| 1 | include "basics/lists/list.ma". |
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| 2 | include "joint/BEGlobalenvs.ma". |
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| 3 | include "common/IO.ma". |
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| 4 | include "common/SmallstepExec.ma". |
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| 5 | include "joint/Joint_paolo.ma". |
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| 6 | include "ASM/I8051bis.ma". |
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| 7 | |
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| 8 | (* CSC: external functions never fail (??) and always return something of the |
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| 9 | size of one register, both in the frontend and in the backend. |
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| 10 | Is this reasonable??? In OCaml it always return a list, but in the frontend |
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| 11 | only the head is kept (or Undef if the list is empty) ??? *) |
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| 12 | |
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| 13 | definition genv ≝ λglobals.λp:params. genv_t Genv (joint_function globals p). |
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| 14 | definition cpointer ≝ Σp.ptype p = Code. |
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| 15 | definition xpointer ≝ Σp.ptype p = XData. |
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| 16 | unification hint 0 ≔ ⊢ cpointer ≡ Sig pointer (λp.ptype p = Code). |
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| 17 | unification hint 0 ≔ ⊢ xpointer ≡ Sig pointer (λp.ptype p = XData). |
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| 18 | |
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| 19 | record sem_state_params : Type[1] ≝ |
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| 20 | { framesT: Type[0] |
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| 21 | ; empty_framesT: framesT |
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| 22 | ; regsT : Type[0] |
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| 23 | ; empty_regsT: xpointer → regsT (* Stack pointer *) |
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| 24 | }. |
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| 25 | |
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| 26 | record state (semp: sem_state_params): Type[0] ≝ |
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| 27 | { st_frms: framesT semp |
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| 28 | ; sp: xpointer |
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| 29 | ; isp: xpointer |
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| 30 | ; carry: beval |
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| 31 | ; regs: regsT semp |
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| 32 | ; m: bemem |
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| 33 | }. |
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| 34 | |
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| 35 | record state_pc (semp : sem_state_params) : Type[0] ≝ |
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| 36 | { st_no_pc :> state semp |
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| 37 | ; pc : cpointer |
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| 38 | }. |
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| 39 | |
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| 40 | definition set_m: ∀p. bemem → state p → state p ≝ |
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| 41 | λp,m,st. mk_state p (st_frms ?…st) (sp … st) (isp … st) (carry … st) (regs … st) m. |
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| 42 | |
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| 43 | definition set_regs: ∀p:sem_state_params. regsT p → state p → state p ≝ |
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| 44 | λp,regs,st. mk_state p (st_frms … st) (sp … st) (isp … st) (carry … st) regs (m … st). |
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| 45 | |
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| 46 | definition set_sp: ∀p. ? → state p → state p ≝ |
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| 47 | λp,sp,st. mk_state … (st_frms … st) sp (isp … st) (carry … st) (regs … st) (m … st). |
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| 48 | |
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| 49 | definition set_isp: ∀p. ? → state p → state p ≝ |
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| 50 | λp,isp,st. mk_state … (st_frms … st) (sp … st) isp (carry … st) (regs … st) (m … st). |
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| 51 | |
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| 52 | definition set_carry: ∀p. beval → state p → state p ≝ |
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| 53 | λp,carry,st. mk_state … (st_frms … st) (sp … st) (isp … st) carry (regs … st) (m … st). |
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| 54 | |
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| 55 | definition set_pc: ∀p. ? → state_pc p → state_pc p ≝ |
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| 56 | λp,pc,st. mk_state_pc … (st_no_pc … st) pc. |
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| 57 | |
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| 58 | definition set_no_pc: ∀p. ? → state_pc p → state_pc p ≝ |
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| 59 | λp,s,st. mk_state_pc … s (pc … st). |
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| 60 | |
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| 61 | definition set_frms: ∀p:sem_state_params. framesT p → state p → state p ≝ |
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| 62 | λp,frms,st. mk_state … frms (sp … st) (isp … st) (carry … st) (regs … st) (m … st). |
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| 63 | |
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| 64 | axiom BadProgramCounter : String. |
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| 65 | (* this can replace graph_fetch function and lin_fetch_function *) |
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| 66 | definition fetch_function: |
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| 67 | ∀pars : params. |
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| 68 | ∀globals. |
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| 69 | genv globals pars → |
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| 70 | cpointer → |
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| 71 | res (joint_internal_function … pars) ≝ |
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| 72 | λpars,globals,ge,p. |
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| 73 | let b ≝ pblock p in |
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| 74 | do def ← opt_to_res ? [MSG BadProgramCounter] (find_funct_ptr … ge b) ; |
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| 75 | match def with |
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| 76 | [ Internal def' ⇒ OK … def' |
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| 77 | | External _ ⇒ Error … [MSG BadProgramCounter]]. |
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| 78 | |
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| 79 | inductive step_flow (p : params) (globals : list ident) : Type[0] ≝ |
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| 80 | | Redirect : label → step_flow p globals (* used for goto-like flow alteration *) |
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| 81 | | Init : Z → joint_internal_function globals p → call_args p → call_dest p → step_flow p globals (* call a function with given id, then proceed *) |
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| 82 | | Proceed : step_flow p globals. (* go to implicit successor *) |
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| 83 | |
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| 84 | inductive stmt_flow (p : params) (globals : list ident) : Type[0] ≝ |
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| 85 | | SRedirect : label → stmt_flow p globals |
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| 86 | | SProceed : succ p → stmt_flow p globals |
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| 87 | | SInit : Z → joint_internal_function globals p → call_args p → call_dest p → succ p → stmt_flow p globals |
