Finset.sup Finset.univ fun i =>\n (Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n Finset.sup Finset.univ fun i =>\n (Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i)", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\n⊢ distortion I = distortion J", "tactic": "simp only [distortion, nndist_pi_def, Real.nndist_eq', h, map_div₀]" }, { "state_after": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\n⊢ (Finset.sup Finset.univ fun i =>\n (Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n Finset.sup Finset.univ fun i =>\n (Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i)", "tactic": "congr 1 with i" }, { "state_after": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nthis : 0 < r\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "state_before": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "tactic": "have : 0 < r := by\n by_contra hr\n have := div_nonpos_of_nonneg_of_nonpos (sub_nonneg.2 <| J.lower_le_upper i) (not_lt.1 hr)\n rw [← h] at this\n exact this.not_lt (sub_pos.2 <| I.lower_lt_upper i)" }, { "state_after": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nthis : 0 < r\nhn0 : ↑Real.nnabs r ≠ 0\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "state_before": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nthis : 0 < r\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "tactic": "have hn0 := (map_ne_zero Real.nnabs).2 this.ne'" }, { "state_after": "no goals", "state_before": "case e_f.h.a\nι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nthis : 0 < r\nhn0 : ↑Real.nnabs r ≠ 0\n⊢ ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b) / ↑Real.nnabs r) /\n (↑Real.nnabs (upper J i - lower J i) / ↑Real.nnabs r)) =\n ↑((Finset.sup Finset.univ fun b => ↑Real.nnabs (upper J b - lower J b)) / ↑Real.nnabs (upper J i - lower J i))", "tactic": "simp_rw [NNReal.finset_sup_div, div_div_div_cancel_right _ hn0]" }, { "state_after": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\n⊢ False", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\n⊢ 0 < r", "tactic": "by_contra hr" }, { "state_after": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\nthis : (upper J i - lower J i) / r ≤ 0\n⊢ False", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\n⊢ False", "tactic": "have := div_nonpos_of_nonneg_of_nonpos (sub_nonneg.2 <| J.lower_le_upper i) (not_lt.1 hr)" }, { "state_after": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\nthis : upper I i - lower I i ≤ 0\n⊢ False", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\nthis : (upper J i - lower J i) / r ≤ 0\n⊢ False", "tactic": "rw [← h] at this" }, { "state_after": "no goals", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI J : Box ι\nr : ℝ\nh : ∀ (i : ι), upper I i - lower I i = (upper J i - lower J i) / r\ni : ι\nhr : ¬0 < r\nthis : upper I i - lower I i ≤ 0\n⊢ False", "tactic": "exact this.not_lt (sub_pos.2 <| I.lower_lt_upper i)" } ]

[ 510, 66 ]

[ 499, 1 ]

[ { "state_after": "α : Sort u_1\nctx : Context α\na b : Expr\nh✝ : (norm ctx a == norm ctx b) = true\nh : evalList α ctx (norm ctx a) = evalList α ctx (norm ctx b)\n⊢ eval α ctx a = eval α ctx b", "state_before": "α : Sort u_1\nctx : Context α\na b : Expr\nh : (norm ctx a == norm ctx b) = true\n⊢ eval α ctx a = eval α ctx b", "tactic": "have h := congrArg (evalList α ctx) (eq_of_beq h)" }, { "state_after": "α : Sort u_1\nctx : Context α\na b : Expr\nh✝ : (norm ctx a == norm ctx b) = true\nh : eval α ctx a = eval α ctx b\n⊢ eval α ctx a = eval α ctx b", "state_before": "α : Sort u_1\nctx : Context α\na b : Expr\nh✝ : (norm ctx a == norm ctx b) = true\nh : evalList α ctx (norm ctx a) = evalList α ctx (norm ctx b)\n⊢ eval α ctx a = eval α ctx b", "tactic": "rw [eval_norm, eval_norm] at h" }, { "state_after": "no goals", "state_before": "α : Sort u_1\nctx : Context α\na b : Expr\nh✝ : (norm ctx a == norm ctx b) = true\nh : eval α ctx a = eval α ctx b\n⊢ eval α ctx a = eval α ctx b", "tactic": "assumption" } ]

[ 320, 13 ]

[ 317, 1 ]

[]

[ 274, 37 ]

[ 273, 1 ]

