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(* ========================================================================= *)
(* First order logic with equality. *)
(* ========================================================================= *)
parse_as_infix("===",(18,"right"));;
(* ------------------------------------------------------------------------- *)
(* Convenient shorthand for equality predicate. *)
(* ------------------------------------------------------------------------- *)
let Equal_DEF = new_definition
`s === t = Atom 0 [s; t]`;;
(* ------------------------------------------------------------------------- *)
(* Universal closure. *)
(* ------------------------------------------------------------------------- *)
let uclose = new_definition
`uclose p = ITLIST (!!) (list_of_set(FV p)) p`;;
(* ------------------------------------------------------------------------- *)
(* Definition of normal model. *)
(* ------------------------------------------------------------------------- *)
let normal = new_definition
`normal fns M <=>
!s t v. valuation(M) v /\
s IN terms fns /\
t IN terms fns
==> (holds M v (s === t) <=> (termval M v s = termval M v t))`;;
(* ------------------------------------------------------------------------- *)
(* Equality "axioms" for functions and predicates. *)
(* ------------------------------------------------------------------------- *)
let Varpairs_DEF = new_recursive_definition num_RECURSION
`(Varpairs 0 = []) /\
(Varpairs (SUC n) = CONS (V(2 * n),V(2 * n + 1)) (Varpairs n))`;;
let Eqaxiom_Func = new_definition
`Eqaxiom_Func (f,n) =
uclose (ITLIST (&&) (MAP (\(s,t). s === t) (Varpairs n)) True
--> Fn f (MAP FST (Varpairs n)) === Fn f (MAP SND (Varpairs n)))`;;
let Eqaxiom_Pred = new_definition
`Eqaxiom_Pred (p,n) =
uclose (ITLIST (&&) (MAP (\(s,t). s === t) (Varpairs n)) True
--> (Atom p (MAP FST (Varpairs n))
<-> Atom p (MAP SND (Varpairs n))))`;;
let Eqaxioms_DEF = new_definition
`Eqaxioms L = { (!! 0 (V 0 === V 0)) } UNION
{ (!!0 (!!1 (!!2
(V 0 === V 1 --> (V 2 === V 1 --> V 0 === V 2))))) } UNION
{ Eqaxiom_Func fa | fa IN (FST L) } UNION
{ Eqaxiom_Pred pa | pa IN (SND L) }`;;
let DOWNFROM = new_recursive_definition num_RECURSION
`(DOWNFROM 0 = []) /\
(DOWNFROM (SUC n) = CONS n (DOWNFROM n))`;;
(* ------------------------------------------------------------------------- *)
(* Various stupid lemmas. *)
(* ------------------------------------------------------------------------- *)
let SATISFIES_UNION = prove
(`!M s t. M satisfies (s UNION t) <=> M satisfies s /\ M satisfies t`,
REWRITE_TAC[satisfies; IN_UNION] THEN MESON_TAC[]);;
let HOLDS_UCLOSE_ALL = prove
(`!M x p. (!v:num->A. valuation(M) v ==> holds M v p)
==> !v. valuation(M) v ==> holds M v (uclose p)`,
REPEAT GEN_TAC THEN REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set(FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[uclose; ITLIST] THEN
REWRITE_TAC[holds] THEN ASM_SIMP_TAC[VALUATION_VALMOD]);;
let HOLDS_UCLOSE_ALL_EQ = prove
(`!M. ~(Dom M :A->bool = {})
==> !p. (!v. valuation(M) v ==> holds M v (uclose p)) <=>
(!v. valuation(M) v ==> holds M v p)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set(FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[uclose; ITLIST] THEN
ASM_REWRITE_TAC[HOLDS_UCLOSE]);;
let UCLOSE_FV_LEMMA = prove
(`!l p. FV(ITLIST !! l p) = FV(p) DIFF (set_of_list l)`,
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC[ITLIST; FV; set_of_list] THEN SET_TAC[]);;
let UCLOSE_CLOSED = prove
(`!p. FV(uclose p) = EMPTY`,
GEN_TAC THEN REWRITE_TAC[uclose; UCLOSE_FV_LEMMA] THEN
SIMP_TAC[SET_OF_LIST_OF_SET; FV_FINITE] THEN SET_TAC[]);;
let HOLDS_UCLOSE_ANY = prove
(`!M (v:num->A) p.
holds M v (uclose p)
==> ~(Dom M = EMPTY)
==> !w. valuation(M) w ==> holds M w p`,
REPEAT GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `!w:num->A. valuation(M) w ==> holds M w (uclose p)`
MP_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN
UNDISCH_TAC `holds M (v:num->A) (uclose p)` THEN
MATCH_MP_TAC EQ_IMP THEN
MATCH_MP_TAC HOLDS_VALUATION THEN
REWRITE_TAC[UCLOSE_CLOSED; NOT_IN_EMPTY];
REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set(FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[ITLIST] THEN
ASM_REWRITE_TAC[HOLDS_UCLOSE]]);;
let SATISFIES_NOT = prove
(`~(Dom M :A->bool = {})
==> (M satisfies E /\ ~(M satisfies {p}) <=>
M satisfies (Not (uclose p) INSERT E))`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[TAUT
`(a /\ (b \/ c) ==> d) <=> (a /\ c ==> d) /\ (a /\ b ==> d)`] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
SIMP_TAC[LEFT_FORALL_IMP_THM; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[EXISTS_REFL] THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV)
[GSYM(MATCH_MP HOLDS_UCLOSE_ALL_EQ th)]) THEN
REWRITE_TAC[HOLDS] THEN
ASM_MESON_TAC[VALUATION_EXISTS; HOLDS_VALUATION;
UCLOSE_CLOSED; NOT_IN_EMPTY]);;
let FUNCTIONS_FORM_UCLOSE = prove
(`functions_form(uclose p) = functions_form p`,
REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set (FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ITLIST; functions_form]);;
let PREDICATES_FORM_UCLOSE = prove
(`predicates_form(uclose p) = predicates_form p`,
REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set (FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ITLIST; predicates_form]);;
let HOLDS_ANDS = prove
(`!l. holds M v (ITLIST (&&) l True) <=> ALL (holds M v) l`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ITLIST; ALL; HOLDS]);;
