1 | (**************************************************************************) |
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2 | (* ___ *) |
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3 | (* ||M|| *) |
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4 | (* ||A|| A project by Andrea Asperti *) |
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5 | (* ||T|| *) |
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6 | (* ||I|| Developers: *) |
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7 | (* ||T|| The HELM team. *) |
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8 | (* ||A|| http://helm.cs.unibo.it *) |
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9 | (* \ / *) |
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10 | (* \ / This file is distributed under the terms of the *) |
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11 | (* v GNU General Public License Version 2 *) |
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12 | (* *) |
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13 | (**************************************************************************) |
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14 | |
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15 | include "variable.ma". |
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16 | |
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17 | inductive guard_combinators : Type[0] ≝ |
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18 | | push_var: variable → guard_combinators |
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19 | | push_const : DeqNat → guard_combinators |
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20 | | push_plus : guard_combinators |
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21 | | push_minus : guard_combinators |
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22 | | push_eq : guard_combinators |
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23 | | push_not : guard_combinators. |
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24 | |
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25 | definition eq_guard_combinator : guard_combinators → guard_combinators → bool ≝ |
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26 | λx.match x with |
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27 | [ push_var v1⇒ λy.match y with [push_var v2 ⇒ eqb v1 v2 | _ ⇒ false] |
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28 | | push_const n1 ⇒ λy.match y with [push_const n2 ⇒ eqb n1 n2 | _ ⇒ false] |
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29 | | push_plus ⇒ λy.match y with [push_plus ⇒ true | _ ⇒ false ] |
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30 | | push_minus ⇒ λy.match y with [push_minus ⇒ true | _⇒ false ] |
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31 | | push_eq ⇒ λy.match y with [push_eq ⇒ true | _ ⇒ false ] |
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32 | | push_not ⇒ λy.match y with [push_not ⇒ true | _ ⇒ false ] |
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33 | ]. |
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34 | |
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35 | lemma eq_guard_combinator_elim : ∀P : bool → Prop. |
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36 | ∀c1,c2.(c1 = c2 → P true) → (c1 ≠ c2 → P false) → |
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37 | P(eq_guard_combinator c1 c2). |
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38 | #P * |
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39 | [#v1 | #n1 ] |
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40 | * |
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41 | [1,7,13,19,25,31: #v2 |2,8,14,20,26,32: #n2] |
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42 | normalize #H1 #H2 try @H1 try @H2 // |
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43 | try(% #EQ destruct) |
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44 | [ @(eqb_elim … v1 v2) [#EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %] |
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45 | | @(eqb_elim … n1 n2) [#EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %] |
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46 | ] |
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47 | qed. |
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48 | |
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49 | inductive stack_seq:Type[0]≝ |
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50 | | pop_Ass: variable → stack_seq |
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51 | | push : guard_combinators → stack_seq |
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52 | | popreg : variable → stack_seq. |
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53 | |
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54 | definition eq_stack_seq : stack_seq → stack_seq → bool ≝ |
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55 | λx.match x with |
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56 | [ pop_Ass v1 ⇒ λy.match y with [pop_Ass v2 ⇒ eqb v1 v2 | _ ⇒ false ] |
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57 | | push c1 ⇒ λy.match y with [push c2 ⇒ eq_guard_combinator c1 c2 | _ ⇒ false] |
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58 | | popreg v⇒ λy.match y with [popreg v' ⇒ eqb v v' | _ ⇒ false] |
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59 | ]. |
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60 | |
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61 | lemma eq_stack_seq_elim : ∀P : bool → Prop. |
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62 | ∀x1,x2.(x1 = x2 → P true) → (x1 ≠ x2 → P false) → |
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63 | P(eq_stack_seq x1 x2). |
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64 | #P * [#v1 | #c1 | #v1 ] * [1,4,7: #v2 |2,5,8: #c2 |*: #v2] #H1 #H2 try(@H2 % #EQ destruct) |
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65 | [ normalize @(eqb_elim … v1 v2) [ #EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %] |
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66 | | whd in match eq_stack_seq; normalize nodelta @eq_guard_combinator_elim /2/ |
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67 | * #H @H2 % #EQ destruct @H % |
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68 | | normalize @(eqb_elim … v1 v2) [ #EQ destruct @H1 % | * #H @H2 % #EQ destruct @H %] |
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69 | ] |
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70 | qed. |
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71 | |
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72 | definition DeqStackSeq ≝ mk_DeqSet stack_seq eq_stack_seq ?. |
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73 | @hide_prf #x #y @eq_stack_seq_elim |
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74 | [ #EQ destruct % // | * #H % [ #EQ destruct | #EQ destruct cases(H (refl …)) ] |
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75 | qed. |
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76 | |
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77 | definition DeqGuardCombinator ≝ mk_DeqSet guard_combinators eq_guard_combinator ?. |
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78 | @hide_prf #x #y @eq_guard_combinator_elim |
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79 | [ #EQ destruct % // | * #H % [ #EQ destruct | #EQ destruct cases(H (refl …)) ] |
