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(* ========================================================================= *)
(* Completeness of first order resolution. *)
(* ========================================================================= *)
(* ------------------------------------------------------------------------- *)
(* We want to restrict ourselves to finite lists of literals. *)
(* ------------------------------------------------------------------------- *)
let atom = new_recursive_definition form_RECURSION
`(atom False <=> F) /\
(atom (Atom p l) <=> T) /\
(atom (q --> r) <=> F) /\
(atom (!!x q) <=> F)`;;
let literal = new_definition
`literal p <=> atom(p) \/ ?q. atom(q) /\ (p = Not q)`;;
let clause = new_definition
`clause cl <=> FINITE cl /\ !p. p IN cl ==> literal p`;;
let LITERAL = prove
(`literal(Atom r args) /\
literal(Not(Atom r args))`,
MESON_TAC[literal; atom]);;
let ATOM = prove
(`atom p <=> ?q l. p = Atom q l`,
MESON_TAC[atom; form_CASES]);;
(* ------------------------------------------------------------------------- *)
(* Negation on literals. *)
(* ------------------------------------------------------------------------- *)
let negative = new_definition
`negative p <=> ?q. p = Not q`;;
let positive = new_definition
`positive p <=> ~(negative p)`;;
let negate = new_definition
`~~p = if negative p then @q. (Not q = p) else Not p`;;
let PHOLDS_NEGATE = prove
(`pholds d (~~p) <=> ~(pholds d p)`,
REWRITE_TAC[negate] THEN
COND_CASES_TAC THEN REWRITE_TAC[PHOLDS] THEN
POP_ASSUM(CHOOSE_THEN SUBST1_TAC o REWRITE_RULE[negative]) THEN
REWRITE_TAC[Not_DEF; form_INJ; SELECT_REFL; PHOLDS]);;
let HOLDS_NEGATE = prove
(`holds M v (~~p) <=> ~(holds M v p)`,
REWRITE_TAC[negate] THEN
COND_CASES_TAC THEN REWRITE_TAC[HOLDS] THEN
POP_ASSUM(CHOOSE_THEN SUBST1_TAC o REWRITE_RULE[negative]) THEN
REWRITE_TAC[Not_DEF; form_INJ; SELECT_REFL; HOLDS]);;
let NEGATE_NEG = prove
(`~~(Not p) = p`,
REWRITE_TAC[negate; negative; Not_DEF; form_INJ; GSYM EXISTS_REFL]);;
let NEGATE_ATOM = prove
(`!p. atom p ==> (~~p = Not p)`,
MATCH_MP_TAC form_INDUCTION THEN REWRITE_TAC[atom] THEN
REWRITE_TAC[negate; negative; Not_DEF; form_DISTINCT]);;
let NEGATE_REFL = prove
(`!p. ~(~~p = p)`,
MESON_TAC[PHOLDS_NEGATE]);;
let PHOLDS_ATOM = prove
(`!p d. atom(p) ==> (pholds d p = d(p))`,
MATCH_MP_TAC form_INDUCTION THEN REWRITE_TAC[atom; pholds]);;
let NEGATE_NEGATE = prove
(`!p. literal p ==> (~~(~~p) = p)`,
REWRITE_TAC[literal] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[NEGATE_ATOM; NEGATE_NEG]);;
(* ------------------------------------------------------------------------- *)
(* Basic resolution step. *)
(* ------------------------------------------------------------------------- *)
let resolve = new_definition
`resolve (p:form) cl1 cl2 = (cl1 DELETE p) UNION (cl2 DELETE ~~p)`;;
(* ------------------------------------------------------------------------- *)
(* Inductive definition of propositional resolution. *)
(* ------------------------------------------------------------------------- *)
let presproof_RULES,presproof_INDUCT,presproof_CASES = new_inductive_definition
`(!cl. cl IN hyps ==> presproof hyps cl) /\
(!p cl1 cl2.
presproof hyps cl1 /\ presproof hyps cl2 /\ p IN cl1 /\ ~~p IN cl2
==> presproof hyps (resolve p cl1 cl2))`;;
(* ------------------------------------------------------------------------- *)
(* Interpretation of clause as formula (meaningful only if it's a clause). *)
(* ------------------------------------------------------------------------- *)
let interp = new_definition
`interp cl = ITLIST (||) (list_of_set cl) False`;;
(* ------------------------------------------------------------------------- *)
(* Compactness variant. *)
(* ------------------------------------------------------------------------- *)
let UNPSATISFIABLE_FINITE_SUBSET = prove
(`!s. ~(psatisfiable s)
==> ?t. FINITE t /\ t SUBSET s /\ ~(psatisfiable t)`,
REWRITE_TAC[psatisfiable] THEN MESON_TAC[COMPACT_PROP]);;
(* ------------------------------------------------------------------------- *)
(* Monotonicity w.r.t. hyps. *)
(* ------------------------------------------------------------------------- *)
let PRESPROOF_MONO = prove
(`!hyps1 hyps2 c.
presproof hyps1 c /\ hyps1 SUBSET hyps2 ==> presproof hyps2 c`,
GEN_TAC THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC presproof_INDUCT THEN MESON_TAC[presproof_RULES; SUBSET]);;
let PRESPROOF_TRANS = prove
(`!hyps hyps' c.
