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p4_e_progressScript.sml
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p4_e_progressScript.sml
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open HolKernel boolLib liteLib simpLib Parse bossLib;
open arithmeticTheory stringTheory containerTheory pred_setTheory
listTheory finite_mapTheory;
open p4Lib;
open blastLib bitstringLib;
open p4Theory;
open p4_auxTheory;
open bitstringTheory;
open wordsTheory;
open optionTheory;
open sumTheory;
open stringTheory;
open ottTheory;
open pairTheory;
open rich_listTheory;
open arithmeticTheory;
open alistTheory;
open numeralTheory;
open intLib;
open p4_e_subject_reductionTheory;
fun OPEN_EXP_RED_TAC exp_term =
(Q.PAT_X_ASSUM `e_red c scope scopest ^exp_term exp2 fr` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once e_red_cases] thm)))
fun OPEN_STMT_RED_TAC stm_term =
(Q.PAT_X_ASSUM `stmt_red ct (ab, gsl,[(fun,[^stm_term],gam)],st) stat` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once stmt_red_cases] thm)))
fun OPEN_V_TYP_TAC v_term =
(Q.PAT_X_ASSUM `v_typ v_term t bll` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once v_typ_cases] thm)))
fun OPEN_EXP_TYP_TAC exp_term =
(Q.PAT_X_ASSUM ` e_typ (t1,t2) t ^exp_term ta bll` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once e_typ_cases] thm)))
val _ = new_theory "p4_e_progress";
(****** Type progress for expressions ******)
val prog_exp_def = Define `
prog_exp (e) (ty:'a itself) =
!gscope (scopest:scope list) t_scope_list t_scope_list_g
T_e tau b (c:'a ctx) order delta_g delta_b (delta_t:delta_t) delta_x f Prs_n.
type_scopes_list gscope t_scope_list_g ∧
type_scopes_list scopest t_scope_list ∧
star_not_in_sl (scopest) ∧
~(is_const e) ∧
e_typ (t_scope_list_g,t_scope_list) T_e e tau b ∧
(T_e = (order, f, (delta_g, delta_b, delta_x, delta_t))) /\
WT_c c order t_scope_list_g delta_g delta_b delta_x delta_t Prs_n ==>
?e' framel. e_red c gscope scopest e e' framel
`;
(****** Type progress for expression list ******)
val prog_exp_list_def = Define `
prog_exp_list (l : e list) (ty:'a itself) =
!(e : e). MEM e l ==> prog_exp (e) (ty:'a itself)
`;
(****** Type progress for strexp list ******)
val prog_strexp_list_def = Define `
prog_strexp_list (l : (string#e) list) (ty:'a itself)
= ! (e:e) . MEM e (SND (UNZIP l)) ==> prog_exp(e) (ty:'a itself)
`;
(****** Type progress for str e tup list ******)
val prog_strexp_tup_def = Define `
prog_strexp_tup (tup : (string#e)) (ty:'a itself)
= prog_exp ((SND tup)) (ty:'a itself)
`;
Theorem INDEX_FIND_concat3:
! P sl gsl.
INDEX_FIND 0 P (sl ⧺ gsl) = NONE <=>
(INDEX_FIND 0 P (sl) = NONE /\
INDEX_FIND 0 P (gsl) = NONE)
Proof
rw[] >>
rpt strip_tac>>
fs[INDEX_FIND_NONE_EXISTS]
QED
Theorem find_topmost_map_concat_none:
! sl gsl v .
find_topmost_map (sl ⧺ gsl) v = NONE <=>
(find_topmost_map (sl) v = NONE /\
find_topmost_map (gsl) v = NONE )
Proof
rpt strip_tac >>
fs[find_topmost_map_def] >>
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc v)) (sl ⧺ gsl)` >>
fs[] >| [
IMP_RES_TAC INDEX_FIND_concat3 >>
gvs[]
,
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc v)) sl` >>
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc v)) gsl` >>
gvs[] >>
IMP_RES_TAC INDEX_FIND_concat3 >>
gvs[]
]
QED
val topmost_map_concat_none = prove ( ``
! sl gsl v .
