feat(Computability): define oracle computability and Turing degrees

Mathlib/Computability/TuringDegrees.lean

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+/-
+Copyright (c) 2024 Tanner Duve. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Tanner Duve, Elan Roth
+-/
+import Mathlib.Computability.Primrec
+import Mathlib.Computability.Partrec
+import Mathlib.Computability.Reduce
+import Mathlib.Data.Part
+import Mathlib.Order.Antisymmetrization
+/-!
+# Oracle Computability and Turing Degrees
+
+This file defines a model of oracle computability, introduces Turing reducibility and equivalence,
+proves that Turing equivalence is an equivalence relation, and defines Turing degrees as the
+quotient under this relation.
+
+## Main Definitions
+
+- `RecursiveIn g f`:
+  An inductive definition representing that a partial function `f` is recursive in oracle `g`.
+- `turing_equivalent`: A relation defining Turing equivalence between partial functions.
+- `TuringDegree`:
+  The type of Turing degrees, defined as equivalence classes under `turing_equivalent`.
+- `join`: Combines two partial functions into one by =
+  mapping even and odd numbers to respective functions.
+
+## Notation
+
+- `f ≤ᵀ g` : `f` is Turing reducible to `g`.
+- `f ≡ᵀ g` : `f` is Turing equivalent to `g`.
+- `f ⊕ g` : The join of partial functions `f` and `g`.
+
+## Implementation Notes
+
+The type of partial functions recursive in an oracle `g` is the smallest type containing
+the constant zero, the successor, projections, the oracle `g`, and is closed under
+pairing, composition, primitive recursion, and μ-recursion. The `join` operation
+combines two partial functions into a single partial function by mapping even and odd
+numbers to the respective functions.
+
+## References
+
+* [Carneiro2018] Carneiro, Mario.
+  *Formalizing Computability Theory via Partial Recursive Functions*.
+  arXiv preprint arXiv:1810.08380, 2018.
+* [Odifreddi1989] Odifreddi, Piergiorgio.
+  *Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers,
+  Vol. I*. Springer-Verlag, 1989.
+* [Soare1987] Soare, Robert I. *Recursively Enumerable Sets and Degrees*. Springer-Verlag, 1987.
+* [Gu2015] Gu, Yi-Zhi. *Turing Degrees*. Institute for Advanced Study, 2015.
+
+## Tags
+
+Computability, Oracle, Turing Degrees, Reducibility, Equivalence Relation
+-/
+
+open Primrec Nat.Partrec Part
+
+/--
+The type of partial functions `f` that are recursive in an oracle `g` is the smallest type
+containing the constant zero, the successor, left and right projections, the oracle `g`,
+and is closed under pairing, composition, primitive recursion, and μ-recursion.
+
+Equivalently one can say that `f` is turing reducible to `g` when `f` is recursive in `g`.
+-/
+inductive RecursiveIn : (ℕ →. ℕ) → (ℕ →. ℕ) → Prop
+  | zero {g} : RecursiveIn (fun _ => 0) g
+  | succ {g} : RecursiveIn Nat.succ g
+  | left {g} : RecursiveIn (fun n => (Nat.unpair n).1) g
+  | right {g} : RecursiveIn (fun n => (Nat.unpair n).2) g
+  | oracle {g} : RecursiveIn g g
+  | pair {f h g : ℕ →. ℕ} (hf : RecursiveIn f g) (hh : RecursiveIn h g) :
+      RecursiveIn (fun n => (Nat.pair <$> f n <*> h n)) g
+  | comp {f h g : ℕ →. ℕ} (hf : RecursiveIn f g) (hh : RecursiveIn h g) :
+      RecursiveIn (fun n => h n >>= f) g
+  | prec {f h g : ℕ →. ℕ} (hf : RecursiveIn f g) (hh : RecursiveIn h g) :
+      RecursiveIn (fun p =>
+        let (a, n) := Nat.unpair p
+        n.rec (f a) (fun y ih => do
+          let i ← ih
+          h (Nat.pair a (Nat.pair y i)))) g
+  | rfind {f g : ℕ →. ℕ} (hf : RecursiveIn f g) :
+      RecursiveIn (fun a =>
+        Nat.rfind (fun n => (fun m => m = 0) <$> f (Nat.pair a n))) g
+
+
+/--
+`f` is Turing equivalent to `g` if `f` is reducible to `g` and `g` is reducible to `f`.
+-/
+def turing_equivalent (f g : ℕ →. ℕ) : Prop :=
+  AntisymmRel RecursiveIn f g
+
+/--
+Custom infix notation for `turing_equivalent`.
+-/
+infix:50 " ≡ᵀ " => turing_equivalent
+
+/--
+If a function is partial recursive, then it is recursive in every partial function.
