1 | |
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2 | (**************************************************************************) |
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3 | (* ___ *) |
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4 | (* ||M|| *) |
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5 | (* ||A|| A project by Andrea Asperti *) |
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6 | (* ||T|| *) |
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7 | (* ||I|| Developers: *) |
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8 | (* ||T|| The HELM team. *) |
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9 | (* ||A|| http://helm.cs.unibo.it *) |
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10 | (* \ / *) |
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11 | (* \ / This file is distributed under the terms of the *) |
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12 | (* v GNU General Public License Version 2 *) |
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13 | (* *) |
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14 | (**************************************************************************) |
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15 | |
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16 | |
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17 | include "joint/StatusSimulationHelper.ma". |
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18 | include "ERTL/ERTLtoERTLptrUtils.ma". |
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19 | |
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20 | |
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21 | |
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22 | (* |
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23 | lemma get_internal_function_from_ident_ok : |
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24 | ∀p : sem_params. ∀globals : list ident. ∀ge : genv_t (joint_function p globals). |
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25 | ∀bl,f,fn. fetch_internal_function ? ge bl = return 〈f,fn〉 → |
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26 | get_internal_function_from_ident p globals ge f= return fn. |
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27 | #p #globals #ge #bl #f #fn #EQf |
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28 | @('bind_inversion EQf) * #f1 * #fn1 whd in match fetch_function; |
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29 | normalize nodelta #H lapply(opt_eq_from_res ???? H) -H #H @('bind_inversion H) -H |
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30 | #f2 #EQf2 #H @('bind_inversion H) -H #fn2 #EQfn2 whd in ⊢ (??%% → ??%% → ?); |
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31 | #EQ1 #EQ2 destruct whd in match get_internal_function_from_ident; normalize nodelta |
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32 | >(symbol_of_block_rev … EQf2) >m_return_bind |
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33 | cut(code_block_of_block bl = return bl) |
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34 | [ whd in match code_block_of_block; normalize nodelta @match_reg_elim |
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35 | [ >(pi2 ?? bl) #ABS destruct] elim bl #bl1 #EQ #prf % ] #EQbl >EQbl |
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36 | >m_return_bind >EQf % |
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37 | qed. |
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38 | *) |
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39 | |
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40 | |
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41 | |
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42 | |
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43 | (* |
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44 | whd in match sigma_frames_opt; whd in match m_list_map; normalize nodelta |
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45 | whd in match sigma_state in ⊢ (???????%%); normalize nodelta |
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46 | cases(st_frms … st) normalize nodelta [ @res_preserve_error1] |
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47 | * #loc_mem #bl #tl inversion(sigma_frames ????) |
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48 | [ #_ normalize nodelta @res_preserve_error1] * #loc_mem1 #bl1 #tl1 |
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49 | whd in match sigma_frames; normalize nodelta inversion(sigma_frames_opt ????) |
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50 | [ #_ normalize nodelta #_ #ABS destruct] #l whd in match sigma_frames_opt; |
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51 | whd in match m_list_map; normalize nodelta whd in match (foldr ?????); |
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52 | normalize nodelta inversion(fetch_internal_function ???) |
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53 | [2: #e #_ whd in ⊢ (??%% → ?); #ABS destruct] * #id1 #fn1 #EQfn1 >m_return_bind |
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54 | normalize nodelta #H @('bind_inversion H) -H #l1 |
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55 | change with (sigma_frames_opt ???? = ? → ?) #EQl1 |
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56 | cut (sigma_frames prog f_lbls f_regs tl = l1) |
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57 | [whd in match sigma_frames; normalize nodelta >EQl1 %] #EQl11 |
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58 | cases l [ whd in ⊢ (??%? → ?); #EQ destruct] #x #y |
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59 | whd in ⊢ (??%? → ?); #EQ1 #_ #EQ2 destruct @mfr_bind1 |
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60 | [2: whd in match set_regs; whd in match set_frms; normalize nodelta |
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61 | >EQl1 in ⊢ (???????%?); normalize nodelta |
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62 | cut(sigma_regs prog f_lbls (added_registers ERTL (prog_var_names … prog) fn1 (f_regs bl1)) |
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63 | 〈loc_mem,\snd (regs … st)〉 = 〈map RegisterTag beval beval loc_mem |
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64 | (λbv:beval.sigma_beval prog f_lbls bv)∖ |
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65 | added_registers ERTL (prog_var_names … prog) fn1 (f_regs bl1), |
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66 | \snd (sigma_regs prog f_lbls restr (regs ERTLptr_semantics st))〉) |
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67 | [ whd in match sigma_regs; normalize nodelta @eq_f2 [ %] cases(regs ? st) |
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68 | #x #y %] #EQ <EQ -EQ |
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69 | change with (sigma_state ???? (set_regs ERTL_semantics 〈loc_mem,\snd (regs … st)〉 |
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70 | (set_frms ERTL_semantics tl st))) in ⊢ (???????(??%)?); |
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71 | @pop_ra_ok | |
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72 | | * #st1 #pc1 @if_elim [2: #_ @res_preserve_error1] normalize nodelta |
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73 | @eq_block_elim [2: #_ *] #EQbl1 * @if_elim |
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74 | [2: >EQbl1 @eq_block_elim [#_ *] * #H @⊥ @H <pc_block_eq [%] % |
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75 | cases bl1 in EQbl1 EQfn1; #p1 #p2 #EQ destruct lapply p2 |
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76 | whd in match sigma_stored_pc; normalize nodelta cases(sigma_pc_opt ???) |
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77 | normalize nodelta [2: #pc2] #p2 [#_ #EQ destruct] |
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78 | >fetch_internal_function_no_zero [2: %] #EQ destruct |
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79 | ] @eq_block_elim [2: #_ *] #EQbl11 * @mfr_return_eq1 normalize nodelta |
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80 | cases bl1 in EQbl11 EQfn1; #p1 #p2 #EQ destruct |
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81 | lapply p2 cases(pc_block pc1) #p11 #p22 #e #EQfn1 >EQfn1 |
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82 | % |
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83 | ] |
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84 | qed. |
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85 | *) |
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86 | |
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87 | |
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88 | definition ERTLptrStatusSimulation : |
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89 | ∀ prog : ertl_program. |
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90 | let trans_prog ≝ ertl_to_ertlptr prog in |
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91 | ∀stack_sizes.∀ f_lbls : lbl_funct. ∀ f_regs : regs_funct. |
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92 | ∀f_bl_r. |
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93 | b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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94 | translate_data prog f_bl_r f_lbls f_regs → |
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95 | status_rel (ERTL_status prog stack_sizes) (ERTLptr_status trans_prog stack_sizes) ≝ |
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96 | λprog,stack_sizes,f_lbls,f_regs,f_lb_r,good. |
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97 | let trans_prog ≝ ertl_to_ertlptr prog in |
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98 | mk_status_rel ?? |
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99 | (* sem_rel ≝ *) (λs1:ERTL_status prog stack_sizes |
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100 | .λs2:ERTLptr_status trans_prog stack_sizes |
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101 | .s1=sigma_state_pc prog f_lbls f_regs s2) |
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102 | (* call_rel ≝ *) |
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103 | (λs1:Σs:ERTL_status prog stack_sizes |
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104 | .as_classifier (ERTL_status prog stack_sizes) s cl_call |
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105 | .λs2:Σs:ERTLptr_status trans_prog stack_sizes |
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106 | .as_classifier (ERTLptr_status trans_prog stack_sizes) s cl_call |
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107 | . |
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108 | pc (mk_prog_params ERTL_semantics prog stack_sizes) s1 |
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109 | =sigma_stored_pc prog f_lbls |
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110 | (pc |
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111 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_sizes) |
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112 | s2)) |
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113 | (* sim_final ≝ *) ?. |
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114 | cases daemon |
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115 | qed. |
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116 | |
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117 | include "joint/semantics_blocks.ma". |
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118 | |
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119 | |
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120 | |
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121 | lemma as_label_ok : ∀ prog : ertl_program. |
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122 | let trans_prog ≝ ertl_to_ertlptr prog in |
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123 | ∀ f_lbls,f_regs,stack_sizes. |
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124 | ∀f_bl_r. |
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125 | b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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126 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs → |
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127 | ∀st,fn,id,stmt. |
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128 | fetch_statement ERTL_semantics (prog_var_names … prog) |
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129 | (globalenv_noinit … prog) (pc … (sigma_state_pc prog f_lbls f_regs st)) = |
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130 | return 〈id,fn,stmt〉 → |
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131 | as_label (ERTLptr_status trans_prog stack_sizes) st = as_label |
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132 | (ERTL_status prog stack_sizes) (sigma_state_pc prog f_lbls f_regs st). |
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133 | #prog #f_lbls #f_regs #stack_size #f_lb_r #good #st #fn #id #stmt #EQfetch |
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134 | whd in match as_label; normalize nodelta change with (pc ? ?) in ⊢ (??(??%)(??%)); |
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135 | cases(b_graph_transform_program_fetch_statement … good … EQfetch) |
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136 | #init_data * #t_fn ** #EQt_fn whd in ⊢ (% → ?); cases(f_lb_r ?) |
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137 | normalize nodelta [2: #r #tl *] #EQinit destruct(EQinit) * #l_labs * #l_regs ** |
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138 | #_ #_ cases stmt in EQfetch; |
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139 | [ * [ #c | * [ #c_id | #c_ptr ] #c_arg #c_dest | #reg #lbl | #s ] #nxt | #fin | * ] |
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140 | #EQfetch normalize nodelta * #bl * [1,2,3,4,5: >if_merge_right in ⊢ (% → ?); try %] |
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141 | whd in ⊢ (%→?); cases l_regs normalize nodelta [2,4,5,8,10,12: [3: *] #x #y *] |
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142 | [1,2,4,5,6:| #r #tl whd in ⊢ (% → ?); cases tl -tl normalize nodelta [2: #r1 #tl1 *]] |
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143 | #EQ destruct(EQ) |
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144 | [1,2,3,4,6: * #pre_l * #mid1 * #mid2 * #post_l *** #EQmid1 |
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145 | whd in match map_eval in ⊢ (% → ?); normalize nodelta |
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146 | whd in ⊢ (?????%?? → ?); |
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147 | whd in match (seq_list_in_code ???????) in ⊢ (%→?); * |
