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pan_to_wordProofScript.sml
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pan_to_wordProofScript.sml
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(*
Correctness proof for combined pan_to_word compilation.
*)
open preamble pan_to_wordTheory
pan_simpProofTheory pan_to_crepProofTheory
crep_to_loopProofTheory loop_to_wordProofTheory
val _ = new_theory "pan_to_wordProof";
Definition pan_simp_st_def:
pan_simp_st (s:('a,'ffi) panSem$state) (pan_code:(mlstring # (mlstring # shape) list # α prog) list) =
s with code := alist_to_fmap (pan_simp$compile_prog pan_code)
End
Definition crep_state_def:
crep_state (s:('a,'ffi) panSem$state) (pan_code:(mlstring # (mlstring # shape) list # α prog) list) =
<| locals := FEMPTY;
globals := FEMPTY;
code := alist_to_fmap (pan_to_crep$compile_prog pan_code);
memory := s.memory;
memaddrs := s.memaddrs;
sh_memaddrs := s.sh_memaddrs;
clock := s.clock;
be := s.be;
ffi := s.ffi;
base_addr := s.base_addr|>
End
(* wlab_wloc should have taken only funcs of context *)
Definition mk_mem_def:
mk_mem funcs smem =
λad. wlab_wloc funcs (smem ad)
End
Definition loop_state_def:
loop_state (s:('a,'ffi) crepSem$state) c crep_code ck =
<| locals := LN;
globals := FEMPTY;
memory := mk_mem (make_funcs crep_code) s.memory;
mdomain := s.memaddrs;
sh_mdomain := s.sh_memaddrs;
code := fromAList (crep_to_loop$compile_prog c crep_code);
clock := ck;
be := s.be;
ffi := s.ffi;
base_addr := s.base_addr|>
End
Definition consistent_labels_def:
consistent_labels (mem:'a word -> 'a word_lab)
(pan_code:(mlstring # (mlstring # shape) list # α prog) list) <=>
∀ad f.
mem ad = Label f ⇒
∃n m. FLOOKUP (make_funcs (compile_prog (pan_simp$compile_prog pan_code))) f = SOME (n,m)
End
Definition distinct_params_def:
distinct_params prog <=>
EVERY (λ(name,params,body). ALL_DISTINCT params) prog
End
Theorem first_compile_prog_all_distinct:
!prog. ALL_DISTINCT (MAP FST prog) ==>
ALL_DISTINCT (MAP FST (pan_to_word$compile_prog c prog))
Proof
rw [] >>
fs [pan_to_wordTheory.compile_prog_def] >>
match_mp_tac loop_to_wordProofTheory.first_compile_all_distinct >>
metis_tac [crep_to_loopProofTheory.first_compile_prog_all_distinct]
QED
Theorem FDOM_get_eids_pan_simp_compile_eq:
!prog. FDOM ((get_eids prog): mlstring |-> α word) =
FDOM ((get_eids (pan_simp$compile_prog prog)):mlstring |-> α word)
Proof
rw [] >>
fs [pan_to_crepTheory.get_eids_def] >>
qmatch_goalsub_abbrev_tac ‘remove_dup (FLAT es)’ >>
qmatch_goalsub_abbrev_tac ‘_ = set (MAP FST (MAP2 (λx y. (x,y))
(remove_dup (FLAT ces)) _ ))’ >>
qsuff_tac ‘es = ces’
>- fs [] >>
fs [Abbr ‘es’, Abbr ‘ces’, pan_simpTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
fs [pan_simpProofTheory.map_snd_f_eq] >>
fs [MAP_EQ_EVERY2, LIST_REL_EL_EQN] >>
rw [] >>
‘EL n (MAP (SND ∘ SND) prog) =
(SND ∘ SND) (EL n prog)’ by (
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
fs [exp_ids_compile_eq]
QED
Theorem flookup_pan_simp_mk_funcs_none_eq:
!p f.
