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diffScript.sml
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diffScript.sml
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(*
Implementation and verification of diff and patch algorithms
*)
open preamble lcsTheory mlintTheory mlstringTheory;
val _ = new_theory "diff";
val _ = temp_delsimps ["NORMEQ_CONV"]
val _ = temp_delsimps ["lift_disj_eq", "lift_imp_disj"]
val _ = diminish_srw_ss ["ABBREV"]
val _ = set_trace "BasicProvers.var_eq_old" 1
fun drule0 th =
first_assum(mp_tac o MATCH_MP (ONCE_REWRITE_RULE[GSYM AND_IMP_INTRO] th))
(* Diff algorithm definition *)
val line_numbers_def = Define `
(line_numbers l n =
if LENGTH l <= 1 then
toString (n:num)
else
strcat (toString n) (strcat(implode ",") (toString (n+LENGTH l))))`
val acd_def = Define `
(acd [] [] = #" ")
/\ (acd l [] = #"d")
/\ (acd [] l = #"a")
/\ (acd l l' = #"c")`
val diff_single_header_def = Define `
(diff_single_header l n l' n' =
strcat (strcat (line_numbers l n) (strcat (implode [acd l l']) (line_numbers l' n'))) (strlit "\n"))`
val diff_add_prefix_def = Define `
diff_add_prefix l h = MAP (strcat h) l `
val diff_single_def = Define `
diff_single l n l' n' =
diff_single_header l n l' n'::
if l = [] then (* add *)
diff_add_prefix l' (strlit "> ")
else if l' = [] then (* delete *)
diff_add_prefix l (strlit "< ")
else (* change *)
diff_add_prefix l (strlit "< ")
++ (strlit "---\n")::diff_add_prefix l' (strlit "> ")`
val diff_with_lcs_def = Define `
(diff_with_lcs [] l n l' n' =
if l = [] /\ l' = [] then
[]
else
diff_single l n l' n') /\
(diff_with_lcs (f::r) l n l' n' =
let (ll,lr) = SPLITP ($= f) l in
let (l'l,l'r) = SPLITP ($= f) l' in
if ll = [] /\ l'l = [] then
diff_with_lcs r (TL lr) (n+1) (TL l'r) (n+1)
else
diff_single ll n l'l n' ++ diff_with_lcs r (TL lr) (n+LENGTH ll+1) (TL l'r) (n'+LENGTH l'l+1))`
val diff_alg_def = Define `
diff_alg l l' = diff_with_lcs (optimised_lcs l l') l 0 l' 0`
val diff_alg_offs_def = Define `
diff_alg_offs l n l' n' = diff_with_lcs (dynamic_lcs l l') l n l' n'`
val diff_alg2_def = Define `
diff_alg2 l l' =
let prefix_length = LENGTH(longest_common_prefix l l');
l = DROP prefix_length l;
l' = DROP prefix_length l';
llength = LENGTH l;
l'length = LENGTH l';
suffix_length = if llength = l'length then
longest_common_suffix_length l l' 0
else if llength < l'length then
longest_common_suffix_length l (DROP (l'length-llength) l') 0
else
longest_common_suffix_length (DROP (llength-l'length) l) l' 0;
l = TAKE (llength - suffix_length) l;
l' = TAKE (l'length - suffix_length) l'
in
diff_with_lcs (dynamic_lcs l l') l prefix_length l' prefix_length`
Theorem diff_alg2_thm:
diff_alg2 l l' =
let prefix_length = LENGTH(longest_common_prefix l l');
l = DROP prefix_length l;
l' = DROP prefix_length l';
llength = LENGTH l;
l'length = LENGTH l';
suffix_length = LENGTH(longest_common_suffix l l');
l = TAKE (llength - suffix_length) l;
l' = TAKE (l'length - suffix_length) l'
in
diff_with_lcs (dynamic_lcs l l') l prefix_length l' prefix_length
Proof
PURE_ONCE_REWRITE_TAC [diff_alg2_def]
>> ntac 5 (PURE_ONCE_REWRITE_TAC [LET_THM])
>> ntac 5 (Ho_Rewrite.PURE_ONCE_REWRITE_TAC [BETA_THM])
>> PURE_ONCE_REWRITE_TAC [longest_common_suffix_length_if]
>> REFL_TAC
QED
(* Diff algorithm properties *)
Theorem diff_with_lcs_refl:
!n n'. diff_with_lcs l l n l n' = []
Proof
Induct_on `l` >> rw[diff_with_lcs_def,SPLITP]
QED
Theorem diff_alg_refl:
diff_alg l l = []
Proof
rw[diff_alg_def,lcs_refl',diff_with_lcs_refl,optimised_lcs_refl]
QED
Theorem diff_alg2_refl:
diff_alg2 l l = []
Proof
rw[diff_alg2_thm,lcs_refl',diff_with_lcs_refl,dynamic_lcs_refl,
longest_common_prefix_refl]
QED
(* Patch algorithm definition *)
val parse_patch_header_def = Define `
parse_patch_header s =
case tokens (\x. x = #"a" \/ x = #"d" \/ x = #"c" \/ x = #"\n") s of
| [l;r] =>
(case tokens ($= #",") l of
| [ll;lr] =>
(case (fromNatString ll, fromNatString lr) of
| (SOME lln, SOME lrn) =>
(case tokens ($= #",") r of
| [rl;rr] =>
(case (fromNatString rl, fromNatString rr) of
| (SOME rln, SOME rrn) =>
SOME (lln,SOME(lrn),strsub s (strlen l),
rln,SOME(rrn))
| _ => NONE)
| [r] =>
(case fromNatString r of
| SOME rn =>
SOME (lln,SOME(lrn),strsub s (strlen l),rn,NONE)
| _ => NONE)
| _ => NONE)
| _ => NONE)
| [l'] =>
(case fromNatString l' of
| (SOME l'n) =>
(case tokens ($= #",") r of
| [rl;rr] =>
(case (fromNatString rl, fromNatString rr) of
| (SOME rln, SOME rrn) =>
SOME (l'n,NONE,strsub s (strlen l),rln,SOME(rrn))
| _ => NONE)
| [r] =>
(case fromNatString r of
| (SOME rn) =>
SOME (l'n,NONE,strsub s (strlen l),rn,NONE)
| _ => NONE)
| _ => NONE)
| _ => NONE)
| _ => NONE)
| _ => NONE`;
val depatch_line_def = Define `
