source: src/utilities/monad.ma @ 2200

Last change on this file since 2200 was 2179, checked in by campbell, 8 years ago

Dependent pair monad binding notation.

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1include "basics/types.ma".
2include "basics/relations.ma".
3include "utilities/setoids.ma".
4
5definition pred_transformer ≝ λA,B : Type[0].(A → Prop) → B → Prop.
6
7definition modus_ponens ≝ λA,B.λPT : pred_transformer A B.
8  ∀P,Q.(∀x.P x → Q x) → ∀y.PT P y → PT Q y.
9 
10lemma mp_transitive :
11  ∀A,B,C,PT1,PT2.modus_ponens A B PT1 → modus_ponens B C PT2 →
12    modus_ponens A C (PT2 ∘ PT1). /4/ qed.
13
14definition rel_transformer ≝ λA,B,C,D : Type[0].
15  (A → B → Prop) → C → D → Prop.
16 
17definition rel_modus_ponens ≝ λA,B,C,D.λRT : rel_transformer A B C D.
18  ∀P,Q.(∀x,y.P x y → Q x y) → ∀z,w.RT P z w → RT Q z w.
19 
20lemma rel_mp_transitive :
21  ∀A,B,C,D,E,F,RT1,RT2.rel_modus_ponens A B C D RT1 → rel_modus_ponens C D E F RT2 →
22    rel_modus_ponens … (RT2 ∘ RT1). /4/ qed.
23 
24record Monad : Type[1] ≝ {
25  monad :1> Type[0] → Type[0] ;
26  m_return : ∀X. X → monad X ;
27  m_bind : ∀X,Y. monad X → (X → monad Y) → monad Y
28}.
29
30notation "m »= f" with precedence 49
31  for @{'m_bind $m $f }.
32
33notation > "!_ e; e'"
34  with precedence 48 for @{'m_bind ${e} (λ_. ${e'})}.
35notation > "! ident v ← e; e'"
36  with precedence 48 for @{'m_bind ${e} (λ${ident v}. ${e'})}.
37notation > "! ident v : ty ← e; e'"
38  with precedence 48 for @{'m_bind ${e} (λ${ident v} : ${ty}. ${e'})}.
39notation < "vbox(! \nbsp ident v ← e ; break e')"
40  with precedence 48 for @{'m_bind ${e} (λ${ident v}.${e'})}.
41notation > "! ident v ← e : ty ; e'"
42  with precedence 48 for @{'m_bind (${e} : ${ty}) (λ${ident v}.${e'})}.
43notation < "vbox(! \nbsp ident v : ty \nbsp ←  \nbsp e ; break e')"
44  with precedence 48 for @{'m_bind ${e} (λ${ident v} : ${ty}. ${e'})}.
45notation > "! ident v : ty ← e : ty' ; e'"
46  with precedence 48 for @{'m_bind (${e} : ${ty'}) (λ${ident v} : ${ty}. ${e'})}.
47notation > "! 〈ident v1, ident v2〉 ← e ; e'"
48  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1}. λ${ident v2}. ${e'})}.
49notation > "! 〈ident v1, ident v2〉 ← e : ty ; e'"
50  with precedence 48 for @{'m_bind2 (${e} : $ty) (λ${ident v1}. λ${ident v2}. ${e'})}.
51notation > "! 〈ident v1 : ty1, ident v2 : ty2〉 ← e ; e'"
52  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. ${e'})}.
53notation < "vbox(! \nbsp 〈ident v1, ident v2〉 ← e ; break e')"
54  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1}. λ${ident v2}. ${e'})}.
55notation < "vbox(! \nbsp 〈ident v1 : ty1, ident v2 : ty2〉  \nbsp ←  \nbsp e ; break e')"
56  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. ${e'})}.
57notation > "! 〈ident v1, ident v2, ident v3〉 ← e ; e'"
58  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1}. λ${ident v2}. λ${ident v3}. ${e'})}.
59notation > "! 〈ident v1, ident v2, ident v3〉 ← e : ty ; e'"
60  with precedence 48 for @{'m_bind3 (${e} : ${ty}) (λ${ident v1}. λ${ident v2}. λ${ident v3}. ${e'})}.
61notation > "! 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e ; e'"
62  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. λ${ident v3} : ${ty3}. ${e'})}.
63notation < "vbox(! \nbsp 〈ident v1, ident v2, ident v3〉 \nbsp ← \nbsp e ; break e')"
64  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1}. λ${ident v2}. λ${ident v3}. ${e'})}.
