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(* ========================================================================= *)
(* Conversion to prenex form. *)
(* ========================================================================= *)
(* ------------------------------------------------------------------------- *)
(* Quantifier-free formulas. *)
(* ------------------------------------------------------------------------- *)
let qfree = new_recursive_definition form_RECURSION
`(qfree False <=> T) /\
(qfree (Atom n l) <=> T) /\
(qfree (p --> q) <=> qfree p /\ qfree q) /\
(qfree (!!x p) <=> F)`;;
let QFREE = prove
(`(qfree False <=> T) /\
(qfree True <=> T) /\
(!a l. qfree (Atom a l) <=> T) /\
(!p. qfree (Not p) <=> qfree p) /\
(!p q. qfree (p || q) <=> qfree p /\ qfree q) /\
(!p q. qfree (p && q) <=> qfree p /\ qfree q) /\
(!p q. qfree (p --> q) <=> qfree p /\ qfree q) /\
(!p q. qfree (p <-> q) <=> qfree p /\ qfree q) /\
(!x p. qfree (!! x p) <=> F) /\
(!x p. qfree (?? x p) <=> F)`,
REWRITE_TAC[Not_DEF; True_DEF; Or_DEF; And_DEF;
Iff_DEF; Exists_DEF; qfree; CONJ_ACI]);;
let QFREE_FORMSUBST = prove
(`!p v. qfree (formsubst v p) <=> qfree p`,
MATCH_MP_TAC form_INDUCTION THEN
ASM_REWRITE_TAC[formsubst; qfree; LET_DEF; LET_END_DEF] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);;
let QFREE_BV_EMPTY = prove
(`!p. qfree(p) <=> (BV(p) = EMPTY)`,
MATCH_MP_TAC form_INDUCTION THEN REWRITE_TAC[qfree; BV] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[EMPTY_UNION] THEN
RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_INSERT; NOT_IN_EMPTY]) THEN
ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Prenex and prenex universal formulas. *)
(* ------------------------------------------------------------------------- *)
let prenex_RULES,prenex_INDUCT,prenex_CASES = new_inductive_definition
`(!p. qfree p ==> prenex p) /\
(!x p. prenex p ==> prenex (!!x p)) /\
(!x p. prenex p ==> prenex (??x p))`;;
let universal_RULES,universal_INDUCT,universal_CASES =
new_inductive_definition
`(!p. qfree p ==> universal p) /\
(!x p. universal p ==> universal (!!x p))`;;
let prenex_INDUCT_NOT = prove
(`!P. (!p. qfree p ==> P p) /\
(!x p. P p ==> P (!! x p)) /\
(!x p. P p ==> P (Not p))
==> !a. prenex a ==> P a`,
GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC prenex_INDUCT THEN
REWRITE_TAC[Exists_DEF] THEN ASM_MESON_TAC[]);;
let PRENEX = prove
(`(prenex False <=> T) /\
(prenex True <=> T) /\
(!a l. prenex (Atom a l) <=> T) /\
(!p. prenex (Not p) <=> qfree p \/ (?q x. (Not p = ??x q) /\ prenex q)) /\
(!p q. prenex (p || q) <=> qfree p /\ qfree q) /\
(!p q. prenex (p && q) <=> qfree p /\ qfree q) /\
(!p q. prenex (p --> q) <=> qfree p /\ qfree q \/
(?r x. (p --> q = ??x r) /\ prenex r)) /\
(!p q. prenex (p <-> q) <=> qfree p /\ qfree q) /\
(!x p. prenex (!! x p) <=> prenex p) /\
(!x p. prenex (?? x p) <=> prenex p)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [prenex_CASES] THEN
REWRITE_TAC[True_DEF; Not_DEF; Or_DEF; And_DEF; Iff_DEF; Exists_DEF] THEN
REWRITE_TAC[form_DISTINCT; form_INJ; QFREE; CONJ_ACI] THEN
MESON_TAC[]);;
let FORMSUBST_STRUCTURE_LEMMA = prove
(`!p i. ((formsubst i p = False) <=> (p = False)) /\
