-- Lean 4 proof step from Mathlib.Computability.PartrecCode (hypothesis)
Goal:
α : Type u_1
σ : Type u_2
inst : Primcodable α
inst_1 : 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
PR : (a : α) → (cf cg : Code) → (hf hg : σ) → σ := fun (a : α) (cf cg : Code) (hf hg : σ) ↦ pr a (cf, cg, hf, hg)
CO : (a : α) → (cf cg : Code) → (hf hg : σ) → σ := fun (a : α) (cf cg : Code) (hf hg : σ) ↦ co a (cf, cg, hf, hg)
PC : (a : α) → (cf cg : Code) → (hf hg : σ) → σ := fun (a : α) (cf cg : Code) (hf hg : σ) ↦ pc a (cf, cg, hf, hg)
RF : (a : α) → (cf : Code) → (hf : σ) → σ := fun (a : α) (cf : Code) (hf : σ) ↦ rf a (cf, hf)
F : (a : α) → (c : Code) → σ :=
  fun (a : α) (c : Code) ↦
    Code.recOn (motive := fun (x : Code) ↦ σ) c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a)
G₁ : Prod.{u_1, u_2} α (List σ) × ℕ × ℕ → Option σ :=
  fun (p : Prod.{u_1, u_2} α (List σ) × ℕ × ℕ) ↦
    (getElem? (valid := fun (as : List σ) (i : ℕ) ↦ i < as.length) p.1.2 p.2.2).bind fun (s : σ) ↦
      (getElem? (valid := fun (as : List σ) (i : ℕ) ↦ i < as.length) p.1.2 (unpair p.2.2).1).bind fun (s₁ : σ) ↦
        Option.map
          (fun (s₂ : σ) ↦
            bif p.2.1.bodd then
              bif p.2.1.div2.bodd then rf p.1.1 (ofNat Code p.2.2, s)
              else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂)
            else
              bif p.2.1.div2.bodd then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂)
              else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂))
          (getElem? (valid := fun (as : List σ) (i : ℕ) ↦ i < as.length) p.1.2 (unpair p.2.2).2)
⊢ Primrec.{u_1, u_2} (β := σ) fun (a : α) ↦ F a (c a)
have this : Primrec G₁
Tactic: 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)))