depatch_line s =
if strlen s > 1 then
if substring s 0 2 = strlit "> " then
SOME(substring s 2 (strlen s - 2))
else
NONE
else
NONE`
val depatch_lines_def = Define `
(depatch_lines [] = SOME []) /\
(depatch_lines (s::r) =
case depatch_line s of
NONE => NONE
| SOME s' =>
case depatch_lines r of
NONE => NONE
| SOME r' => SOME(s'::r'))`
val patch_aux_def = tDefine "patch_aux" `
(patch_aux [] file remfl n = SOME file) /\
(patch_aux (h::patch) file remfl n =
case parse_patch_header h of
| NONE => NONE
| SOME(orig,NONE,#"a",dest,opt) =>
let dest' = case opt of NONE => dest+1 | SOME dest' => dest' in
if orig < n \/ remfl < (orig-n) \/ dest' <= dest \/ LENGTH patch < (dest' - dest) then
NONE
else
(case depatch_lines(TAKE (dest'-dest) patch) of
NONE => NONE
| SOME p =>
let lf = TAKE (orig-n) file in
let rf = DROP (orig-n) file in
(case patch_aux (DROP (dest'-dest) patch) rf (remfl-(orig-n)) orig of
SOME rf' => SOME(lf++p++rf')
| NONE => NONE))
| SOME(orig,opt,#"d",dest,NONE) =>
let orig' = case opt of NONE => orig+1 | SOME orig' => orig' in
if orig < n \/ orig' < orig \/ remfl < (orig'-n) then
NONE
else
let lf = TAKE (orig-n) file in
let rf = DROP (orig'-n) file in
(case patch_aux (DROP (orig'-orig) patch) rf (remfl-(orig'-n)) orig' of
SOME rf' => SOME(lf++rf')
| NONE => NONE)
| SOME(orig,opt,#"c",dest,opt') =>
let orig' = case opt of NONE => orig+1 | SOME orig' => orig' in
let dest' = case opt' of NONE => dest+1 | SOME dest' => dest' in
if orig < n \/ orig' < orig \/ remfl < (orig'-n) \/ dest' <= dest \/ LENGTH patch < (dest' - dest) then
NONE
else
let relevant_patch = DROP (1+orig'-orig) patch in
(case depatch_lines(TAKE (dest'-dest) relevant_patch) of
NONE => NONE
| SOME p =>
let lf = TAKE (orig-n) file in
let rf = DROP (orig'-n) file in
(case patch_aux (DROP (dest'-dest) relevant_patch) rf (remfl-(orig'-n)) orig' of
SOME rf' => SOME(lf++p++rf')
| NONE => NONE))
| SOME _ => NONE
)`
(WF_REL_TAC `inv_image $< (LENGTH o FST)` >> fs[]);
val patch_alg_def = Define `
patch_alg patch file = patch_aux patch file (LENGTH file) 0`
val patch_alg_offs_def = Define `
patch_alg_offs n patch file = patch_aux patch file (LENGTH file) n`
(* Patch cancels diff *)
Theorem string_concat_empty:
!s. s ^ strlit "" = s /\ strlit "" ^ s = s
Proof
fs[strcat_thm,implode_explode]
QED
Theorem tokens_append_strlit:
∀P s1 x s2.
P x ⇒ tokens P (s1 ^ strlit [x] ^ s2) = tokens P s1 ++ tokens P s2
Proof
rpt strip_tac >> drule0 tokens_append >> fs[str_def,implode_def]
QED
Theorem tokens_append_right_strlit:
∀P s x.
P x ⇒ tokens P (s ^ strlit [x]) = tokens P s
Proof
rpt strip_tac >> drule0 tokens_append_strlit
>> disch_then (qspecl_then [`s`,`strlit ""`] assume_tac)
>> fs[string_concat_empty,tokens_def,tokens_aux_def]
QED
Theorem one_to_ten:
!P. P 0 /\ P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5 /\ P 6 /\ P 7 /\ P 8 /\ P 9 /\ (!n. (n:num) >= 10 ==> P n) ==> !n. P n
Proof
rpt strip_tac >> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC n`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC n)`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC n))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC n)))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC(SUC n))))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC(SUC(SUC n)))))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC(SUC(SUC(SUC n))))))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC(SUC(SUC(SUC(SUC n)))))))`
>> Cases_on `n` >> fs[]
>> qmatch_goalsub_rename_tac `SUC(SUC(SUC(SUC(SUC(SUC(SUC(SUC(SUC n))))))))`
>> Cases_on `n` >> fs[]
QED
val SPLITP_HEX = Q.prove(
`!n. n < 10 ==> SPLITP (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(STRING (HEX n) acc) =
let (l,r) = SPLITP (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n") acc in
(STRING (HEX n) l,r)`,
recInduct one_to_ten >> rpt strip_tac >> fs[] >> pairarg_tac >> fs[SPLITP]);
Overload ml_num_toString[local] = ``mlint$num_to_str``
Overload hol_int_toString[local] = ``integer_word$toString``
Overload num_toString[local] = ``num_to_dec_string``
val SPLITP_num_toString = Q.prove(
`!i.
SPLITP (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(toString (i:num)) = (toString i,[])`,
recInduct COMPLETE_INDUCTION >> rpt strip_tac
>> fs[ASCIInumbersTheory.num_to_dec_string_def]
>> fs[ASCIInumbersTheory.n2s_def]
>> PURE_ONCE_REWRITE_TAC[numposrepTheory.n2l_def] >> rw[] >> fs[SPLITP,SPLITP_HEX]
>> first_x_assum (qspec_then `n DIV 10` assume_tac) >> rfs[]
>> fs[SPLITP_APPEND,SPLITP_NIL_SND_EVERY]
>> `n MOD 10 < 10` by fs[] >> rename [`HEX n`]
>> pop_assum mp_tac >> rpt(pop_assum kall_tac)
>> Q.SPEC_TAC (`n`,`n`) >> recInduct one_to_ten >> fs[]);
val SPLITP_int_toString = Q.prove(
`!i.
