source: src/common/IOMonad.ma @ 1298

include "utilities/extralib.ma".
include "common/Errors.ma".

(* IO monad *)

inductive IO (output:Type[0]) (input:output → Type[0]) (T:Type[0]) : Type[0] ≝
| Interact : ∀o:output. (input o → IO output input T) → IO output input T
| Value : T → IO output input T
| Wrong : errmsg → IO output input T.

let rec bindIO (O:Type[0]) (I:O → Type[0]) (T,T':Type[0]) (v:IO O I T) (f:T → IO O I T') on v : IO O I T' ≝
match v with
[ Interact out k ⇒ (Interact ??? out (λres. bindIO O I T T' (k res) f))
| Value v' ⇒ (f v')
| Wrong m ⇒ Wrong O I T' m
].

let rec bindIO2 (O:Type[0]) (I:O → Type[0]) (T1,T2,T':Type[0]) (v:IO O I (T1×T2)) (f:T1 → T2 → IO O I T') on v : IO O I T' ≝
match v with
[ Interact out k ⇒ (Interact ??? out (λres. bindIO2 ?? T1 T2 T' (k res) f))
| Value v' ⇒ match v' with [ pair v1 v2 ⇒ f v1 v2 ]
| Wrong m ⇒ Wrong ?? T' m
].

definition err_to_io : ∀O,I,T. res T → IO O I T ≝
λO,I,T,v. match v with [ OK v' ⇒ Value O I T v' | Error m ⇒ Wrong O I T m ].
coercion err_to_io : ∀O,I,A.∀c:res A.IO O I A ≝ err_to_io on _c:res ? to IO ???.
definition err_to_io_sig : ∀O,I,T.∀P:T → Prop. res (Sig T P) → IO O I (Sig T P) ≝
λO,I,T,P,v. match v with [ OK v' ⇒ Value O I (Sig T P) v' | Error m ⇒ Wrong O I (Sig T P) m ].
(*coercion err_to_io_sig : ∀O,I,A.∀P:A → Prop.∀c:res (Sig A P).IO O I (Sig A P) ≝ err_to_io_sig on _c:res (Sig ??) to IO ?? (Sig ??).*)


(* If the original definitions are vague enough, do I need to do this? *)
notation > "! ident v ← e; e'" with precedence 40 for @{'bindIO ${e} (λ${ident v}.${e'})}.
notation > "! ident v : ty ← e; e'" with precedence 40 for @{'bindIO ${e} (λ${ident v} : ${ty}.${e'})}.
notation < "vbox(! \nbsp ident v ← e; break e')" with precedence 40 for @{'bindIO ${e} (λ${ident v}.${e'})}.
notation < "vbox(! \nbsp ident v : ty ← e; break e')" with precedence 40 for @{'bindIO ${e} (λ${ident v} : ${ty}.${e'})}.
notation > "! 〈ident v1, ident v2〉 ← e; e'" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}.
notation > "! 〈ident v1 : ty1, ident v2 : ty2〉 ← e; e'" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}.
notation < "vbox(! \nbsp 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}.
notation < "vbox(! \nbsp 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}.
interpretation "IO monad bind" 'bindIO e f = (bindIO ???? e f).
interpretation "IO monad Prod bind" 'bindIO2 e f = (bindIO2 ????? e f).
(**)
let rec P_io O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝
match v return λ_.Prop with
[ Wrong _ ⇒ True
| Value z ⇒ P z
| Interact out k ⇒ ∀v'.P_io O I A P (k v')
].

let rec P_io' O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝
match v return λ_.Prop with
[ Wrong _ ⇒ False
| Value z ⇒ P z
| Interact out k ⇒ ∀v'.P_io' O I A P (k v')
].

definition P_to_P_option_io : ∀O,I,A.∀P:A → Prop.option (IO O I A) → Prop ≝
  λO,I,A,P,a.match a with
   [ None ⇒ False
   | Some y ⇒ P_io O I A P y
   ].

