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0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.NamedActions
namespace OAI
namespace SimpleFInftyOvergroups.BoundedEncoding.NamedAction.Operations
variable {G X : Type*} [Primcodable G] [Primcodable X] (N : Operations G X)
attribute [local fun_prop] Primrec
attribute [local fun_prop] Primrec.id Primrec.const Primrec.fst Primrec.snd Primrec.pair Primrec.comp
attribute [local fun_prop] Primrec.option_some Primrec.sumInl Primrec.sumInr
private theorem primrec_cons {A : Type*} [Primcodable A] :
Primrec (fun p : A × List A => p.1 :: p.2) := Primrec.list_cons
private theorem primrec_append {A : Type*} [Primcodable A] :
Primrec (fun p : List A × List A => p.1 ++ p.2) := Primrec.list_append
attribute [local fun_prop] primrec_cons primrec_append
private theorem primrec_match_list {A B R : Type*} [Primcodable A] [Primcodable B] [Primcodable R]
{f : A → List B} {g : A → R} {h : A → B × List B → R}
(hf : Primrec f) (hg : Primrec g) (hh : Primrec₂ h) :
Primrec (fun a => match f a with | [] => g a | b::l => h a (b,l)) :=
(Primrec.list_casesOn hf hg hh).of_eq (fun a => by cases f a <;> rfl)
private theorem primrec_match_option {A B R : Type*} [Primcodable A] [Primcodable B] [Primcodable R]
{f : A → Option B} {g : A → R} {h : A → B → R}
(hf : Primrec f) (hg : Primrec g) (hh : Primrec₂ h) :
Primrec (fun a => match f a with | none => g a | some b => h a b) :=
(Primrec.option_casesOn hf hg hh).of_eq (fun a => by cases f a <;> rfl)
private theorem primrec_match_sum {A B C R : Type*}
[Primcodable A] [Primcodable B] [Primcodable C] [Primcodable R]
{f : A → B ⊕ C} {g : A → B → R} {h : A → C → R}
(hf : Primrec f) (hg : Primrec₂ g) (hh : Primrec₂ h) :
Primrec (fun a => match f a with | .inl b => g a b | .inr c => h a c) :=
(Primrec.sumCasesOn hf hg hh).of_eq (fun a => by cases f a <;> rfl)
theorem primrec_letterCost (hg : Primrec N.gcost) (hx : Primrec N.xcost) :
Primrec N.letterCost := by
have ho : Primrec (fun p : Letter G X × Option X =>
match p.2 with | none => 0 | some x => N.xcost x) :=
primrec_match_option Primrec.snd (Primrec.const 0) (hx.comp Primrec.snd).to₂
have hs : Primrec (fun p : Letter G X × (X ⊕ Option X) =>
match p.2 with
| .inl x => N.xcost x
| .inr o => match o with | none => 0 | some x => N.xcost x) :=
(primrec_match_sum Primrec.snd (hx.comp Primrec.snd).to₂
(ho.comp ((Primrec.fst.comp Primrec.fst).pair Primrec.snd)).to₂).of_eq
(fun p => by cases p.2 <;> rfl)
exact (primrec_match_sum Primrec.id (hg.comp Primrec.snd).to₂ hs.to₂).of_eq
(fun a => by rcases a with g | (x | y); rfl; rfl; cases y <;> rfl)
theorem primrec_plan (hm : Primrec₂ N.mul) (hi : Primrec N.inv) (ha : Primrec₂ N.act) :
Primrec₂ N.plan := by
have hm' : Primrec (fun p : G × G => N.mul p.1 p.2) := hm
have ha' : Primrec (fun p : G × X => N.act p.1 p.2) := ha
let T := Letter G X × Form G X
let B := List X × G × List X
let R := Form G X ⊕ (X × X × Form G X)
have hgrp : Primrec (fun p : (T × B) × G =>
match p.1.2.1 with
