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divScript.sml
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(*
Examples of non-termination.
*)
open preamble basis
open integerTheory cfDivTheory cfDivLib
val _ = temp_delsimps ["NORMEQ_CONV"]
val _ = new_theory "div";
val _ = translation_extends "basisProg";
(* A simple pure non-terminating loop *)
val _ = process_topdecs `
fun pureLoop x = pureLoop x;
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem pureLoop_spec:
!xv s u ns.
limited_parts ns p ==>
app (p:'ffi ffi_proj) ^(fetch_v "pureLoop" st) [xv]
(one (FFI_part s u ns [])) (POSTd io. io = [||])
Proof
xcf_div "pureLoop" st
\\ MAP_EVERY qexists_tac [`K emp`, `K []`, `K(K T)`, `K s`, `u`]
\\ xsimpl \\ rw [lprefix_lub_def]
\\ xvar \\ xsimpl
QED
(* Lemma needed for examples with integers *)
val eq_v_INT_thm = Q.prove(`(INT --> INT --> BOOL) $= eq_v`,
metis_tac[DISCH_ALL mlbasicsProgTheory.eq_v_thm,EqualityType_NUM_BOOL]);
(* A conditionally terminating loop *)
val _ = process_topdecs `
fun condLoop x = if x = 0 then 0 else condLoop (x - 1);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem condLoop_spec:
!x xv s u ns.
limited_parts ns p /\ INT x xv ==>
app (p:'ffi ffi_proj) ^(fetch_v "condLoop" st) [xv]
(one (FFI_part s u ns []))
(POSTvd
(\v. &(0 <= x /\ INT 0 v) * one (FFI_part s u ns []))
(\io. x < 0 /\ io = [||]))
Proof
strip_tac \\ Cases_on `x`
THEN1 (
pop_assum (K ALL_TAC) \\ qid_spec_tac `n`
\\ Induct_on `n` \\ rpt strip_tac
THEN1 (
xcf "condLoop" st
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlit \\ xsimpl)
\\ xcf "condLoop" st
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlet_auto THEN1 xsimpl
\\ xapp
\\ MAP_EVERY qexists_tac [`emp`, `u`, `s`, `ns`]
\\ xsimpl \\ fs [INT_def, NUM_def])
THEN1 (
fs [SEP_CLAUSES] \\ fs [SEP_F_to_cond, POSTvd_def, GSYM POSTd_def]
\\ xcf_div "condLoop" st
\\ MAP_EVERY qexists_tac
[`K emp`, `K []`, `\n v. ?i. INT i v /\ i < 0`, `K s`, `u`]
\\ xsimpl \\ qexists_tac `-&n` \\ simp[lprefix_lub_def]
\\ rw[]
\\ xlet_auto >- xsimpl
\\ xif \\ fs[]
\\ xlet_auto >- xsimpl
\\ xvar \\ xsimpl \\ fs[INT_def] \\ intLib.COOPER_TAC)
\\ rpt strip_tac
\\ xcf "condLoop" st
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlit \\ xsimpl
QED
(* Another conditionally terminating loop, using FFI_full *)
val _ = process_topdecs `
fun oddLoop x = if x = 0 then () else oddLoop(x-2);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem oddLoop_spec:
!i iv.
INT i iv /\ ¬(2 int_divides i) ==>
app (p:'ffi ffi_proj) ^(fetch_v "oddLoop" st) [iv]
(one (FFI_full [])) (POSTd io. io = [||])
Proof
xcf_div_FFI_full "oddLoop" st
\\ MAP_EVERY qexists_tac [`K emp`,`K []`,`(\n. Litv(IntLit(i - 2 * &n)))`]
\\ simp[lprefix_lub_def]
\\ conj_tac >- (fs[ml_translatorTheory.INT_def])
\\ conj_tac >- xsimpl
\\ fs[SEP_CLAUSES]
\\ strip_tac
\\ rename1 `2 * SUC j`
\\ xlet `POSTv bv. &BOOL F bv * one(FFI_full [])`
>- (xapp_spec eq_v_INT_thm \\ xsimpl
\\ fs[ml_translatorTheory.BOOL_def,semanticPrimitivesTheory.Boolv_def]
\\ rw[] \\ intLib.COOPER_TAC)
\\ xif
\\ asm_exists_tac \\ simp[]
\\ xlet `POSTv iv2. &INT (i − &(2 * SUC j)) iv2 * one(FFI_full [])`
>- (xapp \\ xsimpl \\ fs[ml_translatorTheory.INT_def]
\\ intLib.COOPER_TAC)
\\ xvar \\ xsimpl \\ fs[ml_translatorTheory.INT_def]
QED
(* A loop containing a divergent function *)
val _ = process_topdecs `
fun outerLoop x = if x = 5000 then pureLoop () else outerLoop (x + 1);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem outerLoop_spec:
!n nv s u ns.