(FLOOKUP (make_funcs (compile_prog p)) f): (num#num) option = NONE ==>
FLOOKUP (crep_to_loop$make_funcs (pan_to_crep$compile_prog (compile_prog p))) f = NONE
Proof
rw [] >>
fs [flookup_thm] >>
qmatch_asmsub_abbrev_tac ‘_ ∉ xs’ >>
qmatch_goalsub_abbrev_tac ‘_ ∉ ys’ >>
qsuff_tac ‘xs = ys’
>- (strip_tac >> fs []) >>
fs [Abbr ‘xs’, Abbr ‘ys’] >>
pop_assum kall_tac >>
fs [crep_to_loopTheory.make_funcs_def] >>
qmatch_goalsub_abbrev_tac ‘set xs = set ys’ >>
qsuff_tac ‘xs = ys’
>- fs [] >>
fs [Abbr ‘xs’, Abbr ‘ys’] >>
fs [MAP_EQ_EVERY2, LIST_REL_EL_EQN] >>
conj_asm1_tac
>- fs [pan_to_crepTheory.compile_prog_def, pan_simpTheory.compile_prog_def] >>
fs [] >>
rw [] >>
qmatch_goalsub_abbrev_tac ‘FST (EL n (MAP2 f ws xs)) = FST (EL n (MAP2 g ys zs))’ >>
‘EL n (MAP2 f ws xs) = f (EL n ws) (EL n xs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP2 g ys zs) = g (EL n ys) (EL n zs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >> rveq >> rfs [] >>
fs [pan_to_crepTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP f _) = EL n (MAP g _)’ >>
‘EL n (MAP f p) = f (EL n p)’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP g (compile_prog p)) = g (EL n (compile_prog p))’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
‘EL n (pan_simp$compile_prog p) =
(λ(name,params,body). (name,params,compile body)) (EL n p)’ by (
fs [pan_simpTheory.compile_prog_def] >>
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
cases_on ‘EL n p’ >> fs [] >>
cases_on ‘r’ >> fs []
QED
Theorem flookup_pan_simp_mk_funcs_some_eq:
!p f x. ALL_DISTINCT (MAP FST p) ∧
(FLOOKUP (make_funcs (compile_prog p)) f): (num#num) option = SOME x ==>
FLOOKUP (crep_to_loop$make_funcs (pan_to_crep$compile_prog (compile_prog p))) f = SOME x
Proof
rw [] >>
fs [crep_to_loopTheory.make_funcs_def] >>
dxrule ALOOKUP_MEM >>
strip_tac >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
reverse conj_tac
>- (
fs [MEM_EL] >>
qexists_tac ‘n’ >>
reverse conj_asm1_tac
>- (
qmatch_goalsub_abbrev_tac ‘EL n (MAP2 ff ws xs) = EL n (MAP2 gg ys zs)’ >>
‘EL n (MAP2 ff ws xs) = ff (EL n ws) (EL n xs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP2 gg ys zs) = gg (EL n ys) (EL n zs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >> rveq >> rfs [] >>
conj_tac
>- (
fs [pan_to_crepTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP ff _) = EL n (MAP gg _)’ >>
‘EL n (MAP ff p) = ff (EL n p)’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP gg (compile_prog p)) = gg (EL n (compile_prog p))’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
‘EL n (pan_simp$compile_prog p) =
(λ(name,params,body). (name,params,compile body)) (EL n p)’ by (
fs [pan_simpTheory.compile_prog_def] >>
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
cases_on ‘EL n p’ >> fs [] >>
cases_on ‘r’ >> fs []) >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP2 ff ws xs) = EL n (MAP2 gg ys zs)’ >>
‘EL n (MAP2 ff ws xs) = ff (EL n ws) (EL n xs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP2 gg ys zs) = gg (EL n ys) (EL n zs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >> rveq >> fs [] >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP ff pp) = EL n (MAP gg qq)’ >>
‘EL n (MAP ff pp) = ff (EL n pp)’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP gg qq) = gg (EL n qq)’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >> rveq >>
fs [pan_to_crepTheory.compile_prog_def] >>
qmatch_goalsub_abbrev_tac
‘LENGTH (FST (SND (EL n (MAP ff _)))) =
LENGTH (FST (SND (EL n (MAP gg _))))’ >>
‘EL n (MAP ff p) = ff (EL n p)’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
‘EL n (MAP gg (compile_prog p)) = gg (EL n (compile_prog p))’ by (
match_mp_tac EL_MAP >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
‘EL n (pan_simp$compile_prog p) =
(λ(name,params,body). (name,params,compile body)) (EL n p)’ by (
fs [pan_simpTheory.compile_prog_def] >>
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
cases_on ‘EL n p’ >> fs [] >>
cases_on ‘r’ >> fs []) >>
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def]) >>
qmatch_goalsub_abbrev_tac ‘MAP _ xs’ >>
‘MAP FST xs = MAP FST (compile_prog (compile_prog p))’ by (
unabbrev_all_tac >> fs [] >>
qmatch_goalsub_abbrev_tac ‘MAP _ xs = MAP _ ys’ >>
fs [MAP_EQ_EVERY2, LIST_REL_EL_EQN] >>
unabbrev_all_tac >> fs [] >>
rw [] >>
qmatch_goalsub_abbrev_tac ‘FST (EL n (MAP2 ff ws xs)) = _’ >>
‘EL n (MAP2 ff ws xs) = ff (EL n ws) (EL n xs)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
match_mp_tac pan_to_crepProofTheory.first_compile_prog_all_distinct >>
match_mp_tac pan_simpProofTheory.first_compile_prog_all_distinct >>
fs []
QED
Theorem flookup_pan_simp_mk_funcs_eq:
!p f x. ALL_DISTINCT (MAP FST p) ==>
(FLOOKUP (make_funcs (compile_prog p)) f): (num#num) option =
FLOOKUP (crep_to_loop$make_funcs (pan_to_crep$compile_prog (compile_prog p))) f
Proof
rpt gen_tac >>
cases_on ‘(FLOOKUP (make_funcs (compile_prog p)) f): (num#num) option’ >>
metis_tac [flookup_pan_simp_mk_funcs_none_eq, flookup_pan_simp_mk_funcs_some_eq]
QED
Theorem crep_to_loop_intermediate_label:
∀pan_code start prog.