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148 | [1,2,3,4: #EQ1 #EQ2 destruct(EQ1 EQ2) whd in EQmid1 : (???%); |
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149 | destruct(EQmid1) * #nxt1 * #EQt_stmt #_ #_ |
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150 | | * #lbl1 * #rest ** #EQ destruct(EQ) * #nxt1 * |
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151 | #EQt_stmt #_ #_ #_ #_ |
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152 | ] |
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153 | | * #pre_l * #mid ** #EQmid whd in ⊢ (% → ?); * #EQ destruct(EQ) |
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154 | whd in EQmid : (???%); destruct(EQmid) #EQ destruct(EQ) |
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155 | whd in ⊢ (% → ?); #EQt_stmt |
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156 | ] |
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157 | whd in ⊢ (??%%); >EQfetch normalize nodelta whd in match fetch_statement; |
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158 | normalize nodelta >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQt_fn; |
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159 | #EQt_fn >EQt_fn >m_return_bind >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQt_stmt; |
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160 | #EQt_stmt >EQt_stmt % |
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161 | qed. |
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162 | |
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163 | include alias "basics/lists/listb.ma". |
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164 | |
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165 | definition state_rel ≝ λprog : ertl_program.λf_lbls. |
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166 | λf_regs : (block → label → list register). |
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167 | λbl.λ_ : label.λst1,st2.∃f,fn. |
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168 | fetch_internal_function ? (globalenv_noinit … prog) bl = return 〈f,fn〉 ∧ |
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169 | let added ≝ added_registers ERTL (prog_var_names … prog) fn (f_regs bl) in |
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170 | st1 = (sigma_state prog f_lbls f_regs added st2). |
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171 | |
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172 | lemma make_ERTL_To_ERTLptr_good_state_relation : |
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173 | ∀prog : ertl_program. |
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174 | let p_out ≝ ertl_to_ertlptr prog in |
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175 | ∀stack_sizes. |
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176 | ∃init_regs : block → list register. |
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177 | ∃f_lbls : block → label → list label. |
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178 | ∃f_regs : block → label → list register. |
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179 | ∃good : b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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180 | (λglobals,fn.«translate_data globals fn,(refl …)») |
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181 | prog init_regs f_lbls f_regs. |
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182 | good_state_relation ERTL_semantics ERTLptr_semantics prog stack_sizes |
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183 | (λglobals,fn.«translate_data globals fn,(refl …)») init_regs f_lbls f_regs |
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184 | good (state_rel prog f_lbls f_regs) |
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185 | (sem_rel … (ERTLptrStatusSimulation prog … stack_sizes … good)). |
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186 | #prog #stack_sizes |
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187 | cases(make_b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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188 | (λglobals,fn.«translate_data globals fn,(refl …)») prog) |
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189 | #init_regs * #f_lbls * #f_regs #good %{init_regs} %{f_lbls} %{f_regs} %{good} |
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190 | % |
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191 | [ #st1 #st2 #f #fn whd in ⊢ (% → ?); #EQ destruct(EQ) #EQfn whd %{f} %{fn} |
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192 | >EQfn %{(refl …)} whd in match sigma_state_pc; normalize nodelta |
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193 | lapply EQfn >(fetch_ok_sigma_pc_ok … EQfn) #EQfn1 >EQfn1 % |
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194 | | #st1 #st2 #f #fn whd in ⊢ (% → ?); #EQ destruct(EQ) #EQfn @fetch_ok_sigma_pc_ok |
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195 | assumption |
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196 | | #st1 #st2 #f #fn whd in ⊢ (% → ?); #EQ destruct(EQ) #EQfn |
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197 | >(fetch_ok_sigma_last_pop_ok … f fn st2 EQfn) % |
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198 | | #st1 #st2 #pc #lp1 #lp2 #f #fn #EQfn * #f1 * #fn1 * >EQfn whd in ⊢ (??%% → ?); |
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199 | #EQ destruct(EQ) normalize nodelta #EQ destruct(EQ) #EQ destruct(EQ) whd |
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200 | whd in match sigma_state_pc; normalize nodelta >EQfn % |
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201 | | #st1 #st2 #f #fn #stmt #EQfetch whd in ⊢ (% → ?); #EQ destruct(EQ) |
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202 | lapply(as_label_ok … stack_sizes … good st2 … EQfetch) #EQ <EQ in ⊢ (??%?); % |
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203 | | normalize nodelta whd in ⊢ (% → % → % → ?); #st1 #st1' #st2 #f #fn |
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204 | #a #ltrue #lfalse #EQfetch cases(fetch_statement_inv … EQfetch) #EQfn #EQstmt |
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205 | whd in match eval_state; whd in match eval_statement_no_pc; |
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206 | normalize nodelta >EQfetch >m_return_bind normalize nodelta >m_return_bind |
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207 | whd in match eval_statement_advance; normalize nodelta |
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208 | change with (ps_reg_retrieve ??) in match (acca_retrieve ?????); |
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209 | #H lapply(err_eq_from_io ????? H) -H #H @('bind_inversion H) -H |
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210 | #bv whd in match set_no_pc; normalize nodelta #EQbv |
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211 | #H @('bind_inversion H) -H * #EQbool normalize nodelta |
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212 | [whd in match goto; normalize nodelta >(pc_of_label_eq … EQfn) |
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213 | >m_return_bind] whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
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214 | whd in ⊢ (% → ?); #EQ destruct(EQ) |
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215 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) in EQbv; |
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216 | whd in match sigma_state; normalize nodelta #EQbv |
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217 | cases(ps_reg_retrieve_ok ?????? EQbv) #bv1 * #EQbv1 #EQ destruct(EQ) |
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218 | cases(bool_of_beval_ok … EQbool) #b * #EQb #EQ destruct(EQ) #t_fn #EQt_fn |
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219 | whd %{(refl …)} #mid #EQt_stmt %{(st_no_pc … st2)} % |
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220 | [1,3: % |
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221 | [1,3: % |
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222 | [1,3: %{(refl …)} |
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223 | |*: % |
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224 | ] |
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225 | |*: % |
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226 | ] |
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227 | |*: normalize nodelta |
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228 | change with (ps_reg_retrieve ??) in match (acca_retrieve ?????); |
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229 | >EQbv1 >m_return_bind >EQb >m_return_bind normalize nodelta |
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230 | [ >(pc_of_label_eq … EQt_fn) >m_return_bind] |
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231 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) % |
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232 | ] |
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233 | whd %{f} %{fn} >EQfn %{(refl …)} >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
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234 | cases st2 * #frms #is #b #regs #m #pc2 #lp2 % |
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235 | | #st1 #st1' #st2 #f #fn #stmt #nxt #EQfetch whd in match eval_state; |
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236 | whd in match eval_statement_no_pc; normalize nodelta >EQfetch >m_return_bind |
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237 | normalize nodelta #H @('bind_inversion H) -H #st1_no_pc' #EQst1_no_pc' |
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238 | whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
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239 | whd in match (succ_pc ???); normalize nodelta |
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240 | whd in match (set_no_pc ???); normalize nodelta |
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241 | whd in ⊢ (%→?); #EQ destruct(EQ) |
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242 | whd in match sigma_state_pc in EQst1_no_pc'; normalize nodelta in EQst1_no_pc'; |
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243 | cases(fetch_statement_inv … EQfetch) >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
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244 | #EQfn #EQstmt >EQfn in EQst1_no_pc'; normalize nodelta #EQst1_no_pc' |
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245 | #t_fn #EQt_fn whd % [2: %{(refl …)} |] lapply(err_eq_from_io ????? EQst1_no_pc') |
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246 | -EQst1_no_pc' #EQst1_no_pc' |
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247 | cases(eval_seq_no_pc_no_call_ok prog f_lbls f_regs ???????? EQst1_no_pc') |
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248 | [ #st2_no_pc' * #EQst2_no_pc' #EQ destruct(EQ) %{st2_no_pc'} % |
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249 | [ whd in ⊢ (??%?); >EQst2_no_pc' % |
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250 | | whd %{f} %{fn} >EQfn %{(refl …)} whd in match sigma_state_pc; normalize nodelta |
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251 | >EQfn % |
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252 | ] |
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253 | | #lbl whd in match code_has_label; whd in match code_has_point; normalize nodelta |
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254 | inversion(stmt_at ????) [#_ *] #stmt1 #EQstmt1 #_ |
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255 | cut(fetch_statement ERTL_semantics (prog_var_names … prog) (globalenv_noinit … prog) |
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256 | (pc_of_point ERTL_semantics (pc_block (pc … st2)) lbl) = |
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257 | return 〈f,fn,stmt1〉) |
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258 | [ whd in ⊢ (??%?); >EQfn normalize nodelta >point_of_pc_of_point >EQstmt1 %] #EQf_stmt1 |
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259 | cases(b_graph_transform_program_fetch_statement_plus … good … EQf_stmt1) |
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260 | #init * #t_fn' ** #EQt_fn' whd in ⊢ (% → ?); cases(init_regs ?) [2: #x #y *] |
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261 | normalize nodelta #EQ destruct(EQ) * #labels * #registers |
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262 | ******* #EQlabels >point_of_pc_of_point change with (pc_block (pc … st2)) in match (pc_block ?); |
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263 | #EQregisters #_ #_ #_ #_ #fresh_regs #_ >EQregisters |
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264 | cases(excluded_middle_list ? (λreg.bool_to_Prop(¬(reg∈registers))) |
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265 | (get_used_registers_from_seq … (functs ERTL) stmt) ?) |
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266 | [3: #a #_ cases(¬a∈registers) // |1: #H assumption] #H @⊥ |
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267 | cases(Exists_All … H (regs_are_in_univers … (pi2 ?? fn) … EQstmt)) |
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268 | #reg ** #H1 #H2 @H1 -H1 @notb_Prop % #H1 lapply(fresh_regs … lbl) >EQregisters |
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269 | #H3 lapply(All_memb … H3 H1) ** #H4 #_ @H4 assumption |
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270 | ] |
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271 | ] |
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272 | qed. |
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273 | |
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274 | include alias "basics/lists/listb.ma". |
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275 | |
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276 | lemma eval_seq_no_call_ok : |
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277 | ∀prog. |
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278 | let trans_prog ≝ ertl_to_ertlptr prog in |
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279 | ∀ f_lbls,f_regs,stack_sizes. |
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280 | ∀f_bl_r. |
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281 | b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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282 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs → |
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283 | ∀st2 : state_pc ERTLptr_semantics. |
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284 | let st1 ≝ sigma_state_pc prog f_lbls f_regs st2 in |
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285 | ∀st1' : state_pc ERTL_semantics. |
