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p4_stmt_subject_reductionScript.sml
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p4_stmt_subject_reductionScript.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 p4_deterTheory;
open p4_e_subject_reductionTheory;
open p4_e_progressTheory;
open bitstringTheory;
open wordsTheory;
open optionTheory;
open sumTheory;
open stringTheory;
open ottTheory;
open pairTheory;
open rich_listTheory;
open arithmeticTheory;
open alistTheory;
open numeralTheory;
(*tactics*)
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_RED_TAC exp_term =
(Q.PAT_X_ASSUM `e_red j scope scopest ^exp_term exp2 fr` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once e_sem_cases] thm)))
fun OPEN_ANY_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_sem_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)))
fun OPEN_ANY_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)))
fun OPEN_STMT_TYP_TAC stmt_term =
(Q.PAT_X_ASSUM ` stmt_typ (t1,t2) q g ^stmt_term` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once stmt_typ_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_sem_cases] thm)))
val OPEN_ANY_STMT_RED_TAC =
(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_sem_cases] thm)))
fun OPEN_FRAME_TYP_TAC frame_term =
(Q.PAT_X_ASSUM ` frame_typ (t1,t2) a b h d ^frame_term` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once frame_typ_cases] thm)))
fun OPEN_LVAL_TYP_TAC lval_term =
(Q.PAT_X_ASSUM `lval_typ (g1,q1) t (^lval_term) (tp)` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once lval_typ_cases] thm)))
val _ = new_theory "p4_stmt_subject_reduction";
(* represent the definition of:
ψlist_G ψlist_local ⊢ Φ
is the programmable blocks names so it is the main *)
(* here in t_scopes_consistent we need the same T_e as the caller, in the state's typing it should be the passed version *)
val res_frame_typ_def = Define ‘
res_frame_typ (order, f, (delta_g, delta_b, delta_x, delta_t)) Prs_n t_scope_list_g t_scope_list gscope framel func_map b_func_map t_scope_list_pre =
∀i. 0 <= i ∧ i < LENGTH framel ⇒
∃stmt_stack f_name local_scope_list passed_gscope passed_delta_b passed_delta_t passed_tslg.
EL i framel = (f_name,stmt_stack,local_scope_list) ∧
order (order_elem_f f_name) (order_elem_f f) ∧
t_passed_elem f_name delta_g delta_b delta_t t_scope_list_g = (SOME passed_delta_b, SOME passed_delta_t , SOME passed_tslg) ∧
scopes_to_pass f_name func_map b_func_map gscope = SOME passed_gscope ∧
t_scopes_consistent (order, f, (delta_g, delta_b, delta_x, delta_t)) (t_scope_list_pre) (t_scope_list_g) (t_scope_list) ∧
frame_typ (passed_tslg,t_scope_list) (order, f_name, (delta_g, passed_delta_b, delta_x, passed_delta_t))
Prs_n passed_gscope local_scope_list stmt_stack
’;
val sr_stmt_def = Define `
sr_stmt (stmt) (ty:'a itself) =
∀ stmtl ascope ascope' gscope gscope' (scopest:scope list) scopest' framel status status' t_scope_list t_scope_list_g T_e (c:'a ctx) order delta_g delta_b delta_t delta_x f Prs_n
apply_table_f ext_map func_map b_func_map pars_map tbl_map.
(c = ( apply_table_f , ext_map , func_map , b_func_map , pars_map , tbl_map ) ) ∧
(WT_c c order t_scope_list_g delta_g delta_b delta_x delta_t Prs_n) ∧
(T_e = (order, f, (delta_g, delta_b, delta_x, delta_t))) ∧
(frame_typ ( t_scope_list_g , t_scope_list ) T_e Prs_n gscope scopest [stmt] ) ∧
(stmt_red c ( ascope , gscope , [ (f, [stmt], scopest )] , status)
( ascope', gscope' , framel ++ [ (f, stmtl , scopest')] , status')) ⇒
(∃ t_scope_list'.
res_frame_typ T_e Prs_n t_scope_list_g t_scope_list' gscope framel func_map b_func_map t_scope_list) ∧
(∃ t_scope_list'' .