ALOOKUP pan_code start = SOME ([],prog) ∧
ALL_DISTINCT (MAP FST pan_code) ⇒
∃n. n < LENGTH pan_code ∧ EL n pan_code = (start,[],prog) ∧
FLOOKUP (crep_to_loop$make_funcs
(pan_to_crep$compile_prog (pan_simp$compile_prog pan_code))) start = SOME (n+first_name,0)
Proof
rw [] >>
dxrule ALOOKUP_MEM >>
strip_tac >>
fs [crep_to_loopTheory.make_funcs_def] >>
fs [ALOOKUP_EXISTS_IFF] >>
fs [MEM_EL] >>
qexists_tac ‘n’ >>
fs [] >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
conj_tac
>- (
qmatch_goalsub_abbrev_tac ‘MAP _ pp’ >>
qsuff_tac ‘MAP FST pp = MAP FST pan_code’
>- fs [] >>
fs [MAP_EQ_EVERY2, LIST_REL_EL_EQN] >>
fs [Abbr ‘pp’] >>
conj_asm1_tac
>- (
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def]) >>
fs [] >>
rw [] >>
qmatch_goalsub_abbrev_tac ‘EL n' (MAP2 f xs ys)’ >>
‘EL n' (MAP2 f xs ys) = f (EL n' xs) (EL n' ys)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘EL n' (MAP ff _)’ >>
‘EL n' (MAP ff pan_code) = ff (EL n' pan_code)’ by (
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
cases_on ‘EL n' pan_code’ >>
fs [] >> cases_on ‘r’ >> rfs []) >>
fs [MEM_EL] >>
qexists_tac ‘n’ >>
conj_asm1_tac
>- (
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def]) >>
qmatch_goalsub_abbrev_tac ‘_ = EL n (MAP2 f xs ys)’ >>
‘EL n (MAP2 f xs ys) = f (EL n xs) (EL n ys)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
conj_tac
>- (
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘_ = EL n (MAP ff _)’ >>
‘EL n (MAP ff pan_code) = ff (EL n pan_code)’ by (
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
cases_on ‘EL n pan_code’ >>
fs [] >> cases_on ‘r’ >> rfs []) >>
qmatch_goalsub_abbrev_tac ‘_ = EL n (MAP2 f xs ys)’ >>
‘EL n (MAP2 f xs ys) = f (EL n xs) (EL n ys)’ by (
match_mp_tac EL_MAP2 >>
unabbrev_all_tac >> fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP f pp) = _’ >>
‘EL n (MAP f pp) = f (EL n pp)’ by (
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
fs [pan_to_crepTheory.compile_prog_def,
pan_simpTheory.compile_prog_def] >>
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘EL n (MAP ff _)’ >>
‘EL n (MAP ff pan_code) = ff (EL n pan_code)’ by (
match_mp_tac EL_MAP >>
fs []) >>
fs [] >>
unabbrev_all_tac >> fs [] >>
cases_on ‘EL n pan_code’ >>
fs [] >> cases_on ‘r’ >>
rfs [pan_to_crepTheory.crep_vars_def, panLangTheory.size_of_shape_def]
QED
Theorem state_rel_imp_semantics:
t.memory = mk_mem (make_funcs (compile_prog pan_code)) s.memory /\
distinct_params pan_code ∧
consistent_labels s.memory pan_code /\
t.mdomain = s.memaddrs ∧
t.sh_mdomain = s.sh_memaddrs ∧
t.be = s.be ∧
t.ffi = s.ffi ∧
ALOOKUP (fmap_to_alist t.store) CurrHeap = SOME (Word s.base_addr) ∧
ALL_DISTINCT (MAP FST pan_code) ∧
ALOOKUP pan_code start = SOME ([],prog) ∧
lc < LENGTH pan_code ∧ EL lc pan_code = (start,[],prog) ∧
s.code = alist_to_fmap pan_code ∧
t.code = fromAList (pan_to_word$compile_prog c pan_code) ∧
s.locals = FEMPTY ∧ size_of_eids pan_code < dimword (:α) ∧
FDOM s.eshapes = FDOM ((get_eids pan_code):mlstring |-> 'a word) ∧
lookup 0 t.locals = SOME (Loc 1 0) /\ good_dimindex (:'a) ∧
semantics s start <> Fail ==>
semantics (t:('a,'b, 'ffi) wordSem$state) (lc+first_name) =
semantics (s:('a,'ffi) panSem$state) start
Proof
rw [] >>
drule crep_to_loop_intermediate_label >>
rfs [] >>
strip_tac >>
‘n = lc’ by (
drule ALL_DISTINCT_EL_IMP >>
disch_then (qspecl_then [‘n’, ‘lc’] mp_tac) >>
fs [] >>
strip_tac >>
‘EL n (MAP FST pan_code) = FST (EL n pan_code)’ by (
match_mp_tac EL_MAP >>
fs []) >>
‘EL lc (MAP FST pan_code) = FST (EL lc pan_code)’ by (
match_mp_tac EL_MAP >>
fs []) >>
gs []) >> rveq >>
(* pan-simp pass *)
‘state_rel s (pan_simp_st s pan_code) (pan_simp_st s pan_code).code’ by (
fs [pan_simpProofTheory.state_rel_def, pan_simp_st_def] >>
conj_tac >> rw []
>- (
fs [pan_simpTheory.compile_prog_def] >>
fs [ALOOKUP_FAILS] >>
rw [] >>
fs [MEM_MAP] >>
rw [] >>
cases_on ‘y’ >>
cases_on ‘r’ >> fs [] >>
CCONTR_TAC >> fs [] >>
rveq >> fs [] >> metis_tac []) >>
fs [pan_simpTheory.compile_prog_def] >>
fs [ALOOKUP_EXISTS_IFF] >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
conj_tac
>- (