((?a l. formsubst i p = Atom a l) <=> (?a l. p = Atom a l)) /\
((?q r. formsubst i p = q --> r) <=> (?q r. p = q --> r)) /\
((?x q. formsubst i p = !!x q) <=> (?x q. p = !!x q))`,
MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[formsubst; LET_DEF; LET_END_DEF] THEN
REWRITE_TAC[form_DISTINCT; form_INJ] THEN REPEAT STRIP_TAC THEN
MESON_TAC[]);;
let FORMSUBST_STRUCTURE_NOT = prove
(`!p i. (?q. formsubst i p = Not q) <=> (?q. p = Not q)`,
REWRITE_TAC[Not_DEF; FORMSUBST_STRUCTURE_LEMMA] THEN
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `q:form`) THENL
[MP_TAC(el 2 (CONJUNCTS (SPEC_ALL FORMSUBST_STRUCTURE_LEMMA))) THEN
DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPECL [`q:form`; `False`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[formsubst; FORMSUBST_STRUCTURE_LEMMA; form_INJ] THEN
MESON_TAC[];
ASM_REWRITE_TAC[formsubst] THEN MESON_TAC[]]);;
let FORMSUBST_STRUCTURE_EXISTS = prove
(`!p i. (?x q. formsubst i p = ??x q) <=> (?x q. p = ??x q)`,
REPEAT GEN_TAC THEN REWRITE_TAC[Exists_DEF] THEN EQ_TAC THEN STRIP_TAC THENL
[ALL_TAC; ASM_REWRITE_TAC[Not_DEF; formsubst; LET_DEF; LET_END_DEF] THEN
MESON_TAC[]] THEN
MP_TAC(fst(EQ_IMP_RULE(SPEC_ALL FORMSUBST_STRUCTURE_NOT))) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPEC `!! x (Not q)`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `r:form` SUBST_ALL_TAC) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[formsubst; Not_DEF; form_INJ; LET_DEF; LET_END_DEF] THEN
REWRITE_TAC[GSYM Not_DEF] THEN DISCH_TAC THEN
MP_TAC(fst(EQ_IMP_RULE(last(CONJUNCTS(SPEC_ALL(SPEC `r:form`
FORMSUBST_STRUCTURE_LEMMA)))))) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPECL [`x:num`; `Not q`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `y:num` (X_CHOOSE_THEN `s:form` SUBST_ALL_TAC)) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[formsubst; Not_DEF; form_INJ; LET_DEF; LET_END_DEF] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[GSYM Not_DEF] THEN MESON_TAC[FORMSUBST_STRUCTURE_NOT]);;
let PRENEX_FORMSUBST_LEMMA = prove
(`!p. prenex p ==> !i q. (p = formsubst i q) ==> prenex q`,
MATCH_MP_TAC prenex_INDUCT THEN REPEAT CONJ_TAC THENL
[MESON_TAC[QFREE_FORMSUBST; prenex_RULES];
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`q:form`; `i:num->term`] FORMSUBST_STRUCTURE_LEMMA) THEN
DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE o last o CONJUNCTS) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPECL [`x:num`; `p:form`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN
(X_CHOOSE_THEN `y:num` (X_CHOOSE_THEN `r:form` SUBST_ALL_TAC)) THEN
UNDISCH_TAC `!! x p = formsubst i (!! y r)` THEN
REWRITE_TAC[formsubst; form_INJ; PRENEX; LET_DEF; LET_END_DEF] THEN
ASM_MESON_TAC[];
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`q:form`; `i:num->term`] FORMSUBST_STRUCTURE_EXISTS) THEN
DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE o last o CONJUNCTS) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPECL [`x:num`; `p:form`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN
(X_CHOOSE_THEN `y:num` (X_CHOOSE_THEN `r:form` SUBST_ALL_TAC)) THEN
UNDISCH_TAC `?? x p = formsubst i (?? y r)` THEN