LENGTH t_scope_list'' = (LENGTH stmtl - LENGTH [stmt]) ∧
frame_typ ( t_scope_list_g , t_scope_list''++t_scope_list ) T_e Prs_n gscope' scopest' stmtl)
`;
val fr_len_exp_def = Define `
fr_len_exp (e) (ty:'a itself) =
∀ e' gscope (scopest:scope list) framel (c:'a ctx).
e_red c gscope scopest e e' framel ⇒
((LENGTH framel = 1 ∧ ∃f_called stmt_called copied_in_scope. framel = [(f_called,[stmt_called],copied_in_scope)]) ∨
(LENGTH framel = 0))
`;
val fr_len_exp_list_def = Define `
fr_len_exp_list (l : e list) (ty:'a itself) =
∀(e : e). MEM e l ==> fr_len_exp (e) (ty:'a itself)
`;
val fr_len_strexp_list_def = Define `
fr_len_strexp_list (l : (string#e) list) (ty:'a itself) =
∀ (e:e) . MEM e (SND (UNZIP l)) ==> fr_len_exp(e) (ty:'a itself)
`;
val fr_len_strexp_tup_def = Define `
fr_len_strexp_tup (tup : (string#e)) (ty:'a itself) =
fr_len_exp ((SND tup)) (ty:'a itself)
`;
fun FR_LEN_IND_CASE exp = OPEN_EXP_RED_TAC (exp) >>
gvs[] >>
METIS_TAC[]
(* The expression reduction can create a frame of length 1 or nothing [] *)
Theorem fr_len_from_e_theorem:
! (ty:'a itself) .
(!e. fr_len_exp e ty) /\
(! (l1: e list). fr_len_exp_list l1 ty) /\
(! (l2: (string#e) list) . fr_len_strexp_list l2 ty) /\
(! tup. fr_len_strexp_tup tup ty)
Proof
STRIP_TAC >>
Induct >>
fs[fr_len_exp_def] >>
REPEAT STRIP_TAC >| [
fs[Once e_sem_cases]
,
fs[Once e_sem_cases]
,
fs[Once e_sem_cases]
,
FR_LEN_IND_CASE “e_acc e s”
,
FR_LEN_IND_CASE “e_unop u e”
,
FR_LEN_IND_CASE “e_cast c e”
,
FR_LEN_IND_CASE “e_binop e b e'”
,
FR_LEN_IND_CASE “e_concat e e'”
,
FR_LEN_IND_CASE “e_slice e e' e''”
,
OPEN_EXP_RED_TAC “e_call f l” >>
gvs[fr_len_exp_list_def] >>
IMP_RES_TAC unred_arg_index_in_range >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`EL i (MAP (λ(e_,e'_,x_,d_). e_) (e_e'_x_d_list : (e # e # string # d) list)) `])) >>
gvs[EL_MEM] >>
fs[fr_len_exp_def] >>
RES_TAC >>
gvs[]
,
FR_LEN_IND_CASE “(e_select e l s)”
,
OPEN_EXP_RED_TAC “(e_struct l2)” >>
gvs[] >>
gvs[fr_len_strexp_list_def] >>
IMP_RES_TAC unred_mem_index_in_range >>
subgoal `fr_len_exp (EL i (MAP (λ(f_,e_,e'_). e_) f_e_e'_list)) ty ` >-
(FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`EL i (MAP (λ(f_,e_,e'_). e_) (f_e_e'_list : (string # e # e) list)) `])) >>
IMP_RES_TAC EL_MEM >>
IMP_RES_TAC mem_el_map2 >>
gvs[] ) >>
fs[fr_len_exp_def] >>
RES_TAC >>
gvs[]
,
OPEN_EXP_RED_TAC “(e_header b l2)” >>
gvs[fr_len_strexp_list_def] >>
IMP_RES_TAC unred_mem_index_in_range >>
subgoal `fr_len_exp (EL i (MAP (λ(f_,e_,e'_). e_) f_e_e'_list)) ty ` >-
(FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL
[`EL i (MAP (λ(f_,e_,e'_). e_) (f_e_e'_list : (string # e # e) list)) `])) >>
IMP_RES_TAC EL_MEM >>
IMP_RES_TAC mem_el_map2 >>
gvs[] ) >>
fs[fr_len_exp_def] >>
RES_TAC >>
gvs[]
,
fs[fr_len_strexp_list_def]
,
fs[fr_len_strexp_list_def, fr_len_strexp_tup_def] >>
REPEAT STRIP_TAC >>
PairCases_on ‘tup’ >> gvs[] >> gvs[]
,
fs[fr_len_strexp_tup_def, fr_len_exp_def] >>
gvs[]
,
fs[fr_len_exp_list_def]
,
fs[fr_len_exp_list_def] >>
REPEAT STRIP_TAC >>
gvs[] >>
gvs[fr_len_exp_def] >>
rfs[]
]
QED
fun FR_LEN_STMT_IND_CASE stm = OPEN_STMT_RED_TAC stm >>
gvs[] >>
ASSUME_TAC fr_len_from_e_theorem >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`e`])) >>
fs[fr_len_exp_def] >> gvs[] >>
RES_TAC >> gvs[]
(* the statement reduction can create a framel of length 1 or 0 *)
Theorem fr_len_from_a_frame_theorem:
∀ stmt stmtl ascope ascope' gscope gscope' scopest scopest' f c status status' framel.