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘MAP ff _’ >>
‘MAP ff pan_code = MAP FST pan_code’ by (
fs [Abbr ‘ff’, MAP_EQ_f] >>
rw [] >>
cases_on ‘e’ >> fs [] >>
cases_on ‘r’ >> fs []) >>
fs []) >>
fs [MEM_MAP] >>
qexists_tac ‘(f,vshs,prog')’ >>
fs [] >>
drule ALOOKUP_MEM >> fs []) >>
drule pan_simpProofTheory.state_rel_imp_semantics >>
disch_then (qspecl_then [‘pan_code’, ‘start’, ‘prog’] mp_tac) >>
fs [] >>
impl_tac >- fs [pan_simp_st_def] >>
strip_tac >>
pop_assum (assume_tac o GSYM) >>
fs [] >>
qmatch_goalsub_abbrev_tac ‘semantics pst start’ >>
(* pan_to_crep pass *)
qmatch_asmsub_abbrev_tac ‘make_funcs (_ pcode)’ >>
‘ALOOKUP pcode start = SOME ([],compile prog)’ by (
fs [Abbr ‘pcode’, pan_simpTheory.compile_prog_def] >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
conj_tac
>- (
fs [MAP_MAP_o] >>
qmatch_goalsub_abbrev_tac ‘MAP ff _’ >>
‘MAP ff pan_code = MAP FST pan_code’ by (
fs [Abbr ‘ff’, MAP_EQ_f] >>
rw [] >>
cases_on ‘e’ >> fs [] >>
cases_on ‘r’ >> fs []) >>
fs []) >>
fs [MEM_MAP] >>
qexists_tac ‘(start,[],prog)’ >>
fs [] >>
drule ALOOKUP_MEM >> fs []) >>
‘state_rel pst (crep_state pst pcode)’ by
fs [Abbr ‘pcode’, pan_to_crepProofTheory.state_rel_def, crep_state_def] >>
drule pan_to_crepProofTheory.state_rel_imp_semantics >>
disch_then (qspecl_then [‘pcode’,
‘start’, ‘pan_simp$compile prog’] mp_tac) >>
fs [] >>
impl_tac
>- (
fs [Abbr ‘pcode’, Abbr ‘pst’, pan_simp_st_def, crep_state_def] >>
conj_tac
>- (
match_mp_tac pan_simpProofTheory.first_compile_prog_all_distinct >>
fs []) >>
fs [size_of_eids_compile_eq] >>
fs [Once FDOM_get_eids_pan_simp_compile_eq]) >>
strip_tac >>
pop_assum (assume_tac o GSYM) >>
fs [] >>
qmatch_goalsub_abbrev_tac ‘semantics cst start’ >>
(* crep_to_loop pass *)
qmatch_asmsub_abbrev_tac ‘make_funcs ccode’ >>
‘ALOOKUP ccode start =
SOME ([],comp_func (make_funcs pcode)
(get_eids pcode) [] (compile prog))’ by (
fs [Abbr ‘ccode’, Abbr ‘pcode’, Abbr ‘pst’, Abbr ‘cst’,
pan_simp_st_def, crep_state_def, loop_state_def] >>
match_mp_tac alookup_compile_prog_code >>
conj_tac
>- (
match_mp_tac pan_simpProofTheory.first_compile_prog_all_distinct >>
fs []) >>
fs [pan_simpTheory.compile_prog_def]) >>
‘cst.memaddrs =
(loop_state cst c ccode t.clock).mdomain’ by
fs [Abbr ‘ccode’, Abbr ‘pcode’, Abbr ‘cst’, Abbr ‘pst’, crep_state_def, loop_state_def] >>
drule crep_to_loopProofTheory.state_rel_imp_semantics >>
disch_then (qspecl_then [‘ccode’,
‘start’, ‘comp_func (make_funcs pcode)
(get_eids pcode) [] (compile prog)’,
‘lc+first_name’,‘c’] mp_tac) >>
impl_tac
>- (
fs [Abbr ‘ccode’, Abbr ‘pcode’, Abbr ‘pst’, Abbr ‘cst’,
pan_simp_st_def, crep_state_def, loop_state_def] >>
conj_tac
>- (
fs [crep_to_loopTheory.mk_ctxt_def, mk_mem_def, mem_rel_def, consistent_labels_def] >>
rw [] >> res_tac >> rfs []) >>
conj_tac >- fs [crep_to_loopProofTheory.globals_rel_def] >>
match_mp_tac pan_to_crepProofTheory.first_compile_prog_all_distinct >>
match_mp_tac pan_simpProofTheory.first_compile_prog_all_distinct >>
fs []) >>
strip_tac >>
pop_assum (assume_tac o GSYM) >>
fs [] >>
pop_assum kall_tac >>
qmatch_goalsub_abbrev_tac ‘_ = semantics lst _’ >>
(* loop_to_word pass *)
qmatch_asmsub_abbrev_tac ‘_ = SOME ([],cprog)’ >>
drule pan_simpProofTheory.first_compile_prog_all_distinct >>
strip_tac >>
drule pan_to_crepProofTheory.first_compile_prog_all_distinct >>
strip_tac >>
‘st_rel lst t (compile_prog c ccode)’ by (
fs [st_rel_def] >>
conj_tac
>- (
fs [loop_removeProofTheory.state_rel_def] >>
qexists_tac ‘fromAList (comp_prog (compile_prog c ccode))’ >>
fs [] >> rw []
>- (
fs [Abbr ‘lst’, loop_state_def] >>
fs [crep_to_loopTheory.compile_prog_def] >>
drule pan_commonPropsTheory.lookup_some_el >>
strip_tac >>
pop_assum mp_tac >>
qmatch_goalsub_abbrev_tac ‘EL m (MAP2 gg xs ys)’ >>
‘EL m (MAP2 gg xs ys) = gg (EL m xs) (EL m ys)’ by (
ho_match_mp_tac EL_MAP2 >>
fs [Abbr ‘xs’, Abbr ‘ys’]) >>
fs [Abbr ‘gg’, Abbr ‘xs’, Abbr ‘ys’] >>
pop_assum kall_tac >>
cases_on ‘EL m ccode’ >> fs [] >>
cases_on ‘r’ >> fs [] >>
strip_tac >> rveq >> gs [] >>
fs [loop_liveTheory.optimise_def] >>
fs [loop_liveTheory.comp_def] >>
fs [loop_liveProofTheory.mark_all_syntax_ok]) >>
