Supporting results for pointwise provable equality.

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.term_primrec {ξ : Type u_1} {k : ℕ} {f : ξ → ℕ} (t : ArithmeticSemiterm ξ k) : Primrec fun (v : List.Vector ℕ k) => Semiterm.val v.get f t

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.primrecPred_ball {ξ : Type u_1} (ε : ξ → ℕ) {n : ℕ} (t : ArithmeticSemiterm ξ n) {ψ : ArithmeticSemiformula ξ (n + 1)} (h : PrimrecPred fun (v : List.Vector ℕ (n + 1)) => (Semiformula.Eval v.get ε) ψ) : PrimrecPred fun (v : List.Vector ℕ n) => ∀ x < Semiterm.val v.get ε t, (Semiformula.Eval (x :> v.get) ε) ψ

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.primrecPred_bexs {ξ : Type u_1} (ε : ξ → ℕ) {n : ℕ} (t : ArithmeticSemiterm ξ n) {ψ : ArithmeticSemiformula ξ (n + 1)} (h : PrimrecPred fun (v : List.Vector ℕ (n + 1)) => (Semiformula.Eval v.get ε) ψ) : PrimrecPred fun (v : List.Vector ℕ n) => ∃ x < Semiterm.val v.get ε t, (Semiformula.Eval (x :> v.get) ε) ψ

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.delta0_primrec {ξ : Type u_1} (ε : ξ → ℕ) {k : ℕ} {φ : ArithmeticSemiformula ξ k} (hp : Hierarchy Σ 0 φ) : PrimrecPred fun (v : List.Vector ℕ k) => (Semiformula.Eval v.get ε) φ

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.sigma1_re {ξ : Type u_1} (ε : ξ → ℕ) {k : ℕ} {φ : ArithmeticSemiformula ξ k} (hp : Hierarchy Σ 1 φ) : REPred fun (v : List.Vector ℕ k) => (Semiformula.Eval v.get ε) φ

def PointwiseComposition.FFL.FirstOrder.Arithmetic.codeAux {k : ℕ} : Nat.ArithPart₁.Code k → ArithmeticFormula (Fin (k + 1))

The code Aux construction in FFL.FirstOrder.Arithmetic.

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• PointwiseComposition.FFL.FirstOrder.Arithmetic.codeAux (PointwiseComposition.Nat.ArithPart₁.Code.zero k) = (“!!(PointwiseComposition.FFL.FirstOrder.Semiterm.fvar 0) = 0”)
• PointwiseComposition.FFL.FirstOrder.Arithmetic.codeAux (PointwiseComposition.Nat.ArithPart₁.Code.one k) = (“!!(PointwiseComposition.FFL.FirstOrder.Semiterm.fvar 0) = 1”)

def PointwiseComposition.FFL.FirstOrder.Arithmetic.code {k : ℕ} (c : Nat.ArithPart₁.Code k) : ArithmeticSemisentence (k + 1)

The code construction in FFL.FirstOrder.Arithmetic.

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.code_sigma_one {k : ℕ} (c : Nat.ArithPart₁.Code k) : Hierarchy Σ 1 (code c)

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.natCast_nat' (n : ℕ) : ↑n = n

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.models_code {k : ℕ} {c : Nat.ArithPart₁.Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) : (Semiformula.Evalb (y :> v)) (code c) ↔ y ∈ f (List.Vector.ofFn v)

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfPartrec' {k : ℕ} (f : List.Vector ℕ k →. ℕ) : ArithmeticSemisentence (k + 1)

The code Of Partrec' construction in FFL.FirstOrder.Arithmetic.

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfPartrec'_spec {k : ℕ} {f : List.Vector ℕ k →. ℕ} (hf : Nat.Partrec' f) {y : ℕ} {v : Fin k → ℕ} : (Semiformula.Evalb (y :> v)) (codeOfPartrec' f) ↔ y ∈ f (List.Vector.ofFn v)

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.code_uniq {M : Type u_1} [ORingStructure M] [M↓[ℒₒᵣ] ⊧* 𝐏𝐀⁻] {k : ℕ} {c : Nat.ArithPart₁.Code k} {v : Fin k → M} {z z' : M} : (Semiformula.Evalb (z :> v)) (code c) → (Semiformula.Evalb (z' :> v)) (code c) → z = z'

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfREPred (p : ℕ → Prop) : ArithmeticSemisentence 1

The code Of REPred construction in FFL.FirstOrder.Arithmetic.

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfREPred_spec {p : ℕ → Prop} (hp : REPred p) {x : ℕ} : (Semiformula.Evalb ![x]) (codeOfREPred p) ↔ p x

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.rePred_weak_representation {T : ArithmeticTheory} [𝐑₀ ≼ T] [T.SoundOnHierarchy Σ 1] {p : ℕ → Prop} (hp : REPred p) {x : ℕ} : p x ↔ T ⊢ (codeOfREPred p)/[↑x]

Weak representation of a r.e. predicate

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfComputablePred (p : ℕ → Prop) : ArithmeticSemisentence 1

The code Of Computable Pred construction in FFL.FirstOrder.Arithmetic.

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfComputablePred_sigma1 (p : ℕ → Prop) : Hierarchy Σ 1 (codeOfComputablePred p)

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfComputablePred_provable {p : ℕ → Prop} {T : ArithmeticTheory} {x : ℕ} [𝐑₀ ≼ T] (hp : ComputablePred p) (h : p x) : T ⊢ (codeOfComputablePred p)/[↑x]

Positive representation of a computable predicate.

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.codeOfComputablePred_provable_neg {p : ℕ → Prop} {T : ArithmeticTheory} {x : ℕ} [𝐏𝐀⁻ ≼ T] (hp : ComputablePred p) (h : ¬p x) : T ⊢ ∼(codeOfComputablePred p)/[↑x]

Negative representation of a computable predicate.

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.rePred_iff_sigma1 {p : ℕ → Prop} : REPred p ↔ Σ₁-Predicate p

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.computablePred_iff_delta1 {p : ℕ → Prop} : ComputablePred p ↔ ∆₁-Predicate p

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.computable_iff_sigma1 {f : ℕ → ℕ} : Computable f ↔ Σ₁-Function₁ f

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.computable₂_iff_sigma1 {f : ℕ → ℕ → ℕ} : Computable₂ f ↔ Σ₁-Function₂ f