(stmt_red c ( ascope , gscope , [ (f, [stmt], scopest )] , status)
( ascope', gscope' , framel ++ [ (f, stmtl , scopest')] , status')) ⇒
((LENGTH framel = 1 ∧ ∃f_called stmt_called copied_in_scope. framel = [(f_called,[stmt_called],copied_in_scope)]) ∨ (LENGTH framel = 0))
Proof
Induct >>
REPEAT GEN_TAC >>
STRIP_TAC >| [
fs[Once stmt_sem_cases]
,
FR_LEN_STMT_IND_CASE “stmt_ass l e”
,
FR_LEN_STMT_IND_CASE “stmt_cond e stmt stmt'”
,
FR_LEN_STMT_IND_CASE “stmt_block l stmt”
,
FR_LEN_STMT_IND_CASE “stmt_ret e”
,
OPEN_STMT_RED_TAC “stmt_seq stmt stmt'” >>
gvs[] >>
RES_TAC >>
gvs[]
,
(*FR_LEN_STMT_IND_CASE “stmt_verify e e0” >>
gvs[] >>
ASSUME_TAC fr_len_from_e_theorem >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
fs[fr_len_exp_def] >> gvs[] >>
RES_TAC >> gvs[]
,*)
FR_LEN_STMT_IND_CASE “stmt_trans e”
,
OPEN_STMT_RED_TAC “stmt_app s l” >>
gvs[] >>
ASSUME_TAC fr_len_from_e_theorem >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
fs[fr_len_exp_def] >> gvs[] >>
RES_TAC >> gvs[]
,
fs[Once stmt_sem_cases]
]
QED
(* NOTE that in the res_frame, we take the previous T_e to calculate the new T_e *)
val frame_typ_imp_res_frame_single = prove (
“ ∀ Prs_n gscope t_scope_list_g t_scope_list_fr f_called copied_in_scope
stmt_called func_map b_func_map passed_gscope passed_delta_b
passed_delta_t passed_tslg order delta_g delta_b delta_x delta_t f t_scope_list.
order (order_elem_f f_called) (order_elem_f f) ∧
scopes_to_pass f_called func_map b_func_map gscope =
SOME passed_gscope ∧
t_passed_elem f_called delta_g delta_b delta_t t_scope_list_g =
(SOME passed_delta_b,SOME passed_delta_t,SOME passed_tslg) ∧
frame_typ (passed_tslg,t_scope_list_fr)
(order,f_called,delta_g,passed_delta_b,delta_x,passed_delta_t)
Prs_n passed_gscope copied_in_scope [stmt_called] ∧
t_scopes_consistent (order,f,delta_g,delta_b,delta_x,delta_t)
t_scope_list t_scope_list_g t_scope_list_fr
⇒
res_frame_typ (order,f,delta_g,delta_b,delta_x,delta_t) Prs_n t_scope_list_g t_scope_list_fr gscope
[(f_called,[stmt_called],copied_in_scope)] func_map b_func_map t_scope_list”,
fs[res_frame_typ_def] >>
REPEAT STRIP_TAC >>
‘i=0’ by fs[] >>
fs[Once EL] >>
Q.EXISTS_TAC ‘Prs_n’ >>
gvs[]
);
Theorem fr_len_from_a_stmtl_theorem:
∀ stmtl stmtl' ascope ascope' gscope gscope' scopest scopest' f c status status' framel.