limited_parts ns p /\ NUM n nv ==>
app (p:'ffi ffi_proj) ^(fetch_v "outerLoop" st) [nv]
(one (FFI_part s u ns [])) (POSTd io. io = [||])
Proof
strip_tac \\ Cases_on `n <= 5000`
THEN1 (
Induct_on `5000 - n` \\ rw []
THEN1 (
xcf "outerLoop" st
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xapp \\ fs [])
\\ xcf "outerLoop" st
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlet_auto THEN1 xsimpl
\\ xapp \\ fs []
\\ qexists_tac `n + 1` \\ fs [])
\\ xcf_div "outerLoop" st
\\ MAP_EVERY qexists_tac
[`K emp`, `K []`, `\i. NUM (n + i)`, `K s`, `u`]
\\ xsimpl \\ rw [lprefix_lub_def]
\\ xlet_auto THEN1 xsimpl
\\ xif \\ instantiate
\\ xlet_auto THEN1 xsimpl
\\ xvar \\ xsimpl \\ fs [ADD_CLAUSES, GSYM ADD1]
QED
(* A small IO model needed for IO examples *)
val names_def = Define `names = ["put_char"; "get_char"]`;
val put_char_event_def = Define `
put_char_event c = IO_event (ExtCall "put_char") [n2w (ORD c)] []`;
val put_str_event_def = Define `
put_str_event cs = IO_event (ExtCall "put_char") (MAP (n2w o ORD) cs) []`;
val get_char_event_def = Define `
get_char_event c = IO_event (ExtCall "get_char") [] [0w, 1w; 0w, n2w (ORD c)]`;
val get_char_eof_event_def = Define `
get_char_eof_event = IO_event (ExtCall "get_char") [] [0w, 0w; 0w, 0w]`;
val update_def = PmatchHeuristics.with_classic_heuristic Define `
(update "put_char" cs [] s = SOME (FFIreturn [] s)) /\
(update "get_char" [] [0w; 0w] s = case destStream s of
| NONE => NONE
| SOME ll => if ll = [||] then
SOME (FFIreturn [0w; 0w] s)
else
SOME (FFIreturn [1w; n2w (THE (LHD ll))]
(Stream (THE (LTL ll)))))`
val State_def = Define `
State input = Stream (LMAP ORD input)`
val SIO_def = Define `
SIO input events =
one (FFI_part (State input) update names events)`
val _ = process_topdecs `
fun put_char c = let
val s = String.implode [c]
val a = Word8Array.array 0 (Word8.fromInt 0)
val _ = #(put_char) s a
in () end
` |> append_prog;
val _ = process_topdecs `
fun put_line l = let
val s = l ^ "\n"
val a = Word8Array.array 0 (Word8.fromInt 0)
val _ = #(put_char) s a
in () end
` |> append_prog;
val _ = process_topdecs `
fun get_char (u:unit) = let
val a = Word8Array.array 2 (Word8.fromInt 0)
val _ = #(get_char) "" a
in if Word8Array.sub a 0 = Word8.fromInt 1 then
Some (Char.chr (Word8.toInt (Word8Array.sub a 1)))
else
None
end
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem put_char_spec:
!c cv input events.
limited_parts names p /\ CHAR c cv ==>
app (p:'ffi ffi_proj) ^(fetch_v "put_char" st) [cv]
(SIO input events)
(POSTv v. &UNIT_TYPE () v *
SIO input (SNOC (put_char_event c) events))
Proof
rpt strip_tac
\\ xcf "put_char" st
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xlet
`POSTv v. &STRING_TYPE (implode [c]) v * SIO input events`
THEN1 (xapp \\ xsimpl \\ qexists_tac `[c]` \\ fs [LIST_TYPE_def])
\\ xlet_auto THEN1 xsimpl
\\ xlet_auto THEN1 xsimpl
\\ rename1 `W8ARRAY av`
\\ xlet
`POSTv v. &UNIT_TYPE () v * W8ARRAY av [] *
SIO input (SNOC (put_char_event c) events)`
THEN1 (
xffi \\ xsimpl \\ fs [SIO_def]
\\ MAP_EVERY qexists_tac
[`[n2w (ORD c)]`, `emp`, `State input`, `update`, `names`, `events`]
\\ fs [update_def, put_char_event_def, names_def, SNOC_APPEND,
implode_def, STRING_TYPE_def, State_def]
\\ xsimpl)
\\ xcon \\ xsimpl
QED
Theorem put_line_spec:
!l lv input events.
limited_parts names p /\ STRING_TYPE (strlit l) lv ==>
app (p:'ffi ffi_proj) ^(fetch_v "put_line" st) [lv]
(SIO input events)
(POSTv v. &UNIT_TYPE () v *
SIO input (SNOC (put_str_event (l ++ "\n")) events))
Proof
rpt strip_tac
\\ xcf "put_line" st
\\ xlet_auto THEN1 xsimpl
\\ xlet_auto THEN1 xsimpl
\\ xlet_auto THEN1 xsimpl
\\ rename1 `W8ARRAY av`
\\ xlet
`POSTv v. &UNIT_TYPE () v * W8ARRAY av [] *
SIO input (SNOC (put_str_event (l ++ "\n")) events)`
THEN1 (
xffi \\ xsimpl \\ fs [SIO_def]
\\ MAP_EVERY qexists_tac
[`MAP (n2w o ORD) (l ++ "\n")`, `emp`, `State input`, `update`,
`names`, `events`]
\\ fs [update_def, put_str_event_def, names_def, SNOC_APPEND,
STRING_TYPE_def, State_def, strlit_STRCAT, MAP_MAP_o, o_DEF,
CHR_ORD, ORD_BOUND]
\\ xsimpl)
\\ xcon \\ xsimpl
QED
Theorem get_char_spec:
!uv c input events.