REWRITE_TAC[PRENEX] THEN
REWRITE_TAC[Exists_DEF; Not_DEF; formsubst; form_INJ; LET_DEF; LET_END_DEF] THEN
ASM_MESON_TAC[]]);;
let PRENEX_FORMSUBST = prove
(`!p i. prenex(formsubst i p) = prenex p`,
REPEAT GEN_TAC THEN EQ_TAC THENL
[MESON_TAC[PRENEX_FORMSUBST_LEMMA];
SPEC_TAC(`i:num->term`,`i:num->term`) THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
SPEC_TAC(`p:form`,`p:form`) THEN
MATCH_MP_TAC prenex_INDUCT THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[formsubst; LET_DEF; LET_END_DEF] THENL
[ASM_MESON_TAC[QFREE_FORMSUBST; prenex_RULES];
ASM_REWRITE_TAC[PRENEX];
REWRITE_TAC[formsubst; Not_DEF; Exists_DEF; LET_DEF; LET_END_DEF] THEN
REWRITE_TAC[GSYM Not_DEF; GSYM Exists_DEF] THEN
ASM_REWRITE_TAC[PRENEX]]]);;
(* ------------------------------------------------------------------------- *)
(* It's more convenient to argue by non-structural induction here. *)
(* ------------------------------------------------------------------------- *)
let size = new_recursive_definition form_RECURSION
`(size False = 1) /\
(size (Atom p l) = 1) /\
(size (q --> r) = size q + size r) /\
(size (!!x q) = 1 + size q)`;;
let SIZE = prove
(`(size False = 1) /\
(size True = 2) /\
(!a l. size (Atom a l) = 1) /\
(!p. size (Not p) = 1 + size p) /\
(!p q. size (p || q) = size p + 2 * size q) /\
(!p q. size (p && q) = size p + 2 * size q + 4) /\
(!p q. size (p --> q) = size p + size q) /\
(!p q. size (p <-> q) = 3 * size p + 3 * size q + 4) /\
(!x p.size (!! x p) = 1 + size p) /\
(!x p. size (?? x p) = 3 + size p)`,
REWRITE_TAC[True_DEF; Not_DEF; Or_DEF; And_DEF; Iff_DEF; Exists_DEF; size] THEN
REPEAT STRIP_TAC THEN ARITH_TAC);;
let SIZE_FORMSUBST = prove
(`!p i. size (formsubst i p) = size p`,
MATCH_MP_TAC form_INDUCTION THEN
ASM_REWRITE_TAC[formsubst; size; LET_END_DEF; LET_DEF] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Constructive prenexing procedure. *)
(* ------------------------------------------------------------------------- *)
let PPAT_DEF = new_definition
`PPAT A B C r =
if ?x p. r = !!x p then
A (@x. ?p. r = !!x p)
(@p. r = !!(@x. ?p. r = !!x p) p)
else if ?x p. r = ??x p then
B (@x. ?p. r = ??x p)
(@p. r = ??(@x. ?p. r = ??x p) p)
else C r`;;
let PRENEX_DISTINCT = prove
(`!x y p q. ((!!x p = !!y q) <=> (x = y) /\ (p = q)) /\
((??x p = ??y q) <=> (x = y) /\ (p = q)) /\
~(!!x p = ??y q)`,
REWRITE_TAC[Exists_DEF; Not_DEF; form_DISTINCT; form_INJ]);;
let PPAT = prove
(`!A B C. (!x p. PPAT A B C (!!x p) = A x p :A) /\
(!x p. PPAT A B C (??x p) = B x p) /\
!r. ~(?x p. r = !!x p) /\ ~(?x p. r = ??x p)
==> (PPAT A B C r = C r)`,
REPEAT GEN_TAC THEN REWRITE_TAC[PPAT_DEF] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[PRENEX_DISTINCT] THEN
(COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[]]) THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL; GSYM EXISTS_REFL;
SELECT_REFL; CONV_RULE(LAND_CONV(ONCE_DEPTH_CONV SYM_CONV))
(SPEC_ALL SELECT_REFL)]);;
let SIZE_REC = prove
(`!H. (!f g x. (!z. size z < size x ==> (f z = g z)) ==> (H f x = H g x))
==> (?f. !x. f x = H f x)`,
MATCH_MP_TAC WF_REC THEN
REWRITE_TAC[REWRITE_RULE[MEASURE] WF_MEASURE]);;
let PRENEX_RIGHT_EXISTENCE = prove
(`?Prenex_right.