(stmt_red c ( ascope , gscope , [ (f, stmtl, scopest )] , status)
( ascope', gscope' , framel ++ [ (f, stmtl' , scopest')] , status')) ⇒
((LENGTH framel = 1 ∧ ∃f_called stmt_called copied_in_scope. framel = [(f_called,[stmt_called],copied_in_scope)]) ∨ (LENGTH framel = 0))
Proof
Cases >>
REPEAT GEN_TAC >-
gvs[Once stmt_sem_cases] >>
REPEAT STRIP_TAC >>
gvs[Once stmt_sem_cases] >>
IMP_RES_TAC fr_len_from_a_frame_theorem >> gvs[]>>
ASSUME_TAC fr_len_from_e_theorem >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`e`])) >>
fs[fr_len_exp_def] >> gvs[] >>
RES_TAC >> gvs[] >>
(*case stmt apply e*)
ASSUME_TAC fr_len_from_e_theorem >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`(EL i (MAP (λ(e_,e'_). e_) (e_e'_list: (e # e) list)))`])) >>
fs[fr_len_exp_def] >> gvs[] >>
RES_TAC >> gvs[]
QED
fun OPEN_FRAME_TYP_TAC frame_term =
(Q.PAT_X_ASSUM ` frame_typ (t1,t2) a b h d ^frame_term` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once frame_typ_cases] thm)))
fun EXP_IS_WT_IN_FRAME_TAC frm = OPEN_FRAME_TYP_TAC frm >>
fs[stmtl_typ_cases, type_ith_stmt_def] >>
gvs[] >>
rfs[Once stmt_typ_cases]
fun ASSUME_SR_EXP_FOR e = ASSUME_TAC SR_e >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ty`])) >>
PAT_ASSUM ``∀e. sr_exp e ty`` ( STRIP_ASSUME_TAC o (Q.SPECL [e])) >>
fs[sr_exp_def]
fun INST_SR_EXP_FOR (e', tau, b) = FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [e', ‘gscope’, ‘scopest’,
‘[(f_called,[stmt_called],copied_in_scope)]’, ‘t_scope_list’, ‘t_scope_list_g’,
tau,b, ‘order’,‘delta_g’,‘delta_b’, ‘delta_t’,‘delta_x’,‘f’,‘f_called’,
‘stmt_called’,‘copied_in_scope’, ‘Prs_n’ ,‘apply_table_f’, ‘ext_map’, ‘func_map’, ‘b_func_map’,
‘pars_map’, ‘tbl_map’])) >>
gvs[]
(* here we know that also the frame we create and trying to type is empty*)
val SR_stmt_newframe = prove (“
∀ stmt stmtl ascope ascope' gscope gscope' (scopest:scope list) scopest' framel status status' t_scope_list t_scope_list_g T_e
(c:'a ctx) order delta_g delta_b delta_t delta_x f Prs_n apply_table_f ext_map func_map b_func_map pars_map tbl_map.
(c = ( apply_table_f , ext_map , func_map , b_func_map , pars_map , tbl_map ) ) ∧
(WT_c c order t_scope_list_g delta_g delta_b delta_x delta_t Prs_n) ∧
(T_e = (order, f, (delta_g, delta_b, delta_x, delta_t))) ∧
(frame_typ ( t_scope_list_g , t_scope_list ) T_e Prs_n gscope scopest [stmt] ) ∧
(stmt_red c ( ascope , gscope , [ (f, [stmt], scopest )] , status)
( ascope', gscope' , framel ++ [ (f, stmtl , scopest')] , status')) ⇒
(∃ t_scope_list'. res_frame_typ (order,f,(delta_g, delta_b, delta_x, delta_t)) Prs_n t_scope_list_g t_scope_list' gscope framel func_map b_func_map t_scope_list) ”,
Induct >>
REPEAT STRIP_TAC >| [
fs[Once stmt_sem_cases]
,
(** assignment case **)
(* we know that the length of the frame framel is either 0 or one from :*)
IMP_RES_TAC fr_len_from_a_frame_theorem >|[
(* if 1, then we also know that e made a reduction*)
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
(* we also know that e is well typed from frame typ *)
EXP_IS_WT_IN_FRAME_TAC “[stmt_ass l e]” >>
(* from sr we know that this frame is well typed, we need to know the
typing scope for the body *)
ASSUME_SR_EXP_FOR ‘e’ >>
INST_SR_EXP_FOR (‘e''’,‘(t_tau tau')’,‘b’) >>
gvs[type_frame_tsl_def] >>
qexistsl_tac [‘t_scope_list_fr’] >>
drule frame_typ_imp_res_frame_single >> gvs[]
,
fs[res_frame_typ_def]
]
,
(* condition case *)
schneiderUtils.POP_NO_TAC 5 >>
schneiderUtils.POP_NO_TAC 5 >>
IMP_RES_TAC fr_len_from_a_frame_theorem >|[
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
EXP_IS_WT_IN_FRAME_TAC “[stmt_cond e stmt stmt']” >>
ASSUME_SR_EXP_FOR ‘e’ >>
INST_SR_EXP_FOR (‘e''’,‘(t_tau tau_bool)’,‘b’) >>