let rec io_inject_0 O I (A:Type[0]) (P:A → Prop) (a:IO O I A) (p:P_io O I A P a) on a : IO O I (Sig A P) ≝
(match a return λa'.P_io O I A P a' → ? with
 [ Wrong m ⇒ λ_. Wrong O I ? m
 | Value c ⇒ λp'. Value ??? (dp A P c p')
 | Interact out k ⇒ λp'. Interact ??? out (λv. io_inject_0 O I A P (k v) (p' v))
 ]) p.

definition io_inject : ∀O,I,A.∀P:A → Prop.∀a:option (IO O I A).∀p:P_to_P_option_io O I A P a.IO O I (Sig A P) ≝ 
  λO,I,A.λP:A → Prop.λa:option (IO O I A).λp:P_to_P_option_io O I A P a. 
  (match a return λa'.P_to_P_option_io O I A P a' → IO O I (Sig A P) with
   [ None ⇒ λp'.?
   | Some b ⇒ λp'. io_inject_0 O I A P b p'
   ]) p.
elim p'; qed.

let rec io_eject O I (A:Type[0]) (P: A → Prop) (a:IO O I (Sig A P)) on a : IO O I A ≝ 
match a with
[ Wrong m ⇒ Wrong ??? m
| Value b ⇒ match b with [ dp w p ⇒ Value ??? w]
| Interact out k ⇒ Interact ??? out (λv. io_eject ?? A P (k v))
].

coercion io_inject : 
  ∀O,I,A.∀P:A → Prop.∀a.∀p:P_to_P_option_io O I ? P a.IO O I (Sig A P) ≝ io_inject 
  on a:option (IO ???) to IO ?? (Sig ? ?).
coercion io_eject : ∀O,I,A.∀P:A → Prop.∀c:IO O I (Sig A P).IO O I A ≝ io_eject 
  on _c:IO ?? (Sig ? ?) to IO ???.

definition opt_to_io : ∀O,I,T.errmsg → option T → IO O I T ≝
λO,I,T,m,v. match v with [ None ⇒ Wrong ?? T m | Some v' ⇒ Value ??? v' ].

lemma sig_bindIO_OK: ∀O,I,A,B. ∀P:A → Prop. ∀P':B → Prop. ∀e:IO O I (Sig A P). ∀f:Sig A P → IO O I B.
  (∀v:A. ∀p:P v. P_io O I ? P' (f (dp A P v p))) →
  P_io O I ? P' (bindIO O I (Sig A P) B e f).
#O #I #A #B #P #P' #e #f elim e;
[ #out #k #IH #IH' whd; #res @IH //;
| #v0 elim v0; #v #Hv #IH whd; @IH
| //;
] qed.

lemma sig_bindIO2_OK: ∀O,I,A,B,C. ∀P:(A×B) → Prop. ∀P':C → Prop. ∀e:IO O I (Sig (A×B) P). ∀f: A → B → IO O I C.
  (∀vA:A.∀vB:B. ∀p:P 〈vA,vB〉. P_io O I ? P' (f vA vB)) →
  P_io O I ? P' (bindIO2 O I A B C e f).
#I #O #A #B #C #P #P' #e #f elim e;
[ #out #k #IH #IH' whd; #res @IH @IH'
| #v0 elim v0; #v elim v; #vA #vB #Hv #IH @IH //;
| //;
] qed.

lemma opt_bindIO_OK: ∀O,I,A,B,m. ∀P:B → Prop. ∀e:option A. ∀f: A → IO O I B.
  (∀v:A. e = Some A v → P_io O I ? P (f v)) →
  P_io O I ? P (bindIO O I A B (opt_to_io ??? m e) f).
#I #O #A #B #m #P #e elim e; //; #v #f #H @H //;
qed.