[ { "state_after": "no goals", "state_before": "K : Type u_1\nE : Type ?u.7115686\ninst✝ : IsROrC K\nx : K\n⊢ ‖↑(starRingEnd K) x + x‖ ^ 2 = ↑re (↑(starRingEnd K) x + x) ^ 2", "tactic": "rw [add_comm, norm_sq_re_add_conj]" } ]

[ 770, 37 ]

[ 769, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl t₁ t₂ : List α\np : t₁ ~ t₂\na : α\nx✝ : a ∈ l\n⊢ decide (a ∈ t₁) = true ↔ decide (a ∈ t₂) = true", "tactic": "simpa using p.mem_iff" } ]

[ 1013, 52 ]

[ 1012, 1 ]

[ { "state_after": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => abs (eval x P)\n⊢ degree P ≤ 0", "state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ (IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => abs (eval x P)) ↔ degree P ≤ 0", "tactic": "refine' ⟨fun h => _, fun h => ⟨|P.coeff 0|, eventually_map.mpr (eventually_of_forall\n (forall_imp (fun _ => le_of_eq) fun x => congr_arg abs <| _root_.trans (congr_arg (eval x)\n (eq_C_of_degree_le_zero h)) eval_C))⟩⟩" }, { "state_after": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : 0 < degree P\n⊢ ¬IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => abs (eval x P)", "state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => abs (eval x P)\n⊢ degree P ≤ 0", "tactic": "contrapose! h" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : 0 < degree P\n⊢ ¬IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop fun x => abs (eval x P)", "tactic": "exact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h)" } ]

[ 100, 68 ]

[ 94, 1 ]

[]

[ 257, 41 ]

[ 256, 1 ]

[]

[ 180, 25 ]

[ 176, 1 ]

[]

[ 738, 9 ]

[ 737, 1 ]

[ { "state_after": "no goals", "state_before": "d✝ : ℤ\nc d x y z w : ℕ\nxy : SqLe x c y d\nzw : SqLe z c w d\n⊢ c * (x * z) * (c * (x * z)) ≤ d * (y * w) * (d * (y * w))", "tactic": "simpa [mul_comm, mul_left_comm] using mul_le_mul xy zw (Nat.zero_le _) (Nat.zero_le _)" } ]

[ 440, 91 ]

[ 437, 1 ]

[]

[ 889, 30 ]

[ 888, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1", "tactic": "simpa only [one_zsmul] using toIcoDiv_add_zsmul' hp a b 1" } ]

[ 305, 60 ]

[ 304, 1 ]

[]

[ 811, 75 ]

[ 810, 1 ]

[ { "state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\ne_2✝ : (forget TopCat).obj Y = ↑Y\n⊢ (forget TopCat).map pullback.snd =\n ↑(homeoOfIso (asIso pullback.snd)) ∘\n (forget TopCat).map (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g))", "state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\n⊢ OpenEmbedding ((forget TopCat).map pullback.snd)", "tactic": "convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶ _))).openEmbedding.comp\n (pullback_map_openEmbedding_of_open_embeddings (i₂ := 𝟙 Y) f g (𝟙 _) g H\n (homeoOfIso (Iso.refl _)).openEmbedding (𝟙 _) rfl (by simp))" }, { "state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\ne_2✝ : (forget TopCat).obj Y = ↑Y\n⊢ (forget TopCat).map pullback.snd =\n (forget TopCat).map\n (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫\n (asIso pullback.snd).hom)", "state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\ne_2✝ : (forget TopCat).obj Y = ↑Y\n⊢ (forget TopCat).map pullback.snd =\n ↑(homeoOfIso (asIso pullback.snd)) ∘\n (forget TopCat).map (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g))", "tactic": "erw [← coe_comp]" }, { "state_after": "no goals", "state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\ne_2✝ : (forget TopCat).obj Y = ↑Y\n⊢ (forget TopCat).map pullback.snd =\n (forget TopCat).map\n (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫\n (asIso pullback.snd).hom)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\nH : OpenEmbedding ((forget TopCat).map f)\ng : Y ⟶ S\n⊢ g ≫ 𝟙 S = 𝟙 Y ≫ g", "tactic": "simp" } ]

[ 322, 7 ]

[ 316, 1 ]

[]

[ 161, 25 ]

[ 159, 1 ]

[]

[ 1328, 24 ]

[ 1326, 1 ]

[]

[ 118, 60 ]