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| 88 | | STailInit : Z → joint_internal_function globals p → call_args p → stmt_flow p globals |
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| 89 | | SEnd : stmt_flow p globals. (* end flow *) |
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| 90 | |
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| 91 | |
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| 92 | record more_sem_unserialized_params (uns_pars : unserialized_params) : Type[1] ≝ |
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| 93 | { st_pars :> sem_state_params |
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| 94 | ; acca_store_ : acc_a_reg uns_pars → beval → regsT st_pars → res (regsT st_pars) |
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| 95 | ; acca_retrieve_ : regsT st_pars → acc_a_reg uns_pars → res beval |
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| 96 | ; acca_arg_retrieve_ : regsT st_pars → acc_a_arg uns_pars → res beval |
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| 97 | ; accb_store_ : acc_b_reg uns_pars → beval → regsT st_pars → res (regsT st_pars) |
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| 98 | ; accb_retrieve_ : regsT st_pars → acc_b_reg uns_pars → res beval |
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| 99 | ; accb_arg_retrieve_ : regsT st_pars → acc_b_arg uns_pars → res beval |
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| 100 | ; dpl_store_ : dpl_reg uns_pars → beval → regsT st_pars → res (regsT st_pars) |
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| 101 | ; dpl_retrieve_ : regsT st_pars → dpl_reg uns_pars → res beval |
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| 102 | ; dpl_arg_retrieve_ : regsT st_pars → dpl_arg uns_pars → res beval |
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| 103 | ; dph_store_ : dph_reg uns_pars → beval → regsT st_pars → res (regsT st_pars) |
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| 104 | ; dph_retrieve_ : regsT st_pars → dph_reg uns_pars → res beval |
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| 105 | ; dph_arg_retrieve_ : regsT st_pars → dph_arg uns_pars → res beval |
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| 106 | ; snd_arg_retrieve_ : regsT st_pars → snd_arg uns_pars → res beval |
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| 107 | ; pair_reg_move_ : regsT st_pars → pair_move uns_pars → res (regsT st_pars) |
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| 108 | ; fetch_ra: state st_pars → res ((state st_pars) × cpointer) |
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| 109 | |
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| 110 | ; allocate_local : localsT uns_pars → regsT st_pars → regsT st_pars |
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| 111 | (* Paolo: save_frame separated from call_setup to factorize tailcall code *) |
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| 112 | ; save_frame: cpointer → call_dest uns_pars → state st_pars → framesT st_pars |
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| 113 | (*CSC: setup_call returns a res only because we can call a function with the wrong number and |
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| 114 | type of arguments. To be fixed using a dependent type *) |
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| 115 | ; setup_call : nat → paramsT … uns_pars → call_args uns_pars → |
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| 116 | state st_pars → res (state st_pars) |
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| 117 | ; fetch_external_args: external_function → state st_pars → |
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| 118 | res (list val) |
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| 119 | ; set_result: list val → state st_pars → res (state st_pars) |
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| 120 | ; call_args_for_main: call_args uns_pars |
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| 121 | ; call_dest_for_main: call_dest uns_pars |
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| 122 | }. |
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| 123 | |
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| 124 | |
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| 125 | definition helper_def_retrieve : |
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| 126 | ∀X : ? → ? → Type[0]. |
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| 127 | (∀up.∀p:more_sem_unserialized_params up. regsT p → X up p → res beval) → |
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| 128 | ∀up.∀p : more_sem_unserialized_params up.state p → X up p → res beval ≝ |
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| 129 | λX,f,up,p,st.f ? p (regs … st). |
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| 130 | |
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| 131 | definition helper_def_store : |
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| 132 | ∀X : ? → ? → Type[0]. |
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| 133 | (∀up.∀p : more_sem_unserialized_params up.X up p → beval → regsT p → res (regsT p)) → |
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| 134 | ∀up.∀p : more_sem_unserialized_params up.X up p → beval → state p → res (state p) ≝ |
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| 135 | λX,f,up,p,x,v,st.! r ← f ? p x v (regs … st) ; return set_regs … r st. |
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| 136 | |
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| 137 | definition acca_retrieve ≝ helper_def_retrieve ? acca_retrieve_. |
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| 138 | definition acca_store ≝ helper_def_store ? acca_store_. |
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| 139 | definition acca_arg_retrieve ≝ helper_def_retrieve ? acca_arg_retrieve_. |
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| 140 | definition accb_store ≝ helper_def_store ? accb_store_. |
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| 141 | definition accb_retrieve ≝ helper_def_retrieve ? accb_retrieve_. |
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| 142 | definition accb_arg_retrieve ≝ helper_def_retrieve ? accb_arg_retrieve_. |
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| 143 | definition dpl_store ≝ helper_def_store ? dpl_store_. |
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| 144 | definition dpl_retrieve ≝ helper_def_retrieve ? dpl_retrieve_. |
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| 145 | definition dpl_arg_retrieve ≝ helper_def_retrieve ? dpl_arg_retrieve_. |
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| 146 | definition dph_store ≝ helper_def_store ? dph_store_. |
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| 147 | definition dph_retrieve ≝ helper_def_retrieve ? dph_retrieve_. |
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| 148 | definition dph_arg_retrieve ≝ helper_def_retrieve ? dph_arg_retrieve_. |
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| 149 | definition snd_arg_retrieve ≝ helper_def_retrieve ? snd_arg_retrieve_. |
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| 150 | definition pair_reg_move : ?→?→?→?→? ≝ |
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| 151 | λup.λp : more_sem_unserialized_params up.λst : state p.λpm : pair_move up. |
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| 152 | ! r ← pair_reg_move_ ? p (regs ? st) pm; |
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| 153 | return set_regs ? r st. |
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| 154 | |
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| 155 | |
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| 156 | axiom BadPointer : String. |
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| 157 | |
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| 158 | (* this is preamble to all calls (including tail ones). The optional argument in |
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| 159 | input tells the function whether it has to save the frame for internal |
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| 160 | calls. |
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| 161 | the first element of the triple is the entry point of the function if |
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| 162 | it is an internal one, or false in case of an external one. |
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| 163 | The actual update of the pc is left to later, as it depends on |
