[3394] | 1 | (**************************************************************************) |
---|
| 2 | (* ___ *) |
---|
| 3 | (* ||M|| *) |
---|
| 4 | (* ||A|| A project by Andrea Asperti *) |
---|
| 5 | (* ||T|| *) |
---|
| 6 | (* ||I|| Developers: *) |
---|
| 7 | (* ||T|| The HELM team. *) |
---|
| 8 | (* ||A|| http://helm.cs.unibo.it *) |
---|
| 9 | (* \ / *) |
---|
| 10 | (* \ / This file is distributed under the terms of the *) |
---|
| 11 | (* v GNU General Public License Version 2 *) |
---|
| 12 | (* *) |
---|
| 13 | (**************************************************************************) |
---|
| 14 | |
---|
| 15 | include "basics/types.ma". |
---|
| 16 | include "Traces.ma". |
---|
| 17 | include "basics/lists/list.ma". |
---|
| 18 | include "../src/utilities/option.ma". |
---|
[3396] | 19 | include "basics/jmeq.ma". |
---|
[3549] | 20 | include "utils.ma". |
---|
[3394] | 21 | |
---|
[3397] | 22 | discriminator option. |
---|
| 23 | |
---|
[3394] | 24 | record instr_params : Type[1] ≝ |
---|
| 25 | { seq_instr : DeqSet |
---|
| 26 | ; io_instr : DeqSet |
---|
| 27 | ; cond_instr : DeqSet |
---|
| 28 | ; loop_instr : DeqSet |
---|
| 29 | ; act_params_type : DeqSet |
---|
| 30 | ; return_type : DeqSet |
---|
| 31 | }. |
---|
| 32 | |
---|
| 33 | |
---|
[3549] | 34 | inductive Instructions (p : instr_params) |
---|
| 35 | (l_p :label_params) : Type[0] ≝ |
---|
| 36 | | EMPTY : Instructions p l_p |
---|
| 37 | | RETURN : return_type p → Instructions p l_p |
---|
| 38 | | SEQ : (seq_instr p) → option (NonFunctionalLabel l_p) → Instructions p l_p → Instructions p l_p |
---|
| 39 | | COND : (cond_instr p) → (NonFunctionalLabel l_p) → Instructions p l_p → |
---|
| 40 | (NonFunctionalLabel l_p) → Instructions p l_p → Instructions p l_p → |
---|
| 41 | Instructions p l_p |
---|
| 42 | | LOOP : (loop_instr p) → NonFunctionalLabel l_p → Instructions p l_p → |
---|
| 43 | NonFunctionalLabel l_p → Instructions p l_p → Instructions p l_p |
---|
| 44 | | CALL : FunctionName → (act_params_type p) → option (ReturnPostCostLabel l_p) → |
---|
| 45 | Instructions p l_p → Instructions p l_p |
---|
| 46 | | IO : NonFunctionalLabel l_p → (io_instr p) → NonFunctionalLabel l_p → Instructions p l_p → |
---|
| 47 | Instructions p l_p. |
---|
[3394] | 48 | |
---|
[3549] | 49 | let rec eq_instructions (p : instr_params) (l_p : label_params) (i : Instructions p l_p) |
---|
| 50 | on i : (Instructions p l_p) → bool ≝ |
---|
[3394] | 51 | match i with |
---|
| 52 | [ EMPTY ⇒ λi'.match i' with [ EMPTY ⇒ true | _ ⇒ false ] |
---|
| 53 | | RETURN x ⇒ λi'.match i' with [ RETURN y ⇒ x == y | _ ⇒ false ] |
---|
| 54 | | SEQ x lab instr ⇒ λi'.match i' with |
---|
| 55 | [ SEQ y lab' instr' ⇒ x == y ∧ eq_instructions … instr instr' ∧ |
---|
| 56 | match lab with [ None ⇒ match lab' with [ None ⇒ true | _ ⇒ false ] |
---|
[3549] | 57 | | Some l1 ⇒ match lab' with [Some l2 ⇒ eq_nf_label … l1 l2 | _ ⇒ false] |
---|
[3394] | 58 | ] |
---|
| 59 | | _ ⇒ false |
---|
| 60 | ] |
---|
| 61 | | COND x ltrue i1 lfalse i2 i3 ⇒ λi'.match i' with |
---|
| 62 | [ COND y ltrue' i1' lfalse' i2' i3' ⇒ |
---|
[3549] | 63 | x == y ∧ eq_nf_label … ltrue ltrue' ∧ |
---|
| 64 | eq_instructions … i1 i1' ∧ eq_nf_label … lfalse lfalse' ∧ |
---|
[3394] | 65 | eq_instructions … i2 i2' ∧ eq_instructions … i3 i3' |
---|
| 66 | | _ ⇒ false |
---|
| 67 | ] |
---|
| 68 | | LOOP x ltrue i1 lfalse i2 ⇒ λi'.match i' with |
---|
| 69 | [ LOOP y ltrue' i1' lfalse' i2' ⇒ x == y ∧ |
---|
[3549] | 70 | eq_instructions … i1 i1' ∧ eq_nf_label … ltrue ltrue' ∧ |
---|
[3394] | 71 | eq_instructions … i2 i2' |
---|
| 72 | | _ ⇒ false |
---|
| 73 | ] |
---|
| 74 | | CALL f act_p r_lb i1 ⇒ λi'.match i' with |
---|
| 75 | [ CALL f' act_p' r_lb' i1' ⇒ eq_function_name f f' ∧ |
---|
| 76 | act_p == act_p' ∧ eq_instructions … i1 i1' ∧ |
---|
| 77 | match r_lb with [ None ⇒ match r_lb' with [None ⇒ true | _ ⇒ false] |
---|
[3549] | 78 | | Some z ⇒ match r_lb' with [Some w ⇒ eq_return_cost_lab … z w | _ ⇒ false ] |
---|
[3394] | 79 | ] |
---|
| 80 | | _ ⇒ false |
---|
| 81 | ] |
---|
| 82 | | IO lin io lout i1 ⇒ λi'.match i' with |
---|
[3549] | 83 | [ IO lin' io' lout' i1' ⇒ eq_nf_label … lin lin' ∧ io == io' ∧ |
---|
| 84 | eq_nf_label … lout lout' ∧ eq_instructions … i1 i1' |
---|
[3394] | 85 | | _ ⇒ false |
---|
| 86 | ] |
---|
| 87 | ]. |
---|
| 88 | |
---|
[3561] | 89 | lemma eq_instructions_elim : ∀ P : bool → Prop.∀p,l_p,i1,i2.(i1 = i2 → P true) → |
---|
| 90 | (i1 ≠ i2 → P false) → P (eq_instructions p l_p i1 i2). |
---|
[3570] | 91 | #P #p #l_p #i1 lapply P -P elim i1 |
---|
| 92 | [ #P * normalize [/2/ ] #x [|*: #y #z [2,3: #w1 #w2 [#w3] |4,5: #w1]] #_ #H2 @H2 % #EQ |
---|
[3401] | 93 | lapply (eq_to_jmeq ??? EQ) #EQ' destruct(EQ') |
---|
[3570] | 94 | | #rt #P * normalize [2: #rt' cases (dec_eq … rt rt') #H [>(\b H) | >(\bf H) ] /2/ |
---|
[3401] | 95 | #_ #K @K % #abs lapply (eq_to_jmeq ??? abs) #abs' destruct(abs') @(absurd ?? H) //] |
---|
| 96 | [|*: #x #y #z [2,3: #w1 #w2 [#w3] |4,5: #w1]] #_ #H2 @H2 % #EQ |
---|
| 97 | lapply (eq_to_jmeq ??? EQ) #EQ' destruct(EQ') |
---|
[3570] | 98 | | #seq * [| #lbl] #i #IH #P * normalize |
---|
[3449] | 99 | [1,8: |2,9: #t_t |3,10: #seq1 #opt_l #i2 |4,11: #cond #ltrue #i_t #lfalse #i_f #i_c |
---|
| 100 | |5,12: #cond #ltrue #i_t #lfalse #i_f |6,13: #f #act_p #ret #i3 |*: #lin #io #lout #i3] |
---|
| 101 | #H1 #H2 try(@H2 % #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct) |
---|
| 102 | inversion(?==?) normalize nodelta |
---|
| 103 | [2,4: #ABS @H2 % #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct |
---|
| 104 | >(\b (refl …)) in ABS; #EQ destruct] |
---|
[3561] | 105 | #EQseq cases daemon |
---|
| 106 | |*: cases daemon |
---|
| 107 | qed. |
---|
[3394] | 108 | |
---|
| 109 | record env_params : Type[1] ≝ |
---|
| 110 | { form_params_type : Type[0] |
---|
| 111 | }. |
---|
| 112 | |
---|
[3397] | 113 | record signature (p : env_params) (p' : instr_params) : Type[0] ≝ |
---|
[3394] | 114 | { f_name : FunctionName |
---|
| 115 | ; f_pars : form_params_type p |
---|
[3397] | 116 | ; f_ret : return_type p' |
---|
| 117 | }. |
---|
| 118 | |
---|
[3549] | 119 | record env_item (p : env_params) (p' : instr_params) (l_p : label_params) : Type[0] ≝ |
---|
[3397] | 120 | { f_sig :> signature p p' |
---|
[3549] | 121 | ; f_lab : CallCostLabel l_p |
---|
| 122 | ; f_body : Instructions p' l_p |
---|
[3394] | 123 | }. |
---|
| 124 | |
---|
| 125 | record state_params : Type[1] ≝ |
---|
[3399] | 126 | { i_pars :> instr_params |
---|
| 127 | ; e_pars :> env_params |
---|
[3549] | 128 | ; l_pars :> label_params |
---|
[3394] | 129 | ; store_type : DeqSet |
---|
| 130 | }. |
---|
| 131 | |
---|
| 132 | record state (p : state_params) : Type[0] ≝ |
---|
[3549] | 133 | { code : Instructions p p |
---|
| 134 | ; cont : list (ActionLabel p × (Instructions p p)) |
---|
[3394] | 135 | ; store : store_type p |
---|
| 136 | ; io_info : bool |
---|
| 137 | }. |
---|
| 138 | |
---|
| 139 | definition is_io : ∀p.state p → Prop ≝ λp,st.io_info … st = true. |
---|
| 140 | |
---|
| 141 | record sem_state_params (p : state_params) : Type[0] ≝ |
---|
[3446] | 142 | { eval_seq : seq_instr p → (store_type p) → option (store_type p) |
---|
| 143 | ; eval_io : io_instr p → store_type p → option (store_type p) |
---|
| 144 | ; eval_cond_cond : cond_instr p → store_type p → option (bool × (store_type p)) |
---|
| 145 | ; eval_loop_cond : loop_instr p → store_type p → option (bool × (store_type p)) |
---|
[3399] | 146 | ; eval_call : signature p p → act_params_type p → store_type p → option (store_type p) |
---|
[3394] | 147 | ; eval_after_return : return_type p → store_type p → option (store_type p) |
---|
| 148 | ; init_store : store_type p |
---|
| 149 | }. |
---|
| 150 | |
---|
| 151 | |
---|
[3549] | 152 | let rec lookup (p : env_params) (p' : instr_params) (l_p : label_params) (l : list (env_item p p' l_p)) |
---|
| 153 | on l : FunctionName → option (env_item p p' l_p) ≝ |
---|
[3394] | 154 | match l with |
---|
| 155 | [ nil ⇒ λ_.None ? |
---|
| 156 | | cons x xs ⇒ λf.if (eq_function_name f (f_name … x)) |
---|
| 157 | then Some ? x |
---|
| 158 | else lookup … xs f |
---|
| 159 | ]. |
---|
| 160 | |
---|
[3549] | 161 | inductive execute_l (p : state_params) (p' : sem_state_params p) (env : list (env_item p p p)) : |
---|
| 162 | ActionLabel p → relation (state p) ≝ |
---|
| 163 | | empty : ∀st,st',hd,tl.(code ? st) = (EMPTY p p)→ (cont ? st) = hd :: tl → |
---|
[3394] | 164 | (code ? st') = \snd hd → (cont … st') = tl → (store … st) = (store … st') → |
---|
[3549] | 165 | (io_info … st = true → is_non_silent_cost_act … (\fst hd)) → (io_info … st') = false → ¬ is_ret_call_act … (\fst hd) → execute_l … (\fst hd) st st' |
---|
[3394] | 166 | | seq_sil : ∀st,st',i,cd,s,opt_l.(code ? st) = SEQ … i opt_l cd → |
---|
[3446] | 167 | eval_seq … p' i (store … st) = return s → (code ? st') = cd → |
---|
[3394] | 168 | (cont … st) = (cont … st') → (store … st') = s → |
---|
[3549] | 169 | io_info … st = false → io_info ? st' = false → execute_l … (cost_act … opt_l) st st' |
---|
[3394] | 170 | | cond_true : ∀st,st',exp,ltrue,i_true,lfalse,i_false,cd,new_m. |
---|
[3446] | 171 | (code ? st) = COND … exp ltrue i_true lfalse i_false cd → eval_cond_cond … p' exp (store … st) = return 〈true,new_m〉 → |
---|
[3549] | 172 | cont ? st' = 〈cost_act … (None ?),cd〉 ::(cont … st) → code … st' = i_true → store … st' = new_m → |
---|
| 173 | io_info … st = false → io_info … st' = false → execute_l … (cost_act … (Some ? ltrue)) st st' |
---|
[3394] | 174 | | cond_false : ∀st,st',exp,ltrue,i_true,lfalse,i_false,cd,new_m. |
---|
[3446] | 175 | (code ? st) = COND … exp ltrue i_true lfalse i_false cd → eval_cond_cond … p' exp (store … st) = return 〈false,new_m〉 → |
---|
[3549] | 176 | cont ? st' = 〈cost_act … (None ?),cd〉 ::(cont … st) → code … st' = i_false → store … st' = new_m → |
---|
| 177 | io_info … st = false → io_info … st' = false → execute_l … (cost_act … (Some ? lfalse)) st st' |
---|
[3394] | 178 | | loop_true : ∀st,st',exp,ltrue,i_true,lfalse,i_false,new_m. |
---|
[3446] | 179 | code ? st = LOOP … exp ltrue i_true lfalse i_false → eval_loop_cond … p' exp (store … st) = return 〈true,new_m〉 → |
---|
[3549] | 180 | cont ? st' = 〈cost_act … (None ?),LOOP … exp ltrue i_true lfalse i_false〉 :: (cont … st) → |
---|
