Update Fri Nov 1 00:41:37 EDT 2024

Math2001/Homework/hw7.lean

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+/- Copyright (c) Heather Macbeth, 2023.  All rights reserved. -/
+import Mathlib.Data.Real.Basic
+import Library.Basic
+import Library.Tactic.ModEq
+import AutograderLib
+
+math2001_init
+
+
+/-! # Homework 7
+
+Don't forget to compare with the text version,
+https://github.com/hrmacbeth/math2001/wiki/Homework-7,
+for clearer statements and any special instructions. -/
+
+
+@[autograded 5]
+theorem problem1 (P Q : Prop) : ¬ (P → Q) ↔ (P ∧ ¬ Q) := by
+  sorry
+
+@[autograded 3]
+theorem problem2 : ¬ (∀ x : ℚ, 2 * x ^ 2 ≥ x) := by
+  push_neg
+  sorry
+
+@[autograded 4]
+theorem problem3 (n : ℕ) : 6 ^ n ≡ 1 [ZMOD 7] ∨ 6 ^ n ≡ 6 [ZMOD 7] := by
+  sorry
+
+@[autograded 4]
+theorem problem4 (n : ℕ) :
+    4 ^ n ≡ 1 [ZMOD 7] ∨ 4 ^ n ≡ 2 [ZMOD 7] ∨ 4 ^ n ≡ 4 [ZMOD 7] := by
+  sorry
+
+@[autograded 5]
+theorem problem5 {a : ℝ} (ha : -1 ≤ a) : ¬ ∃ n : ℕ, (1 + a) ^ n < 1 + n * a := by
+  push_neg
+  sorry
+
+@[autograded 4]
+theorem problem6 : forall_sufficiently_large n : ℕ, (3:ℤ) ^ n ≥ 2 ^ n + 100 := by
+  dsimp
+  sorry