let LENGTH_VARPAIRS = prove
(`!n. LENGTH (Varpairs n) = n`,
INDUCT_TAC THEN ASM_REWRITE_TAC[Varpairs_DEF; LENGTH]);;
let MAP_FST_VARPAIRS = prove
(`!n. MAP FST (Varpairs n) = MAP (\n. V(2 * n)) (DOWNFROM n)`,
INDUCT_TAC THEN ASM_REWRITE_TAC[Varpairs_DEF; DOWNFROM; MAP]);;
let MAP_SND_VARPAIRS = prove
(`!n. MAP SND (Varpairs n) = MAP (\n. V(2 * n + 1)) (DOWNFROM n)`,
INDUCT_TAC THEN ASM_REWRITE_TAC[Varpairs_DEF; DOWNFROM; MAP]);;
let MULT_DIV_LEMMA = prove
(`(!n. (2 * n) DIV 2 = n) /\
(!n. (2 * n + 1) DIV 2 = n)`,
SIMP_TAC[DIV_MULT; ARITH_EQ] THEN
GEN_TAC THEN MATCH_MP_TAC DIV_UNIQ THEN
EXISTS_TAC `1` THEN REWRITE_TAC[MULT_AC; ARITH]);;
let PAIR_LEMMA = prove
(`!z. (\(x,y). P x y) z = P (FST z) (SND z)`,
REWRITE_TAC[FORALL_PAIR_THM]);;
let FORALL_VARPAIRS = prove
(`!n. ALL P (Varpairs n) <=>
ALL (\n. P (V (2 * n),V (2 * n + 1))) (DOWNFROM n)`,
INDUCT_TAC THEN ASM_REWRITE_TAC[ALL; Varpairs_DEF; DOWNFROM]);;
let FORALL2_VARPAIRS = prove
(`!n. ALL (\p. Pred M a [termval M v (FST p); termval M v (SND p)])
(Varpairs n) <=>
ALL2 (\x y. Pred M a [x;y])
(MAP (termval M v) (MAP FST (Varpairs n)))
(MAP (termval M v) (MAP SND (Varpairs n)))`,
INDUCT_TAC THEN ASM_REWRITE_TAC[Varpairs_DEF; ALL; ALL2; MAP]);;
let FORALL2_FORALL = prove
(`!l. ALL2 (\x y. P x y) (MAP f l) (MAP g l) <=>
ALL (\x. P (f x) (g x)) l`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ALL; MAP; ALL2]);;
let FORALL_DOWNFROM = prove
(`!n. ALL P (DOWNFROM n) <=> !k. k < n ==> P k`,
INDUCT_TAC THEN ASM_REWRITE_TAC[DOWNFROM; ALL; LT] THEN MESON_TAC[]);;
let MAP_INDEXED = prove
(`!l:A list. MAP (\n. EL ((LENGTH l - 1) - n) l) (DOWNFROM (LENGTH l)) = l`,
let lemma = prove
(`!k n. k < n ==> (SUC n - 1 - k = SUC (n - 1 - k))`,
REPEAT GEN_TAC THEN REWRITE_TAC[ARITH_RULE `SUC n - 1 = n`] THEN
REWRITE_TAC[LT_EXISTS] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[ARITH_RULE `(m + SUC n) - 1 = m + n`; ADD_SUB2]) in
ONCE_REWRITE_TAC[lemma] THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[MAP; LENGTH; DOWNFROM; EL] THEN
REWRITE_TAC[ARITH_RULE `(SUC n - 1) - n = 0`; EL; HD] THEN
AP_TERM_TAC THEN
POP_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
MATCH_MP_TAC MAP_EQ THEN REWRITE_TAC[FORALL_DOWNFROM] THEN
GEN_TAC THEN SPEC_TAC(`LENGTH (t:A list)`,`n:num`) THEN
MESON_TAC[EL; TL; lemma]);;
let RECONSTRUCT_TERMVAL = prove
(`!M a f1 f2.
(MAP (termval M (\n. if 2 * LENGTH l <= n then a
else if EVEN(n)
then f1 (EL ((LENGTH l - 1) - (n DIV 2)) l)
else f2 (EL ((LENGTH l - 1) - (n DIV 2)) l)))
(MAP FST (Varpairs (LENGTH l))) = MAP f1 l) /\
(MAP (termval M (\n. if 2 * LENGTH l <= n then a
else if EVEN(n)
then f1 (EL ((LENGTH l - 1) - (n DIV 2)) l)
else f2 (EL ((LENGTH l - 1) - (n DIV 2)) l)))
(MAP SND (Varpairs (LENGTH l))) = MAP f2 l)`,
let lemma = prove
(`!L K. L <= K
==> (MAP (\n. if K <= n then a else f n) (DOWNFROM L) =
MAP f (DOWNFROM L))`,
INDUCT_TAC THEN REWRITE_TAC[MAP; DOWNFROM] THEN GEN_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP (ARITH_RULE
`SUC L <= K ==> L <= K /\ ~(K <= L)`)) THEN
DISCH_THEN(CONJUNCTS_THEN2 (ANTE_RES_THEN SUBST1_TAC) ASSUME_TAC) THEN
ASM_REWRITE_TAC[]) in
REPEAT GEN_TAC THEN
REWRITE_TAC[MAP_FST_VARPAIRS; MAP_SND_VARPAIRS] THEN
REWRITE_TAC[GSYM MAP_o; o_DEF; termval] THEN
REWRITE_TAC[LE_MULT_LCANCEL; ARITH] THEN
REWRITE_TAC[EVEN_MULT; EVEN_ADD; MULT_DIV_LEMMA; ARITH] THEN
REWRITE_TAC[ARITH_RULE `2 * x <= 2 * y + 1 <=> x <= y`] THEN
SIMP_TAC[lemma; LE_REFL] THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN
REWRITE_TAC[MAP_o; MAP_INDEXED]);;
let FORALL_TERMS_STRONG = prove
(`!s t. terms (functions s) t <=> functions_term t SUBSET (functions s)`,
let lemma0 = prove
(`(a INSERT b) SUBSET s <=> a IN s /\ b SUBSET s`,
SET_TAC[]) in
let lemma1 = prove
(`s UNION t SUBSET u <=> s SUBSET u /\ t SUBSET u`,
SET_TAC[]) in
let lemma2 = prove
(`!l. LIST_UNION l SUBSET s <=> ALL (\x. x SUBSET s) l`,
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; LIST_UNION; EMPTY_SUBSET] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[lemma1]) in
GEN_TAC THEN MATCH_MP_TAC term_INDUCT THEN
REWRITE_TAC[EMPTY_SUBSET; functions_term] THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[terms_CASES] THEN
REWRITE_TAC[term_INJ; term_DISTINCT] THEN MESON_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[terms_CASES] THEN
REWRITE_TAC[term_INJ; term_DISTINCT] THEN
REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[UNWIND_THM1; lemma0] THEN AP_TERM_TAC THEN
REWRITE_TAC[lemma2] THEN REWRITE_TAC[ALL_MAP; o_DEF] THEN
POP_ASSUM MP_TAC THEN
SPEC_TAC(`l:term list`,`l:term list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL] THEN
ASM_MESON_TAC[]);;
let FORALL_TERMS_RAW = prove
(`!s t. terms (functions s) t ==> functions_term t SUBSET (functions s)`,
REWRITE_TAC[FORALL_TERMS_STRONG]);;
let FORALL_TERMS_INDEXED = prove
(`!l n. ALL (terms (functions s)) l /\ n < LENGTH l
==> functions_term (EL n l) SUBSET functions s`,
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; LENGTH; CONJUNCT1 LT] THEN
INDUCT_TAC THEN REWRITE_TAC[LT_SUC; LT_0; EL; HD] THENL
[MESON_TAC[FORALL_TERMS_RAW];
REWRITE_TAC[TL] THEN ASM_MESON_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* The key theorem. *)
(* ------------------------------------------------------------------------- *)
let NORMAL_THM = prove
(`(?M:(A->bool)#(num->A list->A)#(num->A list->bool).
interpretation (language s) M /\
~(Dom M = EMPTY) /\
normal (functions s) M /\
M satisfies s) <=>
(?M:(A->bool)#(num->A list->A)#(num->A list->bool).