[ 117, 1 ]

[ { "state_after": "case mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.24956\nδ : Type ?u.24959\ne' : LocalEquiv β γ\ntoFun✝ : α → β\ninvFun✝ : β → α\nsource✝ : Set α\ntarget✝ : Set β\nmap_source'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → toFun✝ x ∈ target✝\nmap_target'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → invFun✝ x ∈ source✝\nleft_inv'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → invFun✝ (toFun✝ x) = x\nright_inv'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → toFun✝ (invFun✝ x) = x\n⊢ LocalEquiv.symm\n (LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }) =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.24956\nδ : Type ?u.24959\ne : LocalEquiv α β\ne' : LocalEquiv β γ\n⊢ LocalEquiv.symm (LocalEquiv.symm e) = e", "tactic": "cases e" }, { "state_after": "no goals", "state_before": "case mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.24956\nδ : Type ?u.24959\ne' : LocalEquiv β γ\ntoFun✝ : α → β\ninvFun✝ : β → α\nsource✝ : Set α\ntarget✝ : Set β\nmap_source'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → toFun✝ x ∈ target✝\nmap_target'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → invFun✝ x ∈ source✝\nleft_inv'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → invFun✝ (toFun✝ x) = x\nright_inv'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → toFun✝ (invFun✝ x) = x\n⊢ LocalEquiv.symm\n (LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }) =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }", "tactic": "rfl" } ]

[ 332, 6 ]

[ 330, 1 ]

[ { "state_after": "R : Type u\nS : Type ?u.4503\ninst✝ : Semiring R\nn : WithBot ℕ\nf : R[X]\n⊢ (∀ (i : ℕ), ↑i > n → ↑(lcoeff R i) f = 0) ↔ ∀ (m : ℕ), n < ↑m → coeff f m = 0", "state_before": "R : Type u\nS : Type ?u.4503\ninst✝ : Semiring R\nn : WithBot ℕ\nf : R[X]\n⊢ f ∈ degreeLE R n ↔ degree f ≤ n", "tactic": "simp only [degreeLE, Submodule.mem_iInf, degree_le_iff_coeff_zero, LinearMap.mem_ker]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type ?u.4503\ninst✝ : Semiring R\nn : WithBot ℕ\nf : R[X]\n⊢ (∀ (i : ℕ), ↑i > n → ↑(lcoeff R i) f = 0) ↔ ∀ (m : ℕ), n < ↑m → coeff f m = 0", "tactic": "rfl" } ]

[ 68, 93 ]

[ 67, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nE : Type ?u.911849\nF : Type u_2\nG : Type ?u.911855\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp0_lt : 0 < q\n⊢ snorm' 0 q μ = 0", "tactic": "simp [snorm', hp0_lt]" } ]

[ 187, 94 ]

[ 187, 1 ]

[ { "state_after": "no goals", "state_before": "n : ℕ\na b : Fin n\n⊢ card (Ioo a b) = ↑b - ↑a - 1", "tactic": "rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]" } ]

[ 95, 56 ]

[ 94, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nr✝ : α → α → Prop\ns : β → β → Prop\nr : α → α → Prop\ninst✝ : IsTrichotomous α r\na b : α\n⊢ Function.swap r a b ∨ a = b ∨ Function.swap r b a", "tactic": "simpa [Function.swap, or_comm, or_left_comm] using trichotomous_of r a b" } ]

[ 96, 91 ]

[ 95, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3743\nf : α → β\nga : α → α → α\ngb : β → β → β\ninst✝ : IsIdempotent α ga\nh : Semiconj₂ f ga gb\nh_surj : Surjective f\nx : α\n⊢ gb (f x) (f x) = f x", "tactic": "simp only [← h.eq, @IsIdempotent.idempotent _ ga]" } ]

[ 176, 81 ]

[ 174, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.6300\nι : Type u_1\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : SupConvergenceClass α\nf : ι → α\na : α\nh_anti : Antitone f\nha : IsLUB (range f) a\n⊢ Tendsto f atBot (𝓝 a)", "tactic": "convert tendsto_atTop_isLUB h_anti.dual_left ha using 1" } ]

[ 106, 88 ]

[ 105, 1 ]

[]

[ 493, 60 ]

[ 491, 1 ]