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286 | ∀f,fn,stmt,nxt. |
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287 | fetch_statement ERTL_semantics |
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288 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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289 | (ev_genv (mk_prog_params ERTL_semantics prog stack_sizes)) |
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290 | (pc … st1) = |
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291 | return 〈f, fn, sequential … (step_seq ERTL … stmt) nxt〉 → |
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292 | eval_state ERTL_semantics |
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293 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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294 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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295 | st1 = |
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296 | return st1' → |
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297 | ∃st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
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298 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) |
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299 | st2 |
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300 | st2'. |
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301 | if taaf_non_empty … taf then True else |
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302 | (¬as_costed (ERTL_status prog stack_sizes) st1 ∨ |
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303 | ¬as_costed (ERTL_status prog stack_sizes) st1'). |
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304 | #prog #f_lbls #f_regs #stack_size #f_bl_r #good #st2 #st1' #f #fn #stmt #nxt |
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305 | #EQfetch whd in match eval_state; normalize nodelta >EQfetch >m_return_bind |
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306 | #H @('bind_inversion H)-H #st_no_pc |
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307 | whd in match eval_statement_no_pc; normalize nodelta #EQnopc |
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308 | whd in match eval_statement_advance; normalize nodelta |
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309 | whd in match set_no_pc; whd in match next; whd in match set_pc; |
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310 | normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct(EQ) lapply EQfetch |
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311 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) #EQfetch' |
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312 | cases(b_graph_transform_program_fetch_statement_plus … good … EQfetch') |
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313 | #init_data * #t_fn1 ** #EQt_fn1 whd in ⊢ (% → ?); cases (f_bl_r ?) normalize nodelta |
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314 | [2: #r #tl *] #EQ destruct(EQ) * #labs * #regs ******* #EQlabs #EQregs #_ #_ #_ #_ |
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315 | #fresh_regs * #bl * >if_merge_right in ⊢ (% → ?); [2: %] |
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316 | whd in ⊢ (% → ?); cases regs in EQregs; normalize nodelta [2: #x #y #_ *] #EQregs |
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317 | #EQ destruct(EQ) #eval_spec |
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318 | lapply(err_eq_from_io ????? EQnopc) -EQnopc >(fetch_stmt_ok_sigma_state_ok … EQfetch) |
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319 | #EQnopc |
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320 | cases(eval_seq_no_pc_no_call_ok prog f_lbls f_regs ???????? EQnopc) |
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321 | [ #stnopc'' * #EQstnopc'' #EQsem % |
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322 | [ % [@stnopc'' | @(succ_pc ERTL_semantics (pc … st2) nxt) | @(last_pop … st2)]] |
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323 | % |
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324 | [ whd in match sigma_state_pc; normalize nodelta @('bind_inversion EQfetch) |
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325 | * #f1 #fn1 >(fetch_stmt_ok_sigma_pc_ok … EQfetch) #EQfn1 |
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326 | #H @('bind_inversion H) -H #stmt1 #_ whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
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327 | >EQfn1 normalize nodelta >EQsem % ] |
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328 | % [ @(produce_trace_any_any_free_coerced) [ @f | @t_fn1 |6: @eval_spec ||| @EQt_fn1] |
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329 | whd in match repeat_eval_seq_no_pc; normalize nodelta whd in ⊢ (??%?); |
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330 | >EQstnopc'' %] @if_elim [#_ @I] #_ %1 % whd in match as_costed; |
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331 | normalize nodelta * #H @H whd in ⊢ (??%?); >EQfetch % |
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332 | | #lbl whd in match code_has_label; whd in match code_has_point; normalize nodelta |
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333 | inversion(stmt_at ????) [#_ *] #stmt1 #EQstmt1 #_ |
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334 | cut(fetch_statement ERTL_semantics (prog_var_names … prog) (globalenv_noinit … prog) |
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335 | (pc_of_point ERTL_semantics (pc_block (pc … st2)) lbl) = |
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336 | return 〈f,fn,stmt1〉) |
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337 | [ lapply(fetch_statement_inv … EQfetch') * #EQfn #_ whd in ⊢ (??%?); |
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338 | >EQfn normalize nodelta >point_of_pc_of_point >EQstmt1 %] #EQf_stmt1 |
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339 | cases(b_graph_transform_program_fetch_statement_plus … good … EQf_stmt1) |
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340 | #init * #t_fn' ** #EQt_fn' whd in ⊢ (% → ?); cases(f_bl_r ?) [2: #x #y *] |
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341 | normalize nodelta #EQ destruct(EQ) * #labels * #registers |
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342 | ******* #EQlabels >point_of_pc_of_point change with (pc_block (pc … st2)) in match (pc_block ?); |
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343 | #EQregisters #_ #_ #_ #_ #_ #_ >EQregisters |
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344 | cases(excluded_middle_list ? (λreg.bool_to_Prop(¬(reg∈registers))) |
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345 | (get_used_registers_from_seq … (functs ERTL) stmt) ?) |
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346 | [3: #a #_ cases(¬a∈registers) // |1: #H assumption] #H @⊥ |
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347 | lapply(fetch_statement_inv … EQfetch') * #_ normalize nodelta #EQstmt |
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348 | cases(Exists_All … H (regs_are_in_univers … (pi2 ?? fn) … EQstmt)) |
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349 | #reg ** #H1 #H2 @H1 -H1 @notb_Prop % #H1 lapply(fresh_regs … lbl) >EQregisters |
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350 | whd in ⊢ (% → ?); #H3 lapply(All_memb … H3 H1) ** #H4 #_ @H4 assumption |
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351 | ] |
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352 | qed. |
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353 | |
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354 | lemma eval_goto_ok : |
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355 | ∀prog : ertl_program. |
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356 | let trans_prog ≝ ertl_to_ertlptr prog in |
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357 | ∀ f_lbls,f_regs,stack_sizes. |
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358 | ∀f_bl_r. |
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359 | b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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360 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs → |
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361 | ∀st2. |
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362 | let st1 ≝ (sigma_state_pc prog f_lbls f_regs st2) in |
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363 | ∀st1',f,fn,nxt. |
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364 | fetch_statement ERTL_semantics … |
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365 | (globalenv_noinit ? prog) (pc … st1) = |
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366 | return 〈f, fn, final … (GOTO ERTL … nxt)〉 → |
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367 | eval_state ERTL_semantics |
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368 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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369 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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370 | st1 = |
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371 | return st1' → |
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372 | ∃ st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
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373 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) |
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374 | st2 |
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375 | st2'. |
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376 | bool_to_Prop (taaf_non_empty … taf). |
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377 | #prog #f_lbls #f_regs #stack_size #f_bl_r #good #st2 #st1' #f #fn #nxt #EQfetch |
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378 | whd in match eval_state; normalize nodelta >EQfetch >m_return_bind |
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379 | whd in match eval_statement_no_pc; normalize nodelta |
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380 | >m_return_bind whd in match eval_statement_advance; whd in match set_no_pc; |
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381 | whd in match goto; normalize nodelta #H lapply(err_eq_from_io ????? H) -H |
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382 | #H @('bind_inversion H) -H #new_pc lapply(fetch_statement_inv … EQfetch) * |
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383 | #EQfn #_ >(pc_of_label_eq … EQfn) whd in ⊢ (??%% → ??%% → ?); |
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384 | #EQ1 #EQ2 destruct(EQ1 EQ2) |
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385 | % [ % [ @st2 | @(mk_program_counter (pc_block (pc … st2)) |
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386 | (offset_of_point ERTL_semantics nxt)) | @(last_pop … st2)]] |
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387 | % [ >(fetch_stmt_ok_sigma_state_ok … EQfetch) >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
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388 | >(fetch_stmt_ok_sigma_last_pop_ok … EQfetch) whd in match sigma_state_pc; |
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389 | normalize nodelta >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQfn; #EQfn |
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390 | >EQfn %] |
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391 | cases(b_graph_transform_program_fetch_statement … good … EQfetch) |
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392 | #init_data * #t_fn1 ** #EQt_fn1 whd in ⊢ (% → ?); cases (f_bl_r ?) normalize nodelta |
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393 | [2: #r #tl *] #EQ destruct(EQ) * #labs ** |
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394 | [2: #hd #tl ** #_ #_ ** #pre #inst * whd in ⊢ (%→?); *] ** #EQlabs #EQf_regs |
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395 | whd in match translate_fin_step; normalize nodelta * #bl * |
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396 | whd in ⊢ (% → ?); #EQ destruct(EQ) ** |
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397 | [2: #lb #tl * #mid ** #EQmid whd in ⊢ (% → ?); * #ABS destruct(ABS)] * #mid ** |
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398 | whd in ⊢ (???% → ?); #EQ destruct(EQ) * #_ #_ change with (stmt_at ???? = ? → ?) |
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399 | #EQstmt >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQt_fn1; #EQt_fn1 |
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400 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQstmt; #EQstmt |
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401 | %{(taaf_step … (taa_base …) …)} |
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402 | [ whd in match as_classifier; normalize nodelta whd in ⊢ (??%?); |
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403 | whd in match fetch_statement; normalize nodelta |
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404 | >EQt_fn1 >m_return_bind >EQstmt % |
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405 | | whd whd in match eval_state; normalize nodelta whd in match fetch_statement; |
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406 | normalize nodelta >EQt_fn1 >m_return_bind >EQstmt >m_return_bind |
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407 | whd in match eval_statement_no_pc; normalize nodelta >m_return_bind |
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408 | whd in match eval_statement_advance; normalize nodelta |
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409 | whd in match goto; normalize nodelta >(pc_of_label_eq … EQt_fn1) % |
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410 | ] @I |
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411 | qed. |
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412 | |
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413 | |
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414 | |
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415 | (* |
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416 | lemma list_elim_on_last : ∀A : Type[0].∀ P : (list A) → Prop. ∀ l. |
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417 | P ([ ]) → (∀pre,last. P (pre@[last])) → P l. |
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418 | #A #P #l #H1 #H2 cut(∀A.∀l : list A.l = [ ] ∨ ∃pre,last. l = pre@[last]) |
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419 | [ #A #l elim l [ %1 % | #a #tl * [ #EQ destruct %2 %{([ ])} %{a} %] |
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420 | * #l1 * #a1 #EQ destruct %2 %{(a::l1)} %{a1} %]] #APP1 |