interpretation (language s) M /\
~(Dom M = EMPTY) /\
M satisfies (s UNION Eqaxioms (language s)))`,
let lemma0 = prove
(`a SUBSET s /\ b SUBSET s ==> (a UNION b) SUBSET s`,SET_TAC[]) in
let lemma1 = prove
(`a IN s /\ b SUBSET s ==> (a INSERT b) SUBSET s`,SET_TAC[]) in
let lemma2 = prove
(`!l. ALL (\x. x SUBSET s) l ==> LIST_UNION l SUBSET s`,
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; LIST_UNION; EMPTY_SUBSET] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma0 THEN ASM_MESON_TAC[]) in
let lemma3 = prove
(`{x} SUBSET s <=> x IN s`,
SET_TAC[]) in
let lemma4a = prove
(`s UNION t SUBSET u <=> s SUBSET u /\ t SUBSET u`,
SET_TAC[]) in
let lemma4 = prove
(`!l. LIST_UNION l SUBSET s <=> ALL (\x. x SUBSET s) l`,
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; LIST_UNION; EMPTY_SUBSET] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[lemma4a]) in
let slemma1 = prove
(`(s UNION t) SUBSET u <=> s SUBSET u /\ t SUBSET u`,
SET_TAC[]) in
let clemma1 = prove
(`canonize o valmod (x,a) v = valmod (x,canonize a) (canonize o v)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; valmod] THEN
GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[]) in
REWRITE_TAC[Eqaxioms_DEF; SATISFIES_UNION] THEN EQ_TAC THEN STRIP_TAC THENL
[EXISTS_TAC `M:(A->bool)#(num->A list->A)#(num->A list->bool)` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[satisfies] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[RULE_ASSUM_TAC(REWRITE_RULE[normal]) THEN
ASM_SIMP_TAC[holds; IN; terms_RULES; VALUATION_VALMOD];
RULE_ASSUM_TAC(REWRITE_RULE[normal]) THEN
REWRITE_TAC[holds; IN] THEN
ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4)
[IN; terms_RULES; VALUATION_VALMOD];
UNDISCH_TAC `valuation M (v:num->A)` THEN
SPEC_TAC(`v:num->A`,`v:num->A`) THEN
SUBST_ALL_TAC(SYM(ISPEC `fa:num#num` PAIR)) THEN
PURE_REWRITE_TAC[Eqaxiom_Func] THEN
MATCH_MP_TAC HOLDS_UCLOSE_ALL THEN
REWRITE_TAC[holds] THEN
RULE_ASSUM_TAC(REWRITE_RULE[language; normal]) THEN
GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[HOLDS_ANDS] THEN
REWRITE_TAC[ALL_MAP] THEN
FIRST_ASSUM(MP_TAC o SPECL
[`Fn (FST fa) (MAP FST (Varpairs (SND fa)))`;
`Fn (FST fa) (MAP SND (Varpairs (SND fa)))`;
`v:num->A`]) THEN
W(C SUBGOAL_THEN (fun t -> REWRITE_TAC[t]) o funpow 2 lhand o snd) THENL
[ASM_REWRITE_TAC[IN] THEN CONJ_TAC THEN
MATCH_MP_TAC(CONJUNCT2(SPEC_ALL terms_RULES)) THEN
ASM_REWRITE_TAC[LENGTH_MAP; LENGTH_VARPAIRS] THEN
SPEC_TAC(`SND(fa:num#num)`,`n:num`) THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[ALL; Varpairs_DEF; MAP] THEN
MESON_TAC[terms_RULES];
DISCH_THEN(fun th -> REWRITE_TAC[th])] THEN
REWRITE_TAC[termval] THEN REWRITE_TAC[GSYM MAP_o] THEN
DISCH_THEN(fun th -> AP_TERM_TAC THEN MP_TAC th) THEN
SPEC_TAC(`SND(fa:num#num)`,`n:num`) THEN
INDUCT_TAC THEN REWRITE_TAC[Varpairs_DEF; ALL; MAP] THEN
REWRITE_TAC[o_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_SIMP_TAC[terms_RULES; IN];
UNDISCH_TAC `valuation M (v:num->A)` THEN
SPEC_TAC(`v:num->A`,`v:num->A`) THEN
SUBST_ALL_TAC(SYM(ISPEC `pa:num#num` PAIR)) THEN
PURE_REWRITE_TAC[Eqaxiom_Pred] THEN
MATCH_MP_TAC HOLDS_UCLOSE_ALL THEN
REWRITE_TAC[HOLDS] THEN
RULE_ASSUM_TAC(REWRITE_RULE[language; normal]) THEN
GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[HOLDS_ANDS] THEN
REWRITE_TAC[ALL_MAP] THEN
REWRITE_TAC[termval] THEN REWRITE_TAC[GSYM MAP_o] THEN
DISCH_THEN(fun th -> AP_TERM_TAC THEN MP_TAC th) THEN
SPEC_TAC(`SND(pa:num#num)`,`n:num`) THEN
INDUCT_TAC THEN REWRITE_TAC[Varpairs_DEF; ALL; MAP] THEN
REWRITE_TAC[o_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_SIMP_TAC[terms_RULES; IN]]; ALL_TAC] THEN
ABBREV_TAC `equiv = \a:A b. a IN Dom(M) /\ b IN Dom(M) /\
Pred(M) 0 [a; b]` THEN
ABBREV_TAC `canonize = \a:A. @b:A. equiv a b` THEN
ABBREV_TAC `M' = { a:A | a IN Dom(M) /\ (a = canonize a) },
(\f l. canonize(Fun(M) f l)),Pred(M)` THEN
EXISTS_TAC `M':(A->bool)#(num->A list->A)#(num->A list->bool)` THEN
REWRITE_TAC[normal] THEN
SUBGOAL_THEN `!a:A. equiv a a <=> a IN Dom(M)` ASSUME_TAC THENL
[GEN_TAC THEN EXPAND_TAC "equiv" THEN
UNDISCH_TAC `(M:(A->bool)#(num->A list->A)#(num->A list->bool))
satisfies { (!!0 (V 0 === V 0)) }` THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN(MP_TAC o SPEC `\x:num. a:A`) THEN
ASM_REWRITE_TAC[valuation] THEN SIMP_TAC[holds] THEN
REWRITE_TAC[Equal_DEF; holds] THEN
DISCH_THEN(MP_TAC o SPEC `!!0 (Atom 0 [V 0; V 0])`) THEN
REWRITE_TAC[] THEN
ASM_CASES_TAC `a:A IN Dom(M)` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `a:A`) THEN
ASM_REWRITE_TAC[MAP; termval; valmod]; ALL_TAC] THEN
SUBGOAL_THEN `!a:A. equiv a (canonize a :A) <=> a IN Dom(M)` ASSUME_TAC THENL
[EXPAND_TAC "canonize" THEN
GEN_TAC THEN CONV_TAC(LAND_CONV SELECT_CONV) THEN
ASM_MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!a:A b c. equiv a b ==> equiv c b ==> equiv a c`
ASSUME_TAC THENL
[REPEAT GEN_TAC THEN EXPAND_TAC "equiv" THEN
STRIP_TAC THEN STRIP_TAC THEN
UNDISCH_TAC `(M:(A->bool)#(num->A list->A)#(num->A list->bool))
satisfies
{ (!!0 (!!1 (!!2
(V 0 === V 1 --> (V 2 === V 1 --> V 0 === V 2))))) }` THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN(MP_TAC o SPEC `valmod (0,a) (valmod (1,b) (\n. c:A))`) THEN
ASM_SIMP_TAC[VALUATION_VALMOD; valuation] THEN
DISCH_THEN(MP_TAC o SPEC
`!!0 (!!1 (!!2 (V 0 === V 1 --> (V 2 === V 1 --> V 0 === V 2))))`) THEN
REWRITE_TAC[holds; valmod; Equal_DEF; MAP; termval; ARITH_EQ] THEN
ASM_MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!a:A b. equiv a b ==> equiv b a`
ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!a:A b. equiv a b ==> (canonize a :A = canonize b)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN EXPAND_TAC "canonize" THEN
AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `~(Dom M' :A->bool = EMPTY)` ASSUME_TAC THENL
[EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
UNDISCH_TAC `~(Dom M :A->bool = EMPTY)` THEN
REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; NOT_FORALL_THM] THEN
DISCH_THEN(X_CHOOSE_TAC `a:A`) THEN
EXISTS_TAC `(canonize:A->A) a` THEN
REWRITE_TAC[IN_ELIM_THM] THEN
SUBGOAL_THEN `(equiv:A->A->bool) a (canonize a)`
(fun th -> ASM_MESON_TAC[th]) THEN
EXPAND_TAC "canonize" THEN CONV_TAC SELECT_CONV THEN
EXISTS_TAC `a:A` THEN EXPAND_TAC "equiv" THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `(M:(A->bool)#(num->A list->A)#(num->A list->bool))
satisfies
{ (!!0 (V 0 === V 0)) }` THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN(MP_TAC o SPECL [`\n:num. a:A`; `!! 0 (V 0 === V 0)`]) THEN
ASM_REWRITE_TAC[valuation] THEN
REWRITE_TAC[holds; Equal_DEF; MAP; termval; valmod] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REWRITE_TAC[interpretation; language] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Fun_DEF; Dom_DEF] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
SUBGOAL_THEN `(equiv:A->A->bool) (Fun M f l) (canonize (Fun M f l))`
(fun th -> ASM_MESON_TAC[th]) THEN
ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[language; interpretation]) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC ALL_IMP THEN
FIRST_ASSUM(fun th -> EXISTS_TAC (lhand(concl th)) THEN
CONJ_TAC THENL [ALL_TAC; FIRST_ASSUM ACCEPT_TAC]) THEN
REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[]; DISCH_TAC] THEN
SUBGOAL_THEN `!t v. valuation M (v:num->A) /\
t IN terms (functions s)
==> (termval M' (canonize o v) t =
(canonize:A->A) (termval M v t))`
ASSUME_TAC THENL
[ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
REWRITE_TAC[IMP_CONJ] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
GEN_TAC THEN EXPAND_TAC "M'" THEN
DISCH_THEN(ASSUME_TAC o GSYM o
REWRITE_RULE[valuation; Dom_DEF; IN_ELIM_THM]) THEN
MATCH_MP_TAC term_INDUCT THEN
ASM_REWRITE_TAC[termval; Fun_DEF] THEN REWRITE_TAC[o_THM] THEN
X_GEN_TAC `f:num` THEN REPEAT STRIP_TAC THEN
FIRST_ASSUM MATCH_MP_TAC THEN
FIRST_ASSUM MATCH_MP_TAC THEN
EXPAND_TAC "equiv" THEN REPEAT CONJ_TAC THENL
[REWRITE_TAC[GSYM(CONJUNCT2 termval)] THEN
MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `functions s` THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_RAW THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN ASM_REWRITE_TAC[];
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[interpretation; language]) THEN
UNDISCH_TAC `Fn f l IN terms (functions s)` THEN
REWRITE_TAC[IN; LENGTH_MAP] THEN ONCE_REWRITE_TAC[terms_CASES] THEN
REWRITE_TAC[term_INJ; term_DISTINCT] THEN
DISCH_THEN(X_CHOOSE_THEN `g:num` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `L:term list` MP_TAC) THEN
DISCH_THEN(fun th -> MP_TAC(CONJUNCT2 th) THEN
REWRITE_TAC[GSYM(CONJUNCT1 th)]) THEN
REWRITE_TAC[IN] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[ALL_MAP] THEN
MATCH_MP_TAC ALL_IMP THEN
FIRST_ASSUM(fun th -> EXISTS_TAC (lhand(concl th)) THEN
CONJ_TAC THENL [ALL_TAC; FIRST_ASSUM ACCEPT_TAC]) THEN
X_GEN_TAC `u:term` THEN REWRITE_TAC[o_THM] THEN STRIP_TAC THEN
SUBGOAL_THEN `(termval M' ((canonize:A->A) o v) u) IN (Dom M')`
MP_TAC THENL
[MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN CONJ_TAC THENL
[UNDISCH_TAC `interpretation (language s)
(M':(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
UNDISCH_TAC `terms (functions s) u` THEN
SPEC_TAC(`functions s`,`fns:num#num->bool`) THEN
SPEC_TAC(`u:term`,`u:term`) THEN
MATCH_MP_TAC term_INDUCT THEN
REWRITE_TAC[functions_term; EMPTY_SUBSET] THEN
REPEAT GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [terms_CASES] THEN
REWRITE_TAC[term_INJ; term_DISTINCT] THEN
DISCH_THEN(X_CHOOSE_THEN `g:num` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `L:term list` MP_TAC) THEN
DISCH_THEN(fun th -> MP_TAC(CONJUNCT2 th) THEN
REWRITE_TAC[GSYM(CONJUNCT1 th)]) THEN
STRIP_TAC THEN MATCH_MP_TAC lemma1 THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC lemma2 THEN
REWRITE_TAC[ALL_MAP] THEN
MATCH_MP_TAC ALL_IMP THEN EXISTS_TAC
`\u. !fns. terms fns u ==> functions_term u SUBSET fns` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[o_THM] THEN
X_GEN_TAC `tt:term` THEN STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
UNDISCH_TAC `MEM (tt:term) l'` THEN
UNDISCH_TAC `ALL (terms fns) l'` THEN
SPEC_TAC(`l':term list`,`ll:term list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; MEM] THEN
ASM_MESON_TAC[];
REWRITE_TAC[valuation; o_THM] THEN
X_GEN_TAC `x:num` THEN EXPAND_TAC "M'" THEN
REWRITE_TAC[Dom_DEF] THEN REWRITE_TAC[IN_ELIM_THM] THEN
ASM_MESON_TAC[]];
REWRITE_TAC[IN] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM] THEN
SIMP_TAC[IN]];
UNDISCH_TAC `(M:(A->bool)#(num->A list->A)#(num->A list->bool))
satisfies {Eqaxiom_Func fa | fa IN FST (language s)}` THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `Eqaxiom_Func (f,LENGTH (l:term list))`) THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `(f:num,LENGTH(l:term list))`) THEN
REWRITE_TAC[] THEN
REWRITE_TAC[Eqaxiom_Func] THEN
SUBGOAL_THEN `?a:A. a IN Dom(M)` CHOOSE_TAC THENL
[ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `\n:num. a:A`) THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `Fn f l IN terms (functions s)` THEN
REWRITE_TAC[IN] THEN ONCE_REWRITE_TAC[terms_CASES] THEN
REWRITE_TAC[term_INJ; term_DISTINCT] THEN
DISCH_THEN(X_CHOOSE_THEN `g:num` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `L:term list` MP_TAC) THEN
DISCH_THEN(fun th -> MP_TAC(CONJUNCT2 th) THEN
REWRITE_TAC[GSYM(CONJUNCT1 th)]) THEN