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| 164 | serialization and on whether the call is a tail one. *) |
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| 165 | definition eval_call_block: |
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| 166 | ∀globals.∀p : params.∀p':more_sem_unserialized_params p. |
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| 167 | genv globals p → state p' → block → call_args p → call_dest p → |
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| 168 | IO io_out io_in ((step_flow p globals)×trace×(state p')) ≝ |
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| 169 | λglobals,p,p',ge,st,b,args,dest. |
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| 170 | ! fd ← (opt_to_res ? [MSG BadPointer] (find_funct_ptr ?? ge b) : IO ? io_in ?); |
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| 171 | match fd with |
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| 172 | [ Internal fn ⇒ |
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| 173 | return 〈Init … (block_id b) fn args dest, E0, st〉 |
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| 174 | | External fn ⇒ |
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| 175 | ! params ← fetch_external_args … p' fn st : IO ???; |
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| 176 | ! evargs ← check_eventval_list params (sig_args (ef_sig fn)) : IO ???; |
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| 177 | ! evres ← do_io (ef_id fn) evargs (proj_sig_res (ef_sig fn)); |
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| 178 | (* CSC: XXX bug here; I think I should split it into Byte-long |
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| 179 | components; instead I am making a singleton out of it. To be |
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| 180 | fixed once we fully implement external functions. *) |
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| 181 | let vs ≝ [mk_val ? evres] in |
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| 182 | ! st ← set_result … p' vs st : IO ???; |
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| 183 | return 〈Proceed ??, Eextcall (ef_id fn) evargs (mk_eventval ? evres), st〉 |
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| 184 | ]. |
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| 185 | |
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| 186 | axiom FailedStore: String. |
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| 187 | |
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| 188 | definition push: ∀p.state p → beval → res (state p) ≝ |
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| 189 | λp,st,v. |
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| 190 | let isp' ≝ isp ? st in |
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| 191 | do m ← opt_to_res ? (msg FailedStore) (bestorev (m … st) isp' v); |
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| 192 | let isp'' ≝ shift_pointer … isp' [[false;false;false;false;false;false;false;true]] in |
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| 193 | return set_m … m (set_isp … isp'' st). |
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| 194 | normalize elim (isp p st) // |
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| 195 | qed. |
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| 196 | |
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| 197 | definition pop: ∀p. state p → res ((state p ) × beval) ≝ |
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| 198 | λp,st. |
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| 199 | let isp' ≝ isp ? st in |
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| 200 | let isp'' ≝ neg_shift_pointer ? isp' [[false;false;false;false;false;false;false;true]] in |
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| 201 | let ist ≝ set_isp … isp'' st in |
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| 202 | do v ← opt_to_res ? (msg FailedStore) (beloadv (m ? st) isp''); |
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| 203 | OK ? 〈ist,v〉. |
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| 204 | normalize elim (isp p st) // |
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| 205 | qed. |
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| 206 | |
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| 207 | definition save_ra : ∀p. state p → cpointer → res (state p) ≝ |
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| 208 | λp,st,l. |
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| 209 | let 〈addrl,addrh〉 ≝ address_of_code_pointer l in |
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| 210 | ! st' ← push … st addrl; |
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| 211 | push … st' addrh. |
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| 212 | |
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| 213 | definition load_ra : ∀p.state p → res ((state p) × cpointer) ≝ |
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| 214 | λp,st. |
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| 215 | do 〈st',addrh〉 ← pop … st; |
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| 216 | do 〈st'',addrl〉 ← pop … st'; |
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| 217 | do pr ← code_pointer_of_address 〈addrh, addrl〉; |
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| 218 | return 〈st'', pr〉. |
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| 219 | |
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| 220 | definition flow_no_seq : ∀p,g.stmt_flow p g → Prop ≝ λp,g,flow. |
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| 221 | match flow with |
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| 222 | [ SProceed _ ⇒ False |
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| 223 | | SInit _ _ _ _ _ ⇒ False |
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| 224 | | _ ⇒ True |
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| 225 | ]. |
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| 226 | |
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| 227 | (* parameters that need full params to have type of functions, |
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| 228 | but are still independent of serialization *) |
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| 229 | record more_sem_genv_params (pp : params) : Type[1] ≝ |
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| 230 | { msu_pars :> more_sem_unserialized_params pp |
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| 231 | ; read_result: ∀globals.genv globals pp → state msu_pars → res (list beval) |
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| 232 | (* change of pc must be left to *_flow execution *) |
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| 233 | ; exec_extended: ∀globals.genv globals pp → ext_step pp → |
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| 234 | state msu_pars → |
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| 235 | IO io_out io_in ((step_flow pp globals)× trace × (state msu_pars)) |
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| 236 | ; exec_fin_extended: ∀globals.genv globals pp → ext_fin_stmt pp → |
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| 237 | state msu_pars → |
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| 238 | IO io_out io_in ((Σs.flow_no_seq pp globals s)× trace × (state msu_pars)) |
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| 239 | ; pop_frame: ∀globals.genv globals pp → state msu_pars → res (state msu_pars) |
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| 240 | }. |
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| 241 | |
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| 242 | (* parameters that are defined by serialization *) |
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| 243 | record more_sem_params (pp : params) : Type[1] ≝ |
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| 244 | { msg_pars :> more_sem_genv_params pp |
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| 245 | ; offset_of_point : code_point pp → offset |
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| 246 | ; point_of_offset : offset → option (code_point pp) |
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| 247 | ; point_of_offset_of_point : ∀pt.point_of_offset (offset_of_point pt) = Some ? pt |
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| 248 | ; offset_of_point_of_offset : ∀o.opt_All ? (eq ? o) (! pt ← point_of_offset o ; return offset_of_point pt) |
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| 249 | }. |