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421 | lapply(APP1 … l) * [ #EQ >EQ assumption] |
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422 | * #pre * #last #EQ >EQ @H2 |
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423 | qed.*) |
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424 | |
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425 | |
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426 | |
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427 | lemma eval_return_ok : |
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428 | ∀prog : ertl_program. |
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429 | let trans_prog ≝ ertl_to_ertlptr prog in |
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430 | ∀ f_lbls,f_regs,stack_sizes. |
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431 | ∀f_bl_r. |
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432 | ∀ good :b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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433 | translate_data prog f_bl_r f_lbls f_regs. |
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434 | ∀st2, st1',f,fn. |
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435 | let st1 ≝ sigma_state_pc prog f_lbls f_regs st2 in |
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436 | fetch_statement ERTL_semantics … |
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437 | (globalenv_noinit ? prog) (pc … st1) = |
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438 | return 〈f, fn, final … (RETURN ERTL … )〉 → |
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439 | eval_state ERTL_semantics |
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440 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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441 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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442 | st1 = |
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443 | return st1' → |
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444 | joint_classify (mk_prog_params ERTLptr_semantics trans_prog stack_sizes) |
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445 | st2 = cl_return ∧ |
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446 | ∃st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
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447 | ∃st2_after_ret. |
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448 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) (* always empty in this case *) |
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449 | st2_after_ret |
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450 | st2'. |
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451 | (if taaf_non_empty … taf then |
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452 | ¬as_costed (ERTLptr_status trans_prog stack_sizes) |
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453 | st2_after_ret |
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454 | else True) ∧ |
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455 | eval_state … (ev_genv … (mk_prog_params ERTLptr_semantics trans_prog stack_sizes)) st2 = |
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456 | return st2_after_ret ∧ |
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457 | ret_rel ?? (ERTLptrStatusSimulation prog stack_sizes ??? good) st1' st2_after_ret. |
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458 | #prog #f_lbls #f_regs #stack_size #f_bl_r #good #st2 #st1' #f #fn #EQfetch |
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459 | whd in match eval_state; normalize nodelta >EQfetch >m_return_bind |
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460 | #H @('bind_inversion H) -H #st1_tmp whd in ⊢ (??%%→?); #EQ destruct |
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461 | whd in match set_no_pc in ⊢ (%→?); whd in match eval_statement_advance in ⊢ (%→?); |
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462 | whd in match eval_return; normalize nodelta #H lapply(err_eq_from_io ????? H) -H |
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463 | #H @('bind_inversion H) -H * #n_st #n_pc change with (ertl_pop_frame ? = ? → ?) |
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464 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) #EQpop_frame |
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465 | cases(pop_frame_ok ?????? EQpop_frame) * #t_n_st #t_n_pc * #EQt_pop_frame |
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466 | normalize nodelta |
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467 | inversion (fetch_internal_function ??) normalize nodelta |
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468 | [ * #id1 #fn1 | #err ] normalize nodelta #EQfetch_fn1 #EQ destruct(EQ) |
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469 | #H @('bind_inversion H) -H #next_of_n_pc |
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470 | [2: >next_of_call_pc_error [2: % %] whd in ⊢ (???% → ?); #EQ destruct(EQ)] |
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471 | #EQnext_of_n_pc cases(next_of_call_pc_ok … good … EQnext_of_n_pc) |
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472 | #pc1 * #EQpc1 cut(pc1 = t_n_pc) |
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473 | [ @(sigma_stored_pc_inj prog f_lbls) |
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474 | lapply EQnext_of_n_pc; <EQpc1 whd in match sigma_stored_pc; normalize nodelta |
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475 | inversion(sigma_pc_opt ???) [1,3: #_ >next_of_call_pc_error [2,4: % %] |
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476 | whd in ⊢ (???% → ?); #EQ destruct| #x #_ #_ % #EQ destruct] #x #EQx #_ |
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477 | lapply EQnext_of_n_pc whd in match sigma_stored_pc; normalize nodelta |
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478 | inversion (sigma_pc_opt ???) [ #_ >next_of_call_pc_error [2: % %] |
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479 | whd in ⊢ (???% → ?); #EQ destruct] #y #EQy #_ lapply EQpc1 |
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480 | whd in match sigma_stored_pc; normalize nodelta >EQx >EQy normalize nodelta |
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481 | #EQ destruct % |
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482 | ] #EQ destruct(EQ) #EQnxt whd in match next; whd in match set_last_pop; |
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483 | whd in match set_pc; normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
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484 | cases(b_graph_transform_program_fetch_statement … good … EQfetch) |
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485 | #init_data * #t_fn1 ** #EQt_fn1 whd in ⊢ (% → ?); cases (f_bl_r ?) normalize nodelta |
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486 | [2: #r #tl *] #EQ destruct(EQ) * #labs ** |
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487 | [2: #hd #tl ** #_ #_ * #ibl * whd in ⊢ (%→?); *] ** #EQlabs #EQf_regs |
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488 | whd in match translate_fin_step; normalize nodelta * #bl * |
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489 | whd in ⊢ (% → ?); #EQ destruct(EQ) ** |
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490 | [2: #lb #tl * #mid ** #EQmid whd in ⊢ (% → ?); * #ABS destruct(ABS)] * #mid ** |
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491 | whd in ⊢ (???% → ?); #EQ destruct(EQ) * #_ #_ change with (stmt_at ???? = ? → ?) |
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492 | #EQstmt % |
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493 | [ whd in match joint_classify; normalize nodelta whd in match fetch_statement; |
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494 | normalize nodelta <(fetch_stmt_ok_sigma_pc_ok … EQfetch) >EQt_fn1 >m_return_bind |
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495 | >EQstmt % |
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496 | | % |
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497 | [ % |
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498 | [ @t_n_st |
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499 | | @(succ_pc ERTL_semantics (sigma_stored_pc prog f_lbls t_n_pc) next_of_n_pc) |
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500 | | @t_n_pc] |
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501 | ] |
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502 | % |
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503 | [ whd in match sigma_state_pc; normalize nodelta |
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504 | lapply(next_of_call_pc_inv … EQnext_of_n_pc) * #id2 * #fn2 * #c_id * #c_arg |
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505 | * #c_dest #EQfetch1 whd in match (succ_pc ???); |
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506 | >(fetch_stmt_ok_sigma_pc_block_ok … EQfetch1) >EQfetch_fn1 |
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507 | normalize nodelta % |
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508 | ] |
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509 | % |
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510 | [ % |
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511 | [ @t_n_st |
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512 | | @(succ_pc ERTL_semantics (sigma_stored_pc prog f_lbls t_n_pc) next_of_n_pc) |
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513 | | @t_n_pc] |
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514 | ] |
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515 | %{(taaf_base … )} normalize nodelta % [% [@I]] |
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516 | [ whd in match fetch_statement; normalize nodelta <(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
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517 | >EQt_fn1 >m_return_bind >EQstmt >m_return_bind whd in match eval_statement_no_pc; |
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518 | normalize nodelta >m_return_bind whd in match eval_statement_advance; |
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519 | whd in match eval_return; normalize nodelta |
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520 | change with (ertl_pop_frame ?) in match (pop_frame ????????); |
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521 | >EQt_pop_frame >m_return_bind >EQnxt >m_return_bind whd in match next; |
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522 | whd in match set_pc; whd in match set_last_pop; whd in match succ_pc; |
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523 | normalize nodelta lapply(next_of_call_pc_inv … EQnext_of_n_pc) |
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524 | * #id2 * #fn2 * #c_id * #c_arg * #c_dest #EQfetch1 whd in match (succ_pc ???); |
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525 | >(fetch_stmt_ok_sigma_pc_block_ok … EQfetch1) % |
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526 | | whd * #s1_pre #s1_call |
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527 | cases (joint_classify_call … s1_call) |
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528 | * #calling_i #calling * #call_spec * #arg * #dest * #nxt' #EQfetch_call |
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529 | * #s2_pre #s2_call whd in ⊢ (% → ?); >EQfetch_call normalize nodelta |
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530 | * #s1_pre_prf #EQpc_s2_pre whd in ⊢ (% → ?); #EQ1 |
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531 | >EQ1 in EQfetch_call; #EQfetch_call |
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532 | cases(fetch_call_commute … good … EQfetch_call) #calling' * #pc1 * |
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533 | #EQ1 cut(pc … s2_pre = pc1) |
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534 | [ @(sigma_stored_pc_inj prog f_lbls) lapply(EQfetch_call) |
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535 | whd in match sigma_stored_pc; normalize nodelta inversion(sigma_pc_opt ???) |
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536 | [1,3: #_ >fetch_statement_no_zero [2,4: %] #EQ destruct(EQ)| #x #_ #_ % #EQ destruct] |
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537 | #pc2 #EQpc2 #_ lapply EQ1 whd in match sigma_stored_pc; |
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538 | normalize nodelta >EQpc2 normalize nodelta cases(sigma_pc_opt ???) |
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539 | [2: #x normalize nodelta #EQ >EQ %] normalize nodelta #EQ <EQ in EQpc2; |
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540 | #EQ1 lapply EQfetch_call >fetch_statement_no_zero [ #ABS destruct(ABS)] |
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541 | whd in match sigma_stored_pc; normalize nodelta >EQ1 % |
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542 | ] #EQ2 destruct(EQ2) #EQt_fetch_call whd >EQt_fetch_call normalize nodelta % |
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543 | [ >EQ1 in s1_pre_prf; #EQ @(sigma_stored_pc_inj prog f_lbls) |
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544 | lapply EQnext_of_n_pc whd in match sigma_stored_pc; normalize nodelta |
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545 | inversion(sigma_pc_opt ???) [1,3: #_ >next_of_call_pc_error [2,4: % %] |
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546 | whd in ⊢ (???% → ?); #EQ destruct] #pc1 #EQpc1 #_ [ % #EQ destruct] |
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547 | lapply EQ whd in match sigma_stored_pc; normalize nodelta |
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548 | >EQpc1 normalize nodelta inversion(sigma_pc_opt ???) |
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549 | [2: #pc2 #_ normalize nodelta #EQ >EQ %] normalize nodelta |
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550 | #ABS #ABS1 lapply EQnext_of_n_pc whd in match sigma_stored_pc; |
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551 | normalize nodelta >EQpc1 >ABS1 normalize nodelta |
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552 | >next_of_call_pc_error [2: % %] whd in ⊢ (???% → ?); #EQ destruct |
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553 | | whd in match succ_pc; normalize nodelta |
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554 | change with next_of_n_pc in match (point_of_succ ???); |
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555 | change with nxt' in match (point_of_succ ???); |
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556 | lapply EQpc_s2_pre whd in match succ_pc; normalize nodelta |
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557 | change with next_of_n_pc in match (point_of_succ ???); |
