(* ========================================================================= *)
(* Positive resolution and semantic resolution.                              *)
(* ========================================================================= *)

let allpositive = new_definition
 `allpositive cl <=> !p. p IN cl ==> positive p`;;

(* ------------------------------------------------------------------------- *)
(* Various simple lemmas.                                                    *)
(* ------------------------------------------------------------------------- *)

let NOT_NEGATIVE_ATOM = prove
 (`!p a. ~(negative (Atom p a))`,
  REWRITE_TAC[negative; Not_DEF; form_DISTINCT]);;

let NEGATIVE_NOT = prove
 (`!p. negative(Not p)`,
  MESON_TAC[negative]);;

let CLAUSE_FINITE = prove
 (`!c. clause c ==> FINITE c`,
  SIMP_TAC[clause]);;

let POSITIVE_LITERAL_ATOM = prove
 (`!p. literal(p) /\ positive(p) <=> atom(p)`,
  REWRITE_TAC[literal; positive; negative] THEN
  MESON_TAC[Not_DEF; form_DISTINCT; ATOM]);;

let PHOLDS_ATOM = prove
 (`!v p. atom(p) ==> (pholds v p <=> v p)`,
  SIMP_TAC[ATOM; LEFT_IMP_EXISTS_THM; PHOLDS]);;

let PHOLDS_ALLTRUE_POSLIT = prove
 (`!p. literal p /\ positive p ==> pholds (\x. T) p`,
  REWRITE_TAC[literal; ATOM; positive; negative] THEN
  REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[PHOLDS] THEN
  ASM_MESON_TAC[atom; Not_DEF; form_DISTINCT]);;

let PHOLDS_ALLFALSE_NEGLIT = prove
 (`!p. literal p /\ negative p ==> pholds (\x. F) p`,
  REWRITE_TAC[literal; ATOM; positive; negative] THEN
  REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[PHOLDS] THEN
  ASM_MESON_TAC[atom; Not_DEF; form_DISTINCT]);;

let PHOLDS_ALLTRUE_POSCLAUSE = prove
 (`!c. clause(c) /\ allpositive c /\ ~(c = {}) ==> pholds (\x. T) (interp c)`,
  SIMP_TAC[clause; PHOLDS_INTERP; allpositive; EXTENSION; NOT_IN_EMPTY] THEN
  MESON_TAC[PHOLDS_ALLTRUE_POSLIT]);;

let PHOLDS_ALLFALSE_NONPOSCLAUSE = prove
 (`!c. clause(c) /\ ~allpositive c ==> pholds (\x. F) (interp c)`,
  SIMP_TAC[clause; PHOLDS_INTERP; allpositive; EXTENSION; NOT_IN_EMPTY] THEN
  MESON_TAC[PHOLDS_ALLFALSE_NEGLIT; positive]);;

(* ------------------------------------------------------------------------- *)
(* Main lemma from Robinson's original proof.                                *)
(* ------------------------------------------------------------------------- *)