[3394] | 181 | code … st' = i_true → store … st' = new_m → io_info … st = false → io_info … st' = false → |
---|
[3549] | 182 | execute_l … (cost_act … (Some ? ltrue)) st st' |
---|
[3394] | 183 | | loop_false : ∀st,st',exp,ltrue,i_true,lfalse,i_false,new_m. |
---|
[3446] | 184 | code ? st = LOOP … exp ltrue i_true lfalse i_false → eval_loop_cond … p' exp (store … st) = return 〈false,new_m〉 → |
---|
| 185 | cont ? st' = cont … st → code … st' = i_false → store … st' = new_m → |
---|
[3549] | 186 | io_info … st = false → io_info … st' = false → execute_l … (cost_act … (Some ? lfalse)) st st' |
---|
[3394] | 187 | | io_in : ∀st,st',lin,io,lout,cd,mem.(code ? st) = IO … lin io lout cd → |
---|
[3549] | 188 | eval_io … p' io (store … st) = return mem → code ? st' = EMPTY p p → |
---|
| 189 | cont … st' = 〈cost_act … (Some ? lout),cd〉 :: (cont … st) → store … st' = mem → |
---|
| 190 | io_info … st' = true → execute_l … (cost_act … (Some ? lin)) st st' |
---|
[3394] | 191 | | call : ∀st,st',f,act_p,r_lb,cd,mem,env_it.(code ? st) = CALL … f act_p r_lb cd → |
---|
| 192 | lookup … env f = return env_it → |
---|
[3399] | 193 | eval_call ? p' env_it act_p (store … st) = return mem → |
---|
[3394] | 194 | store ? st' = mem → code … st' = f_body … env_it → |
---|
| 195 | cont … st' = |
---|
[3549] | 196 | 〈(ret_act … r_lb),cd〉 :: (cont … st) → |
---|
[3394] | 197 | io_info … st = false → (io_info … st') = false → |
---|
[3549] | 198 | execute_l … (call_act … f (f_lab … env_it)) st st' |
---|
[3394] | 199 | | ret_instr : ∀st,st',r_t,mem,tl',rb,cd.code ? st = RETURN … r_t → |
---|
[3549] | 200 | cont … st = 〈ret_act … rb,cd〉 :: tl' → cont ? st' = tl' → |
---|
[3394] | 201 | io_info … st = false → io_info ? st' = false → |
---|
| 202 | eval_after_return … p' r_t (store … st) = return mem → code … st' = cd → |
---|
[3549] | 203 | store … st' = mem → execute_l … (ret_act … rb) st st'. |
---|
[3400] | 204 | |
---|
[3549] | 205 | let rec get_labels_of_code (p : instr_params) (l_p : label_params) (i : Instructions p l_p) on i : list (CostLabel l_p) ≝ |
---|
[3400] | 206 | match i with |
---|
| 207 | [ EMPTY ⇒ [ ] |
---|
| 208 | | RETURN x ⇒ [ ] |
---|
| 209 | | SEQ x lab instr ⇒ let ih ≝ get_labels_of_code … instr in |
---|
[3549] | 210 | match lab with [ None ⇒ ih | Some lbl ⇒ a_non_functional_label … lbl :: ih ] |
---|
[3400] | 211 | | COND x ltrue i1 lfalse i2 i3 ⇒ |
---|
| 212 | let ih3 ≝ get_labels_of_code … i3 in |
---|
| 213 | let ih2 ≝ get_labels_of_code … i2 in |
---|
| 214 | let ih1 ≝ get_labels_of_code … i1 in |
---|
| 215 | ltrue :: lfalse :: (ih1 @ ih2 @ih3) |
---|
| 216 | | LOOP x ltrue i1 lfalse i2 ⇒ |
---|
| 217 | let ih2 ≝ get_labels_of_code … i2 in |
---|
| 218 | let ih1 ≝ get_labels_of_code … i1 in |
---|
[3549] | 219 | a_non_functional_label … ltrue :: a_non_functional_label … lfalse :: (ih1 @ ih2) |
---|
[3400] | 220 | | CALL f act_p r_lb i1 ⇒ |
---|
| 221 | let ih1 ≝ get_labels_of_code … i1 in |
---|
[3549] | 222 | match r_lb with [ None ⇒ ih1 | Some lbl ⇒ a_return_post … lbl :: ih1] |
---|
[3400] | 223 | | IO lin io lout i1 ⇒ |
---|
| 224 | let ih1 ≝ get_labels_of_code … i1 in |
---|
[3549] | 225 | a_non_functional_label … lin :: a_non_functional_label … lout :: ih1 |
---|
[3400] | 226 | ]. |
---|
| 227 | |
---|
[3549] | 228 | record Program (p : env_params) (p' : instr_params) (l_p : label_params) : Type[0] ≝ |
---|
| 229 | { env : list (env_item p p' l_p) |
---|
| 230 | ; main : Instructions p' l_p |
---|
[3394] | 231 | }. |
---|
| 232 | |
---|
[3400] | 233 | |
---|
[3549] | 234 | |
---|
| 235 | definition no_duplicates_labels : ∀p,p',l_p.Program p p' l_p → Prop ≝ |
---|
| 236 | λp,p',l_p,prog. |
---|
[3400] | 237 | no_duplicates … |
---|
[3549] | 238 | (foldr … |
---|
| 239 | (λitem,acc.((a_call … (f_lab … item)) :: get_labels_of_code … (f_body … item)) @ acc) |
---|
| 240 | (get_labels_of_code … (main … prog)) (env … prog)). |
---|
[3400] | 241 | |
---|
| 242 | lemma no_duplicates_domain_of_fun: |
---|
[3549] | 243 | ∀p,p',l_p,prog.no_duplicates_labels … prog → |
---|
| 244 | ∀f,env_it.lookup p p' l_p (env … prog) f = return env_it → |
---|
[3400] | 245 | no_duplicates … (get_labels_of_code … (f_body … env_it)). |
---|
[3549] | 246 | #p #p' #l_p * #env elim env [ #main normalize #_ #f #env_it #EQ destruct(EQ)] |
---|
[3400] | 247 | #x #xs #IH #main whd in ⊢ (% → ?); whd in match (foldr ?????); #H #f #env_it |
---|
| 248 | whd in ⊢ (??%? → ?); @eq_function_name_elim normalize nodelta |
---|
| 249 | [ whd in ⊢ (? → ???% → ?); #EQ1 #EQ2 destruct(EQ1 EQ2) cases H #_ /2/ ] |
---|
| 250 | #H1 #EQenv_it @IH cases H /2/ |
---|
| 251 | qed. |
---|
| 252 | |
---|
| 253 | |
---|
[3447] | 254 | definition is_synt_succ : ∀p.relation (state p) ≝ λp,s1,s2.cont … s1 = cont … s2 ∧ |
---|
[3394] | 255 | match (code … s1) with |
---|
| 256 | [ CALL f act_p r_lb i1 ⇒ code … s2 = i1 |
---|
| 257 | | _ ⇒ False |
---|
| 258 | ]. |
---|
| 259 | |
---|
[3549] | 260 | definition operational_semantics : ∀p : state_params.∀p'.Program p p p → abstract_status p ≝ |
---|
| 261 | λp,p',prog.mk_abstract_status … |
---|
[3394] | 262 | (state p) |
---|
| 263 | (execute_l ? p' (env … prog)) |
---|
| 264 | (is_synt_succ …) |
---|
| 265 | (λs.match (code … s) with |
---|
| 266 | [ COND _ _ _ _ _ _ ⇒ cl_jump |
---|
| 267 | | LOOP _ _ _ _ _ ⇒ cl_jump |
---|
| 268 | | EMPTY ⇒ if io_info … s then cl_io else cl_other |
---|
| 269 | | _ ⇒ cl_other |
---|
| 270 | ]) |
---|
| 271 | (λs.match (code … s) with |
---|
| 272 | [CALL _ _ m _ ⇒ match m with [ Some _ ⇒ true | None ⇒ false ] |
---|
| 273 | | _ ⇒ false |
---|
| 274 | ]) |
---|
| 275 | (λs.eq_instructions … (code … s) (main … prog) ∧ isnilb … (cont … s) ∧ store … s == init_store … p' ∧ io_info … s) |
---|
| 276 | (λs.match (cont … s) with |
---|
| 277 | [ nil ⇒ match (code … s) with |
---|
| 278 | [ EMPTY ⇒ true |
---|
| 279 | | RETURN _ ⇒ true |
---|
| 280 | | _ ⇒ false |
---|
| 281 | ] |
---|
| 282 | | _ ⇒ false |
---|
| 283 | ]) |
---|
| 284 | ???. |
---|
| 285 | @hide_prf |
---|
| 286 | [ #s1 #s2 #l #H #H1 inversion H1 #st #st' |
---|
| 287 | [ #hd #tl |
---|
| 288 | | #i #cd #s #opt_l |
---|
| 289 | |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m |
---|
| 290 | |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m |
---|
| 291 | | #lin #io #lout #cd #mem |
---|
| 292 | | #f #act_p #r_lb #cd #mem #env_it |
---|
| 293 | | #r_t #mem #tl #rb #cd |
---|
| 294 | ] |
---|
| 295 | #EQcode |
---|
| 296 | [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQ5 #H2 #EQ' #EQ6 #EQ7 |
---|
| 297 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 |
---|
| 298 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 |
---|
| 299 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 |
---|
| 300 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 |
---|
| 301 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 |
---|
| 302 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 |
---|
| 303 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 |
---|
| 304 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 |
---|
| 305 | ] |
---|
| 306 | #_ destruct >EQcode in H; normalize nodelta /2 by ex_intro/ |
---|
| 307 | [ cases(io_info ??) normalize nodelta] #EQ destruct |
---|
| 308 | | #s1 #s2 #l #H #H1 inversion H1 #st #st' |
---|
| 309 | [ #hd #tl |
---|
| 310 | | #i #cd #s #opt_l |
---|
| 311 | |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m |
---|
| 312 | |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m |
---|
| 313 | | #lin #io #lout #cd #mem |
---|
| 314 | | #f #act_p #r_lb #cd #mem #env_it |
---|
| 315 | | #r_t #mem #tl #rb #cd |
---|
| 316 | ] |
---|
| 317 | #EQcode |
---|
| 318 | [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQiost' #H2 #EQ' #EQ6 #EQ7 |
---|
| 319 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 320 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 321 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 322 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 323 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 324 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost' #EQ6 #EQ7 #EQ8 |
---|
| 325 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQiost' #EQ8 #EQ9 #EQ10 |
---|
| 326 | | #EQ1 #EQ2 #EQ3 #EQiost' #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 |
---|
| 327 | ] |
---|
| 328 | #_ destruct |
---|
| 329 | cases(code ? st') in H; normalize nodelta >EQiost' normalize nodelta |
---|
| 330 | #eq destruct try (#eq1 destruct) try (#eq2 destruct) try (#eq3 destruct) |
---|
| 331 | try (#eq4 destruct) try (#eq5 destruct) try (#eq6 destruct) %{lin} % |
---|
| 332 | | #s1 #s2 #l #H #H1 inversion H1 #st #st' |
---|
| 333 | [ #hd #tl |
---|
| 334 | | #i #cd #s #opt_l |
---|
| 335 | |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m |
---|
| 336 | |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m |
---|
| 337 | | #lin #io #lout #cd #mem |
---|
| 338 | | #f #act_p #r_lb #cd #mem #env_it |
---|
| 339 | | #r_t #mem #tl #rb #cd |
---|
| 340 | ] |
---|
| 341 | #EQcode |
---|
| 342 | [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQiost' #H2 #EQ' #EQ6 #EQ7 |
---|
| 343 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 344 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 345 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 346 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 347 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 |
---|
| 348 | | #EQ1 #EQ2 #EQ3 #EQiost #EQiost' #EQ6 #EQ7 #EQ8 |
---|
| 349 | | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost #EQiost' #EQ8 #EQ9 #EQ10 |
---|
| 350 | | #EQ1 #EQ2 #EQiost #EQiost' #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 |
---|
| 351 | ] |
---|
| 352 | #_ destruct >EQcode in H; normalize nodelta [|*: #EQ destruct] |
---|
| 353 | cases(io_info … st) in x; normalize nodelta [2: #_ #EQ destruct] |
---|
| 354 | #H3 #_ @H3 % |
---|
| 355 | ] |
---|
| 356 | qed. |
---|
| 357 | |
---|
[3549] | 358 | record call_post_info (p : instr_params) (l_p : label_params) : Type[0] ≝ |
---|
| 359 | { gen_labels : list (CostLabel l_p) |
---|
| 360 | ; t_code : Instructions p l_p |
---|
| 361 | ; fresh : l_p |
---|
| 362 | ; lab_map : associative_list (DEQCostLabel l_p) (CostLabel l_p) |
---|
| 363 | ; lab_to_keep : list (ReturnPostCostLabel l_p) |
---|
[3396] | 364 | }. |
---|
| 365 | |
---|