limited_parts names p /\ UNIT_TYPE () uv ==>
app (p:'ffi ffi_proj) ^(fetch_v "get_char" st) [uv]
(SIO input events)
(POSTv v. &OPTION_TYPE CHAR (LHD input) v *
if input = [||] then
SIO input (SNOC (get_char_eof_event) events)
else
SIO (THE (LTL input))
(SNOC (get_char_event (THE (LHD input))) events))
Proof
rpt strip_tac
\\ xcf "get_char" st
\\ qmatch_goalsub_abbrev_tac `_ * sio`
\\ qabbrev_tac `a:word8 list = if input = [||] then
[0w; 0w]
else
[1w; n2w (ORD (THE (LHD input)))]`
\\ xmatch \\ rpt (xlet_auto THEN1 xsimpl)
\\ Cases_on `input` \\ (
fs [] \\ rename1 `W8ARRAY av`
\\ xlet `POSTv v. &UNIT_TYPE () v * W8ARRAY av a * sio`
THEN1 (
xffi \\ xsimpl \\ fs [SIO_def]
\\ qpat_abbrev_tac `s = State _`
\\ MAP_EVERY qexists_tac [`emp`, `s`, `update`, `names`, `events`]
\\ unabbrev_all_tac
\\ fs [update_def, get_char_event_def, get_char_eof_event_def,
names_def, SNOC_APPEND, EVAL ``REPLICATE 2 0w``, State_def]
\\ xsimpl)
\\ rpt (xlet_auto THEN1 xsimpl)
\\ xlet_auto THEN1 (xsimpl \\ fs [WORD_def])
\\ xif \\ instantiate
\\ rpt (xlet_auto THEN1 xsimpl)
\\ xcon \\ xsimpl \\ fs [OPTION_TYPE_def, CHR_ORD, ORD_BOUND])
QED
(* TODO: Move REPLICATE_LIST and lemmas to an appropriate theory *)
val REPLICATE_LIST_def = Define `
(REPLICATE_LIST l 0 = []) /\
(REPLICATE_LIST l (SUC n) = REPLICATE_LIST l n ++ l)`
Theorem REPLICATE_LIST_SNOC:
!x n. SNOC x (REPLICATE_LIST [x] n) = REPLICATE_LIST [x] (SUC n)
Proof
rw [REPLICATE_LIST_def]
QED
Theorem REPLICATE_LIST_APPEND:
!l n. REPLICATE_LIST l n ++ l = REPLICATE_LIST l (SUC n)
Proof
rw [REPLICATE_LIST_def]
QED
Theorem REPLICATE_LIST_APPEND_SYM:
!l n. REPLICATE_LIST l n ++ l = l ++ REPLICATE_LIST l n
Proof
strip_tac \\ Induct_on `n` \\ fs [REPLICATE_LIST_def]
QED
Theorem REPLICATE_LIST_LENGTH:
!l n. LENGTH (REPLICATE_LIST l n) = LENGTH l * n
Proof
Induct_on `n` THEN1 (EVAL_TAC \\ fs [])
\\ rw [REPLICATE_LIST_def, MULT_CLAUSES]
QED
Theorem LPREFIX_REPLICATE_LIST_LREPEAT:
!l n. LPREFIX (fromList (REPLICATE_LIST l n)) (LREPEAT l)
Proof
strip_tac \\ Induct_on `n`
\\ fs [REPLICATE_LIST_def, REPLICATE_LIST_APPEND_SYM, GSYM LAPPEND_fromList]
\\ rw [Once LREPEAT_thm] \\ fs [LPREFIX_APPEND]
\\ qexists_tac `ll` \\ fs [LAPPEND_ASSOC]
QED
Theorem LTAKE_EQ_MULT:
!ll1 ll2 m.
0 < m /\ ~LFINITE ll1 /\
(!n. LTAKE (m * n) ll1 = LTAKE (m * n) ll2) ==>
(!n. LTAKE n ll1 = LTAKE n ll2)
Proof
rw []
\\ first_x_assum (qspec_then `n` mp_tac) \\ strip_tac
\\ drule NOT_LFINITE_TAKE
\\ disch_then (qspec_then `m * n` mp_tac) \\ strip_tac \\ fs []
\\ rename1 `SOME l`
\\ `n <= m * n` by fs []
\\ `LTAKE n ll1 = SOME (TAKE n l)` by (
irule LTAKE_TAKE_LESS \\ qexists_tac `m * n` \\ fs [])
\\ `LTAKE n ll2 = SOME (TAKE n l)` by (
irule LTAKE_TAKE_LESS \\ qexists_tac `m * n` \\ fs [])
\\ fs []
QED
Theorem LTAKE_LAPPEND_fromList:
!ll l n.
LTAKE (n + LENGTH l) (LAPPEND (fromList l) ll) =
OPTION_MAP (APPEND l) (LTAKE n ll)
Proof
rw [] \\ Cases_on `LTAKE n ll` \\ fs []
THEN1 (
`LFINITE ll` by (fs [LFINITE] \\ instantiate)
\\ drule LFINITE_HAS_LENGTH \\ strip_tac \\ rename1 `SOME m`
\\ irule LTAKE_LLENGTH_NONE
\\ qexists_tac `m + LENGTH l` \\ rw []
THEN1 (
drule LTAKE_LLENGTH_SOME \\ strip_tac
\\ Cases_on `n ≤ m` \\ fs []
\\ drule (GEN_ALL LTAKE_TAKE_LESS)
\\ disch_then drule \\ fs [])
\\ fs [LLENGTH_APPEND, LFINITE_fromList])
\\ Induct_on `l` \\ rw []
\\ fs [LTAKE_CONS_EQ_SOME]
\\ instantiate
QED
Theorem REPLICATE_LIST_LREPEAT:
!l ll.
