zero

formal source

  74
  75/-- Zero is the identity comparison: a thing compared with itself,
  76costing zero in J. -/
  77@[simp] def zero : LogicNat := .identity
  78
  79/-- Successor is one more application of the generator. -/
  80@[simp] def succ (n : LogicNat) : LogicNat := .step n
  81
  82/-! ## 3. Peano Axioms as Theorems
  83
  84Each axiom is a theorem of the inductive structure. None is posited.
  85-/
  86
  87/-- **Peano P1 (zero is not a successor)**: the identity is
  88distinguishable from any iterate of the generator. -/
  89theorem zero_ne_succ (n : LogicNat) : zero ≠ succ n := by
  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}