[3549] | 366 | let rec call_post_trans (p : instr_params) (l_p : label_params) (i : Instructions p l_p) (n : l_p) on i : |
---|
| 367 | list (CostLabel l_p) → call_post_info p l_p ≝ |
---|
[3396] | 368 | λabs. |
---|
[3394] | 369 | match i with |
---|
[3549] | 370 | [ EMPTY ⇒ mk_call_post_info … abs (EMPTY …) n (nil ?) (nil ?) |
---|
| 371 | | RETURN x ⇒ mk_call_post_info … abs (RETURN … x) n (nil ?) (nil ?) |
---|
[3394] | 372 | | SEQ x lab instr ⇒ |
---|
[3396] | 373 | let ih ≝ call_post_trans … instr n abs in |
---|
[3394] | 374 | match lab with |
---|
[3549] | 375 | [ None ⇒ mk_call_post_info … (gen_labels ?? ih) (SEQ … x (None ?) (t_code … ih)) |
---|
[3396] | 376 | (fresh … ih) (lab_map … ih) (lab_to_keep … ih) |
---|
| 377 | | Some lbl ⇒ |
---|
[3549] | 378 | mk_call_post_info … (nil ?) (SEQ … x (Some ? lbl) (t_code … ih)) (fresh … ih) |
---|
| 379 | (〈a_non_functional_label … lbl,((a_non_functional_label … lbl) :: (gen_labels … ih))〉 :: (lab_map … ih)) |
---|
[3396] | 380 | (lab_to_keep … ih) |
---|
[3394] | 381 | ] |
---|
| 382 | | COND x ltrue i1 lfalse i2 i3 ⇒ |
---|
[3396] | 383 | let ih3 ≝ call_post_trans … i3 n abs in |
---|
| 384 | let ih2 ≝ call_post_trans … i2 (fresh … ih3) (gen_labels … ih3) in |
---|
| 385 | let ih1 ≝ call_post_trans … i1 (fresh … ih2) (gen_labels … ih3) in |
---|
[3549] | 386 | mk_call_post_info p l_p (nil ?) (COND … x ltrue (t_code … ih1) lfalse (t_code … ih2) (t_code … ih3)) |
---|
[3396] | 387 | (fresh … ih1) |
---|
[3549] | 388 | (〈a_non_functional_label … ltrue,(a_non_functional_label … ltrue :: (gen_labels … ih1))〉:: |
---|
| 389 | 〈a_non_functional_label … lfalse,(a_non_functional_label … lfalse :: (gen_labels … ih2))〉:: |
---|
[3396] | 390 | ((lab_map … ih1) @ (lab_map … ih2) @ (lab_map … ih3))) |
---|
| 391 | ((lab_to_keep … ih1) @ (lab_to_keep … ih2) @ (lab_to_keep … ih3)) |
---|
[3394] | 392 | | LOOP x ltrue i1 lfalse i2 ⇒ |
---|
[3396] | 393 | let ih2 ≝ call_post_trans … i2 n abs in |
---|
| 394 | let ih1 ≝ call_post_trans … i1 (fresh … ih2) (nil ?) in |
---|
[3549] | 395 | mk_call_post_info p l_p (nil ?) (LOOP … x ltrue (t_code … ih1) lfalse (t_code … ih2)) (fresh … ih1) |
---|
| 396 | (〈a_non_functional_label … lfalse,(a_non_functional_label … lfalse :: (gen_labels … ih2))〉 :: |
---|
| 397 | 〈a_non_functional_label … ltrue,(a_non_functional_label … ltrue :: (gen_labels … ih1))〉 :: |
---|
[3396] | 398 | ((lab_map … ih1) @ (lab_map … ih2))) |
---|
| 399 | ((lab_to_keep … ih1) @ (lab_to_keep … ih2)) |
---|
[3394] | 400 | | CALL f act_p r_lb i1 ⇒ |
---|
[3396] | 401 | let ih ≝ call_post_trans … i1 n abs in |
---|
[3394] | 402 | match r_lb with |
---|
[3549] | 403 | [ None ⇒ let 〈l',f''〉 ≝ fresh_rc_label l_p (fresh … ih) in |
---|
| 404 | mk_call_post_info p l_p ((a_return_post … l')::(gen_labels … ih)) |
---|
[3396] | 405 | (CALL … f act_p (Some ? l') (t_code … ih)) f'' (lab_map … ih) (lab_to_keep … ih) |
---|
| 406 | | Some lbl ⇒ |
---|
[3549] | 407 | mk_call_post_info p l_p (nil ?) (CALL … f act_p (Some ? lbl) (t_code … ih)) (fresh … ih) |
---|
| 408 | (〈a_return_post … lbl,(a_return_post … lbl :: (gen_labels … ih))〉 :: (lab_map … ih)) |
---|
[3414] | 409 | (lbl :: lab_to_keep … ih) |
---|
[3394] | 410 | ] |
---|
| 411 | | IO lin io lout i1 ⇒ |
---|
[3396] | 412 | let ih ≝ call_post_trans … i1 n abs in |
---|
[3549] | 413 | mk_call_post_info p l_p (nil ?) (IO … lin io lout (t_code … ih)) (fresh … ih) |
---|
| 414 | (〈a_non_functional_label … lout,(a_non_functional_label … lout :: (gen_labels … ih))〉 :: |
---|
| 415 | 〈a_non_functional_label … lin,[a_non_functional_label … lin]〉 :: (lab_map … ih)) (lab_to_keep … ih) |
---|
[3394] | 416 | ]. |
---|
| 417 | |
---|
[3396] | 418 | |
---|
[3549] | 419 | let rec call_post_clean (p : instr_params) (l_p : label_params) (i : Instructions p l_p) on i : |
---|
| 420 | associative_list (DEQCostLabel l_p) (CostLabel l_p) → list (ReturnPostCostLabel l_p) → list (CostLabel l_p) → |
---|
| 421 | option ((list (CostLabel l_p)) × (Instructions p l_p)) ≝ |
---|
[3396] | 422 | λm,keep,abs. |
---|
| 423 | match i with |
---|
| 424 | [ EMPTY ⇒ Some ? 〈abs,EMPTY …〉 |
---|
| 425 | | RETURN x ⇒ Some ? 〈abs,RETURN … x〉 |
---|
| 426 | | SEQ x lab instr ⇒ |
---|
| 427 | ! 〈l,i1〉 ← call_post_clean … instr m keep abs; |
---|
| 428 | match lab with |
---|
| 429 | [ None ⇒ return 〈l,SEQ … x (None ?) i1〉 |
---|
| 430 | | Some lbl ⇒ if ((get_element … m lbl) == lbl :: l) |
---|
| 431 | then return 〈nil ?,SEQ … x (Some ? lbl) i1〉 |
---|
| 432 | else None ? |
---|
| 433 | ] |
---|
| 434 | | COND x ltrue i1 lfalse i2 i3 ⇒ |
---|
| 435 | ! 〈l3,instr3〉 ← call_post_clean … i3 m keep abs; |
---|
| 436 | ! 〈l2,instr2〉 ← call_post_clean … i2 m keep l3; |
---|
| 437 | ! 〈l1,instr1〉 ← call_post_clean … i1 m keep l3; |
---|
| 438 | if ((get_element … m ltrue) == ltrue :: l1) ∧ |
---|
| 439 | ((get_element … m lfalse) == lfalse :: l2) |
---|
| 440 | then return 〈nil ?,COND … x ltrue instr1 lfalse instr2 instr3〉 |
---|
| 441 | else None ? |
---|
| 442 | | LOOP x ltrue i1 lfalse i2 ⇒ |
---|
| 443 | ! 〈l2,instr2〉 ← call_post_clean … i2 m keep abs; |
---|
| 444 | ! 〈l1,instr1〉 ← call_post_clean … i1 m keep (nil ?); |
---|
| 445 | if ((get_element … m ltrue) == ltrue :: l1) ∧ |
---|
| 446 | ((get_element … m lfalse) == lfalse :: l2) |
---|
| 447 | then return 〈nil ?,LOOP … x ltrue instr1 lfalse instr2〉 |
---|
| 448 | else None ? |
---|
| 449 | | CALL f act_p r_lb i1 ⇒ |
---|
| 450 | ! 〈l1,instr1〉 ← call_post_clean … i1 m keep abs; |
---|
| 451 | match r_lb with |
---|
[3410] | 452 | [ None ⇒ None ? |
---|
[3414] | 453 | | Some lbl ⇒ if (lbl ∈ keep) |
---|
[3396] | 454 | then if ((get_element … m lbl) == lbl :: l1) |
---|
| 455 | then return 〈nil ?,CALL … f act_p (Some ? lbl) instr1〉 |
---|
| 456 | else None ? |
---|
[3549] | 457 | else return 〈(a_return_post l_p lbl) :: l1,CALL … f act_p (None ?) instr1〉 |
---|
[3396] | 458 | ] |
---|
| 459 | | IO lin io lout i1 ⇒ |
---|
| 460 | ! 〈l1,instr1〉 ← call_post_clean … i1 m keep abs; |
---|
| 461 | if ((get_element … m lout) == lout :: l1) ∧ ((get_element … m lin) == [lin]) |
---|
| 462 | then return 〈nil ?,IO … lin io lout instr1〉 |
---|
| 463 | else None ? |
---|
| 464 | ]. |
---|
| 465 | |
---|
| 466 | |
---|
[3549] | 467 | definition ret_costed_abs : ∀p.list (ReturnPostCostLabel p) → option (ReturnPostCostLabel p) → |
---|
| 468 | option (CostLabel p) ≝ |
---|
| 469 | λp,keep,x. |
---|
[3396] | 470 | match x with |
---|
[3549] | 471 | [ Some lbl ⇒ if lbl ∈ keep then return (a_return_post … lbl) |
---|
[3396] | 472 | else None ? |
---|
| 473 | | None ⇒ None ? |
---|
| 474 | ]. |
---|
| 475 | |
---|
[3399] | 476 | |
---|
[3549] | 477 | definition check_continuations : ∀p : instr_params.∀l_p : label_params. |
---|
| 478 | ∀l1,l2 : list ((ActionLabel l_p) × (Instructions p l_p)). |
---|
| 479 | associative_list (DEQCostLabel l_p) (CostLabel l_p) → |
---|
| 480 | list (ReturnPostCostLabel l_p) → option (Prop × (list (CostLabel l_p)) × (list (CostLabel l_p))) ≝ |
---|
| 481 | λp,l_p,cont1,cont2,m,keep. |
---|
[3399] | 482 | foldr2 ??? 〈True,nil ?,nil ?〉 cont1 cont2 |
---|
| 483 | (λx,y,z. |
---|
| 484 | let 〈cond,abs_top',abs_tail'〉 ≝ x in |
---|
[3549] | 485 | match call_post_clean p l_p (\snd z) m keep abs_top' with |
---|
[3399] | 486 | [ None ⇒ 〈False,nil ?,nil ?〉 |
---|
| 487 | | Some w ⇒ |
---|
| 488 | match \fst z with |
---|
| 489 | [ ret_act opt_x ⇒ |
---|
[3549] | 490 | match ret_costed_abs … keep opt_x with |
---|
[3399] | 491 | [ Some lbl ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ |
---|
| 492 | get_element … m lbl = lbl :: (\fst w),(nil ?),abs_tail'〉 |
---|
| 493 | | None ⇒ |
---|
[3549] | 494 | 〈\fst y = ret_act … (None ?) ∧ cond ∧ \snd w = \snd y,(nil ?),(\fst w) @ abs_tail'〉 |
---|
[3399] | 495 | ] |
---|
| 496 | | cost_act opt_x ⇒ |
---|
| 497 | match opt_x with |
---|
| 498 | [ None ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y,\fst w,abs_tail'〉 |
---|
| 499 | | Some xx ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ |
---|
| 500 | get_element … m xx = xx :: (\fst w),(nil ?),abs_tail'〉] |
---|
| 501 | | _ ⇒ (* dummy *) 〈False,nil ?,nil ?〉]]). |
---|
| 502 | |
---|
[3398] | 503 | (* in input : |
---|
| 504 | abs_top is the list of labels to be propageted to the deepest level of the call stack |
---|
[3525] | 505 | equivalently (?) is the list of labels I need to pay now |
---|
| 506 | |
---|
[3398] | 507 | abs_tail are the lists of labels to be propagated to the levels "below" the deepest level |
---|
[3525] | 508 | equivalently (?) is the list of labels I must have already payid in the |
---|
| 509 | code already executed; generated by the continuations below me in the stack |
---|
[3398] | 510 | in output : |
---|
| 511 | abs_top is the list of labels to be propageted from the current level of the call stack |
---|
[3525] | 512 | non empty only in the case of non-call stack frames (whiles, ifs, etc; but in |
---|
| 513 | practice it is always nil!) |
---|
| 514 | abs_tail are the lists of labels to be propagated from the levels "below" the current level |
---|
| 515 | or equivalently (?) the list of labels I must have already paied in the code |
---|
| 516 | already executed; generated by this level of the stack |
---|
| 517 | *) |
---|
[3398] | 518 | |
---|
[3396] | 519 | |
---|
[3549] | 520 | definition state_rel : ∀p : state_params. |
---|
| 521 | associative_list (DEQCostLabel p) (CostLabel p) → list (ReturnPostCostLabel p) → |
---|
| 522 | list (CostLabel p) → list (CostLabel p) → |
---|
[3398] | 523 | relation (state p) ≝ λp,m,keep,abs_top,abs_tail,st1,st2. |
---|
[3549] | 524 | match check_continuations … (cont ? st1) (cont … st2) m keep with |
---|
[3398] | 525 | [ Some x ⇒ let 〈prf1,abs_top',abs_tail'〉 ≝ x in |
---|
| 526 | prf1 ∧ call_post_clean … (code … st2) m keep abs_top' = return 〈abs_top,(code … st1)〉 |
---|
| 527 | ∧ store … st1 = store … st2 ∧ io_info … st1 = io_info … st2 ∧ abs_tail = abs_tail' |
---|
[3396] | 528 | | None ⇒ False |
---|
| 529 | ]. |
---|
| 530 | |
---|
[3549] | 531 | let rec compute_max_n (p : instr_params) (l_p : label_params) (i : Instructions p l_p) on i : l_p ≝ |
---|
[3396] | 532 | match i with |
---|
[3549] | 533 | [ EMPTY ⇒ e … l_p |
---|
| 534 | | RETURN x ⇒ e … l_p |
---|
[3396] | 535 | | SEQ x lab instr ⇒ let n ≝ compute_max_n … instr in |
---|
| 536 | match lab with |
---|
| 537 | [ None ⇒ n |
---|
| 538 | | Some l ⇒ |