[ { "state_after": "case h\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\n⊢ ↑n < trailingDegree p", "state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\n⊢ coeff p n = 0", "tactic": "apply coeff_eq_zero_of_trailingDegree_lt" }, { "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : p = 0\n⊢ ↑n < trailingDegree p\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : ¬p = 0\n⊢ ↑n < trailingDegree p", "state_before": "case h\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\n⊢ ↑n < trailingDegree p", "tactic": "by_cases hp : p = 0" }, { "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : p = 0\n⊢ ↑n < ⊤", "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : p = 0\n⊢ ↑n < trailingDegree p", "tactic": "rw [hp, trailingDegree_zero]" }, { "state_after": "no goals", "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : p = 0\n⊢ ↑n < ⊤", "tactic": "exact WithTop.coe_lt_top n" }, { "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : ¬p = 0\n⊢ ↑n < ↑(natTrailingDegree p)", "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : ¬p = 0\n⊢ ↑n < trailingDegree p", "tactic": "rw [trailingDegree_eq_natTrailingDegree hp]" }, { "state_after": "no goals", "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : n < natTrailingDegree p\nhp : ¬p = 0\n⊢ ↑n < ↑(natTrailingDegree p)", "tactic": "exact WithTop.coe_lt_coe.2 h" } ]

[ 276, 33 ]

[ 269, 1 ]

[ { "state_after": "no goals", "state_before": "R✝ : Type u\nA✝ : Type v\nB : Type w\ninst✝⁷ : CommSemiring R✝\ninst✝⁶ : Semiring A✝\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R✝ B\nS✝ : Subalgebra R✝ A✝\nα : Type ?u.1950480\nβ : Type ?u.1950483\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ RingHom.rangeS (algebraMap { x // x ∈ S } A) = S.toSubsemiring", "tactic": "rw [algebraMap_eq, Algebra.id.map_eq_id, RingHom.id_comp, ← toSubsemiring_subtype,\n Subsemiring.rangeS_subtype]" } ]

[ 1316, 32 ]

[ 1313, 1 ]

[ { "state_after": "case a\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ (p * q).toFinsupp = (∑ i in support p, sum q fun j a => ↑(monomial (i + j)) (coeff p i * a)).toFinsupp", "state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ p * q = ∑ i in support p, sum q fun j a => ↑(monomial (i + j)) (coeff p i * a)", "tactic": "apply toFinsupp_injective" }, { "state_after": "case a.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\nq : R[X]\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝ } * q).toFinsupp =\n (∑ i in support { toFinsupp := toFinsupp✝ },\n sum q fun j a => ↑(monomial (i + j)) (coeff { toFinsupp := toFinsupp✝ } i * a)).toFinsupp", "state_before": "case a\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ (p * q).toFinsupp = (∑ i in support p, sum q fun j a => ↑(monomial (i + j)) (coeff p i * a)).toFinsupp", "tactic": "rcases p with ⟨⟩" }, { "state_after": "case a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp =\n (∑ i in support { toFinsupp := toFinsupp✝¹ },\n sum { toFinsupp := toFinsupp✝ } fun j a =>\n ↑(monomial (i + j)) (coeff { toFinsupp := toFinsupp✝¹ } i * a)).toFinsupp", "state_before": "case a.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\nq : R[X]\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝ } * q).toFinsupp =\n (∑ i in support { toFinsupp := toFinsupp✝ },\n sum q fun j a => ↑(monomial (i + j)) (coeff { toFinsupp := toFinsupp✝ } i * a)).toFinsupp", "tactic": "rcases q with ⟨⟩" }, { "state_after": "case a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ toFinsupp✝¹ * toFinsupp✝ =\n ∑ x in toFinsupp✝¹.support, ∑ x_1 in toFinsupp✝.support, Finsupp.single (x + x_1) (↑toFinsupp✝¹ x * ↑toFinsupp✝ x_1)", "state_before": "case a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp =\n (∑ i in support { toFinsupp := toFinsupp✝¹ },\n sum { toFinsupp := toFinsupp✝ } fun j a =>\n ↑(monomial (i + j)) (coeff { toFinsupp := toFinsupp✝¹ } i * a)).toFinsupp", "tactic": "simp [support, sum, coeff, toFinsupp_sum]" }, { "state_after": "no goals", "state_before": "case a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ toFinsupp✝¹ * toFinsupp✝ =\n ∑ x in toFinsupp✝¹.support, ∑ x_1 in toFinsupp✝.support, Finsupp.single (x + x_1) (↑toFinsupp✝¹ x * ↑toFinsupp✝ x_1)", "tactic": "rfl" } ]