(* has_code *)
match_mp_tac loop_removeProofTheory.comp_prog_has_code >>
reverse conj_tac
>- (
fs [Abbr ‘lst’, loop_state_def] >>
fs [lookup_fromAList] >>
drule ALOOKUP_MEM >>
fs []) >>
fs [crep_to_loopProofTheory.first_compile_prog_all_distinct]) >>
conj_tac
>- (
fs [loop_to_wordProofTheory.state_rel_def] >>
fs [Abbr ‘lst’, Abbr ‘cst’, Abbr ‘pst’, pan_simp_st_def,
loop_state_def, crep_state_def] >>
conj_tac
>- (
fs [mk_mem_def, crep_to_loopTheory.mk_ctxt_def] >>
fs [FUN_EQ_THM] >>
rw [] >>
cases_on ‘s.memory ad’ >> fs [wlab_wloc_def, Once flookup_pan_simp_mk_funcs_eq]) >>
fs [globals_rel_def] >>
fs [loop_to_wordProofTheory.code_rel_def] >>
rw []
>- (
fs [lookup_fromAList] >>
dxrule ALOOKUP_MEM >>
strip_tac >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
conj_tac
>- (
match_mp_tac first_compile_prog_all_distinct >>
fs []) >>
fs [pan_to_wordTheory.compile_prog_def] >>
fs [loop_to_wordTheory.compile_def] >>
drule mem_prog_mem_compile_prog >> fs []) >>
fs [lookup_fromAList] >>
drule ALOOKUP_MEM >>
strip_tac >>
rfs []
>- (drule loop_removeProofTheory.comp_prog_no_loops >> fs []) >>
drule loop_removeProofTheory.compile_prog_distinct_params >>
impl_tac
>- (
ho_match_mp_tac crep_to_loopProofTheory.compile_prog_distinct_params >>
fs [Abbr ‘ccode’] >>
ho_match_mp_tac pan_to_crepProofTheory.compile_prog_distinct_params >>
fs [Abbr ‘pcode’] >>
ho_match_mp_tac pan_simpProofTheory.compile_prog_distinct_params >>
fs [distinct_params_def]) >>
fs []) >>
fs [loop_to_wordProofTheory.code_rel_def] >>
rw []
>- (
fs [lookup_fromAList] >>
dxrule ALOOKUP_MEM >>
strip_tac >>
match_mp_tac ALOOKUP_ALL_DISTINCT_MEM >>
conj_tac
>- (
match_mp_tac first_compile_prog_all_distinct >>
fs []) >>
fs [pan_to_wordTheory.compile_prog_def] >>
fs [loop_to_wordTheory.compile_def] >>
drule mem_prog_mem_compile_prog >> fs []
)
>- (
drule pan_commonPropsTheory.lookup_some_el >> rw [] >>
imp_res_tac EL_MEM >>
gs [] >>
drule loop_removeProofTheory.comp_prog_no_loops >> fs []
)
>- (
drule pan_commonPropsTheory.lookup_some_el >> rw [] >>
imp_res_tac EL_MEM >>
gs [] >>
drule_then irule loop_removeProofTheory.compile_prog_distinct_params >>
irule crep_to_loopProofTheory.compile_prog_distinct_params >>
fs [Abbr ‘ccode’] >>
ho_match_mp_tac pan_to_crepProofTheory.compile_prog_distinct_params >>
fs [Abbr ‘pcode’] >>
ho_match_mp_tac pan_simpProofTheory.compile_prog_distinct_params >>
fs [distinct_params_def])
) >>
drule fstate_rel_imp_semantics >>
disch_then irule >>
fs [Abbr ‘lst’, loop_state_def,
Abbr ‘ccode’, Abbr ‘pcode’,
pan_to_wordTheory.compile_prog_def] >>
fs [lookup_fromAList] >>
irule_at Any ALOOKUP_ALL_DISTINCT_MEM >>
irule_at Any crep_to_loopProofTheory.first_compile_prog_all_distinct >>
fs [crep_to_loopTheory.compile_prog_def] >>
simp [MAP2_MAP, MEM_MAP, EXISTS_PROD] >>
csimp [MEM_ZIP, EL_GENLIST] >>
fs [pan_to_crepTheory.compile_prog_def, pan_simpTheory.compile_prog_def] >>
simp [EL_MAP] >>
simp [pan_to_crepTheory.crep_vars_def, panLangTheory.size_of_shape_def]
QED
(*** no_install/no_alloc/no_mt lemmas ***)
Theorem pan_to_word_compile_prog_no_install_code:
compile_prog c prog = prog' ⇒
no_install_code (fromAList prog')
Proof
gs[compile_prog_def]>>strip_tac>>
metis_tac[loop_to_wordProofTheory.loop_compile_no_install_code]
QED
Theorem pan_to_word_compile_prog_no_alloc_code:
compile_prog c prog = prog' ⇒
no_alloc_code (fromAList prog')
Proof
gs[compile_prog_def]>>strip_tac>>
metis_tac[loop_to_wordProofTheory.loop_compile_no_alloc_code]
QED
Theorem pan_to_word_compile_prog_no_mt_code:
compile_prog c prog = prog' ⇒
no_mt_code (fromAList prog')
Proof
gs[compile_prog_def]>>strip_tac>>
metis_tac[loop_to_wordProofTheory.loop_compile_no_mt_code]
QED
(*** pan_to_word good_handlers ***)
Theorem pan_to_word_good_handlers:
compile_prog c prog = prog' ⇒
EVERY (λ(n,m,pp). good_handlers n pp) prog'
Proof
gs[compile_prog_def,
loop_to_wordTheory.compile_def]>>
strip_tac>>
irule loop_to_wordProofTheory.loop_to_word_good_handlers>>metis_tac[]
QED
(* lab_pres *)
Theorem pan_to_word_compile_lab_pres:
pan_to_word$compile_prog c prog = prog' ⇒
EVERY
(λ(n,m,p).