---|
| 539 | match l with |
---|
[3549] | 540 | [ a_non_functional_label n' ⇒ op … l_p n n' ] |
---|
[3396] | 541 | ] |
---|
| 542 | | COND x ltrue i1 lfalse i2 i3 ⇒ |
---|
| 543 | let n1 ≝ compute_max_n … i1 in |
---|
| 544 | let n2 ≝ compute_max_n … i2 in |
---|
| 545 | let n3 ≝ compute_max_n … i3 in |
---|
[3549] | 546 | let mx ≝ op … l_p (op … l_p n1 n2) n3 in |
---|
[3396] | 547 | match ltrue with |
---|
| 548 | [ a_non_functional_label lt ⇒ |
---|
| 549 | match lfalse with |
---|
[3549] | 550 | [a_non_functional_label lf ⇒ op … l_p (op … l_p mx lt) lf ] ] |
---|
[3396] | 551 | | LOOP x ltrue i1 lfalse i2 ⇒ |
---|
| 552 | let n1 ≝ compute_max_n … i1 in |
---|
| 553 | let n2 ≝ compute_max_n … i2 in |
---|
[3549] | 554 | let mx ≝ op … l_p n1 n2 in |
---|
[3396] | 555 | match ltrue with |
---|
| 556 | [ a_non_functional_label lt ⇒ |
---|
| 557 | match lfalse with |
---|
[3549] | 558 | [a_non_functional_label lf ⇒ op … l_p (op … l_p mx lt) lf ] ] |
---|
[3396] | 559 | | CALL f act_p r_lb i1 ⇒ |
---|
| 560 | let n ≝ compute_max_n … i1 in |
---|
| 561 | match r_lb with |
---|
| 562 | [ None ⇒ n |
---|
[3549] | 563 | | Some lbl ⇒ match lbl with [a_return_cost_label l ⇒ op … l_p l n ] |
---|
[3396] | 564 | ] |
---|
| 565 | | IO lin io lout i1 ⇒ |
---|
| 566 | let n ≝ compute_max_n … i1 in |
---|
| 567 | match lin with |
---|
| 568 | [a_non_functional_label l1 ⇒ |
---|
| 569 | match lout with |
---|
[3549] | 570 | [a_non_functional_label l2 ⇒ op … l_p (op … l_p n l1) l2 ] ] |
---|
[3396] | 571 | ]. |
---|
| 572 | |
---|
| 573 | |
---|
[3549] | 574 | definition same_fresh_map_on : ∀p.list (CostLabel p) → |
---|
| 575 | relation (associative_list (DEQCostLabel p) (CostLabel p)) ≝ |
---|
| 576 | λp,dom,m1,m2.∀x.bool_to_Prop (x ∈ dom) → get_element … m1 x = get_element … m2 x. |
---|
[3400] | 577 | |
---|
[3549] | 578 | definition same_to_keep_on : ∀p. list (CostLabel p) → relation (list (ReturnPostCostLabel p)) ≝ |
---|
| 579 | λp,dom,keep1,keep2.∀x. bool_to_Prop (a_return_post … x ∈ dom) → (x ∈ keep1) = (x ∈ keep2). |
---|
[3400] | 580 | |
---|
| 581 | |
---|
[3549] | 582 | lemma same_to_keep_on_append : ∀p.∀dom1,dom2,dom3 : list (CostLabel p). |
---|
| 583 | ∀l1,l2,l3,l : list (ReturnPostCostLabel p). |
---|
| 584 | no_duplicates … (dom1@dom2@dom3) → (∀x.x ∈ l1 → a_return_post … x ∈ dom1) → |
---|
| 585 | (∀x.x ∈ l3 → a_return_post … x ∈ dom3) → |
---|
| 586 | same_to_keep_on … (dom1@dom2@dom3) (l1@l2@l3) l → |
---|
| 587 | same_to_keep_on … dom2 l2 l. |
---|
| 588 | #p #dom1 #dom2 #dom3 #l1 #l2 #l3 #l #no_dup #sub_set1 #sub_set3 #H2 |
---|
[3403] | 589 | #x #Hx inversion (x ∈ l2) |
---|
| 590 | [ #EQkeep <H2 [> memb_append_l2 // >memb_append_l1 // ] |
---|
| 591 | >memb_append_l2 // >memb_append_l1 // >Hx // |
---|
| 592 | | #EQno_keep <H2 |
---|
| 593 | [2: >memb_append_l2 // >memb_append_l1 // >Hx // |
---|
| 594 | | @sym_eq @memb_not_append [2: @memb_not_append //] |
---|
| 595 | [ <associative_append in no_dup; #no_dup ] |
---|
[3549] | 596 | lapply(memb_no_duplicates_append … (a_return_post … x) … no_dup) #H |
---|
[3403] | 597 | inversion(memb ???) // #H1 cases H |
---|
| 598 | [1,4: [>memb_append_l2 | >memb_append_l1] // >Hx // |
---|
| 599 | | @sub_set3 >H1 // |
---|
| 600 | | @sub_set1 >H1 // |
---|
| 601 | ] |
---|
| 602 | ] |
---|
| 603 | ] |
---|
| 604 | qed. |
---|
| 605 | |
---|
[3549] | 606 | lemma same_fresh_map_on_append : ∀p.∀dom1,dom2,dom3,l1,l2,l3,l. |
---|
[3403] | 607 | no_duplicates … (dom1 @dom2 @ dom3) → (∀x.x ∈ domain_of_associative_list … l1 → x ∈ dom1) → |
---|
| 608 | (∀x.x ∈ domain_of_associative_list … l3 → x ∈ dom3) → |
---|
[3549] | 609 | same_fresh_map_on p … (dom1 @dom2 @dom3) (l1 @l2 @ l3) l → |
---|
| 610 | same_fresh_map_on p … dom2 l2 l. |
---|
| 611 | #p #dom1 #dom2 #dom3 #l1 #l2 #l3 #l #no_dup #subset1 #subset3 whd in ⊢ (% → ?); #H1 |
---|
[3403] | 612 | whd #x #Hx <H1 |
---|
| 613 | [2: >memb_append_l2 // >memb_append_l1 // >Hx //] |
---|
| 614 | >get_element_append_r [ >get_element_append_l1 // ] % #K |
---|
| 615 | [ lapply(subset3 … K) | lapply(subset1 … K) ] #ABS |
---|
| 616 | [ <associative_append in no_dup; #no_dup] @(memb_no_duplicates_append … x … no_dup) |
---|
| 617 | // [>memb_append_l2 | >memb_append_l1 ] // >Hx // |
---|
| 618 | qed. |
---|
| 619 | |
---|
| 620 | |
---|
[3549] | 621 | lemma lab_to_keep_in_domain : ∀p,l_p.∀i : Instructions p l_p. |
---|
[3400] | 622 | ∀x,n,l. |
---|
[3549] | 623 | x ∈ lab_to_keep … (call_post_trans … i n l) → a_return_post … x ∈ get_labels_of_code … i. |
---|
| 624 | #p #l_p #i elim i // |
---|
[3400] | 625 | [ #seq #opt_l #instr #IH #x #n #l whd in match (call_post_trans ????); |
---|
| 626 | cases opt_l -opt_l normalize nodelta [|#lbl] |
---|
| 627 | whd in match (get_labels_of_code ??); #H [2: @orb_Prop_r] /2/ |
---|
| 628 | | #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #x #n #l |
---|
| 629 | whd in match (call_post_trans ????); whd in match (get_labels_of_code ??); |
---|
| 630 | #H cases(memb_append … H) -H #H @orb_Prop_r @orb_Prop_r |
---|
| 631 | [ >memb_append_l1 // @IH1 [3: >H // |*: ] |
---|
| 632 | | >memb_append_l2 // cases(memb_append … H) -H #H |
---|
| 633 | [>memb_append_l1 // @IH2 [3: >H // |*: ] |
---|
| 634 | | >memb_append_l2 // @IH3 [3: >H // |*: ] |
---|
| 635 | ] |
---|
| 636 | ] |
---|
| 637 | | #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #x #n #l |
---|
| 638 | whd in match (call_post_trans ????); whd in match (get_labels_of_code ??); |
---|
| 639 | #H cases(memb_append … H) -H #H @orb_Prop_r @orb_Prop_r |
---|
| 640 | [ >memb_append_l1 | >memb_append_l2 ] // [ @IH1 | @IH2 ] [3,6: >H |*: ] // |
---|
| 641 | | #f #act_p * [|#lbl] #i1 #IH #x #n #l whd in match (call_post_trans ????); |
---|
| 642 | whd in match (get_labels_of_code ??); /2/ whd in match (memb ???); |
---|
| 643 | inversion(x == lbl) #Hlbl normalize nodelta |
---|
| 644 | [* >(\P Hlbl) @orb_Prop_l @eq_costlabel_elim // * #H @H % |
---|
| 645 | | #H @orb_Prop_r @IH // |
---|
| 646 | ] |
---|
| 647 | | #lin #io #lout #i1 #IH #x #n #l whd in match (call_post_trans ????); |
---|
| 648 | whd in match (get_labels_of_code ??); #H @orb_Prop_r @orb_Prop_r @IH // |
---|
| 649 | ] |
---|
| 650 | qed. |
---|
[3401] | 651 | |
---|
[3549] | 652 | lemma lab_map_in_domain: ∀p,l_p.∀i: Instructions p l_p. |
---|
[3401] | 653 | ∀x,n,l. |
---|
[3549] | 654 | x ∈ domain_of_associative_list … (lab_map … (call_post_trans … i n l)) → |
---|
[3401] | 655 | x ∈ get_labels_of_code … i. |
---|
[3549] | 656 | #p #l_p #i elim i // |
---|
[3402] | 657 | [ #seq * [|#lbl] #i1 #IH #x #n #l whd in match(call_post_trans ????); |
---|
| 658 | whd in match (get_labels_of_code ??); /2/ whd in match (memb ???); |
---|
| 659 | inversion(x==lbl) #Hlbl normalize nodelta [#_ whd in match (memb ???); >Hlbl % ] |
---|
| 660 | #H >memb_cons // @IH // |
---|
| 661 | | #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #x #n #l |
---|
| 662 | whd in match (call_post_trans ????); whd in match (memb ???); |
---|
| 663 | whd in match (get_labels_of_code ??); inversion(x == ltrue) #Hlbl normalize nodelta |
---|
| 664 | [ #_ whd in match (memb ???); >Hlbl % ] whd in match (memb ???); |
---|
| 665 | inversion(x == lfalse) #Hlbl1 normalize nodelta |
---|
| 666 | [ #_ whd in match (memb ???); >Hlbl normalize nodelta whd in match (memb ???); |
---|
| 667 | >Hlbl1 % ] #H >memb_cons // >memb_cons // >domain_of_associative_list_append in H; |
---|
| 668 | #H cases(memb_append … H) [ #H1 >memb_append_l1 // @IH1 [3: >H1 // |*:] ] |
---|
| 669 | >domain_of_associative_list_append #H1 cases(memb_append … H1) |
---|
| 670 | #H2 >memb_append_l2 // [ >memb_append_l1 | >memb_append_l2 ] // |
---|
| 671 | [ @IH2 | @IH3] /2 by eq_true_to_b/ |
---|
| 672 | | #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #x #n #l whd in match (call_post_trans ????); |
---|
| 673 | whd in match (get_labels_of_code ??); whd in match (memb ???); inversion(x == lfalse) |
---|
| 674 | #Hlfalse normalize nodelta [ #_ >memb_cons // whd in match (memb ???); >Hlfalse // ] |
---|
| 675 | whd in match (memb ???); inversion(x==ltrue) normalize nodelta #Hltrue |
---|
| 676 | [ #_ whd in match (memb ???); >Hltrue %] >domain_of_associative_list_append #H |
---|
| 677 | cases(memb_append … H) #H1 >memb_cons // >memb_cons // [ >memb_append_l1 | >memb_append_l2 ] |
---|
| 678 | // [ @IH1 | @IH2] /2/ |
---|
| 679 | | #f #act_p * [|#lbl] #i1 #IH #x #n #l whd in match (call_post_trans ????); |
---|
| 680 | whd in match (get_labels_of_code ??); /2/ whd in match (memb ???); inversion (x == lbl) |
---|
| 681 | #Hlbl normalize nodelta [ #_ whd in match (memb ???); >Hlbl % ] #H >memb_cons // @IH /2/ |
---|
| 682 | | #lin #io #lout #i1 #IH #x #n #l whd in match (memb ???); inversion(x == lout) #Hlout |
---|
| 683 | normalize nodelta [ #_ >memb_cons // whd in match (memb ???); >Hlout % ] |
---|
| 684 | whd in match (memb ???); inversion(x==lin) #Hlin normalize nodelta |
---|
| 685 | [ #_ whd in match (memb ???); >Hlin % ] #H >memb_cons // >memb_cons // @IH /2/ |
---|
| 686 | ] |
---|
[3401] | 687 | qed. |
---|
| 688 | |
---|
[3549] | 689 | let rec is_fresh_for_return (p : label_params) (keep : list (CostLabel p)) (n : p) on keep : Prop ≝ |
---|
[3414] | 690 | match keep with |
---|
| 691 | [ nil ⇒ True |
---|
[3549] | 692 | | cons x xs ⇒ let ih ≝ is_fresh_for_return … xs n in |
---|
[3414] | 693 | match x with |
---|
[3549] | 694 | [ a_return_post y ⇒ match y with [a_return_cost_label m ⇒ m ⊑^{p} n ∧ ih ] |
---|
[3416] | 695 | | _ ⇒ ih |
---|
| 696 | ] |
---|
[3414] | 697 | ]. |
---|
| 698 | |
---|
[3549] | 699 | lemma fresh_ok_call_post_trans : ∀p : instr_params.∀l_p.∀i1 : Instructions p l_p.∀n : l_p.∀l. |
---|
| 700 | n ⊑^{l_p} fresh … (call_post_trans … i1 n l). |
---|
| 701 | #p #l_p #i1 elim i1 normalize /2 by trans_po_rel, refl_po_rel/ |
---|
[3414] | 702 | [ #seq * [|#lbl] #i2 #IH #n #l normalize /2 by / |
---|
| 703 | | #cond #ltrue #i_true #lfalse #i_false #i2 #IH1 #IH2 #IH3 #n #l |
---|
[3549] | 704 | @(trans_po_rel … (IH3 …)) [2: @(trans_po_rel … (IH2 …)) [2: @(trans_po_rel … (IH1 …)) ]] // |
---|
| 705 | | #f #act_p * [|#lbl] normalize #i2 #IH /2 by / #n #l @(trans_po_rel … (IH … l …)) @lab_po_rel_succ |
---|
[3414] | 706 | ] |
---|
| 707 | qed. |
---|
| 708 | |
---|
[3549] | 709 | lemma fresh_keep_n_ok : ∀p : label_params.∀n,m.∀l. |
---|