REWRITE_TAC[language; IN] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o MATCH_MP HOLDS_UCLOSE_ANY) THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[holds; HOLDS_ANDS; Equal_DEF; MAP; GSYM MAP_o] THEN
REWRITE_TAC[termval] THEN
REWRITE_TAC[ALL_MAP; o_DEF; PAIR_LEMMA; holds] THEN
REWRITE_TAC[MAP] THEN
REWRITE_TAC[FORALL2_VARPAIRS] THEN
DISCH_THEN(MP_TAC o SPEC
`\n:num. if 2 * LENGTH l <= n then a
else if EVEN(n)
then termval M v
(EL ((LENGTH l - 1) - (n DIV 2)) l) :A
else termval M' (canonize o v)
(EL ((LENGTH l - 1) - (n DIV 2)) l)`) THEN
REWRITE_TAC[RECONSTRUCT_TERMVAL] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN
REWRITE_TAC[IMP_IMP] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [o_DEF] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[FORALL2_FORALL] THEN CONJ_TAC THENL
[REWRITE_TAC[valuation] THEN X_GEN_TAC `k:num` THEN
ASM_CASES_TAC `2 * LENGTH (l:term list) <= k` THEN
ASM_REWRITE_TAC[] THEN COND_CASES_TAC THENL
[MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_INDEXED THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(2 * LENGTH (l:term list) <= k)` THEN
REWRITE_TAC[ARITH_RULE `~(2 * n <= k) ==> n - 1 - m < n`];
SUBGOAL_THEN `termval M' ((canonize:A->A) o v)
(EL (LENGTH l - 1 - k DIV 2) l) IN
(Dom M' :A->bool)` MP_TAC THENL
[MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN
ASM_REWRITE_TAC[valuation] THEN CONJ_TAC THENL
[UNDISCH_TAC `interpretation (language s)
(M':(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_INDEXED THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(2 * LENGTH (l:term list) <= k)` THEN
REWRITE_TAC[ARITH_RULE `~(2 * n <= k) ==> n - 1 - m < n`];
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF; o_THM] THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[]];
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]]];
MATCH_MP_TAC ALL_IMP THEN
EXISTS_TAC `\t. terms (functions s) t /\
(t IN terms (functions s)
==> (termval M' (canonize o v) t =
(canonize:A->A) (termval M v t)))` THEN
REWRITE_TAC[GSYM AND_ALL] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN
ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `(equiv:A->A->bool) (termval M v x)
(termval M' (canonize o v) x)`
MP_TAC THENL [ALL_TAC; EXPAND_TAC "equiv" THEN SIMP_TAC[]] THEN
SUBGOAL_THEN `!a b. (a = canonize b) /\ b IN Dom(M)
==> (equiv:A->A->bool) b a`
MATCH_MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
CONJ_TAC THENL [ASM_SIMP_TAC[IN]; ALL_TAC] THEN
MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `functions s` THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_RAW THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN ASM_REWRITE_TAC[]]];
ALL_TAC] THEN
SUBGOAL_THEN
`!p v. valuation M (v:num->A) /\
(functions_form p) SUBSET (functions s) /\
(predicates_form p) SUBSET (predicates s)
==> (holds M' ((canonize:A->A) o v) p <=> holds M v p)`
ASSUME_TAC THENL
[MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[holds; functions_form; predicates_form] THEN
REWRITE_TAC[slemma1] THEN REPEAT CONJ_TAC THENL
[ALL_TAC;
MESON_TAC[];
REPEAT GEN_TAC THEN
DISCH_THEN(fun th -> SIMP_TAC[GSYM th; VALUATION_VALMOD]) THEN
REWRITE_TAC[clemma1] THEN GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `!P. (!a:A. a IN Dom(M') ==> P a) <=>
(!a:A. a IN Dom(M) ==> P(canonize a))`
(fun th -> REWRITE_TAC[th]) THEN
GEN_TAC THEN EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[]] THEN
X_GEN_TAC `p:num` THEN X_GEN_TAC `l:term list` THEN GEN_TAC THEN
REWRITE_TAC[lemma3; lemma4; ALL_MAP] THEN
DISCH_THEN(STRIP_ASSUME_TAC o REWRITE_RULE[o_DEF]) THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Pred_DEF] THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `(M:(A->bool)#(num->A list->A)#(num->A list->bool))
satisfies {Eqaxiom_Pred pa | pa IN SND (language s)}` THEN
REWRITE_TAC[satisfies; IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `Eqaxiom_Pred (p,LENGTH (l:term list))`) THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `(p:num,LENGTH(l:term list))`) THEN
REWRITE_TAC[] THEN
REWRITE_TAC[Eqaxiom_Pred] THEN
SUBGOAL_THEN `?a:A. a IN Dom(M)` CHOOSE_TAC THENL
[ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `\n:num. a:A`) THEN
ASM_REWRITE_TAC[valuation] THEN
ASM_REWRITE_TAC[language] THEN
DISCH_THEN(MP_TAC o MATCH_MP HOLDS_UCLOSE_ANY) THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[HOLDS; HOLDS_ANDS; Equal_DEF; MAP; GSYM MAP_o] THEN
REWRITE_TAC[MAP_o] THEN
DISCH_THEN(MP_TAC o SPEC
`\n:num. if 2 * LENGTH l <= n then a
else if EVEN(n)
then termval M v
(EL ((LENGTH l - 1) - (n DIV 2)) l) :A
else termval M' (canonize o v)
(EL ((LENGTH l - 1) - (n DIV 2)) l)`) THEN
REWRITE_TAC[RECONSTRUCT_TERMVAL] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN
REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(fun th -> CONV_TAC SYM_CONV THEN MATCH_MP_TAC th) THEN
REWRITE_TAC[FORALL2_FORALL] THEN CONJ_TAC THENL
[REWRITE_TAC[valuation] THEN X_GEN_TAC `k:num` THEN
ASM_CASES_TAC `2 * LENGTH (l:term list) <= k` THEN
ASM_REWRITE_TAC[] THEN COND_CASES_TAC THENL
[MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_INDEXED THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC ALL_IMP THEN
EXISTS_TAC `\x. functions_term x SUBSET functions s` THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[FORALL_TERMS_STRONG];
UNDISCH_TAC `~(2 * LENGTH (l:term list) <= k)` THEN
REWRITE_TAC[ARITH_RULE `~(2 * n <= k) ==> n - 1 - m < n`]];
SUBGOAL_THEN `termval M' ((canonize:A->A) o v)
(EL (LENGTH l - 1 - k DIV 2) l) IN
(Dom M' :A->bool)` MP_TAC THENL
[MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN
ASM_REWRITE_TAC[valuation] THEN CONJ_TAC THENL
[UNDISCH_TAC `interpretation (language s)
(M':(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_INDEXED THEN
