  34  LogicNat.toNat n
  35
  36/-- Musical realization as interval-step comparison. -/
  37def musicRealization : LogicRealization where
  38  Carrier := MusicalIntervalStep
  39  Cost := Nat
  40  zeroCost := inferInstance
  41  compare := musicCost
  42  zero := 0
  43  step := Nat.succ
  44  Orbit := LogicNat
  45  orbitZero := LogicNat.zero
  46  orbitStep := LogicNat.succ
  47  interpret := musicInterpret
  48  interpret_zero := by rfl
  49  interpret_step := by
  50    intro n
  51    show LogicNat.toNat (LogicNat.succ n) = Nat.succ (LogicNat.toNat n)
  52    rfl
  53  orbit_no_confusion := by intro n h; exact LogicNat.zero_ne_succ n h
  54  orbit_step_injective := LogicNat.succ_injective
  55  orbit_induction := by
  56    intro P h0 hs n
  57    exact LogicNat.induction (motive := P) h0 hs n
  58  orbitEquivLogicNat := Equiv.refl LogicNat
  59  orbitEquiv_zero := rfl
  60  orbitEquiv_step := by intro n; rfl
  61  identity := musicCost_self
  62  nonContradiction := musicCost_symm
  63  excludedMiddle := True
  64  composition := True
  65  actionInvariant := True
  66  nontrivial := by
  67    refine ⟨1, ?_⟩

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