SPLITP (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(toString (i:int)) = (toString i,[])`,
rpt strip_tac >> fs[integer_wordTheory.toString_def] >> rw[] >> fs[SPLITP,SPLITP_num_toString]);
val TOKENS_tostring = Q.prove(
`TOKENS (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n") (toString(n:num)) = [toString n]`,
Cases_on `num_toString n` >> fs[TOKENS_def]
>> qpat_x_assum `_ = STRING _ _` (assume_tac o GSYM) >> fs[]
>> pairarg_tac >> pop_assum (assume_tac o GSYM)
>> fs[SPLITP_num_toString,TOKENS_def]
>> qpat_x_assum `STRING _ _ = _` (assume_tac o GSYM) >> fs[]);
val num_le_10 = Q.prove(
`!n. 0 ≤ n /\ n < 10 ==> Num n < 10`,
Cases >> fs[]);
val tokens_toString = Q.prove(
`tokens (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n") (toString (n:num)) = [toString n]`,
simp [toString_thm, num_to_str_thm, TOKENS_eq_tokens_sym, TOKENS_tostring]);
val tokens_strcat = Q.prove(
`l ≠ [] ==>
(tokens (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(toString (n:num) ^
strlit (STRING (acd l r) "") ^ toString (m:num) ^ strlit "\n")
= [toString n; toString m])`,
Cases_on `l` >> Cases_on `r` >> fs[acd_def] >>
fs[tokens_append_strlit,strcat_assoc,tokens_append_right_strlit,tokens_toString]);
val tokens_strcat' = Q.prove(
`r ≠ [] ==>
(tokens (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(toString (n:num) ^
strlit (STRING (acd l r) "") ^ toString (m:num) ^ strlit "\n")
= [toString n; toString m])`,
Cases_on `l` >> Cases_on `r` >> fs[acd_def] >>
fs[tokens_append_strlit,strcat_assoc,tokens_append_right_strlit,tokens_toString]);
val strsub_strcat =
Q.prove(`!s s'. strsub(s ^ s') n = if n < strlen s then strsub s n else strsub s' (n - strlen s)`,
Induct >> simp[strcat_thm,implode_def,strsub_def,EL_APPEND_EQN]
\\ gen_tac \\ Cases \\ simp[]);
Theorem strsub_str:
strsub (str c) 0 = c
Proof
rw[str_def,implode_def,strsub_def]
QED
val acd_simps =
Q.prove(`l ≠ [] ==> (acd [] l = #"a" /\ acd l [] = #"d")`,
Cases_on `l` >> fs[acd_def])
val acd_more_simps =
Q.prove(`l ≠ [] /\ r ≠ [] ==> (acd l r = #"c")`,
Cases_on `l` >> Cases_on `r` >> fs[acd_def])
val HEX_isDigit = Q.prove(`!n. n < 10 ==> isDigit(HEX n)`,
recInduct one_to_ten >> fs[isDigit_def]);
(* TODO: move at least these (and probably others in this file) *)
Theorem toString_isDigit:
!n. EVERY isDigit (toString(n:num))
Proof
recInduct COMPLETE_INDUCTION
>> rpt strip_tac
>> fs[ASCIInumbersTheory.num_to_dec_string_def]
>> fs[ASCIInumbersTheory.n2s_def]
>> PURE_ONCE_REWRITE_TAC[numposrepTheory.n2l_def]
>> rw[] >> fs[HEX_isDigit]
QED
(* -- *)
(*`!n. explode (toString n) = toString n`*)
Theorem int_abs_toString_num:
!n. toString (&n) = toString n
Proof
recInduct COMPLETE_INDUCTION >> strip_tac
>> fs[integer_wordTheory.toString_def]
QED
val substring_adhoc_simps = Q.prove(`!h.
(substring (strlit "> " ^ h) 0 2 = strlit "> ")
/\ (substring (strlit "> " ^ h) 2 (strlen h) = h)
/\ (substring (strlit "< " ^ h) 0 2 = strlit "< ")
/\ (substring (strlit "< " ^ h) 2 (strlen h) = h)
`,
Induct >> rpt strip_tac >> fs[strcat_thm,implode_def,substring_def,strlen_def]
>> fs[ADD1,MIN_DEF] >> fs[SEG_compute] >> simp_tac pure_ss [ONE,TWO,SEG_SUC_CONS]
>> fs[SEG_LENGTH_ID])
val depatch_lines_strcat_cancel = Q.prove(
`!r. depatch_lines (MAP (strcat (strlit "> ")) r) = SOME r`,
Induct >> fs[depatch_lines_def,depatch_line_def,strlen_strcat,substring_adhoc_simps])
Theorem depatch_lines_diff_add_prefix_cancel:
depatch_lines (diff_add_prefix l (strlit "> ")) = SOME l
Proof
fs[diff_add_prefix_def,depatch_lines_strcat_cancel]
QED
val patch_aux_nil = Q.prove(`patch_aux [] file remfl n = SOME file`,fs[patch_aux_def]);
val line_numbers_not_empty = Q.prove(
`!l n . line_numbers l n <> strlit ""`,
fs[line_numbers_def, num_to_str_thm, implode_def]
\\ rw []
\\ simp_tac std_ss [GSYM explode_11, explode_strcat]
\\ simp []
);
Theorem tokens_eq_sing:
!s f. EVERY ($~ o f) (explode s) /\ s <> strlit "" ==> tokens f s = [s]
Proof
Cases
\\ fs[TOKENS_eq_tokens_sym,toString_thm,explode_implode,implode_def]
\\ Cases_on `s'` \\ fs [TOKENS_def] \\ rw []
\\ fs [o_DEF,SPLITP_EVERY,TOKENS_def]
QED
val tokens_toString_comma =
Q.prove(`tokens ($= #",") (toString (n:num)) = [toString n]`,
rw [] \\ match_mp_tac tokens_eq_sing
\\ fs [num_to_str_thm,implode_def]
\\ fs [num_to_str_def]
\\ match_mp_tac (MP_CANON EVERY_MONOTONIC)
\\ qexists_tac `isDigit`
\\ fs [EVERY_isDigit_num_to_dec_string] \\ EVAL_TAC);