(let
labs = extract_labels p
in
EVERY (λ(l1,l2). l1 = n ∧ l2 ≠ 0 ∧ l2 ≠ 1) labs ∧
ALL_DISTINCT labs)) prog'
Proof
strip_tac>>gs[pan_to_wordTheory.compile_prog_def]>>
gs[loop_to_wordTheory.compile_def]>>
drule loop_to_word_compile_prog_lab_pres>>gs[]
QED
(* first_name offset : lab_min *)
Theorem pan_to_word_compile_prog_lab_min:
compile_prog c pprog = wprog ⇒
EVERY (λprog. 60 ≤ FST prog) wprog
Proof
gs[pan_to_wordTheory.compile_prog_def]>>
strip_tac>>
drule_then irule loop_to_word_compile_lab_min>>
irule crep_to_loop_compile_prog_lab_min>>metis_tac[]
QED
(* inst_ok_less *)
Theorem every_inst_ok_loop_call:
∀l prog.
every_prog (loop_inst_ok c) prog ⇒
every_prog (loop_inst_ok c) (FST(loop_call$comp l prog))
Proof
ho_match_mp_tac loop_callTheory.comp_ind \\
rw[loopPropsTheory.every_prog_def,loop_callTheory.comp_def,
loop_inst_ok_def] \\
rpt(pairarg_tac \\ gvs[]) \\
rw[loopPropsTheory.every_prog_def,loop_callTheory.comp_def,
loop_inst_ok_def] \\
every_case_tac \\
rw[loopPropsTheory.every_prog_def,loop_callTheory.comp_def,
loop_inst_ok_def]
QED
val prog = “prog:'a loopLang$prog”
Theorem every_inst_ok_loop_live:
∀prog.
every_prog (loop_inst_ok c) ^prog ⇒
every_prog (loop_inst_ok c) (loop_live$comp prog)
Proof
rw[loop_liveTheory.comp_def] \\
‘(∀p prog q. every_prog (loop_inst_ok c) ^prog ⇒ every_prog (loop_inst_ok c) (FST $ shrink p prog q)) ∧
(∀p q r prog. every_prog (loop_inst_ok c) ^prog ⇒ OPTION_ALL (every_prog (loop_inst_ok c) o FST) (fixedpoint p q r prog))
’
by(pop_assum kall_tac \\
ho_match_mp_tac loop_liveTheory.shrink_ind \\ rw[] \\
gvs $ loopPropsTheory.every_prog_def::loop_inst_ok_def::
butlast(CONJUNCTS loop_liveTheory.shrink_def) \\
rpt(pairarg_tac \\ gvs[]) \\
gvs $ loopPropsTheory.every_prog_def::loop_inst_ok_def::
butlast(CONJUNCTS loop_liveTheory.shrink_def) \\
rpt TOP_CASE_TAC \\ gvs[loopPropsTheory.every_prog_def,loop_inst_ok_def] \\
rpt(pairarg_tac \\ gvs[]) \\
gvs $ loopPropsTheory.every_prog_def::loop_inst_ok_def::
butlast(CONJUNCTS loop_liveTheory.shrink_def) \\
simp[Once loop_liveTheory.shrink_def] \\
rpt(pairarg_tac \\ gvs[]) \\
rw[]) \\
‘∀prog. every_prog (loop_inst_ok c) ^prog ⇒
every_prog (loop_inst_ok c) (FST $ mark_all prog)’
by(rpt $ pop_assum kall_tac \\
ho_match_mp_tac loop_liveTheory.mark_all_ind \\
rw[loopPropsTheory.every_prog_def,loop_inst_ok_def,
loop_liveTheory.mark_all_def] \\
rpt(pairarg_tac \\ gvs[]) \\
every_case_tac \\
gvs[loopPropsTheory.every_prog_def,loop_inst_ok_def,
loop_liveTheory.mark_all_def] \\
rpt(pairarg_tac \\ gvs[]) \\
rw[loopPropsTheory.every_prog_def,loop_inst_ok_def,
loop_liveTheory.mark_all_def]) \\
metis_tac[]
QED
Theorem every_inst_ok_less_optimise:
every_prog (loop_inst_ok c) prog ⇒
every_prog (loop_inst_ok c) (optimise prog)
Proof
rw[loop_liveTheory.optimise_def] \\
metis_tac[every_inst_ok_loop_call,every_inst_ok_loop_live]
QED
Theorem every_inst_ok_less_crep_to_loop_compile_exp:
(∀ctxt n ns (e:'a crepLang$exp).
