awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
⊢ Primrec (β := Option ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ do
    let x ← lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) a.1.1.2
    lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x

Tactic:
Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_

Premises used: Primrec.pair, _private.0.Nat.Partrec.Code.hlup, Primrec.comp, Primrec.option_bind
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
hco :
  Primrec₂ (σ := Option ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) (x : ℕ) ↦
    lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x
⊢ Primrec (β := Option ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ do
    let x ← lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) a.1.1.2
    lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x

Tactic:
refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_

Premises used: Primrec.pair, _private.0.Nat.Partrec.Code.hlup, Primrec.comp, Primrec.option_bind
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
⊢ Primrec (β := Option ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    lup (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.2

Tactic:
have h :=
      hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd)
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
h :
  Primrec (β := Option ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    lup (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.2
⊢ Primrec (β := Option ℕ) fun (p : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) (x : ℕ) ↦
        lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x)
      p.1 p.2

Tactic:
h
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
h :
  Primrec (β := Option ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    lup (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.1
      (a.1.1.1.1, ((ofNat (ℕ × Code) a.1.1.1.1.length).1, a.1.2.1), a.2).2.2
⊢ Primrec (β := Option ℕ) fun (p : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) (x : ℕ) ↦
        lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x)
      p.1 p.2

Tactic:
exact h
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
⊢ Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1

Tactic:
have z := Primrec.fst.comp (Primrec.unpair.comp n)
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
⊢ Primrec (β := Option ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    have z := (unpair a.1.1.2).1;
    Nat.casesOn (motive := fun (x : ℕ) ↦ Option ℕ) (unpair a.1.1.2).2
      (lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) z) fun (y : ℕ) ↦ do
      let i ← lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z y)
      lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair z (Nat.pair y i))

Tactic:
Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))
        (hlup.comp <| L.pair <| (k.pair cf).pair z)
        (?_ : Primrec _)

Premises used: Primrec.snd, Primrec.pair, Primrec.unpair, _private.0.Nat.Partrec.Code.hlup, Primrec.comp, Primrec.nat_casesOn
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
hpc :
  Primrec (β := Option ℕ) fun (p : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) (y : ℕ) ↦
        (do
          let i ← lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair (unpair a.1.1.2).1 y)
          lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)) :
          Option ℕ))
      p.1 p.2
⊢ Primrec (β := Option ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    have z := (unpair a.1.1.2).1;
    Nat.casesOn (motive := fun (x : ℕ) ↦ Option ℕ) (unpair a.1.1.2).2
      (lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) z) fun (y : ℕ) ↦ do
      let i ← lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z y)
      lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair z (Nat.pair y i))

Tactic:
refine
      Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))
        (hlup.comp <| L.pair <| (k.pair cf).pair z)
        (?_ : Primrec _)

Premises used: Primrec.snd, Primrec.pair, Primrec.unpair, _private.0.Nat.Partrec.Code.hlup, Primrec.comp, Primrec.nat_casesOn
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
⊢ Primrec (β := List (List (Option ℕ)))
    fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦ a.1.1.1.1

Tactic:
have L := L.comp (Primrec.fst (β := ℕ))
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L_1 :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦ a.1.1.1.1
⊢ Primrec (β := ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (unpair a.1.1.1.2).1

Tactic:
have z := z.comp (Primrec.fst (β := ℕ))
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L_1 :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z_1 :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦ a.1.1.1.1
z :
  Primrec (β := ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (unpair a.1.1.1.2).1
⊢ Primrec (α := (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) Prod.snd

Tactic:
have y := Primrec.snd
      (α := ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) (β := ℕ)
-- Module: Mathlib.Computability.PartrecCode
Goal:
a : Primrec (β := ℕ × Code) fun (a : List (List (Option ℕ))) ↦ ofNat (ℕ × Code) a.length
k_1 : Primrec (β := ℕ) fun (a : List (List (Option ℕ)) × ℕ) ↦ (ofNat (ℕ × Code) a.1.length).1
n_1 : Primrec (α := List (List (Option ℕ)) × ℕ) Prod.snd
k_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).1
n_2 : Primrec (β := ℕ) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ a.1.2
k' : Primrec (α := (List (List (Option ℕ)) × ℕ) × ℕ) Prod.snd
c : Primrec (β := Code) fun (a : (List (List (Option ℕ)) × ℕ) × ℕ) ↦ (ofNat (ℕ × Code) a.1.1.length).2
L_1 :
  Primrec (β := List (List (Option ℕ)))
    fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.1
k :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦
    (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.1.1.2
cf : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.1
cg : Primrec (β := Code) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ a.2.2.1
z_1 :
  Primrec (β := ℕ) fun (a : ((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) ↦ (unpair a.1.1.2).1
L :
  Primrec (β := List (List (Option ℕ)))
    fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦ a.1.1.1.1
z :
  Primrec (β := ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    (unpair a.1.1.1.2).1
y : Primrec (α := (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) Prod.snd
⊢ Primrec (β := Option ℕ) fun (a : (((List (List (Option ℕ)) × ℕ) × ℕ) × Code × Code × Option ℕ × Option ℕ) × ℕ) ↦
    lup (a.1.1.1.1, (a.1.1.2, (ofNat (ℕ × Code) a.1.1.1.1.length).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1
      (a.1.1.1.1, (a.1.1.2, (ofNat (ℕ × Code) a.1.1.1.1.length).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1
      (a.1.1.1.1, (a.1.1.2, (ofNat (ℕ × Code) a.1.1.1.1.length).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2

Tactic:
have h₁ := hlup.comp <| L.pair <| (((k'.pair c).comp Primrec.fst).comp Primrec.fst).pair
      (Primrec₂.natPair.comp z y)