topmost_map (sl ⧺ gsl) v = NONE <=>
(topmost_map (sl) v = NONE /\
topmost_map (gsl) v = NONE ) ``,
rpt strip_tac >>
fs[topmost_map_def] >>
fs[] >>
Cases_on `find_topmost_map (sl ⧺ gsl) v` >>
fs[] >| [
IMP_RES_TAC find_topmost_map_concat_none >>
gvs[]
,
PairCases_on `x` >> fs[] >>
Cases_on `find_topmost_map sl v` >>
Cases_on `find_topmost_map gsl v` >>
fs[] >>
TRY (PairCases_on `x` >> fs[] ) >>
IMP_RES_TAC find_topmost_map_concat_none >>
gvs[]
]
);
val topmost_map_not_none = prove ( ``
! l v x .
topmost_map l v = SOME x ==>
~(ALOOKUP x v = NONE) ``,
fs[topmost_map_def] >>
fs[find_topmost_map_def] >>
REPEAT STRIP_TAC >>
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc v)) l` >>
fs[] >>
PairCases_on `x'` >>
gvs[] >>
gvs[INDEX_FIND_EQ_SOME_0]
);
Theorem lookup_map_concat_none:
! sl gsl v .
lookup_map (sl ⧺ gsl) (v) = NONE <=>
(lookup_map (sl) (v) = NONE /\
lookup_map (gsl) (v) = NONE)
Proof
rpt strip_tac >>
fs[lookup_map_def] >>
fs[] >>
Cases_on `topmost_map (sl ⧺ gsl) v` >>
fs[] >| [
IMP_RES_TAC topmost_map_concat_none >>
gvs[]
,
Cases_on `topmost_map sl v` >>
Cases_on `topmost_map gsl v` >>
fs[] >>
IMP_RES_TAC topmost_map_concat_none >>
gvs[] >>
Cases_on `ALOOKUP x v` >>
Cases_on `ALOOKUP x' v` >>
gvs[] >>
IMP_RES_TAC topmost_map_not_none
]
QED
Theorem R_implies_ALOOKUP_scopes:
! R x v t sc tc.
similar R tc sc ==>
(((SOME t = ALOOKUP tc x) ==>
(? v . SOME v = ALOOKUP sc x /\ R t v )) /\
((SOME v = ALOOKUP sc x) ==>
(? t . SOME t = ALOOKUP tc x /\ R t v )))
Proof
Induct_on `tc` >>
Induct_on `sc` >>
fs[ALOOKUP_def, similar_def] >>
rpt strip_tac >| [
fs[similar_def] >>
PairCases_on `h` >>
PairCases_on `h'` >>
gvs[] >>
Cases_on `h'0 = x` >> fs[]
,
PairCases_on `h` >>
PairCases_on `h'` >>
gvs[] >>
fs[ALOOKUP_def, similar_def] >>
Cases_on `h'0 = x` >>
fs[] >> gvs[] >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`R`, `x`, `v`, `t'`, `sc`])) >> gvs[] >>
srw_tac [SatisfySimps.SATISFY_ss][]
]
QED
val R_ALOOKUP_NONE_scopelist = prove (``
! R v x t scl tcl.