+-/
+lemma partrec_implies_recursive_in_everything
+  (f : ℕ →. ℕ) : Nat.Partrec f → (∀ g, RecursiveIn f g) := by
+    intro pF
+    intro g
+    induction pF
+    case zero =>
+      apply RecursiveIn.zero
+    case succ =>
+      apply RecursiveIn.succ
+    case left =>
+      apply RecursiveIn.left
+    case right =>
+      apply RecursiveIn.right
+    case pair _ _ _ _ ih1 ih2 =>
+      apply RecursiveIn.pair ih1 ih2
+    case comp _ _ _ _ ih1 ih2 =>
+      apply RecursiveIn.comp ih1 ih2
+    case prec _ _ _ _ ih1 ih2 =>
+      apply RecursiveIn.prec ih1 ih2
+    case rfind _ _ ih =>
+      apply RecursiveIn.rfind ih
+
+/--
+If a function is recursive in the constant zero function,
+then it is partial recursive.
+-/
+lemma partrec_in_zero_implies_partrec
+(f : ℕ →. ℕ) : RecursiveIn f (fun _ => Part.some 0) → Nat.Partrec f := by
+  intro fRecInZero
+  generalize h : (fun _ => Part.some 0) = fp at *
+  induction fRecInZero
+  case zero =>
+    apply Nat.Partrec.zero
+  case succ =>
+    apply Nat.Partrec.succ
+  case left =>
+    apply Nat.Partrec.left
+  case right =>
+    apply Nat.Partrec.right
+  case oracle g =>
+    rw [← h]
+    apply Nat.Partrec.zero
+  case pair _ _ _ _ ih1 ih2 =>
+    apply Nat.Partrec.pair
+    · apply ih1
+      rw [← h]
+    · apply ih2
+      rw [← h]
+  case comp _ _ _ _ ih1 ih2 =>
+    apply Nat.Partrec.comp
+    · apply ih1
+      rw [← h]
+    · apply ih2
+      rw [← h]
+  case prec _ _ _ _ ih1 ih2 =>
+    apply Nat.Partrec.prec
+    · apply ih1
+      rw [← h]
+    · apply ih2
+      rw [← h]
+  case rfind _ _ ih =>
+    apply Nat.Partrec.rfind
+    apply ih
+    rw [← h]
+
+/--
+A partial function `f` is partial recursive if and only if it is recursive in
+every partial function `g`.
+-/
+theorem partrec_iff_partrec_in_everything
+  (f : ℕ →. ℕ) : Nat.Partrec f ↔ (∀ g, RecursiveIn f g) := by
+  constructor
+  · exact partrec_implies_recursive_in_everything f
+  · intro H
+    have lem : RecursiveIn f (fun _ => Part.some 0) := H (fun _ => Part.some 0)
+    have lem : Nat.Partrec f := partrec_in_zero_implies_partrec f lem
+    exact lem
+
+/--
+Proof that turing reducibility is reflexive.
+-/
+theorem RecursiveIn.refl (f : ℕ →. ℕ) : RecursiveIn f f :=
+  RecursiveIn.oracle
+
+/--
+Proof that `turing_equivalent` is reflexive.
+-/
+@[refl]
+theorem turing_equivalent_refl (f : ℕ →. ℕ) : f ≡ᵀ f :=
+  ⟨recursive_in_refl f, recursive_in_refl f⟩
+
+/--
+Proof that `turing_equivalent` is symmetric.
+-/
+@[symm]
+theorem turing_equivalent_symm {f g : ℕ →. ℕ} (h : f ≡ᵀ g) : g ≡ᵀ f :=
+  ⟨h.2, h.1⟩
+
+/--
+Proof that turing reducibility is transitive.
+-/
+@[trans]
+theorem recursive_in_trans {f g h : ℕ →. ℕ} :
+  RecursiveIn f g → RecursiveIn g h → RecursiveIn f h := by
+    intro hg hh
+    induction hg
+    case zero =>
+      apply RecursiveIn.zero
+    case succ =>
+      apply RecursiveIn.succ
+    case left =>
+      apply RecursiveIn.left
+    case right =>
+      apply RecursiveIn.right
+    case oracle =>
+      exact hh
+    case pair f' h' _ _ hf_ih hh_ih =>
+      apply RecursiveIn.pair
+      · apply hf_ih
+        apply hh
+      · apply hh_ih
+        apply hh
+    case comp f' h' _ _ hf_ih hh_ih =>
+      apply RecursiveIn.comp
+      · apply hf_ih
+        apply hh
+      · apply hh_ih
+        apply hh
+    case prec f' h' _ _ hf_ih hh_ih =>
+      apply RecursiveIn.prec
+      · apply hf_ih
+        apply hh
+      · apply hh_ih
+        apply hh
+    case rfind f' _ hf_ih =>
+      apply RecursiveIn.rfind
+      · apply hf_ih
+        apply hh
+
+/--
+Proof that `turing_equivalent` is transitive.
+-/
+theorem turing_equivalent_trans :
+  Transitive turing_equivalent :=
+  fun _ _ _ ⟨fg₁, fg₂⟩ ⟨gh₁, gh₂⟩ =>
+    ⟨recursive_in_trans fg₁ gh₁, recursive_in_trans gh₂ fg₂⟩
+
+/--
+Instance declaring that `turing_equivalent` is an equivalence relation.
+-/
+instance : Equivalence turing_equivalent :=
+  {
+    refl := turing_equivalent_refl,
+    symm := turing_equivalent_symm,
+    trans := @turing_equivalent_trans
+  }
+
+/--
+Instance declaring that `RecursiveIn` is a preorder.
+-/
+instance : IsPreorder (ℕ →. ℕ) RecursiveIn where
+  refl := recursive_in_refl
+  trans := @recursive_in_trans
+
+/--
+The Turing degrees as the set of equivalence classes under Turing equivalence.
+-/
+def TuringDegree :=
+  Antisymmetrization _ RecursiveIn
+
+/--
+The `join` function combines two partial functions `f` and `g` into a single partial function.
+For a given input `n`:
+
+- **If `n` is even**:
+  - It checks if `f` is defined at `n / 2`.
+  - If so, `join f g` is defined at `n` with the value `2 * f(n / 2)`.
+
+- **If `n` is odd**:
+  - It checks if `g` is defined at `n / 2`.
+  - If so, `join f g` is defined at `n` with the value `2 * g(n / 2) + 1`.
+-/
+def join (f g : ℕ →. ℕ) : ℕ →. ℕ :=
+  fun n =>
+    if n % 2 = 0 then
+      (f (n / 2)).map (fun x => 2 * x)
+    else
+      (g (n / 2)).map (fun y => 2 * y + 1)
+
+/-- Join notation `f ⊕ g` is the "join" of partial functions `f` and `g`. -/
+infix:99 "⊕" => join
+
+/--
+For any partial functions `a`, `b₁`, and `b₂`, if `b₁` is Turing equivalent to `b₂`,
+then `a` is Turing reducible to `b₁` if and only if `a` is Turing reducible to `b₂`.
+-/
+lemma reduce_lifts₁ : ∀ (a b₁ b₂ : ℕ →. ℕ), b₁≡ᵀb₂ → (RecursiveIn a b₁) = (RecursiveIn a b₂) := by
+  intros a b₁ b₂ bEqb
+  apply propext
+  constructor
+  · intro aRedb₁
+    apply recursive_in_trans aRedb₁ bEqb.1
+  · intro aRedb₂
+    apply recursive_in_trans aRedb₂ bEqb.2
+
+/--
+For any partial functions `f`, `g`, and `h`, if `f` is Turing equivalent to `g`,
+then `f` is Turing reducible to `h` if and only if `g` is Turing reducible to `h`.
+-/
+lemma reduce_lifts₂ : ∀ (f g h : ℕ →. ℕ),
+f ≡ᵀ g → (RecursiveIn f h = RecursiveIn g h) := by
+  intros f g h fEqg
+  apply propext
+  constructor
+  · intro fRedh
+    apply recursive_in_trans fEqg.2 fRedh
+  · intro gRedh
+    apply recursive_in_trans fEqg.1 gRedh
+
+/--
+Here we show how to lift the Turing reducibility relation from
+partial functions to their Turing degrees, using the above lemmas.
+-/
+def TuringDegree.turing_red (d₁ d₂ : TuringDegree) : Prop :=
+  @Quot.lift₂ _ _ Prop (turing_equivalent)
+  (turing_equivalent) (RecursiveIn) (reduce_lifts₁) (reduce_lifts₂) d₁ d₂
+
+/--
+Instance declaring that `TuringDegree.turing_red` is a partial order.
+-/
+instance : PartialOrder TuringDegree where
+  le := TuringDegree.turing_red
+  le_refl := by
+    apply Quot.ind
+    intro a
+    apply recursive_in_refl
+  le_trans := by
+    apply Quot.ind
+    intro a
+    apply Quot.ind
+    intro b
+    apply Quot.ind
+    intro c
+    exact recursive_in_trans
+  le_antisymm := by
+    apply Quot.ind
+    intro a
+    apply Quot.ind
+    intro b
+    intros aRedb bReda
+    apply Quot.sound
+    constructor
+    · exact aRedb
+    · exact bReda