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| 250 | |
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| 251 | definition pointer_of_point : ∀p.more_sem_params p → |
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| 252 | cpointer→ code_point p → cpointer ≝ |
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| 253 | λp,msp,ptr,pt. |
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| 254 | mk_pointer Code (pblock ptr) ? (offset_of_point p msp pt). |
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| 255 | [ elim ptr #ptr' #EQ <EQ @pok |
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| 256 | | %] qed. |
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| 257 | |
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| 258 | axiom BadOffsetForCodePoint : String. |
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| 259 | |
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| 260 | definition point_of_pointer : |
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| 261 | ∀p.more_sem_params p → cpointer → res (code_point p) ≝ |
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| 262 | λp,msp,ptr.opt_to_res … (msg BadOffsetForCodePoint) |
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| 263 | (point_of_offset p msp (poff ptr)). |
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| 264 | |
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| 265 | lemma point_of_pointer_of_point : |
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| 266 | ∀p,msp.∀ptr : cpointer.∀pt.point_of_pointer p msp (pointer_of_point p msp ptr pt) = return pt. |
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| 267 | #p #msp #ptr #pt normalize |
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| 268 | >point_of_offset_of_point % |
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| 269 | qed. |
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| 270 | |
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| 271 | lemma pointer_of_point_block : |
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| 272 | ∀p,msp,ptr,pt.pblock (pointer_of_point p msp ptr pt) = pblock ptr. |
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| 273 | / by refl/ |
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| 274 | qed. |
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| 275 | |
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| 276 | lemma pointer_of_point_uses_block : |
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| 277 | ∀p,msp.∀ptr1,ptr2 : cpointer.∀pt.pblock ptr1 = pblock ptr2 → pointer_of_point p msp ptr1 pt = pointer_of_point p msp ptr2 pt. |
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| 278 | #p #msp ** #ty1 #bl1 #H1 #o1 #EQ1 |
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| 279 | ** #ty2 #bl2 #H2 #o2 #EQ2 |
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| 280 | #pt #EQ3 destruct normalize // |
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| 281 | qed. |
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| 282 | |
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| 283 | lemma pointer_of_point_of_pointer : |
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| 284 | ∀p,msp.∀ptr1,ptr2 : cpointer.∀pt. |
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| 285 | pblock ptr1 = pblock ptr2 → point_of_pointer p msp ptr1 = OK … pt → |
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| 286 | pointer_of_point p msp ptr2 pt = ptr1. |
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| 287 | #p #msp ** #ty1 #bl1 #H1 #o1 #EQ1 |
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| 288 | ** #ty2 #bl2 #H2 #o2 #EQ2 #pt #EQ |
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| 289 | destruct |
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| 290 | lapply (offset_of_point_of_offset p msp o1) |
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| 291 | normalize |
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| 292 | elim (point_of_offset ???) normalize [* #ABS destruct(ABS)] |
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| 293 | #pt' #EQ #EQ' destruct % |
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| 294 | qed. |
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| 295 | |
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| 296 | |
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| 297 | axiom ProgramCounterOutOfCode : String. |
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| 298 | axiom PointNotFound : String. |
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| 299 | axiom LabelNotFound : String. |
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| 300 | |
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| 301 | definition fetch_statement: ∀p : params.∀ msp : more_sem_params p.∀globals. |
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| 302 | genv globals p → cpointer → res (joint_statement p globals) ≝ |
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| 303 | λp,msp,globals,ge,ptr. |
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| 304 | ! pt ← point_of_pointer ? msp ptr ; |
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| 305 | ! fn ← fetch_function … ge ptr ; |
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| 306 | opt_to_res … (msg ProgramCounterOutOfCode) (stmt_at … (joint_if_code … fn) pt). |
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| 307 | |
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| 308 | definition pointer_of_label : ∀p : params.∀ msp : more_sem_params p.∀globals. |
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| 309 | genv globals p → cpointer → label → res cpointer ≝ |
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| 310 | λp,msp,globals,ge,ptr,lbl. |
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| 311 | ! fn ← fetch_function … ge ptr ; |
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| 312 | ! pt ← opt_to_res … [MSG LabelNotFound ; CTX ? lbl] |
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| 313 | (point_of_label … (joint_if_code … fn) lbl) ; |
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| 314 | return pointer_of_point ? msp ptr pt. |
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| 315 | |
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| 316 | definition succ_pc : ∀p : params.∀ msp : more_sem_params p. |
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| 317 | cpointer → succ p → res cpointer ≝ |
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| 318 | λp,msp,ptr,nxt. |
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| 319 | ! curr ← point_of_pointer p msp ptr ; |
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| 320 | return pointer_of_point p msp ptr (point_of_succ p curr nxt). |
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| 321 | |
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| 322 | record sem_params : Type[1] ≝ |
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| 323 | { spp :> params |
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| 324 | ; more_sem_pars :> more_sem_params spp |
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| 325 | }. |
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| 326 | |
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| 327 | (* definition address_of_label: ∀globals. ∀p:sem_params. |
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| 328 | genv globals p → pointer → label → res address ≝ |
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| 329 | λglobals,p,ge,ptr,l. |
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| 330 | do newptr ← pointer_of_label … p ? ge … ptr l ; |
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| 331 | OK … (address_of_code_pointer newptr). *) |
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| 332 | |
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| 333 | definition goto: ∀globals. ∀p:sem_params. |
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| 334 | genv globals p → label → state_pc p → res (state_pc p) ≝ |
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| 335 | λglobals,p,ge,l,st. |
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| 336 | ! newpc ← pointer_of_label ? p … ge (pc … st) l ; |
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| 337 | return set_pc … newpc st. |
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| 338 | |
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| 339 | (* |
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| 340 | definition empty_state: ∀p. more_sem_params p → state p ≝ |
