 133@[simp] theorem zero_def : (0 : LogicNat) = zero := rfl
 134@[simp] theorem one_def : (1 : LogicNat) = succ zero := rfl
 135
 136@[simp] theorem add_zero (n : LogicNat) : n + zero = n := rfl
 137
 138@[simp] theorem add_succ (n m : LogicNat) : n + succ m = succ (n + m) := rfl
 139
 140theorem zero_add (n : LogicNat) : zero + n = n := by
 141  induction n with
 142  | identity => rfl
 143  | step n ih =>
 144    show zero + succ n = succ n
 145    rw [add_succ, ih]
 146
 147theorem succ_add (n m : LogicNat) : succ n + m = succ (n + m) := by
 148  induction m with
 149  | identity => rfl
 150  | step m ih =>
 151    show succ n + succ m = succ (n + succ m)
 152    rw [add_succ, add_succ, ih]
 153
 154theorem add_assoc (a b c : LogicNat) : (a + b) + c = a + (b + c) := by
 155  induction c with
 156  | identity => rfl
 157  | step c ih =>
 158    show (a + b) + succ c = a + (b + succ c)
 159    rw [add_succ, add_succ, add_succ, ih]
 160
 161theorem add_comm (a b : LogicNat) : a + b = b + a := by
 162  induction b with
 163  | identity =>
 164    show a + zero = zero + a
 165    rw [add_zero, zero_add]
 166  | step b ih =>

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