Update Thu Oct 3 23:54:58 EDT 2024

Math2001/Homework/hw5.lean

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+/- Copyright (c) Heather Macbeth, 2023-4.  All rights reserved. -/
+import Mathlib.Data.Real.Basic
+import Library.Basic
+import Library.Tactic.ModEq
+import AutograderLib
+
+math2001_init
+
+/-! # Homework 5
+
+Don't forget to compare with the text version,
+https://github.com/hrmacbeth/math2001/wiki/Homework-5,
+for clearer statements and any special instructions. -/
+
+@[autograded 2]
+theorem problem1 : ∃ k : ℤ, k > 10 ∧ 3 * k ≡ 2 [ZMOD 5] ∧ k ∣ 72 := by
+  sorry
+
+@[autograded 3]
+theorem problem2 {a : ℤ} (ha : a ≡ 4 [ZMOD 5]) :
+    a ^ 3 + 2 * a ^ 2 + 3 ≡ 4 [ZMOD 5] := by
+  sorry
+
+@[autograded 5]
+theorem problem3 {x : ℤ} : x ^ 5 ≡ x [ZMOD 5] := by
+  sorry
+
+@[autograded 3]
+theorem problem4 {a : ℚ} (h : ∀ b : ℚ, a + b ^ 2 ≥ 0) : a ≥ 0 := by
+  sorry
+
+@[autograded 5]
+theorem problem5 (n : ℕ) (h : ∀ a : ℕ, 6 ≤ a → a ≤ 10 → a ∣ n) :
+    ∀ b : ℕ, 1 ≤ b → b ≤ 5 → b ∣ n := by
+  sorry
+
+@[autograded 3]
+theorem problem6 : ∃ a : ℝ, ∀ b : ℝ, a ≤ b ^ 2 := by
+  sorry
+
+@[autograded 4]
+theorem problem7 : forall_sufficiently_large x : ℝ, x ^ 3 - 5 * x ≥ 11 * x ^ 2 := by
+  dsimp
+  sorry