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558 | change with nxt' in match (point_of_succ ???); #EQ >EQ |
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559 | cut(pc_block (pc … s1_pre) = pc_block (pc … s2_pre)) |
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560 | [2: #EQ >EQ %] >EQ1 <(pc_block_eq prog f_lbls …) [%] |
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561 | lapply EQfetch_call whd in match sigma_stored_pc; normalize nodelta |
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562 | cases(sigma_pc_opt ???) [ >fetch_statement_no_zero [2: %] #EQ destruct] |
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563 | #x #_ % #EQ destruct |
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564 | ] |
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565 | ] |
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566 | ] |
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567 | qed. |
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568 | |
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569 | |
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570 | |
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571 | lemma eval_cond_ok : |
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572 | ∀prog. |
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573 | let trans_prog ≝ ertl_to_ertlptr prog in |
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574 | ∀ f_lbls,f_regs,stack_sizes. |
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575 | ∀f_bl_r. |
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576 | ∀ good :b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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577 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs. |
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578 | ∀st2,st1',f,fn,a,ltrue,lfalse. |
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579 | let st1 ≝ sigma_state_pc prog f_lbls f_regs st2 in |
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580 | fetch_statement ERTL_semantics … |
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581 | (globalenv_noinit ? prog) (pc … st1) = |
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582 | return 〈f, fn, sequential … (COND ERTL … a ltrue) lfalse〉 → |
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583 | eval_state ERTL_semantics |
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584 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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585 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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586 | st1 = return st1' → |
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587 | as_costed (ERTL_status prog stack_sizes) st1' → |
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588 | ∃ st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
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589 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) |
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590 | st2 st2'. |
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591 | bool_to_Prop (taaf_non_empty … taf). |
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592 | #prog #f_lbls #f_regs #stack_size #f_bl_r #good #st2 #st1' #f #fn #a #ltrue #lfalse |
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593 | #EQfetch whd in match eval_state; normalize nodelta >EQfetch >m_return_bind |
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594 | whd in match eval_statement_no_pc; normalize nodelta >m_return_bind |
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595 | whd in match eval_statement_advance; normalize nodelta |
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596 | change with (ps_reg_retrieve ??) in match (acca_retrieve ?????); |
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597 | #H lapply(err_eq_from_io ????? H) -H #H @('bind_inversion H) -H |
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598 | #bv >(fetch_stmt_ok_sigma_state_ok … EQfetch) in ⊢ (% → ?); whd in match set_no_pc; |
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599 | normalize nodelta #EQbv #H @('bind_inversion H) -H * #EQbool normalize nodelta |
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600 | lapply(fetch_statement_inv … EQfetch) * #EQfn #_ |
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601 | [ whd in match goto; normalize nodelta >(pc_of_label_eq ??????? EQfn) >m_return_bind |
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602 | | whd in match next; normalize nodelta |
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603 | ] whd in match set_pc; normalize nodelta |
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604 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) whd in match set_no_pc; normalize nodelta |
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605 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) >(fetch_stmt_ok_sigma_last_pop_ok … EQfetch) |
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606 | whd in ⊢ (??%% → ?); #EQ destruct #n_cost |
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607 | %{(mk_state_pc ? (st_no_pc … st2) |
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608 | (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) ?) |
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609 | (last_pop … st2))} [ @ltrue |3: @lfalse] |
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610 | % [1,3: whd in match sigma_state_pc; normalize nodelta |
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611 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQfn; #EQfn >EQfn %] |
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612 | cases(b_graph_transform_program_fetch_statement … good … EQfetch) |
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613 | #init_data * #t_fn1 ** #EQt_fn1 whd in ⊢ (% → ?); cases (f_bl_r ?) normalize nodelta |
---|
614 | [2,4: #r #tl *] #EQ destruct(EQ) >if_merge_right in ⊢ (% → ?); [2,4: %] * #labs ** |
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615 | [2,4: #hd #tl ** #_ #_ *** #pre #inst #post * whd in ⊢ (%→?); *] ** #EQlabs #EQf_regs |
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616 | whd in match translate_step; normalize nodelta * #bl * |
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617 | whd in ⊢ (% → ?); #EQ destruct(EQ) whd in ⊢ (% → ?); * #l1 * #mid1 * #mid2 * #l2 |
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618 | *** #EQmid1 whd in ⊢ (% → ?); * #EQ1 #EQ2 destruct(EQ1 EQ2) whd in ⊢ (% → ?); * |
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619 | #nxt1 * #EQcond #EQ destruct(EQ) whd in ⊢ (% → ?); * #EQ1 #EQ2 destruct(EQ1 EQ2) |
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620 | whd in EQmid1 : (??%%); destruct(EQmid1) |
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621 | %{(taaf_step_jump … (taa_base …) …) I} |
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622 | [2,5: whd whd in ⊢ (??%?); whd in match fetch_statement; normalize nodelta |
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623 | <(fetch_stmt_ok_sigma_pc_ok … EQfetch) >EQt_fn1 >m_return_bind |
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624 | >EQcond % |
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625 | |3,6: whd whd in match eval_state; normalize nodelta whd in match eval_statement_no_pc; |
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626 | normalize nodelta whd in match fetch_statement; normalize nodelta |
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627 | <(fetch_stmt_ok_sigma_pc_ok … EQfetch) >EQt_fn1 >m_return_bind |
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628 | >EQcond >m_return_bind normalize nodelta >m_return_bind |
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629 | whd in match eval_statement_advance; normalize nodelta |
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630 | change with (ps_reg_retrieve ??) in match (acca_retrieve ?????); |
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631 | cases(ps_reg_retrieve_ok … EQbv) #bv1 * #EQbv1 #EQsem >EQbv1 |
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632 | >m_return_bind >EQsem in EQbool; #EQbool cases(bool_of_beval_ok … EQbool) |
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633 | #bool1 * #EQbool1 #EQ destruct(EQ) >EQbool1 >m_return_bind normalize nodelta |
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634 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) [2: %] whd in match goto; |
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635 | normalize nodelta >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQt_fn1; #EQt_fn1 |
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636 | >(pc_of_label_eq … EQt_fn1) >m_return_bind % |
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637 | |*: lapply n_cost whd in match as_costed; normalize nodelta |
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638 | [ cut((mk_state_pc ERTL_semantics |
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639 | (sigma_state prog f_lbls f_regs |
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640 | (added_registers ERTL (prog_var_names (joint_function ERTL) ℕ prog) fn |
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641 | (f_regs (pc_block (pc ERTLptr_semantics st2)))) st2) |
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642 | (pc_of_point ERTL_semantics (pc_block (pc ERTLptr_semantics st2)) ltrue) |
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643 | (sigma_stored_pc prog f_lbls (last_pop ERTLptr_semantics st2))) = |
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644 | sigma_state_pc prog f_lbls f_regs (mk_state_pc ERTLptr_semantics st2 |
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645 | (pc_of_point ERTLptr_semantics (pc_block (pc ERTLptr_semantics st2)) ltrue) |
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646 | (last_pop ERTLptr_semantics st2))) [ whd in match sigma_state_pc; normalize nodelta |
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647 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQfn; #EQfn >EQfn %] |
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648 | #EQ >EQ >(as_label_ok … good … (mk_state_pc ERTLptr_semantics st2 |
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649 | (pc_of_point ERTLptr_semantics (pc_block (pc ERTLptr_semantics st2)) ltrue) |
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650 | (last_pop ERTLptr_semantics st2))) [#H @H] cases daemon (*needs lemma see below *) |
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651 | | cut((mk_state_pc ERTL_semantics |
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652 | (sigma_state prog f_lbls f_regs |
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653 | (added_registers ERTL (prog_var_names (joint_function ERTL) ℕ prog) fn |
---|
654 | (f_regs (pc_block (pc ERTLptr_semantics st2)))) st2) |
---|
655 | (succ_pc ERTL_semantics (pc ERTLptr_semantics st2) |
---|
656 | (point_of_succ ERTLptr_semantics |
---|
657 | (point_of_pc ERTLptr_semantics |
---|
658 | (pc ERTL_semantics (sigma_state_pc prog f_lbls f_regs st2))) nxt1)) |
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659 | (sigma_stored_pc prog f_lbls (last_pop ERTLptr_semantics st2))) = |
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660 | sigma_state_pc prog f_lbls f_regs (mk_state_pc ERTLptr_semantics st2 |
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661 | (pc_of_point ERTLptr_semantics (pc_block (pc ERTLptr_semantics st2)) |
---|
662 | (point_of_succ ERTLptr_semantics |
---|
663 | (point_of_pc ERTLptr_semantics |
---|
664 | (pc ERTL_semantics (sigma_state_pc prog f_lbls f_regs st2))) nxt1)) |
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665 | (last_pop ERTLptr_semantics st2))) [ whd in match sigma_state_pc; normalize nodelta |
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666 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in EQfn; #EQfn >EQfn %] #EQ >EQ |
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667 | >(as_label_ok … good … (mk_state_pc ERTLptr_semantics st2 |
---|
668 | (pc_of_point ERTLptr_semantics (pc_block (pc ERTLptr_semantics st2)) |
---|
669 | (point_of_succ ERTLptr_semantics |
---|
670 | (point_of_pc ERTLptr_semantics |
---|
671 | (pc ERTL_semantics (sigma_state_pc prog f_lbls f_regs st2))) nxt1)) |
---|
672 | (last_pop ERTLptr_semantics st2))) [#H @H] cases daemon (*needs lemma see below ! *) |
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673 | ] |
---|
674 | qed. |
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675 | |
---|
676 | lemma eval_cost_ok : |
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677 | ∀prog. |
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678 | let trans_prog ≝ ertl_to_ertlptr prog in |
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679 | ∀ f_lbls,f_regs,stack_sizes. |
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680 | ∀f_bl_r. |
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681 | ∀ good :b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
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682 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs. |
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683 | ∀st2,st1',f,fn,c,nxt. |
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684 | let st1 ≝ sigma_state_pc prog f_lbls f_regs st2 in |
---|
685 | fetch_statement ERTL_semantics … |
---|
686 | (globalenv_noinit ? prog) (pc … st1) = |
---|
687 | return 〈f, fn, sequential … (COST_LABEL ERTL … c) nxt〉 → |
---|
688 | eval_state ERTL_semantics |
---|
689 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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690 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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691 | st1 = return st1' → |
---|
692 | ∃ st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
---|
693 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) |
---|
694 | st2 st2'. |
---|
695 | bool_to_Prop (taaf_non_empty … taf). |
---|
696 | #prog #f_lbls #f_regs #stack_Size #f_bl_r #good #st2 #st1' #f #fn #c #nxt |
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697 | #EQfetch whd in match eval_state; normalize nodelta >EQfetch >m_return_bind |
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698 | whd in match eval_statement_no_pc; normalize nodelta >m_return_bind |
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699 | whd in match eval_statement_advance; normalize nodelta |
---|
700 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) whd in match set_no_pc; |
---|
701 | normalize nodelta >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
---|
702 | >(fetch_stmt_ok_sigma_last_pop_ok … EQfetch) whd in match next; |
---|
703 | whd in match set_pc; normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
---|