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| 341 | mk_state … (empty_framesT …) |
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| 342 | *) |
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| 343 | |
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| 344 | definition next : ∀p : sem_params.succ p → state_pc p → res (state_pc p) ≝ |
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| 345 | λp,l,st. |
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| 346 | ! nw ← succ_pc ? p … (pc ? st) l ; |
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| 347 | return set_pc … nw st. |
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| 348 | |
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| 349 | axiom MissingSymbol : String. |
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| 350 | axiom FailedLoad : String. |
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| 351 | axiom BadFunction : String. |
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| 352 | axiom SuccessorNotProvided : String. |
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| 353 | |
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| 354 | (* the optional point must be given only for calls *) |
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| 355 | definition eval_step : |
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| 356 | ∀globals.∀p:sem_params. genv globals p → state p → |
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| 357 | joint_step p globals → |
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| 358 | IO io_out io_in ((step_flow p globals)×trace×(state p)) ≝ |
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| 359 | λglobals,p,ge,st,seq. |
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| 360 | match seq with |
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| 361 | [ extension c ⇒ |
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| 362 | exec_extended … ge c st |
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| 363 | | COST_LABEL cl ⇒ |
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| 364 | return 〈Proceed ??, Echarge cl, st〉 |
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| 365 | | COMMENT c ⇒ |
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| 366 | return 〈Proceed ??, E0, st〉 |
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| 367 | | LOAD dst addrl addrh ⇒ |
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| 368 | ! vaddrh ← dph_arg_retrieve … st addrh ; |
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| 369 | ! vaddrl ← dpl_arg_retrieve … st addrl; |
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| 370 | ! vaddr ← pointer_of_address 〈vaddrl,vaddrh〉; |
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| 371 | ! v ← opt_to_res … (msg FailedLoad) (beloadv (m … st) vaddr); |
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| 372 | ! st ← acca_store p … dst v st ; |
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| 373 | return 〈Proceed ??, E0, st〉 |
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| 374 | | STORE addrl addrh src ⇒ |
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| 375 | ! vaddrh ← dph_arg_retrieve … st addrh; |
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| 376 | ! vaddrl ← dpl_arg_retrieve … st addrl; |
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| 377 | ! vaddr ← pointer_of_address 〈vaddrl,vaddrh〉; |
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| 378 | ! v ← acca_arg_retrieve … st src; |
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| 379 | ! m' ← opt_to_res … (msg FailedStore) (bestorev (m … st) vaddr v); |
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| 380 | return 〈Proceed ??, E0, set_m … m' st〉 |
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| 381 | | COND src ltrue ⇒ |
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| 382 | ! v ← acca_retrieve … st src; |
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| 383 | ! b ← bool_of_beval v; |
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| 384 | if b then |
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| 385 | return 〈Redirect ?? ltrue, E0, st〉 |
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| 386 | else |
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| 387 | return 〈Proceed ??, E0, st〉 |
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| 388 | | ADDRESS ident prf ldest hdest ⇒ |
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| 389 | let addr ≝ opt_safe ? (find_symbol ?? ge ident) ? in |
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| 390 | ! 〈laddr,haddr〉 ← address_of_pointer (mk_pointer (block_region addr) addr ? zero_offset) ; |
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| 391 | ! st' ← dpl_store … ldest laddr st; |
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| 392 | ! st'' ← dph_store … hdest haddr st'; |
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| 393 | return 〈Proceed ??, E0, st''〉 |
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| 394 | | OP1 op dacc_a sacc_a ⇒ |
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| 395 | ! v ← acca_retrieve … st sacc_a; |
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| 396 | ! v ← Byte_of_val v; |
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| 397 | let v' ≝ BVByte (op1 eval op v) in |
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| 398 | ! st ← acca_store … dacc_a v' st; |
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| 399 | return 〈Proceed ??, E0, st〉 |
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| 400 | | OP2 op dacc_a sacc_a src ⇒ |
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| 401 | ! v1 ← acca_arg_retrieve … st sacc_a; |
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| 402 | ! v1 ← Byte_of_val v1; |
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| 403 | ! v2 ← snd_arg_retrieve … st src; |
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| 404 | ! v2 ← Byte_of_val v2; |
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| 405 | ! carry ← bool_of_beval (carry … st); |
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| 406 | let 〈v',carry〉 ≝ op2 eval carry op v1 v2 in |
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| 407 | let v' ≝ BVByte v' in |
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| 408 | let carry ≝ beval_of_bool carry in |
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| 409 | ! st' ← acca_store … dacc_a v' st; |
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| 410 | let st'' ≝ set_carry … carry st' in |
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| 411 | return 〈Proceed ??, E0, st''〉 |
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| 412 | | CLEAR_CARRY ⇒ |
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| 413 | let st' ≝ set_carry … BVfalse st in |
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| 414 | return 〈Proceed ??, E0, st'〉 |
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| 415 | | SET_CARRY ⇒ |
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| 416 | let st' ≝ set_carry … BVtrue st in |
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| 417 | return 〈Proceed ??, E0, st'〉 |
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| 418 | | OPACCS op dacc_a_reg dacc_b_reg sacc_a_reg sacc_b_reg ⇒ |
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| 419 | ! v1 ← acca_arg_retrieve … st sacc_a_reg; |
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| 420 | ! v1 ← Byte_of_val v1; |
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| 421 | ! v2 ← accb_arg_retrieve … st sacc_b_reg; |
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| 422 | ! v2 ← Byte_of_val v2; |
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| 423 | let 〈v1',v2'〉 ≝ opaccs eval op v1 v2 in |
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| 424 | let v1' ≝ BVByte v1' in |
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| 425 | let v2' ≝ BVByte v2' in |
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| 426 | ! st' ← acca_store … dacc_a_reg v1' st; |
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| 427 | ! st'' ← accb_store … dacc_b_reg v2' st'; |
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| 428 | return 〈Proceed ??, E0, st''〉 |
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| 429 | | POP dst ⇒ |
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| 430 | ! 〈st',v〉 ← pop p st; |
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| 431 | ! st'' ← acca_store p p dst v st'; |
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| 432 | return 〈Proceed ??, E0, st''〉 |
