Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code × Code × σ × σ → σ
hpr : Primrec₂ pr
co : α → Code × Code × σ × σ → σ
hco : Primrec₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Primrec₂ pc
rf : α → Code × σ → σ
hrf : Primrec₂ rf
⊢ let PR := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg);
  let CO := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg);
  let PC := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg);
  let RF := fun a cf hf ↦ rf a (cf, hf);
  let F := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a);
  Primrec fun a ↦ F a (c a)

Tactic:
by
  intro _ _ _ _ F
  let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>
    letI a := p.1.1; letI IH := p.1.2; letI n := p.2.1; letI m := p.2.2
    IH[m]?.bind fun s =>
    IH[m.unpair.1]?.bind fun s₁ =>
    IH[m.unpair.2]?.map fun s₂ =>
    cond n.bodd
      (cond n.div2.bodd (rf a (ofNat Code m, s))
        (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s₁, s₂)))
      (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s₁, s₂))
        (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s₁, s₂)))
  have : Primrec G₁ :=
    option_bind (list_getElem?.comp (snd.comp fst) (snd.comp snd)) <| .mk <|
    option_bind ((list_getElem?.comp (snd.comp fst)
      (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) <| .mk <|
    option_map ((list_getElem?.comp (snd.comp fst)
      (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) <| .mk <|
    have a := fst.comp (fst.comp <| fst.comp <| fst.comp fst)
    have n := fst.comp (snd.comp <| fst.comp <| fst.comp fst)
    have m := snd.comp (snd.comp <| fst.comp <| fst.comp fst)
    have m₁ := fst.comp (Primrec.unpair.comp m)
    have m₂ := snd.comp (Primrec.unpair.comp m)
    have s := snd.comp (fst.comp fst)
    have s₁ := snd.comp fst
    have s₂ := snd
    (nat_bodd.comp n).cond
      ((nat_bodd.comp <| nat_div2.comp n).cond
        (hrf.comp a (((Primrec.ofNat Code).comp m).pair s))
        (hpc.comp a (((Primrec.ofNat Code).comp m₁).pair <|
          ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)))
      (Primrec.cond (nat_bodd.comp <| nat_div2.comp n)
        (hco.comp a (((Primrec.ofNat Code).comp m₁).pair <|
          ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂))
        (hpr.comp a (((Primrec.ofNat Code).comp m₁).pair <|
          ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)))
  let G : α → List σ → Option σ := fun a IH =>
    IH.length.casesOn (some (z a)) fun n =>
    n.casesOn (some (s a)) fun n =>
    n.casesOn (some (l a)) fun n =>
    n.casesOn (some (r a)) fun n =>
    G₁ ((a, IH), n, n.div2.div2)
  have : Primrec₂ G := .mk <|
    nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) <| .mk <|
    nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) <| .mk <|
    nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) <| .mk <|
    nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) <| .mk <|
    this.comp <|
      ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|
      snd.pair <| nat_div2.comp <| nat_div2.comp snd
  refine (nat_strong_rec (fun a n => F a (ofNat Code n)) this.to₂ fun a n => ?_)
    |>.comp .id (encode_iff.2 hc) |>.of_eq fun a => by simp
  iterate 4 rcases n with - | n; · simp [ofNatCode_eq, ofNatCode]; rfl
  simp only [G]; rw [List.length_map, List.length_range]
  let m := n.div2.div2
  change G₁ ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m)
    = some (F a (ofNat Code (n + 4)))
  have hm : m < n + 4 := by
    simp only [m, div2_val]
    exact lt_of_le_of_lt
      (le_trans (Nat.div_le_self ..) (Nat.div_le_self ..))
      (Nat.succ_le_succ (Nat.le_add_right ..))
  have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm
  have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm
  simp [G₁, m, hm, m1, m2]
  rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]
  simp [ofNatCode]
  cases n.bodd <;> cases n.div2.bodd <;> rfl

Goal state after: