Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Computable₂ pc
rf : α → Code × σ → σ
hrf : Computable₂ rf
PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg)
CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg)
PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg)
RF✝ : α → Code → σ → σ := fun a cf hf ↦ rf a (cf, hf)
F : α → Code → σ := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p ↦
    p.1.2[p.2.2]?.bind fun s ↦
      p.1.2[(unpair p.2.2).1]?.bind fun s₁ ↦
        Option.map
          (fun s₂ ↦
            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))
          p.1.2[(unpair p.2.2).2]?
this✝ : Computable G₁
G : α → List σ → Option σ :=
  fun a IH ↦
    Nat.casesOn IH.length (some (z a)) fun n ↦
      Nat.casesOn n (some (s a)) fun n ↦
        Nat.casesOn n (some (l a)) fun n ↦ Nat.casesOn n (some (r a)) fun n ↦ G₁ ((a, IH), n, n.div2.div2)
this : Computable₂ G
a : α
n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
m1 : (unpair m).1 < n + 4
m2 : (unpair m).2 < n + 4
⊢ (bif n.bodd then
      bif n.div2.bodd then rf a (ofNat Code n.div2.div2, F a (ofNat Code n.div2.div2))
      else
        pc a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
    else
      bif n.div2.bodd then
        co a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
      else
        pr a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))) =
    F a
      (match n.bodd, n.div2.bodd with
      | false, false => (ofNatCode (unpair n.div2.div2).1).pair (ofNatCode (unpair n.div2.div2).2)
      | false, true => (ofNatCode (unpair n.div2.div2).1).comp (ofNatCode (unpair n.div2.div2).2)
      | true, false => (ofNatCode (unpair n.div2.div2).1).prec (ofNatCode (unpair n.div2.div2).2)
      | true, true => (ofNatCode n.div2.div2).rfind')

Tactic:
cases n.bodd <;> cases n.div2.bodd

Goal state after:
case succ.succ.succ.succ.false.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Computable₂ pc
rf : α → Code × σ → σ
hrf : Computable₂ rf
PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg)
CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg)
PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg)
RF✝ : α → Code → σ → σ := fun a cf hf ↦ rf a (cf, hf)
F : α → Code → σ := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p ↦
    p.1.2[p.2.2]?.bind fun s ↦
      p.1.2[(unpair p.2.2).1]?.bind fun s₁ ↦
        Option.map
          (fun s₂ ↦
            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))
          p.1.2[(unpair p.2.2).2]?
this✝ : Computable G₁
G : α → List σ → Option σ :=
  fun a IH ↦
    Nat.casesOn IH.length (some (z a)) fun n ↦
      Nat.casesOn n (some (s a)) fun n ↦
        Nat.casesOn n (some (l a)) fun n ↦ Nat.casesOn n (some (r a)) fun n ↦ G₁ ((a, IH), n, n.div2.div2)
this : Computable₂ G
a : α
n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
m1 : (unpair m).1 < n + 4
m2 : (unpair m).2 < n + 4
⊢ (bif false then
      bif false then rf a (ofNat Code n.div2.div2, F a (ofNat Code n.div2.div2))
      else
        pc a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
    else
      bif false then
        co a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
      else
        pr a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))) =
    F a
      (match false, false with
      | false, false => (ofNatCode (unpair n.div2.div2).1).pair (ofNatCode (unpair n.div2.div2).2)
      | false, true => (ofNatCode (unpair n.div2.div2).1).comp (ofNatCode (unpair n.div2.div2).2)
      | true, false => (ofNatCode (unpair n.div2.div2).1).prec (ofNatCode (unpair n.div2.div2).2)
      | true, true => (ofNatCode n.div2.div2).rfind')

case succ.succ.succ.succ.false.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Computable₂ pc
rf : α → Code × σ → σ
hrf : Computable₂ rf
PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg)
CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg)
PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg)
RF✝ : α → Code → σ → σ := fun a cf hf ↦ rf a (cf, hf)
F : α → Code → σ := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p ↦
    p.1.2[p.2.2]?.bind fun s ↦
      p.1.2[(unpair p.2.2).1]?.bind fun s₁ ↦
        Option.map
          (fun s₂ ↦
            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))
          p.1.2[(unpair p.2.2).2]?
this✝ : Computable G₁
G : α → List σ → Option σ :=
  fun a IH ↦
    Nat.casesOn IH.length (some (z a)) fun n ↦
      Nat.casesOn n (some (s a)) fun n ↦
        Nat.casesOn n (some (l a)) fun n ↦ Nat.casesOn n (some (r a)) fun n ↦ G₁ ((a, IH), n, n.div2.div2)
this : Computable₂ G
a : α
n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
m1 : (unpair m).1 < n + 4
m2 : (unpair m).2 < n + 4
⊢ (bif false then
      bif true then rf a (ofNat Code n.div2.div2, F a (ofNat Code n.div2.div2))
      else
        pc a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
    else
      bif true then
        co a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
      else
        pr a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))) =
    F a
      (match false, true with
      | false, false => (ofNatCode (unpair n.div2.div2).1).pair (ofNatCode (unpair n.div2.div2).2)
      | false, true => (ofNatCode (unpair n.div2.div2).1).comp (ofNatCode (unpair n.div2.div2).2)
      | true, false => (ofNatCode (unpair n.div2.div2).1).prec (ofNatCode (unpair n.div2.div2).2)
      | true, true => (ofNatCode n.div2.div2).rfind')