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| 433 | | PUSH src ⇒ |
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| 434 | ! v ← acca_arg_retrieve … st src; |
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| 435 | ! st ← push … st v; |
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| 436 | return 〈Proceed ??, E0, st〉 |
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| 437 | | MOVE dst_src ⇒ |
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| 438 | ! st ← pair_reg_move … st dst_src ; |
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| 439 | return 〈Proceed ??, E0, st〉 |
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| 440 | | CALL_ID id args dest ⇒ |
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| 441 | ! b ← opt_to_res … [MSG MissingSymbol; CTX ? id] (find_symbol ?? ge id) : IO ???; |
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| 442 | eval_call_block … ge st b args dest |
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| 443 | ]. |
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| 444 | [ cases addr // |
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| 445 | | (* TODO: to prove *) |
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| 446 | elim daemon |
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| 447 | ] qed. |
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| 448 | |
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| 449 | definition eval_step_flow : |
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| 450 | ∀globals.∀p:sem_params.step_flow p globals → |
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| 451 | succ p → stmt_flow p globals ≝ |
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| 452 | λglobals,p,cmd,nxt. |
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| 453 | match cmd with |
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| 454 | [ Redirect l ⇒ SRedirect … l |
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| 455 | | Init id fn args dest ⇒ SInit … id fn args dest nxt |
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| 456 | | Proceed ⇒ SProceed … nxt |
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| 457 | ]. |
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| 458 | |
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| 459 | definition eval_statement : ∀globals: list ident.∀p:sem_params. genv globals p → |
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| 460 | state p → joint_statement … p globals → |
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| 461 | IO io_out io_in ((stmt_flow p globals) × trace × (state p)) ≝ |
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| 462 | λglobals,p,ge,st,s. |
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| 463 | match s with |
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| 464 | [ GOTO l ⇒ return 〈SRedirect … l, E0, (st : state ?)〉 |
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| 465 | | RETURN ⇒ return 〈SEnd ??, E0, (st : state ?) 〉 |
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| 466 | | sequential seq l ⇒ |
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| 467 | ! 〈flow, tr, st〉 ← eval_step … ge st seq ; |
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| 468 | return 〈eval_step_flow … flow l, tr, st〉 |
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| 469 | | extension_fin c ⇒ |
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| 470 | ! 〈flow, tr, st〉 ← exec_fin_extended … ge c st ; |
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| 471 | return 〈pi1 … flow, tr, st〉 |
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| 472 | ]. |
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| 473 | |
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| 474 | definition eval_statement_no_seq : ∀globals: list ident.∀p:sem_params. genv globals p → |
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| 475 | state p → (Σs : joint_statement … p globals.no_seq … s) → |
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| 476 | IO io_out io_in ((Σf.flow_no_seq p globals f) × trace × (state p)) ≝ |
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| 477 | λglobals,p,ge,st,s_sig. |
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| 478 | match s_sig with |
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| 479 | [ mk_Sig s s_prf ⇒ |
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| 480 | match s return λx.no_seq p globals x → ? with |
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| 481 | [ GOTO l ⇒ λ_.return 〈«SRedirect … l,?», E0, (st : state ?)〉 |
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| 482 | | RETURN ⇒ λ_.return 〈«SEnd ??,?», E0, (st : state ?) 〉 |
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| 483 | | sequential seq l ⇒ Ⓧ |
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| 484 | | extension_fin c ⇒ λ_.exec_fin_extended … ge c st |
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| 485 | ] s_prf |
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| 486 | ]. % qed. |
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| 487 | |
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| 488 | let rec rel_io O I A B (P : A → B → Prop) (v1 : IO O I A) |
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| 489 | (v2 : IO O I B) on v1 : Prop ≝ |
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| 490 | match v1 with |
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| 491 | [ Value x ⇒ |
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| 492 | match v2 with |
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| 493 | [ Value y ⇒ |
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| 494 | P x y |
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| 495 | | _ ⇒ False |
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| 496 | ] |
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| 497 | | Wrong _ ⇒ |
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| 498 | match v2 with |
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| 499 | [ Wrong _ ⇒ True |
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| 500 | | _ ⇒ False |
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| 501 | ] |
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| 502 | | Interact o1 f1 ⇒ |
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| 503 | match v2 with |
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| 504 | [ Interact o2 f2 ⇒ |
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| 505 | ∃prf:o1 = o2.∀i.rel_io … P (f1 i) (f2 ?) |
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| 506 | | _ ⇒ False |
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| 507 | ] |
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| 508 | ]. <prf @i qed. |
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| 509 | |
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| 510 | definition IORel ≝ λO,I. |
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| 511 | mk_MonadRel (IOMonad O I) (IOMonad O I) |
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| 512 | (rel_io O I) ???. |
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| 513 | [// |
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| 514 | |#X #Y #Z #W #relin #relout #m elim m |
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| 515 | [ #o #f #IH * [#o' #f' | #v | #e] * #EQ destruct(EQ) #G |
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| 516 | #f1 #f2 #G' whd %{(refl …)} #i |
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| 517 | @(IH … (G i) f1 f2 G') |
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| 518 | | #v * [#o #f * | #v' | #e *] |
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| 519 | #Pvv' #f #g #H normalize @H @Pvv' |
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| 520 | | #e1 * [#o #f * | #v' *| #e2] * #f #g #_ % |
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| 521 | ] |
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| 522 | | #X #Y #P #Q #H #m elim m [#o #f #IH | #v | #e] * |
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| 523 | [1,4,7: #o' #f' [2,3: *] |
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| 524 | normalize * #prf destruct(prf) normalize #G |
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| 525 | % [%] normalize #i @IH @G |
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| 526 | |2,5,8: #v' [1,3: *] |
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| 527 | @H |
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| 528 | |*: #e' [1,2: *] // |
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| 529 | ] |