| [] => (Sum.inl (some ([],N.mul p.2 p.1.2.2.1,p.1.2.2.2)) : Form G X ⊕ (X × X × Form G X))
| x::l => .inl (some (N.act p.2 x::l,p.1.2.2.1,p.1.2.2.2))) := by
have h0 : Primrec (fun p : (T × B) × G => (Sum.inl
(some ([],N.mul p.2 p.1.2.2.1,p.1.2.2.2)) : R)) := by
dsimp only [B,T,R]
fun_prop (disch := assumption)
have hc : Primrec₂ (fun (p : (T × B) × G) (q : X × List X) =>
(Sum.inl (some (N.act p.2 q.1::q.2,p.1.2.2.1,p.1.2.2.2)) : R)) := by
change Primrec _
dsimp only [B,T,R]
fun_prop (disch := assumption)
exact (primrec_match_list (f := fun p : (T × B) × G => p.1.2.1)
(by dsimp only [B,T,R]; fun_prop) h0 hc).of_eq (fun p => by cases p.1.2.1 <;> rfl)
have hdel : Primrec (fun p : (T × B) × X =>
match p.1.2.1 with
| [] => (Sum.inl (some ([],N.one,p.1.2.2.2++[N.act (N.inv p.1.2.2.1) p.2])) :
Form G X ⊕ (X × X × Form G X))
| x::l => .inr (p.2,x,some (l,p.1.2.2.1,p.1.2.2.2))) := by
have h0 : Primrec (fun p : (T × B) × X => (Sum.inl
(some ([],N.one,p.1.2.2.2++[N.act (N.inv p.1.2.2.1) p.2])) : R)) := by
dsimp only [B,T,R]
fun_prop (disch := assumption)
have hc : Primrec₂ (fun (p : (T × B) × X) (q : X × List X) =>
(Sum.inr (p.2,q.1,some (q.2,p.1.2.2.1,p.1.2.2.2)) : R)) := by
change Primrec _
dsimp only [B,T,R]
fun_prop
exact (primrec_match_list (f := fun p : (T × B) × X => p.1.2.1)
(by dsimp only [B,T,R]; fun_prop) h0 hc).of_eq (fun p => by cases p.1.2.1 <;> rfl)
have hopt : Primrec (fun p : (T × B) × Option X =>
match p.2 with
| none => (Sum.inl none : Form G X ⊕ (X × X × Form G X))
| some x => match p.1.2.1 with
| [] => .inl (some ([],N.one,p.1.2.2.2++[N.act (N.inv p.1.2.2.1) x]))
| z::l => .inr (x,z,some (l,p.1.2.2.1,p.1.2.2.2))) := by
apply primrec_match_option Primrec.snd (Primrec.const (Sum.inl none))
exact (hdel.comp (Primrec.fst.comp Primrec.fst |>.pair Primrec.snd)).to₂
have hnon : Primrec (fun p : (T × B) × (X ⊕ Option X) =>
match p.2 with
| .inl x => (Sum.inl (some (x::p.1.2.1,p.1.2.2.1,p.1.2.2.2)) :
Form G X ⊕ (X × X × Form G X))
| .inr y => match y with
| none => .inl none
| some x => match p.1.2.1 with
| [] => .inl (some ([],N.one,p.1.2.2.2++[N.act (N.inv p.1.2.2.1) x]))
| z::l => .inr (x,z,some (l,p.1.2.2.1,p.1.2.2.2))) := by
have hl : Primrec₂ (fun (p : (T × B) × (X ⊕ Option X)) (x : X) =>
(Sum.inl (some (x::p.1.2.1,p.1.2.2.1,p.1.2.2.2)) : R)) := by
change Primrec _
dsimp only [B,T,R]
fun_prop
exact (primrec_match_sum Primrec.snd hl
(hopt.comp (Primrec.fst.comp Primrec.fst |>.pair Primrec.snd)).to₂).of_eq
(fun p => by cases p.2 <;> rfl)
have hs : Primrec (fun p : T × B => N.plan p.1.1 (some p.2)) := by
apply (Primrec.sumCasesOn (Primrec.fst.comp Primrec.fst)
(hgrp.to₂) (hnon.to₂)).of_eq
intro p
rcases p with ⟨⟨a,f⟩,l,g,r⟩
rcases a with h | (x|y)
· cases l <;> rfl
· rfl
· cases y with
| none => rfl
| some y => cases l <;> rfl
exact (Primrec.option_casesOn Primrec.snd (Primrec.const (.inl none)) hs.to₂).of_eq
(fun p => by rcases p with ⟨a,f⟩; cases f <;> rfl)
end SimpleFInftyOvergroups.BoundedEncoding.NamedAction.Operations
end OAI
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