(!p x q. Prenex_right p (!!x q) =
let y = VARIANT(FV(p) UNION FV(!!x q)) in
!!y (Prenex_right p (formsubst (valmod (x,V y) V) q))) /\
(!p x q. Prenex_right p (??x q) =
let y = VARIANT(FV(p) UNION FV(??x q)) in
??y (Prenex_right p (formsubst (valmod (x,V y) V) q))) /\
(!p q. qfree q ==> (Prenex_right p q = p --> q))`,
SUBGOAL_THEN
`!p. ?Prenex_right. !r. Prenex_right r =
PPAT (\x q. let y = VARIANT(FV(p) UNION FV(!!x q)) in
!!y (Prenex_right (formsubst (valmod (x,V y) V) q)))
(\x q. let y = VARIANT(FV(p) UNION FV(??x q)) in
??y (Prenex_right (formsubst (valmod (x,V y) V) q)))
(\q. p --> q) r`
MP_TAC THENL
[GEN_TAC THEN MATCH_MP_TAC SIZE_REC THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[PPAT_DEF] THEN
ASM_CASES_TAC `?x p. r = !!x p` THEN ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(CHOOSE_THEN (CHOOSE_THEN SUBST_ALL_TAC)) THEN
REWRITE_TAC[form_DISTINCT; form_INJ] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL; GSYM EXISTS_REFL;
SELECT_REFL; CONV_RULE(LAND_CONV(ONCE_DEPTH_CONV SYM_CONV))
(SPEC_ALL SELECT_REFL)] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
AP_TERM_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[SIZE_FORMSUBST; SIZE] THEN ARITH_TAC; ALL_TAC] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(CHOOSE_THEN (CHOOSE_THEN SUBST_ALL_TAC)) THEN
REWRITE_TAC[PRENEX_DISTINCT] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL; GSYM EXISTS_REFL;
SELECT_REFL; CONV_RULE(LAND_CONV(ONCE_DEPTH_CONV SYM_CONV))
(SPEC_ALL SELECT_REFL)] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
AP_TERM_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[SIZE_FORMSUBST; SIZE] THEN ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [SKOLEM_THM]) THEN
DISCH_THEN(X_CHOOSE_TAC `Prenex_right:form->form->form`) THEN
EXISTS_TAC `Prenex_right:form->form->form` THEN
REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN TRY DISCH_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [th]) THEN
REWRITE_TAC[PPAT] THEN
UNDISCH_TAC `qfree q` THEN REWRITE_TAC[PPAT_DEF] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
COND_CASES_TAC THENL [POP_ASSUM(STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[QFREE]; ALL_TAC] THEN
COND_CASES_TAC THENL [POP_ASSUM(STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[QFREE]; ALL_TAC] THEN
REWRITE_TAC[]);;
let PRENEX_RIGHT = new_specification ["Prenex_right"] PRENEX_RIGHT_EXISTENCE;;
let PRENEX_LEFT_EXISTENCE = prove
(`?Prenex_left.
(!p x q. Prenex_left (!!x q) p =
let y = VARIANT(FV(!!x q) UNION FV(p)) in
??y (Prenex_left (formsubst (valmod (x,V y) V) q) p)) /\
(!p x q. Prenex_left (??x q) p =
let y = VARIANT(FV(??x q) UNION FV(p)) in
!!y (Prenex_left (formsubst (valmod (x,V y) V) q) p)) /\
(!p q. qfree q ==> (Prenex_left q p = Prenex_right q p))`,
SUBGOAL_THEN
`!p. ?Prenex_left. !r. Prenex_left r =
PPAT (\x q. let y = VARIANT(FV(!!x q) UNION FV(p)) in
??y (Prenex_left (formsubst (valmod (x,V y) V) q)))
(\x q. let y = VARIANT(FV(??x q) UNION FV(p)) in
!!y (Prenex_left (formsubst (valmod (x,V y) V) q)))
(\q. Prenex_right q p) r`
MP_TAC THENL
[GEN_TAC THEN MATCH_MP_TAC SIZE_REC THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[PPAT_DEF] THEN
ASM_CASES_TAC `?x p. r = !!x p` THEN ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(CHOOSE_THEN (CHOOSE_THEN SUBST_ALL_TAC)) THEN
REWRITE_TAC[form_DISTINCT; form_INJ] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL; GSYM EXISTS_REFL;
SELECT_REFL; CONV_RULE(LAND_CONV(ONCE_DEPTH_CONV SYM_CONV))
(SPEC_ALL SELECT_REFL)] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