704 | %{(set_pc ? (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) nxt) st2)} |
---|
705 | % [ whd in match sigma_state_pc; normalize nodelta |
---|
706 | lapply(fetch_statement_inv … EQfetch) >(fetch_stmt_ok_sigma_pc_ok … EQfetch) * |
---|
707 | #EQfn #_ >EQfn %] lapply EQfetch >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
---|
708 | #EQfetch' |
---|
709 | cases(b_graph_transform_program_fetch_statement … good … EQfetch') |
---|
710 | #init_data * #t_fn ** #EQt_fn whd in ⊢ (% → ?); cases(f_bl_r ?) [2: #x #y *] |
---|
711 | normalize nodelta #EQ destruct(EQ) * #lbls * #regs ** #_ #_ whd in ⊢ (% → ?); |
---|
712 | * #bl * >if_merge_right [2: %] whd in ⊢ (% → ?); cases regs [2: #x #y *] |
---|
713 | normalize nodelta #EQ destruct(EQ) whd in ⊢ (% → ?); * #l1 * #mid1 * #mid2 * #l2 |
---|
714 | *** #EQmid whd in ⊢ (% → ?); * #EQ1 #EQ2 destruct(EQ1 EQ2) whd in EQmid : (???%); |
---|
715 | destruct(EQmid) whd in ⊢ (% → ?); * #nxt1 * #EQstmt #EQ destruct(EQ) |
---|
716 | whd in ⊢ (% → ?); * #EQ1 #EQ2 destruct(EQ1 EQ2) %{(taaf_step … (taa_base …) …)} |
---|
717 | [1,2: whd [ whd in ⊢ (??%?); | whd in match eval_state; ] |
---|
718 | whd in match fetch_statement; normalize nodelta >EQt_fn >m_return_bind |
---|
719 | >EQstmt [%] >m_return_bind whd in match eval_statement_no_pc; |
---|
720 | normalize nodelta >m_return_bind %] @I |
---|
721 | qed. |
---|
722 | |
---|
723 | |
---|
724 | lemma eval_call_ok : |
---|
725 | ∀prog : ertl_program. |
---|
726 | let trans_prog ≝ ertl_to_ertlptr prog in |
---|
727 | ∀ f_lbls,f_regs,stack_sizes. |
---|
728 | ∀f_bl_r. |
---|
729 | ∀ good :b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
---|
730 | (λglobals,fn.«translate_data globals fn,(refl …)») prog f_bl_r f_lbls f_regs. |
---|
731 | ∀st2,st1',f,fn,called,args,dest,nxt. |
---|
732 | let st1 ≝ sigma_state_pc prog f_lbls f_regs st2 in |
---|
733 | fetch_statement ERTL_semantics … |
---|
734 | (globalenv_noinit ? prog) (pc … st1) = |
---|
735 | return 〈f, fn, |
---|
736 | sequential … (CALL ERTL … called args dest ) nxt〉 → |
---|
737 | eval_state … (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) st1 = |
---|
738 | return st1' → |
---|
739 | ∃is_call,st2_pre_call,is_call'. |
---|
740 | as_call_ident (ERTLptr_status trans_prog stack_sizes) |
---|
741 | («st2_pre_call,is_call'») = as_call_ident (ERTL_status prog stack_sizes) |
---|
742 | («st1, is_call») ∧ |
---|
743 | (pc … st1) = sigma_stored_pc prog f_lbls (pc … st2_pre_call) ∧ |
---|
744 | ∃taa2 : trace_any_any (ERTLptr_status trans_prog stack_sizes) st2 st2_pre_call. |
---|
745 | ∃ st2'. st1' = sigma_state_pc prog f_lbls f_regs st2' ∧ |
---|
746 | eval_state ERTLptr_semantics … |
---|
747 | (ev_genv … (mk_prog_params ERTLptr_semantics trans_prog stack_sizes)) st2_pre_call |
---|
748 | =return st2'. |
---|
749 | #prog #f_lbls #f_regs #stack_size #f_bl_r #good #st2 #st1' #f #fn * [#c_id | * #c_ptr1 #c_ptr2 ] |
---|
750 | #args #dest #nxt #EQfetch whd in match eval_state; normalize nodelta >EQfetch |
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751 | >m_return_bind whd in match eval_statement_no_pc; normalize nodelta >m_return_bind |
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752 | whd in match eval_statement_advance; whd in match eval_call; normalize nodelta |
---|
753 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
---|
754 | >(fetch_stmt_ok_sigma_last_pop_ok … EQfetch) #H @('bind_inversion H) -H |
---|
755 | #c_bl whd in match set_no_pc; normalize nodelta #H lapply(err_eq_from_io ????? H) -H |
---|
756 | #EQc_bl cases(block_of_call_ok ??????? EQc_bl) #c_bl' * #EQc_bl' #EQ destruct(EQ) |
---|
757 | #H @('bind_inversion H) -H * #f1 * [1,3: #fn1 |*: #ext_f] #H |
---|
758 | lapply(err_eq_from_io ????? H) -H #EQfn1 normalize nodelta |
---|
759 | [3,4: #H @('bind_inversion H) -H #st whd in match eval_external_call; normalize nodelta |
---|
760 | #H @('bind_inversion H) -H #list_val #_ #H @('bind_inversion H) -H #x |
---|
761 | #_ #H @('bind_inversion H) -H #y whd in match do_io; normalize nodelta |
---|
762 | whd in ⊢ (???% → ?); #ABS destruct(ABS) ] #H lapply(err_eq_from_io ????? H) -H |
---|
763 | #H @('bind_inversion H) -H #st1'' >(fetch_stmt_ok_sigma_pc_ok … EQfetch) |
---|
764 | >(fetch_stmt_ok_sigma_last_pop_ok … EQfetch) whd in match kind_of_call; |
---|
765 | normalize nodelta change with (ertl_save_frame ? it ?) in ⊢ (??%? → ?); |
---|
766 | [2: @ID |4: @PTR] #EQst1'' #H @('bind_inversion H) -H #st1''' |
---|
767 | whd in match eval_internal_call; normalize nodelta #H @('bind_inversion H) -H |
---|
768 | #s_size #H lapply(opt_eq_from_res ???? H) -H |
---|
769 | change with (stack_size ?) in ⊢ (??%? → ?); #EQs_size whd in ⊢ (??%? → ?); |
---|
770 | whd in ⊢ (???% → ??%% → ?); #EQ #EQ1 destruct(EQ EQ1) |
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771 | lapply EQfetch >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in ⊢ (% → ?); |
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772 | #EQfetch' lapply(fetch_statement_inv … EQfetch') * #EQfn normalize nodelta #EQstmt |
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773 | cases(b_graph_transform_program_fetch_statement_plus … good … EQfetch') |
---|
774 | #init_data * #calling' ** #EQcalling' whd in ⊢ (% → ?); cases(f_bl_r ?) |
---|
775 | [2,4: #x #y *] normalize nodelta #EQ destruct(EQ) * #labels * #registers |
---|
776 | ******* #EQlabels #EQregisters #_ #_ #_ #_ #fresh_regs |
---|
777 | normalize nodelta >if_merge_right [2,4: %] whd in match translate_step; |
---|
778 | normalize nodelta * #bl * cases registers in EQregisters; -registers |
---|
779 | normalize nodelta [2,3: [ #x #y] #_ *|4: #r #tl] #EQregisters |
---|
780 | [ whd in ⊢ (% → ?); cases tl in EQregisters; -tl [2: #x #y #_ *] normalize nodelta |
---|
781 | #EQregisters | whd in ⊢ (% → ?);] #EQ destruct(EQ) |
---|
782 | * #pre_l * #mid1 * #mid2 * #post_l *** #EQmid1 #m_fetch_pre whd in ⊢ (% → ?); |
---|
783 | * #nxt1 * #EQcall change with nxt1 in ⊢ (??%? → ?); #EQ destruct(EQ) |
---|
784 | whd in ⊢ (% → ?); * #EQ1 #EQ2 destruct(EQ1 EQ2) cases(pre_l) in m_fetch_pre EQmid1; |
---|
785 | [* #x * #y ** #ABS destruct(ABS) |
---|
786 | |4: #lab #tl * #ABS destruct(ABS) |
---|
787 | | #l1 #tl #m_fetch_pre whd in ⊢ (???% → ?); #EQ destruct(EQ) |
---|
788 | | * #_ #EQ destruct(EQ) whd in ⊢ (???% → ?); #EQ destruct(EQ) |
---|
789 | ] |
---|
790 | % |
---|
791 | [1,3: @hide_prf whd in ⊢ (??%?); >EQfetch %] |
---|
792 | cases(ertl_save_frame_ok prog f_lbls f_regs ? |
---|
793 | (added_registers ERTL (prog_var_names … prog) fn1 (f_regs c_bl')) ? st1''' ?) |
---|
794 | [2: @PTR |
---|
795 | |3,7: [ %{(mk_state_pc ? (st_no_pc … st2) |
---|
796 | (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) mid1) |
---|
797 | (last_pop … st2))} |
---|
798 | |%{(st2)} |
---|
799 | ] @hide_prf whd in match sigma_pc_opt; normalize nodelta @if_elim |
---|
800 | [1,3: @eqZb_elim [2,4: #_ *] #EQbl >fetch_internal_function_no_minus_one in EQfn; |
---|
801 | [1,3: #EQ destruct(EQ) |*: assumption]] |
---|
802 | #_ [>point_of_pc_of_point >(get_sigma_last … good … EQfn EQstmt EQlabels) |
---|
803 | | >(get_sigma_idempotent … good … EQfn EQstmt EQlabels) |
---|
804 | ] >m_return_bind % whd in ⊢ (??%? → ?); #ABS destruct(ABS) |
---|
805 | |6: @ID |
---|
806 | |4,8: [change with (pc_block (pc … st2)) in match (pc_block ?); |
---|
807 | >EQfn in ⊢ (??(???match % with [_ ⇒ ?| _ ⇒ ?])?); |
---|
808 | |>EQfn in ⊢ (??(???match % with [_ ⇒ ?| _ ⇒ ?])?); |
---|
809 | ] normalize nodelta whd in match (st_no_pc ??); |
---|
810 | whd in match sigma_stored_pc in ⊢ (??(???(???%?))?); |
---|
811 | whd in match sigma_pc_opt; normalize nodelta @if_elim |
---|
812 | [1,3: @eqZb_elim [2,4: #_ *] #EQbl >fetch_internal_function_no_minus_one in EQfn; |
---|
813 | [1,3: #EQ destruct(EQ) |*: assumption]] |
---|
814 | #_ [>point_of_pc_of_point >(get_sigma_last … good … EQfn EQstmt EQlabels) |
---|
815 | | >(get_sigma_idempotent … good … EQfn EQstmt EQlabels) |
---|
816 | ] >m_return_bind normalize nodelta |
---|
817 | change with (pc_block (pc … st2)) in match (pc_block ?); |
---|
818 | >pc_of_point_of_pc assumption |
---|
819 | ] #st_no_pc_after_call normalize nodelta * |
---|
820 | [ #H @('bind_inversion H) -H #st_no_pc_pre_call #EQst_no_pc_pre_call |
---|
821 | whd in match set_no_pc; normalize nodelta] |
---|
822 | #EQst_no_pc_after_call #EQ destruct(EQ) |
---|
823 | [ letin pairpc ≝ (beval_pair_of_pc |
---|
824 | (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) mid1)) |
---|
825 | %{(mk_state_pc ? (set_regs ERTL_state |
---|
826 | (〈add ?? (\fst (regs … st2)) r (\snd pairpc),\snd(regs … st2)〉) |
---|
827 | st_no_pc_pre_call) |
---|
828 | (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) mid1) |
---|
829 | (last_pop … st2))} |
---|
830 | | %{st2} |
---|
831 | ] |
---|
832 | % |
---|
833 | [1,3: @hide_prf whd in ⊢ (??%?); whd in match fetch_statement; normalize nodelta |
---|
834 | >EQcalling' >m_return_bind [>point_of_pc_of_point ] >EQcall >m_return_bind %] |
---|
835 | % |
---|
836 | [1,3: % |
---|
837 | [1,3: whd in ⊢ (??%%); whd in match fetch_statement in ⊢ (??%?); |
---|
838 | normalize nodelta >EQcalling' in ⊢ (??(match % with [_ ⇒ ? | _ ⇒ ?])?); |
---|
839 | >m_return_bind in ⊢ (??(match % with [_ ⇒ ? | _ ⇒ ?])?); |
---|
840 | [ >point_of_pc_of_point in ⊢ (??(match % with [_ ⇒ ? | _ ⇒ ?])?);] |
---|
841 | >EQcall in ⊢ (??(match % with [_ ⇒ ? | _ ⇒ ?])?); normalize nodelta |
---|
842 | >(fetch_stmt_ok_sigma_pc_ok … EQfetch) in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?])); |
---|
843 | >EQfetch' in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?])); normalize nodelta |
---|
844 | >(fetch_stmt_ok_sigma_state_ok … EQfetch) >EQc_bl >m_return_bind |
---|
845 | lapply EQc_bl' whd in match block_of_call; normalize nodelta |
---|
846 | cases daemon (* CSC: The proof used to be %, why did it change? *) |
---|
847 | (* This point will be generalized when the call case |
---|
848 | will be available in StatusSimulationHelper*) |
---|
849 | (* |
---|
850 | XXX |
---|
851 | |
---|
852 | [cut (block_of_call |
---|
853 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size) |
---|
854 | (globals |
---|
855 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size)) |
---|
856 | (ev_genv |
---|
857 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size)) |
---|
858 | (inr ident … 〈c_ptr1,c_ptr2〉) |
---|
859 | (set_regs ERTL_state |
---|
860 | 〈add RegisterTag beval (\fst (regs ERTLptr_semantics st2)) r |
---|
861 | (\snd (beval_pair_of_pc |
---|
862 | (pc_of_point ERTLptr_semantics |
---|
863 | (pc_block (pc ERTLptr_semantics st2)) mid1))), |
---|
864 | \snd (regs ERTLptr_semantics st2)〉 |
---|
865 | st_no_pc_pre_call) = |
---|
866 | block_of_call (mk_prog_params ERTL_semantics prog stack_size) |
---|
867 | (globals (mk_prog_params ERTL_semantics prog stack_size)) |
---|
868 | (ev_genv (mk_prog_params ERTL_semantics prog stack_size)) |
---|
869 | (inr ident … 〈c_ptr1,c_ptr2〉) |
---|
870 | (sigma_state_pc prog f_lbls f_regs st2)) |
---|
871 | |
---|
872 | [2: #URCA <URCA cases block_of_call #xyz |
---|
873 | [2: whd in ⊢ (??%%); %] |
---|
874 | >m_return_bind >m_return_bind |
---|
875 | whd in match fetch_internal_function; normalize nodelta |
---|
876 | whd in match fetch_function; normalize nodelta |
---|
877 | XXX |
---|
878 | %*) |
---|
879 | |*: whd in match sigma_stored_pc; whd in match sigma_pc_opt; normalize nodelta |
---|
880 | @if_elim change with (pc_block(pc … st2)) in match (pc_block ?); |
---|
881 | [1,3: @eqZb_elim [2,4: #_ *] #EQbl #_ |
---|
882 | >fetch_internal_function_no_minus_one in EQcalling'; [2,4: assumption] |
---|
883 | whd in ⊢ (???% → ?); #ABS destruct(ABS) ] #_ |
---|
884 | [ >point_of_pc_of_point >(get_sigma_last … good … EQfn EQstmt EQlabels) |
---|
885 | | >(get_sigma_idempotent … good … EQfn EQstmt EQlabels) |
---|
886 | ] |
---|
887 | >m_return_bind >pc_of_point_of_pc % |
---|
888 | ] |
---|
889 | |4: %{(taa_base …)} |
---|
890 | | %{(taaf_to_taa ??? |
---|
891 | (produce_trace_any_any_free |
---|
892 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size) |
---|
893 | st2 ????? |
---|
894 | (mk_state_pc ? |
---|
895 | (set_regs ERTL_state |
---|
896 | (〈add ?? (\fst (regs … st2)) r (\snd pairpc),\snd(regs … st2)〉) |
---|
897 | st_no_pc_pre_call) |
---|
898 | (pc_of_point ERTLptr_semantics (pc_block (pc … st2)) mid1) |
---|
899 | (last_pop … st2)) |
---|
900 | EQcalling' m_fetch_pre ?) ?)} |
---|
901 | [ @if_elim [2: #_ @I] #_ % whd in ⊢ (% → ?); whd in match (as_label ??); |
---|
902 | whd in match (as_pc_of ??); whd in match fetch_statement; normalize nodelta |
---|
903 | >EQcalling' >m_return_bind whd in match point_of_pc; normalize nodelta |
---|
904 | >point_of_offset_of_point >EQcall >m_return_bind normalize nodelta |
---|
905 | whd in match cost_label_of_stmt; normalize nodelta * #H @H % |
---|
906 | | whd in match repeat_eval_seq_no_pc; normalize nodelta whd in ⊢ (??%?); |
---|
907 | whd in match eval_seq_no_pc in ⊢ (??match % with [_ ⇒ ?| _ ⇒ ?]?); |
---|
908 | normalize nodelta whd in match (eval_ext_seq ????????); |
---|
909 | whd in match (get_pc_from_label ?????); whd in match block_of_funct_id; |
---|
910 | normalize nodelta @('bind_inversion EQfn) * #f2 * #fn2 |
---|
911 | whd in match fetch_function; normalize nodelta #H |
---|
912 | lapply(opt_eq_from_res ???? H) -H #H @('bind_inversion H) -H #f3 #EQf3 |
---|
913 | #H @('bind_inversion H) -H #fn4 #_ whd in ⊢ (??%%→ ??%% → ?); #EQ1 #EQ2 |
---|
914 | destruct(EQ1 EQ2) |
---|
915 | >(find_symbol_transf … |
---|
916 | (λvars.transf_fundef … (λfn.(b_graph_translate … fn))) prog f) |
---|
917 | >(symbol_of_block_rev … EQf3) >m_return_bind >code_block_of_block_eq |
---|
918 | normalize nodelta >(pc_of_label_eq … EQcalling') normalize nodelta |
---|
919 | whd in ⊢ (??%?); |
---|
920 | whd in match eval_seq_no_pc in ⊢ (??match % with [_ ⇒ ?| _ ⇒ ?]?); |
---|
921 | normalize nodelta whd in match (eval_ext_seq ????????); |
---|
922 | whd in match acca_arg_retrieve; normalize nodelta |
---|
923 | change with (ps_reg_retrieve ??) in match (acca_arg_retrieve_ ?????); |
---|
924 | whd in match ps_reg_retrieve; whd in match reg_retrieve; normalize nodelta |
---|
925 | >lookup_add_hit >m_return_bind whd in match set_regs; normalize nodelta |
---|
926 | @('bind_inversion EQst_no_pc_pre_call) #new_st whd in match push in ⊢ (%→ ?); |
---|
927 | normalize nodelta #H @('bind_inversion H) -H #is1 #EQis1 whd in ⊢ (??%% → ?); |
---|
928 | #EQ destruct(EQ) #H @('bind_inversion H) -H #is2 #EQis2 whd in match set_istack; |
---|
929 | normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct(EQ) whd in match push; |
---|
930 | normalize nodelta >EQis1 >m_return_bind whd in match set_istack; normalize nodelta |
---|
931 | whd in ⊢ (??%?); whd in match eval_seq_no_pc in ⊢ (??match % with [_ ⇒ ?| _ ⇒ ?]?); |
---|
932 | normalize nodelta whd in match (eval_ext_seq ????????); |
---|
933 | whd in match (get_pc_from_label ?????); whd in match block_of_funct_id; |