(!c'. c' IN hyps' ==> presproof hyps c') /\ presproof hyps' c
==> presproof hyps c`,
GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN GEN_TAC THEN
ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC presproof_INDUCT THEN MESON_TAC[presproof_RULES]);;
(* ------------------------------------------------------------------------- *)
(* Lemmas. *)
(* ------------------------------------------------------------------------- *)
let IMAGE_CLAUSE = prove
(`{interp cl | cl IN hyps} = IMAGE interp hyps`,
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; CONJ_ACI]);;
let PHOLDS_INTERP = prove
(`!cl d. FINITE cl ==> (pholds d (interp cl) <=> ?p. p IN cl /\ pholds d p)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[interp] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
[GSYM(CONJUNCT1(MATCH_MP LIST_OF_SET_PROPERTIES th))]) THEN
SPEC_TAC(`list_of_set(cl:form->bool)`,`l:form list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[ITLIST; PHOLDS; set_of_list; NOT_IN_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT] THEN MESON_TAC[]);;
let HOLDS_INTERP = prove
(`!cl M v. FINITE cl
==> (holds M v (interp cl) <=> ?p. p IN cl /\ holds M v p)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[interp] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
[GSYM(CONJUNCT1(MATCH_MP LIST_OF_SET_PROPERTIES th))]) THEN
SPEC_TAC(`list_of_set(cl:form->bool)`,`l:form list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[ITLIST; HOLDS; set_of_list; NOT_IN_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Assuming finite set of hypotheses. *)
(* ------------------------------------------------------------------------- *)
let PRESPROOF_REFUTATION_COMPLETE_FINITE = prove
(`FINITE hyps /\
(!cl. cl IN hyps ==> clause cl) /\
~(psatisfiable {interp cl | cl IN hyps})
==> presproof hyps {}`,
REWRITE_TAC[IMAGE_CLAUSE] THEN
SUBGOAL_THEN
`!n hyps. FINITE hyps /\
(CARD(UNIONS hyps) = n) /\
(!cl. cl IN hyps ==> clause cl) /\
~(psatisfiable (IMAGE interp hyps))
==> presproof hyps {}`
(fun th -> MESON_TAC[th]) THEN
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
X_GEN_TAC `hyps:(form->bool)->bool` THEN
ASM_CASES_TAC `{}:form->bool IN hyps` THENL
[ASM_SIMP_TAC[presproof_RULES]; ALL_TAC] THEN
ASM_CASES_TAC `hyps:(form->bool)->bool = {}` THENL
[ASM_REWRITE_TAC[psatisfiable; IMAGE_CLAUSES; NOT_IN_EMPTY]; ALL_TAC] THEN
STRIP_TAC THEN SUBGOAL_THEN `FINITE(UNIONS hyps :form->bool)`
ASSUME_TAC THENL
[ASM_MESON_TAC[clause; FINITE_UNIONS]; ALL_TAC] THEN
SUBGOAL_THEN `?p. atom p /\ p IN UNIONS hyps /\ ~~p IN UNIONS hyps`
(X_CHOOSE_THEN `p:form` STRIP_ASSUME_TAC)
THENL
[SUBGOAL_THEN `?p. literal p /\ p IN UNIONS hyps /\ ~~p IN UNIONS hyps`
MP_TAC THENL
[ALL_TAC;
REWRITE_TAC[literal] THEN STRIP_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
EXISTS_TAC `~~p` THEN ASM_REWRITE_TAC[NEGATE_NEG] THEN
ASM_MESON_TAC[NEGATE_ATOM]] THEN
GEN_REWRITE_TAC I [TAUT `p <=> ~ ~ p`] THEN
PURE_ONCE_REWRITE_TAC[NOT_EXISTS_THM] THEN
REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a ==> ~b \/ ~c`] THEN
REWRITE_TAC[IN_UNIONS; NOT_EXISTS_THM] THEN
REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [psatisfiable]) THEN
REWRITE_TAC[] THEN EXISTS_TAC `\p:form. p IN UNIONS hyps` THEN
REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
GEN_TAC THEN X_GEN_TAC `cl:form->bool` THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST1_TAC ASSUME_TAC) THEN
SUBGOAL_THEN `FINITE(cl:form->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[clause]; ALL_TAC] THEN
ASM_SIMP_TAC[PHOLDS_INTERP] THEN
SUBGOAL_THEN `?p:form. p IN cl` MP_TAC THENL
[ASM_MESON_TAC[EXTENSION; NOT_IN_EMPTY]; ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `q:form` THEN DISCH_TAC THEN
SUBGOAL_THEN `literal q` MP_TAC THENL
[ASM_MESON_TAC[clause]; ALL_TAC] THEN
REWRITE_TAC[literal] THEN
STRIP_TAC THEN ASM_SIMP_TAC[PHOLDS; PHOLDS_ATOM] THEN
ASM_MESON_TAC[IN_UNIONS; NEGATE_ATOM; literal];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE BINDER_CONV [RIGHT_IMP_FORALL_THM]) THEN
ONCE_REWRITE_TAC[TAUT `(a ==> b /\ c /\ d ==> e) <=>
(c ==> a /\ b /\ d ==> e)`] THEN
ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN SIMP_TAC[] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
UNDISCH_THEN `CARD (UNIONS hyps :form->bool) = n` (SUBST1_TAC o SYM) THEN
ABBREV_TAC `hyps' = {cl | cl IN hyps /\ ~(p IN cl) /\ ~(~~p IN cl)} UNION
{cl | ?cl1 cl2. cl1 IN hyps /\ cl2 IN hyps /\
p IN cl1 /\ ~(~~p IN cl1) /\
~~p IN cl2 /\ ~(p IN cl2) /\