similarl R tcl scl ==>
((NONE = find_topmost_map scl x) <=>
(NONE = find_topmost_map tcl x ) )``,
reverse (
Induct_on `scl` >>
Induct_on `tcl`
) >- (
rpt strip_tac >>
SIMP_TAC list_ss [find_topmost_map_def] >>
fs[INDEX_FIND_def] >>
Cases_on `(ALOOKUP h' x)` >>
Cases_on `(ALOOKUP h x)` >>
fs[] >| [
(* use IH *)
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`R`, `x`, `tcl`])) >>
`similarl R tcl scl` by fs[similarl_def] >>
gvs[] >>
fs[find_topmost_map_def] >>
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc x)) scl` >> fs[] >>
Cases_on `INDEX_FIND 1 (λsc. IS_SOME (ALOOKUP sc x)) scl` >> fs[] >>
Cases_on `INDEX_FIND 0 (λsc. IS_SOME (ALOOKUP sc x)) tcl` >> fs[] >>
Cases_on `INDEX_FIND 1 (λsc. IS_SOME (ALOOKUP sc x)) tcl` >> fs[] >>
IMP_RES_TAC P_NONE_hold >>
IMP_RES_TAC P_NONE_hold2 >> gvs[]
,
`similar R h h'` by fs[similarl_def] >>
IMP_RES_TAC R_ALOOKUP_NONE_scopes >> gvs[]
,
`similar R h h'` by fs[similarl_def] >>
IMP_RES_TAC R_ALOOKUP_NONE_scopes >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`x`])) >> gvs[]
]
)>>
RW_TAC list_ss [similarl_def, similar_def] >>
EVAL_TAC
);
Theorem R_implies_topmost_map_scopesl:
!R x l1 l2 scl tcl.
similarl R tcl scl ⇒
(SOME l1 = topmost_map tcl x ⇒
? l2'. SOME l2' = topmost_map scl x ∧ similar R l1 l2') /\
(SOME l2 = topmost_map scl x ⇒
? l1'. SOME l1' = topmost_map tcl x ∧ similar R l1' l2)
Proof
reverse (Induct_on `tcl` >>
Induct_on `scl` ) >- (
rpt strip_tac >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`R`, `x`,`l1`, `l2`, `scl`])) >>
fs[topmost_map_def] >>
gvs[] >>
Cases_on `find_topmost_map (h'::tcl) x` >> gvs[] >>
Cases_on `find_topmost_map (h::scl) x` >> gvs[] >| [
IMP_RES_TAC R_ALOOKUP_NONE_scopelist >> gvs[]
,
PairCases_on `x'` >>
PairCases_on `x''` >>
gvs[] >>
IMP_RES_TAC R_find_topmost_map_scopesl >> gvs[] >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`(x'0,l1)`])) >>
gvs[]
,
IMP_RES_TAC R_ALOOKUP_NONE_scopelist >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`])) >>
gvs[]
,
PairCases_on `x'` >>
PairCases_on `x''` >>
gvs[] >>
IMP_RES_TAC R_find_topmost_map_scopesl >> gvs[] >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`(x'0,x'1)`])) >>
gvs[]
]
) >>
fs[topmost_map_def, find_topmost_map_def, similar_def, similarl_def] >>
fs[INDEX_FIND_def]
QED
Theorem R_implies_lookup_map_scopesl:
! R v x t (scl :(varn # β) list list) (tcl :(varn # α) list list).
similarl R tcl scl ==>
((SOME t = lookup_map tcl x ==> ? v . SOME v = lookup_map scl x /\ (R t v)) /\
(SOME v = lookup_map scl x ==> ? t . SOME t = lookup_map tcl x /\ (R t v)))
Proof
REPEAT STRIP_TAC >>
fs[lookup_map_def] >>
Cases_on `topmost_map tcl x` >>
Cases_on `topmost_map scl x` >>
gvs[] >| [
IMP_RES_TAC R_implies_topmost_map_scopesl >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`x'`])) >>
gvs[]
,
(* we need to show that x'' and x' have a relation between them*)
IMP_RES_TAC R_implies_topmost_map_scopesl >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`x'`])) >>
gvs[] >>
(* then if we look up, it should also have a relation *)
Cases_on `ALOOKUP x' x` >>
Cases_on `ALOOKUP x'' x` >> gvs[] >| [
drule R_ALOOKUP_NONE_scopes >>
strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`t`,`x`])) >>
gvs[]
,
drule R_ALOOKUP_scopes >>
strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x''''`, `x`, `t`])) >>
gvs[]
]
,
(* third subgoal *)
IMP_RES_TAC R_implies_topmost_map_scopesl >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`x'`])) >>
gvs[]
,
(* we need to show that x'' and x' have a relation between them*)
IMP_RES_TAC R_implies_topmost_map_scopesl >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`,`x'`])) >>
gvs[] >>
(* then if we look up, it should also have a relation *)
Cases_on `ALOOKUP x' x` >>
Cases_on `ALOOKUP x'' x` >> gvs[] >| [
drule R_ALOOKUP_NONE_scopes >>
strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`t`,`x`])) >>
gvs[]
,
drule R_ALOOKUP_scopes >>
strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`v`, `x`, `x'''`])) >>
gvs[]
]
]
QED
Theorem varn_tau_exists_in_scl_tscl:
! gsl gtsl sl tsl varn v tau .