ctxt.target = c.ISA ∧
every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2) e
⇒
EVERY (every_prog (loop_inst_ok c)) (FST $ crep_to_loop$compile_exp ctxt n ns e)) ∧
(∀ctxt n ns (es: 'a crepLang$exp list).
ctxt.target = c.ISA ∧
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) es
⇒
EVERY (every_prog (loop_inst_ok c)) (FST $ crep_to_loop$compile_exps ctxt n ns es))
Proof
ho_match_mp_tac crep_to_loopTheory.compile_exp_ind \\
rw $ [loopPropsTheory.every_prog_def,loop_inst_ok_def,crep_to_loopTheory.prog_if_def,crepPropsTheory.every_exp_def] @ butlast(CONJUNCTS crep_to_loopTheory.compile_exp_def) \\
rpt(pairarg_tac \\ gvs[]) \\
gvs[loopPropsTheory.every_prog_def,loop_inst_ok_def,crep_to_loopTheory.prog_if_def]
THEN1 (gvs[DefnBase.one_line_ify NONE crep_to_loopTheory.compile_crepop_def,AllCaseEqs(),
loopPropsTheory.every_prog_def,loop_inst_ok_def] \\
rw[EVERY_MEM,MEM_MAPi] \\
rw[loopPropsTheory.every_prog_def,loop_inst_ok_def] \\
Cases_on ‘es’ \\ gvs[quantHeuristicsTheory.LIST_LENGTH_1,
crep_to_loopProofTheory.compile_exps_alt] \\
rpt(pairarg_tac \\ gvs[]) \\
metis_tac[EVERY_MEM])
THEN1 (gvs[DefnBase.one_line_ify NONE crep_to_loopTheory.compile_crepop_def,AllCaseEqs(),
loopPropsTheory.every_prog_def,loop_inst_ok_def] \\
rw[EVERY_MEM,MEM_MAPi] \\
rw[loopPropsTheory.every_prog_def,loop_inst_ok_def] \\
Cases_on ‘es’ \\ gvs[quantHeuristicsTheory.LIST_LENGTH_1,
crep_to_loopProofTheory.compile_exps_alt] \\
rpt(pairarg_tac \\ gvs[]) \\
metis_tac[EVERY_MEM]) \\
Cases_on ‘es’ \\
gvs[crep_to_loopProofTheory.compile_exps_alt] \\
rpt(pairarg_tac \\ gvs[])
QED
Theorem every_prog_loop_inst_ok_nested_seq:
∀c ps.
every_prog (loop_inst_ok c) $ nested_seq ps = EVERY (every_prog (loop_inst_ok c)) ps
Proof
strip_tac \\ Induct \\
gvs[loopPropsTheory.every_prog_def,loopLangTheory.nested_seq_def,
loop_to_wordProofTheory.loop_inst_ok_def]
QED
Theorem call_label_eq_if:
call_label ctxt e = (dest, indirect_dest) ==>
indirect_dest = (if dest = NONE then [e] else [])
Proof
simp [crep_to_loopTheory.call_label_def]
\\ rw [CaseEq "crepLang$exp"] \\ simp []
QED
Theorem every_inst_ok_less_crep_to_loop_compile:
∀ctxt ns body.
ctxt.target = c.ISA ∧
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) (exps_of body)
⇒
every_prog (loop_inst_ok c) (crep_to_loop$compile ctxt ns body)
Proof
ho_match_mp_tac crep_to_loopTheory.compile_ind \\
rw[loopPropsTheory.every_prog_def,loop_inst_ok_def,
crep_to_loopTheory.compile_def,crepPropsTheory.exps_of_def] \\
rpt(pairarg_tac \\ gvs[]) \\
gvs[loopPropsTheory.every_prog_def,loop_inst_ok_def,
crep_to_loopTheory.compile_def,
every_prog_loop_inst_ok_nested_seq] \\
gvs[EVERY_APPEND,loopPropsTheory.every_prog_def,loop_inst_ok_def,
crep_to_loopTheory.compile_def]
>~ [‘MAP2’] >-
(drule $ cj 2 every_inst_ok_less_crep_to_loop_compile_exp \\
fs [MAP2_ZIP, crep_to_loopTheory.gen_temps_def, EVERY_MAP, UNCURRY,
loopPropsTheory.every_prog_def, loop_inst_ok_def] \\
disch_then $ qspecl_then [‘ctxt.vmax + 1’,‘ns’,‘es ++ indirect_dest’] mp_tac \\
drule_then assume_tac call_label_eq_if \\
simp [] \\
gs [CaseEq "prod", CaseEq "option", crepPropsTheory.exps_of_def] \\
gvs [] \\
rpt (TOP_CASE_TAC \\ fs []) \\
gs [crepPropsTheory.exps_of_def] \\
rw[MAP2_ZIP,EVERY_MEM,crep_to_loopTheory.gen_temps_def,MEM_MAP,MEM_ZIP,PULL_EXISTS,
loopPropsTheory.every_prog_def,loop_inst_ok_def
]) \\
imp_res_tac (cj 1 every_inst_ok_less_crep_to_loop_compile_exp) \\
every_case_tac \\
gvs[loopPropsTheory.every_prog_def,loopLangTheory.nested_seq_def,
loop_to_wordProofTheory.loop_inst_ok_def,
every_prog_loop_inst_ok_nested_seq] \\
metis_tac[FST,SND,PAIR]
QED
Theorem every_inst_ok_less_comp_func:
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) (exps_of body) ⇒
every_prog (loop_inst_ok c) (comp_func c.ISA (crep_to_loop$make_funcs prog) params body)
Proof
rw[crep_to_loopTheory.comp_func_def,crep_to_loopTheory.make_funcs_def] \\
match_mp_tac every_inst_ok_less_crep_to_loop_compile \\
rw[crep_to_loopTheory.mk_ctxt_def]
QED
Theorem every_inst_ok_arith_simp_exp:
! exp.