ASM_REWRITE_TAC[] THEN
RULE_ASSUM_TAC(REWRITE_RULE[GSYM FORALL_TERMS_STRONG]) THEN
RULE_ASSUM_TAC(CONV_RULE(ONCE_DEPTH_CONV ETA_CONV)) THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(2 * LENGTH (l:term list) <= k)` THEN
REWRITE_TAC[ARITH_RULE `~(2 * n <= k) ==> n - 1 - m < n`];
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF; o_THM] THEN
RULE_ASSUM_TAC(REWRITE_RULE[valuation]) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[]];
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[]]];
REWRITE_TAC[ALL_MAP] THEN
REWRITE_TAC[PAIR_LEMMA; o_DEF] THEN
REWRITE_TAC[holds; MAP; termval] THEN
REWRITE_TAC[FORALL2_VARPAIRS] THEN
REWRITE_TAC[GSYM MAP_o] THEN
REWRITE_TAC[FORALL2_FORALL] THEN
REWRITE_TAC[o_DEF; FORALL2_VARPAIRS] THEN
REWRITE_TAC[RECONSTRUCT_TERMVAL] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN
REWRITE_TAC[FORALL2_FORALL] THEN
REWRITE_TAC[GSYM o_DEF] THEN
ONCE_REWRITE_TAC[o_DEF] THEN REWRITE_TAC[] THEN
MATCH_MP_TAC ALL_IMP THEN
EXISTS_TAC `\x. functions_term x SUBSET functions s` THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[GSYM FORALL_TERMS_STRONG] THEN
X_GEN_TAC `t:term` THEN STRIP_TAC THEN
ASM_SIMP_TAC[IN] THEN
SUBGOAL_THEN `(equiv:A->A->bool) (termval M v t)
(canonize (termval M v t))`
MP_TAC THENL
[SUBGOAL_THEN `termval M v t IN (Dom(M):A->bool)`
(fun th -> ASM_MESON_TAC[th]) THEN
MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `functions s` THEN
ASM_REWRITE_TAC[valuation] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_RAW THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN ASM_REWRITE_TAC[];
EXPAND_TAC "equiv" THEN SIMP_TAC[]]]; ALL_TAC] THEN
CONJ_TAC THENL
[X_GEN_TAC `t:term` THEN X_GEN_TAC `u:term` THEN
X_GEN_TAC `v:num->A` THEN STRIP_TAC THEN
REWRITE_TAC[Equal_DEF; holds; MAP] THEN ASM_SIMP_TAC[] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Pred_DEF] THEN
SUBGOAL_THEN `termval M v t IN (Dom(M):A->bool) /\
termval M v u IN Dom(M)`
ASSUME_TAC THENL
[CONJ_TAC THEN
MATCH_MP_TAC INTERPRETATION_TERMVAL THEN
EXISTS_TAC `ppp:num#num->bool` THEN
UNDISCH_TAC `valuation M' (v:num->A)` THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[valuation; Dom_DEF] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `interpretation (language s)
(M:(A->bool)#(num->A list->A)#(num->A list->bool))` THEN
REWRITE_TAC[language] THEN
MATCH_MP_TAC INTERPRETATION_SUBLANGUAGE THEN
MATCH_MP_TAC FORALL_TERMS_RAW THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[]] THEN
SUBGOAL_THEN `canonize o (v:num->A) = v`
(fun th -> ONCE_REWRITE_TAC[GSYM th]) THENL
[REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN
REWRITE_TAC[o_THM] THEN
UNDISCH_TAC `valuation M' (v:num->A)` THEN
REWRITE_TAC[valuation] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `valuation M (v:num->A)` ASSUME_TAC THENL
[UNDISCH_TAC `valuation M' (v:num->A)` THEN
REWRITE_TAC[valuation] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `(equiv:A->A->bool) ((canonize:A->A) (termval M v t))
(canonize (termval M v u))` THEN
CONJ_TAC THENL
[EXPAND_TAC "equiv" THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `(equiv:A->A->bool) (termval M v t) (termval M v u)` THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[satisfies] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `canonize o (v:num->A) = v`
(fun th -> ONCE_REWRITE_TAC[GSYM th]) THENL
[REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN
REWRITE_TAC[o_THM] THEN
UNDISCH_TAC `valuation M' (v:num->A)` THEN
REWRITE_TAC[valuation] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `valuation M (v:num->A)` ASSUME_TAC THENL
[UNDISCH_TAC `valuation M' (v:num->A)` THEN
REWRITE_TAC[valuation] THEN
EXPAND_TAC "M'" THEN REWRITE_TAC[Dom_DEF] THEN
REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `functions_form p SUBSET functions s /\
predicates_form p SUBSET predicates s`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[functions; predicates] THEN
UNDISCH_TAC `p:form IN s` THEN
REWRITE_TAC[SUBSET; IN_UNIONS; IN_ELIM_THM] THEN MESON_TAC[];
RULE_ASSUM_TAC(REWRITE_RULE[satisfies]) THEN ASM_SIMP_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Compactness and Lowenheim-Skolem for normal models. *)
(* ------------------------------------------------------------------------- *)
let FUNCTIONS_SUBSET = prove
(`!s t. t SUBSET s ==> functions t SUBSET functions s`,
REWRITE_TAC[SUBSET; functions; IN_ELIM_THM; IN_UNIONS] THEN MESON_TAC[]);;
let INTERPRETATION_MODIFY_LANGUAGE = prove
(`!s s' t.
t SUBSET s /\
(?M. interpretation (language s) M /\
~(Dom M :A->bool = EMPTY) /\
M satisfies t)
==> ?M. interpretation (language s') M /\
~(Dom M :A->bool = EMPTY) /\
M satisfies t`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
EXISTS_TAC `Dom(M):A->bool,
(\g zs. if (g,LENGTH zs) IN functions s then Fun(M) g zs
else @a. a IN Dom(M)),
Pred(M)` THEN
REWRITE_TAC[Dom_DEF; Fun_DEF; Pred_DEF; interpretation; language] THEN CONJ_TAC THENL
[REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE [language; interpretation]) THEN
ASM_REWRITE_TAC[];
CONV_TAC SELECT_CONV THEN ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]];
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [satisfies]) THEN
REWRITE_TAC[satisfies] THEN
ASM_REWRITE_TAC[valuation; Dom_DEF] THEN
MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a ==> b) <=> (a ==> c))`) THEN
STRIP_TAC THEN SPEC_TAC(`v:num->A`,`v:num->A`) THEN
MATCH_MP_TAC HOLDS_FUNCTIONS THEN
REWRITE_TAC[Dom_DEF; Fun_DEF; Pred_DEF] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_UNIONS; functions; SUBSET;
IN_ELIM_THM]) THEN
ASM_MESON_TAC[]]);;
let INTERPRETATION_RESTRICT_LANGUAGE = prove
(`!s s' t.