val tokens_comma_lemma = Q.prove(
`tokens (λx. x = #"a" ∨ x = #"d" ∨ x = #"c" ∨ x = #"\n")
(line_numbers l n) = [line_numbers l n]`,
`EVERY (λx. isDigit x \/ x = #",") (explode(line_numbers l n))`
by(fs[line_numbers_def] >> rw[]
\\ fs[toString_thm,num_to_str_def]
\\ fs[explode_implode,strcat_thm]
\\ match_mp_tac (MP_CANON EVERY_MONOTONIC)
\\ qexists_tac `isDigit` \\ fs [toString_isDigit])
\\ match_mp_tac tokens_eq_sing
\\ conj_tac THEN1
(match_mp_tac (MP_CANON EVERY_MONOTONIC)
\\ goal_assum (first_x_assum o mp_then Any mp_tac)
\\ fs [] \\ CCONTR_TAC \\ fs [] \\ rveq \\ fs [isDigit_def])
\\ rw [line_numbers_def,num_to_str_thm,implode_def]
\\ fs [strcat_def,concat_def]);
Theorem parse_header_cancel:
l <> [] \/ l' <> [] ==>
(parse_patch_header(diff_single_header l n l' n') =
SOME(n,if LENGTH l <= 1 then NONE else SOME(n+LENGTH l),
if l = [] then #"a" else if l' = [] then #"d" else #"c",
n',if LENGTH l' <= 1 then NONE else SOME(n'+LENGTH l')))
Proof
rw[diff_single_header_def,parse_patch_header_def,
option_case_eq,list_case_eq,PULL_EXISTS,
strsub_strcat,tokens_append_right_strlit,GSYM str_def,
tokens_append,acd_simps,acd_more_simps,tokens_comma_lemma,
tokens_comma_lemma]
\\ rw[line_numbers_def,tokens_toString_comma,
fromNatString_toString,
GSYM str_def,tokens_append,strsub_str]
QED
val patch_aux_cancel_base_case = Q.prove(
`patch_aux (diff_with_lcs [] r n r' m) r (LENGTH r) n = SOME r'`,
fs[diff_with_lcs_def,diff_single_def] >> rw[]
>> fs[patch_aux_def]
>> fs[patch_aux_def,parse_header_cancel,
diff_add_prefix_def,depatch_lines_strcat_cancel,
GSYM MAP_TAKE,GSYM MAP_DROP,DROP_APPEND,TAKE_APPEND]
>> rw[]
>> fs[DROP_LENGTH_TOO_LONG,TAKE_LENGTH_TOO_LONG,patch_aux_def,
quantHeuristicsTheory.LIST_LENGTH_0,
DROP_APPEND,depatch_lines_strcat_cancel]
>> `LENGTH r = 1` by (Cases_on `LENGTH r` >> fs[])
>> fs[depatch_lines_strcat_cancel]);
Theorem SPLITP_NIL_FST:
∀ls P r. SPLITP P ls = ([],r) ⇔ (r = ls ∧ ((ls <> []) ==> P(HD ls)))
Proof
Cases >> rpt strip_tac >> fs[SPLITP,EQ_IMP_THM] >> IF_CASES_TAC
>> strip_tac >> fs[]
QED
Theorem diff_add_prefix_length:
!l s. LENGTH (diff_add_prefix l s) = LENGTH l
Proof
fs[diff_add_prefix_def]
QED
Theorem diff_add_prefix_TAKE:
!l n s. TAKE n (diff_add_prefix l s) = diff_add_prefix (TAKE n l) s
Proof
fs[diff_add_prefix_def,MAP_TAKE]
QED
Theorem diff_add_prefix_DROP:
!l n s. DROP n (diff_add_prefix l s) = diff_add_prefix (DROP n l) s
Proof
fs[diff_add_prefix_def,MAP_DROP]
QED
Theorem diff_add_prefix_nil:
!s. (diff_add_prefix [] s) = []
Proof
fs[diff_add_prefix_def]
QED
val ONE_MINUS_SUCC = Q.prove(`1 - SUC x = 0`,intLib.COOPER_TAC);
val SUCC_LE_ONE = Q.prove(`(SUC n ≤ 1) = (n = 0)`,intLib.COOPER_TAC);
val patch_aux_keep_init = Q.prove(
`!l p t n t' m.
common_subsequence l t t' ==>
patch_aux (diff_with_lcs l t (n + LENGTH p) t' (m + LENGTH p)) (p ++ t) (LENGTH t + LENGTH p) n
=
case patch_aux (diff_with_lcs l t (n + LENGTH p) t' (m + LENGTH p)) t (LENGTH t) (n+LENGTH p) of
SOME r => SOME(p++r)
| NONE => NONE`,
Induct >> rpt strip_tac
>- (fs[patch_aux_cancel_base_case]
>> fs[diff_with_lcs_def,diff_single_def,diff_add_prefix_def] >> rw[]
>> fs[patch_aux_def,parse_header_cancel] >> rw[]
>> fs[quantHeuristicsTheory.LIST_LENGTH_0,GSYM MAP_TAKE,GSYM MAP_DROP,
depatch_lines_strcat_cancel,DROP_LENGTH_NIL,DROP_LENGTH_TOO_LONG,
TAKE_LENGTH_TOO_LONG,patch_aux_nil,DROP_APPEND,TAKE_APPEND,
ONE_MINUS_SUCC]
>> Cases_on `t` >> fs[ONE_MINUS_SUCC])
>> fs[diff_with_lcs_def]
>> rpt(pairarg_tac>>fs[]) >> rw[]
>- (fs[SPLITP_NIL_FST] >> rveq
>> Cases_on `lr` >> fs[common_subsequence_empty']
>> Cases_on `l'r` >> fs[common_subsequence_empty']
>> rveq >> first_assum(qspecl_then [`p++[h]`,`t`,`n`,`t'`,`n`] mp_tac)
>> first_x_assum(qspecl_then [`[h]`,`t`,`n+LENGTH p`,`t'`,`n+LENGTH p`] mp_tac)
>> fs[cons_common_subsequence]
>> full_simp_tac pure_ss [GSYM APPEND_ASSOC,APPEND,ADD1]
>> rpt strip_tac >> fs[] >> TOP_CASE_TAC)
>> Cases_on `ll` >> Cases_on `l'l`
>> fs[diff_single_def,patch_aux_def,parse_header_cancel,SPLITP_NIL_FST]
>> rveq >> fs[]
>> IF_CASES_TAC >> fs[]
>> rw[]
>> fs[TAKE_APPEND,DROP_APPEND,diff_add_prefix_length,diff_add_prefix_TAKE,
depatch_lines_diff_add_prefix_cancel,ONE_MINUS_SUCC,SUCC_LE_ONE,
quantHeuristicsTheory.LIST_LENGTH_0,diff_add_prefix_DROP,diff_add_prefix_nil,
DROP_LENGTH_NIL,DROP_LENGTH_TOO_LONG]
>> TOP_CASE_TAC >> fs[]);
val patch_aux_keep_init_cons = Q.prove(`
!l t n t' h m.