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| 530 | ] |
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| 531 | qed. |
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| 532 | |
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| 533 | lemma IORel_refl : ∀O,I,X,rel.reflexive X rel → |
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| 534 | reflexive ? (m_rel ?? (IORel O I) ?? rel). |
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| 535 | #O #I #X #rel #H #m elim m |
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| 536 | [ #o #f #IH whd %[%] #i normalize @IH |
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| 537 | | #v @H |
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| 538 | | #e % |
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| 539 | ] |
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| 540 | qed. |
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| 541 | |
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| 542 | lemma IORel_transitive : ∀O,I,X,Y,Z,rel1,rel2,rel3. |
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| 543 | (∀x : X.∀y : Y.∀z : Z.rel1 x y → rel2 y z → rel3 x z) → |
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| 544 | ∀m,n,o. |
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| 545 | m_rel ?? (IORel O I) … rel1 m n → |
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| 546 | m_rel ?? (IORel O I) … rel2 n o → |
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| 547 | m_rel ?? (IORel O I) … rel3 m o. |
|---|
| 548 | #O #I #X #Y #Z #rel1 #rel2 #rel3 #H #m elim m |
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| 549 | [ #o #f #IH * [#o' #f' * [#o'' #f'' | #v #_ * | #e #_ * ] | #v #x * | #e #x * ] |
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| 550 | normalize * #EQ #H1 * #EQ' #H2 destruct %[%] normalize #i @(IH ? (f' i)) // |
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| 551 | | #v * [#o #f #x * | #v' * [#o #f #_ * | #v'' |#e #_ *] | #e #x *] |
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| 552 | @H |
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| 553 | | #e * [#o #f #x * | #v #x * | #e' * [#o #f #_ * | #v #_ * | #e'']] // |
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| 554 | ] |
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| 555 | qed. |
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| 556 | |
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| 557 | lemma mr_bind' : ∀M1,M2.∀Rel : MonadRel M1 M2. |
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| 558 | ∀X,Y,Z,W,relin,relout,m,n.m_rel ?? Rel X Y relin m n → |
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| 559 | ∀f,g.(∀x,y.relin x y → m_rel ?? Rel Z W relout (f x) (g y)) → |
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| 560 | m_rel ?? Rel ?? relout (! x ← m ; f x) (! y ← n ; g y). |
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| 561 | #M1 #M2 #Rel #X #Y #Z #W #relin #relout #m #n #H #f #g #G |
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| 562 | @(mr_bind … H) @G qed. |
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| 563 | |
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| 564 | lemma m_bind_ext_eq' : ∀M : MonadProps.∀X,Y.∀m1,m2 : monad M X.∀f1,f2 : X → monad M Y. |
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| 565 | m1 = m2 → (∀x.f1 x = f2 x) → ! x ← m1 ; f1 x = ! x ← m2 ; f2 x. |
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| 566 | #M #X #Y #m1 #m2 #f1 #f2 #EQ >EQ @m_bind_ext_eq |
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| 567 | qed. |
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| 568 | |
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| 569 | lemma eval_statement_no_seq_to_normal : ∀globals,p,ge,st,s1,s2. |
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| 570 | pi1 ?? s1 = s2 → |
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| 571 | m_rel ?? (IORel ??) ?? |
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| 572 | (λx,y.pi1 ?? (\fst (\fst x)) = \fst (\fst y) ∧ |
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| 573 | \snd (\fst x) = \snd (\fst y) ∧ \snd x = \snd y) |
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| 574 | (eval_statement_no_seq globals p ge st s1) |
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| 575 | (eval_statement globals p ge st s2). |
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| 576 | #globals #p #ge #st * * [#s #n | #lbl || #c ]* |
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| 577 | #s2 #EQ destruct(EQ) |
|---|
| 578 | whd in match eval_statement; |
|---|
| 579 | whd in match eval_statement_no_seq; |
|---|
| 580 | normalize nodelta |
|---|
| 581 | [1,2: @(mr_return … (IORel ??)) /3 by pair_destruct, conj/] |
|---|
| 582 | <(m_bind_return … (exec_fin_extended ??????)) in ⊢ (???????%?); |
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| 583 | @(mr_bind … (IORel ??)) [@eq | @IORel_refl // | |
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| 584 | #x #y #EQ destruct(EQ) cases y ** #a #prf #b #c whd /3 by pair_destruct, conj/ |
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| 585 | qed. |
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| 586 | |
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| 587 | definition do_call : ∀globals: list ident.∀p:sem_params. genv globals p → |
|---|
| 588 | state p → Z → joint_internal_function globals p → call_args p → |
|---|
| 589 | res (state_pc p) ≝ |
|---|
| 590 | λglobals,p,ge,st,id,fn,args. |
|---|
| 591 | ! st' ← setup_call … (joint_if_stacksize … fn) (joint_if_params … fn) |
|---|
| 592 | args st ; |
|---|
| 593 | let regs ≝ foldr ?? (allocate_local p p) (regs … st) (joint_if_locals … fn) in |
|---|
| 594 | let l' ≝ joint_if_entry … fn in |
|---|
| 595 | let st' ≝ set_regs p regs st in |
|---|
| 596 | let pointer_in_fn ≝ mk_pointer Code (mk_block Code id) ? (mk_offset 0) in |
|---|
| 597 | let pc ≝ pointer_of_point ? p … pointer_in_fn l' in |
|---|
| 598 | return mk_state_pc ? st' pc. % qed. |
|---|
| 599 | |
|---|
| 600 | definition eval_stmt_flow : ∀globals: list ident.∀p:sem_params. genv globals p → |
|---|
| 601 | state_pc p → stmt_flow p globals → IO io_out io_in (state_pc p) ≝ |
|---|
| 602 | λglobals,p,ge,st,flow. |
|---|
| 603 | match flow with |
|---|
| 604 | [ SRedirect l ⇒ goto … ge l st |
|---|
| 605 | | SProceed nxt ⇒ next ? nxt st |
|---|
| 606 | | SInit id fn args dest nxt ⇒ |
|---|
| 607 | ! ra ← succ_pc ? p … (pc … st) nxt ; |
|---|
| 608 | let st' ≝ set_no_pc … (set_frms … (save_frame … ra dest st) st) st in |
|---|
| 609 | do_call ?? ge st' id fn args |
|---|
| 610 | | STailInit id fn args ⇒ |
|---|
| 611 | do_call … ge st id fn args |
|---|
| 612 | | SEnd ⇒ |
|---|
| 613 | ! 〈st,ra〉 ← fetch_ra … st ; |
|---|
| 614 | ! st' ← pop_frame … ge st; |
|---|
| 615 | return mk_state_pc ? st' ra |
|---|
| 616 | ]. |
|---|
| 617 | |
|---|
| 618 | definition eval_stmt_flow_no_seq : ∀globals: list ident.∀p:sem_params. genv globals p → |
|---|
| 619 | state p → Z → (Σf:stmt_flow p globals.flow_no_seq … f) → |
|---|
| 620 | IO io_out io_in (state_pc p) ≝ |
|---|
| 621 | λglobals,p,ge,st,id,flow_sig. |
|---|
| 622 | match flow_sig with |
|---|
| 623 | [ mk_Sig flow prf ⇒ |
|---|
| 624 | match flow return λx.flow_no_seq p globals x → ? with |
|---|
| 625 | [ SRedirect l ⇒ λ_. |
|---|
| 626 | ! newpc ← pointer_of_label ? p … ge |
|---|
| 627 | (mk_pointer Code (mk_block Code id) ? (mk_offset 0)) l ; |
|---|
| 628 | return mk_state_pc … st newpc |
|---|
| 629 | | STailInit id fn args ⇒ λ_. |
|---|
| 630 | do_call … ge st id fn args |
|---|
| 631 | | SEnd ⇒ λ_. |
|---|
| 632 | ! 〈st,ra〉 ← fetch_ra … st ; |
|---|
| 633 | ! st' ← pop_frame … ge st; |
|---|
| 634 | return mk_state_pc ? st' ra |
|---|
| 635 | | _ ⇒ Ⓧ |
|---|
| 636 | ] prf |
|---|
| 637 | ]. % qed. |
|---|
| 638 | |
|---|
| 639 | lemma pointer_of_label_eq_with_id : |
|---|
| 640 | ∀g.∀p : sem_params.∀ge : genv g p.∀ptr1,ptr2 : cpointer.∀lbl. |
|---|
| 641 | block_id (pblock ptr1) = block_id (pblock ptr2) → |
|---|
| 642 | pointer_of_label ? p ? ge ptr1 lbl = pointer_of_label ? p ? ge ptr2 lbl. |
|---|
| 643 | #g #p #ge |
|---|
| 644 | ** #ty1 * #r1 #id1 #H1 inversion H1 [#s || #s] #id #EQ1 #EQ2 #EQ3 #o #EQ4 destruct |
|---|
| 645 | ** #ty2 * #r2 #id2 #H2 inversion H2 |
|---|
| 646 | [1,3: #s] #id2' #EQ5 #EQ6 #EQ7 #o2 #EQ8 #lbl #EQ9 destruct % |
|---|
| 647 | qed. |
|---|
| 648 | |
|---|
| 649 | |
|---|
| 650 | lemma eval_stmt_flow_no_seq_to_normal : ∀g.∀p : sem_params.∀ge.∀st : state_pc p. |
|---|
| 651 | ∀id.∀s1 : Σs.flow_no_seq p g s.∀s2. |