AP_TERM_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[SIZE_FORMSUBST; SIZE] THEN ARITH_TAC; ALL_TAC] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(CHOOSE_THEN (CHOOSE_THEN SUBST_ALL_TAC)) THEN
REWRITE_TAC[PRENEX_DISTINCT] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL; GSYM EXISTS_REFL;
SELECT_REFL; CONV_RULE(LAND_CONV(ONCE_DEPTH_CONV SYM_CONV))
(SPEC_ALL SELECT_REFL)] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
AP_TERM_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[SIZE_FORMSUBST; SIZE] THEN ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [SKOLEM_THM]) THEN
DISCH_THEN(X_CHOOSE_TAC `Prenex_left:form->form->form`) THEN
EXISTS_TAC `\q p. (Prenex_left:form->form->form) p q` THEN
REWRITE_TAC[] THEN
REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN TRY DISCH_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [th]) THEN
REWRITE_TAC[PPAT] THEN
UNDISCH_TAC `qfree q` THEN REWRITE_TAC[PPAT_DEF] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
COND_CASES_TAC THENL [POP_ASSUM(STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[QFREE]; ALL_TAC] THEN
COND_CASES_TAC THENL [POP_ASSUM(STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[QFREE]; ALL_TAC] THEN
REWRITE_TAC[]);;
let PRENEX_LEFT = new_specification ["Prenex_left"] PRENEX_LEFT_EXISTENCE;;
let Prenex_DEF = new_recursive_definition form_RECURSION
`(Prenex False = False) /\
(Prenex (Atom a l) = Atom a l) /\
(Prenex (p --> q) = Prenex_left (Prenex p) (Prenex q)) /\
(Prenex (!!x p) = !!x (Prenex p))`;;
(* ------------------------------------------------------------------------- *)
(* Proof that it does indeed prenex. *)
(* ------------------------------------------------------------------------- *)
let PRENEX_RIGHT_FORALL = prove
(`~(Dom M :A->bool = EMPTY)
==> (holds M v (p --> !!x q) <=>
holds M v (!! (VARIANT (FV(p) UNION FV(!!x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(!!x q)))) V) q)))`,
DISCH_TAC THEN
ABBREV_TAC `y = VARIANT (FV(p) UNION FV(!!x q))` THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `holds M (v:num->A)
(p --> !!y (formsubst (valmod (x,V y) V) q))` THEN
SUBGOAL_THEN `~(y IN FV(p)) /\ ~(y IN FV(!!x q))` STRIP_ASSUME_TAC THENL
[REWRITE_TAC[GSYM DE_MORGAN_THM; GSYM IN_UNION] THEN
EXPAND_TAC "y" THEN REWRITE_TAC[VARIANT_THM; GSYM FV]; ALL_TAC] THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[holds] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a ==> b) <=> (a ==> c))`) THEN
DISCH_TAC THEN REWRITE_TAC[HOLDS; HOLDS_FORMSUBST] THEN
AP_TERM_TAC THEN ABS_TAC THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a ==> b) <=> (a ==> c))`) THEN
DISCH_TAC THEN MATCH_MP_TAC HOLDS_VALUATION THEN
X_GEN_TAC `z:num` THEN DISCH_TAC THEN
REWRITE_TAC[valmod; o_THM] THEN
COND_CASES_TAC THEN REWRITE_TAC[termval] THEN
COND_CASES_TAC THEN REWRITE_TAC[] THEN
UNDISCH_TAC `~(y IN FV(!!x q))` THEN
ASM_REWRITE_TAC[FV; IN_DELETE] THEN ASM_MESON_TAC[];
REWRITE_TAC[HOLDS] THEN
SUBGOAL_THEN `!v a:A. holds M (valmod (y,a) v) p = holds M v p`
(fun th -> REWRITE_TAC[th]) THENL
[ALL_TAC; ASM_CASES_TAC `holds M (v:num->A) p` THEN ASM_REWRITE_TAC[]] THEN
REPEAT GEN_TAC THEN MATCH_MP_TAC HOLDS_VALUATION THEN
GEN_TAC THEN REWRITE_TAC[valmod] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[]]);;
let PRENEX_RIGHT_EXISTS = prove
(`~(Dom M :A->bool = EMPTY)
==> (holds M v (p --> ??x q) <=>
holds M v (?? (VARIANT (FV(p) UNION FV(??x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(??x q)))) V) q)))`,
DISCH_TAC THEN
ABBREV_TAC `y = VARIANT (FV(p) UNION FV(??x q))` THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `holds M (v:num->A)
