Documentation

Foundation.FirstOrder.Arith.Representation

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theorem Part.unit_dom_iff (x : Part Unit) :
x.Dom ↔ () ∈ x
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theorem Mathlib.Vector.cons_get {α : Type u_1} {k : ℕ} (a : α) (v : Mathlib.Vector α k) :
(a ::ᵥ v).get = a :> v.get
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theorem Nat.Partrec.projection {f : ℕ →. ℕ} (hf : Nat.Partrec f) (unif : ∀ {m n₁ n₂ a₁ a₂ : ℕ}, a₁ ∈ f (Nat.pair m n₁) → a₂ ∈ f (Nat.pair m n₂) → a₁ = a₂) :
∃ (g : ℕ →. ℕ), Nat.Partrec g ∧ ∀ (a m : ℕ), a ∈ g m ↔ ∃ (z : ℕ), a ∈ f (Nat.pair m z)
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theorem Partrec.projection {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Primcodable α] [Primcodable β] [Primcodable γ] {f : α → β →. γ} (hf : Partrec₂ f) (unif : ∀ {a : α} {b₁ b₂ : β} {c₁ c₂ : γ}, c₁ ∈ f a b₁ → c₂ ∈ f a b₂ → c₁ = c₂) :
∃ (g : α →. γ), Partrec g ∧ ∀ (c : γ) (a : α), c ∈ g a ↔ ∃ (b : β), c ∈ f a b
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@[simp]
theorem RePred.const {α : Type u_1} [Primcodable α] (p : Prop) :
RePred fun (x : α) => p
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theorem RePred.iff {α : Type u_1} [Primcodable α] {p : α → Prop} :
RePred p ↔ ∃ (f : α →. Unit), Partrec f ∧ p = fun (x : α) => (f x).Dom
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theorem RePred.iff' {α : Type u_1} [Primcodable α] {p : α → Prop} :
RePred p ↔ ∃ (f : α →. Unit), Partrec f ∧ ∀ (x : α), p x ↔ (f x).Dom
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theorem RePred.and {α : Type u_1} [Primcodable α] {p : α → Prop} {q : α → Prop} (hp : RePred p) (hq : RePred q) :
RePred fun (x : α) => p x ∧ q x
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theorem RePred.or {α : Type u_1} [Primcodable α] {p : α → Prop} {q : α → Prop} (hp : RePred p) (hq : RePred q) :
RePred fun (x : α) => p x ∨ q x
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theorem RePred.projection {α : Type u_1} {β : Type u_2} [Primcodable α] [Primcodable β] {p : α × β → Prop} (hp : RePred p) :
RePred fun (x : α) => ∃ (y : β), p (x, y)
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theorem RePred.comp {α : Type u_1} {β : Type u_2} [Primcodable α] [Primcodable β] {f : α → β} (hf : Computable f) {p : β → Prop} (hp : RePred p) :
RePred fun (x : α) => p (f x)
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theorem LO.FirstOrder.Arith.term_primrec {ξ : Type u_1} {k : ℕ} {f : ξ → ℕ} (t : LO.FirstOrder.Semiterm ℒₒᵣ ξ k) :
Primrec fun (v : Mathlib.Vector ℕ k) => LO.FirstOrder.Semiterm.valm ℕ v.get f t
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theorem LO.FirstOrder.Arith.sigma1_re {ξ : Type u_1} (ε : ξ → ℕ) {k : ℕ} {p : LO.FirstOrder.Semiformula ℒₒᵣ ξ k} (hp : LO.FirstOrder.Arith.Hierarchy 𝚺 1 p) :
RePred fun (v : Mathlib.Vector ℕ k) => (LO.FirstOrder.Semiformula.Evalm ℕ v.get ε) p
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def LO.FirstOrder.Arith.codeAux {k : ℕ} :
Nat.ArithPart₁.Code k → LO.FirstOrder.Formula ℒₒᵣ (Fin (k + 1))
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    def LO.FirstOrder.Arith.code {k : ℕ} (c : Nat.ArithPart₁.Code k) :
    LO.FirstOrder.Semisentence ℒₒᵣ (k + 1)
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      @[simp]
      theorem LO.FirstOrder.Arith.code_sigma_one {k : ℕ} (c : Nat.ArithPart₁.Code k) :
      LO.FirstOrder.Arith.Hierarchy 𝚺 1 (LO.FirstOrder.Arith.code c)
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      @[simp]
      theorem LO.FirstOrder.Arith.natCast_nat (n : ℕ) :
      ↑n = n
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      theorem LO.FirstOrder.Arith.models_code {k : ℕ} {c : Nat.ArithPart₁.Code k} {f : Mathlib.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :
      ℕ ⊧/(y :> v) (LO.FirstOrder.Arith.code c) ↔ y ∈ f (Mathlib.Vector.ofFn v)
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      noncomputable def LO.FirstOrder.Arith.codeOfPartrec' {k : ℕ} (f : Mathlib.Vector ℕ k →. ℕ) :
      LO.FirstOrder.Semisentence ℒₒᵣ (k + 1)
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        theorem LO.FirstOrder.Arith.codeOfPartrec'_spec {k : ℕ} {f : Mathlib.Vector ℕ k →. ℕ} (hf : Nat.Partrec' f) {y : ℕ} {v : Fin k → ℕ} :
        ℕ ⊧/(y :> v) (LO.FirstOrder.Arith.codeOfPartrec' f) ↔ y ∈ f (Mathlib.Vector.ofFn v)
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        noncomputable def LO.FirstOrder.Arith.codeOfRePred (p : ℕ → Prop) :
        LO.FirstOrder.Semisentence ℒₒᵣ 1
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          theorem LO.FirstOrder.Arith.codeOfRePred_spec {p : ℕ → Prop} (hp : RePred p) {x : ℕ} :
          ℕ ⊧/![x] (LO.FirstOrder.Arith.codeOfRePred p) ↔ p x
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          theorem LO.FirstOrder.Arith.re_complete {T : LO.FirstOrder.Theory ℒₒᵣ} [𝐑₀ ≼ T] [LO.FirstOrder.Arith.Sigma1Sound T] {p : ℕ → Prop} (hp : RePred p) {x : ℕ} :
          p x ↔ T ⊢! ↑((LO.FirstOrder.Rew.substs ![LO.FirstOrder.Semiterm.Operator.const (LO.FirstOrder.Semiterm.Operator.numeral ℒₒᵣ x)]).hom (LO.FirstOrder.Arith.codeOfRePred p))