l <> [] /\ (!n. LPREFIX (fromList (REPLICATE_LIST l n)) ll) ==>
ll = LREPEAT l
Proof
rw [LTAKE_EQ]
\\ Cases_on `toList ll`
THEN1 (
irule LTAKE_EQ_MULT
\\ `~LFINITE ll` by fs [LFINITE_toList_SOME] \\ fs []
\\ qexists_tac `LENGTH l`
\\ `0 < LENGTH l` by (Cases_on `l` \\ fs []) \\ rw []
\\ rpt (pop_assum mp_tac) \\ qid_spec_tac `ll`
\\ Induct_on `n` \\ rw []
\\ `?ll1. ll = LAPPEND (fromList l) ll1` by (
first_x_assum (qspec_then `1` mp_tac) \\ strip_tac
\\ fs [LPREFIX_APPEND, EVAL ``REPLICATE_LIST l 1``]
\\ rename1 `LAPPEND _ ll1`
\\ qexists_tac `ll1` \\ fs [])
\\ `~LFINITE ll1` by fs [LFINITE_APPEND, LFINITE_fromList]
\\ `toList ll1 = NONE` by fs [LFINITE_toList_SOME]
\\ `!n. LPREFIX (fromList (REPLICATE_LIST l n)) ll1` by (
strip_tac \\ rename1 `REPLICATE_LIST _ n1`
\\ first_x_assum (qspec_then `SUC n1` mp_tac) \\ strip_tac
\\ fs [LPREFIX_fromList] \\ rfs []
\\ fs [REPLICATE_LIST_LENGTH, MULT_CLAUSES]
\\ qspecl_then [`ll1`, `l`, `n1 * LENGTH l`] mp_tac
LTAKE_LAPPEND_fromList \\ strip_tac
\\ fs [REPLICATE_LIST_def, REPLICATE_LIST_APPEND_SYM])
\\ first_x_assum (qspec_then `ll1` drule)
\\ rpt (disch_then drule) \\ strip_tac \\ rveq
\\ `LENGTH l * SUC n = LENGTH l * n + LENGTH l` by fs [MULT_CLAUSES]
\\ qspecl_then [`ll1`, `l`, `LENGTH l * n`] mp_tac
LTAKE_LAPPEND_fromList \\ strip_tac
\\ rw [Once LREPEAT_thm]
\\ qspecl_then [`LREPEAT l`, `l`, `LENGTH l * n`] mp_tac
LTAKE_LAPPEND_fromList \\ strip_tac
\\ fs [])
\\ first_x_assum (qspec_then `SUC (LENGTH x)` mp_tac) \\ strip_tac
\\ fs [LPREFIX_fromList] \\ rfs []
\\ drule IS_PREFIX_LENGTH \\ strip_tac
\\ fs [REPLICATE_LIST_LENGTH]
\\ Cases_on `l` \\ Cases_on `LENGTH x` \\ fs [MULT_CLAUSES]
QED
(* A non-terminating loop with side effects *)
val _ = process_topdecs `
fun printLoop c = (put_char c; printLoop c);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
Theorem printLoop_spec:
!c cv.
limited_parts names p /\ CHAR c cv ==>
app (p:'ffi ffi_proj) ^(fetch_v "printLoop" st) [cv]
(SIO [||] []) (POSTd io. io = LREPEAT [put_char_event c])
Proof
xcf_div_rule IMP_app_POSTd_one_FFI_part_FLATTEN "printLoop" st
\\ MAP_EVERY qexists_tac
[`K emp`, `\i. if i = 0 then [] else [put_char_event c]`, `K ($= cv)`,
`K (State [||])`, `update`]
\\ SIMP_TAC std_ss [LFLATTEN_LGENLIST_REPEAT,o_DEF,K_DEF,Once LGENLIST_NONE_UNFOLD,
LFLATTEN_THM, CONJUNCT1 llistTheory.fromList_def]
\\ fs [GSYM SIO_def, REPLICATE_LIST_def]
\\ xsimpl
\\ xlet `POSTv v. &UNIT_TYPE () v *
SIO [||] [put_char_event c]`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac [`emp`, `[||]`, `[]`, `c`]
\\ xsimpl)
\\ xvar \\ xsimpl
QED
(* The Unix yes program *)
val _ = process_topdecs `
fun yes u = (put_line "y"; yes u);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
val io_events_def = Define `
io_events = SIO [||]`;
Overload yes = ``yes_v``
Theorem yes_spec_lemma:
!uv.
limited_parts names p ==>
app (p:'ffi ffi_proj) ^(fetch_v "yes" st) [arg]
(io_events []) (POSTd io. io = LREPEAT [put_str_event "y\n"])
Proof
xcf_div "yes" st
\\ MAP_EVERY qexists_tac
[`K emp`, `\i. REPLICATE_LIST [put_str_event "y\n"] i`, `K ($= arg)`,
`K (State [||])`, `update`]
\\ fs [GSYM SIO_def, REPLICATE_LIST_def, io_events_def]
\\ xsimpl \\ rw [lprefix_lub_def]
THEN1 (
xlet `POSTv v. &UNIT_TYPE () v *
SIO [||]
(REPLICATE_LIST [put_str_event "y\n"] (SUC i))`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac
[`emp`, `"y"`, `[||]`, `REPLICATE_LIST [put_str_event "y\n"] i`]
\\ fs [REPLICATE_LIST_SNOC] \\ xsimpl)
\\ xvar \\ fs [REPLICATE_LIST_APPEND] \\ xsimpl)
THEN1 fs [LPREFIX_REPLICATE_LIST_LREPEAT]
\\ fs [PULL_EXISTS]
\\ qmatch_goalsub_abbrev_tac `LPREFIX ll`
\\ `ub = ll` suffices_by simp []
\\ unabbrev_all_tac
\\ irule REPLICATE_LIST_LREPEAT \\ fs []
QED
val yes_spec = save_thm("yes_spec", yes_spec_lemma |> SPEC_ALL |> UNDISCH_ALL);
(* An IO-conditional loop with side effects *)
val _ = process_topdecs `
fun catLoop u = case get_char () of
None => ()
| Some c => (put_char c; catLoop u);
` |> append_prog;
val st = ml_translatorLib.get_ml_prog_state();
val cat_def = Define `
cat ll = LFLATTEN (LMAP (\c. fromList [get_char_event c;
put_char_event c]) ll)`
Theorem cat_LCONS:
!h t. cat (h ::: t) = LAPPEND (fromList [get_char_event h;
put_char_event h]) (cat t)
Proof
rw [cat_def]
QED
Theorem LMAP_EQ_LGENLIST:
¬LFINITE ll
==>
LMAP f ll =
LGENLIST (\x. f(THE(LNTH x ll))) NONE
Proof
rw[LNTH_EQ,LNTH_LMAP,LNTH_LGENLIST,LFINITE_LNTH_NONE,
GSYM quantHeuristicsTheory.IS_SOME_EQ_NOT_NONE,IS_SOME_EXISTS] >>
metis_tac[THE_DEF]
QED
(* TODO: move LFINITE_LFLATTEN and every_LNTH to llistTheory or similar *)
Theorem LFINITE_LFLATTEN:
!lll:'a llist llist.
