English
Primrec_recOn' asserts that the recursion principle for Nat.Partrec.Code is primitive recursive.
Русский
Primrec_recOn' утверждает примитивную рекурсивность принципа рекурсии для Nat.Partrec.Code.
LaTeX
$$$\\operatorname{Primrec\_recOn'}$$$
Lean4
theorem primrec_recOn' {α σ} [Primcodable α] [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 (a) cf cg hf hg := pr a (cf, cg, hf, hg)
let CO (a) cf cg hf hg := co a (cf, cg, hf, hg)
let PC (a) cf cg hf hg := pc a (cf, cg, hf, hg)
let RF (a) cf hf := rf a (cf, hf)
let F (a : α) (c : Code) : σ := Nat.Partrec.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) : α → σ) :=
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
