Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul_const	[36, 1]	[54, 53]	ring	case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul_const	[36, 1]	[54, 53]	rw [h]	case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0	case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul_const	[36, 1]	[54, 53]	ring	case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul_const	[36, 1]	[54, 53]	apply div_pos εpos acpos	s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < \|c\| ε : ℝ εpos : ε > 0 ⊢ 0 < ε / \|c\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < \|c\| ε : ℝ εpos : ε > 0 ⊢ 0 < ε / \|c\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul_const	[36, 1]	[54, 53]	rw [← abs_mul, mul_sub]	s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < \|c\| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / \|c\| Ns : ℕ hs : ∀ n ≥ Ns, \|s n - a\| < ε / \|c\| n : ℕ ngt : n ≥ Ns ⊢ \|c * s n - c * a\| = \|c\| * \|s n - a\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < \|c\| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / \|c\| Ns : ℕ hs : ∀ n ≥ Ns, \|s n - a\| < ε / \|c\| n : ℕ ngt : n ≥ Ns ⊢ \|c * s n - c * a\| = \|c\| * \|s n - a\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	rcases cs 1 zero_lt_one with ⟨N, h⟩	s : ℕ → ℝ a : ℝ cs : ConvergesTo s a ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → \|s n\| < b	case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → \|s n\| < b	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a : ℝ cs : ConvergesTo s a ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → \|s n\| < b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	use N, \|a\| + 1	case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → \|s n\| < b	case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∀ (n : ℕ), N ≤ n → \|s n\| < \|a\| + 1	Please generate a tactic in lean4 to solve the state. STATE: case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → \|s n\| < b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	intro n ngt	case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∀ (n : ℕ), N ≤ n → \|s n\| < \|a\| + 1	case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n\| < \|a\| + 1	Please generate a tactic in lean4 to solve the state. STATE: case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 ⊢ ∀ (n : ℕ), N ≤ n → \|s n\| < \|a\| + 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	calc \|s n\| = \|s n - a + a\| := by congr abel _ ≤ \|s n - a\| + \|a\| := (abs_add _ _) _ < \|a\| + 1 := by linarith [h n ngt]	case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n\| < \|a\| + 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n\| < \|a\| + 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	congr	s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n\| = \|s n - a + a\|	case e_a s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ s n = s n - a + a	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n\| = \|s n - a + a\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	abel	case e_a s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ s n = s n - a + a	no goals	Please generate a tactic in lean4 to solve the state. STATE: case e_a s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ s n = s n - a + a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.exists_abs_le_of_convergesTo	[56, 1]	[66, 41]	linarith [h n ngt]	s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n - a\| + \|a\| < \|a\| + 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, \|s n - a\| < 1 n : ℕ ngt : N ≤ n ⊢ \|s n - a\| + \|a\| < \|a\| + 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	intro ε εpos	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ⊢ ConvergesTo (fun n => s n * t n) 0	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|(fun n => s n * t n) n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ⊢ ConvergesTo (fun n => s n * t n) 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	dsimp	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|(fun n => s n * t n) n - 0\| < ε	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|(fun n => s n * t n) n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _))	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	have pos₀ : ε / B > 0 := div_pos εpos Bpos	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	rcases ct _ pos₀ with ⟨N₁, h₁⟩	case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	case intro.intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	use max N₀ N₁	case intro.intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∀ n ≥ max N₀ N₁, \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∃ N, ∀ n ≥ N, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	intro n ngt	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∀ n ≥ max N₀ N₁, \|s n * t n - 0\| < ε	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ⊢ \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B ⊢ ∀ n ≥ max N₀ N₁, \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ⊢ \|s n * t n - 0\| < ε	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ⊢ \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ⊢ \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ⊢ \|s n * t n - 0\| < ε	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ngeN₁ : n ≥ N₁ ⊢ \|s n * t n - 0\| < ε	Please generate a tactic in lean4 to solve the state. STATE: case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ⊢ \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	calc \|s n * t n - 0\| = \|s n\| * \|t n - 0\| := by