case succ.succ.succ.succ.true.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Computable₂ pc
rf : α → Code × σ → σ
hrf : Computable₂ rf
PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg)
CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg)
PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg)
RF✝ : α → Code → σ → σ := fun a cf hf ↦ rf a (cf, hf)
F : α → Code → σ := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p ↦
    p.1.2[p.2.2]?.bind fun s ↦
      p.1.2[(unpair p.2.2).1]?.bind fun s₁ ↦
        Option.map
          (fun s₂ ↦
            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))
          p.1.2[(unpair p.2.2).2]?
this✝ : Computable G₁
G : α → List σ → Option σ :=
  fun a IH ↦
    Nat.casesOn IH.length (some (z a)) fun n ↦
      Nat.casesOn n (some (s a)) fun n ↦
        Nat.casesOn n (some (l a)) fun n ↦ Nat.casesOn n (some (r a)) fun n ↦ G₁ ((a, IH), n, n.div2.div2)
this : Computable₂ G
a : α
n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
m1 : (unpair m).1 < n + 4
m2 : (unpair m).2 < n + 4
⊢ (bif true then
      bif false then rf a (ofNat Code n.div2.div2, F a (ofNat Code n.div2.div2))
      else
        pc a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
    else
      bif false then
        co a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
      else
        pr a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))) =
    F a
      (match true, false with
      | false, false => (ofNatCode (unpair n.div2.div2).1).pair (ofNatCode (unpair n.div2.div2).2)
      | false, true => (ofNatCode (unpair n.div2.div2).1).comp (ofNatCode (unpair n.div2.div2).2)
      | true, false => (ofNatCode (unpair n.div2.div2).1).prec (ofNatCode (unpair n.div2.div2).2)
      | true, true => (ofNatCode n.div2.div2).rfind')

case succ.succ.succ.succ.true.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ co
pc : α → Code × Code × σ × σ → σ
hpc : Computable₂ pc
rf : α → Code × σ → σ
hrf : Computable₂ rf
PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pr a (cf, cg, hf, hg)
CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ co a (cf, cg, hf, hg)
PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg ↦ pc a (cf, cg, hf, hg)
RF✝ : α → Code → σ → σ := fun a cf hf ↦ rf a (cf, hf)
F : α → Code → σ := fun a c ↦ Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p ↦
    p.1.2[p.2.2]?.bind fun s ↦
      p.1.2[(unpair p.2.2).1]?.bind fun s₁ ↦
        Option.map
          (fun s₂ ↦
            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))
          p.1.2[(unpair p.2.2).2]?
this✝ : Computable G₁
G : α → List σ → Option σ :=
  fun a IH ↦
    Nat.casesOn IH.length (some (z a)) fun n ↦
      Nat.casesOn n (some (s a)) fun n ↦
        Nat.casesOn n (some (l a)) fun n ↦ Nat.casesOn n (some (r a)) fun n ↦ G₁ ((a, IH), n, n.div2.div2)
this : Computable₂ G
a : α
n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
m1 : (unpair m).1 < n + 4
m2 : (unpair m).2 < n + 4
⊢ (bif true then
      bif true then rf a (ofNat Code n.div2.div2, F a (ofNat Code n.div2.div2))
      else
        pc a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
    else
      bif true then
        co a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))
      else
        pr a
          (ofNat Code (unpair n.div2.div2).1, ofNat Code (unpair n.div2.div2).2,
            F a (ofNat Code (unpair n.div2.div2).1), F a (ofNat Code (unpair n.div2.div2).2))) =
    F a
      (match true, true with
      | false, false => (ofNatCode (unpair n.div2.div2).1).pair (ofNatCode (unpair n.div2.div2).2)
      | false, true => (ofNatCode (unpair n.div2.div2).1).comp (ofNatCode (unpair n.div2.div2).2)
      | true, false => (ofNatCode (unpair n.div2.div2).1).prec (ofNatCode (unpair n.div2.div2).2)
      | true, true => (ofNatCode n.div2.div2).rfind')