Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: case succ.zero
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode zero) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode zero))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode zero))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode zero))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode zero))))
                  (k', zero) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode zero))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode zero))))
                (k', zero) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      zero =
    evaln (k' + 1) zero n

pp: case succ.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode succ) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode succ))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode succ))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode succ))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode succ))))
                  (k', succ) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode succ))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode succ))))
                (k', succ) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      succ =
    evaln (k' + 1) succ n

pp: case succ.left
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode left) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode left))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode left))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode left))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode left))))
                  (k', left) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode left))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode left))))
                (k', left) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      left =
    evaln (k' + 1) left n

pp: case succ.right
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode right) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode right))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode right))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode right))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode right))))
                  (k', right) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode right))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode right))))
                (k', right) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      right =
    evaln (k' + 1) right n

pp: case succ.pair
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.pair cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.pair cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                  (k', cf.pair cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                (k', cf.pair cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.pair cg) =
    evaln (k' + 1) (cf.pair cg) n

pp: case succ.comp
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.comp cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.comp cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                  (k', cf.comp cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                (k', cf.comp cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.comp cg) =
    evaln (k' + 1) (cf.comp cg) n

pp: case succ.prec
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.prec cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.prec cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                  (k', cf.prec cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                (k', cf.prec cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.prec cg) =
    evaln (k' + 1) (cf.prec cg) n

pp: case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode cf.rfind'))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf_1 cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                  (k', cf.rfind') (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                (k', cf.rfind') (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      cf.rfind' =
    evaln (k' + 1) cf.rfind' n

Tactic:
<;>

Goal state after:
case succ.zero
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode zero) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode zero))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode zero))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode zero))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode zero))))
                  (k', zero) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode zero))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode zero))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode zero))))
                (k', zero) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      zero =
    evaln (k' + 1) zero n

case succ.succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode succ) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode succ))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode succ))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode succ))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode succ))))
                  (k', succ) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode succ))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode succ))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode succ))))
                (k', succ) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      succ =
    evaln (k' + 1) succ n

case succ.left
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode left) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode left))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode left))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode left))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode left))))
                  (k', left) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode left))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode left))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode left))))
                (k', left) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      left =
    evaln (k' + 1) left n

case succ.right
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode right) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode right))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cf) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode right))))
            (k' + 1, cf) x)
      (fun cf cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode right))))
            (k' + 1, cf) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode right))))
                  (k', right) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode right))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode right))))
              (k' + 1, cf) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode right))))
                (k', right) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      right =
    evaln (k' + 1) right n

case succ.pair
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.pair cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.pair cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                  (k', cf.pair cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.pair cg)))))
                (k', cf.pair cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.pair cg) =
    evaln (k' + 1) (cf.pair cg) n

case succ.comp
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.comp cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.comp cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                  (k', cf.comp cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.comp cg)))))
                (k', cf.comp cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.comp cg) =
    evaln (k' + 1) (cf.comp cg) n

case succ.prec
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf cg : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode (cf.prec cg)) →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode (cf.prec cg)))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cg_1) n
        some (Nat.pair x y))
      (fun cf_1 cg_1 x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cg_1) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg_1 x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                  (k', cf.prec cg) (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                (k' + 1, cg_1) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode (cf.prec cg)))))
                (k', cf.prec cg) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      (cf.prec cg) =
    evaln (k' + 1) (cf.prec cg) n

case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode cf.rfind'))))
          (k', c') n =
        evaln k' c' n
⊢ rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)
      (fun cf_1 cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cf_1) n
        let y ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cg) n
        some (Nat.pair x y))
      (fun cf_1 cg x x_1 ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cg) n
        lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k' + 1, cf_1) x)
      (fun cf_1 cg x x_1 ↦
        Nat.rec
          (lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k' + 1, cf_1) (unpair n).1)
          (fun n_1 n_ih ↦ do
            let i ←
              lup
                  (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                    (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                  (k', cf.rfind') (Nat.pair (unpair n).1 n_1)
            lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
          (unpair n).2)
      (fun cf_1 x ↦ do
        let x ←
          lup
              (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
              (k' + 1, cf_1) n
        Nat.rec (some (unpair n).2)
            (fun n_1 n_ih ↦
              lup
                (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                  (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
                (k', cf.rfind') (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
            x)
      cf.rfind' =
    evaln (k' + 1) cf.rfind' n