(cl = resolve p cl1 cl2)}` THEN
DISCH_THEN(MP_TAC o SPEC `hyps':(form->bool)->bool`) THEN
MATCH_MP_TAC(TAUT
`(e ==> f) /\ b /\ c /\ a /\ (c ==> d)
==> (a /\ b /\ c /\ d ==> e) ==> f`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ]
PRESPROOF_TRANS) THEN
X_GEN_TAC `cl:form->bool` THEN EXPAND_TAC "hyps'" THEN
REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN MESON_TAC[presproof_RULES];
ALL_TAC] THEN
CONJ_TAC THENL
[EXPAND_TAC "hyps'" THEN REWRITE_TAC[FINITE_UNION] THEN CONJ_TAC THENL
[MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `hyps:(form->bool)->bool` THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM]; ALL_TAC] THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `IMAGE (\(cl1,cl2). resolve p cl1 cl2)
{(cl1,cl2) | cl1 IN hyps /\ cl2 IN hyps}` THEN
ASM_SIMP_TAC[FINITE_IMAGE; FINITE_PRODUCT] THEN
REWRITE_TAC[SUBSET; IN_IMAGE; IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[EXISTS_PAIR_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN MESON_TAC[PAIR_EQ];
ALL_TAC] THEN
CONJ_TAC THENL
[GEN_TAC THEN EXPAND_TAC "hyps'" THEN
REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[] THEN REWRITE_TAC[clause; resolve] THEN
REWRITE_TAC[FINITE_UNION; FINITE_DELETE; IN_UNION; IN_DELETE] THEN
ASM_MESON_TAC[clause]; ALL_TAC] THEN
CONJ_TAC THENL
[MATCH_MP_TAC LET_TRANS THEN
EXISTS_TAC `CARD(((UNIONS hyps) DELETE p) DELETE ~~p)` THEN CONJ_TAC THENL
[MATCH_MP_TAC CARD_SUBSET THEN CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[FINITE_UNIONS; FINITE_DELETE] THEN
ASM_MESON_TAC[clause]] THEN
EXPAND_TAC "hyps'" THEN
REWRITE_TAC[SUBSET; IN_UNIONS; IN_UNION; IN_DELETE; IN_ELIM_THM] THEN
REWRITE_TAC[EXTENSION; resolve; IN_UNION; IN_DELETE] THEN MESON_TAC[];
ASM_SIMP_TAC[CARD_DELETE; FINITE_DELETE] THEN
ASM_REWRITE_TAC[IN_DELETE; NEGATE_REFL] THEN
ASM_MESON_TAC[HAS_SIZE; HAS_SIZE_0; MEMBER_NOT_EMPTY;
ARITH_RULE `~(n = 0) ==> n - 1 - 1 < n`]];
ALL_TAC] THEN
DISCH_TAC THEN
UNDISCH_TAC `~psatisfiable (IMAGE interp hyps)` THEN
REWRITE_TAC[TAUT `~a ==> ~b <=> b ==> a`] THEN
SIMP_TAC[psatisfiable; IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d:form->bool` THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
ONCE_REWRITE_TAC[IMP_CONJ] THEN
REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[clause]) THEN
ASM_SIMP_TAC[PHOLDS_INTERP] THEN DISCH_TAC THEN
ASM_CASES_TAC
`!cl. cl IN hyps /\ p IN cl /\ ~(~~p IN cl)
==> ?q. q IN (cl DELETE p) /\ pholds d q`
THENL
[EXISTS_TAC `\x:form. if x = p then F else d(x)` THEN
X_GEN_TAC `cl:form->bool` THEN DISCH_TAC THEN
ASM_CASES_TAC `~~p IN cl` THENL
[EXISTS_TAC `~~p` THEN ASM_SIMP_TAC[PHOLDS_NEGATE; PHOLDS_ATOM];
ALL_TAC] THEN
ASM_CASES_TAC `p:form IN cl` THENL
[UNDISCH_TAC
`!cl. cl IN hyps /\ p IN cl /\ ~(~~p IN cl)
==> (?q. q IN cl DELETE p /\ pholds d q)` THEN
DISCH_THEN(MP_TAC o SPEC `cl:form->bool`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:form` THEN
REWRITE_TAC[IN_DELETE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `literal q` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[literal] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[ASM_SIMP_TAC[PHOLDS_ATOM] THEN ASM_MESON_TAC[PHOLDS_ATOM];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `r:form`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
ASM_SIMP_TAC[PHOLDS; PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[PHOLDS; PHOLDS_ATOM]; ALL_TAC] THEN
UNDISCH_TAC `!x. x IN hyps' ==> (?p. p IN x /\ pholds d p)` THEN
DISCH_THEN(MP_TAC o SPEC `cl:form->bool`) THEN
EXPAND_TAC "hyps'" THEN REWRITE_TAC[IN_UNION] THEN
MATCH_MP_TAC(TAUT `((a ==> c) ==> d) ==> (a \/ b ==> c) ==> d`) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `q:form` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `literal q` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[literal] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[ASM_SIMP_TAC[PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_MESON_TAC[PHOLDS_ATOM];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `r:form`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
ASM_SIMP_TAC[PHOLDS; PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[PHOLDS; PHOLDS_ATOM]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
REWRITE_TAC[NOT_IMP; NOT_EXISTS_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `cl1:form->bool` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC
`!cl. cl IN hyps /\ ~~p IN cl /\ ~(p IN cl)
==> ?q. q IN (cl DELETE ~~p) /\ pholds d q`
THENL