65notation < "vbox(! \nbsp 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 \nbsp ← e \nbsp ; break e')"
66  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. λ${ident v3} : ${ty3}. ${e'})}.
67
68(* dependent pair versions *)
69notation > "! «ident v1, ident v2» ← e ; e'"
70  with precedence 48 for
71  @{'m_bind ${e} (λ${fresh p_sig}.
72    match ${fresh p_sig} with [mk_Sig ${ident v1} ${ident v2} ⇒ ${e'}])}.
73notation < "vbox(! \nbsp «ident v1, ident v2» \nbsp ← \nbsp e ; break e')"
74  with precedence 48 for
75  @{'m_bind ${e} (λ${fresh p_sig}.
76    match ${fresh p_sig} with [mk_Sig ${ident v1} ${ident v2} ⇒ ${e'}])}.
77   
78notation > "! «ident v1, ident v2, ident H» ← e ; e'"
79  with precedence 48 for
80  @{'m_sigbind2 ${e} (λ${ident v1},${ident v2},${ident H}. ${e'})}.
81notation < "vbox(! \nbsp «ident v1, ident v2, ident H» \nbsp ← \nbsp e ; e')"
82  with precedence 48 for
83  @{'m_sigbind2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.
84    λ${ident H} : ${ty3}. ${e'})}.
85   
86 
87(*alternative do notation for backward compatibility *)
88notation > "'do'_ e; e'"
89  with precedence 48 for @{'m_bind ${e} (λ_. ${e'})}.
90notation > "'do' ident v ← e; e'"
91  with precedence 48 for @{'m_bind ${e} (λ${ident v}. ${e'})}.
92notation > "'do' ident v : ty ← e; e'"
93  with precedence 48 for @{'m_bind ${e} (λ${ident v} : ${ty}. ${e'})}.
94notation > "'do' ident v ← e : ty ; e'"
95  with precedence 48 for @{'m_bind (${e} : ${ty}) (λ${ident v}.${e'})}.
96notation > "'do' ident v : ty ← e : ty' ; e'"
97  with precedence 48 for @{'m_bind (${e} : ${ty'}) (λ${ident v} : ${ty}. ${e'})}.
98notation > "'do' 〈ident v1, ident v2〉 ← e ; e'"
99  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1}. λ${ident v2}. ${e'})}.
100notation > "'do' 〈ident v1 : ty1, ident v2 : ty2〉 ← e ; e'"
101  with precedence 48 for @{'m_bind2 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. ${e'})}.
102notation > "'do' 〈ident v1, ident v2, ident v3〉 ← e ; e'"
103  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1}. λ${ident v2}. λ${ident v3}. ${e'})}.
104notation > "'do' 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e ; e'"
105  with precedence 48 for @{'m_bind3 ${e} (λ${ident v1} : ${ty1}. λ${ident v2} : ${ty2}. λ${ident v3} : ${ty3}. ${e'})}.
106
107(* dependent pair versions *)
108notation > "'do' «ident v1, ident v2» ← e ; e'"
109  with precedence 48 for
110  @{'m_bind ${e} (λ${fresh p_sig}.
111    match ${fresh p_sig} with [mk_Sig ${ident v1} ${ident v2} ⇒ ${e'}])}.
112
113notation > "'do' «ident v1, ident v2, ident H» ← e ; e'"
114  with precedence 48 for
115  @{'m_sigbind2 ${e} (λ${ident v1},${ident v2},${ident H}. ${e'})}.
116
117notation > "'do' ❬ident v1, ident v2❭ ← e ; e'"
118  with precedence 48 for
119  @{'m_bind ${e} (λ${fresh p_sig}.
120    match ${fresh p_sig} with [mk_DPair ${ident v1} ${ident v2} ⇒ ${e'}])}.
121
122notation > "'return' t" with precedence 49 for @{'m_return $t}.
123notation < "'return' \nbsp t" with precedence 49 for @{'m_return $t}.
124
125interpretation "monad bind" 'm_bind m f = (m_bind ? ? ? m f).
126interpretation "monad return" 'm_return x = (m_return ? ? x).
127
128
129record MonadProps : Type[1] ≝
130  { max_def :> Monad
131  ; m_return_bind : ∀X,Y.∀x : X.∀f : X → max_def Y. ! y ← return x ; f y = f x
132  ; m_bind_return : ∀X.∀m : max_def X.! x ← m ; return x = m
133  ; m_bind_bind : ∀X,Y,Z. ∀m : max_def X.∀f : X → max_def Y.
