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)

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

@[simp]

theorem REPred.const {α : Type u_1} [Primcodable α] (p : Prop) : REPred fun (x : α) => p

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

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

theorem REPred.and {α : Type u_1} [Primcodable α] {p q : α → Prop} (hp : REPred p) (hq : REPred q) : REPred fun (x : α) => p x ∧ q x

theorem REPred.or {α : Type u_1} [Primcodable α] {p q : α → Prop} (hp : REPred p) (hq : REPred q) : REPred fun (x : α) => p x ∨ q x

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)

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)

@[simp]

theorem ComputablePred.const {α : Type u_1} [Primcodable α] (p : Prop) : ComputablePred fun (x : α) => p

theorem ComputablePred.and {α : Type u_1} [Primcodable α] {p q : α → Prop} : ComputablePred p → ComputablePred q → ComputablePred fun (x : α) => p x ∧ q x

theorem ComputablePred.or {α : Type u_1} [Primcodable α] {p q : α → Prop} : ComputablePred p → ComputablePred q → ComputablePred fun (x : α) => p x ∨ q x

theorem ComputablePred.of_graph_rePred {α : Type u_1} [Primcodable α] {g : α → ℕ} (h : REPred fun (p : α × ℕ) => p.2 = g p.1) : Computable g

theorem Primrec.nat_natToList : Primrec Nat.natToList