common_subsequence l t t' ==>
patch_aux (diff_with_lcs l t (n + 1) t' (m + 1)) (h::t) (SUC (LENGTH t)) n
=
case patch_aux (diff_with_lcs l t (n + 1) t' (m + 1)) t (LENGTH t) (n+1) of
SOME r => SOME(h::r)
| NONE => NONE`,
rpt strip_tac >> drule0 patch_aux_keep_init
>> disch_then (qspecl_then [`[h]`,`n`,`m`] assume_tac) >> fs[ADD1]);
val list_nil_sub_length = Q.prove(`l ≠ [] ==> (1 - LENGTH l = 0)`,
Cases_on `l` >> fs[])
val list_length_1_lemma = Q.prove(`l ≠ [] /\ LENGTH l <= 1 ==> LENGTH l = 1`,
Cases_on `LENGTH l` >> fs[])
val minus_add_too_large = Q.prove(`a - ((a:num) + n) = 0`,intLib.COOPER_TAC);
val minus_add_too_large' = Q.prove(`(a + 1) - ((a:num) + 2) = 0`,intLib.COOPER_TAC);
Theorem patch_aux_diff_cancel:
!l r n r' m.
common_subsequence l r r' ==>
(patch_aux (diff_with_lcs l r n r' m) r (LENGTH r) n = SOME r')
Proof
Induct
>> rpt strip_tac
>-fs[patch_aux_cancel_base_case]
>> fs[diff_with_lcs_def,diff_single_def,
diff_add_prefix_def]
>> rpt(pairarg_tac >> fs[])
>> rw[]
>- (fs[SPLITP_NIL_FST] >> rveq
>> Cases_on `lr` >> fs[common_subsequence_empty']
>> Cases_on `l'r` >> fs[common_subsequence_empty']
>> rveq >> fs[cons_common_subsequence,patch_aux_keep_init_cons])
>- (fs[SPLITP_NIL_FST] >> rveq
>> Cases_on `lr` >> fs[common_subsequence_empty']
>> Cases_on `l'r` >> fs[common_subsequence_empty',SPLITP_NIL_SND_EVERY]
>> rveq
>> fs[cons_common_subsequence,patch_aux_keep_init_cons]
>> drule0 common_subsequence_split_css2
>> fs[SPLITP_EVERY,o_DEF,common_subsequence_empty',SPLITP]
>> drule0 SPLITP_IMP >> rpt strip_tac >> fs[] >> rveq >> fs[cons_common_subsequence]
>> fs[patch_aux_def] >> rw[]
>> fs[parse_header_cancel,TAKE_APPEND]
>> rw[] >> fs[quantHeuristicsTheory.LIST_LENGTH_0,TAKE_LENGTH_TOO_LONG,list_nil_sub_length,
depatch_lines_strcat_cancel,DROP_LENGTH_TOO_LONG,DROP_APPEND]
>> drule0 patch_aux_keep_init_cons
>> disch_then(qspecl_then [`n`,`h`,`m + LENGTH l'l`] assume_tac)
>> drule0 SPLITP_JOIN
>> fs[])
>- (fs[SPLITP_NIL_FST] >> rveq
>> Cases_on `lr` >> fs[common_subsequence_empty']
>> Cases_on `l'r` >> fs[common_subsequence_empty',SPLITP_NIL_SND_EVERY]
>> rveq
>> fs[cons_common_subsequence,patch_aux_keep_init_cons]
>> drule0 common_subsequence_split_css
>> fs[SPLITP_EVERY,o_DEF,common_subsequence_empty',SPLITP]
>> drule0 SPLITP_IMP >> rpt strip_tac >> fs[] >> rveq >> fs[cons_common_subsequence]
>> fs[patch_aux_def] >> rw[]
>> fs[parse_header_cancel,TAKE_APPEND]
>> rw[]
>> drule0 SPLITP_JOIN >> strip_tac >> fs[]
>> fs[quantHeuristicsTheory.LIST_LENGTH_0,TAKE_LENGTH_TOO_LONG,list_nil_sub_length,
depatch_lines_strcat_cancel,DROP_LENGTH_TOO_LONG,DROP_APPEND,list_length_1_lemma,
minus_add_too_large]
>> drule0 patch_aux_keep_init_cons
>> disch_then(qspecl_then [`n + LENGTH ll`,`h`,`m`] mp_tac)
>> fs[list_length_1_lemma,ADD1])
>- (fs[SPLITP_NIL_FST] >> rveq
>> Cases_on `lr` >> fs[common_subsequence_empty']
>> Cases_on `l'r` >> fs[common_subsequence_empty',SPLITP_NIL_SND_EVERY]
>> rveq
>> fs[cons_common_subsequence,patch_aux_keep_init_cons]
>> drule0 common_subsequence_split_css >> strip_tac >> drule0 common_subsequence_split_css2
>> fs[SPLITP_EVERY,o_DEF,common_subsequence_empty',SPLITP]
>> drule0 SPLITP_IMP >> qpat_x_assum `SPLITP _ _ = _ ` mp_tac
>> drule0 SPLITP_IMP >> rpt strip_tac >> fs[] >> rveq >> fs[cons_common_subsequence]
>> fs[patch_aux_def] >> rw[]
>> fs[parse_header_cancel,TAKE_APPEND]
>> rw[]
>> drule0 SPLITP_JOIN >> qpat_x_assum `SPLITP _ _ = _` mp_tac
>> drule0 SPLITP_JOIN >> ntac 3 strip_tac
>> fs[quantHeuristicsTheory.LIST_LENGTH_0,TAKE_LENGTH_TOO_LONG,list_nil_sub_length,
depatch_lines_strcat_cancel,DROP_LENGTH_TOO_LONG,DROP_APPEND,list_length_1_lemma,
minus_add_too_large,TAKE_APPEND,minus_add_too_large',ONE_MINUS_SUCC]
>> drule0 patch_aux_keep_init_cons