134      ∀g : Y → max_def Z.! y ← (! x ← m ; f x) ; g y = ! x ← m; ! y ← f x ; g y
135  ; m_bind_ext_eq : ∀X,Y.∀m : max_def X.∀f,g : X → max_def Y.
136      (∀x.f x = g x) → ! x ← m ; f x = ! x ← m ; g x
137  }.
138
139record SetoidMonadProps : Type[1] ≝
140  { smax_def :> Monad
141  ; sm_eq : ∀X.relation (smax_def X)
142  ; sm_eq_refl : ∀X.reflexive ? (sm_eq X)
143  ; sm_eq_trans : ∀X.transitive ? (sm_eq X)
144  ; sm_eq_sym : ∀X.symmetric ? (sm_eq X)
145  ; sm_return_proper : ∀X,x.sm_eq X (return x) (return x)
146  ; sm_bind_proper : ∀X,Y,x,y,f,g.sm_eq X x y → (∀x.sm_eq Y (f x) (g x)) → sm_eq Y (x »= f) (y »= g)
147  ; sm_return_bind : ∀X,Y.∀x : X.∀f : X → smax_def Y.
148      sm_eq Y (! y ← return x ; f y) (f x)
149  ; sm_bind_return : ∀X.∀m : smax_def X.sm_eq X (! x ← m ; return x) m
150  ; sm_bind_bind : ∀X,Y,Z. ∀m : smax_def X.∀f : X → smax_def Y.
151      ∀g : Y → smax_def Z.sm_eq Z (! y ← (! x ← m ; f x) ; g y) (! x ← m; ! y ← f x ; g y)
152  }.
153
154definition setoid_of_monad : ∀M : SetoidMonadProps.∀X : Type[0].
155  Setoid ≝
156  λM,X.mk_Setoid (M X) (sm_eq M X) (sm_eq_refl M X) (sm_eq_trans M X) (sm_eq_sym M X).
157
158include "hints_declaration.ma".
159alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
160unification hint 0 ≔ M, X;
161M' ≟ smax_def M, S ≟ setoid_of_monad M X
162(*-----------------------------*)⊢
163monad M' X ≡ std_supp S.
164
165include "basics/lists/list.ma".
166
167definition m_map ≝ λM : Monad.λX,Y.λf : X → Y.λm : M X.
168  ! x ← m ; return f x.
169
170definition m_map2 ≝ λM : Monad.λX,Y,Z.λf : X → Y → Z.λm : M X.λn : M Y.
171  ! x ← m ; ! y ← n ; return f x y.
172 
173definition m_bind2 ≝ λM : Monad.λX,Y,Z.λm : M (X × Y).λf : X → Y → M Z.
174  ! p ← m ; f (\fst p) (\snd p).
175
176interpretation "monad bind2" 'm_bind2 m f = (m_bind2 ? ? ? ? m f).
177
178definition m_bind3 :
179  ∀M : Monad.∀X,Y,Z,W.M (X×Y×Z) → (X → Y → Z → M W) → M W ≝
180  λM,X,Y,Z,W,m,f.
181  ! p ← m ; f (\fst (\fst p)) (\snd (\fst p)) (\snd p).
182
183interpretation "monad bind3" 'm_bind3 m f = (m_bind3 ? ? ? ? ? m f).
184
185definition m_join : ∀M : Monad.∀X.M (M X) → M X ≝
186  λM,X,m.! x ← m ; x.
187
188definition m_sigbind2 :
189  ∀M : Monad.∀A,B,C.∀P:A×B → Prop. M (Σx:A×B.P x) →
190      (∀a,b. P 〈a,b〉 → M C) → M C ≝
191λM,A,B,C,P,e,f.
192    ! e_sig ← e ;
193    match e_sig with
194    [ mk_Sig p p_prf ⇒
195      match p return λx.x = p → ? with
196      [ mk_Prod a b ⇒
197      λeq_p.  f a b (eq_ind_r ?? (λx.λ_.P x) p_prf ? eq_p)
198      ](refl …)
199    ].
200
201interpretation "monad sigbind2" 'm_sigbind2 m f = (m_sigbind2 ????? m f).