every (\ll. LFINITE ll /\ ll <> LNIL) lll ==>
LFINITE (LFLATTEN lll) = LFINITE lll
Proof
`!lll.
LFINITE lll ==> llist$every (\ll. LFINITE ll /\ ll <> LNIL) lll ==>
LFINITE (LFLATTEN lll)` by (ho_match_mp_tac LFINITE_ind \\ fs [LFINITE_APPEND])
\\ qsuff_tac `!x.
LFINITE x ==>
!lll. x = LFLATTEN lll /\ llist$every (\ll. LFINITE ll /\ ll <> LNIL) lll ==>
LFINITE lll` THEN1 (metis_tac [])
\\ ho_match_mp_tac LFINITE_ind
\\ fs [PULL_FORALL] \\ rw []
THEN1 (Cases_on `lll` \\ fs [])
\\ rename [`_ = LFLATTEN lll2`]
\\ Cases_on `lll2` \\ fs []
\\ rename [`h2 <> _`]
\\ Cases_on `h2` \\ fs [] \\ rw []
\\ rename [`LAPPEND t2`]
\\ Cases_on `t2` \\ fs []
\\ rename [`LAPPEND t1`]
\\ first_x_assum (qspec_then `(h:::t1) ::: t` mp_tac) \\ fs []
QED
Theorem every_LNTH:
!P ll. every P ll <=> !n e. LNTH n ll = SOME e ==> P e
Proof
fs [every_def,exists_LNTH] \\ metis_tac []
QED
Theorem cat_LFINITE:
!ll. LFINITE ll <=> LFINITE (cat ll)
Proof
rw [cat_def] \\ qmatch_goalsub_abbrev_tac `LFLATTEN ll'`
\\ `LFINITE (LFLATTEN ll') <=> LFINITE ll'`
suffices_by (unabbrev_all_tac \\ fs [LFINITE_MAP])
\\ irule LFINITE_LFLATTEN
\\ rw [every_LNTH] \\ unabbrev_all_tac \\ fs []
QED
Theorem cat_LTAKE_SUC:
!ll c n.
~LFINITE ll /\ LNTH n ll = SOME c ==>
THE (LTAKE (2 * SUC n) (cat ll)) =
THE (LTAKE (2 * n) (cat ll)) ++
[get_char_event c; put_char_event c]
Proof
Induct_on `n` \\ rw []
\\ qmatch_goalsub_abbrev_tac `[g; p]`
THEN1 (
Cases_on `ll` \\ fs [cat_LCONS] \\ rveq
\\ `2 = LENGTH [g; p]` by EVAL_TAC
\\ `IS_SOME (LTAKE (LENGTH [g; p]) (fromList [g; p]))`
by fs [LTAKE_fromList]
\\ drule LTAKE_LAPPEND1
\\ disch_then (qspec_then `cat t` mp_tac) \\ strip_tac
\\ fs [LTAKE_fromList])
\\ Cases_on `ll` \\ fs [cat_LCONS]
\\ qmatch_goalsub_abbrev_tac `g' ::: p' ::: _`
\\ `2 * SUC n = SUC (SUC (2 * n))` by fs [MULT_CLAUSES]
\\ `2 * SUC (SUC n) = SUC (SUC (2 * SUC n))` by fs [MULT_CLAUSES]
\\ rw [] \\ fs []
\\ `SUC (SUC (2 * n)) = 2 * SUC n` by fs [MULT_CLAUSES]
\\ rw [] \\ fs [cat_LFINITE]
\\ drule NOT_LFINITE_TAKE \\ strip_tac
\\ first_assum (qspec_then `2 * SUC n` mp_tac)
\\ first_x_assum (qspec_then `2 * n` mp_tac)
\\ rpt strip_tac \\ rw []
\\ first_x_assum (qspecl_then [`t`, `c`] drule)
\\ disch_then drule \\ fs []
QED
Theorem LPREFIX_LTAKE:
!ll1 ll2.
~LFINITE ll1 /\ (!n. LPREFIX (fromList (THE (LTAKE n ll1))) ll2) ==>
ll1 = ll2
Proof
rw [LTAKE_EQ]
\\ Cases_on `toList ll2`
THEN1 (
drule NOT_LFINITE_TAKE
\\ disch_then (qspec_then `n` mp_tac) \\ strip_tac
\\ drule LTAKE_LENGTH \\ strip_tac
\\ first_x_assum (qspec_then `n` mp_tac) \\ strip_tac
\\ fs [LPREFIX_fromList] \\ rfs [])
\\ rename1 `SOME l`
\\ first_x_assum (qspec_then `SUC (LENGTH l)` mp_tac) \\ strip_tac
\\ fs [LPREFIX_fromList] \\ rfs []
\\ drule NOT_LFINITE_TAKE
\\ disch_then (qspec_then `SUC (LENGTH l)` mp_tac) \\ strip_tac
\\ drule LTAKE_LENGTH \\ strip_tac \\ fs []
\\ drule IS_PREFIX_LENGTH \\ strip_tac \\ fs []
QED
Theorem IS_PREFIX_TAKE:
!l n. TAKE n l ≼ l
Proof
Induct_on `l` \\ Cases_on `n` \\ rw []
QED
Theorem LPREFIX_LTAKE_MULT:
!ll1 ll2 m.
