succ_ne_zero

formal source

  90  intro h; cases h
  91
  92/-- **Peano P1, contrapositive**: every successor differs from zero. -/
  93theorem succ_ne_zero (n : LogicNat) : succ n ≠ zero := by
  94  intro h; cases h
  95
  96/-- **Peano P2 (successor injectivity)**: forced by the constructor
  97disjointness of the inductive type, which itself reflects the
  98injectivity of multiplication by the generator on the orbit. -/
  99theorem succ_injective : Function.Injective succ := by
 100  intro a b h
 101  cases h
 102  rfl
 103
 104/-- **Peano P3 (induction)**: any property closed under successor and
 105holding at zero holds for every `LogicNat`. -/
 106theorem induction
 107    {motive : LogicNat → Prop}
 108    (h0 : motive zero)
 109    (hs : ∀ n, motive n → motive (succ n)) :
 110    ∀ n, motive n := by
 111  intro n
 112  induction n with
 113  | identity => exact h0
 114  | step n ih => exact hs n ih
 115
 116/-! ## 4. Addition and Multiplication
 117
 118Addition is repeated successor; multiplication is repeated addition.
 119Both are defined by recursion on the second argument. We register
 120them as `Add` and `Mul` instances so Lean's standard `+` and `*`
 121notation work on `LogicNat` directly. -/
 122
 123/-- Addition by recursion on the second argument. -/