`LogicRealization`

definition

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module: IndisputableMonolith.Foundation.LogicRealization
domain: Foundation
line: 32 · github
papers citing: 1 paper (below)

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open lean source

IndisputableMonolith.Foundation.LogicRealization on GitHub at line 32.

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formal source

  29order, category, or discrete structure through the propositions carried here.
  30The invariant target is not the ambient carrier; it is the arithmetic object
  31extracted from the identity/step data. -/
  32structure LogicRealization where
  33  Carrier : Type u
  34  Cost : Type v
  35  zeroCost : Zero Cost
  36  compare : Carrier → Carrier → Cost
  37  zero : Carrier
  38  step : Carrier → Carrier
  39  Orbit : Type u
  40  orbitZero : Orbit
  41  orbitStep : Orbit → Orbit
  42  interpret : Orbit → Carrier
  43  interpret_zero : interpret orbitZero = zero
  44  interpret_step : ∀ n : Orbit, interpret (orbitStep n) = step (interpret n)
  45  orbit_no_confusion : ∀ n : Orbit, orbitZero ≠ orbitStep n
  46  orbit_step_injective : Function.Injective orbitStep
  47  orbit_induction :
  48    ∀ P : Orbit → Prop,
  49      P orbitZero →
  50      (∀ n, P n → P (orbitStep n)) →
  51      ∀ n, P n
  52  orbitEquivLogicNat : Orbit ≃ LogicNat
  53  orbitEquiv_zero : orbitEquivLogicNat orbitZero = LogicNat.zero
  54  orbitEquiv_step : ∀ n : Orbit,
  55    orbitEquivLogicNat (orbitStep n) = LogicNat.succ (orbitEquivLogicNat n)
  56  identity : ∀ x : Carrier, compare x x = 0
  57  nonContradiction : ∀ x y : Carrier, compare x y = compare y x
  58  excludedMiddle : Prop
  59  composition : Prop
  60  actionInvariant : Prop
  61  nontrivial : ∃ x : Carrier, compare x zero ≠ 0
  62

papers checked against this theorem (showing 1 of 1)

TTD homomorphism yields low-memory split-beam dictionary
echoes · 2605.06930

"Corollary 1. ∀Φ1, Φ2 ∈ V1, PΦ1+Φ2 = PΦ1 ⋆ PΦ2; <PT, ⋆> defines a group structure"