[EXISTS_TAC `\x:form. if x = p then T else d(x)` THEN
X_GEN_TAC `cl:form->bool` THEN DISCH_TAC THEN
ASM_CASES_TAC `p:form IN cl` THENL
[EXISTS_TAC `p:form` THEN ASM_SIMP_TAC[PHOLDS_NEGATE; PHOLDS_ATOM];
ALL_TAC] THEN
ASM_CASES_TAC `~~p IN cl` THENL
[UNDISCH_TAC
`!cl. cl IN hyps /\ ~~p IN cl /\ ~(p IN cl)
==> (?q. q IN cl DELETE ~~p /\ pholds d q)` THEN
DISCH_THEN(MP_TAC o SPEC `cl:form->bool`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:form` THEN
REWRITE_TAC[IN_DELETE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `literal q` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[literal] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[ASM_SIMP_TAC[PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[PHOLDS_ATOM];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `r:form`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
ASM_SIMP_TAC[PHOLDS; PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[PHOLDS; PHOLDS_ATOM; NEGATE_ATOM]; ALL_TAC] THEN
UNDISCH_TAC `!x. x IN hyps' ==> (?p. p IN x /\ pholds d p)` THEN
DISCH_THEN(MP_TAC o SPEC `cl:form->bool`) THEN
EXPAND_TAC "hyps'" THEN REWRITE_TAC[IN_UNION] THEN
MATCH_MP_TAC(TAUT `((a ==> c) ==> d) ==> (a \/ b ==> c) ==> d`) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `q:form` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `literal q` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[literal] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[ASM_SIMP_TAC[PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_MESON_TAC[PHOLDS_ATOM];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `r:form`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
ASM_SIMP_TAC[PHOLDS; PHOLDS_ATOM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[NEGATE_ATOM];
ASM_MESON_TAC[PHOLDS; PHOLDS_ATOM]]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
REWRITE_TAC[NOT_IMP; NOT_EXISTS_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `cl2:form->bool` STRIP_ASSUME_TAC) THEN
UNDISCH_TAC `!x. x IN hyps' ==> (?p. p IN x /\ pholds d p)` THEN
MATCH_MP_TAC(TAUT `~a ==> a ==> b`) THEN
DISCH_THEN(MP_TAC o SPEC `resolve p cl1 cl2`) THEN
EXPAND_TAC "hyps'" THEN REWRITE_TAC[IN_UNION] THEN
MATCH_MP_TAC(TAUT `b /\ ~c ==> ~(a \/ b ==> c)`) THEN
CONJ_TAC THENL
[REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[];
REWRITE_TAC[resolve; IN_UNION] THEN ASM_MESON_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Via compactness etc., avoid the finiteness assumption for simplicity. *)
(* ------------------------------------------------------------------------- *)
let PRESPROOF_REFUTATION_COMPLETE = prove
(`(!cl. cl IN hyps ==> clause cl) /\
~(psatisfiable {interp cl | cl IN hyps})
==> presproof hyps {}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC PRESPROOF_MONO THEN
FIRST_ASSUM(MP_TAC o MATCH_MP UNPSATISFIABLE_FINITE_SUBSET) THEN
DISCH_THEN(X_CHOOSE_THEN `t:form->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?h. FINITE h /\ h SUBSET hyps /\ t SUBSET {interp cl | cl IN h}`
MP_TAC THENL
[REWRITE_TAC[IMAGE_CLAUSE] THEN MATCH_MP_TAC FINITE_SUBSET_IMAGE_IMP THEN
ASM_REWRITE_TAC[GSYM IMAGE_CLAUSE]; ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC PRESPROOF_REFUTATION_COMPLETE_FINITE THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
MAP_EVERY UNDISCH_TAC
[`~(psatisfiable t)`; `t SUBSET {interp cl | cl IN h}`] THEN
REWRITE_TAC[PSATISFIABLE_MONO; TAUT `b ==> ~c ==> ~a <=> a /\ b ==> c`]);;
(* ------------------------------------------------------------------------- *)
(* The key lifting lemma. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("instance_of",(12,"right"));;
let instance_of = new_definition
`cl1 instance_of cl2 <=> ?i. cl1 = IMAGE (formsubst i) cl2`;;
let FVS = new_definition
`FVS(cl) = UNIONS {FV(p) | p IN cl}`;;
let NEGATIVE_FORMSUBST = prove
(`!p. negative (formsubst i p) = negative p`,
MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[negative; form_DISTINCT; Not_DEF; formsubst] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[form_DISTINCT] THEN
SIMP_TAC[form_INJ; LEFT_EXISTS_AND_THM; GSYM EXISTS_REFL] THEN
GEN_TAC THEN X_GEN_TAC `r:form` THEN STRIP_TAC THEN
SPEC_TAC(`r:form`,`r:form`) THEN MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[formsubst; LET_DEF; LET_END_DEF; form_DISTINCT]);;
let FORMSUBST_NEGATE = prove
(`!p i. formsubst i (~~p) = ~~(formsubst i p)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[negate; NEGATIVE_FORMSUBST] THEN
COND_CASES_TAC THEN REWRITE_TAC[Not_DEF; formsubst] THEN