type_scopes_list gsl gtsl ∧
type_scopes_list sl tsl ==>
((SOME v = lookup_vexp2 sl gsl varn ==> ?tau. SOME tau = lookup_tau tsl gtsl varn /\
v_typ v (t_tau tau) F)
/\
(SOME tau = lookup_tau tsl gtsl varn ==> ?v. SOME v = lookup_vexp2 sl gsl varn /\
v_typ v (t_tau tau) F))
Proof
gvs[lookup_tau_def] >>
fs[lookup_vexp2_def] >>
REPEAT STRIP_TAC >>
(subgoal `type_scopes_list (sl ⧺ gsl) (tsl ⧺ gtsl)` >-
IMP_RES_TAC type_scopes_list_APPEND) >>
Cases_on `lookup_map (sl ⧺ gsl) varn` >>
Cases_on `lookup_map (tsl ⧺ gtsl) varn` >>
fs[type_scopes_list_def] >> rw[] >| [
IMP_RES_TAC R_implies_lookup_map_scopesl >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`varn`, `x`])) >>
gvs[]
,
PairCases_on `x` >> gvs[] >>
drule R_lookup_map_scopesl >>
rpt strip_tac >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x'`, `varn`, `(v,x1)`])) >>
gvs[] >>
PairCases_on ‘x'’ >> gvs[]
,
drule R_implies_lookup_map_scopesl >>
rpt strip_tac >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x`, `varn`, `x`])) >>
gvs[]
,
PairCases_on `x` >> gvs[] >>
drule R_lookup_map_scopesl >>
rpt strip_tac >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x'`, `varn`, `(x0,x1)`])) >>
PairCases_on ‘x'’ >> gvs[]
]
QED
Theorem star_tau_exists_in_scl_tscl:
! gsl gtsl sl tsl v tau f.
star_not_in_sl (sl) /\
type_scopes_list gsl gtsl ==>
(SOME tau = find_star_in_globals gtsl (varn_star f) ⇒
?v. SOME v = lookup_vexp2 sl gsl (varn_star f))
/\
(SOME v = lookup_vexp2 sl gsl (varn_star f) ⇒
?tau. SOME tau = find_star_in_globals gtsl (varn_star f))
Proof
gvs[lookup_tau_def] >>
fs[lookup_vexp2_def] >>
fs[find_star_in_globals_def] >>
rpt strip_tac >| [
Cases_on `lookup_map gtsl (varn_star f)` >>
Cases_on `lookup_map (sl ⧺ gsl) (varn_star f)` >>
gvs[] >| [
`lookup_map (sl) (varn_star f) = NONE /\
lookup_map (gsl) (varn_star f) = NONE` by
(IMP_RES_TAC lookup_map_concat_none >> gvs[] ) >>
fs[star_not_in_sl_def] >>
gvs[type_scopes_list_def] >>
drule R_implies_lookup_map_scopesl >>
rpt strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`x`, `(varn_star f)`, `(x0,x1)`])) >>
gvs[]
,
PairCases_on `x'` >> gvs[]
]
,
Cases_on `lookup_map gtsl (varn_star f)` >>
Cases_on `lookup_map (sl ⧺ gsl) (varn_star f)` >>
gvs[] >>
PairCases_on `x` >>
gvs[] >>
IMP_RES_TAC lookup_map_in_gsl_lemma >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`v`, `x1`, `gsl`, `f`])) >> gvs[] >>
gvs[type_scopes_list_def] >>
drule R_implies_lookup_map_scopesl >>
rpt strip_tac >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`v`, `(varn_star f)`, `(v,x1)`])) >>
gvs[]
]
QED
val FIND_in_typl_imp_rec = prove ( ``
!l x0 tau s.