(p --> ??y (formsubst (valmod (x,V y) V) q))` THEN
SUBGOAL_THEN `~(y IN FV(p)) /\ ~(y IN FV(??x q))` STRIP_ASSUME_TAC THENL
[REWRITE_TAC[GSYM DE_MORGAN_THM; GSYM IN_UNION] THEN
EXPAND_TAC "y" THEN REWRITE_TAC[VARIANT_THM; GSYM FV]; ALL_TAC] THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[holds] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a ==> b) <=> (a ==> c))`) THEN
DISCH_TAC THEN REWRITE_TAC[HOLDS; HOLDS_FORMSUBST] THEN
AP_TERM_TAC THEN ABS_TAC THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a /\ b) <=> (a /\ c))`) THEN
DISCH_TAC THEN MATCH_MP_TAC HOLDS_VALUATION THEN
X_GEN_TAC `z:num` THEN DISCH_TAC THEN
REWRITE_TAC[valmod; o_THM] THEN
COND_CASES_TAC THEN REWRITE_TAC[termval] THEN
COND_CASES_TAC THEN REWRITE_TAC[] THEN
UNDISCH_TAC `~(y IN FV(??x q))` THEN
ASM_REWRITE_TAC[FV; Not_DEF; Exists_DEF; UNION_EMPTY; IN_DELETE] THEN
ASM_MESON_TAC[];
REWRITE_TAC[HOLDS] THEN
SUBGOAL_THEN `!v a:A. holds M (valmod (y,a) v) p = holds M v p`
(fun th -> REWRITE_TAC[th]) THENL
[REPEAT GEN_TAC THEN MATCH_MP_TAC HOLDS_VALUATION THEN
GEN_TAC THEN REWRITE_TAC[valmod] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[];
ASM_CASES_TAC `holds M (v:num->A) p` THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]]]);;
let PRENEX_DUALITY_LEMMAS = prove
(`(holds M v (??x p --> q) <=> holds M v (Not q --> !!x (Not p))) /\
(holds M v (!!x p --> q) <=> holds M v (Not q --> ??x (Not p)))`,
REWRITE_TAC[HOLDS] THEN MESON_TAC[]);;
let PRENEX_LEFT_FORALL = prove
(`~(Dom M :A->bool = EMPTY)
==> (holds M v (!!x p --> q) <=>
holds M v (?? (VARIANT (FV(!!x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(!!x p) UNION FV(q)))) V) p --> q)))`,
DISCH_TAC THEN REWRITE_TAC[PRENEX_DUALITY_LEMMAS] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP PRENEX_RIGHT_EXISTS th]) THEN
REWRITE_TAC[Exists_DEF; Not_DEF; FV; UNION_EMPTY; UNION_COMM] THEN
REWRITE_TAC[holds; formsubst; TAUT `(~a ==> ~b) <=> (b ==> a)`]);;
let PRENEX_LEFT_EXISTS = prove
(`~(Dom M :A->bool = EMPTY)
==> (holds M v (??x p --> q) <=>
holds M v (!! (VARIANT (FV(??x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(??x p) UNION FV(q)))) V) p --> q)))`,
DISCH_TAC THEN REWRITE_TAC[PRENEX_DUALITY_LEMMAS] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP PRENEX_RIGHT_FORALL th]) THEN
REWRITE_TAC[Exists_DEF; Not_DEF; FV; UNION_EMPTY; UNION_COMM] THEN
REWRITE_TAC[holds; formsubst; TAUT `(~a ==> ~b) <=> (b ==> a)`]);;
(* ------------------------------------------------------------------------- *)
(* Extras about free variables. *)
(* ------------------------------------------------------------------------- *)
let PRENEX_RIGHT_FORALL_FV = prove
(`FV(p --> !!x q) = FV(!! (VARIANT (FV(p) UNION FV(!!x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(!!x q)))) V) q))`,
let lemma = prove
(`(s UNION t) DELETE x = (s DELETE x) UNION (t DELETE x)`,SET_TAC[]) in
REWRITE_TAC[FV; FORMSUBST_RENAME; lemma] THEN
MP_TAC(SPEC `p --> !!x q` VARIANT_THM) THEN
REWRITE_TAC[EXTENSION; FV; IN_UNION; IN_DELETE] THEN MESON_TAC[]);;
let PRENEX_RIGHT_EXISTS_FV = prove
(`FV(p --> ??x q) =
FV(?? (VARIANT (FV(p) UNION FV(??x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(??x q)))) V) q))`,
let lemma = prove
(`(s UNION t) DELETE x = (s DELETE x) UNION (t DELETE x)`,SET_TAC[]) in
REWRITE_TAC[FV; FV_EXISTS; FORMSUBST_RENAME; lemma] THEN
MP_TAC(SPEC `p --> ??x q` VARIANT_THM) THEN
REWRITE_TAC[EXTENSION; FV; FV_EXISTS; IN_UNION; IN_DELETE] THEN
MESON_TAC[]);;
let PRENEX_LEFT_FORALL_FV = prove
(`FV(!!x p --> q) =
FV(?? (VARIANT (FV(!!x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(!!x p) UNION FV(q)))) V) p --> q))`,