every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2) exp ==>
every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2) (simp_exp exp)
Proof
ho_match_mp_tac crep_arithTheory.simp_exp_ind
\\ simp [crep_arithTheory.simp_exp_def, crepPropsTheory.every_exp_def]
\\ rw [crep_arithTheory.mul_const_def]
\\ every_case_tac \\ fs []
\\ gvs [listTheory.MAP_EQ_CONS]
\\ fs [crepPropsTheory.every_exp_def]
\\ simp [EVERY_MAP]
\\ fs [EVERY_MEM]
QED
Theorem every_inst_ok_arith_simp_prog:
! prog.
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2))
(exps_of prog) ==>
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2))
(exps_of (simp_prog prog))
Proof
ho_match_mp_tac crep_arithTheory.simp_prog_ind
\\ simp [crep_arithTheory.simp_prog_def, crepPropsTheory.exps_of_def]
\\ simp [every_inst_ok_arith_simp_exp]
\\ rw []
\\ every_case_tac \\ fs []
\\ fs [crepPropsTheory.exps_of_def, every_inst_ok_arith_simp_exp]
\\ fs [EVERY_MAP]
\\ fs [EVERY_MEM, every_inst_ok_arith_simp_exp]
QED
Theorem every_inst_ok_nested_decs:
∀ns ps p.
LENGTH ns = LENGTH ps ⇒
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) (exps_of(nested_decs ns ps p)) =
(EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) ps ∧
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) (exps_of p))
Proof
ho_match_mp_tac crepLangTheory.nested_decs_ind \\
rw[crepLangTheory.nested_decs_def,crepPropsTheory.exps_of_def] \\
metis_tac[]
QED
Theorem every_inst_ok_less_pan_to_crep_comp_field:
∀index shapes es es' sh.
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) es ∧
comp_field index shapes es = (es',sh) ⇒
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) es'
Proof
Induct_on ‘shapes’ \\
rw[pan_to_crepTheory.comp_field_def,crepPropsTheory.every_exp_def,EVERY_TAKE] >>
res_tac >>
gvs[EVERY_DROP,EVERY_TAKE]
QED
Theorem every_inst_ok_less_pan_to_crep_load_shape:
∀w n e.
every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2) e ⇒
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2))
(load_shape w n e)
Proof
Induct_on ‘n’ \\
rw[crepLangTheory.load_shape_def,crepPropsTheory.every_exp_def]
QED
Theorem every_inst_ok_less_pan_to_crep_cexp_heads:
∀ces es.
EVERY (EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2))) ces ∧
cexp_heads ces = SOME es
⇒
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) es
Proof
Induct_on ‘ces’ \\
rw[pan_to_crepTheory.cexp_heads_def] \\
gvs[AllCaseEqs()]
QED
Theorem every_inst_ok_less_pan_to_crep_compile_exp:
∀ctxt e es sh.
every_exp (λx. ∀op es. x = Panop op es ⇒ LENGTH es = 2) e ∧
compile_exp ctxt e = (es,sh) ⇒
EVERY (every_exp (λx. ∀op es. x = Crepop op es ⇒ LENGTH es = 2)) es
Proof
ho_match_mp_tac pan_to_crepTheory.compile_exp_ind \\
rw[pan_to_crepTheory.compile_exp_def,
crepPropsTheory.every_exp_def,
panPropsTheory.every_exp_def
] \\
gvs[crepPropsTheory.every_exp_def,
panPropsTheory.every_exp_def,
AllCaseEqs(),EVERY_MAP,EVERY_FLAT] \\
rpt(pairarg_tac \\ gvs[]) \\
gvs[AllCaseEqs(),crepPropsTheory.every_exp_def,
panPropsTheory.every_exp_def] \\
imp_res_tac every_inst_ok_less_pan_to_crep_comp_field \\
imp_res_tac every_inst_ok_less_pan_to_crep_load_shape
>~ [‘cexp_heads’] >-
(drule_at (Pos last) every_inst_ok_less_pan_to_crep_cexp_heads \\
reverse impl_tac >- metis_tac[] \\
gvs[EVERY_MEM,MEM_MAP,PULL_EXISTS] \\
metis_tac[FST,SND,PAIR])
>~ [‘cexp_heads’] >-
(drule_at (Pos last) every_inst_ok_less_pan_to_crep_cexp_heads \\
impl_keep_tac >-
(gvs[EVERY_MEM,MEM_MAP,PULL_EXISTS] \\
metis_tac[FST,SND,PAIR]) \\
reverse $ rw[]
>- metis_tac[ETA_AX] \\
gvs[crepPropsTheory.cexp_heads_simp_def,pan_to_crepProofTheory.cexp_heads_eq]) \\