s' SUBSET s /\
(?M. interpretation (language s) M /\
~(Dom M :A->bool = EMPTY) /\
M satisfies t)
==> ?M. interpretation (language s') M /\
~(Dom M :A->bool = EMPTY) /\
M satisfies t`,
REPEAT GEN_TAC THEN
REWRITE_TAC[SUBSET; language; interpretation] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC MONO_EXISTS THEN
GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[FUNCTIONS_SUBSET; SUBSET]);;
let FUNCTIONS_INSERT = prove
(`functions(p INSERT s) = functions_form p UNION functions s`,
REWRITE_TAC[EXTENSION; functions; IN_UNION; IN_INSERT;
IN_UNIONS; IN_ELIM_THM] THEN
MESON_TAC[]);;
let FUNCTIONS_UNION = prove
(`functions (s UNION t) = functions s UNION functions t`,
REWRITE_TAC[EXTENSION; functions; IN_UNION; IN_UNIONS; IN_ELIM_THM] THEN
MESON_TAC[]);;
let PREDICATES_UNION = prove
(`predicates (s UNION t) = predicates s UNION predicates t`,
REWRITE_TAC[EXTENSION; predicates; IN_UNION; IN_UNIONS; IN_ELIM_THM] THEN
MESON_TAC[]);;
let FUNCTIONS_FORM_UCLOSE = prove
(`functions_form(uclose p) = functions_form p`,
REWRITE_TAC[uclose] THEN
SPEC_TAC(`list_of_set(FV p)`,`l:num list`) THEN
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ITLIST; functions_form]);;
let FUNCTIONS_TERM_FNV1 = prove
(`functions_term (Fn p (MAP FST (Varpairs n))) = {(p,n)}`,
REWRITE_TAC[functions_term; LENGTH_MAP; LENGTH_VARPAIRS] THEN
AP_TERM_TAC THEN SPEC_TAC(`n:num`,`n:num`) THEN
INDUCT_TAC THEN
REWRITE_TAC[MAP; Varpairs_DEF; functions_term; LIST_UNION] THEN
ASM_REWRITE_TAC[UNION_EMPTY]);;
let FUNCTIONS_FORM_FNV1 = prove
(`!n. LIST_UNION (MAP functions_term (MAP FST (Varpairs n))) = EMPTY`,
INDUCT_TAC THEN
REWRITE_TAC[MAP; Varpairs_DEF; functions_term; LIST_UNION] THEN
ASM_REWRITE_TAC[UNION_EMPTY]);;
let FUNCTIONS_FORM_FNV2 = prove
(`!n. LIST_UNION (MAP functions_term (MAP SND (Varpairs n))) = EMPTY`,
INDUCT_TAC THEN
REWRITE_TAC[MAP; Varpairs_DEF; functions_term; LIST_UNION] THEN
ASM_REWRITE_TAC[UNION_EMPTY]);;
let FUNCTIONS_TERM_FNV2 = prove
(`functions_term (Fn p (MAP SND (Varpairs n))) = {(p,n)}`,
REWRITE_TAC[functions_term; LENGTH_MAP; LENGTH_VARPAIRS] THEN
AP_TERM_TAC THEN SPEC_TAC(`n:num`,`n:num`) THEN
INDUCT_TAC THEN
REWRITE_TAC[MAP; Varpairs_DEF; functions_term; LIST_UNION] THEN
ASM_REWRITE_TAC[UNION_EMPTY]);;
let FUNCTIONS_EQCONJ = prove
(`!n. functions_form(ITLIST (&&) (MAP (\(s,t). Atom 0 [s; t])
(Varpairs n)) True) = EMPTY`,
INDUCT_TAC THEN
REWRITE_TAC[Varpairs_DEF; ITLIST; MAP; True_DEF;
Not_DEF; functions_form; UNION_EMPTY; PAIR_LEMMA; And_DEF; Or_DEF] THEN
RULE_ASSUM_TAC(REWRITE_RULE[True_DEF; Not_DEF]) THEN
ASM_REWRITE_TAC[LIST_UNION; functions_term; UNION_EMPTY]);;
let FUNCTIONS_EQAXIOM_FUNC = prove
(`!fa. functions_form (Eqaxiom_Func fa) = {fa}`,
REWRITE_TAC[FORALL_PAIR_THM; Eqaxiom_Func] THEN
REWRITE_TAC[FUNCTIONS_FORM_UCLOSE] THEN
REWRITE_TAC[functions_form; Equal_DEF; MAP; LIST_UNION] THEN
REWRITE_TAC[FUNCTIONS_TERM_FNV1; FUNCTIONS_TERM_FNV2] THEN
REWRITE_TAC[FUNCTIONS_EQCONJ] THEN
REWRITE_TAC[UNION_EMPTY; UNION_IDEMPOT]);;
let FUNCTIONS_EQAXIOM_PRED = prove
(`!pa. functions_form (Eqaxiom_Pred pa) = EMPTY`,
REWRITE_TAC[FORALL_PAIR_THM; Eqaxiom_Pred] THEN
REWRITE_TAC[FUNCTIONS_FORM_UCLOSE] THEN
REWRITE_TAC[functions_form; Equal_DEF; MAP; LIST_UNION] THEN
REWRITE_TAC[FUNCTIONS_TERM_FNV1; FUNCTIONS_TERM_FNV2] THEN
REWRITE_TAC[FUNCTIONS_EQCONJ] THEN
REWRITE_TAC[functions_form; Iff_DEF; And_DEF; Not_DEF; Or_DEF] THEN
REWRITE_TAC[FUNCTIONS_FORM_FNV1; FUNCTIONS_FORM_FNV2] THEN
REWRITE_TAC[UNION_EMPTY; UNION_IDEMPOT]);;
let FUNCTIONS_EQAXIOM = prove
(`functions (Eqaxioms (language s)) = functions s`,
ONCE_REWRITE_TAC[EXTENSION] THEN
REWRITE_TAC[functions; Eqaxioms_DEF; IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[language] THEN GEN_TAC THEN
REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV o LAND_CONV o BINDER_CONV o
LAND_CONV o ONCE_DEPTH_CONV) [CONJ_SYM] THEN
REWRITE_TAC[OR_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_OR_EXISTS_THM] THEN
REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`] THEN
REWRITE_TAC[EXISTS_OR_THM; GSYM CONJ_ASSOC; UNWIND_THM2] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[UNWIND_THM2] THEN
REWRITE_TAC[functions_form; Equal_DEF; LIST_UNION; MAP] THEN
REWRITE_TAC[functions_term; IN_UNION; NOT_IN_EMPTY] THEN
REWRITE_TAC[FUNCTIONS_EQAXIOM_FUNC; FUNCTIONS_EQAXIOM_PRED] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[functions; IN_UNIONS; IN_ELIM_THM] THEN MESON_TAC[]);;
let INTERPRETATION_EQAXIOMS = prove
(`interpretation (language s) M =
interpretation (language (s UNION (Eqaxioms (language s)))) M`,
REWRITE_TAC[interpretation; language] THEN
REWRITE_TAC[FUNCTIONS_UNION] THEN
REWRITE_TAC[GSYM language; FUNCTIONS_EQAXIOM] THEN
REWRITE_TAC[UNION_IDEMPOT]);;
let TERMS_SUBSET = prove
(`!f1 f2. f1 SUBSET f2 ==> terms f1 SUBSET terms f2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET; IN] THEN
MATCH_MP_TAC terms_INDUCT THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[terms_CASES] THEN MESON_TAC[];
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[terms_CASES] THEN
DISJ2_TAC THEN RULE_ASSUM_TAC(CONV_RULE(ONCE_DEPTH_CONV ETA_CONV)) THEN
ASM_MESON_TAC[SUBSET]]);;
let EQAXIOMS_UNION = prove
(`Eqaxioms (language (s UNION t)) =
Eqaxioms (language s) UNION Eqaxioms (language t)`,