[ 956, 6 ]

[ 951, 1 ]

[ { "state_after": "case pos\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : i₁ = i₂\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁\n\ncase neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "tactic": "by_cases h : i₁ = i₂" }, { "state_after": "no goals", "state_before": "case pos\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : i₁ = i₂\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "tactic": "simp [h]" }, { "state_after": "case neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(card {i₁, i₂}) ≠ 0\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "state_before": "case neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "tactic": "have hc : (card ({i₁, i₂} : Finset ι) : k) ≠ 0 := by\n rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]\n norm_num\n exact nonzero_of_invertible _" }, { "state_after": "case neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(card {i₁, i₂}) ≠ 0\n⊢ (↑(weightedVSubOfPoint {i₁, i₂} p (p i₁)) fun i => centroidWeights k {i₁, i₂} i) +ᵥ p i₁ =\n 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "state_before": "case neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(card {i₁, i₂}) ≠ 0\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "tactic": "rw [centroid_def,\n affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ _ _\n (sum_centroidWeights_eq_one_of_cast_card_ne_zero _ hc) (p i₁)]" }, { "state_after": "no goals", "state_before": "case neg\nk : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(card {i₁, i₂}) ≠ 0\n⊢ (↑(weightedVSubOfPoint {i₁, i₂} p (p i₁)) fun i => centroidWeights k {i₁, i₂} i) +ᵥ p i₁ =\n 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁", "tactic": "simp [h, one_add_one_eq_two]" }, { "state_after": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ↑(1 + 1) ≠ 0", "state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ↑(card {i₁, i₂}) ≠ 0", "tactic": "rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]" }, { "state_after": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ¬2 = 0", "state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ↑(1 + 1) ≠ 0", "tactic": "norm_num" }, { "state_after": "no goals", "state_before": "k : Type u_2\nV : Type u_4\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.484128\ns₂ : Finset ι₂\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ¬2 = 0", "tactic": "exact nonzero_of_invertible _" } ]

[ 875, 33 ]

[ 864, 1 ]

[]

[ 95, 75 ]

[ 94, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.621442\nE : Type u_3\nEₗ : Type ?u.621448\nF : Type u_4\nFₗ : Type ?u.621454\nG : Type ?u.621457\nGₗ : Type ?u.621460\n𝓕 : Type ?u.621463\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\nf : E →SL[σ₁₂] F\nM : ℝ≥0\nhM : ∀ (x : E), ‖x‖₊ ≠ 0 → ‖↑f x‖₊ ≤ M * ‖x‖₊\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖x‖₊ ≠ 0", "tactic": "rwa [← NNReal.coe_ne_zero]" } ]

[ 423, 84 ]

[ 421, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : Set β\nhs : Set.Countable s\nht : Set.Countable t\nthis : Countable ↑s\n⊢ Set.Countable (s ×ˢ t)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : Set β\nhs : Set.Countable s\nht : Set.Countable t\n⊢ Set.Countable (s ×ˢ t)", "tactic": "haveI : Countable s := hs.to_subtype" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : Set β\nhs : Set.Countable s\nht : Set.Countable t\nthis✝ : Countable ↑s\nthis : Countable ↑t\n⊢ Set.Countable (s ×ˢ t)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : Set β\nhs : Set.Countable s\nht : Set.Countable t\nthis : Countable ↑s\n⊢ Set.Countable (s ×ˢ t)", "tactic": "haveI : Countable t := ht.to_subtype" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nt : Set β\nhs : Set.Countable s\nht : Set.Countable t\nthis✝ : Countable ↑s\nthis : Countable ↑t\n⊢ Set.Countable (s ×ˢ t)", "tactic": "exact (Countable.of_equiv _ <| (Equiv.Set.prod _ _).symm).to_set" } ]

[ 282, 67 ]

[ 278, 11 ]

[ { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ns : Set (Set A)\n⊢ zeroLocus 𝒜 (⋃ (s' : Set A) (_ : s' ∈ s), s') = ⋂ (s' : Set A) (_ : s' ∈ s), zeroLocus 𝒜 s'", "tactic": "simp only [zeroLocus_iUnion]" } ]

[ 270, 34 ]

[ 268, 1 ]