rw [sub_zero, abs_mul, sub_zero] _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm	case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ngeN₁ : n ≥ N₁ ⊢ \|s n * t n - 0\| < ε	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ngeN₁ : n ≥ N₁ ⊢ \|s n * t n - 0\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.aux	[68, 1]	[83, 52]	rw [sub_zero, abs_mul, sub_zero]	s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ngeN₁ : n ≥ N₁ ⊢ \|s n * t n - 0\| = \|s n\| * \|t n - 0\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → \|s n\| < B Bpos : 0 < B pos₀ : ε / B > 0 N₁ : ℕ h₁ : ∀ n ≥ N₁, \|t n - 0\| < ε / B n : ℕ ngt : n ≥ max N₀ N₁ ngeN₀ : n ≥ N₀ ngeN₁ : n ≥ N₁ ⊢ \|s n * t n - 0\| = \|s n\| * \|t n - 0\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	have h₁ : ConvergesTo (fun n ↦ s n * (t n + -b)) 0 := by apply aux cs convert convergesTo_add ct (convergesTo_const (-b)) ring	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n * t n) (a * b)	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 ⊢ ConvergesTo (fun n => s n * t n) (a * b)	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n * t n) (a * b) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	have := convergesTo_add h₁ (convergesTo_mul_const b cs)	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 ⊢ ConvergesTo (fun n => s n * t n) (a * b)	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ ConvergesTo (fun n => s n * t n) (a * b)	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 ⊢ ConvergesTo (fun n => s n * t n) (a * b) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ ConvergesTo (fun n => s n * t n) (a * b)	case h.e'_1 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ a * b = 0 + b * a	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ ConvergesTo (fun n => s n * t n) (a * b) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	ring	case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ a * b = 0 + b * a	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ a * b = 0 + b * a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	apply aux cs	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n * (t n + -b)) 0	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => t n + -b) 0	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n * (t n + -b)) 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	convert convergesTo_add ct (convergesTo_const (-b))	s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => t n + -b) 0	case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ 0 = b + -b	Please generate a tactic in lean4 to solve the state. STATE: s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => t n + -b) 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	ring	case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ 0 = b + -b	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_2 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ 0 = b + -b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	ext	case h.e'_1 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n	case h.e'_1.h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) x✝ : ℕ ⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_1 s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) ⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_mul	[85, 1]	[95, 7]	ring	case h.e'_1.h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) x✝ : ℕ ⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e'_1.h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a) x✝ : ℕ ⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	by_contra abne	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b ⊢ a = b	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b ⊢ a = b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	let ε := \|a - b\| / 2	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ⊢ False	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	have εpos : ε > 0 := by change \|a - b\| / 2 > 0 linarith	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ False	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	rcases sa ε εpos with ⟨Na, hNa⟩	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 ⊢ False	case intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	rcases sb ε εpos with ⟨Nb, hNb⟩	case intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε ⊢ False	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	let N := max Na Nb	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε ⊢ False	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	have absa : \|s N - a\| < ε := by apply hNa apply le_max_left	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ False	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	have absb : \|s N - b\| < ε := by apply hNb apply le_max_right	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ False	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	have : \|a - b\| < \|a - b\|	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ False	case this s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|a - b\| < \|a - b\| case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	calc \|a - b\| = \|(-(s N - a)) + (s N - b)\| := by congr ring _ ≤ \|(-(s N - a))\| + \|s N - b\| := (abs_add _ _) _ = \|s N - a\| + \|s N - b\| := by rw [abs_neg] _ < ε + ε := (add_lt_add absa absb) _ = \|a - b\| := by norm_num	case this s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|a - b\| < \|a - b\| case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case this s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|a - b\| < \|a - b\| case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	exact lt_irrefl _ this	case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this✝ : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε this : \|a - b\| < \|a - b\| ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply lt_of_le_of_ne	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ \|a - b\| > 0	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≤ \|a - b\| case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≠ \|a - b\|	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ \|a - b\| > 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	intro h''	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≠ \|a - b\|	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≠ \|a - b\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply abne	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ False	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ a = b	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply eq_of_abs_sub_eq_zero h''.symm	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ a = b	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b h'' : 0 = \|a - b\| ⊢ a = b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply abs_nonneg	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≤ \|a - b\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b ⊢ 0 ≤ \|a - b\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	change \|a - b\| / 2 > 0	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ ε > 0	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ \|a - b\| / 2 > 0	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ ε > 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	linarith	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ \|a - b\| / 2 > 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 ⊢ \|a - b\| / 2 > 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply hNa	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ \|s N - a\| < ε	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ N ≥ Na	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ \|s N - a\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply le_max_left	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ N ≥ Na	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb ⊢ N ≥ Na TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply hNb	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ \|s N - b\| < ε	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ N ≥ Nb	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ \|s N - b\| < ε TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	apply le_max_right	case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ N ≥ Nb	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε ⊢ N ≥ Nb TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	congr	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|a - b\| = \|-(s N - a) + (s N - b)\|	case e_a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ a - b = -(s N - a) + (s N - b)	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|a - b\| = \|-(s N - a) + (s N - b)\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	ring	case e_a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ a - b = -(s N - a) + (s N - b)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case e_a s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ a - b = -(s N - a) + (s N - b) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	rw [abs_neg]	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|(-(s N - a))\| + \|s N - b\| = \|s N - a\| + \|s N - b\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ \|(-(s N - a))\| + \|s N - b\| = \|s N - a\| + \|s N - b\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean	C03S06.convergesTo_unique	[97, 1]	[130, 25]	norm_num	s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ ε + ε = \|a - b\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: s : ℕ → ℝ a b : ℝ sa : ConvergesTo s a sb : ConvergesTo s b abne : ¬a = b this : \|a - b\| > 0 ε : ℝ := \|a - b\| / 2 εpos : ε > 0 Na : ℕ hNa : ∀ n ≥ Na, \|s n - a\| < ε Nb : ℕ hNb : ∀ n ≥ Nb, \|s n - b\| < ε N : ℕ := max Na Nb absa : \|s N - a\| < ε absb : \|s N - b\| < ε ⊢ ε + ε = \|a - b\| TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.div'_add_mod'	[158, 1]	[160, 45]	rw [div', mod']	a b : ℤ ⊢ b * div' a b + mod' a b = a	a b : ℤ ⊢ b * ((a + b / 2) / b) + ((a + b / 2) % b - b / 2) = a	Please generate a tactic in lean4 to solve the state. STATE: a b : ℤ ⊢ b * div' a b + mod' a b = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.div'_add_mod'	[158, 1]	[160, 45]	linarith [Int.ediv_add_emod (a + b / 2) b]	a b : ℤ ⊢ b * ((a + b / 2) / b) + ((a + b / 2) % b - b / 2) = a	no goals	Please generate a tactic in lean4 to solve the state. STATE: a b : ℤ ⊢ b * ((a + b / 2) / b) + ((a + b / 2) % b - b / 2) = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	rw [mod', abs_le]	a b : ℤ h : 0 < b ⊢ \|mod' a b\| ≤ b / 2	a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 ∧ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: a b : ℤ h : 0 < b ⊢ \|mod' a b\| ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	constructor	a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 ∧ (a + b / 2) % b - b / 2 ≤ b / 2	case left a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 case right a b : ℤ h : 0 < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 ∧ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	have := Int.emod_lt_of_pos (a + b / 2) h	case right a b : ℤ h : 0 < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	case right a b : ℤ h : 0 < b this : (a + b / 2) % b < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	have := Int.ediv_add_emod b 2	case right a b : ℤ h : 0 < b this : (a + b / 2) % b < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b this : (a + b / 2) % b < b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	have := Int.emod_lt_of_pos b zero_lt_two	case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	revert this	case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ b % 2 < 2 → (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	intro this	case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ b % 2 < 2 → (a + b / 2) % b - b / 2 ≤ b / 2	case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b this✝ : (a + b / 2) % b < b this : 2 * (b / 2) + b % 2 = b ⊢ b % 2 < 2 → (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	linarith	case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right a b : ℤ h : 0 < b this✝¹ : (a + b / 2) % b < b this✝ : 2 * (b / 2) + b % 2 = b this : b % 2 < 2 ⊢ (a + b / 2) % b - b / 2 ≤ b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.abs_mod'_le	[162, 1]	[170, 11]	linarith [Int.emod_nonneg (a + b / 2) h.ne']	case left a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case left a b : ℤ h : 0 < b ⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	Int.mod'_eq	[172, 1]	[172, 91]	linarith [div'_add_mod' a b]	a b : ℤ ⊢ mod' a b = a - b * div' a b	no goals	Please generate a tactic in lean4 to solve the state. STATE: a b : ℤ ⊢ mod' a b = a - b * div' a b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	sq_add_sq_eq_zero	[176, 1]	[178, 8]	sorry	α : Type u_1 inst✝ : LinearOrderedRing α x y : α ⊢ x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝ : LinearOrderedRing α x y : α ⊢ x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_nonneg	[185, 1]	[186, 8]	sorry	x : gaussInt ⊢ 0 ≤ norm x	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : gaussInt ⊢ 0 ≤ norm x TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_eq_zero	[187, 1]	[188, 8]	sorry	x : gaussInt ⊢ norm x = 0 ↔ x = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : gaussInt ⊢ norm x = 0 ↔ x = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_pos	[189, 1]	[190, 8]	sorry	x : gaussInt ⊢ 0 < norm x ↔ x ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : gaussInt ⊢ 0 < norm x ↔ x ≠ 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mul	[191, 1]	[192, 8]	sorry	x y : gaussInt ⊢ norm (x * y) = norm x * norm y	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt ⊢ norm (x * y) = norm x * norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_conj	[204, 1]	[204, 76]	simp [norm]	x : gaussInt ⊢ norm (conj x) = norm x	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : gaussInt ⊢ norm (conj x) = norm x TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	have norm_y_pos : 0 < norm y := by rwa [norm_pos]	x y : gaussInt hy : y ≠ 0 ⊢ norm (x % y) < norm y	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ norm (x % y) < norm y	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ norm (x % y) < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ norm (x % y) < norm y	case H1 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) < norm y	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ norm (x % y) < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) < norm y	case H2 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) * norm y ≤ norm y / 2 * norm y x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm (x % y) < norm y	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	rwa [norm_pos]	x y : gaussInt hy : y ≠ 0 ⊢ 0 < norm y	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ 0 < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring	case H1 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) }	no goals	Please generate a tactic in lean4 to solve the state. STATE: case H1 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y ⊢ x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	calc norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] _ = \|Int.mod' (x.re * y.re + x.im * y.im) (norm y)\| ^ 2 + \|Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)\| ^ 2 := by simp [H1, norm, sq_abs] _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos _ = norm y / 2 * (norm y / 2 * 2) := by ring _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num	case H2 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) * norm y ≤ norm y / 2 * norm y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case H2 x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) * norm y ≤ norm y / 2 * norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	simp only [norm_mul, norm_conj]	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) * norm y = norm (x % y * conj y)	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y) * norm y = norm (x % y * conj y) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	simp [H1, norm, sq_abs]	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y * conj y) = \|Int.mod' (x.re * y.re + x.im * y.im) (norm y)\| ^ 2 + \|Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)\| ^ 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm (x % y * conj y) = \|Int.mod' (x.re * y.re + x.im * y.im) (norm y)\| ^ 2 + \|Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)\| ^ 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ \|Int.mod' (x.re * y.re + x.im * y.im) (norm y)\| ^ 2 + \|Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)\| ^ 2 ≤ (norm y / 2) ^ 2 + (norm y / 2) ^ 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ \|Int.mod' (x.re * y.re + x.im * y.im) (norm y)\| ^ 2 + \|Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)\| ^ 2 ≤ (norm y / 2) ^ 2 + (norm y / 2) ^ 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	ring	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ (norm y / 2) ^ 2 + (norm y / 2) ^ 2 = norm y / 2 * (norm y / 2 * 2)	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ (norm y / 2) ^ 2 + (norm y / 2) ^ 2 = norm y / 2 * (norm y / 2 * 2) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	gcongr	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm y / 2 * (norm y / 2 * 2) ≤ norm y / 2 * norm y	case h x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm y / 2 * 2 ≤ norm y	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm y / 2 * (norm y / 2 * 2) ≤ norm y / 2 * norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	apply Int.ediv_mul_le	case h x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm y / 2 * 2 ≤ norm y	case h.H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ 2 ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case h x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ norm y / 2 * 2 ≤ norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	norm_num	case h.H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ 2 ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } ⊢ 2 ≠ 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	apply Int.ediv_lt_of_lt_mul	x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm y / 2 < norm y	case H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ 0 < 2 case H' x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm y < norm y * 2	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm y / 2 < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	norm_num	case H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ 0 < 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case H x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ 0 < 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.norm_mod_lt	[219, 1]	[236, 19]	linarith	case H' x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm y < norm y * 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case H' x y : gaussInt hy : y ≠ 0 norm_y_pos : 0 < norm y H1 : x % y * conj y = { re := Int.mod' (x * conj y).re (norm y), im := Int.mod' (x * conj y).im (norm y) } H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y ⊢ norm y < norm y * 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.natAbs_norm_mod_lt	[241, 1]	[245, 25]	apply Int.ofNat_lt.1	x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm (x % y)) < Int.natAbs (norm y)	x y : gaussInt hy : y ≠ 0 ⊢ ↑(Int.natAbs (norm (x % y))) < ↑(Int.natAbs (norm y))	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm (x % y)) < Int.natAbs (norm y) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.natAbs_norm_mod_lt	[241, 1]	[245, 25]	simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg]	x y : gaussInt hy : y ≠ 0 ⊢ ↑(Int.natAbs (norm (x % y))) < ↑(Int.natAbs (norm y))	x y : gaussInt hy : y ≠ 0 ⊢ norm (x % y) < norm y	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ ↑(Int.natAbs (norm (x % y))) < ↑(Int.natAbs (norm y)) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.natAbs_norm_mod_lt	[241, 1]	[245, 25]	apply norm_mod_lt x hy	x y : gaussInt hy : y ≠ 0 ⊢ norm (x % y) < norm y	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ norm (x % y) < norm y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.not_norm_mul_left_lt_norm	[247, 1]	[254, 51]	apply not_lt_of_ge	x y : gaussInt hy : y ≠ 0 ⊢ ¬Int.natAbs (norm (x * y)) < Int.natAbs (norm x)	case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm (x * y)) ≥ Int.natAbs (norm x)	Please generate a tactic in lean4 to solve the state. STATE: x y : gaussInt hy : y ≠ 0 ⊢ ¬Int.natAbs (norm (x * y)) < Int.natAbs (norm x) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.not_norm_mul_left_lt_norm	[247, 1]	[254, 51]	rw [norm_mul, Int.natAbs_mul]	case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm (x * y)) ≥ Int.natAbs (norm x)	case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm x) * Int.natAbs (norm y) ≥ Int.natAbs (norm x)	Please generate a tactic in lean4 to solve the state. STATE: case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm (x * y)) ≥ Int.natAbs (norm x) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.not_norm_mul_left_lt_norm	[247, 1]	[254, 51]	apply le_mul_of_one_le_right (Nat.zero_le _)	case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm x) * Int.natAbs (norm y) ≥ Int.natAbs (norm x)	case h x y : gaussInt hy : y ≠ 0 ⊢ 1 ≤ Int.natAbs (norm y)	Please generate a tactic in lean4 to solve the state. STATE: case h x y : gaussInt hy : y ≠ 0 ⊢ Int.natAbs (norm x) * Int.natAbs (norm y) ≥ Int.natAbs (norm x) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git	7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d	LeanCourse/MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean	gaussInt.not_norm_mul_left_lt_norm	[247, 1]	[254, 51]	apply Int.ofNat_le.1	case h x y : gaussInt hy : y ≠ 0 ⊢ 1 ≤ Int.natAbs (norm y)	case h x y : gaussInt hy : y ≠ 0 ⊢ ↑1 ≤ ↑(Int.natAbs (norm y))	Please generate a tactic in lean4 to solve the state. STATE: case h x y : gaussInt hy : y ≠ 0 ⊢ 1 ≤ Int.natAbs (norm y) TACTIC:

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