`natCost`

definition

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module: IndisputableMonolith.Foundation.OrderedLogicRealization
domain: Foundation
line: 14 · github
papers citing: none yet

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IndisputableMonolith.Foundation.OrderedLogicRealization on GitHub at line 14.

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formal source

  11namespace OrderedLogicRealization
  12
  13/-- Equality cost on `Nat`. -/
  14def natCost (m n : Nat) : Nat :=
  15  if m = n then 0 else 1
  16
  17@[simp] theorem natCost_self (n : Nat) : natCost n n = 0 := by
  18  simp [natCost]
  19
  20theorem natCost_symm (m n : Nat) : natCost m n = natCost n m := by
  21  by_cases h : m = n
  22  · subst h
  23    simp [natCost]
  24  · have h' : n ≠ m := by intro hnm; exact h hnm.symm
  25    simp [natCost, h, h']
  26
  27/-- The ordered natural-number realization. -/
  28def natOrderedRealization : LogicRealization where
  29  Carrier := Nat
  30  Cost := Nat
  31  zeroCost := inferInstance
  32  compare := natCost
  33  zero := 0
  34  step := Nat.succ
  35  Orbit := ArithmeticFromLogic.LogicNat
  36  orbitZero := ArithmeticFromLogic.LogicNat.zero
  37  orbitStep := ArithmeticFromLogic.LogicNat.succ
  38  interpret := ArithmeticFromLogic.LogicNat.toNat
  39  interpret_zero := ArithmeticFromLogic.LogicNat.toNat_zero
  40  interpret_step := by
  41    intro n
  42    exact ArithmeticFromLogic.LogicNat.toNat_succ n
  43  orbit_no_confusion := by
  44    intro n h

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