let lemma = prove
(`(s UNION t) DELETE x = (s DELETE x) UNION (t DELETE x)`,SET_TAC[]) in
REWRITE_TAC[FV; FV_EXISTS; FORMSUBST_RENAME; lemma] THEN
MP_TAC(SPEC `!!x p --> q` VARIANT_THM) THEN
REWRITE_TAC[EXTENSION; FV; FV_EXISTS; IN_UNION; IN_DELETE] THEN
MESON_TAC[]);;
let PRENEX_LEFT_EXISTS_FV = prove
(`FV(??x p --> q) =
FV(!! (VARIANT (FV(??x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(??x p) UNION FV(q)))) V) p --> q))`,
let lemma = prove
(`(s UNION t) DELETE x = (s DELETE x) UNION (t DELETE x)`,SET_TAC[]) in
REWRITE_TAC[FV; FV_EXISTS; FORMSUBST_RENAME; lemma] THEN
MP_TAC(SPEC `??x p --> q` VARIANT_THM) THEN
REWRITE_TAC[EXTENSION; FV; FV_EXISTS; IN_UNION; IN_DELETE] THEN
MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* And extras about the language. *)
(* ------------------------------------------------------------------------- *)
let PRENEX_RIGHT_FORALL_LANGUAGE = prove
(`language {(p --> !!x q)} = language {(!! (VARIANT (FV(p) UNION FV(!!x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(!!x q)))) V) q))}`,
REWRITE_TAC[LANGUAGE_1; functions_form; predicates_form] THEN
REWRITE_TAC[REWRITE_RULE[PAIR_EQ; LANGUAGE_1] FORMSUBST_LANGUAGE_RENAME]);;
let PRENEX_RIGHT_EXISTS_LANGUAGE = prove
(`language {(p --> ??x q)} =
language {(?? (VARIANT (FV(p) UNION FV(??x q)))
(p --> formsubst (valmod
(x,V(VARIANT (FV(p) UNION FV(??x q)))) V) q))}`,
REWRITE_TAC[LANGUAGE_1; functions_form; predicates_form;
Exists_DEF; Not_DEF; UNION_EMPTY] THEN
REWRITE_TAC[REWRITE_RULE[PAIR_EQ; LANGUAGE_1] FORMSUBST_LANGUAGE_RENAME]);;
let PRENEX_LEFT_FORALL_LANGUAGE = prove
(`language {(!!x p --> q)} =
language {(?? (VARIANT (FV(!!x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(!!x p) UNION FV(q)))) V) p --> q))}`,
REWRITE_TAC[LANGUAGE_1; functions_form; predicates_form;
Exists_DEF; Not_DEF; UNION_EMPTY] THEN
REWRITE_TAC[REWRITE_RULE[PAIR_EQ; LANGUAGE_1] FORMSUBST_LANGUAGE_RENAME]);;
let PRENEX_LEFT_EXISTS_LANGUAGE = prove
(`language {(??x p --> q)} =
language {(!! (VARIANT (FV(??x p) UNION FV(q)))
(formsubst (valmod
(x,V(VARIANT (FV(??x p) UNION FV(q)))) V) p --> q))}`,
REWRITE_TAC[LANGUAGE_1; functions_form; predicates_form;
Exists_DEF; Not_DEF; UNION_EMPTY] THEN
REWRITE_TAC[REWRITE_RULE[PAIR_EQ; LANGUAGE_1] FORMSUBST_LANGUAGE_RENAME]);;
(* ------------------------------------------------------------------------- *)
(* Proofs that things work properly. *)
(* ------------------------------------------------------------------------- *)
let PRENEX_LEMMA_FORALL = prove
(`P /\
(FV r1 = FV r2) /\
(language {r1} = language {r2}) /\
(!M v. ~(Dom M :A->bool = EMPTY) ==> (holds M v p <=> holds M v q))
==> P /\
(FV(!!z r1) = FV(!!z r2)) /\
(language {(!!z r1)} = language {(!!z r2)}) /\
!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (!!x p) <=> holds M v (!!x q))`,
REWRITE_TAC[LANGUAGE_1; PAIR_EQ] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[FV; functions_form; predicates_form] THEN
ASM_SIMP_TAC[HOLDS]);;
let PRENEX_LEMMA_EXISTS = prove
(`P /\
(FV r1 = FV r2) /\
(language {r1} = language {r2}) /\
(!M v. ~(Dom M :A->bool = EMPTY) ==> (holds M v p <=> holds M v q))
==> P /\
(FV(??z r1) = FV(??z r2)) /\
(language {(??z r1)} = language {(??z r2)}) /\
!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (??x p) <=> holds M v (??x q))`,
REWRITE_TAC[LANGUAGE_1; PAIR_EQ] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[FV; FV_EXISTS] THEN
ASM_SIMP_TAC[HOLDS] THEN
ASM_REWRITE_TAC[Exists_DEF; Not_DEF; functions_form; predicates_form]);;
let PRENEX_RIGHT_THM = prove
(`!p q. qfree p /\ prenex q
==> prenex (Prenex_right p q) /\