202
203definition m_list_map :
204  ∀M : Monad.∀X,Y.(X → M Y) → list X → M (list Y) ≝
205  λM,X,Y,f,l.foldr … (λel,macc.! r ← f el; !acc ← macc; return (r :: acc)) (return [ ]) l.
206
207definition m_list_map_sigma :
208  ∀M : Monad.∀X,Y,P.(X → M (Σy : Y.P y)) → list X → M (Σl : list Y.All ? P l) ≝
209  λM,X,Y,P,f,l.foldr … (λel,macc.
210    ! «r, r_prf» ← f el ;
211    ! «acc, acc_prf» ← macc ;
212    return (mk_Sig ?? (r :: acc) ?))
213    (return (mk_Sig … [ ] ?)) l. % assumption qed.
214
215definition m_bin_op :
216  ∀M : Monad.∀X,Y,Z.(X → Y → Z) → M X → M Y → M Z ≝
217  λM,X,Y,Z,op,m,n. ! x ← m ; ! y ← n ; return op x y.
218
219unification hint 0 ≔ M, X, Y, Z, m, f ;
220  P ≟ Prod X Y, F ≟ λp.f (\fst p) (\snd p) ⊢
221m_bind M P Z m F ≡ m_bind2 M X Y Z m f.
222
223unification hint 0 ≔ M, X, Y, Z, W, m, f ;
224  P' ≟ Prod X Y, P ≟ Prod P' Z, F ≟ λp.f (\fst (\fst p)) (\snd (\fst p)) (\snd p) ⊢
225m_bind M P W m F ≡ m_bind3 M X Y Z W m f.
226
227definition MakeMonadProps : ?→?→?→?→?→?→?→MonadProps ≝ λmonad,m_return,m_bind.
228mk_MonadProps (mk_Monad monad m_return m_bind).
229
230definition MakeSetoidMonadProps : ?→?→?→?→?→?→?→?→?→?→?→?→SetoidMonadProps ≝
231  λmonad,m_return,m_bind.
232  mk_SetoidMonadProps (mk_Monad monad m_return m_bind).
233 
234include "basics/russell.ma".
235 
236record MonadPred (M : Monad) : Type[1] ≝
237  { m_pred :2> ∀X.(X → Prop) → (M X → Prop)
238  ; mp_return : ∀X,P,x.P x → m_pred X P (return x)
239  ; mp_bind : ∀X,Y,Pin,Pout,m.m_pred X Pin m →
240      ∀f.(∀x.Pin x → m_pred Y Pout (f x)) →
241                  m_pred ? Pout (m_bind … m f)
242  ; m_pred_mp : ∀X.modus_ponens ?? (m_pred X)
243  }.
244
245record InjMonadPred (M : Monad) : Type[1] ≝
246  { im_pred :> MonadPred M
247  ; mp_inject : ∀X.∀P : X → Prop.(Σm.im_pred P m) → M (Σx.P x)
248  ; mp_inject_eq : ∀X,P,m.pi1 ?? m = ! x ← mp_inject X P m ; return pi1 … x
249  }.
250
251coercion coerc_mp_inject : ∀M.∀MP:InjMonadPred M.
252  ∀X.∀P : X → Prop.∀m : Σm.MP P m.M (Σx.P x) ≝
253  mp_inject on _m : Sig (monad ??) (λm.im_pred ??? m) to monad ? (Sig ? (λx.? x)).
254
255lemma mp_inject_bind : ∀M : MonadProps.∀MP,X,P,Y.∀m.∀f : X → M Y.
256  ! x ← mp_inject M MP X P m ; f (pi1 … x) = ! x ← pi1 … m ; f x.
257#M#MP#X#P#Y#m#f >mp_inject_eq >m_bind_bind @m_bind_ext_eq #x >m_return_bind % qed.
258
259record MonadRel (M1 : Monad) (M2 : Monad) : Type[1] ≝
260  { m_rel :3> ∀X,Y.(X → Y → Prop) → (M1 X → M2 Y → Prop)
261  ; mr_return : ∀X,Y,rel,x,y.rel x y → m_rel X Y rel (return x) (return y)
262  ; mr_bind : ∀X,Y,Z,W,relin,relout,m,n.m_rel X Y relin m n → ∀f,g.(∀x,y.relin x y → m_rel Z W relout (f x) (g y)) →
263                  m_rel ?? relout (m_bind … m f) (m_bind … n g)
264  ; m_rel_mp : ∀X,Y.rel_modus_ponens ???? (m_rel X Y)
265  }.
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