>> disch_then(qspecl_then [`n + LENGTH ll`,`h`,`m + LENGTH l'l`] mp_tac)
>> fs[ADD1,list_length_1_lemma])
QED
Theorem patch_diff_cancel:
patch_alg (diff_alg l r) l = SOME r
Proof
fs[patch_alg_def,diff_alg_def]
>> mp_tac (GEN_ALL (INST_TYPE [alpha|->``:mlstring``] optimised_lcs_correct))
>> disch_then (qspecl_then [`r`,`l`] assume_tac)
>> fs[patch_aux_diff_cancel,lcs_def]
QED
val headers_within_def = Define `
headers_within n m l =
EVERY (OPTION_ALL (λ(n':num,m':num option,c,_,_).
(n <= n' /\ n' <= m /\ (IS_NONE m' /\ (c = #"d" \/ c = #"c") ==>
n'+1 <= m) /\
(IS_SOME m' ==> (n <= THE m' /\ THE m' <= m)))))
(MAP parse_patch_header l)`
Theorem headers_within_IMP:
headers_within n m (h::t) /\ parse_patch_header h = SOME(q,NONE,c,tup)
==> n <= q /\ q <= m /\ ((c= #"d" \/ c = #"c") ==> q+1 <= m)
Proof
rpt strip_tac >> fs[headers_within_def] >> rfs[pairTheory.ELIM_UNCURRY]
QED
Theorem headers_within_IMP_SOME:
headers_within n m (h::t) /\ parse_patch_header h = SOME(q,SOME q',c,tup)
==> n <= q /\ q <= m /\ n <= q' /\ q' <= m
Proof
rpt strip_tac >> fs[headers_within_def] >> rfs[pairTheory.ELIM_UNCURRY]
QED
Theorem headers_within_grow:
headers_within n' m' l /\ n <= n' /\ m' <= m ==> headers_within n m l
Proof
Induct_on `l` >> rpt strip_tac >> fs[headers_within_def]
>> Cases_on `parse_patch_header h` >> fs[pairTheory.ELIM_UNCURRY]
>> rw[] >> fs[]
QED
Theorem headers_within_append:
headers_within n m (l++l') = (headers_within n m l /\ headers_within n m l')
Proof
simp[headers_within_def]
QED
Theorem headers_within_dest_cons:
headers_within n m (e::l') ==> headers_within n m l'
Proof
simp[headers_within_def]
QED
(* todo: move to richlist? *)
Theorem EVERY_DROP_T:
!P l m. EVERY P l ==> EVERY P (DROP m l)
Proof
Induct_on `l` >> rw[DROP_def]
QED
Theorem headers_within_drop:
headers_within n m (l) ==> headers_within n m (DROP x l)
Proof
simp[headers_within_def,MAP_DROP,EVERY_DROP_T]
QED
Theorem fromString_gt:
fromString (implode (STRING #">" x)) = NONE /\
fromString (implode (STRING #"<" x)) = NONE
Proof
rw [] \\ match_mp_tac fromString_EQ_NONE \\ EVAL_TAC
QED
Theorem fromNatString_gt:
fromNatString (implode (STRING #">" x)) = NONE /\
fromNatString (implode (STRING #"<" x)) = NONE
Proof
rw [fromNatString_def,fromString_gt]
QED
val parse_nonheader_lemma = Q.prove(
`!f r. EVERY (OPTION_ALL f) (MAP parse_patch_header (diff_add_prefix r (strlit "> ")))`,
strip_tac >> Induct
>- fs[diff_add_prefix_def]
>> strip_tac >> Cases_on `h`
>> fs[parse_patch_header_def,diff_add_prefix_def]
>> every_case_tac
>> fs[strcat_def,mlstringTheory.concat_def]
>> fs[tokens_append_strlit,TOKENS_eq_tokens_sym]
>> fs[TOKENS_def,pairTheory.ELIM_UNCURRY,SPLITP] >> rveq
>> fs[explode_implode,TOKENS_def,SPLITP,pairTheory.ELIM_UNCURRY]
>> rveq
>> fs[explode_implode,isDigit_def,fromNatString_gt]);
val parse_nonheader_lemma2 = Q.prove(
`!f r. EVERY (OPTION_ALL f) (MAP parse_patch_header (diff_add_prefix r (strlit "< ")))`,
strip_tac >> Induct
>- fs[diff_add_prefix_def]
>> strip_tac >> Cases_on `h`
>> fs[parse_patch_header_def,diff_add_prefix_def]
>> every_case_tac
>> fs[strcat_def,mlstringTheory.concat_def]
>> fs[tokens_append_strlit,TOKENS_eq_tokens_sym]
>> fs[TOKENS_def,pairTheory.ELIM_UNCURRY,SPLITP] >> rveq
>> fs[explode_implode,TOKENS_def,SPLITP,pairTheory.ELIM_UNCURRY]
>> rveq
>> fs[explode_implode,isDigit_def,fromNatString_gt]);
val parse_nonheader_lemma3 = Q.prove(
`parse_patch_header (strlit "---\n") = NONE`,
fs[parse_patch_header_def]
>> every_case_tac
>> fs[strcat_def,mlstringTheory.concat_def]
>> fs[tokens_append_strlit,TOKENS_eq_tokens_sym]
>> fs[TOKENS_def,pairTheory.ELIM_UNCURRY,SPLITP] >> rveq);
Theorem diff_with_lcs_headers_within:
!l r n r' m. common_subsequence l r r' ==>
headers_within n (n + LENGTH r) (diff_with_lcs l r n r' m)
Proof