FIRST_ASSUM(X_CHOOSE_THEN `q:form` SUBST1_TAC o
GEN_REWRITE_RULE I [negative]) THEN
REWRITE_TAC[Not_DEF; form_INJ; formsubst]);;
let FORMSUBST_ATOM = prove
(`!p i. atom (formsubst i p) = atom p`,
MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[formsubst; atom; LET_DEF; LET_END_DEF]);;
let FORMSUBST_NOT = prove
(`!i p. formsubst i (Not p) = Not(formsubst i p)`,
REWRITE_TAC[Not_DEF; formsubst]);;
let NOT_NOT_ATOM = prove
(`!p. ~atom(Not p)`,
REWRITE_TAC[Not_DEF; atom; form_INJ]);;
let FORMSUBST_LITERAL = prove
(`!p i. literal (formsubst i p) = literal p`,
MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[formsubst] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[literal; form_DISTINCT; atom; Not_DEF] THEN
MAP_EVERY X_GEN_TAC [`s:form`; `r:form`] THEN STRIP_TAC THEN
SPEC_TAC(`r:form`,`r:form`) THEN MATCH_MP_TAC form_INDUCTION THEN
REWRITE_TAC[formsubst; LET_DEF; LET_END_DEF; form_DISTINCT; form_INJ] THEN
MESON_TAC[FORMSUBST_NOT; FORMSUBST_ATOM]);;
let QFREE_NOT = prove
(`!p. qfree(Not p) = qfree p`,
REWRITE_TAC[Not_DEF; qfree]);;
let QFREE_ATOM = prove
(`!p. atom p ==> qfree p`,
MATCH_MP_TAC form_INDUCTION THEN REWRITE_TAC[atom; qfree]);;
let QFREE_LITERAL = prove
(`!p. literal p ==> qfree p`,
REWRITE_TAC[literal] THEN MESON_TAC[QFREE_ATOM; QFREE_NOT]);;
let QFREE_NEGATE = prove
(`!p. qfree(~~p) = qfree p`,
GEN_TAC THEN REWRITE_TAC[negate] THEN
COND_CASES_TAC THEN REWRITE_TAC[QFREE_NOT] THEN
FIRST_ASSUM(CHOOSE_THEN SUBST1_TAC o GEN_REWRITE_RULE I [negative]) THEN
REWRITE_TAC[QFREE_NOT; Not_DEF; form_INJ]);;
let LIFTING_LEMMA = prove
(`!A B A' B' C' p.
clause A /\ clause B /\ (FVS(A) INTER FVS(B) = {}) /\
A' instance_of A /\ B' instance_of B /\
p IN A' /\ ~~p IN B' /\ (C' = resolve p A' B')
==> ?A1 B1. A1 SUBSET A /\ B1 SUBSET B /\ ~(A1 = {}) /\ ~(B1 = {}) /\
(?i. Unifies i (A1 UNION {~~l | l IN B1})) /\
!j. ismgu (A1 UNION {~~l | l IN B1}) j
==> C' instance_of (IMAGE (formsubst j)
((A DIFF A1) UNION (B DIFF B1)))`,
REWRITE_TAC[clause] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?i. (A' = IMAGE (formsubst i) A) /\
(B' = IMAGE (formsubst i) B)`
(X_CHOOSE_THEN `i:num->term` (STRIP_ASSUME_TAC o GSYM))
THENL
[UNDISCH_TAC `A' instance_of A` THEN REWRITE_TAC[instance_of] THEN
DISCH_THEN(X_CHOOSE_THEN `ia:num->term` SUBST1_TAC) THEN
UNDISCH_TAC `B' instance_of B` THEN REWRITE_TAC[instance_of] THEN
DISCH_THEN(X_CHOOSE_THEN `ib:num->term` SUBST1_TAC) THEN
EXISTS_TAC `\x. if x IN FVS(A) then ia(x) else ib(x):term` THEN
REWRITE_TAC[EXTENSION; IN_IMAGE] THEN CONJ_TAC THEN
X_GEN_TAC `r:form` THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
X_GEN_TAC `q:form` THEN
MATCH_MP_TAC(TAUT `(b ==> (a <=> a')) ==> (a /\ b <=> a' /\ b)`) THEN
DISCH_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC FORMSUBST_VALUATION THEN
X_GEN_TAC `x:num` THEN DISCH_TAC THEN REWRITE_TAC[] THENL
[SUBGOAL_THEN `x IN FVS(A)` (fun th -> REWRITE_TAC[th]) THEN
REWRITE_TAC[FVS; IN_UNIONS; IN_ELIM_THM] THEN ASM_MESON_TAC[];
SUBGOAL_THEN `~(x IN FVS(A))` (fun th -> REWRITE_TAC[th]) THEN
UNDISCH_TAC `FVS(A) INTER FVS(B) = {}` THEN
REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
REWRITE_TAC[FVS; IN_UNIONS; IN_ELIM_THM] THEN ASM_MESON_TAC[]];
ALL_TAC] THEN
ABBREV_TAC `A1 = {q | q IN A /\ (formsubst i q = p)}` THEN
ABBREV_TAC `B1 = {r | r IN B /\ (formsubst i r = ~~p)}` THEN
MAP_EVERY EXISTS_TAC [`A1:form->bool`; `B1:form->bool`] THEN
SUBGOAL_THEN `Unifies i (A1 UNION {~~l | l IN B1})` ASSUME_TAC THENL
[MAP_EVERY EXPAND_TAC ["A1"; "B1"] THEN
REWRITE_TAC[Unifies_DEF; IN_UNION; IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[FORMSUBST_NEGATE] THEN
ASM_MESON_TAC[NEGATE_NEGATE; FORMSUBST_LITERAL];
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ c /\ d /\ e /\
(a /\ b /\ c /\ d ==> f)
==> a /\ b /\ c /\ d /\ e /\ f`) THEN
REPEAT CONJ_TAC THENL
[EXPAND_TAC "A1" THEN SIMP_TAC[SUBSET; IN_ELIM_THM];
EXPAND_TAC "B1" THEN SIMP_TAC[SUBSET; IN_ELIM_THM];
EXPAND_TAC "A1" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
UNDISCH_TAC `p:form IN A'` THEN EXPAND_TAC "A'" THEN
REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[];
EXPAND_TAC "B1" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
UNDISCH_TAC `~~p IN B'` THEN EXPAND_TAC "B'" THEN
REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[];
EXISTS_TAC `i:num->term` THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
STRIP_TAC THEN X_GEN_TAC `j:num->term` THEN
