Documentation

Foundation.FirstOrder.Arith.CobhamR0

theorem LO.Arith.lt_numeral_iff {M : Type u_1} [ORingStruc M] [M βŠ§β‚˜* 𝐑₀] {x : M} {n : β„•} :
x < ORingStruc.numeral n ↔ βˆƒ (i : Fin n), x = ORingStruc.numeral ↑i
theorem LO.Arith.val_numeral {M : Type u_1} [ORingStruc M] [M βŠ§β‚˜* 𝐑₀] {ΞΎ : Type u_2} {n : β„•} (t : FirstOrder.Semiterm β„’β‚’α΅£ ΞΎ n) (v : Fin n β†’ β„•) (f : ΞΎ β†’ β„•) :
theorem LO.Arith.bold_sigma_one_completeness {M : Type u_1} [ORingStruc M] [M βŠ§β‚˜* 𝐑₀] {ΞΎ : Type u_2} {n : β„•} {Ο† : FirstOrder.Semiformula β„’β‚’α΅£ ΞΎ n} (hp : FirstOrder.Arith.Hierarchy 𝚺 1 Ο†) {e : Fin n β†’ β„•} {f : ΞΎ β†’ β„•} :
(FirstOrder.Semiformula.Evalm β„• e f) Ο† β†’ (FirstOrder.Semiformula.Evalm M (fun (x : Fin n) => ORingStruc.numeral (e x)) fun (x : ΞΎ) => ORingStruc.numeral (f x)) Ο†