def foo (_ : Nat) : Int := 10

example : foo x = 8 + 2 := by
  simp
  trace_state
  rw [foo]

example : foo x = 12 + (- 2) := by
  simp
  trace_state
  rw [foo]

example : foo x = 5 * 2 := by
  simp
  trace_state
  rw [foo]

example : foo x = 12 - 2 := by
  simp
  trace_state
  rw [foo]

example : foo x = 20 / 2 := by
  simp
  trace_state
  rw [foo]

example : foo x = 110 % 100 := by
  simp
  trace_state
  rw [foo]

example : foo x = (3+2)*2 := by
  simp
  trace_state
  rw [foo]

example : foo x * foo x = 10 ^ 2 := by
  simp
  trace_state
  simp [foo]

example : foo x = - (- 10) := by
  simp
  trace_state
  simp [foo]

example : Int.natAbs (foo x) = Int.natAbs (- (8 + 2)) := by
  simp
  trace_state
  simp [foo]

def boo (_ : Nat) := True

example : boo x ↔ (2 : Int) < 3 := by
  simp
  trace_state
  trivial

example : boo x ↔ (3 : Int) > 2 := by
  simp
  trace_state
  trivial

example : boo x ↔ (2 : Int) ≤ 3 := by
  simp
  trace_state
  trivial

example : boo x ↔ (3 : Int) ≥ 2 := by
  simp
  trace_state
  trivial

example (h : x = false) : x = ((3 : Int) == 4) := by
  simp
  trace_state
  assumption

example (h : x = true) : x = ((3 : Int) != 4) := by
  simp
  trace_state
  assumption

example (h : ¬x) : x = ((3 : Int) = 4) := by
  simp
  trace_state
  assumption

example (h : x) : x = ((3 : Int) ≠ 4) := by
  simp
  trace_state
  assumption