source: LTS/stack.ma @ 3560

Last change on this file since 3560 was 3560, checked in by piccolo, 4 years ago

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1(**************************************************************************)
2(*       ___                                                              *)
3(*      ||M||                                                             *)
4(*      ||A||       A project by Andrea Asperti                           *)
5(*      ||T||                                                             *)
6(*      ||I||       Developers:                                           *)
7(*      ||T||         The HELM team.                                      *)
8(*      ||A||         http://helm.cs.unibo.it                             *)
9(*      \   /                                                             *)
10(*       \ /        This file is distributed under the terms of the       *)
11(*        v         GNU General Public License Version 2                  *)
12(*                                                                        *)
13(**************************************************************************)
14
15include "variable.ma".
16
17inductive guard_combinators : Type[0] ≝
18| push_var: variable → guard_combinators
19| push_const : DeqNat → guard_combinators
20| push_plus : guard_combinators
21| push_minus : guard_combinators
22| push_eq : guard_combinators
23| push_not : guard_combinators.
24
25definition eq_guard_combinator : guard_combinators → guard_combinators → bool ≝
26λx.match x with
27[ push_var v1⇒ λy.match y with [push_var v2 ⇒ eqb v1 v2 | _ ⇒ false]
28| push_const n1 ⇒ λy.match y with [push_const n2 ⇒ eqb n1 n2 | _ ⇒ false]
29| push_plus ⇒ λy.match y with [push_plus ⇒ true | _ ⇒ false ]
30| push_minus ⇒  λy.match y with [push_minus ⇒ true | _⇒  false ]
31| push_eq ⇒  λy.match y with [push_eq ⇒ true | _ ⇒ false ]
32| push_not ⇒ λy.match y with [push_not ⇒ true | _ ⇒ false ]
33].
34
35lemma eq_guard_combinator_elim : ∀P : bool → Prop.
36∀c1,c2.(c1 = c2 → P true) → (c1 ≠ c2 → P false) →
37P(eq_guard_combinator c1 c2).
38#P *
39[#v1 | #n1 ]
40*
41[1,7,13,19,25,31: #v2 |2,8,14,20,26,32: #n2]
42normalize #H1 #H2 try @H1 try @H2 //
43try(% #EQ destruct)
44[ @(eqb_elim … v1 v2) [#EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %]
45| @(eqb_elim … n1 n2) [#EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %]
46]
47qed.
48
49inductive stack_seq:Type[0]≝
50| pop_Ass: variable → stack_seq
51| push : guard_combinators → stack_seq.
52
53definition eq_stack_seq : stack_seq → stack_seq → bool ≝
54λx.match x with
55[ pop_Ass v1 ⇒ λy.match y with [pop_Ass v2 ⇒ eqb v1 v2 | _ ⇒ false ]
56| push c1 ⇒ λy.match y with [push c2 ⇒ eq_guard_combinator c1 c2 | _ ⇒ false]
57].
58
59lemma eq_stack_seq_elim : ∀P : bool → Prop.
60∀x1,x2.(x1 = x2 → P true) → (x1 ≠ x2 → P false) →
61P(eq_stack_seq x1 x2).
62#P * [#v1 | #c1] * [1,3: #v2 |*: #c2] #H1 #H2 try(@H2 % #EQ destruct)
63[ normalize @(eqb_elim … v1 v2) [ #EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %]
64| whd in match eq_stack_seq; normalize nodelta @eq_guard_combinator_elim /2/
65 * #H @H2 % #EQ destruct @H %
66]
67qed.
68
69definition DeqStackSeq ≝ mk_DeqSet stack_seq eq_stack_seq ?.
70@hide_prf #x #y @eq_stack_seq_elim
71[ #EQ destruct % // | * #H % [ #EQ destruct | #EQ destruct cases(H (refl …)) ]
72qed.
73
74definition DeqGuardCombinator ≝ mk_DeqSet guard_combinators eq_guard_combinator ?.