---|
934 | normalize nodelta |
---|
935 | >(find_symbol_transf … |
---|
936 | (λvars.transf_fundef … (λfn.(b_graph_translate … fn))) prog f) |
---|
937 | >(symbol_of_block_rev … EQf3) >m_return_bind >code_block_of_block_eq |
---|
938 | normalize nodelta >(pc_of_label_eq … EQcalling') normalize nodelta |
---|
939 | whd in ⊢ (??%?); |
---|
940 | whd in match eval_seq_no_pc in ⊢ (??match % with [_ ⇒ ?| _ ⇒ ?]?); |
---|
941 | normalize nodelta whd in match (eval_ext_seq ????????); |
---|
942 | whd in match acca_arg_retrieve; normalize nodelta |
---|
943 | change with (ps_reg_retrieve ??) in match (acca_arg_retrieve_ ?????); |
---|
944 | whd in match ps_reg_retrieve; whd in match reg_retrieve; normalize nodelta |
---|
945 | >lookup_add_hit >m_return_bind whd in match set_regs; normalize nodelta |
---|
946 | whd in match push; normalize nodelta >EQis2 >m_return_bind normalize nodelta |
---|
947 | whd in ⊢ (??%?); @eq_f whd in match set_istack; normalize nodelta |
---|
948 | whd in match reg_store; normalize nodelta >add_idempotent % |
---|
949 | | cut(∀st,st',kind.ertlptr_save_frame kind it st = return st' → |
---|
950 | ∃hd,tl.st_frms … st' = return (cons ? hd tl) ∧ |
---|
951 | \snd hd = pc_block (pc … st)) |
---|
952 | [ #st * #frms #is #b #regs #m * whd in match ertlptr_save_frame; |
---|
953 | normalize nodelta [ >m_return_bind | #H @('bind_inversion H) -H #st'' #EQst''] |
---|
954 | #H @('bind_inversion H) -H #frames #H lapply(opt_eq_from_res ???? H) -H |
---|
955 | #EQframes whd in ⊢ (??%% → ?); whd in match set_frms; whd in match set_regs; |
---|
956 | normalize nodelta #EQ destruct(EQ) %{(〈\fst (regs ERTLptr_semantics ?),pc_block (pc … st)〉)} |
---|
957 | [2,4: %{frames} % [1,3: %] |1,3:] %] |
---|
958 | ] #save_frame_non_empty cases(save_frame_non_empty … EQst_no_pc_after_call) |
---|
959 | * #fst_hd #snd_hd * #rest * #EQrest #EQhd |
---|
960 | ] |
---|
961 | cases(b_graph_transform_program_fetch_internal_function … good c_bl' f1 fn1 ?) |
---|
962 | [2,4: whd in match fetch_internal_function; normalize nodelta >EQfn1 %] |
---|
963 | #init * #t_fn1 ** #EQt_fn1 #_ * #_ #_ #_ #EQentry #_ #_ #_ #_ #_ #_ #_ |
---|
964 | [ %{(mk_state_pc ? st_no_pc_after_call |
---|
965 | (pc_of_point |
---|
966 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size) |
---|
967 | c_bl' (joint_if_entry … t_fn1)) |
---|
968 | (last_pop … st2))} |
---|
969 | | |
---|
970 | %{(mk_state_pc ? (set_frms ERTL_state (〈(add ?? (fst_hd) r (\snd pairpc)),snd_hd〉 |
---|
971 | :: rest) |
---|
972 | st_no_pc_after_call) |
---|
973 | (pc_of_point |
---|
974 | (mk_prog_params ERTLptr_semantics (ertl_to_ertlptr prog) stack_size) |
---|
975 | c_bl' (joint_if_entry … t_fn1)) |
---|
976 | (last_pop … st2))} |
---|
977 | ] |
---|
978 | % |
---|
979 | [1,3: whd in match sigma_state_pc; normalize nodelta |
---|
980 | whd in match pc_of_point in ⊢ (???%); normalize nodelta |
---|
981 | whd in match fetch_internal_function; normalize nodelta >EQfn1 |
---|
982 | >m_return_bind normalize nodelta @eq_f3 |
---|
983 | [1,3,6: % |2,5: >EQentry %] |
---|
984 | whd in match sigma_state; normalize nodelta |
---|
985 | cut(∀ a1,a2,b1,b2,c1,c2,d1,d2,e1,e2. |
---|
986 | a1=a2 → b1 = b2 → c1 = c2 → d1 = d2 → e1 = e2 → |
---|
987 | mk_state ERTL_semantics a1 b1 c1 d1 e1 = |
---|
988 | mk_state ERTL_semantics a2 b2 c2 d2 e2) |
---|
989 | [1,3: #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 |
---|
990 | #H15 //] #APP @APP try % |
---|
991 | whd in match set_frms; normalize nodelta >EQrest |
---|
992 | whd in match sigma_frames; normalize nodelta >m_return_bind |
---|
993 | >m_return_bind whd in match sigma_frames_opt; whd in match m_list_map; |
---|
994 | normalize nodelta whd in match (foldr ?????) in ⊢ (???%); |
---|
995 | normalize nodelta >EQhd >EQfn >m_return_bind normalize nodelta |
---|
996 | whd in match (foldr ?????); normalize nodelta |
---|
997 | >EQfn >m_return_bind normalize nodelta |
---|
998 | change with (sigma_frames_opt prog f_lbls f_regs ?) |
---|
999 | in match (foldr ?????); |
---|
1000 | change with (sigma_frames_opt prog f_lbls f_regs ?) |
---|
1001 | in match (foldr ?????); |
---|
1002 | cases(sigma_frames_opt ????) [%] #t_rest >m_return_bind >m_return_bind |
---|
1003 | @eq_f @eq_f2 [2: %] @eq_f2 [2: %] |
---|
1004 | change with (pc_block (pc … st2)) in match (pc_block ?); |
---|
1005 | change with (pc_block (pc … st2)) in match (pc_block ?); |
---|
1006 | whd in match sigma_register_env; normalize nodelta |
---|
1007 | >map_add >set_minus_add [%] @list_as_set_mem @in_added_registers |
---|
1008 | [@(point_of_pc ERTL_semantics (pc ERTLptr_semantics st2)) |
---|
1009 | | whd in match code_has_label; whd in match code_has_point; normalize nodelta |
---|
1010 | >EQstmt % | >EQregisters % % ] |
---|
1011 | |*: whd in match fetch_statement; normalize nodelta; >EQcalling' >m_return_bind |
---|
1012 | [2: >point_of_pc_of_point ] >EQcall >m_return_bind normalize nodelta |
---|
1013 | >m_return_bind |
---|
1014 | [2: >EQc_bl' >m_return_bind |
---|
1015 | | whd in match block_of_call; normalize nodelta |
---|
1016 | @('bind_inversion EQst_no_pc_pre_call) #st3 #H @('bind_inversion H) -H |
---|
1017 | #is3 inversion(istack … st2) [2: #bv |3: #bv1 #bv2] #EQis3 |
---|
1018 | whd in match is_push; normalize nodelta whd in ⊢ (??%% → ??%% → ?); |
---|
1019 | #EQ1 #EQ2 destruct(EQ1 EQ2) #H @('bind_inversion H) -H #is3 |
---|
1020 | whd in ⊢ (??%% → ??%% → ?); #EQ1 #EQ2 destruct(EQ1 EQ2) |
---|
1021 | whd in match set_regs; whd in match dpl_arg_retrieve; normalize nodelta |
---|
1022 | change with (ps_arg_retrieve ??) in match (dpl_arg_retrieve_ ?????); |
---|
1023 | @('bind_inversion EQc_bl') #c_bl'' whd in match dpl_arg_retrieve; normalize nodelta |
---|
1024 | change with (ps_arg_retrieve ??) in match (dpl_arg_retrieve_ ?????); |
---|
1025 | #H @('bind_inversion H) -H #bv_ptr1 whd in match ps_arg_retrieve; |
---|
1026 | normalize nodelta cases (c_ptr1) in EQstmt; [#r_ptr1 | #b_ptr1] #EQstmt |
---|
1027 | normalize nodelta |
---|
1028 | [ whd in match ps_reg_retrieve; whd in match reg_retrieve; normalize nodelta |
---|
1029 | #H lapply(opt_eq_from_res ???? H) -H #EQbv_ptr1 |
---|
1030 | | whd in ⊢ (??%% → ?); #EQ destruct(EQ) |
---|
1031 | ] |
---|
1032 | whd in match dph_arg_retrieve; normalize nodelta |
---|
1033 | change with (ps_arg_retrieve ??) in match (dph_arg_retrieve_ ?????); |
---|
1034 | whd in match ps_arg_retrieve; normalize nodelta cases c_ptr2 in EQstmt; |
---|
1035 | [1,3: #r_ptr2 |*: #b_ptr2] #EQstmt normalize nodelta |
---|
1036 | [1,2: whd in match ps_reg_retrieve; whd in match reg_retrieve; |
---|
1037 | normalize nodelta #H @('bind_inversion H) -H |
---|
1038 | #bv_ptr2 #H lapply(opt_eq_from_res ???? H) -H #EQbv_ptr2 |
---|
1039 | |*: >m_return_bind |
---|
1040 | ] #H @('bind_inversion H) -H #ptr #EQptr @if_elim #ptr_ok whd in ⊢ (??%% → ?); |
---|
1041 | #EQ destruct(EQ) #H lapply(opt_eq_from_res ???? H) -H #EQptr_code |
---|
1042 | [2,4: >m_return_bind |
---|
1043 | |*: >lookup_add_miss |
---|
1044 | [1,3: >EQbv_ptr1 >m_return_bind |
---|
1045 | |*: % #ABS destruct(ABS) |
---|
1046 | lapply(fresh_regs (point_of_pc ERTL_semantics (pc … st2))) |
---|
1047 | >EQregisters whd in ⊢ (% → ?); *** #ABS #_ #_ @ABS |
---|
1048 | lapply(regs_are_in_univers … (pi2 ?? fn) … EQstmt) whd in match stmt_registers; |
---|
1049 | whd in match get_used_registers_from_step; normalize nodelta |
---|
1050 | whd in ⊢ (???% → ?); * // |
---|
1051 | ] |
---|
1052 | ] |
---|
1053 | change with (ps_arg_retrieve ??) in match (dph_arg_retrieve_ ?????); |
---|
1054 | whd in match ps_arg_retrieve; normalize nodelta |
---|
1055 | [2,4: >m_return_bind |
---|
1056 | |*: whd in match ps_reg_retrieve; whd in match reg_retrieve; |
---|
1057 | normalize nodelta >lookup_add_miss |
---|
1058 | [1,3: >EQbv_ptr2 >m_return_bind |
---|
1059 | |*: % #ABS destruct(ABS) |
---|
1060 | lapply(fresh_regs (point_of_pc ERTL_semantics (pc … st2))) |
---|
1061 | >EQregisters whd in ⊢ (% → ?); *** #ABS #_ #_ @ABS |
---|
1062 | lapply(regs_are_in_univers … (pi2 ?? fn) … EQstmt) whd in match stmt_registers; |
---|
1063 | whd in match get_used_registers_from_step; normalize nodelta |
---|
1064 | whd in ⊢ (???% → ?); * // #_ * // |
---|
1065 | ] |
---|
1066 | ] |
---|
1067 | >EQptr >m_return_bind @if_elim [2,4,6,8: >ptr_ok *] #_ >m_return_bind |
---|
1068 | >EQptr_code >m_return_bind |
---|
1069 | ] |
---|
1070 | @('bind_inversion EQt_fn1) * #f2 * #fn2 #EQfn2 normalize nodelta |
---|
1071 | whd in ⊢ (??%% → ?); #EQ destruct(EQ) >EQfn2 >m_return_bind normalize nodelta |
---|
1072 | [ change with (ertlptr_save_frame ? it ?) in match (save_frame ??????); |
---|
1073 | cut(st2 = mk_state_pc ? st2 (pc … st2) (last_pop … st2)) |
---|
1074 | [ cases st2 #H1 #H2 #H3 %] #EQ <EQ -EQ >EQst_no_pc_after_call |
---|
1075 | >m_return_bind |
---|
1076 | |*: change with (ertlptr_save_frame PTR it ?) in match (save_frame ??????); |
---|
1077 | @('bind_inversion EQst_no_pc_after_call) #st4 whd in ⊢ (??%% → ?); |
---|
1078 | #EQ destruct(EQ) #H @('bind_inversion H) -H #frms4 #H |
---|
1079 | lapply(opt_eq_from_res ???? H) -H #EQfrms4 whd in ⊢ (??%% → ?); |
---|
1080 | #EQ destruct(EQ) whd in match ertlptr_save_frame; normalize nodelta |
---|
1081 | >m_return_bind whd in EQrest : (???%); destruct(EQrest) >EQfrms4 |
---|
1082 | >m_return_bind |
---|
1083 | ] |
---|
1084 | change with (stack_size ?) in match (stack_sizes ????); >EQs_size |
---|
1085 | >m_return_bind % |
---|
1086 | ] |
---|
1087 | qed. |
---|
1088 | |
---|
1089 | |
---|
1090 | inductive ex_Type1 (A:Type[1]) (P:A → Prop) : Prop ≝ |
---|
1091 | ex1_intro: ∀ x:A. P x → ex_Type1 A P. |
---|
1092 | (*interpretation "exists in Type[1]" 'exists x = (ex_Type1 ? x).*) |
---|
1093 | |
---|
1094 | lemma ertl_to_ertlptr_ok: |
---|
1095 | ∀prog.∀stack_sizes. |
---|
1096 | let trans_prog ≝ ertl_to_ertlptr prog in |
---|
1097 | ex_Type1 … (λR. |
---|
1098 | status_simulation |
---|
1099 | (ERTL_status prog stack_sizes) (ERTLptr_status trans_prog stack_sizes) R). |
---|
1100 | #prog #stack_size |
---|
1101 | cases(make_b_graph_transform_program_props ERTL_semantics ERTLptr_semantics |
---|
1102 | (λglobals,fn.«translate_data globals fn,(refl …)») prog) |
---|
1103 | #fregs * #f_lbls * #f_bl_r #good % |
---|
1104 | [@ERTLptrStatusSimulation assumption] |
---|
1105 | whd in match status_simulation; normalize nodelta |
---|
1106 | whd in match ERTL_status; whd in match ERTLptr_status; normalize nodelta |
---|
1107 | whd in ⊢ (% → % → % → % → ?); #st1 #st1' #st2 |
---|
1108 | change with |
---|
1109 | (eval_state ERTL_semantics (prog_var_names ???) ?? = ? → ?) |
---|
1110 | #EQeval @('bind_inversion EQeval) |
---|
1111 | ** #id #fn #stmt #H lapply (err_eq_from_io ????? H) -H #EQfetch |
---|
1112 | #_ whd in match ERTLptrStatusSimulation; normalize nodelta #EQst2 destruct |
---|
1113 | cases stmt in EQfetch; -stmt |
---|
1114 | [ * [ #cost | #called_id #args #dest| #reg #lbl | #seq] #nxt | #fin_step | *] |
---|
1115 | #EQfetch |
---|
1116 | change with (joint_classify ??) in |
---|
1117 | ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? | _ ⇒ ? | _ ⇒ ? | _ ⇒ ? ]); |
---|
1118 | [ (*COST*) whd in match joint_classify; normalize nodelta >EQfetch normalize nodelta |
---|
1119 | cases(eval_cost_ok … good … EQfetch EQeval) #st2' * #EQst2' * #taf #tafne |
---|
1120 | %{st2'} %{taf} >tafne normalize nodelta % [ % [@I | assumption]] |
---|
1121 | whd >EQst2' >(as_label_ok … good … st2') [%] cases daemon (* needs lemma see below!! *) |
---|
1122 | | (*CALL*) whd in match joint_classify; normalize nodelta >EQfetch |
---|
1123 | normalize nodelta #is_call_st1 |
---|
1124 | cases(eval_call_ok … good … EQfetch EQeval) #is_call_st1' |
---|
1125 | * #st2_pre_call * #is_call_st2_pre_call * * #Hcall |
---|
1126 | #call_rel * #taa * #st2' * #sem_rel #eval_rel |
---|
1127 | %{(«st2_pre_call,is_call_st2_pre_call»)} % [ % assumption] |
---|
1128 | %{st2'} %{st2'} %{taa} %{(taa_base …)} % [ % assumption] |
---|
1129 | whd >sem_rel >(as_label_ok … good … st2') [%] cases daemon (*TODO*) |
---|
1130 | | (*COND*) whd in match joint_classify; normalize nodelta >EQfetch |
---|
1131 | normalize nodelta #n_costed |
---|
1132 | cases(eval_cond_ok … good … EQfetch EQeval) [2: @n_costed] |
---|
1133 | #st2' * #EQst2' * #taf #tafne %{st2'} %{taf} >tafne % [% [@I|assumption]] |
---|
1134 | whd >EQst2' >(as_label_ok … good … st2') [%] cases daemon (*TODO*) |
---|
1135 | | (*seq*) whd in match joint_classify; normalize nodelta >EQfetch |
---|
1136 | normalize nodelta |
---|
1137 | cases (eval_seq_no_call_ok … good … EQfetch EQeval) |
---|
1138 | #st3 * #EQ destruct * #taf #taf_spec %{st3} %{taf} |
---|
1139 | % [% //] whd >(as_label_ok … good … st3) [%] |
---|
1140 | cases daemon (*needs lemma about preservation of fetch_statement *) |
---|
1141 | | cases fin_step in EQfetch; |
---|
1142 | [ (*GOTO*) #lbl #EQfetch whd in match joint_classify; normalize nodelta |
---|
1143 | >EQfetch normalize nodelta |
---|
1144 | cases (eval_goto_ok … good … EQfetch EQeval) |
---|
1145 | #st3 * #EQ destruct * #taf #tarne %{st3} %{taf} >tarne normalize nodelta |
---|
1146 | % [% //] whd >(as_label_ok … good … st3) [%] |
---|
1147 | cases daemon (*needs lemma about preservation of fetch_statement *) |
---|
1148 | | (*RETURN*) #EQfetch |
---|
1149 | whd in match joint_classify; normalize nodelta |
---|
1150 | >EQfetch normalize nodelta |
---|
1151 | cases (eval_return_ok … good … EQfetch EQeval) #is_ret |
---|
1152 | * #st3 * #EQ destruct * #after_ret * #taf ** #taf_prf #EQeval' #ret_prf |
---|
1153 | % [2: % [2: % [2: %{(taa_base …)} %{taf}] |*: ] |*:] |
---|
1154 | % [2: whd >(as_label_ok … good … st3) [%] cases daemon (*needs lemma *)] |
---|
1155 | % [2: assumption] % [2: %] % [2: assumption] % assumption |
---|
1156 | | (*TAILCALL*) * |
---|
1157 | ] |
---|
1158 | ] |
---|
1159 | qed. |
---|
1160 | |
---|
1161 | (* |
---|
1162 | lemma foo : |
---|
1163 | ∀P1_unser,P2_unser: unserialized_params. |
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1164 | ∀P1_sem_unser,P2_sem_unser. |
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1165 | let P1_sem ≝ make_sem_graph_params P1_unser P1_sem_unser in |
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1166 | let P2_sem ≝ make_sem_graph_params P2_unser P2_sem_unser in |
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1167 | ∀init,translate_step. |
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1168 | ∀translate_fin_step. |
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1169 | ∀closed_graph_translate. |
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1170 | let translate_internal : |
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1171 | ∀globals. |
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1172 | joint_closed_internal_function P1_sem globals → |
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1173 | joint_closed_internal_function P2_sem globals |