0 < m /\ ~LFINITE ll1 /\
(!n. LPREFIX (fromList (THE (LTAKE (m * n) ll1))) ll2) ==>
(!n. LPREFIX (fromList (THE (LTAKE n ll1))) ll2)
Proof
rw []
\\ first_x_assum (qspec_then `n` mp_tac) \\ strip_tac
\\ irule LPREFIX_TRANS \\ instantiate
\\ fs [LPREFIX_fromList_fromList]
\\ drule NOT_LFINITE_TAKE
\\ disch_then (qspec_then `m * n` mp_tac) \\ strip_tac
\\ `n <= m * n` by simp []
\\ drule (GEN_ALL LTAKE_TAKE_LESS)
\\ disch_then drule \\ rw [IS_PREFIX_TAKE]
QED
Theorem catLoop_spec:
!uv input.
limited_parts names p ==>
app (p:'ffi ffi_proj) ^(fetch_v "catLoop" st) [uv]
(SIO input []) (POSTvd
(\v. &(LFINITE input /\ UNIT_TYPE () v) *
SIO [||]
(SNOC get_char_eof_event (THE(toList(cat input)))))
(\io. ~LFINITE input /\ io = cat input))
Proof
rw [] \\ Cases_on `LFINITE input` \\ fs [POSTvd_def, SEP_CLAUSES]
\\ fs [SEP_F_to_cond, GSYM POSTv_def, GSYM POSTd_def]
THEN1 (
qmatch_goalsub_abbrev_tac `app _ f`
\\ qsuff_tac `
(\input.
!uv events.
app p f [uv] (SIO input events)
(POSTv v. &UNIT_TYPE () v *
SIO [||]
(events ++ SNOC get_char_eof_event
(THE (toList (cat input))))))
input`
THEN1 (
rw [] \\ pop_assum (qspecl_then [`uv`, `[]`] mp_tac) \\ fs [])
\\ irule LFINITE_STRONG_INDUCTION \\ rw []
\\ unabbrev_all_tac
THEN1 (
xcf "catLoop" st
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xlet `POSTv v. &OPTION_TYPE CHAR NONE v *
SIO [||] (SNOC (get_char_eof_event) events)`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac [`emp`, `[||]`, `events`]
\\ xsimpl)
\\ xmatch \\ fs [OPTION_TYPE_def]
\\ reverse (rw []) THEN1 EVAL_TAC
THEN1 (xcon \\ fs [toList, cat_def, SNOC_APPEND] \\ xsimpl)
\\ EVAL_TAC \\ simp [] \\ EVAL_TAC)
\\ xcf "catLoop" st
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xlet `POSTv v. &OPTION_TYPE CHAR (SOME h) v *
SIO t (SNOC (get_char_event h) events)`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac [`emp`, `h ::: t`, `events`]
\\ xsimpl)
\\ xmatch \\ fs [OPTION_TYPE_def] \\ reverse (rpt strip_tac)
THEN1 EVAL_TAC THEN1 EVAL_TAC
\\ xlet `POSTv v. &UNIT_TYPE () v *
SIO t
(SNOC (put_char_event h)
(SNOC (get_char_event h) events))`
THEN1 (xapp \\ fs [])
\\ qmatch_goalsub_abbrev_tac `SIO _ events1`
\\ qmatch_goalsub_abbrev_tac `_ * SIO _ events2`
\\ `events2 = events1 ++ SNOC get_char_eof_event
(THE (toList (cat t)))` by (
unabbrev_all_tac \\ fs [cat_LFINITE, cat_LCONS, toList_THM]
\\ drule LFINITE_toList \\ strip_tac \\ fs [])
\\ xapp
\\ MAP_EVERY qexists_tac [`emp`, `events1`]
\\ xsimpl)
\\ xcf_div_rule IMP_app_POSTd_one_FFI_part_FLATTEN "catLoop" st
\\ MAP_EVERY qexists_tac
[`K emp`, `\i. if i = 0 then [] else
[get_char_event(THE(LNTH (i-1) input));put_char_event(THE(LNTH (i-1) input))]`, `K ($= uv)`,
`\i. State (THE (LDROP i input))`, `update`]
\\ fs [GSYM SIO_def] \\ xsimpl \\ rw [lprefix_lub_def]
THEN1 (
qmatch_goalsub_abbrev_tac `SIO input1 events1`
\\ qmatch_goalsub_abbrev_tac `&_ * _ * SIO input2 events2`
\\ Cases_on `LNTH i input` THEN1 metis_tac [LFINITE_LNTH_NONE]
\\ rename1 `LNTH _ _ = SOME c`
\\ `input1 = c ::: input2` by (
unabbrev_all_tac \\
simp[LDROP_SUC]
\\ Cases_on `LDROP i input`
\\ TRY (metis_tac [LDROP_NONE_LFINITE])
\\ drule LNTH_LDROP
\\ fs[] \\ rw[] \\ rename1 `ll = _`
\\ Cases_on `ll` \\ fs[])
\\ xlet_auto THEN1 (xcon \\ xsimpl)
\\ xlet `POSTv v. &OPTION_TYPE CHAR (SOME c) v *
SIO input2 (SNOC (get_char_event c) events1)`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac [`emp`, `input1`, `events1`]
\\ xsimpl)
\\ xmatch \\ fs [OPTION_TYPE_def] \\ reverse (rpt strip_tac)
THEN1 EVAL_TAC THEN1 EVAL_TAC
\\ xlet `POSTv v. &UNIT_TYPE () v * SIO input2 events2`
THEN1 (
xapp
\\ MAP_EVERY qexists_tac
[`emp`, `input2`, `SNOC (get_char_event c) events1`, `c`]
\\ qmatch_goalsub_abbrev_tac `&_ * SIO _ events'`
\\ `events' = events2` by (
unabbrev_all_tac \\ fs [SNOC_APPEND])
\\ fs [SNOC_APPEND] \\ xsimpl)
\\ xvar \\ xsimpl)
\\ simp[Once LGENLIST_NONE_UNFOLD,o_DEF,cat_def,LMAP_EQ_LGENLIST]
QED
(* Infinite lists encoded as cyclic pointer structures in the heap *)
val REF_LIST_def = Define `
(REF_LIST rv [] A [] = SEP_EXISTS loc. cond(rv=Loc loc))
/\
(REF_LIST rv (rv2::rvs) A (x::l) =
(SEP_EXISTS loc v1.