FIND (λ(xm,tm). xm = s) (MAP (λ(x_,v_,tau_). (x_,tau_)) l) = SOME (x0,tau) ==>
(∃v'. FIND (λ(k,v). k = s) (MAP (λ(x_,v_,tau_). (x_,v_)) l) = SOME (s,v') /\
s=x0 )
``,
Induct_on `l` >>
fs[FIND_def] >>
fs[INDEX_FIND_def] >>
REPEAT STRIP_TAC >>
PairCases_on `z` >>
PairCases_on `h` >>
gvs[] >>
Cases_on `h0 = s` >>
gvs[] >>
ASSUME_TAC (INST_TYPE [``:'a`` |-> ``:('a#γ)``] P_hold_on_next) >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`0`,`(MAP (λ(x_,v_,tau_). (x_,tau_)) l)`,
`(λ(xm,tm). xm = s)`, `(z0,x0,tau)`])) >>
gvs[GSYM ADD1] >>
RES_TAC >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x0`,`tau`])) >>
fs[] >>
PairCases_on `z` >> gvs[] >>
IMP_RES_TAC P_implies_next >>
srw_tac [SatisfySimps.SATISFY_ss][]
);
Theorem correct_field_type_FIND:
! s tau l .
correct_field_type (MAP (λ(x_,v_,tau_). (x_,tau_)) l) s tau ==>
∃v'. FIND (λ(k,v). k = s) (MAP (λ(x_,v_,tau_). (x_,v_)) l) = SOME (s,v')
Proof
gvs[correct_field_type_def] >>
fs[tau_in_rec_def] >>
REPEAT STRIP_TAC >>
Cases_on `FIND (λ(xm,tm). xm = s) (MAP (λ(x_,v_,tau_). (x_,tau_)) l)` >>
fs[] >>
PairCases_on `x` >> fs[] >>
Cases_on `x1 = tau` >> gvs[] >>
IMP_RES_TAC FIND_in_typl_imp_rec >>
srw_tac [SatisfySimps.SATISFY_ss][]
QED
Theorem is_const_val_exsist:
! e . is_const e <=> (?v . e = e_v v)
Proof
Cases_on `e` >>
fs[is_const_def]
QED
(*
(*val oper = ``binop_mul``;*)
fun bv_bv_prog (oper) = prove (``
! bitv bitv'.
bs_width bitv > 0 /\ bs_width bitv <= 64 /\
bs_width bitv = bs_width bitv' /\
is_bv_op(^oper) ==>
∃bitv''. bitv_binop (^oper) bitv bitv' = SOME bitv'' ``,
fs[is_bv_op_def] >>
Cases_on `bitv` >>
Cases_on `bitv'` >>
rpt strip_tac >>
gvs[bs_width_def] >>
IMP_RES_TAC bit_range >>
fs[] >>
fs[bitv_binop_def] >>
RW.ONCE_RW_TAC [bitv_binop_inner_def] >>
fs[] >>
srw_tac [numSimps.SUC_FILTER_ss][]
);
val bv_bv_opl = map bv_bv_prog [ ``binop_mul``, ``binop_add``, ``binop_div``, ``binop_mod``,
``binop_sub``, ``binop_shl``, ``binop_shr``, `` binop_and``,
``binop_or``, ``binop_xor`` ];
*)
val bv_bv_prog = prove (``
! bitv bitv' oper.