gvs[type_frame_tsl_def] >>
qexistsl_tac [‘t_scope_list_fr’] >>
drule frame_typ_imp_res_frame_single >> gvs[]
,
fs[res_frame_typ_def]
]
,
(* stmt_block can never create a frame *)
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
fs[res_frame_typ_def]
,
(* return *)
IMP_RES_TAC fr_len_from_a_frame_theorem >|[
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
EXP_IS_WT_IN_FRAME_TAC “[stmt_ret e]” >>
ASSUME_SR_EXP_FOR ‘e’ >>
INST_SR_EXP_FOR (‘e''’,‘(t_tau tau')’,‘b’) >>
gvs[type_frame_tsl_def] >>
qexistsl_tac [‘t_scope_list_fr’] >>
drule frame_typ_imp_res_frame_single >> gvs[]
,
fs[res_frame_typ_def]
]
,
(* only seq 1 create a frame *)
schneiderUtils.POP_NO_TAC 5 >>
IMP_RES_TAC fr_len_from_a_frame_theorem >| [
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
(* use IH *)
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘stmt_stack' ⧺ [stmt1']’, ‘ascope’, ‘ascope'’, ‘gscope’, ‘gscope'’,
‘scopest’, ‘scopest'’,‘[(f_called,[stmt_called],copied_in_scope)]’,‘status_running’,
‘status_running’,‘t_scope_list’,‘t_scope_list_g’, ‘order’,‘delta_g’,‘delta_b’,
‘delta_t’,‘delta_x’,‘f’,‘Prs_n’])) >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘apply_table_f’, ‘ext_map’, ‘func_map’, ‘b_func_map’, ‘pars_map’, ‘tbl_map’])) >> gvs[] >>
(* we know that [stmt] is well typed from frame_typ*)
subgoal ‘frame_typ (t_scope_list_g,t_scope_list)
(order,f,delta_g,delta_b,delta_x,delta_t) Prs_n gscope scopest [stmt]’ >-
( OPEN_FRAME_TYP_TAC “[stmt_seq stmt stmt']” >>
SIMP_TAC list_ss [Once frame_typ_cases] >>
fs[stmtl_typ_cases, type_ith_stmt_def] >>
gvs[] >>
OPEN_STMT_TYP_TAC “stmt_seq stmt stmt'” >>
gvs[] >>
qexistsl_tac [‘tau_x_d_list’,‘tau’] >> gvs[] >>
REPEAT STRIP_TAC >> gvs[] >>
REPEAT STRIP_TAC >>
‘i=0’ by fs[] >>
fs[Once EL] ) >>
gvs[] >>
srw_tac [SatisfySimps.SATISFY_ss][]
,
fs[res_frame_typ_def]
]
,
(** statement trans **)
IMP_RES_TAC fr_len_from_a_frame_theorem >| [
OPEN_ANY_STMT_RED_TAC >> gvs[] >>
EXP_IS_WT_IN_FRAME_TAC “[stmt_trans e]” >>
ASSUME_SR_EXP_FOR ‘e’ >>
INST_SR_EXP_FOR (‘e''’,‘t_string_names_a x_list’,‘b’) >>
gvs[type_frame_tsl_def] >>
qexistsl_tac [‘t_scope_list_fr’] >>
drule frame_typ_imp_res_frame_single >> gvs[]
,
fs[res_frame_typ_def]
]
,
(* statement apply s l *)
IMP_RES_TAC fr_len_from_a_frame_theorem >| [
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
EXP_IS_WT_IN_FRAME_TAC “[stmt_app s l]” >>
(* we know that i is indeed less than the list *)
subgoal ‘i < LENGTH e_tau_b_list’ >- (
IMP_RES_TAC index_not_const_in_range >>
gvs[LIST_EQ_REWRITE] )>>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`i`])) >>
gvs[] >>
ASSUME_SR_EXP_FOR ‘(EL i (MAP (λ(e_,e'_). e_) (e_e'_list : (e # e) list)))’ >>
INST_SR_EXP_FOR (‘e'’,‘(t_tau (EL i (MAP (λ(e_,tau_,b_). tau_) (e_tau_b_list: (e # tau # bool) list))))’,
‘(EL i (MAP (λ(e_,tau_,b_). b_) (e_tau_b_list : (e # tau # bool) list)))’) >>
gvs[type_frame_tsl_def] >>
qexistsl_tac [‘t_scope_list_fr’] >>
drule frame_typ_imp_res_frame_single >> gvs[]
,
fs[res_frame_typ_def]
]
,
(* statement extern would never create a frame *)
OPEN_ANY_STMT_RED_TAC >>
gvs[] >>
fs[res_frame_typ_def]
]
);
(*
EVAL “declare_list_in_fresh_scope [(varn_name "x",tau_bot)]”
EVAL “ type_scopes_list [[(varn_name "x",v_bot,NONE)]] [[(varn_name "x",tau_bot)]] ”
*)
val tsl_singletone_exsists = prove (“
∀ scopest ts .
type_scopes_list scopest [ts] ⇒
(LENGTH scopest = 1 ∧ ∃ sc . scopest = [sc] ∧
type_scopes_list [sc] [ts] )
”,
REPEAT GEN_TAC >>
STRIP_TAC >>
IMP_RES_TAC type_scopes_list_LENGTH >>
‘LENGTH [ts] = 1’ by fs[] >>
gvs[] >>
fs[type_scopes_list_def] >>
gvs[similarl_def]
);
val similar_normalize2 = prove ( ``
∀ R l l' h h'.