REWRITE_TAC[ISMGU] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `i:num->term`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `k:num->term` (ASSUME_TAC o GSYM)) THEN
ASM_REWRITE_TAC[instance_of] THEN EXISTS_TAC `k:num->term` THEN
MAP_EVERY EXPAND_TAC ["A'"; "B'"] THEN
REWRITE_TAC[IMAGE_UNION; GSYM IMAGE_o; resolve] THEN
SUBGOAL_THEN `IMAGE (formsubst i) A DELETE p =
IMAGE (formsubst i) (A DIFF A1)`
SUBST1_TAC THENL
[EXPAND_TAC "A1" THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DELETE; IN_DIFF; IN_ELIM_THM] THEN
MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `IMAGE (formsubst i) B DELETE ~~p =
IMAGE (formsubst i) (B DIFF B1)`
SUBST1_TAC THENL
[EXPAND_TAC "B1" THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DELETE; IN_DIFF; IN_ELIM_THM] THEN
MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[GSYM IMAGE_UNION] THEN
SUBGOAL_THEN `!q. q IN (A DIFF A1 UNION B DIFF B1)
==> (formsubst k (formsubst j q) = formsubst i q)`
MP_TAC THENL
[REWRITE_TAC[IN_UNION; IN_DIFF] THEN ASM_MESON_TAC[QFREE_LITERAL];
ALL_TAC] THEN
ABBREV_TAC `s:form->bool = A DIFF A1 UNION B DIFF B1` THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; o_THM] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Method for renaming away (we rely on nothing but the basic property). *)
(* ------------------------------------------------------------------------- *)
let FVS_CLAUSE_FINITE = prove
(`!cl. clause(cl) ==> FINITE(FVS cl)`,
REWRITE_TAC[clause; FVS] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(MATCH_MP (TAUT `(a ==> (b <=> c)) ==> a /\ c ==> b`)
(SPEC_ALL FINITE_FINITE_UNIONS)) THEN
CONJ_TAC THENL
[SUBGOAL_THEN `FINITE (IMAGE FV cl)` MP_TAC THENL
[ASM_SIMP_TAC[FINITE_IMAGE]; ALL_TAC] THEN
MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM] THEN
MESON_TAC[];
SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM; FV_FINITE]]);;
let RENAME_AWAY = prove
(`!cl s. FINITE(s) /\ clause(cl)
==> ?i. renaming(i) /\ (FVS(IMAGE (formsubst i) cl) INTER s = {})`,
let lemma = prove
(`((UNIONS s) INTER y = {}) <=> !x. x IN s ==> (x INTER y = {})`,
SET_TAC[]) in
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?n. !x. x IN s \/ x IN FVS cl ==> x < n` MP_TAC THENL
[REWRITE_TAC[GSYM IN_UNION] THEN
SUBGOAL_THEN `FINITE(s UNION FVS(cl))` MP_TAC THENL
[ASM_SIMP_TAC[FINITE_UNION; FVS_CLAUSE_FINITE]; ALL_TAC] THEN
SPEC_TAC(`s UNION FVS(cl)`,`s:num->bool`) THEN
MATCH_MP_TAC FINITE_INDUCT THEN REWRITE_TAC[NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_INSERT] THEN
MESON_TAC[ARITH_RULE `(x < a ==> x < a + b + 1) /\ b < a + b + 1`];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
EXISTS_TAC `\x. if x < N then V(x + N) else if x < 2 * N then V(x - N)
else V(x)` THEN
CONJ_TAC THENL
[REWRITE_TAC[renaming] THEN
EXISTS_TAC `\x. if x < N then V(x + N) else if x < 2 * N then V(x - N)
else V(x)` THEN
REWRITE_TAC[] THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_DEF] THEN
REWRITE_TAC[TERMSUBST_TERMSUBST] THEN GEN_TAC THEN
GEN_REWRITE_TAC RAND_CONV [GSYM TERMSUBST_TRIV] THEN
MATCH_MP_TAC TERMSUBST_VALUATION THEN
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[termsubst; o_THM] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
REPEAT(COND_CASES_TAC THEN REWRITE_TAC[termsubst]) THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[ADD_SUB]) THEN
AP_TERM_TAC THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[FVS; lemma] THEN
SIMP_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM; IN_ELIM_THM;
LEFT_IMP_EXISTS_THM] THEN
GEN_TAC THEN GEN_TAC THEN X_GEN_TAC `r:form` THEN STRIP_TAC THEN
SUBGOAL_THEN `!x. x IN FV(r) ==> x < N:num` ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[FVS; IN_UNIONS; IN_ELIM_THM] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
X_GEN_TAC `x:num` THEN REWRITE_TAC[FORMSUBST_FV] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `y:num` MP_TAC) THEN
REPEAT(COND_CASES_TAC THEN
ASM_REWRITE_TAC[FVT; IN_INSERT; NOT_IN_EMPTY]) THEN
ASM_MESON_TAC[ARITH_RULE `~(x + N < N)`;
ARITH_RULE `y < 2 * N ==> y - N < N`]);;
let rename = new_specification ["rename"]
(REWRITE_RULE[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] RENAME_AWAY);;
(* ------------------------------------------------------------------------- *)
(* General resolution. *)
(* ------------------------------------------------------------------------- *)
let resproof_RULES,resproof_INDUCT,resproof_CASES = new_inductive_definition
`(!cl. cl IN hyps ==> resproof hyps cl) /\
(!cl1 cl2 cl2' ps1 ps2 i.