bs_width bitv > 0 /\ bs_width bitv <= 64 /\
bs_width bitv = bs_width bitv' /\
is_bv_op(oper) ==>
∃bitv''. bitv_binop (oper) bitv bitv' = SOME bitv'' ``,
fs[is_bv_op_def] >>
Cases_on `bitv` >>
Cases_on `bitv'` >>
rpt strip_tac >>
gvs[bs_width_def] >>
drule bit_range >>
rpt strip_tac >>
RES_TAC >>
gvs[NOT_CLAUSES] >>
fs[bitv_binop_def] >>
RW.ONCE_RW_TAC [bitv_binop_inner_def] >>
fs[] >>
srw_tac [numSimps.SUC_FILTER_ss][]
);
val bv_bv_bool_prog = prove (``
! bitv bitv' op.
bs_width bitv > 0 /\
bs_width bitv <= 64 /\
bs_width bitv = bs_width bitv' /\
is_bv_bool_op(op) ==>
∃b. bitv_binpred op bitv bitv' = SOME b
``,
fs[is_bv_bool_op_def] >>
Cases_on `bitv` >>
Cases_on `bitv'` >>
rpt strip_tac >>
gvs[bitv_binpred_def, bs_width_def] >>
rpt strip_tac >>
gvs[bs_width_def] >>
drule bit_range >>
rpt strip_tac >>
RES_TAC >>
gvs[NOT_CLAUSES] >>
fs[] >>
RW.ONCE_RW_TAC [bitv_binpred_inner_def] >>
fs[] >>
srw_tac [numSimps.SUC_FILTER_ss][]
);
val not_const_unred_mem = prove ( `` ! el .
¬is_consts el ==>
(? i . i <LENGTH el /\ unred_mem_index el = SOME i) ``,
REPEAT STRIP_TAC>>
FULL_SIMP_TAC std_ss [unred_mem_index_def] >>
Cases_on `unred_mem el` >>
fs[is_consts_def, unred_mem_def] >>
FULL_SIMP_TAC (std_ss) [combinTheory.o_DEF] >>
IMP_RES_TAC not_index_none_exist >>
PairCases_on `x` >>
gvs[INDEX_FIND_EQ_SOME_0]
);
val unred_mem_is_const = prove (``
!l i. i < LENGTH l /\ unred_mem_index l = SOME i ==>
~is_const (EL i (l)) `` ,
REPEAT STRIP_TAC >>
fs[unred_mem_index_def] >>
Cases_on `unred_mem l` >> gvs[] >>
PairCases_on `x` >> gvs[] >>
gvs[unred_mem_def] >>
IMP_RES_TAC INDEX_FIND_EQ_SOME_0 >>
gvs[]
);
val notlval_case = prove ( ``
!e .
¬is_e_lval (e) ==>
(¬is_const (e) \/ (?v. e = e_v v)) ``,
Induct_on `e` >>
gvs[is_e_lval_def, is_const_def, get_lval_of_e_def] >>
gvs[]
)
val unred_arg_eq = prove ( ``! dl el .
unred_arg_index dl el ≠ NONE <=>
? i . unred_arg_index dl el = SOME i ``,
fs[unred_arg_index_def] >>
rpt strip_tac >>
Cases_on `find_unred_arg dl el` >>
gvs[] >>
PairCases_on `x` >> gvs[]
);
val correct_field_access = prove ( ``
! x_v_tau_list s tau' v boolv.
correct_field_type (MAP (λ(x_,v_,tau_). (x_,tau_)) x_v_tau_list) s tau' /\
(v = v_header boolv \/ v = v_struct )==>
∃v'. acc_f (v (MAP (λ(x_,v_,tau_). (x_,v_)) x_v_tau_list)) s =
SOME v' ``,
REPEAT STRIP_TAC >>
gvs[] >>
fs[acc_f_def] >>
CASE_TAC >>
IMP_RES_TAC correct_field_type_FIND >>
gvs[]
);
val e_T_get_lval = prove ( ``
! e gtsl tsl gscope scopest T_e tau .