similar R (h::l) (h'::l') ⇔
similar R [h] [h'] ∧
similar R l l'``,
REPEAT STRIP_TAC >>
fs[similar_def]
);
val similar_ext_out_scope_lemma = prove (“
∀ sc sc' varnl varnl' tl.
similar (λsi so. SND si = SND so) sc sc' ∧
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc' (ZIP (varnl,tl)) ∧
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc (ZIP (varnl',tl)) ⇒
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc' (ZIP (varnl',tl))
”,
reverse (
Induct_on ‘sc’ >>
Induct_on ‘sc'’ >>
Induct_on ‘tl’ >>
Induct_on ‘varnl’ >>
Induct_on ‘varnl'’ >>
REPEAT STRIP_TAC) >-
(
SIMP_TAC list_ss [Once similar_normalize2] >> gvs[] >> CONJ_TAC >| [
(*head case*)
gvs[Once similar_normalize2] >>
PairCases_on ‘h'’ >>
PairCases_on ‘h''''’ >>
gvs[similar_def]
,
(*IH case*)
rfs[Once similar_normalize2] >>
RES_TAC
]
) >>
srw_tac [][ZIP, ZIP_def] >>
gvs[similar_def] >>
TRY (PairCases_on ‘h’) >>
TRY (PairCases_on ‘h'’) >>
TRY(PairCases_on ‘y’) >>
fsrw_tac [][LENGTH_ZIP_MIN] >>
rfs[ZIP_def]
);
val similar_ext_out_scope_lemma2 = prove (“
∀ sc sc' varnl varnl' tl.
similar (λsi so. SND si = SND so ∧ SND si = NONE) sc sc' ∧
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc' (ZIP (varnl,tl)) ∧
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc (ZIP (varnl',tl)) ⇒
similar (λ(v,lop1) (t,lop2). v_typ v (t_tau t) F ∧ lop1 = lop2) sc' (ZIP (varnl',tl))
”,
reverse (
Induct_on ‘sc’ >>
Induct_on ‘sc'’ >>
Induct_on ‘tl’ >>
Induct_on ‘varnl’ >>
Induct_on ‘varnl'’ >>
REPEAT STRIP_TAC) >-
(
SIMP_TAC list_ss [Once similar_normalize2] >> gvs[] >> CONJ_TAC >| [
(*head case*)
gvs[Once similar_normalize2] >>
PairCases_on ‘h'’ >>
PairCases_on ‘h''''’ >>
gvs[similar_def]
,
(*IH case*)
rfs[Once similar_normalize2] >>
RES_TAC
]
) >>
srw_tac [][ZIP, ZIP_def] >>
gvs[similar_def] >>
TRY (PairCases_on ‘h’) >>
TRY (PairCases_on ‘h'’) >>
TRY(PairCases_on ‘y’) >>
fsrw_tac [][LENGTH_ZIP_MIN] >>
rfs[ZIP_def]
);
val typ_ext_out_scope_lemma = prove (“
∀ sc sc' tl varnl varnl'.
ext_sc_same_as_input [sc] [sc'] ∧
type_scopes_list [sc'] [ZIP (varnl, tl)] ∧
type_scopes_list [sc] [ZIP(varnl', tl)] ⇒
type_scopes_list [sc'] [ZIP(varnl', tl)] ”,
REPEAT STRIP_TAC >>
gvs[ext_sc_same_as_input_def] >>
gvs[type_scopes_list_def] >>
gvs[similarl_def] >>
IMP_RES_TAC similar_ext_out_scope_lemma
);
val ext_sc_same_as_input_LENGTH = prove (“
∀ scopest tsl .
ext_sc_same_as_input scopest tsl ⇒
LENGTH scopest = LENGTH tsl”,
gvs[ext_sc_same_as_input_def] >>
fs[type_scopes_list_def, similarl_def, similar_def] >>
REPEAT STRIP_TAC >>
IMP_RES_TAC LIST_REL_LENGTH
);
val LOL_ext_sc_same_as_input = prove (“
∀ sc outsc tsc' tsc lol.
ext_sc_same_as_input [outsc] [sc] ∧
MAP (λx. SND (SND x)) tsc' = lol ∧
type_scopes_list [sc] [tsc'] ∧
type_scopes_list [outsc] [tsc] ⇒
MAP (λx. SND (SND x)) tsc = lol ”,
gvs[type_scopes_list_def]>>
gvs[ext_sc_same_as_input_def] >>
gvs[similarl_def] >>
reverse (
Induct_on ‘sc’ >>
Induct_on ‘outsc’ >>
Induct_on ‘tsc'’ >>
Induct_on ‘tsc’) >- (
REPEAT STRIP_TAC >>
PairCases_on ‘h’ >>
PairCases_on ‘h'’ >>
PairCases_on ‘h''’ >>
PairCases_on ‘h'''’ >>
gvs[] >>
gvs[similar_def] >>
METIS_TAC []) >>
gvs[similar_def]
);
val map_lemma_local_tmp = prove (“
∀ txdl lol .