resproof hyps cl1 /\ resproof hyps cl2 /\
(IMAGE (formsubst (rename cl2 (FVS cl1))) cl2 = cl2') /\
ps1 SUBSET cl1 /\ ps2 SUBSET cl2' /\ ~(ps1 = {}) /\ ~(ps2 = {}) /\
(?i. Unifies i (ps1 UNION {~~p | p IN ps2})) /\
(mgu (ps1 UNION {~~p | p IN ps2}) = i)
==> resproof hyps
(IMAGE (formsubst i) ((cl1 DIFF ps1) UNION (cl2' DIFF ps2))))`;;
(* ------------------------------------------------------------------------- *)
(* Refutation completeness of resolution. *)
(* ------------------------------------------------------------------------- *)
let PHOLDS_FORMSUBST = prove
(`!p i d. qfree(p)
==> (pholds d (formsubst i p) <=> pholds (d o formsubst i) p)`,
MATCH_MP_TAC form_INDUCTION THEN
SIMP_TAC[o_THM; qfree; pholds; formsubst]);;
let QFREE_INTERP = prove
(`!cl. clause cl ==> qfree(interp cl)`,
REWRITE_TAC[clause; interp] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `!p. MEM p (list_of_set cl) ==> qfree p` MP_TAC THENL
[ASM_MESON_TAC[MEM_LIST_OF_SET; QFREE_LITERAL]; ALL_TAC] THEN
SPEC_TAC(`list_of_set cl :form list`,`l:form list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[MEM; ITLIST; Or_DEF; qfree] THEN
ASM_MESON_TAC[]);;
let PHOLDS_INTERP_IMAGE = prove
(`!cl v d. clause cl
==> (pholds d (interp (IMAGE (formsubst v) cl)) <=>
pholds d (formsubst v (interp cl)))`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o REWRITE_RULE[clause]) THEN
ASM_SIMP_TAC[QFREE_INTERP; PHOLDS_FORMSUBST] THEN
ASM_SIMP_TAC[PHOLDS_INTERP; QFREE_LITERAL; FINITE_IMAGE] THEN
REWRITE_TAC[IN_IMAGE] THEN ASM_MESON_TAC[PHOLDS_FORMSUBST; QFREE_LITERAL]);;
let IMAGE_FORMSUBST_CLAUSE = prove
(`!v cl. clause(cl) ==> clause(IMAGE (formsubst v) cl)`,
SIMP_TAC[clause; IN_IMAGE; FINITE_IMAGE] THEN
MESON_TAC[FORMSUBST_LITERAL]);;
let INSTANCE_OF_EMPTY = prove
(`!cl. {} instance_of cl ==> (cl = {})`,
REWRITE_TAC[instance_of; EXTENSION; NOT_IN_EMPTY; IN_IMAGE] THEN
MESON_TAC[]);;
let RESPROOF_CLAUSE = prove
(`(!cl. cl IN hyps ==> clause cl)
==> !cl. resproof hyps cl ==> clause cl`,
let lemma = prove (`s DIFF t SUBSET s`,SET_TAC[]) in
DISCH_TAC THEN MATCH_MP_TAC resproof_INDUCT THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[clause; IMAGE_UNION; FINITE_UNION] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN CONJ_TAC THENL
[ASM_MESON_TAC[clause; FINITE_IMAGE; lemma; FINITE_SUBSET]; ALL_TAC] THEN
EXPAND_TAC "cl2'" THEN REWRITE_TAC[IN_IMAGE; IN_UNION; IN_DIFF] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[FORMSUBST_LITERAL]);;
let RESOLUTION_COMPLETE = prove
(`(!cl. cl IN hyps ==> clause cl) /\
~(?M:(term->bool)#(num->term list->term)#(num->term list->bool).