|---|
| 652 | pi1 … s1 = s2 → block_id (pblock (pc … st)) = id → |
|---|
| 653 | (eval_stmt_flow_no_seq g p ge st id s1) = |
|---|
| 654 | (eval_stmt_flow g p ge st s2). |
|---|
| 655 | #g#p#ge#st#id' |
|---|
| 656 | **[#lbl|#nxt*|#id#fn#args#dest#n*|#id#fn#args]* |
|---|
| 657 | #s2 #EQ #EQ' destruct(EQ) |
|---|
| 658 | whd in match eval_stmt_flow_no_seq; normalize nodelta |
|---|
| 659 | whd in match eval_stmt_flow; normalize nodelta |
|---|
| 660 | [2,3: %] |
|---|
| 661 | whd in match goto; normalize nodelta |
|---|
| 662 | whd in match set_pc; normalize nodelta |
|---|
| 663 | >pointer_of_label_eq_with_id [% | >EQ' % |] |
|---|
| 664 | qed. |
|---|
| 665 | |
|---|
| 666 | definition eval_state : ∀globals: list ident.∀p:sem_params. genv globals p → |
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| 667 | state_pc p → IO io_out io_in (trace × (state_pc p)) ≝ |
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| 668 | λglobals,p,ge,st. |
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| 669 | ! s ← fetch_statement ? p … ge (pc … st) : IO ???; |
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| 670 | ! 〈flow, tr, st_npc〉 ← eval_statement … ge st s; |
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| 671 | let st ≝ set_no_pc … st_npc st in |
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| 672 | ! st ← eval_stmt_flow … ge st flow; |
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| 673 | return 〈tr, st〉. |
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| 674 | |
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| 675 | definition is_final: ∀globals:list ident. ∀p: sem_params. |
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| 676 | genv globals p → cpointer → state_pc p → option int ≝ |
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| 677 | λglobals,p,ge,exit,st.res_to_opt ? ( |
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| 678 | do s ← fetch_statement ? p … ge (pc … st); |
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| 679 | match s with |
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| 680 | [ RETURN ⇒ |
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| 681 | do 〈st,ra〉 ← fetch_ra … st; |
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| 682 | if eq_pointer ra exit then |
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| 683 | do vals ← read_result … ge st ; |
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| 684 | Word_of_list_beval vals |
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| 685 | else |
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| 686 | Error ? [ ] |
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| 687 | | _ ⇒ Error ? [ ]]). |
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| 688 | |
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| 689 | record evaluation_parameters : Type[1] ≝ |
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| 690 | { globals: list ident |
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| 691 | ; sparams:> sem_params |
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| 692 | ; exit: cpointer |
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| 693 | ; genv2: genv globals sparams |
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| 694 | }. |
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| 695 | |
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| 696 | definition joint_exec: trans_system io_out io_in ≝ |
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| 697 | mk_trans_system … evaluation_parameters (λG. state_pc G) |
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| 698 | (λG.is_final (globals G) G (genv2 G) (exit G)) |
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| 699 | (λG.eval_state (globals G) G (genv2 G)). |
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| 700 | |
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| 701 | definition make_global : |
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| 702 | ∀pars: sem_params. |
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| 703 | joint_program pars → evaluation_parameters |
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| 704 | ≝ |
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| 705 | λpars. |
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| 706 | (* Invariant: a -1 block is unused in common/Globalenvs *) |
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| 707 | let b ≝ mk_block Code (-1) in |
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| 708 | let ptr ≝ mk_pointer Code b ? (mk_offset 0) in |
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| 709 | (λp. mk_evaluation_parameters |
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| 710 | (prog_var_names … p) |
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| 711 | (mk_sem_params … pars) |
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| 712 | ptr |
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| 713 | (globalenv Genv … p) |
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| 714 | ). |
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| 715 | % qed. |
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| 716 | |
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| 717 | axiom ExternalMain: String. |
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| 718 | |
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| 719 | definition make_initial_state : |
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| 720 | ∀pars: sem_params. |
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| 721 | ∀p: joint_program … pars. res (state_pc pars) ≝ |
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| 722 | λpars,p. |
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| 723 | let sem_globals ≝ make_global pars p in |
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| 724 | let ge ≝ genv2 sem_globals in |
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| 725 | let m ≝ init_mem Genv … p in |
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| 726 | let 〈m,spb〉 ≝ alloc … m 0 external_ram_size XData in |
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| 727 | let 〈m,ispb〉 ≝ alloc … m 0 internal_ram_size XData in |
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| 728 | let dummy_pc ≝ mk_pointer Code (mk_block Code (-1)) ? (mk_offset 0) in |
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| 729 | let spp ≝ mk_pointer XData spb ? (mk_offset external_ram_size) in |
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| 730 | let ispp ≝ mk_pointer XData ispb ? (mk_offset 47) in |
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| 731 | let main ≝ prog_main … p in |
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| 732 | let st0 ≝ mk_state pars (empty_framesT …) spp ispp BVfalse (empty_regsT … spp) m in |
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| 733 | (* use exit sem_globals as ra and call_dest_for_main as dest *) |
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| 734 | let st0' ≝ set_frms ? (save_frame … (exit sem_globals) (call_dest_for_main … pars) st0) st0 in |
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| 735 | let st_pc0 ≝ mk_state_pc ? st0' dummy_pc in |
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| 736 | ! main_block ← opt_to_res … [MSG MissingSymbol; CTX ? main] (find_symbol ?? ge main) ; |
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| 737 | ! main_fd ← opt_to_res ? [MSG BadPointer] (find_funct_ptr ?? ge main_block) ; |
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| 738 | match main_fd with |
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| 739 | [ Internal fn ⇒ |
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| 740 | do_call ?? ge st_pc0 (block_id main_block) fn (call_args_for_main … pars) |
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| 741 | | External _ ⇒ Error … [MSG ExternalMain] (* External main not supported: why? *) |
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| 742 | ]. [5: cases ispb * |6: cases spb * ] (* without try it fails! *) try // |
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| 743 | qed. |
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| 744 | |
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| 745 | definition joint_fullexec : |
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| 746 | ∀pars:sem_params.fullexec io_out io_in ≝ |
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| 747 | λpars.mk_fullexec ??? joint_exec (make_global pars) (make_initial_state pars). |
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