| 710 | is_fresh_for_return p l n → n ⊑^{p} m → is_fresh_for_return p l m. |
---|
| 711 | #p #n #m #l lapply n -n lapply m -m elim l // * |
---|
[3478] | 712 | [1,2,3: * #x] #xs #IH #n #m |
---|
[3549] | 713 | normalize [2: * #H1] #H2 #H3 [ %] /2 by trans_po_rel/ @(IH … H2) assumption |
---|
[3414] | 714 | qed. |
---|
| 715 | |
---|
[3549] | 716 | definition cast_return_to_cost_labels ≝ λp.map … (a_return_post p …). |
---|
[3416] | 717 | coercion cast_return_to_cost_labels. |
---|
| 718 | |
---|
[3549] | 719 | lemma succ_label_leq_absurd : ∀p : label_params.∀x : p.succ_label … p … x ⊑^{p} x → False. |
---|
| 720 | #p #x #ABS @(absurd ?? (no_maps_in_id … p x)) @(antisym_po_rel … (po_label … p)) // |
---|
| 721 | qed. |
---|
| 722 | |
---|
| 723 | lemma fresh_false : ∀p.∀n.∀l: list (ReturnPostCostLabel p).is_fresh_for_return … l n → |
---|
| 724 | a_return_cost_label p (succ_label … n) ∈ l = false. |
---|
| 725 | #p #n #l lapply n -n elim l // * #x #xs #IH #n whd in ⊢ (% → ??%?); * #H1 |
---|
[3416] | 726 | #H2 @eq_return_cost_lab_elim |
---|
[3549] | 727 | [ #EQ destruct @⊥ @succ_label_leq_absurd // |
---|
[3414] | 728 | | #_ >IH // |
---|
| 729 | ] |
---|
| 730 | qed. |
---|
| 731 | |
---|
[3549] | 732 | lemma inverse_call_post_trans : ∀p : instr_params.∀l_p : label_params.∀i1 : Instructions p l_p.∀n : l_p. |
---|
[3400] | 733 | let dom ≝ get_labels_of_code … i1 in |
---|
| 734 | no_duplicates … dom → |
---|
[3399] | 735 | ∀m,keep. |
---|
[3549] | 736 | ∀info,l.call_post_trans … i1 n l = info → |
---|
| 737 | same_fresh_map_on … dom (lab_map … info) m → |
---|
| 738 | same_to_keep_on … dom (lab_to_keep … info) keep → |
---|
| 739 | is_fresh_for_return … keep n → |
---|
| 740 | call_post_clean … (t_code … info) m keep l |
---|
[3396] | 741 | = return 〈gen_labels … info,i1〉. |
---|
[3549] | 742 | #p #l_p #i1 elim i1 |
---|
[3400] | 743 | [ #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?); |
---|
| 744 | #EQ destruct(EQ) // |
---|
| 745 | | #x #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?); |
---|
| 746 | #EQ destruct(EQ) // |
---|
| 747 | | #seq * [|#lbl] #instr #IH #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep |
---|
[3414] | 748 | #l whd in ⊢ (??%? → ?); #EQ destruct(EQ) #H1 #H2 #H3 whd in ⊢ (??%%); normalize nodelta |
---|
[3400] | 749 | >IH // |
---|
| 750 | [1,4: whd whd in H2; #x #Hx @H2 whd in match (get_labels_of_code ??); // |
---|
| 751 | |2,5: whd whd in H1; #x #Hx [ @H1 // ] cases no_dup #H3 #_ <H1 |
---|
| 752 | [2: whd in match (get_labels_of_code ??); @orb_Prop_r // ] |
---|
| 753 | whd in ⊢ (???%); cases(eqb_true … x lbl) inversion(x == lbl) normalize nodelta |
---|
| 754 | [2: //] #_ #H4 >H4 in Hx; // #H5 >H5 in H3; * #ABS @⊥ @ABS % |
---|
| 755 | |6: cases no_dup // |
---|
| 756 | ] |
---|
| 757 | normalize nodelta <(H1 lbl) |
---|
[3549] | 758 | [2: whd in match (get_labels_of_code ??); @orb_Prop_l cases(eqb_true … (a_non_functional_label … lbl) lbl) |
---|
[3400] | 759 | #H3 #H4 >H4 % ] |
---|
| 760 | whd in match (get_element ????); >(\b (refl …)) normalize nodelta |
---|
| 761 | >(\b (refl …)) % |
---|
| 762 | | #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #n normalize nodelta |
---|
| 763 | #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?); |
---|
[3414] | 764 | #EQ destruct(EQ) #H1 #H2 #H3 whd in ⊢ (??%?); >IH3 // |
---|
[3403] | 765 | [2: whd in match (get_labels_of_code ??) in H2; |
---|
| 766 | change with ([?;?]@?) in match (?::?) in H2; |
---|
| 767 | <associative_append in H2; <associative_append |
---|
[3549] | 768 | <(append_nil … (?@?)) <associative_append in ⊢ (???%? → ?); |
---|
| 769 | <(append_nil … (?@?)) in ⊢ (???%? → ?); >associative_append |
---|
| 770 | >associative_append in ⊢ (???%? → ?); #H2 |
---|
[3403] | 771 | @(same_to_keep_on_append … H2) // [ >append_nil |
---|
| 772 | whd in ⊢ (??%); whd in no_dup:(??%); >associative_append // ] |
---|
| 773 | #x #Hx cases (memb_append … Hx) -Hx #Hx @orb_Prop_r @orb_Prop_r |
---|
| 774 | [ >memb_append_l1 | >memb_append_l2 ] // |
---|
| 775 | @(lab_to_keep_in_domain … (eq_true_to_b … Hx)) |
---|
| 776 | |3: -H2 whd in match (get_labels_of_code ??) in H1; |
---|
| 777 | change with ([?;?]@?) in match (?::?) in H1; |
---|
| 778 | <associative_append in H1; <associative_append |
---|
| 779 | <(append_nil … (?@?)) >associative_append |
---|
[3549] | 780 | change with ([?;?]@?) in match (?::?::?) in ⊢ (???%? → ?); |
---|
| 781 | <associative_append in ⊢ (???%? → ?); |
---|
| 782 | <associative_append in ⊢ (???%? → ?); |
---|
| 783 | <(append_nil … (?@?)) in ⊢ (???%? → ?); |
---|
| 784 | >associative_append in ⊢ (???%? → ?); #H1 |
---|
[3403] | 785 | @(same_fresh_map_on_append … H1) // |
---|
| 786 | [ >append_nil >associative_append // ] |
---|
| 787 | #x whd in match (memb ???); inversion(x == ltrue) |
---|
| 788 | [ #Hltrue normalize nodelta #_ whd in match (memb ???); >Hltrue % |
---|
| 789 | | #Hltrue normalize nodelta whd in match (memb ???); inversion(x == lfalse) |
---|
| 790 | [ #Hlfalse #_ @orb_Prop_r @orb_Prop_l >Hlfalse % |
---|
| 791 | | #Hlfalse normalize nodelta #Hx @orb_Prop_r @orb_Prop_r |
---|
| 792 | >domain_of_associative_list_append in Hx; #H |
---|
| 793 | cases(memb_append … H) #H2 [ >memb_append_l1 | >memb_append_l2 ] |
---|
| 794 | // @(lab_map_in_domain … (eq_true_to_b … H2)) |
---|
[3400] | 795 | ] |
---|
[3403] | 796 | ] |
---|
[3400] | 797 | |4: cases no_dup #_ * #_ #H @no_duplicates_append_r [2: @(no_duplicates_append_r … H) |] |
---|
| 798 | ] |
---|
[3414] | 799 | normalize nodelta >IH2 |
---|
| 800 | [5: % |
---|
| 801 | |2: /2 by fresh_keep_n_ok/ |
---|
| 802 | |3: whd in match (get_labels_of_code ??) in H2; |
---|
[3403] | 803 | change with ([?;?]@?) in match (?::?) in H2; |
---|
| 804 | <associative_append in H2; #H2 |
---|
| 805 | @(same_to_keep_on_append … H2) // #x #Hx [ @orb_Prop_r @orb_Prop_r ] |
---|
| 806 | @(lab_to_keep_in_domain … (eq_true_to_b … Hx)) |
---|
[3414] | 807 | |4: whd in match (get_labels_of_code ??) in H1; |
---|
[3403] | 808 | change with ([?;?]@?) in match (?::?) in H1; |
---|
[3549] | 809 | change with ([?;?]@?) in match (?::?::?) in H1 : (???%?); |
---|
| 810 | <associative_append in H1; <associative_append in ⊢ (???%? → ?); #H1 |
---|
[3403] | 811 | @(same_fresh_map_on_append … H1) // [2: /2 by lab_map_in_domain/ ] |
---|
| 812 | #x >domain_of_associative_list_append #H cases(memb_append … H) |
---|
| 813 | [ whd in ⊢ (??%? → ?%); cases(x == ltrue) // normalize nodelta |
---|
| 814 | whd in ⊢ (??%? → ?%); cases(x == lfalse) // normalize nodelta |
---|
| 815 | normalize #EQ destruct |
---|
| 816 | | #H1 @orb_Prop_r @orb_Prop_r |
---|
| 817 | @(lab_map_in_domain … (eq_true_to_b … H1)) |
---|
| 818 | ] |
---|
[3414] | 819 | |6: cases no_dup #_ * #_ #K lapply (no_duplicates_append_r … K) @no_duplicates_append_l |
---|
| 820 | |*: |
---|
[3400] | 821 | ] |
---|
[3414] | 822 | >m_return_bind >IH1 |
---|
| 823 | [5: % |
---|
| 824 | |2: /3 by fresh_keep_n_ok/ |
---|
| 825 | |3: whd in match (get_labels_of_code ??) in H2; |
---|
[3403] | 826 | change with ([?;?]@?) in match (?::?) in H2; |
---|
[3549] | 827 | change with ([ ] @ ?) in match (lab_to_keep ???) in H2; |
---|
| 828 | >associative_append in H2 : (???%?); #H2 |
---|
[3414] | 829 | @(same_to_keep_on_append … H2) // #x #Hx cases(memb_append … Hx) |
---|
[3403] | 830 | -Hx #Hx [ >memb_append_l1 | >memb_append_l2] // |
---|
| 831 | @(lab_to_keep_in_domain … (eq_true_to_b … Hx)) |
---|
[3414] | 832 | |4: whd in match (get_labels_of_code ??) in H1; |
---|
[3403] | 833 | change with ([?;?]@?) in match (?::?) in H1; |
---|
[3549] | 834 | change with ([?;?]@?) in match (?::?::?) in H1 : (???%?); |
---|
[3403] | 835 | @(same_fresh_map_on_append … H1) // #x >domain_of_associative_list_append |
---|
| 836 | #Hx cases(memb_append … Hx) -Hx #Hx [ >memb_append_l1 | >memb_append_l2 ] |
---|
| 837 | // @(lab_map_in_domain … (eq_true_to_b … Hx)) |
---|
[3414] | 838 | |6: cases no_dup #_ * #_ @no_duplicates_append_l |
---|
| 839 | |*: |
---|
[3401] | 840 | ] |
---|
[3400] | 841 | >m_return_bind normalize nodelta whd in H1; <H1 |
---|
| 842 | [2: whd in match (get_labels_of_code ??); whd in match (memb ???); |
---|
| 843 | >(\b (refl …)) % ] whd in match (get_element ????); >(\b (refl …)) |
---|
| 844 | normalize nodelta >(\b (refl …)) <H1 |
---|
| 845 | [2: whd in match (get_labels_of_code ??); >memb_cons // |
---|
| 846 | whd in match (memb ???); >(\b (refl …)) % ] |
---|
| 847 | whd in match (get_element ????); @eq_costlabel_elim normalize nodelta |
---|
[3404] | 848 | [ #ABS @⊥ cases no_dup >ABS * #H #_ @H @orb_Prop_l |
---|
[3549] | 849 | >(\b (refl ? (a_non_functional_label … ltrue))) % ] #_ |
---|
[3404] | 850 | whd in match (get_element ????); >(\b (refl …)) normalize nodelta |
---|
[3414] | 851 | >(\b (refl …)) % |
---|
[3406] | 852 | | #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #n #no_dup #m #keep #info #l whd in ⊢ (??%? → ?); |
---|
[3414] | 853 | #EQ destruct(EQ) whd in match (get_labels_of_code ??); #fresh_map #keep_on #f_k |
---|
| 854 | whd in ⊢ (??%?); >(IH2 … (refl …)) |
---|
| 855 | [ normalize nodelta >(IH1 … (refl …)) |
---|
[3406] | 856 | [ >m_return_bind <fresh_map [2: @orb_Prop_l >(\b (refl …)) % ] |
---|
| 857 | whd in match (get_element ????); |
---|
[3549] | 858 | inversion(a_non_functional_label … ltrue == a_non_functional_label … lfalse) |
---|
[3406] | 859 | #Hltrue normalize nodelta |
---|
| 860 | [ cases no_dup whd in match (memb ???); |
---|
[3549] | 861 | cases(eqb_true … (a_non_functional_label … ltrue) (a_non_functional_label … lfalse)) |
---|
[3406] | 862 | #H1 #_ lapply(H1 Hltrue) #EQ destruct(EQ) >(\b (refl …)) * #ABS @⊥ @ABS % ] |
---|
| 863 | whd in match (get_element ????); >(\b (refl …)) normalize nodelta >(\b (refl …)) |
---|
| 864 | <fresh_map [2: @orb_Prop_r @orb_Prop_l >(\b (refl …)) % ] |
---|
| 865 | whd in match (get_element ????); >(\b (refl …)) normalize nodelta |
---|
| 866 | >(\b (refl …)) % |
---|
[3414] | 867 | | /2 by fresh_keep_n_ok/ |
---|
[3549] | 868 | | change with ([?;?]@?@?) in keep_on : (??%??); change with ((nil ?) @ ? @ ?) in keep_on : (???%?); |
---|
[3414] | 869 | @(same_to_keep_on_append … keep_on) // #x /2 by lab_to_keep_in_domain/ |
---|
[3549] | 870 | | change with ([?;?]@?@?) in fresh_map : (??%%?); @(same_fresh_map_on_append … fresh_map) |
---|