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1174 | ≝ |
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1175 | λglobals,fun. |
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1176 | «b_graph_translate (mk_graph_params P1_sem) (mk_graph_params P2_sem) globals |
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1177 | (init globals) |
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1178 | (translate_step globals) |
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1179 | (translate_fin_step globals) |
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1180 | (pi1 … fun), closed_graph_translate globals fun» in |
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1181 | ∀prog. |
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1182 | let trans_prog ≝ transform_program ??? prog (λvarnames. transf_fundef ?? (translate_internal varnames)) in |
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1183 | ∀stack_size. |
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1184 | ∀sigma_state_pc. |
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1185 | (∀s. pc_block (pc … (sigma_state_pc s)) = pc_block … (pc … s)) → |
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1186 | ∀st : state_pc P2_sem. |
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1187 | ∀st' : state_pc P1_sem. |
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1188 | ∀f. |
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1189 | ∀fn: joint_closed_internal_function P1_sem (globals (mk_prog_params P1_sem prog stack_size)). |
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1190 | luniverse_ok … fn → |
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1191 | ∀stmt,nxt. |
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1192 | ∀P : (state_pc P2_sem) → Prop. |
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1193 | (∀pre_Instrs',last_Instrs',dst. |
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1194 | ∃st''.∃st'''.∃st''''. |
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1195 | repeat_eval_seq_no_pc (mk_prog_params P2_sem trans_prog stack_size) |
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1196 | f (translate_internal ? fn) (map_eval ?? pre_Instrs' dst) st = return st''' ∧ |
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1197 | eval_seq_no_pc ? (prog_var_names … trans_prog) (ev_genv … (mk_prog_params P2_sem trans_prog stack_size)) |
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1198 | f (translate_internal … fn) (last_Instrs' dst) st''' = return st'''' ∧ |
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1199 | st'' = (mk_state_pc (mk_prog_params P2_sem trans_prog stack_size) |
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1200 | st'''' (pc_of_point P2_sem (pc_block (pc … (sigma_state_pc st))) dst) (last_pop … st)) ∧ |
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1201 | P st'' (*st' = sigma_state_pc st''*) ∧ |
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1202 | let P2_prog_params ≝ mk_prog_params P2_sem trans_prog stack_size in |
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1203 | let P2_globals ≝ globals P2_prog_params in |
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1204 | All |
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1205 | (joint_seq … P2_globals) |
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1206 | (no_cost_label … P2_globals) |
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1207 | (map_eval (code_point P2_sem) (joint_seq … P2_globals) pre_Instrs' dst)) → |
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1208 | fetch_statement … |
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1209 | (prog_var_names (joint_function P1_sem) ℕ prog) |
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1210 | (ev_genv (mk_prog_params P1_sem prog stack_size)) |
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1211 | (pc … (sigma_state_pc st)) = |
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1212 | return 〈f, fn, sequential … (step_seq P1_sem … stmt) nxt〉 → |
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1213 | eval_state … |
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1214 | (prog_var_names (joint_function P1_sem) ℕ prog) |
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1215 | (ev_genv … (mk_prog_params P1_sem prog stack_size)) |
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1216 | (sigma_state_pc st) = return st' → |
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1217 | ∃st''. (*CSC: this needs to be linked back again st' = sigma_state_pc ? good st'' ∧*) |
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1218 | P st'' ∧ |
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1219 | ∃taf : trace_any_any_free (joint_abstract_status (mk_prog_params P2_sem trans_prog stack_size)) st st''. |
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1220 | (if taaf_non_empty … taf then True else |
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1221 | (¬as_costed (joint_abstract_status (mk_prog_params P1_sem prog stack_size)) (sigma_state_pc st) ∨ |
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1222 | ¬as_costed (joint_abstract_status (mk_prog_params P1_sem prog stack_size)) st')) |
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1223 | (* the length of taf is the same of the length of ??? *). |
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1224 | #P1_unser #P2_unser #P1_sem_unser #P2_sem_unser |
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1225 | letin P1_sem ≝ (make_sem_graph_params P1_unser P1_sem_unser) |
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1226 | letin P2_sem ≝ (make_sem_graph_params P2_unser P2_sem_unser) |
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1227 | #init #translate_step #translate_fin_step #closed_graph_translate. |
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1228 | letin translate_internal ≝ |
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1229 | (λglobals,fun. |
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1230 | «b_graph_translate (mk_graph_params P1_sem) (mk_graph_params P2_sem) globals |
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1231 | (init globals) |
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1232 | (translate_step globals) |
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1233 | (translate_fin_step globals) |
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1234 | (pi1 … fun), closed_graph_translate globals fun») |
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1235 | #prog |
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1236 | letin trans_prog ≝ (transform_program ??? prog (λvarnames. transf_fundef ?? (translate_internal varnames))) |
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1237 | #stack_size #sigma_state_pc #sigma_state_pc_ok #st #st' #f #fn #fn_universe_ok #stmt #nxt #P |
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1238 | #Hpass #EQfetch lapply(fetch_statement_inv … EQfetch) * #EQfn #EQstmt |
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1239 | cases |
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1240 | (b_graph_translate_ok … (init …) (translate_step …) (translate_fin_step …) … fn_universe_ok) |
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1241 | #_ * #MK_fresh_labels * #MK_fresh_registers **** #_ #_ #_ #_ (*CSC: useful things thrown out here *) |
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1242 | #MULTI_FETCH_OK cases (MULTI_FETCH_OK … EQstmt) |
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1243 | #list_b_last * #fresh_registers ** #EQlist_b_last #EQfresh_registers |
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1244 | normalize nodelta * #Instrs * #fresh_registers_spec whd in ⊢ (% → ?); |
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1245 | @pair_elim #pre_Instrs #last_Instrs #EQInstrs * #dst * #Multi_fetch #STEP_in_code |
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1246 | #EQeval |
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1247 | cut((list |
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1248 | (code_point P2_sem |
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1249 | →joint_seq P2_sem (prog_var_names (joint_function P1_sem) ℕ prog)))) [ cases daemon (*Paolo should fix the lemma*)] |
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1250 | #pre_Instrs' |
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1251 | cut((code_point P2_sem |
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1252 | →joint_seq P2_sem (prog_var_names (joint_function P1_sem) ℕ prog))) [ cases daemon (*Paolo should fix the lemma*)] |
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1253 | #last_Instrs' |
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1254 | cases (Hpass pre_Instrs' last_Instrs' dst (* MANY MORE HERE *)) |
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1255 | #st'' * #st''' * #st'''' **** #REPEAT #EVAL_NO_PC_Last #EQst'' #EQst' #NO_COSTED |
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1256 | lapply(produce_trace_any_any_free … NO_COSTED … REPEAT) |
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1257 | [ cases daemon (* should be @Multi_fetch *) |
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1258 | | <sigma_state_pc_ok @(fetch_internal_function_transf … (λvars.translate_internal vars)) |
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1259 | assumption |
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1260 | | @dst |
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1261 | | @list_b_last (*wrong, should dst be destination or the last of list_b_last *) |
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1262 | ] #TAAF |
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1263 | lapply |
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1264 | (produce_step_trace (mk_prog_params P2_sem trans_prog stack_size) |
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1265 | (mk_state_pc (mk_prog_params P2_sem trans_prog stack_size) |
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1266 | st''' (pc_of_point P2_sem (pc_block (pc … st)) dst) (last_pop … st)) |
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1267 | f (translate_internal ? fn) (last_Instrs' dst) nxt st'''' … EVAL_NO_PC_Last) |
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1268 | [ cases daemon (* should be @STEP_In_code *) |
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1269 | | <sigma_state_pc_ok |
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1270 | @(fetch_internal_function_transf … (λvars. translate_internal vars) … EQfn) ] |
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1271 | #LAST_STEP |
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1272 | cases daemon |
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1273 | qed. |
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1274 | |
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1275 | lemma eval_seq_no_call_ok : |
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1276 | ∀prog. |
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1277 | let trans_prog ≝ ertl_to_ertlptr prog in |
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1278 | ∀good : (∀fn.good_state_transformation prog fn). |
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1279 | ∀stack_sizes. |
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1280 | (*? →*) |
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1281 | ∀st : state_pc ERTLptr_semantics. |
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1282 | ∀st' : state_pc ERTL_semantics. |
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1283 | ∀f,fn,stmt,nxt. |
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1284 | fetch_statement ERTL_semantics |
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1285 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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1286 | (ev_genv (mk_prog_params ERTL_semantics prog stack_sizes)) |
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1287 | (pc … (sigma_state_pc ? good st)) = |
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1288 | return 〈f, fn, sequential … (step_seq ERTL … stmt) nxt〉 → |
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1289 | eval_state ERTL_semantics |
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1290 | (prog_var_names (joint_function ERTL_semantics) ℕ prog) |
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1291 | (ev_genv … (mk_prog_params ERTL_semantics prog stack_sizes)) |
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1292 | (sigma_state_pc ? good st) = |
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1293 | return st' → |
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1294 | ∃st''. st' = sigma_state_pc ? good st'' ∧ |
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1295 | ∃taf : trace_any_any_free (ERTLptr_status trans_prog stack_sizes) |
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1296 | st |
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1297 | st''. |
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1298 | bool_to_Prop (taaf_non_empty … taf). |
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1299 | #prog #good #stack_size #st #st' #f #fn #stmt #nxt #EQfetch #EQeval |
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1300 | cases (foo … translate_step translate_fin_step … EQfetch EQeval) |
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1301 | [2,3: cases daemon (* first one, take out init definition from ERTLtoERTLptr; |
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1302 | second one to be taken from somewhere *) |
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1303 | |4: #x cases daemon (* TO BE DONE, EASY *) |
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1304 | |5: cases daemon (* to be taken in input *) |
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1305 | |6: @(λst''.st' = sigma_state_pc ? good st'') |
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1306 | |7: cases daemon |
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1307 | | #st'' * #EQ destruct * change with ERTL_uns in match (mk_unserialized_params ???????????????); |
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1308 | change with ERTLptr_uns in match (mk_unserialized_params ???????????????); |
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1309 | (* change with ERTL_semantics in match (make_sem_graph_params ??); |
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1310 | change with ERTLptr_semantics in match (make_sem_graph_params ??); *) |
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1311 | #taaf #Htaaf %{st''} |
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1312 | % [%] cases daemon |
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1313 | qed. *) |
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