MAP (λx. SND (SND x))
(ZIP
(mk_varn (MAP (λx. FST (SND x)) txdl),
ZIP (MAP FST txdl,MAP (λtxd. lol) txdl))) =
MAP (λtxd. lol) txdl ”,
Induct >>
REPEAT STRIP_TAC >>
gvs[mk_varn_def]
);
val typ_scope_list_ext_out_scope_lemma = prove (
“ ∀ f apply_table_f ext_map func_map b_func_map pars_map tbl_map
order tslg delta_g delta_b delta_x ascope ascope'
gscope scopest scopest' v ext_fun tsl tau txdl delta_t Prs_n.
WT_c (apply_table_f,ext_map,func_map,b_func_map,pars_map,tbl_map)
order tslg delta_g delta_b delta_x delta_t Prs_n ∧
SOME (txdl,tau) = t_lookup_funn f delta_g delta_b delta_x ∧
args_t_same (MAP FST txdl) tsl ∧
type_scopes_list scopest tsl ∧
type_scopes_list gscope tslg ∧
star_not_in_sl scopest ∧
SOME ext_fun = lookup_ext_fun f ext_map ∧
SOME (ascope',scopest',v) = ext_fun (ascope,gscope,scopest) ⇒
(type_scopes_list scopest' tsl ∧ star_not_in_sl scopest')”,
Cases_on ‘f’ >>
fs[lookup_ext_fun_def] >>
REPEAT GEN_TAC >>
STRIP_TAC >| [
(* we know that the length of scopest is one,*)
(* and if the length is one, we know that there exists only one element there via *)
(*inst*)
Cases_on ‘ALOOKUP ext_map s’ >> gvs[] >>
PairCases_on ‘x’ >> gvs[] >>
Cases_on ‘x0’ >> gvs[] >>
PairCases_on ‘x’ >> gvs[] >>
fs[WT_c_cases] >>
fs[WTX_cases] >>
fs[extern_map_IoE_typed_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x0`, `s`, ‘ext_fun’,‘x1’])) >>
gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ascope`, `gscope`, ‘scopest’])) >>
gvs[] >>
Cases_on ‘ext_fun (ascope,gscope,scopest)’ >> gvs[] >>
(* the output scope sc is only of length 1*)
IMP_RES_TAC tsl_singletone_exsists >>
gvs[]
(*
(* show that txdl and txdl ' are the same *)
IMP_RES_TAC t_lookup_funn_ext_lemma >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`delta_g`, `delta_b`])) >>
gvs[] >>
Cases_on ‘t_lookup_funn (funn_inst s) delta_g delta_b delta_x’ >> gvs[] >>
(* we know that the scopest (initial one) must be of size 1 *)
IMP_RES_TAC ext_sc_same_as_input_LENGTH >>
(*‘LENGTH scopest = 1’ by gvs[] >>
‘∃ outsc. scopest = [outsc]’ by fs[quantHeuristicsTheory.LIST_LENGTH_1] >>
(* same for the typing scope*)
‘LENGTH tsl = 1’ by (IMP_RES_TAC type_scopes_list_LENGTH >> gvs[]) >>
‘∃ tsc. tsl = [tsc]’ by fs[quantHeuristicsTheory.LIST_LENGTH_1] >>
gvs[] >>
*)
fs[args_t_same_def, same_dir_x_def] >>
gvs[mk_tscope_def] >>
srw_tac [] [] >>
(* we know that ts contains the same types as the input*)
subgoal ‘ [tsc] = [ZIP (MAP FST tsc ,ZIP(MAP (λ(t,x,d). t) txdl, MAP (λtxd. lol) txdl ))]’ >-
( gvs[ELIM_UNCURRY, map_fst_EQ ] >>
ASSUME_TAC LOL_ext_sc_same_as_input >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘sc’, ‘outsc’,
‘ZIP (mk_varn (MAP (λx. FST (SND x)) (txdl : (tau # string # d) list)),
ZIP (MAP FST (txdl : (tau # string # d) list),
MAP (λtxd. lol) txdl ))’,
‘tsc’, ‘MAP (λtxd. lol) (txdl : (tau # string # d) list)’])) >>
gvs[] >>
METIS_TAC [ZIP_tri_id1, map_lemma_local_tmp ]
) >>
IMP_RES_TAC typ_ext_out_scope_lemma >>
rfs[] >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`MAP FST (tsc : (varn # tau) list)`])) >>