[3406] | 871 | /2 by lab_map_in_domain/ #x whd in match (memb ???); inversion(x==lfalse) #Hlfalse |
---|
| 872 | normalize nodelta |
---|
| 873 | [ #_ @orb_Prop_r whd in match (memb ???); >Hlfalse % |
---|
| 874 | | whd in match (memb ???); inversion(x==ltrue) #Hltrue normalize nodelta [2: *] #_ |
---|
| 875 | @orb_Prop_l >Hltrue % |
---|
| 876 | ] |
---|
| 877 | | cases no_dup #_ * #_ /2/ |
---|
| 878 | ] |
---|
[3414] | 879 | | // |
---|
[3549] | 880 | | change with ([?;?]@?@?) in keep_on : (??%??); <associative_append in keep_on; |
---|
| 881 | <(append_nil … (?@?)) <(append_nil … (?@?)) in ⊢ (???%? → ?); |
---|
| 882 | >associative_append in ⊢ (??%%? → ?); >associative_append in ⊢ (???%? → ?); |
---|
[3408] | 883 | #keep_on @(same_to_keep_on_append … keep_on) // |
---|
| 884 | [ >associative_append >append_nil // |
---|
| 885 | | #x #Hx @orb_Prop_r @orb_Prop_r /2 by lab_to_keep_in_domain/ |
---|
| 886 | ] |
---|
[3549] | 887 | | change with ([?;?]@?@?) in fresh_map : (??%??); <associative_append in fresh_map; |
---|
| 888 | <(append_nil … (?@?)) change with ([?;?]@?@?) in ⊢ (???%? → ?); |
---|
| 889 | <associative_append in ⊢ (???%? → ?); <(append_nil … (?@?)) in ⊢ (???%? → ?); |
---|
| 890 | >associative_append in ⊢ (??%%? → ?); >associative_append in ⊢ (???%? → ?); |
---|
[3408] | 891 | #fresh_map @(same_fresh_map_on_append … fresh_map) // |
---|
| 892 | [ >append_nil // |
---|
| 893 | | #x >domain_of_associative_list_append #Hx cases(memb_append … Hx) |
---|
| 894 | [2: #Hx1 @orb_Prop_r @orb_Prop_r @(lab_map_in_domain … (eq_true_to_b … Hx1)) ] |
---|
| 895 | whd in match (memb ???); inversion(x == lfalse) normalize nodelta #Hlfalse |
---|
| 896 | [ #_ @orb_Prop_r @orb_Prop_l >Hlfalse % |
---|
| 897 | | whd in match (memb ???); inversion (x==ltrue) normalize nodelta #Hltrue |
---|
| 898 | [ #_ @orb_Prop_l >Hltrue % |
---|
| 899 | | whd in match (memb ???); #EQ destruct |
---|
| 900 | ] |
---|
| 901 | ] |
---|
| 902 | ] |
---|
| 903 | | cases no_dup #_ * #_ /2 by no_duplicates_append_r/ |
---|
| 904 | ] |
---|
[3410] | 905 | | #f #act_p * [|#r_lb] #i #IH #n #no_dup #m #keep #info #l whd in ⊢ (??%? → ?); |
---|
[3414] | 906 | #EQ destruct(EQ) #fresh_map #same_keep #f_k whd in ⊢ (??%?); |
---|
| 907 | >(IH … (refl …)) |
---|
| 908 | [1,6: normalize nodelta |
---|
[3416] | 909 | [ >fresh_false [2: /2 by fresh_keep_n_ok/] % |
---|
[3414] | 910 | | <same_keep |
---|
| 911 | [ whd in match (memb ???); >(\b (refl …)) normalize nodelta |
---|
| 912 | <fresh_map |
---|
| 913 | [ whd in match (get_element ????); >(\b (refl …)) normalize nodelta |
---|
| 914 | >(\b (refl …)) % |
---|
| 915 | | whd in match (memb ???); >(\b (refl …)) % |
---|
| 916 | ] |
---|
| 917 | | whd in match (memb ???); >(\b (refl …)) % |
---|
| 918 | ] |
---|
| 919 | ] |
---|
| 920 | |2,7: // |
---|
[3549] | 921 | |3,8: whd in match (get_labels_of_code ???) in same_keep; // #x #Hx <same_keep |
---|
[3414] | 922 | [2: >memb_cons // >Hx // ] cases no_dup * #ABS #_ whd in ⊢ (???%); |
---|
| 923 | inversion(x==?) [2: #_ //] #ABS1 @⊥ @ABS <(\P ABS1) >Hx // |
---|
[3549] | 924 | |4,9: whd in match (get_labels_of_code ???) in fresh_map; // #x #Hx <fresh_map |
---|
[3414] | 925 | [2: >memb_cons // >Hx //] cases no_dup * #ABS #_ whd in ⊢ (???%); |
---|
| 926 | inversion(x==?) [2: #_ //] #ABS1 @⊥ @ABS <(\P ABS1) // |
---|
| 927 | |5,10: [ @no_dup | cases no_dup // ] |
---|
| 928 | ] |
---|
[3549] | 929 | | #lin #io #lout #i #IH #n whd in match (get_labels_of_code ???); #no_dup |
---|
[3414] | 930 | #m #keep #info #l whd in ⊢ (??%? → ?); #EQ destruct(EQ) #fresh_map #same_keep |
---|
| 931 | #f_k whd in ⊢ (??%?); >(IH … (refl …)) |
---|
| 932 | [ normalize nodelta <fresh_map [2: >memb_cons // >memb_hd // ] |
---|
| 933 | whd in match (get_element ????); >(\b (refl …)) normalize nodelta |
---|
| 934 | >(\b (refl …)) <fresh_map [2: >memb_hd //] whd in match (get_element ????); |
---|
| 935 | inversion(lin==lout) |
---|
| 936 | [ #ABS @⊥ cases no_dup * #ABS1 #_ @ABS1 whd in match (memb ???); >(\P ABS) |
---|
| 937 | >(\b (refl …)) // |
---|
[3549] | 938 | | #H inversion (a_non_functional_label … lin== ? lout) |
---|
[3414] | 939 | [ #ABS lapply(\P ABS) #EQ destruct >(\b (refl …)) in H; #EQ destruct |
---|
| 940 | | #_ normalize nodelta whd in match (get_element ????); >(\b (refl …)) |
---|
| 941 | normalize nodelta >(\b (refl …)) % |
---|
| 942 | ] |
---|
| 943 | ] |
---|
| 944 | | // |
---|
| 945 | | #x #Hx >same_keep [2: >memb_cons // >memb_cons // >Hx % ] % |
---|
| 946 | | #x #Hx <fresh_map [2: >memb_cons // >memb_cons // >Hx %] |
---|
| 947 | cases no_dup * #ABS1 ** #ABS2 #_ whd in ⊢ (???%); inversion(x == lout) |
---|
| 948 | normalize nodelta |
---|
| 949 | [2: #_ whd in ⊢ (???%); inversion(x==lin) normalize nodelta // |
---|
| 950 | #H @⊥ @ABS1 >memb_cons // <(\P H) >Hx // |
---|
| 951 | | #H @⊥ @ABS2 <(\P H) >Hx // |
---|
| 952 | ] |
---|
| 953 | | cases no_dup #_ * #_ // |
---|
| 954 | ] |
---|
[3408] | 955 | ] |
---|
[3416] | 956 | qed. |
---|
[3396] | 957 | |
---|
[3549] | 958 | definition fresh_for_prog_aux : ∀p,p',l_p.Program p p' l_p → l_p → l_p ≝ |
---|
| 959 | λp,p',l_p,prog,n.foldl … (λn,i.op … l_p … n (compute_max_n … (f_body … i))) n (env … prog). |
---|
[3445] | 960 | |
---|
| 961 | |
---|
[3549] | 962 | lemma fresh_aux_ok : ∀p,p',l_p.∀prog : Program p p' l_p.∀n,m.n ⊑^{l_p} m → |
---|
| 963 | fresh_for_prog_aux … prog n ⊑^{l_p} fresh_for_prog_aux … prog m. |
---|
| 964 | #p #p' #l_p * #env #main elim env // #hd #tl #IH #n #m #H whd in ⊢ (???%%); |
---|
| 965 | @IH whd in ⊢ (???%?); @(monotonic_magg … l_p … H) |
---|
[3445] | 966 | qed. |
---|
| 967 | |
---|
[3549] | 968 | definition fresh_for_prog : ∀p,p',l_p.Program p p' l_p → l_p ≝ |
---|
| 969 | λp,p',l_p,prog.fresh_for_prog_aux … prog |
---|
[3445] | 970 | (compute_max_n … (main … prog)). |
---|
[3400] | 971 | |
---|
[3416] | 972 | definition translate_env ≝ |
---|
[3549] | 973 | λp,p',l_p.λenv : list (env_item p p' l_p).λmax_all.(foldr ?? |
---|
[3396] | 974 | (λi,x.let 〈t_env,n,m,keep〉 ≝ x in |
---|
| 975 | let info ≝ call_post_trans … (f_body … i) n (nil ?) in |
---|
[3549] | 976 | 〈(mk_env_item ??? |
---|
[3397] | 977 | (mk_signature ??(f_name ?? i) (f_pars … i) (f_ret … i)) |
---|
[3400] | 978 | (f_lab … i) (t_code … info)) :: t_env, |
---|
[3549] | 979 | fresh … info, 〈a_call … (f_lab … i),(a_call … (f_lab … i)) :: (gen_labels ?? info)〉 :: |
---|
[3396] | 980 | ((lab_map … info) @ m),(lab_to_keep … info) @ keep〉) |
---|
[3416] | 981 | (〈nil ?,max_all,nil ?,nil ?〉) env). |
---|
[3396] | 982 | |
---|
[3549] | 983 | definition trans_prog : ∀p,p',l_p.Program p p' l_p → |
---|
| 984 | ((Program p p' l_p) × (associative_list (DEQCostLabel l_p) (CostLabel l_p)) × ((list (ReturnPostCostLabel l_p))))≝ |
---|
| 985 | λp,p',l_p,prog. |
---|
[3416] | 986 | let max_all ≝ fresh_for_prog … prog in |
---|
| 987 | let info_main ≝ (call_post_trans … (main … prog) max_all (nil ?)) in |
---|
| 988 | let 〈t_env,n,m,keep〉 ≝ translate_env … (env … prog) (fresh … info_main) in |
---|
[3549] | 989 | 〈mk_Program ??? t_env (t_code … info_main),m @ (lab_map … info_main),keep @ (lab_to_keep … info_main)〉. |
---|
[3416] | 990 | |
---|
[3549] | 991 | definition map_labels_on_trace : ∀p : label_params. |
---|
| 992 | (associative_list (DEQCostLabel p) (CostLabel p)) → list (CostLabel p) → list (CostLabel p) ≝ |
---|
| 993 | λp,m,l.foldr … (λlab,t.(get_element … m lab) @ t) (nil ?) l. |
---|
[3396] | 994 | |
---|
| 995 | lemma map_labels_on_trace_append: |
---|
[3549] | 996 | ∀p,m,l1,l2. map_labels_on_trace p m (l1@l2) = |
---|
| 997 | map_labels_on_trace p m l1 @ map_labels_on_trace p m l2. |
---|
| 998 | #p #m #l1 elim l1 // #hd #tl #IH #l2 >associative_append <IH // |
---|
[3396] | 999 | qed. |
---|
| 1000 | |
---|
[3397] | 1001 | include "../src/common/Errors.ma". |
---|
[3527] | 1002 | include "Permutation.ma". |
---|
[3397] | 1003 | |
---|
[3527] | 1004 | (* |
---|
| 1005 | |
---|
[3399] | 1006 | axiom is_permutation: ∀A.list A → list A → Prop. |
---|
| 1007 | axiom is_permutation_eq : ∀A.∀l : list A.is_permutation … l l. |
---|
| 1008 | axiom is_permutation_cons : ∀A.∀l1,l2,x.is_permutation A l1 l2 → |
---|
| 1009 | is_permutation A (x :: l1) (x :: l2). |
---|
[3527] | 1010 | *) |
---|
[3399] | 1011 | (* |
---|
| 1012 | inductive is_permutation (A : Type[0]) : list A → list A → Prop ≝ |
---|
| 1013 | | p_empty : is_permutation A (nil ?) (nil ?) |
---|
| 1014 | | p_append : ∀x,x1,x2,y,y1,y2. |
---|
| 1015 | x = y → is_permutation A (x1 @ x2) (y1 @ y2) → |
---|
| 1016 | is_permutation A (x1 @ [x] @ x2) (y1 @ [y] @ y2). |
---|
| 1017 | |
---|
| 1018 | lemma is_permutation_eq : ∀A.∀l : list A.is_permutation … l l. |
---|
| 1019 | #A #l elim l // #x #xs #IH |
---|
| 1020 | change with ((nil ?) @ (x :: xs)) in ⊢ (??%%); |
---|
| 1021 | >append_cons >associative_append |
---|
| 1022 | @(p_append ? x (nil ?) xs x (nil ?) xs (refl …)) @IH |
---|
| 1023 | qed. |
---|
| 1024 | |
---|
| 1025 | lemma is_permutation_append : ∀A.∀l1,l2,l3,l4 : list A. |
---|
| 1026 | is_permutation A l1 l3 → is_permutation A l2 l4 → |
---|
| 1027 | is_permutation A (l1 @ l2) (l3 @ l4). |
---|
| 1028 | #A #l1 inversion (|l1|) [2: #n lapply l1 elim n |
---|
| 1029 | [ #l2 #l3 #l4 #H inversion H // #x #x1 #x2 #y #y1 #y2 #EQ #H1 #_ |
---|
| 1030 | #ABS cases(nil_to_nil … (sym_eq ??? ABS)) -ABS #_ #ABS |
---|
| 1031 | cases(nil_to_nil … ABS) #EQ1 destruct(EQ1) ] |
---|
| 1032 | #x #xs #IH #l2 #l3 #l4 #H inversion H |
---|
| 1033 | [#EQ lapply(jmeq_to_eq ??? EQ) -EQ #EQ destruct(EQ) ] |
---|
| 1034 | #y #y1 #y2 #z #z1 #z2 #EQ destruct(EQ) #H1 #_ #EQx_xs #EQ destruct(EQ) #_ |
---|
| 1035 | *) |
---|
| 1036 | |
---|
[3405] | 1037 | |
---|
| 1038 | |
---|
[3549] | 1039 | lemma lookup_ok_append : ∀p,p',l_p,l,f,env_it. |
---|
| 1040 | lookup p p' l_p l f = return env_it → ∃l1,l2. l = l1 @ [env_it] @ l2 ∧ |
---|
[3405] | 1041 | f_name … env_it = f. |
---|
[3549] | 1042 | #p #p' #l_p #l elim l [ #f #env_it normalize #EQ destruct] |
---|
[3405] | 1043 | #x #xs #IH #f #env_it whd in ⊢ (??%? → ?); @eq_function_name_elim |
---|
| 1044 | [ #EQ destruct(EQ) normalize nodelta whd in ⊢ (???% → ?); #EQ destruct |
---|
| 1045 | %{(nil ?)} %{xs} /2/ |
---|
| 1046 | | #Hno_f normalize nodelta #EQ_env_it cases(IH … EQ_env_it) |
---|
| 1047 | #l1 * #l2 * #EQ1 #EQ2 %{(x :: l1)} %{l2} >EQ1 /2/ |
---|
| 1048 | ] |
---|
| 1049 | qed. |
---|
| 1050 | (* |
---|