(FV(Prenex_right p q) = FV(p --> q)) /\
(language {(Prenex_right p q)} = language {(p --> q)}) /\
(!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (Prenex_right p q) <=>
holds M v (p --> q)))`,
SUBGOAL_THEN
`!p. qfree p
==> !n q. prenex(q) /\ (size(q) = n)
==> prenex (Prenex_right p q) /\
(FV(Prenex_right p q) = FV(p --> q)) /\
(language {(Prenex_right p q)} = language {(p --> q)}) /\
(!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (Prenex_right p q) <=>
holds M v (p --> q)))`
(fun th -> MESON_TAC[th]) THEN
GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC num_WF THEN
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN STRIP_TAC THEN
FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [prenex_CASES]) THENL
[ASM_SIMP_TAC [PRENEX_RIGHT; PRENEX]; ALL_TAC; ALL_TAC] THEN
ASM_REWRITE_TAC[PRENEX_RIGHT] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF; PRENEX] THEN
SIMP_TAC [PRENEX_RIGHT_FORALL; PRENEX_RIGHT_EXISTS] THEN
REWRITE_TAC[PRENEX_RIGHT_FORALL_FV; PRENEX_RIGHT_FORALL_LANGUAGE] THEN
REWRITE_TAC[PRENEX_RIGHT_EXISTS_FV; PRENEX_RIGHT_EXISTS_LANGUAGE] THENL
[MATCH_MP_TAC PRENEX_LEMMA_FORALL; MATCH_MP_TAC PRENEX_LEMMA_EXISTS] THEN
RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM;
IMP_IMP]) THEN
FIRST_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `size p'` THEN
REWRITE_TAC[PRENEX_FORMSUBST; SIZE_FORMSUBST] THEN
UNDISCH_TAC `size q = n` THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
ASM_REWRITE_TAC[SIZE] THEN ARITH_TAC);;
let PRENEX_LEFT_THM = prove
(`!p q. prenex p /\ prenex q
==> prenex (Prenex_left p q) /\
(FV(Prenex_left p q) = FV(p --> q)) /\
(language {(Prenex_left p q)} = language {(p --> q)}) /\
(!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (Prenex_left p q) <=> holds M v (p --> q)))`,
SUBGOAL_THEN
`!q. prenex(q)
==> !n p. prenex(p) /\ (size(p) = n)
==> prenex (Prenex_left p q) /\
(FV(Prenex_left p q) = FV(p --> q)) /\
(language {(Prenex_left p q)} = language {(p --> q)}) /\
(!M v. ~(Dom M :A->bool = EMPTY)
==> (holds M v (Prenex_left p q) <=>
holds M v (p --> q)))`
(fun th -> MESON_TAC[th]) THEN
GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC num_WF THEN
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN STRIP_TAC THEN
FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [prenex_CASES]) THENL
[ASM_SIMP_TAC [PRENEX_LEFT; PRENEX] THEN
MATCH_MP_TAC PRENEX_RIGHT_THM THEN ASM_REWRITE_TAC[];
ALL_TAC; ALL_TAC] THEN
ASM_REWRITE_TAC[PRENEX_LEFT] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF; PRENEX] THEN
SIMP_TAC [PRENEX_LEFT_FORALL; PRENEX_LEFT_EXISTS] THEN
REWRITE_TAC[PRENEX_LEFT_FORALL_FV; PRENEX_LEFT_FORALL_LANGUAGE] THEN
REWRITE_TAC[PRENEX_LEFT_EXISTS_FV; PRENEX_LEFT_EXISTS_LANGUAGE] THENL
[MATCH_MP_TAC PRENEX_LEMMA_EXISTS; MATCH_MP_TAC PRENEX_LEMMA_FORALL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM;
IMP_IMP]) THEN
FIRST_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `size p'` THEN
REWRITE_TAC[PRENEX_FORMSUBST; SIZE_FORMSUBST] THEN
UNDISCH_TAC `size p = n` THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
ASM_REWRITE_TAC[SIZE] THEN ARITH_TAC);;
let PRENEX_THM = prove
(`!p. prenex(Prenex p) /\
(FV(Prenex p) = FV(p)) /\
(language {(Prenex p)} = language {p}) /\
!M v. ~(Dom M = EMPTY)
==> (holds M (v:num->A) (Prenex p) <=> holds M v p)`,
MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[PRENEX; Prenex_DEF] THEN REWRITE_TAC[HOLDS] THEN
MP_TAC PRENEX_LEFT_THM THEN
REWRITE_TAC[HOLDS; FV; LANGUAGE_1; PAIR_EQ] THEN
REWRITE_TAC[functions_form; predicates_form] THEN
MESON_TAC[]);;
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