srw_tac [][] >>
METIS_TAC [] *)
,
(*ext methods *)
Cases_on ‘ALOOKUP ext_map s’ >> gvs[] >>
PairCases_on ‘x’ >> gvs[] >>
Cases_on ‘ALOOKUP x1 s0’ >> gvs[] >>
PairCases_on ‘x’ >> gvs[] >>
fs[WT_c_cases] >>
fs[WTX_cases] >>
fs[extern_MoE_typed_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`x0'`, `s`, ‘s0’,‘x0’,‘x1’,‘ext_fun’])) >>
gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`ascope`, `gscope`, ‘scopest’])) >>
gvs[] >>
Cases_on ‘ext_fun (ascope,gscope,scopest)’ >> gvs[]
(*
IMP_RES_TAC tsl_singletone_exsists >>
gvs[] >>
IMP_RES_TAC t_lookup_funn_ext_lemma >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`delta_g`, `delta_b`])) >>
gvs[] >>
Cases_on ‘t_lookup_funn (funn_ext s s0) delta_g delta_b delta_x’ >> gvs[] >>
(* we know that the scopest (initial one) must be of size 1 *)
IMP_RES_TAC ext_sc_same_as_input_LENGTH >>
‘LENGTH scopest = 1’ by gvs[] >>
‘∃ outsc. scopest = [outsc]’ by fs[quantHeuristicsTheory.LIST_LENGTH_1] >>
(* same for the typing scope*)
‘LENGTH tsl = 1’ by (IMP_RES_TAC type_scopes_list_LENGTH >> gvs[]) >>
‘∃ tsc. tsl = [tsc]’ by fs[quantHeuristicsTheory.LIST_LENGTH_1] >>
gvs[] >>
fs[args_t_same_def, same_dir_x_def] >>
gvs[mk_tscope_def] >>
srw_tac [] [] >>
(* we know that ts contains the same types as the input*)
subgoal ‘ [tsc] = [ZIP (MAP FST tsc ,ZIP(MAP (λ(t,x,d). t) txdl, MAP (λtxd. lol) txdl ))]’ >-
( gvs[ELIM_UNCURRY, map_fst_EQ ] >>
ASSUME_TAC LOL_ext_sc_same_as_input >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘sc’, ‘outsc’,
‘ZIP (mk_varn (MAP (λx. FST (SND x)) (txdl : (tau # string # d) list)),
ZIP (MAP FST (txdl : (tau # string # d) list),
MAP (λtxd. lol) txdl))’,
‘tsc’, ‘MAP (λtxd. lol) (txdl : (tau # string # d) list)’])) >>
gvs[] >>
METIS_TAC [ZIP_tri_id1, map_lemma_local_tmp ]
) >>
IMP_RES_TAC typ_ext_out_scope_lemma >>
rfs[] >>
LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`MAP FST (tsc : (varn # tau) list)`])) >>
srw_tac [][] >>
METIS_TAC [] *)
]
);
val directionless_lval_f = prove (“∀ dl.
is_directionless (dl) ⇒ out_is_lval (dl) (MAP (λd. F) dl)”,
Induct >>
gvs[is_d_out_def, out_is_lval_def] >>
REPEAT STRIP_TAC >>
fs[Once EL_compute] >> Cases_on ‘i=0’ >> gvs[EL_CONS] >| [
Cases_on ‘h’ >> gvs[is_d_out_def, is_directionless_def]
,
‘is_directionless dl’ by fs[is_directionless_def] >>
gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘i - 1’])) >>
gvs[numeralTheory.numeral_pre, PRE_SUB1, PRE_SUC_EQ] >>
Cases_on `i = 1` >>
fs[] >>
`~ (i ≤ 1)` by fs[] >>
gvs[] >>
fs[GSYM EL] >> fs[ADD1] (* TODO add this everywherre i needed to show something about the tail*) ]
);
Theorem ALOOKUP_not_empty:
∀ l s t . ALOOKUP l s = SOME t ⇒
l ≠ []
Proof
Induct >>
fs[]
QED
Theorem assign_to_null_same_lemma:
∀ scl scl' v .
SOME scl' = assign (scl) v lval_null ⇔
scl' = scl
Proof
gvs[assign_def]
QED
(*EVAL “oDROP 3 [a;b;c]”*)
(*EVAL “oTAKE 3 [a;b;c;d]”*)
Theorem oDROP_empty:
∀l . oDROP (LENGTH l) l = SOME []
Proof
Induct >>
gvs[oDROP_def]
QED