| 1051 | lemma foldr_append : |
---|
| 1052 | ∀A,B:Type[0]. ∀l1, l2 : list A. ∀f:A → B → B. ∀seed. foldr ?? f seed (l1 @ l2) = foldr ?? f (foldr ?? f seed l2) l1. |
---|
| 1053 | #A #B #l1 elim l1 // |
---|
| 1054 | #hd #tl #Hind #l2 #f #seed normalize >Hind @refl |
---|
| 1055 | qed. |
---|
| 1056 | *) |
---|
| 1057 | |
---|
| 1058 | |
---|
| 1059 | |
---|
[3414] | 1060 | (* aggiungere fresh_to_keep al lemma seguente??*) |
---|
[3409] | 1061 | |
---|
[3549] | 1062 | lemma fresh_for_subset : ∀p : label_params.∀l1,l2,n.l1 ⊆ l2 → is_fresh_for_return p … l2 n → |
---|
| 1063 | is_fresh_for_return p … l1 n. |
---|
| 1064 | #p #l1 elim l1 // * [1,2,3: * #x] #xs #IH #l2 #n * #H1 #H2 #H3 whd |
---|
[3416] | 1065 | try(@IH) // % [2: @IH //] elim l2 in H1 H3; normalize [*] |
---|
[3478] | 1066 | * [1,2,3: * #y] #ys #IH normalize |
---|
[3416] | 1067 | [2,3: * [2,4: #H3 [2: @IH //] * #H4 #H5 @IH // |
---|
| 1068 | |*: #EQ destruct * // |
---|
| 1069 | ]] |
---|
| 1070 | * |
---|
[3478] | 1071 | [1,3: #EQ destruct ] #H3 #H4 @IH // |
---|
[3416] | 1072 | qed. |
---|
| 1073 | |
---|
[3549] | 1074 | lemma fresh_append : ∀p.∀n,l1,l2.is_fresh_for_return p l1 n → is_fresh_for_return p l2 n → |
---|
| 1075 | is_fresh_for_return p (l1@l2) n. |
---|
| 1076 | #p #n #l1 lapply n -n elim l1 // * [1,2,3: * #x] #xs #IH #n #l2 [2: * #H1 ] #H2 #H3 |
---|
[3416] | 1077 | [ % // @IH //] @IH // |
---|
| 1078 | qed. |
---|
| 1079 | |
---|
[3549] | 1080 | definition labels_of_prog : ∀p,p',l_p.Program p p' l_p → ? ≝ |
---|
| 1081 | λp,p',l_p,prog.foldr … (λx,l.l @ (get_labels_of_code … (f_body … x))) |
---|
[3445] | 1082 | (get_labels_of_code … (main … prog)) (env … prog). |
---|
[3416] | 1083 | |
---|
[3549] | 1084 | lemma cast_return_append : ∀p.∀l1,l2.cast_return_to_cost_labels p … (l1 @ l2) = |
---|
| 1085 | (cast_return_to_cost_labels p … l1) @ (cast_return_to_cost_labels p … l2). |
---|
| 1086 | #p #l1 #l2 @(sym_eq … (map_append …)) qed. |
---|
[3416] | 1087 | |
---|
[3445] | 1088 | include alias "arithmetics/nat.ma". |
---|
| 1089 | |
---|
| 1090 | |
---|
[3549] | 1091 | lemma is_fresh_code : ∀p,l_p.∀i : Instructions p l_p. |
---|
| 1092 | is_fresh_for_return l_p (get_labels_of_code … i) (compute_max_n … i). |
---|
| 1093 | #p #l_p #main elim main // |
---|
| 1094 | [ #seq * [| * #lbl] #i #IH normalize // @(fresh_keep_n_ok … IH) // |
---|
| 1095 | | #cond * #ltrue #i1 * #lfalse #i2 #i3 #IH1 #IH2 #IH3 whd in ⊢ (??%%); |
---|
[3445] | 1096 | @fresh_append |
---|
[3549] | 1097 | [ @(fresh_keep_n_ok … IH1) @max_1 @max_1 @max_1 @max_1 // |
---|
[3445] | 1098 | | @fresh_append |
---|
[3549] | 1099 | [ @(fresh_keep_n_ok … IH2) @max_1 @max_1 @max_1 @max_2 // |
---|
| 1100 | | @(fresh_keep_n_ok … IH3) @max_1 @max_1 @max_2 // |
---|
[3445] | 1101 | ] |
---|
| 1102 | ] |
---|
[3549] | 1103 | | #cond * #ltrue #i1 * #lfalse #i2 #IH1 #IH2 whd in ⊢ (??%%); @fresh_append |
---|
| 1104 | [ @(fresh_keep_n_ok … IH1) @max_1 @max_1 @max_1 // |
---|
| 1105 | | @(fresh_keep_n_ok … IH2) @max_1 @max_1 @max_2 // |
---|
[3445] | 1106 | ] |
---|
[3549] | 1107 | | #f #act_p * [| * #lbl] #i #IH whd in ⊢ (??%%); // |
---|
| 1108 | change with ([?]@?) in ⊢ (??%?); @fresh_append |
---|
| 1109 | [ whd % // |
---|
| 1110 | | @(fresh_keep_n_ok … IH) @max_2 // |
---|
[3445] | 1111 | ] |
---|
[3549] | 1112 | | * #lin #io * #lout #i #IH whd in ⊢ (??%%); @(fresh_keep_n_ok … IH) |
---|
| 1113 | @max_1 @max_1 // |
---|
[3445] | 1114 | ] |
---|
| 1115 | qed. |
---|
| 1116 | |
---|
[3549] | 1117 | lemma is_fresh_fresh_for_prog : ∀p,p',l_p.∀prog : Program p p' l_p. |
---|
| 1118 | is_fresh_for_return … (labels_of_prog … prog) (fresh_for_prog … prog). |
---|
| 1119 | #p #p' #l_p * #env #main whd in match fresh_for_prog; normalize nodelta whd in ⊢ (??%?); |
---|
| 1120 | elim env // * #sig #cost #i #tail #IH whd in ⊢ (??%?); @fresh_append |
---|
[3445] | 1121 | [ @(fresh_keep_n_ok … IH) @fresh_aux_ok @max_1 // |
---|
| 1122 | | @(fresh_keep_n_ok … (is_fresh_code … i)) whd in match fresh_for_prog_aux; normalize nodelta |
---|
[3549] | 1123 | whd in ⊢ (????%); elim tail [ @max_2 // ] #hd1 #tl1 #IH1 @(trans_po_rel … IH1) |
---|
| 1124 | whd in ⊢ (????%); change with (fresh_for_prog_aux ??? (mk_Program ??? tl1 main) ?) in ⊢ (???%%); |
---|
[3445] | 1125 | @fresh_aux_ok @max_1 // |
---|
| 1126 | ] |
---|
| 1127 | qed. |
---|
| 1128 | |
---|
[3549] | 1129 | lemma memb_cast_return : ∀p.∀keep,x.x ∈ cast_return_to_cost_labels p keep → |
---|
| 1130 | ∃ y.x = a_return_post … y ∧ bool_to_Prop (y ∈ keep). |
---|
| 1131 | #p #keep elim keep |
---|
[3445] | 1132 | [ #x *] #x #xs #IH #y whd in match cast_return_to_cost_labels; |
---|
| 1133 | whd in match (map ????); whd in match (memb ???); inversion(y==x) |
---|
| 1134 | [ #Hx #_ %{x} >(\P Hx) %{(refl …)} >memb_hd // |
---|
| 1135 | | #Hx normalize nodelta #H cases(IH … H) #z * #H1 #H2 %{z} %{H1} >memb_cons // >H2 // |
---|
| 1136 | ] |
---|
| 1137 | qed. |
---|
| 1138 | |
---|
[3549] | 1139 | lemma lab_to_keep_in_prog : ∀p,p',l_p.∀prog : Program p p' l_p. |
---|
[3445] | 1140 | ∀t_prog,m,keep.trans_prog … prog = 〈t_prog,m,keep〉 → |
---|
[3549] | 1141 | (cast_return_to_cost_labels l_p keep) ⊆ (labels_of_prog p p' l_p prog). |
---|
| 1142 | #p #p' #l_p * #env #main #t_prog #m #keep whd in match trans_prog; normalize nodelta |
---|
[3445] | 1143 | @pair_elim * #env1 #fresh * #m1 #keep1 #EQenv1 normalize nodelta #EQ destruct |
---|
| 1144 | lapply EQenv1 -EQenv1 lapply keep1 -keep1 lapply m1 -m1 lapply fresh -fresh |
---|
[3549] | 1145 | lapply env1 -env1 generalize in match (fresh_for_prog ????); elim env |
---|
[3445] | 1146 | [ #n #t_env #n1 #m #keep whd in ⊢ (??%? → ?); #EQ destruct whd in match (append ???); |
---|
| 1147 | @subset_def #x #H whd in match (labels_of_prog); normalize nodelta |
---|
| 1148 | whd in match (foldr ?????); cases(memb_cast_return … H) -H #x1 * #EQ1 #H destruct |
---|
| 1149 | @(lab_to_keep_in_domain … H) |
---|
| 1150 | | #x #xs #IH #n #t_env #n1 #m #keep whd in ⊢ (??%? → ?); @pair_elim |
---|
| 1151 | * #t_env_tail #fresh_tail * #t_m_tail #t_keep_tail |
---|
[3549] | 1152 | change with (translate_env ?????) in match (foldr ?????); #EQt_env_tail |
---|
[3445] | 1153 | normalize nodelta #EQ1 destruct >cast_return_append @subset_append |
---|
| 1154 | [ >cast_return_append @subset_append |
---|
| 1155 | [ whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????); |
---|
| 1156 | @subset_def #y #H cases(memb_cast_return … H) -H #y1 * #EQ destruct #H |
---|
| 1157 | >memb_append_l2 // @(lab_to_keep_in_domain … H) |
---|
| 1158 | | whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????); |
---|
[3549] | 1159 | change with (labels_of_prog ??? (mk_Program ??? xs ?)) in match (foldr ?????); |
---|
[3445] | 1160 | @subset_append_h1 @(transitive_subset … (IH … EQt_env_tail)) |
---|
| 1161 | >cast_return_append @subset_append_h1 // |
---|
| 1162 | ] |
---|
| 1163 | | whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????); |
---|
[3549] | 1164 | change with (labels_of_prog ??? (mk_Program ??? xs ?)) in match (foldr ?????); |
---|
[3445] | 1165 | @subset_append_h1 @(transitive_subset … (IH … EQt_env_tail)) |
---|
| 1166 | >cast_return_append @subset_append_h2 // |
---|
| 1167 | ] |
---|
| 1168 | ] |
---|
| 1169 | qed. |
---|
| 1170 | |
---|
[3549] | 1171 | lemma fresh_call_post_trans_ok : ∀p,l_p.∀i : Instructions p l_p.∀n,l. |
---|
| 1172 | n ⊑^{l_p} fresh … (call_post_trans … i n l). |
---|
| 1173 | #p #l_p #i elim i // |
---|
[3445] | 1174 | qed. |
---|
| 1175 | |
---|
[3549] | 1176 | lemma fresh_translate_env_ok : ∀p,p',l_p.∀env,t_env : list (env_item p p' l_p).∀n,n1,m,keep. |
---|
| 1177 | translate_env … env n = 〈t_env,n1,m,keep〉 → n ⊑^{l_p} n1. |
---|
| 1178 | #p #p' #l_p #env elim env |
---|
[3445] | 1179 | [ #t_env #n #n1 #m #keep whd in ⊢ (??%? → ?); #EQ destruct // ] |
---|
| 1180 | #x #xs #IH #t_env #n #n1 #m #keep whd in ⊢ (??%? → ?); |
---|
[3549] | 1181 | change with (translate_env ?????) in match (foldr ?????); @pair_elim |
---|
[3445] | 1182 | * #t_env_tail #fresh_tail * #t_m_tail #t_keep_tail #EQt_env_tail normalize nodelta |
---|
[3549] | 1183 | #EQ destruct @(trans_po_rel … (IH … EQt_env_tail)) @fresh_call_post_trans_ok |
---|
[3445] | 1184 | qed. |
---|
| 1185 | |
---|
| 1186 | |
---|
[3400] | 1187 | lemma trans_env_ok : ∀p : state_params.∀ prog. |
---|
| 1188 | no_duplicates_labels … prog → |
---|
| 1189 | let 〈t_prog,m,keep〉 ≝ trans_prog … prog in |
---|
[3549] | 1190 | ∀f,env_it.lookup p p p (env … prog) f = return env_it → |
---|
[3416] | 1191 | let dom ≝ get_labels_of_code … (f_body … env_it) in |
---|
[3549] | 1192 | ∃env_it',n.is_fresh_for_return p keep n ∧lookup p p p (env … t_prog) f = return env_it' ∧ |
---|
[3400] | 1193 | let info ≝ call_post_trans … (f_body … env_it) n (nil ?) in |
---|
| 1194 | t_code … info = f_body … env_it' ∧ |
---|
[3549] | 1195 | get_element … m (a_call … (f_lab … env_it')) = (a_call … (f_lab … env_it')) :: gen_labels … info ∧ |
---|
[3400] | 1196 | f_sig … env_it = f_sig … env_it' ∧ f_lab … env_it = f_lab … env_it' ∧ |
---|
[3549] | 1197 | same_fresh_map_on … dom m (lab_map … info) ∧ same_to_keep_on … dom keep (lab_to_keep … info). |
---|
[3416] | 1198 | #p #prog inversion(trans_prog … prog) * #t_prog0 #m0 #keep0 #EQt_prog |
---|
| 1199 | lapply EQt_prog normalize nodelta |
---|
| 1200 | generalize in match keep0 in ⊢ (% → ? → ? → ? → ? → ??(λ_.??(λ_.?%?))); |
---|
| 1201 | #keep1 #EQkeep1 inversion prog in EQt_prog; #env #main #EQprog |
---|
| 1202 | whd in match trans_prog; normalize nodelta |
---|
| 1203 | @pair_elim |
---|
[3549] | 1204 | cut(fresh_for_prog ??? prog ⊑^{p} fresh_for_prog ??? (mk_Program … env main)) [ >EQprog //] |
---|
| 1205 | generalize in match (fresh_for_prog ????) in ⊢ (????% → %); |
---|
[3416] | 1206 | lapply t_prog0 lapply m0 lapply keep0 |
---|
| 1207 | elim env in ⊢ (?→ ? → ? → ? → ? → %); |
---|
| 1208 | [ #keep #m #t_prog #n #_ * #env' #fresh * #x #y #_ #_ #_ #f #env_it normalize in ⊢ (% → ?); #ABS destruct] |
---|
| 1209 | * #hd_sig #hd_lab #hd_code #tail #IH #keep #m #t_prog #fresh1 #Hfresh1 * #env' #fresh * #m' #keep' |
---|
[3400] | 1210 | normalize in ⊢ (% → ?); normalize nodelta @pair_elim * #env_tail #fresh_tail |
---|
[3549] | 1211 | * #m_tail #keep_tail change with (translate_env ?????) in ⊢ (??%? → ?); #EQtail normalize nodelta #EQ1 destruct(EQ1) #EQ2 destruct(EQ2) |
---|