Induct
>> rpt strip_tac
>- (fs[diff_with_lcs_def,headers_within_def,diff_single_def]
>> rw[] >> fs[parse_header_cancel]
>> rw[]
>> TRY(MATCH_ACCEPT_TAC parse_nonheader_lemma)
>> TRY(MATCH_ACCEPT_TAC parse_nonheader_lemma2)
>> fs[parse_nonheader_lemma3]
>> Cases_on `r` >> fs[])
>> fs[diff_with_lcs_def]
>> rpt(pairarg_tac >> fs[])
>> IF_CASES_TAC
>- (drule0 split_common_subsequence
>> strip_tac
>> first_x_assum drule0 >> fs[]
>> disch_then(qspecl_then [`n+1`,`n+1`] assume_tac)
>> drule0(GEN_ALL headers_within_grow)
>> disch_then match_mp_tac
>> fs[SPLITP_NIL_FST]
>> Cases_on `r` >> fs[common_subsequence_empty'])
>> simp[headers_within_append]
>> strip_tac
>- (fs[headers_within_def,diff_single_def]
>> fs[parse_header_cancel]
>> rw[]
>> Q.ISPECL_THEN [`($= h)`,`r`] assume_tac (GEN_ALL (GSYM SPLITP_LENGTH))
>> fs[]
>> TRY(MATCH_ACCEPT_TAC parse_nonheader_lemma)
>> TRY(MATCH_ACCEPT_TAC parse_nonheader_lemma2)
>> fs[parse_nonheader_lemma3] >> rfs[]
>> drule0 common_subsequence_split >> strip_tac >> fs[] >> rveq >> fs[])
>> drule0 split_common_subsequence
>> drule0 common_subsequence_split
>> drule0 common_subsequence_split2
>> rpt strip_tac >> fs[] >> rveq
>> fs[] >> rfs[] >> first_x_assum drule0
>> disch_then(qspecl_then [`n + (LENGTH ll + 1)`,`m + (LENGTH l'l + 1)`] assume_tac)
>> drule0(GEN_ALL headers_within_grow)
>> disch_then match_mp_tac
>> Q.ISPECL_THEN [`($= h)`,`r`] assume_tac (GEN_ALL (GSYM SPLITP_LENGTH))
>> fs[]
QED
val highly_specific_implication = Q.prove(
`¬(q < n + 1) /\ ¬(m < q − n) ==> n + (SUC m) - (q + 1) = (n + m - q)`,
fs[]);
val highly_specific_implication2 = Q.prove(
`¬(SUC m < (q+1) − n) ==> n + (SUC m) - (q + 1) = (n + m - q)`,
fs[]);
Theorem headers_within_cons:
headers_within (n+1) m p1 /\ m >= n+1 ==>
patch_alg_offs n p1 (e::l) = OPTION_MAP (CONS e) (patch_alg_offs (n+1) p1 l)
Proof
Cases_on `p1` >> rpt strip_tac
>> fs[patch_alg_offs_def,patch_aux_def]
>> every_case_tac
>> fs[headers_within_def] >> rfs[]
>> TRY(drule0(GEN_ALL highly_specific_implication) >> disch_then drule0 >> disch_then (fn x => fs[x]))
>> TRY(drule0(GEN_ALL highly_specific_implication2) >> disch_then (fn x => fs[x]))
>> fs[GSYM ADD1]
QED
val IS_SUFFIX_induct_aux =
Q.prove(`!P l. P [] /\ (!h l. (!sl. IS_SUFFIX l sl ==> P sl) ==> P (h::l)) ==> (!sl. IS_SUFFIX l sl ==> P sl)`,
strip_tac >> Induct_on `l`
>> rpt strip_tac
>> fs[]
>- (qspec_then `sl` assume_tac SNOC_CASES >> fs[] >> fs[IS_SUFFIX_compute,REVERSE_APPEND])
>> Cases_on `sl` >> fs[IS_SUFFIX]
>> Cases_on `IS_SUFFIX l (h'::t)` >> fs[]
>> fs[IS_SUFFIX_compute]
>> fs[] >> fs[IS_SUFFIX_compute,REVERSE_APPEND]
>> FULL_SIMP_TAC std_ss [GSYM SNOC_APPEND, IS_PREFIX_SNOC]
>> fs[])
Theorem IS_SUFFIX_induct:
!P. P [] /\ (!h l. (!sl. IS_SUFFIX l sl ==> P sl) ==> P (h::l)) ==> !l. P l
Proof
metis_tac[IS_SUFFIX_induct_aux,IS_SUFFIX_REFL]
QED
Theorem IS_SUFFIX_DROP:
!l n. IS_SUFFIX l (DROP n l)
Proof
Induct >> rpt strip_tac >> rw[DROP_def]
>> metis_tac[IS_SUFFIX_CONS]
QED
Theorem headers_within_snoc:
!p1 n l m e. headers_within m (n + LENGTH l) p1 /\ m <= (n + LENGTH l) ==>
patch_alg_offs n p1 (SNOC e l) = OPTION_MAP (SNOC e) (patch_alg_offs n p1 l)
Proof
ho_match_mp_tac IS_SUFFIX_induct
>> rpt strip_tac
>> fs[patch_alg_offs_def,patch_aux_def]
>> every_case_tac >> fs[] >> rfs[]
>> fs[DROP_DROP_T]
>> TRY(qmatch_goalsub_abbrev_tac `F`
>> qmatch_asmsub_abbrev_tac `patch_aux (DROP a1 _) _ _ a2`
>> `IS_SUFFIX p1 (DROP a1 p1)`
by(MATCH_ACCEPT_TAC IS_SUFFIX_DROP)
>> first_assum drule0
>> qunabbrev_tac `a1`
>> disch_then(qspecl_then [`a2`,`DROP (a2 - n) l`,`m`,`e`] mp_tac)
>> impl_tac
>- (qunabbrev_tac `a2`
>> match_mp_tac(GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n + LENGTH l`]
>> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> fs[]