e_typ (gtsl,tsl) T_e e (tau) T ==>
? v . get_lval_of_e e = SOME (v) ``,
REPEAT STRIP_TAC >>
IMP_RES_TAC e_lval_tlval >>
fs[is_e_lval_def] >>
Cases_on `get_lval_of_e e` >> gvs[]
);
val is_get_lval = prove ( ``
!e . is_e_lval e <=> ? lval . get_lval_of_e e = SOME lval ``,
fs[is_e_lval_def] >>
STRIP_TAC >>
Cases_on `get_lval_of_e e` >> gvs[]
);
Theorem INDEX_FIND_same_list_2:
! l s i i' s' s'' v t .
INDEX_FIND 0 (λ(k,v). k = s) (MAP (λ(x,v,t). (x,v)) l) = SOME (i,s',v) /\
INDEX_FIND 0 (λ(xm,tm). xm = s) (MAP (λ(x,v,t). (x,t)) l) = SOME (i',s'',t) ==>
(i = i' /\ s = s'' /\ s = s /\ i < LENGTH l /\
v = EL i (MAP (λ(x,v,t). v) l) /\
t = EL i (MAP (λ(x,v,t). t) l)
)
Proof
REPEAT GEN_TAC >>
STRIP_TAC >>
subgoal `
i' < LENGTH (MAP (λ(x,v,t). (x,t)) l) /\
i < LENGTH (MAP (λ(x,v,t). (x,v)) l) /\
EL i' (MAP (λ(x,v,t). (x,t)) l) = (s'',t) /\
EL i (MAP (λ(x,v,t). (x,v)) l) = (s' ,v) /\
(λ(xm,tm). xm = s) (s'',t) /\
(λ(k ,v ). k = s) (s' ,v)` >- (IMP_RES_TAC INDEX_FIND_EQ_SOME_0 >> gvs[]) >>
gvs[] >>
IMP_RES_TAC INDEX_FIND_same_list >> gvs[] >>
gvs[EL_simp5]
QED
val FIND_same_list = prove (``
! l s s' s'' v t .
FIND (λ(k,v). k = s) (MAP (λ(x,v,t). (x,v)) l) =
SOME (s',v) /\
FIND (λ(xm,tm). xm = s) (MAP (λ(x,v,t). (x,t)) l) =
SOME (s'',t) ==>
(s = s'' /\ s = s /\
? i i' . i < LENGTH l /\ i = i' /\
v = EL i (MAP (λ(x,v,t). v) l) /\
t = EL i (MAP (λ(x,v,t). t) l)) ``,
REPEAT GEN_TAC >>
STRIP_TAC >>
fs[FIND_def] >>
PairCases_on `z` >>
PairCases_on `z'` >>
gvs[] >>
IMP_RES_TAC INDEX_FIND_same_list_2 >>
gvs[] >>
Q.EXISTS_TAC `z'0` >>
gvs[]
);
val var_lval_is_typed = prove ( ``
! e v tau gtsl gscope tsl scopest T_e .
star_not_in_sl scopest /\
type_scopes_list gscope gtsl /\
type_scopes_list scopest tsl /\
e_typ (gtsl,tsl) T_e (e_var v) tau T ==>
∃lval v'.