interpretation (language(IMAGE interp hyps)) M /\ ~(Dom M = {}) /\
M satisfies (IMAGE interp hyps))
==> resproof hyps {}`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `IMAGE interp hyps` HERBRAND_THEOREM) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[REWRITE_TAC[IN_IMAGE] THEN ASM_MESON_TAC[QFREE_INTERP]; ALL_TAC] THEN
DISCH_TAC THEN
SUBGOAL_THEN
`~(psatisfiable
{interp cl |
cl IN {IMAGE(formsubst v) cl | cl,v | cl IN hyps}})`
MP_TAC THENL
[REWRITE_TAC[psatisfiable] THEN
FIRST_X_ASSUM(fun th -> MP_TAC th THEN
MATCH_MP_TAC(TAUT `(b ==> a) ==> ~a ==> ~b`)) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:form->bool` THEN
REWRITE_TAC[psatisfies] THEN
SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM;
RIGHT_AND_EXISTS_THM; IN_IMAGE] THEN
ASM_SIMP_TAC[PHOLDS_INTERP_IMAGE] THEN MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ_ALT]
PRESPROOF_REFUTATION_COMPLETE)) THEN
ANTS_TAC THENL
[SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
ASM_SIMP_TAC[IMAGE_FORMSUBST_CLAUSE]; ALL_TAC] THEN
SUBGOAL_THEN
`!cl0. presproof {IMAGE (formsubst v) cl | cl,v | cl IN hyps} cl0
==> ?cl. resproof hyps cl /\ cl0 instance_of cl`
MP_TAC THENL
[ALL_TAC;
DISCH_THEN(MP_TAC o SPEC `{}:form->bool`) THEN
MATCH_MP_TAC(TAUT `(b ==> c) ==> (a ==> b) ==> (a ==> c)`) THEN
MESON_TAC[INSTANCE_OF_EMPTY]] THEN
MATCH_MP_TAC presproof_INDUCT THEN CONJ_TAC THENL
[REWRITE_TAC[IN_IMAGE; instance_of; IN_ELIM_THM] THEN
MESON_TAC[resproof_RULES]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`p:form`; `A':form->bool`; `B':form->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `A:form->bool` STRIP_ASSUME_TAC)
MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `B:form->bool` STRIP_ASSUME_TAC)
STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL
[`A:form->bool`; `IMAGE (formsubst (rename B (FVS A))) B`;
`A':form->bool`; `B':form->bool`; `resolve p A' B'`; `p:form`]
LIFTING_LEMMA) THEN
ABBREV_TAC `C = IMAGE (formsubst (rename B (FVS A))) B` THEN
MP_TAC(SPECL [`B:form->bool`; `FVS(A)`] rename) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[FVS_CLAUSE_FINITE; RESPROOF_CLAUSE]; ALL_TAC] THEN
ASM_REWRITE_TAC[renaming] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
[FUN_EQ_THM; o_THM; I_DEF; BETA_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `j:num->term` (ASSUME_TAC o CONJUNCT1)) THEN
ANTS_TAC THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[RESPROOF_CLAUSE];
ASM_MESON_TAC[IMAGE_FORMSUBST_CLAUSE; RESPROOF_CLAUSE];
ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[];
UNDISCH_TAC `B' instance_of B` THEN REWRITE_TAC[instance_of] THEN
DISCH_THEN(X_CHOOSE_THEN `k:num->term` SUBST1_TAC) THEN
EXPAND_TAC "C" THEN REWRITE_TAC[GSYM IMAGE_o] THEN
EXISTS_TAC `termsubst k o (j:num->term)` THEN
SUBGOAL_THEN
`termsubst k = termsubst (termsubst k o j) o termsubst (rename B (FVS A))`
MP_TAC THENL
[REWRITE_TAC[FUN_EQ_THM] THEN MATCH_MP_TAC term_INDUCT THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[termsubst; GSYM TERMSUBST_TERMSUBST; o_THM];
SIMP_TAC[termsubst; term_INJ; o_THM; GSYM MAP_o] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC MAP_EQ THEN ASM_REWRITE_TAC[o_THM]];
ALL_TAC] THEN
REWRITE_TAC[GSYM FORMSUBST_TERMSUBST_LEMMA] THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; o_THM] THEN
ASM_MESON_TAC[RESPROOF_CLAUSE; clause; QFREE_LITERAL]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `A1:form->bool` (X_CHOOSE_THEN `B1:form->bool`
MP_TAC)) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPEC `mgu (A1 UNION {~~ l | l IN B1})`) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[MATCH_MP_TAC ISMGU_MGU THEN ASM_REWRITE_TAC[FINITE_UNION] THEN
REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[RESPROOF_CLAUSE; clause; FINITE_SUBSET];
SUBGOAL_THEN `{~~l | l IN B1} = IMAGE (~~) B1` SUBST1_TAC THENL
[REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM] THEN
MESON_TAC[]; ALL_TAC] THEN
ASM_MESON_TAC[RESPROOF_CLAUSE; clause; FINITE_SUBSET; FINITE_IMAGE];
REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN
ASM_MESON_TAC[RESPROOF_CLAUSE; clause; QFREE_LITERAL; SUBSET;
IMAGE_FORMSUBST_CLAUSE; QFREE_NEGATE]];
ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN EXISTS_TAC (rand(concl th))) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(CONJUNCT2(SPEC_ALL resproof_RULES)) THEN
EXISTS_TAC `B:form->bool` THEN ASM_REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Variant with an explicit "is a resolvent" predicate. *)
(* ------------------------------------------------------------------------- *)
let isaresolvent = new_definition
`isaresolvent cl (cl1,cl2) <=>
let cl2' = IMAGE (formsubst (rename cl2 (FVS cl1))) cl2 in
?ps1 ps2.
ps1 SUBSET cl1 /\ ps2 SUBSET cl2' /\ ~(ps1 = {}) /\ ~(ps2 = {}) /\
(?i. Unifies i (ps1 UNION {~~p | p IN ps2})) /\
let i = mgu (ps1 UNION {~~p | p IN ps2}) in
cl = IMAGE (formsubst i) ((cl1 DIFF ps1) UNION (cl2' DIFF ps2))`;;
let RESPROOF_RULES = prove
(`!hyps. (!cl. cl IN hyps ==> resproof hyps cl) /\
(!cl1 cl2 cl.
resproof hyps cl1 /\ resproof hyps cl2 /\
isaresolvent cl (cl1,cl2)
==> resproof hyps cl)`,
REWRITE_TAC[isaresolvent] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[resproof_RULES] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(CONJUNCT2(SPEC_ALL resproof_RULES)) THEN
EXISTS_TAC `cl2:form->bool` THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `i:num->term` THEN ASM_REWRITE_TAC[]);;
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