`ordered_arithmetic_invariant`

definition

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module: IndisputableMonolith.Foundation.OrderedLogicRealization
domain: Foundation
line: 63 · github
papers citing: none yet

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IndisputableMonolith.Foundation.OrderedLogicRealization on GitHub at line 63.

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formal source

  60    simp [natCost]
  61
  62/-- Ordered arithmetic is invariant with every realization. -/
  63noncomputable def ordered_arithmetic_invariant (R : LogicRealization.{0, 0}) :
  64    (UniversalForcing.arithmeticOf natOrderedRealization).peano.carrier
  65      ≃ (UniversalForcing.arithmeticOf R).peano.carrier :=
  66  ArithmeticOf.equivOfInitial
  67    (UniversalForcing.arithmeticOf natOrderedRealization)
  68    (UniversalForcing.arithmeticOf R)
  69
  70/-- The ordered realization interprets arithmetic faithfully. -/
  71theorem ordered_faithful :
  72    LogicRealization.FaithfulArithmeticInterpretation natOrderedRealization where
  73  injective := by
  74    intro a b h
  75    exact (ArithmeticFromLogic.LogicNat.eq_iff_toNat_eq).mpr h
  76  zero_step_noncollapse := by
  77    intro n h
  78    have hnat := congrArg id h
  79    simp [natOrderedRealization] at hnat
  80    exact Nat.succ_ne_zero _ hnat.symm
  81
  82/-- Order on the carrier matches the recovered Peano order. -/
  83theorem ordered_interpret_le_iff (a b : ArithmeticFromLogic.LogicNat) :
  84    ArithmeticFromLogic.LogicNat.toNat a ≤ ArithmeticFromLogic.LogicNat.toNat b ↔ a ≤ b := by
  85  exact (ArithmeticFromLogic.LogicNat.toNat_le a b).symm
  86
  87end OrderedLogicRealization
  88end Foundation
  89end IndisputableMonolith

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