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80 | qed. |
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81 | |
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82 | definition stack_instr_params ≝ mk_instr_params DeqStackSeq DeqFalse |
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83 | DeqUnit (DeqSet_List DeqGuardCombinator) DeqUnit DeqUnit. |
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84 | |
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85 | definition stack_env_params ≝ mk_env_params ℕ. |
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86 | |
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87 | definition stack_state_params:state_params≝ |
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88 | mk_state_params stack_instr_params stack_env_params flat_labels frame_store_t (*DeqEnv*). |
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89 | |
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90 | definition push : frame_store_t → DeqNat → option frame_store_t ≝ |
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91 | λs,n.match \fst s with |
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92 | [nil ⇒ None ? |
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93 | |cons x xs ⇒ return 〈(n::x) :: xs,\snd s〉 |
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94 | ]. |
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95 | |
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96 | definition pop : frame_store_t → option (DeqNat × frame_store_t) ≝ |
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97 | λs.match \fst s with |
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98 | [nil ⇒ None ? |
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99 | |cons x xs ⇒ match x with |
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100 | [ nil ⇒ None ? |
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101 | | cons y ys ⇒ return 〈y,〈ys::xs,\snd s〉〉 |
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102 | ] |
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103 | ]. |
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104 | |
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105 | definition eval_combinators : guard_combinators → frame_store_t→ option frame_store_t≝ |
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106 | λc,s.match c with |
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107 | [ push_var x ⇒ ! curr_env ← frame_current_env s; |
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108 | ! n ← read_frame curr_env (to_shift+x) O; |
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109 | push s n |
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110 | | push_const n ⇒ push s n |
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111 | | push_plus ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (val2 + val1) |
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112 | | push_minus ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (val2 - val1) |
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113 | | push_eq ⇒ ! 〈val1,st1〉 ← pop s; ! 〈val2,st2〉 ← pop st1; push st2 (if eqb val2 val1 then 1 else O) |
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114 | | push_not ⇒ ! 〈val1,st1〉 ← pop s;push st1 (1-val1) |
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115 | ]. |
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116 | |
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117 | |
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118 | definition stack_eval_seq: stack_seq→frame_store_t→ option frame_store_t≝ |
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119 | λi,s.match i with |
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120 | [ pop_Ass v ⇒ ! 〈val1,st1〉 ← pop s;frame_assign_var st1 (to_shift +v) val1 |
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121 | | push c ⇒ eval_combinators c s |
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122 | | popreg v ⇒ frame_assign_var s (to_shift +v) (\snd (\snd s)) |
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123 | ]. |
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124 | |
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125 | definition stack_eval_cond_cond: |
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126 | (list guard_combinators) → frame_store_t→ option (bool × frame_store_t)≝ |
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127 | λl,st.! fin_St ← m_fold Option … eval_combinators l st; |
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128 | !〈val,st2〉← pop fin_St; |
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129 | return 〈eqb val 1,st2〉. |
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130 | |
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131 | |
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132 | definition stack_signature≝signature stack_state_params stack_state_params. |
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133 | |
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134 | definition stack_eval_call:stack_signature→ |
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135 | frame_store_t → (option frame_store_t)≝ |
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136 | λsgn.λst. |
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137 | !〈n,st1〉 ← pop st; |
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138 | frame_assign_var 〈(list_n … O (to_shift + (f_pars … sgn)))::(\fst st1),(\snd st1)〉 to_shift n. |
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139 | |
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140 | definition stack_return_call:frame_store_t→ (option frame_store_t)≝ |
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141 | λs.match (\fst s) with |
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142 | [nil⇒ None ? |
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143 | |cons hd tl⇒ !〈n,st1〉 ← pop s; |
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144 | return 〈tl,〈(\fst (\snd s)),n〉〉 |
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145 | ]. |
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146 | |
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147 | definition stack_init_store ≝ λn : ℕ.〈[list_n … O (to_shift + n)],〈O,O〉〉. |
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148 | |
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149 | |
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150 | definition stack_sem_state_params : ℕ → sem_state_params stack_state_params ≝ |
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151 | λn.mk_sem_state_params stack_state_params stack_eval_seq frame_eval_io |
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152 | (λ_.stack_eval_cond_cond (nil ?)) stack_eval_cond_cond (λsgn.λ_.stack_eval_call sgn) (λ_.stack_return_call) (stack_init_store n). |
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153 | |
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154 | (* Abitare tipo Instructions *) |
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155 | |
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156 | definition stack_Instructions≝Instructions stack_state_params flat_labels. |
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157 | |
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158 | |
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159 | definition stack_envitem≝ (env_item stack_env_params stack_instr_params flat_labels). |
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