cond(rv = Loc loc)
* cell loc (Refv(Conv NONE [v1;rv2]))
* cond(A x v1)
* REF_LIST rv2 rvs A l
) /\
(REF_LIST _ _ _ _ = &F)
)
`
Theorem REF_LIST_extend:
!rv rvs A l x v1.
(REF_LIST rv rvs A l *
SEP_EXISTS v1 loc loc'.
cond(LAST(rv::rvs) = Loc loc)
* cell loc (Refv(Conv NONE [v1;rv2]))
* cond(rv2 = Loc loc')
* cond(A x v1))
= (REF_LIST rv (SNOC rv2 rvs) A (SNOC x l))
Proof
PURE_ONCE_REWRITE_TAC[FUN_EQ_THM] >>
ho_match_mp_tac (fetch "-" "REF_LIST_ind") >>
rpt strip_tac >-
(simp[REF_LIST_def] >>
simp[SEP_CLAUSES] >>
simp[AC STAR_COMM STAR_ASSOC] >>
simp[SEP_EXISTS,cond_STAR] >>
metis_tac[]) >-
(pop_assum(assume_tac o REWRITE_RULE[GSYM FUN_EQ_THM] o GSYM) >>
simp[REF_LIST_def] >>
simp[SEP_CLAUSES] >>
simp[AC STAR_COMM STAR_ASSOC] >>
simp[SEP_EXISTS,cond_STAR]) >-
(simp[REF_LIST_def,SNOC_APPEND] >>
rename1 `REF_LIST _ (a1 ++ _)` >>
Cases_on `a1` >> simp[REF_LIST_def,SEP_CLAUSES]) >-
(simp[REF_LIST_def,SNOC_APPEND] >>
rename1 `REF_LIST _ _ _ (a1 ++ _)` >>
Cases_on `a1` >> simp[REF_LIST_def,SEP_CLAUSES])
QED
(* TODO: lots of these lemmas should probably live in characteristic/ or llistTheory *)
Inductive llist_upto:
(llist_upto R x x) /\
(R x y ==> llist_upto R x y) /\
(llist_upto R x y /\ llist_upto R y z ==> llist_upto R x z) /\
(llist_upto R x y ==> llist_upto R (LAPPEND z x) (LAPPEND z y))
End
val [llist_upto_eq,llist_upto_rel,llist_upto_trans,llist_upto_context] =
llist_upto_rules |> SPEC_ALL |> CONJUNCTS |> map GEN_ALL
|> curry (ListPair.map save_thm)
["llist_upto_eq","llist_upto_rel",
"llist_upto_trans","llist_upto_context"]
Theorem LLIST_BISIM_UPTO:
∀ll1 ll2 R.
R ll1 ll2 ∧
(∀ll3 ll4.
R ll3 ll4 ⇒
ll3 = [||] ∧ ll4 = [||] ∨
LHD ll3 = LHD ll4 ∧
llist_upto R (THE (LTL ll3)) (THE (LTL ll4)))
==> ll1 = ll2
Proof
rpt strip_tac
>> PURE_ONCE_REWRITE_TAC[LLIST_BISIMULATION]
>> qexists_tac `llist_upto R`
>> conj_tac >- rw[llist_upto_rules]
>> ho_match_mp_tac llist_upto_ind
>> rpt conj_tac
>- rw[llist_upto_rules]
>- first_x_assum ACCEPT_TAC
>- (rw[]
>> match_mp_tac OR_INTRO_THM2
>> conj_tac >- simp[]
>> metis_tac[llist_upto_rules])
>- (rw[llist_upto_rules]
>> Cases_on `ll3 = [||]`
>- (Cases_on `ll4` >> fs[llist_upto_rules])
>> match_mp_tac OR_INTRO_THM2
>> conj_tac
>- (Cases_on `z` >> simp[])
>> Cases_on `z` >- simp[]
>> simp[]
>> Cases_on `ll3` >> Cases_on `ll4`
>> fs[] >> rveq
>> CONV_TAC(RAND_CONV(RAND_CONV(RAND_CONV(PURE_ONCE_REWRITE_CONV [GSYM(CONJUNCT1 LAPPEND)]))))
>> CONV_TAC(RATOR_CONV(RAND_CONV(RAND_CONV(RAND_CONV(PURE_ONCE_REWRITE_CONV [GSYM(CONJUNCT1 LAPPEND)])))))
>> PURE_ONCE_REWRITE_TAC[GSYM(CONJUNCT2 LAPPEND)]
>> simp[GSYM LAPPEND_ASSOC]
>> metis_tac[llist_upto_rules])
QED
Theorem REF_cell_eq:
loc ~~>> Refv v = Loc loc ~~> v
Proof
rw[FUN_EQ_THM,cell_def,REF_def,SEP_EXISTS,cond_STAR]
QED
val LTAKE_LNTH_EQ = Q.prove(
`!x ll y. LTAKE (LENGTH x) ll = SOME x
/\ y < LENGTH x
==> LNTH y ll = SOME(EL y x)`,
Induct_on `x` >> rw[LTAKE] >>
Cases_on `ll` >> fs[] >>
PURE_FULL_CASE_TAC >> fs[] >> rveq >>
Cases_on `y` >> fs[]);
Theorem LTAKE_LPREFIX:
!x ll.