REWRITE_TAC[Eqaxioms_DEF; language; FUNCTIONS_UNION; PREDICATES_UNION] THEN
REWRITE_TAC[EXTENSION; IN_INSERT; IN_UNION; NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[]);;
let EQAXIOMS_SUBSET = prove
(`s SUBSET t ==> Eqaxioms (language s) SUBSET Eqaxioms (language t)`,
REWRITE_TAC[SUBSET_UNION_ABSORPTION; GSYM EQAXIOMS_UNION] THEN SIMP_TAC[]);;
let IN_EQAXIOMS = prove
(`p IN Eqaxioms (language s)
==> (s = EMPTY) \/ ?q. q IN s /\ p IN Eqaxioms (language {q})`,
REWRITE_TAC[Eqaxioms_DEF; language] THEN
REWRITE_TAC[IN_INSERT; IN_UNION; NOT_IN_EMPTY] THEN
REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN
REWRITE_TAC[NOT_IN_EMPTY] THEN
REWRITE_TAC[functions; predicates; IN_UNIONS; IN_ELIM_THM] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN MESON_TAC[]);;
let COMPACT_LS_NORM_LEMMA1 = prove
(`(!t. FINITE t /\ t SUBSET s
==> ?M. interpretation (language s) M /\
~(Dom M :A->bool = EMPTY) /\
normal (functions s) M /\
M satisfies t)
==> (!t. FINITE t /\ t SUBSET s
==> ?M. interpretation (language t) M /\
~(Dom M :A->bool = EMPTY) /\
normal (functions t) M /\
M satisfies t)`,
let lemma = prove
(`(!t. P t ==> !M. Q t M ==> R t M)
==> (!t. P t ==> ?M. Q t M) ==> (!t. P t ==> ?M. R t M)`,
MESON_TAC[]) in
MATCH_MP_TAC lemma THEN GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN
REWRITE_TAC[interpretation; normal; language] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[SUBSET; FUNCTIONS_SUBSET; TERMS_SUBSET]);;
let COMPACT_LS_NORM_LEMMA2a = prove
(`FINITE t /\ t SUBSET Eqaxioms (language s)
==> ?s0. FINITE s0 /\ s0 SUBSET s /\ t SUBSET Eqaxioms (language s0)`,
let lemma = prove
(`(x INSERT s SUBSET t <=> x IN t /\ s SUBSET t) /\
(u UNION s SUBSET t <=> u SUBSET t /\ s SUBSET t)`,
SET_TAC[]) in
REWRITE_TAC[IMP_CONJ] THEN
SPEC_TAC(`t:form->bool`,`t:form->bool`) THEN
MATCH_MP_TAC FINITE_INDUCT THEN
REWRITE_TAC[EMPTY_SUBSET] THEN CONJ_TAC THENL
[EXISTS_TAC `EMPTY:form->bool` THEN
REWRITE_TAC[EMPTY_SUBSET; FINITE_RULES]; ALL_TAC] THEN
REWRITE_TAC[lemma] THEN
X_GEN_TAC `p:form` THEN X_GEN_TAC `t:form->bool` THEN DISCH_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (ANTE_RES_THEN MP_TAC)) THEN
POP_ASSUM(K ALL_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s0:form->bool` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `s:form->bool = EMPTY` THEN ASM_REWRITE_TAC[] THENL
[DISCH_TAC THEN EXISTS_TAC `s0:form->bool` THEN
ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[EQAXIOMS_SUBSET; SUBSET; EMPTY_SUBSET]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP IN_EQAXIOMS) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `q:form` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `{(q:form)} UNION s0` THEN
ASM_REWRITE_TAC[FINITE_INSERT; FINITE_UNION; lemma] THEN
REWRITE_TAC[FINITE_RULES; EMPTY_SUBSET] THEN
ASM_REWRITE_TAC[EQAXIOMS_UNION; IN_UNION] THEN
UNDISCH_TAC `t SUBSET Eqaxioms (language s0)` THEN SET_TAC[]);;
let COMPACT_LS_NORM_LEMMA2 = prove
(`!t. FINITE t /\ t SUBSET s UNION (Eqaxioms (language s))
==> ?u. t SUBSET (u UNION (Eqaxioms (language u))) /\
FINITE u /\ u SUBSET s`,
let lemma = prove
(`(s:A->bool) SUBSET (t UNION u)
==> ?s0 s1. (s = s0 UNION s1) /\ s0 SUBSET t /\ s1 SUBSET u`,
DISCH_THEN(fun th ->
EXISTS_TAC `(s:A->bool) INTER t` THEN
EXISTS_TAC `(s:A->bool) INTER u` THEN
MP_TAC th) THEN
SET_TAC[]) in
let slemma1 = prove
(`(s UNION t) SUBSET u <=> s SUBSET u /\ t SUBSET u`,
SET_TAC[]) in
GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN
DISCH_THEN(X_CHOOSE_THEN `b:form->bool` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `e:form->bool` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[FINITE_UNION] THEN
DISCH_THEN(fun th1 -> DISCH_THEN(fun th2 ->
ASSUME_TAC(CONJUNCT1 th1) THEN
ASSUME_TAC(CONJUNCT1 th2) THEN
MP_TAC(CONJ (CONJUNCT2 th1) (CONJUNCT2 th2)))) THEN
DISCH_THEN(MP_TAC o MATCH_MP COMPACT_LS_NORM_LEMMA2a) THEN
DISCH_THEN(X_CHOOSE_THEN `s0:form->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(b:form->bool) UNION s0` THEN
ASM_REWRITE_TAC[slemma1; FINITE_UNION] THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[EQAXIOMS_UNION] THEN
UNDISCH_TAC `e SUBSET Eqaxioms (language s0)` THEN SET_TAC[]);;
let COMPACT_LS_NORM = prove
(`(!t. FINITE t /\ t SUBSET s ==> ?M. interpretation(language s) M /\
~(Dom(M):A->bool = EMPTY) /\
normal (functions s) M /\
M satisfies t)
==> ?C. interpretation (language s) C /\
~(Dom(C):term->bool = EMPTY) /\
normal (functions s) C /\
C satisfies s`,
DISCH_THEN(MP_TAC o MATCH_MP COMPACT_LS_NORM_LEMMA1) THEN
REWRITE_TAC[NORMAL_THM] THEN DISCH_TAC THEN
MATCH_MP_TAC INTERPRETATION_RESTRICT_LANGUAGE THEN
EXISTS_TAC `s UNION (Eqaxioms (language s))` THEN
REWRITE_TAC[SUBSET_UNION] THEN
MATCH_MP_TAC COMPACT_LS THEN
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP COMPACT_LS_NORM_LEMMA2) THEN
DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC th) THEN
DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [INTERPRETATION_EQAXIOMS] THEN
DISCH_THEN(MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ_ALT]
INTERPRETATION_MODIFY_LANGUAGE)) THEN
DISCH_THEN(MP_TAC o SPEC `s:form->bool`) THEN
REWRITE_TAC[SUBSET_REFL] THEN
REWRITE_TAC[GSYM INTERPRETATION_EQAXIOMS] THEN
REWRITE_TAC[satisfies] THEN ASM_MESON_TAC[SUBSET]);;
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