[3400] | 1212 | whd in ⊢ (% → ?); whd in match (foldr ?????); * #Hhd_lab #H lapply(no_duplicates_append_r … H) |
---|
[3549] | 1213 | change with (no_duplicates_labels p p p (mk_Program p p p tail main)) in match |
---|
| 1214 | (no_duplicates_labels p p p (mk_Program p p p tail main)); #no_dup_tail |
---|
[3400] | 1215 | lapply(no_duplicates_append_l … H) #no_dup_head normalize nodelta |
---|
| 1216 | #f #env_it whd in ⊢ (??%? → ?); @eq_function_name_elim normalize nodelta |
---|
| 1217 | [ #EQ destruct(EQ) whd in ⊢ (???% → ?); #EQ destruct(EQ) |
---|
| 1218 | inversion (call_post_trans … hd_code fresh_tail []) |
---|
| 1219 | #gen_labs #t_hd_code #t_fresh #t_lab_map #t_lab_to_keep #EQ_trans_code |
---|
[3416] | 1220 | %{(mk_env_item … hd_sig hd_lab t_hd_code)} %{fresh_tail} % |
---|
| 1221 | [ % |
---|
| 1222 | [ @(fresh_keep_n_ok … fresh1) |
---|
| 1223 | [ @(fresh_keep_n_ok … Hfresh1) |
---|
| 1224 | @(fresh_for_subset … (labels_of_prog … prog)) |
---|
[3445] | 1225 | [ @(lab_to_keep_in_prog … EQkeep1) | @is_fresh_fresh_for_prog ] |
---|
[3549] | 1226 | | @(trans_po_rel … (fresh_translate_env_ok … EQtail)) // |
---|
[3416] | 1227 | ] |
---|
| 1228 | | whd in ⊢ (??%?); @eq_function_name_elim [2: * #H @⊥ @H %] #_ normalize nodelta |
---|
| 1229 | @eq_f cases hd_sig // ]] >EQ_trans_code % [% [ % [ % [% // whd in ⊢ (??%?); >(\b (refl …)) %] % ] % | whd |
---|
[3405] | 1230 | #x #Hx whd in ⊢ (??%?); >(? : (x == hd_lab) = false) |
---|
| 1231 | [2: inversion(x==hd_lab) // #EQx_hdlab cases Hhd_lab -Hhd_lab #Hhd_lab cases Hhd_lab |
---|
| 1232 | >memb_append_l1 // <(\P EQx_hdlab) >Hx // ] |
---|
[3416] | 1233 | normalize nodelta >get_element_append_l1 |
---|
[3448] | 1234 | [2: % #ABS @(memb_no_duplicates_append … x … H) // elim tail |
---|
| 1235 | [ whd in match (foldr ?????); @lab_map_in_domain // ] |
---|
| 1236 | #x #xs #IH whd in match (foldr ?????); @orb_Prop_r |
---|
| 1237 | >memb_append_l2 // >IH % |
---|
| 1238 | ] @get_element_append_l1 |
---|
[3416] | 1239 | % #H1 |
---|
[3409] | 1240 | (* subproof with no nice statement *) |
---|
| 1241 | lapply H1 -H1 lapply H -H lapply Hhd_lab -Hhd_lab lapply EQtail -EQtail |
---|
[3416] | 1242 | generalize in match fresh1; lapply env_tail -env_tail lapply fresh_tail |
---|
| 1243 | -fresh_tail lapply m_tail -m_tail lapply keep_tail -keep_tail elim tail |
---|
[3409] | 1244 | normalize nodelta |
---|
| 1245 | [ #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?); |
---|
| 1246 | #EQ destruct(EQ) whd in match (foldr ?????); |
---|
| 1247 | #H1 #H2 * ] |
---|
| 1248 | #head #tail #IH #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?); |
---|
| 1249 | whd in match (foldr ?????); |
---|
| 1250 | @pair_elim * #env_tail_res #fresh_tail_res * #lab_map_res #keep_res #EQres |
---|
| 1251 | normalize nodelta #EQ destruct(EQ) whd in match (foldr ?????); |
---|
| 1252 | #H1 #H2 whd in match (memb ???); inversion(x == ?) |
---|
| 1253 | [ #H3 #_ <(\P H3) in H2; change with ([?]@?) in match (?::?); #H2 |
---|
| 1254 | lapply(no_duplicates_append_commute … H2) -H2 ** #ABS #_ @ABS |
---|
| 1255 | >memb_append_l2 // >Hx % |
---|
| 1256 | | #H3 normalize nodelta #H4 @(IH … EQres) |
---|
| 1257 | [3: >domain_of_associative_list_append in H4; #H4 cases(memb_append … H4) [2: #EQ >EQ %] |
---|
| 1258 | #ABS @⊥ @(memb_no_duplicates_append … x … H2) // @orb_Prop_r >memb_append_l1 // |
---|
| 1259 | @(lab_map_in_domain … (eq_true_to_b … ABS)) |
---|
| 1260 | | % #ABS elim H1 -H1 #H1 @H1 cases(memb_append … ABS) |
---|
| 1261 | [ #H5 >memb_append_l1 // |
---|
| 1262 | | #H5 >memb_append_l2 // @orb_Prop_r >memb_append_l2 // |
---|
| 1263 | ] |
---|
| 1264 | | lapply(no_duplicates_append_commute … H2) * #_ >associative_append |
---|
| 1265 | #h @no_duplicates_append_commute @(no_duplicates_append_r … h) |
---|
| 1266 | ] |
---|
| 1267 | ] |
---|
| 1268 | ] |
---|
[3448] | 1269 | | whd #x #Hx >memb_append_l12 |
---|
| 1270 | [2: @notb_Prop % #ABS @(memb_no_duplicates_append … H … Hx) elim tail |
---|
| 1271 | [ whd in match (foldr ?????); @lab_to_keep_in_domain // ] |
---|
| 1272 | #x #xs #IH whd in match (foldr ?????); @orb_Prop_r |
---|
| 1273 | >memb_append_l2 // >IH % |
---|
| 1274 | ] |
---|
[3549] | 1275 | >memb_append_l12 // inversion(memb ???) // #ABS @(memb_no_duplicates_append … (a_return_post … x) … H) |
---|
[3409] | 1276 | // @⊥ |
---|
| 1277 | (* subproof with no nice statement *) |
---|
| 1278 | lapply ABS -ABS lapply H -H lapply Hhd_lab -Hhd_lab lapply EQtail -EQtail |
---|
[3416] | 1279 | generalize in match fresh1; lapply env_tail -env_tail lapply fresh_tail |
---|
| 1280 | -fresh_tail lapply m_tail -m_tail lapply keep_tail -keep_tail elim tail |
---|
[3409] | 1281 | normalize nodelta |
---|
| 1282 | [ #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?); |
---|
| 1283 | #EQ destruct(EQ) whd in match (foldr ?????); |
---|
| 1284 | #H1 #H2 whd in ⊢ (??%? → ?); #EQ destruct ] |
---|
| 1285 | #head #tail #IH #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?); |
---|
| 1286 | whd in match (foldr ?????); |
---|
| 1287 | @pair_elim * #env_tail_res #fresh_tail_res * #lab_map_res #keep_res #EQres |
---|
| 1288 | normalize nodelta #EQ destruct(EQ) whd in match (foldr ?????); |
---|
| 1289 | #H1 #H2 #H3 cases(memb_append … H3) -H3 |
---|
| 1290 | [ #H3 change with ([?]@?) in match (?::?) in H2; |
---|
[3549] | 1291 | lapply(no_duplicates_append_commute … H2) -H2 * #_ #H4 @(memb_no_duplicates_append … (a_return_post … x) … H4) |
---|
[3409] | 1292 | [ whd in match (append ???); >memb_append_l1 // >(lab_to_keep_in_domain … (eq_true_to_b … H3)) % |
---|
| 1293 | | // |
---|
| 1294 | ] |
---|
| 1295 | | #H3 normalize nodelta @(IH … EQres) |
---|
| 1296 | [3: // |
---|
| 1297 | | % #ABS elim H1 -H1 #H1 @H1 cases(memb_append … ABS) |
---|
| 1298 | [ #H5 >memb_append_l1 // |
---|
| 1299 | | #H5 >memb_append_l2 // @orb_Prop_r >memb_append_l2 // |
---|
| 1300 | ] |
---|
| 1301 | | lapply(no_duplicates_append_commute … H2) * #_ >associative_append |
---|
| 1302 | #h @no_duplicates_append_commute @(no_duplicates_append_r … h) |
---|
| 1303 | ] |
---|
| 1304 | ] |
---|
[3400] | 1305 | ] |
---|
| 1306 | | #Hf #Henv_it cases(IH … no_dup_tail … Henv_it) |
---|
[3416] | 1307 | [9: >EQtail in ⊢ (??%?); % |
---|
| 1308 | |13: % |
---|
| 1309 | |6: assumption |
---|
| 1310 | |10: % |
---|
[3404] | 1311 | |*: |
---|
| 1312 | ] |
---|
[3416] | 1313 | #new_env_it * #new_fresh ** #is_fresh_newfresh #EQlook_new_env_it ***** #EQt_code #EQ_get_el |
---|
[3404] | 1314 | #EQsign_env_it #EQ_f_lab #same_fresh_map #same_to_keep %{new_env_it} %{new_fresh} |
---|
| 1315 | % |
---|
[3416] | 1316 | [ % |
---|
| 1317 | [ assumption |
---|
| 1318 | | whd in ⊢ (??%?); @eq_function_name_elim [ #ABS >ABS in Hf; * #H @⊥ @H %] |
---|
| 1319 | #_ normalize nodelta assumption ]] |
---|
| 1320 | % [2: #x #Hx <same_to_keep // >associative_append @memb_append_l22 |
---|
[3405] | 1321 | inversion(memb ???) // #ABS lapply(lab_to_keep_in_domain … (eq_true_to_b … ABS)) |
---|
[3549] | 1322 | #ABS1 @(memb_no_duplicates_append … (a_return_post … x) … H) // |
---|
[3405] | 1323 | cases(lookup_ok_append … Henv_it) #l1 * #l2 * #EQ1 #EQ2 destruct(EQ1 EQ2) |
---|
| 1324 | >foldr_map_append >memb_append_l2 // >foldr_map_append >memb_append_l1 // |
---|
| 1325 | whd in match (foldr ?????); @orb_Prop_r >memb_append_l1 // >Hx % ] |
---|
[3416] | 1326 | % [2: #x #Hx <same_fresh_map // >cons_append <associative_append |
---|
| 1327 | <associative_append in ⊢ (??(???(??%?)?)?); >associative_append |
---|
| 1328 | @(get_element_append_r1) |
---|
[3408] | 1329 | % >domain_of_associative_list_append #ABS cases(memb_append … ABS) |
---|
| 1330 | [ whd in match (memb ???); inversion(x==hd_lab) normalize nodelta |
---|
| 1331 | [2: #_ whd in match (memb ???); #EQ destruct ] #EQx_hdlab #_ |
---|
| 1332 | <(\P EQx_hdlab) in Hhd_lab; cases(lookup_ok_append … Henv_it) |
---|
| 1333 | #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append |
---|
| 1334 | * #ABS1 @ABS1 >memb_append_l2 // >memb_append_l2 // |
---|
| 1335 | >memb_append_l1 // whd in ⊢ (??%?); cases(x==?) // |
---|
| 1336 | normalize nodelta >memb_append_l1 // >Hx % |
---|
| 1337 | | #ABS1 @(memb_no_duplicates_append … x … H) |
---|
| 1338 | [ @(lab_map_in_domain … (eq_true_to_b … ABS1)) |
---|
| 1339 | | cases(lookup_ok_append … Henv_it) |
---|
| 1340 | #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append |
---|
| 1341 | >memb_append_l2 // >memb_append_l1 // |
---|
| 1342 | whd in ⊢ (??%?); cases(x==?) // |
---|
| 1343 | normalize nodelta >memb_append_l1 // >Hx % |
---|
| 1344 | ] |
---|
| 1345 | ] |
---|
| 1346 | ] |
---|
[3416] | 1347 | % // % // % // <EQ_get_el >cons_append <associative_append <associative_append in ⊢ (??(???(??%?)?)?); |
---|
| 1348 | >associative_append |
---|
[3408] | 1349 | @get_element_append_r1 % >domain_of_associative_list_append #ABS cases(memb_append … ABS) |
---|
[3549] | 1350 | [ whd in match (memb ???); inversion(a_call … (f_lab … new_env_it)== a_call … hd_lab) |
---|
[3408] | 1351 | #EQ_hdlab normalize nodelta |
---|
| 1352 | [2: whd in ⊢ (??%? → ?); #EQ destruct ] |
---|
| 1353 | #_ <(\P EQ_hdlab) in Hhd_lab; cases(lookup_ok_append … Henv_it) |
---|
| 1354 | #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append |
---|
| 1355 | * #ABS1 @ABS1 >memb_append_l2 // >memb_append_l2 // |
---|
| 1356 | >memb_append_l1 // whd in ⊢ (??%?); >EQ_f_lab >(\b (refl …)) // |
---|
[3549] | 1357 | | #ABS1 @(memb_no_duplicates_append … (a_call … (f_lab … new_env_it)) … H) |
---|
[3408] | 1358 | [ @(lab_map_in_domain … (eq_true_to_b … ABS1)) |
---|
| 1359 | | cases(lookup_ok_append … Henv_it) |
---|
| 1360 | #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append |
---|
| 1361 | >memb_append_l2 // >memb_append_l1 // |
---|
| 1362 | whd in ⊢ (??%?); >EQ_f_lab >(\b (refl …)) // |
---|
| 1363 | ] |
---|
| 1364 | ] |
---|
[3405] | 1365 | ] |
---|
[3409] | 1366 | qed. |
---|
[3399] | 1367 | |
---|
[3527] | 1368 | (* |
---|
[3445] | 1369 | axiom permute_ok : ∀A.∀l1,l2,l3,l4,l5,l6,l7,l8,l9,x,y. |
---|
| 1370 | (is_permutation A ((l5 @l1) @l6@l7) ((l4 @[]) @l6@l7) |
---|
| 1371 | →is_permutation A ((l6 @l2) @l7) ((l3 @l8) @l9) |
---|
| 1372 | →is_permutation A |
---|
| 1373 | (y ::((l6 @((x ::l5) @(l1 @l2))) @l7)) |
---|
| 1374 | (((x ::l4 @y ::l3) @l8) @l9)). |
---|
[3527] | 1375 | *) |
---|
[3531] | 1376 | |
---|