>> FULL_SIMP_TAC bool_ss [NOT_LESS, GSYM DROP_SNOC,LE]
>> fs[ADD1]
>> fs[SNOC_APPEND,DROP_APPEND,DROP_LENGTH_TOO_LONG]
>> rfs[DROP_LENGTH_TOO_LONG]
>> fs[DROP_def] >> imp_res_tac headers_within_IMP >> fs[]
>> qunabbrev_tac `a2` >> fs[]
>> qmatch_asmsub_abbrev_tac `patch_aux _ _ _ a2`
>> (first_assum drule0
>> disch_then(qspecl_then [`a2`,`[]`,`m`,`e`] mp_tac)
>> impl_tac
>- (match_mp_tac(GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n+LENGTH l`]
>> qunabbrev_tac `a2` >> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> TRY(
`(n + (LENGTH l + 1) − a2) = 0` by(intLib.COOPER_TAC)
>> pop_assum(fn x => fs[x])
>> `(n + LENGTH l - a2) = 0` by(intLib.COOPER_TAC)
>> pop_assum(fn x => fs[x]) >> rveq >> fs[]
>> (first_assum drule0
>> disch_then(qspecl_then [`a2`,`[]`,`m`,`e`] mp_tac)
>> impl_tac
>- (match_mp_tac(GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n+LENGTH l`]
>> qunabbrev_tac `a2` >> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> impl_tac >> fs[]
>> rpt strip_tac >> fs[]
>> imp_res_tac headers_within_IMP_SOME >> fs[] >> NO_TAC))
>> `n + LENGTH l - a2 = 0` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])
>> qunabbrev_tac `a2`
>> `n + LENGTH l - q = 0` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])
>> imp_res_tac headers_within_IMP >> fs[]
>> `n + (LENGTH l + 1) - q = 1` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])
>> imp_res_tac headers_within_IMP
>> `q - (n + LENGTH l) = 0` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])))
>> TRY(qmatch_goalsub_abbrev_tac `F`
>> qmatch_asmsub_abbrev_tac `DROP 1 _`
>> `IS_SUFFIX p1 (DROP 1 p1)`
by(MATCH_ACCEPT_TAC IS_SUFFIX_DROP)
>> first_assum drule0
>> disch_then(qspecl_then [`q`,`DROP (q - n) l`,`m`,`e`] mp_tac)
>> impl_tac
>- (match_mp_tac(GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n + LENGTH l`]
>> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> fs[]
>> FULL_SIMP_TAC bool_ss [NOT_LESS, GSYM DROP_SNOC,LE]
>> fs[ADD1]
>> fs[SNOC_APPEND,DROP_APPEND,DROP_LENGTH_TOO_LONG]
>> rfs[DROP_LENGTH_TOO_LONG]
>> imp_res_tac headers_within_IMP
>> `q = n + LENGTH l - 1` by intLib.COOPER_TAC
>> rveq >> fs[]
>> (first_assum drule0
>> disch_then(qspecl_then [`q`,`[]`,`m`,`e`] mp_tac)
>> impl_tac
>- (match_mp_tac(GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n+LENGTH l`]
>> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> `n + LENGTH l - (q + 1) = 0` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])
>> `n + LENGTH l - q = 0` by(intLib.COOPER_TAC)
>> pop_assum (fn x => fs[x])
>> imp_res_tac headers_within_IMP >> fs[]))
>> TRY((qmatch_goalsub_abbrev_tac `TAKE (q − n) (l ⧺ [e]) ⧺ a1 ⧺ a2 = TAKE (q − n) l ⧺ a1 ⧺ a3 ⧺ [e]`
ORELSE qmatch_goalsub_abbrev_tac `TAKE (q − n) (l ⧺ [e]) ⧺ a1 = TAKE (q − n) l ⧺ a2 ⧺ [e]`)
>> qmatch_asmsub_abbrev_tac `patch_aux (DROP a4 _) _ _ a5 = _`
>> imp_res_tac headers_within_IMP >> fs[]
>> `a5 - n <= LENGTH l` by(unabbrev_all_tac >> intLib.COOPER_TAC)
>> drule0 TAKE_SNOC
>> simp[SNOC_APPEND]
>> disch_then kall_tac
>> `IS_SUFFIX p1 (DROP a4 p1)` by(MATCH_ACCEPT_TAC IS_SUFFIX_DROP)
>> first_assum drule0
>> qunabbrev_tac `a4`
>> disch_then(qspecl_then [`a5`,`DROP (a5 - n) l`,`m`,`e`] mp_tac)
>> impl_tac
>- (qunabbrev_tac `a5`
>> match_mp_tac (GEN_ALL headers_within_grow)
>> MAP_EVERY qexists_tac [`m`,`n+LENGTH l`]
>> fs[]
>> match_mp_tac headers_within_drop
>> imp_res_tac headers_within_dest_cons)
>> fs[ADD1]
>> `a5 - n <= LENGTH l` by(unabbrev_all_tac >> intLib.COOPER_TAC)
>> simp[GSYM DROP_SNOC]
>> qunabbrev_tac `a5`
>> `q - n <= LENGTH l` by intLib.COOPER_TAC
>> simp[TAKE_APPEND,SNOC_APPEND,DROP_APPEND]
>> simp[DROP_def]
>> fs[SNOC_APPEND,DROP_APPEND,DROP_def] >> NO_TAC)
QED
Theorem headers_within_append1:
!l' l. headers_within (n+LENGTH l') m p1 /\ m >= n+LENGTH l' ==>