get_lval_of_e (e_var v) = SOME lval ∧
lookup_lval (scopest ⧺ gscope) lval = SOME v' ∧ v_typ v' tau F ``,
rw[] >>
`∃v'. get_lval_of_e (e_var v) = SOME v'` by (IMP_RES_TAC e_T_get_lval >> gvs[]) >>
gvs[] >>
Cases_on `v` >>
fs[Once e_typ_cases] >>
gvs[] >| [
(`? v'' . SOME v'' = lookup_vexp2 scopest gscope (varn_name s) ` by
(IMP_RES_TAC varn_tau_exists_in_scl_tscl >>
srw_tac [SatisfySimps.SATISFY_ss][])) >>
fs[get_lval_of_e_def] >>
fs[lookup_vexp2_def] >>
Cases_on `lookup_map (scopest ⧺ gscope) (varn_name s)` >>
gvs[] >>
fs[lookup_lval_def, lookup_v_def] >>
PairCases_on `x` >>
gvs[] >>
fs[lookup_tau_def] >>
Cases_on `lookup_map (tsl ⧺ gtsl) (varn_name s)` >>
gvs[] >>
`type_scopes_list (scopest ⧺ gscope) (tsl ⧺ gtsl)` by
(IMP_RES_TAC type_scopes_list_APPEND) >>
gvs[type_scopes_list_def] >>
ASSUME_TAC (INST_TYPE [``:'a`` |-> ``:(v#lval option)``,
``:'b`` |-> ``:(tau#lval option)``] R_lookup_map_scopesl) >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[‘(λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2)’ , `(x)`, `(varn_name s)`,
`(v'',x1)`, `(tsl ⧺ gtsl)`,`(scopest ⧺ gscope)`])) >>
PairCases_on ‘x’ >> gvs[]
,
(`? v'' . SOME v'' = lookup_vexp2 scopest gscope (varn_star f) ` by
(IMP_RES_TAC star_tau_exists_in_scl_tscl >>
srw_tac [SatisfySimps.SATISFY_ss][])) >>
fs[get_lval_of_e_def] >>
fs[lookup_vexp2_def] >>
Cases_on `lookup_map (scopest ⧺ gscope) (varn_star f)` >>
gvs[] >>
fs[lookup_lval_def, lookup_v_def] >>
PairCases_on `x` >>
gvs[] >>
fs[find_star_in_globals_def] >>
Cases_on `lookup_map gtsl (varn_star f)` >>
gvs[] >>
IMP_RES_TAC lookup_map_in_gsl_lemma >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`v''`, `x1`, `gscope`, `f`])) >>
gvs[] >>
ASSUME_TAC (INST_TYPE [``:'a`` |-> ``:(v#lval option)``,
``:'b`` |-> ``:(tau#lval option)``] R_lookup_map_scopesl) >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[‘(λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2)’, `x`, `(varn_star f)`,
`(v'',x1)`, `(gtsl)`,`(gscope)`])) >>
gvs[type_scopes_list_def] >>
PairCases_on ‘x’ >> gvs[]
]
);
Theorem e_lval_WT:
! e x tau gtsl gscope tsl scopest T_e .
star_not_in_sl scopest /\
type_scopes_list gscope gtsl /\
type_scopes_list scopest tsl /\
is_e_lval e /\
e_typ (gtsl,tsl) T_e e tau T ==>
(∃lval v.
get_lval_of_e e = SOME lval ∧
lookup_lval (scopest ⧺ gscope) lval = SOME v /\
v_typ v tau F)
Proof
Induct >>
REPEAT STRIP_TAC >~ [`∃lval v. get_lval_of_e (e_acc e s) = SOME lval ∧
lookup_lval (scopest ⧺ gscope) lval = SOME v ∧
v_typ v tau F`] >- (
fs[] >>
(subgoal `? v . get_lval_of_e (e_acc e s) = SOME v` >-
( IMP_RES_TAC e_T_get_lval >>
gvs[]) >>
gvs[]) >>
OPEN_EXP_TYP_TAC ``(e_acc e s)`` >>
fs[] >>
Cases_on `struct_ty` >>
gvs[] >>
(subgoal `? v . get_lval_of_e (e) = SOME v` >-
(IMP_RES_TAC e_T_get_lval >> gvs[]) >>
gvs[clause_name_def]) >>
fs[is_get_lval] >| [