75@hide_prf #x #y @eq_guard_combinator_elim
76[ #EQ destruct % // | * #H % [ #EQ destruct | #EQ destruct cases(H (refl …)) ]
77qed.
78 
79definition stack_instr_params ≝ mk_instr_params DeqStackSeq (mk_DeqSet io_i io_i_eq io_i_eq_equality)
80 (mk_DeqSet unit (λ_.λ_.true) ?) (DeqSet_List DeqGuardCombinator) variable (mk_DeqSet unit (λ_.λ_.true) ?).
81@hide_prf ** % // qed.
82
83definition stack_state_params:state_params≝
84mk_state_params stack_instr_params imp_env_params flat_labels frame_store_t (*DeqEnv*).
85
86definition push : (store_type stack_state_params) → DeqNat → option (store_type stack_state_params) ≝
87λs,n.match \fst s with
88[nil ⇒ None ?
89|cons x xs ⇒ return 〈〈(n::(\fst x)),\snd x〉 :: xs,\snd s〉
90].
91
92definition pop : (store_type stack_state_params) → option (DeqNat × (store_type stack_state_params)) ≝
93λs.match \fst s with
94[nil ⇒ None ?
95|cons x xs ⇒ match \fst x with
96             [ nil ⇒ None ?
97             | cons y ys ⇒ return 〈y,〈〈ys,\snd x〉::xs,\snd s〉〉
98             ]
99             ].
100
101definition eval_combinators : guard_combinators → (store_type stack_state_params)→ option (store_type stack_state_params)≝
102λc,s.match c with
103[ push_var x ⇒ ! curr_env ← frame_current_env s;
104               ! n ← read_frame curr_env (to_shift+x) O;
105                 push s n
106| push_const n ⇒ push s n
107| push_plus ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (val1 + val2)
108| push_minus ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (val1 - val2)
109| push_eq ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (if eqb val1 val2 then 1 else O)
110| push_not ⇒ ! 〈val1,st1〉 ← pop s;push st1 (1-val1)
111].
112
113
114definition stack_eval_seq: (seq_instr stack_state_params)→(store_type stack_state_params)→ option (store_type stack_state_params)≝
115λi,s.match i with
116[ pop_Ass v ⇒ ! 〈val1,st1〉 ← pop s;frame_assign_var st1 (to_shift +v) val1
117| push c ⇒ eval_combinators c s
118].
119
120definition stack_eval_cond_cond:
121(list guard_combinators) → (store_type stack_state_params)→ option (bool × (store_type stack_state_params))≝
122λl,st.! fin_St ← m_fold Option … eval_combinators l st;
123      !〈val,st2〉← pop fin_St;
124      return 〈eqb val 1,st2〉.
125
126
127definition stack_eval_call:(signature stack_state_params stack_state_params)→ act_params_type stack_state_params →
128frame_store_t → (option frame_store_t)≝
129λsgn,var,st.
130  match sgn with
131  [ mk_signature fun fpt rt ⇒
132    !〈n,st1〉 ← pop st;
133    frame_assign_var 〈(〈(nil ?),var〉::(\fst st1)),(\snd st1)〉 fpt n
134  ].
135
136definition stack_return_call:frame_store_t→ (option frame_store_t)≝
137λs.match (\fst s) with
138[nil⇒ None ?
139|cons hd tl⇒ let 〈env,v〉≝ hd in
140             !〈n,st1〉 ← pop s;
141             frame_assign_var 〈tl,\snd s〉 v n
142].
143 
144
145
146definition stack_sem_state_params : sem_state_params stack_state_params ≝ mk_sem_state_params stack_state_params stack_eval_seq frame_eval_io
147(λ_.stack_eval_cond_cond (nil ?)) stack_eval_cond_cond stack_eval_call (λ_.stack_return_call) frame_init_store.
148
149  (* Abitare tipo Instructions *)
150 
151definition stack_Instructions≝Instructions stack_state_params flat_labels.
152
153definition stack_signature≝signature stack_state_params stack_state_params.
154
155
156definition stack_envitem≝ (env_item imp_env_params stack_instr_params flat_labels).
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