~LFINITE ll ==> ?l. LTAKE x ll = SOME l /\
LPREFIX (fromList l) ll
Proof
Induct >> rw[] >>
Cases_on `ll` >> fs[] >>
first_x_assum(drule_then strip_assume_tac) >>
fs[LPREFIX_LCONS]
QED
Theorem LMAP_fromList:
LMAP f (fromList l) = fromList(MAP f l)
Proof
Induct_on `l` >> fs[]
QED
Theorem LTAKE_LMAP:
!n f ll. LTAKE n (LMAP f ll) =
OPTION_MAP (MAP f) (LTAKE n ll)
Proof
Induct_on `n` >> rw[] >>
Cases_on `ll` >> fs[OPTION_MAP_COMPOSE,o_DEF]
QED
Theorem LNTH_LREPEAT:
!i x l.
LNTH i (LREPEAT l) = SOME x
==> x = EL (i MOD LENGTH l) l
Proof
Induct_on `i DIV LENGTH l` >> rw[]
>- (Cases_on `l = []`
>> fs[Once LREPEAT_thm]
>> fs[LNTH_LAPPEND]
>> `0 < LENGTH l` by(Cases_on `l` >> fs[])
>> qpat_x_assum `0 = _` (assume_tac o GSYM)
>> rfs[RatProgTheory.DIV_EQ_0]
>> fs[LNTH_fromList])
>> fs[ADD1]
>> `LENGTH l <= i`
by(CCONTR_TAC >> fs[LESS_DIV_EQ_ZERO])
>> `0 < LENGTH l` by(Cases_on `l` >> fs[])
>> `v = (i - LENGTH l) DIV LENGTH l`
by(fs[Q.INST[`q`|->`1`] DIV_SUB |> REWRITE_RULE [MULT_CLAUSES]])
>> first_x_assum drule
>> drule lnth_some_down_closed
>> disch_then(qspec_then `i - LENGTH l` mp_tac)
>> impl_tac >- simp[]
>> strip_tac
>> disch_then drule
>> strip_tac
>> rveq
>> qpat_x_assum `LNTH (_ - _) _ = _`
(fn thm => qpat_x_assum `LNTH _ _ = _` mp_tac >> assume_tac thm)
>> simp[Once LREPEAT_thm]
>> fs[LNTH_LAPPEND]
>> fs[SUB_MOD]
QED
Theorem REF_LIST_is_loc:
!rv rvs A l h. REF_LIST rv rvs A l h ==> ?loc. rv = Loc loc
Proof
ho_match_mp_tac (fetch "-" "REF_LIST_ind") >>
rw[REF_LIST_def,SEP_CLAUSES,SEP_F_def,STAR_def,SEP_EXISTS,cond_def]
QED
Theorem REF_LIST_LENGTH:
!rv rvs A l h. REF_LIST rv rvs A l h ==> LENGTH rvs = LENGTH l
Proof
ho_match_mp_tac (fetch "-" "REF_LIST_ind") >>
rw[REF_LIST_def,SEP_CLAUSES,SEP_F_def,STAR_def,SEP_EXISTS,cond_def] >>
metis_tac[]
QED
Theorem REF_LIST_rotate_1:
REF_LIST rv (SNOC rv (rv2::rvs)) A (x::l) =
REF_LIST rv2 (SNOC rv2 (SNOC rv rvs)) A (SNOC x l)
Proof
rw[FUN_EQ_THM] >>
simp[REF_LIST_def,GSYM REF_LIST_extend,Once LAST_DEF] >>
simp[SEP_CLAUSES,AC STAR_COMM STAR_ASSOC] >>
simp[SEP_EXISTS,cond_STAR] >>
rw[EQ_IMP_THM] >-
(asm_exists_tac >> simp[] >>
fs[STAR_def] >> Cases_on `l` >>
fs[REF_LIST_def] >> metis_tac[REF_LIST_is_loc]) >>
metis_tac[]
QED
val push_cond = Q.prove(`
m ~~>> v * (&C * B) = cond C * (m ~~>> v * B)
/\ m ~~>> v * &C = &C * m ~~>> v
/\ REF_LIST rv rvs A l * (&C * B) = cond C * (REF_LIST rv rvs A l * B)
/\ REF_LIST rv rvs A l * &C = &C * REF_LIST rv rvs A l
`,
simp[AC STAR_COMM STAR_ASSOC]);
val EL_LENGTH_TAKE = Q.prove(
`!h e. EL (LENGTH l) (h::TAKE (LENGTH l) (e::l))
= EL(LENGTH l) (h::e::l)`,
Induct_on `l` >> fs[]);
val EL_LENGTH_TAKE2 = Q.prove(
`!h e l. n < LENGTH l ==>