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def definition def or abbrev

modularRealization

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formal statement (Lean)

  67def modularRealization (k : ℕ) : LogicRealization where
  68  Carrier := Fin (modulus k)

proof body

Definition body.

  69  Cost := Nat
  70  zeroCost := inferInstance
  71  compare := finCost
  72  zero := ⟨0, modulus_pos k⟩
  73  step := cycStep k
  74  Orbit := ArithmeticFromLogic.LogicNat
  75  orbitZero := ArithmeticFromLogic.LogicNat.zero
  76  orbitStep := ArithmeticFromLogic.LogicNat.succ
  77  interpret := modularInterpret k
  78  interpret_zero := modularInterpret_zero k
  79  interpret_step := modularInterpret_step k
  80  orbit_no_confusion := by
  81    intro n h
  82    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n h
  83  orbit_step_injective := ArithmeticFromLogic.LogicNat.succ_injective
  84  orbit_induction := by
  85    intro P h0 hs n
  86    exact ArithmeticFromLogic.LogicNat.induction (motive := P) h0 hs n
  87  orbitEquivLogicNat := Equiv.refl ArithmeticFromLogic.LogicNat
  88  orbitEquiv_zero := rfl
  89  orbitEquiv_step := by intro n; rfl
  90  identity := finCost_self
  91  nonContradiction := finCost_symm
  92  excludedMiddle := True
  93  composition := True
  94  actionInvariant := True
  95  nontrivial := by
  96    refine ⟨⟨1, one_lt_modulus k⟩, ?_⟩
  97    simp [finCost]
  98
  99/-- Modular realization has invariant extracted arithmetic. -/

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