lemma opt_bindIO2_OK: ∀O,I,A,B,C,m. ∀P:C → Prop. ∀e:option (A×B). ∀f: A → B → IO O I C.
  (∀vA:A.∀vB:B. e = Some (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) →
  P_io O I ? P (bindIO2 O I A B C (opt_to_io ??? m e) f).
#I #O #A #B #C #m #P #e elim e; //; #v cases v; #vA #vB #f #H @H //;
qed.

lemma res_bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:res A. ∀f: A → IO O I B.
  (∀v:A. e = OK A v → P_io O I ? P (f v)) →
  P_io O I ? P (bindIO O I A B e f).
#I #O #A #B #P #e elim e; //; #v #f #H @H //;
qed.

lemma res_bindIO2_OK: ∀O,I,A,B,C. ∀P:C → Prop. ∀e:res (A×B). ∀f: A → B → IO O I C.
  (∀vA:A.∀vB:B. e = OK (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) →
  P_io O I ? P (bindIO2 O I A B C e f).
#I #O #A #B #C #P #e elim e; //; #v cases v; #vA #vB #f #H @H //;
qed.

lemma bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:IO O I A. ∀f: A → IO O I B.
  (∀v:A. P_io O I ? P (f v)) →
  P_io O I ? P (bindIO O I A B e f).
#I #O #A #B #P #e elim e;
[ #out #k #IH #f #H whd; #res @IH //;
| #v #f #H @H
| //;
] qed.

lemma bindIO2_OK: ∀O,I,A,B,C. ∀P:C → Prop. ∀e:IO O I (A×B). ∀f: A → B → IO O I C.
  (∀v1:A.∀v2:B. P_io O I ? P (f v1 v2)) →
  P_io O I ? P (bindIO2 O I A B C e f).
#I #O #A #B #C #P #e elim e;
[ #out #k #IH #f #H whd; #res @IH //;
| #v cases v; #v1 #v2 #f #H @H
| //;
] qed.

lemma P_bindIO_OK: ∀O,I,A,B. ∀P':A → Prop. ∀P:B → Prop. ∀e:IO O I A. ∀f: A → IO O I B.
  P_io … P' e →
  (∀v:A. P' v → P_io O I ? P (f v)) →
  P_io O I ? P (bindIO O I A B e f).
#I #O #A #B #P' #P #e elim e;
[ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2/;
| #v #f #He #H @H @He
| //;
] qed.

lemma P_bindIO2_OK: ∀O,I,A,B,C. ∀P':A×B → Prop. ∀P:C → Prop. ∀e:IO O I (A×B). ∀f: A → B → IO O I C.
  P_io … P' e →
  (∀v1:A.∀v2:B. P' 〈v1,v2〉 → P_io O I ? P (f v1 v2)) →
  P_io O I ? P (bindIO2 O I A B C e f).
#I #O #A #B #C #P' #P #e elim e;
[ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2/;
| #v cases v; #v1 #v2 #f #He #H @H @He
| //;
] qed.


(* Is there a way to prove this without extensionality? *)

lemma bind_assoc_r: ∀O,I,A,B,C,e,f,g.
  ∀ext:(∀T1,T2:Type[0].∀f,f':T1 → T2.(∀x.f x = f' x) → f = f').
  bindIO O I B C (bindIO O I A B e f) g = bindIO O I A C e (λx.bindIO O I B C (f x) g).
#O #I #A #B #C #e #f #g #ext elim e;
[ #o #k #IH whd in ⊢ (??%%); @eq_f
    @ext @IH
| #v @refl
| #m @refl
] qed.

(*
lemma extract_subset_pair_io: ∀O,I,A,B,C,P. ∀e:{e:A×B | P e}. ∀Q:A→B→IO O I C. ∀R:C→Prop.
  (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → P_io O I ? R (Q a b)) →
  P_io O I ? R (match eject ?? e with [ pair a b ⇒ Q a b ]).
#I #O #A #B #C #P #e #Q #R cases e; #e' cases e'; normalize;
[ *;
| #e'' cases e''; #a #b #Pab #H normalize; /2/;
] qed.
*)