let PRESOLUTION_LEMMA = prove
 (`!s. FINITE s /\ (!c. c IN s ==> clause c) /\
       ~psatisfiable (IMAGE interp s) /\ ~({} IN s)
       ==> ?c1 c2 p. c1 IN s /\ c2 IN s /\
                     (allpositive c1 \/ allpositive c2) /\
                     p IN c1 /\ ~~p IN c2 /\
                     ~((resolve p c1 c2) IN s)`,
  REPEAT STRIP_TAC THEN
  ABBREV_TAC `P = {c | c IN s /\ allpositive c}` THEN
  ABBREV_TAC `N = {c | c IN s /\ ~(allpositive c)}` THEN
  SUBGOAL_THEN `~(P:(form->bool)->bool = {})` ASSUME_TAC THENL
   [EXPAND_TAC "P" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
    REWRITE_TAC[NOT_IN_EMPTY; TAUT `~(a /\ b) <=> a ==> ~b`] THEN
    DISCH_TAC THEN
    UNDISCH_TAC `~psatisfiable (IMAGE interp s)` THEN
    REWRITE_TAC[psatisfiable] THEN EXISTS_TAC `\p:form. F` THEN
    ASM_SIMP_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM; PHOLDS_ALLFALSE_NONPOSCLAUSE];
    ALL_TAC] THEN
  SUBGOAL_THEN `~(N:(form->bool)->bool = {})` ASSUME_TAC THENL
   [EXPAND_TAC "N" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
    REWRITE_TAC[NOT_IN_EMPTY; TAUT `~(a /\ b) <=> a ==> ~b`] THEN
    DISCH_TAC THEN
    UNDISCH_TAC `~psatisfiable (IMAGE interp s)` THEN
    REWRITE_TAC[psatisfiable] THEN EXISTS_TAC `\p:form. T` THEN
    SIMP_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
    REPEAT STRIP_TAC THEN MATCH_MP_TAC PHOLDS_ALLTRUE_POSCLAUSE THEN
    ASM_MESON_TAC[]; ALL_TAC] THEN
  SUBGOAL_THEN `?n v. v psatisfies (IMAGE interp P) /\ v HAS_SIZE n`
  MP_TAC THENL
   [EXISTS_TAC `CARD((UNIONS P):form->bool)` THEN
    EXISTS_TAC `(UNIONS P):form->bool` THEN
    REWRITE_TAC[HAS_SIZE] THEN CONJ_TAC THENL
     [REWRITE_TAC[psatisfies; IN_IMAGE; IN_UNIONS; LEFT_IMP_EXISTS_THM] THEN
      GEN_TAC THEN X_GEN_TAC `c:form->bool` THEN
      DISCH_THEN(CONJUNCTS_THEN2 SUBST1_TAC ASSUME_TAC) THEN
      SUBGOAL_THEN `FINITE(c:form->bool)` ASSUME_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[clause]; ALL_TAC] THEN
      ASM_SIMP_TAC[PHOLDS_INTERP] THEN
      SUBGOAL_THEN `~(c:form->bool = {})` MP_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[]; ALL_TAC] THEN
      REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; NOT_FORALL_THM] THEN
      MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:form` THEN DISCH_TAC THEN
      ASM_REWRITE_TAC[] THEN
      SUBGOAL_THEN `positive q` ASSUME_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[allpositive]; ALL_TAC] THEN
      SUBGOAL_THEN `atom q` MP_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[clause; literal; positive; negative]; ALL_TAC] THEN
      SIMP_TAC[ATOM; LEFT_IMP_EXISTS_THM; PHOLDS] THEN
      REPEAT GEN_TAC THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
      GEN_REWRITE_TAC I [GSYM IN] THEN REWRITE_TAC[IN_UNIONS] THEN
      ASM_MESON_TAC[];
      ALL_TAC] THEN
    SUBGOAL_THEN `FINITE(P:(form->bool)->bool)` MP_TAC THENL
     [EXPAND_TAC "P" THEN MATCH_MP_TAC FINITE_SUBSET THEN
      EXISTS_TAC `s:(form->bool)->bool` THEN
      ASM_SIMP_TAC[SUBSET; IN_ELIM_THM]; ALL_TAC] THEN
    SIMP_TAC[FINITE_UNIONS] THEN
    RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[clause]; ALL_TAC] THEN
  GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
  REWRITE_TAC[NOT_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN
  REWRITE_TAC[TAUT `a ==> ~(b /\ c) <=> a /\ c ==> ~b`] THEN
  DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN
  DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
  DISCH_THEN(X_CHOOSE_THEN `v:form->bool` STRIP_ASSUME_TAC) THEN
  SUBGOAL_THEN `?m c. c IN N /\
                      ~(pholds v (interp c)) /\
                      {p | p IN c /\ negative p} HAS_SIZE m`
  MP_TAC THENL
   [GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
    UNDISCH_TAC `~psatisfiable (IMAGE interp s)` THEN
    REWRITE_TAC[psatisfiable; NOT_EXISTS_THM; NOT_FORALL_THM] THEN
    DISCH_THEN(MP_TAC o SPEC `v:form->bool`) THEN
    SIMP_TAC[LEFT_IMP_EXISTS_THM; IN_IMAGE; NOT_FORALL_THM] THEN
    REWRITE_TAC[NOT_IMP] THEN GEN_TAC THEN
    REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN
    MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:form->bool` THEN
    STRIP_TAC THEN EXISTS_TAC `CARD {p | p IN k /\ negative p}` THEN
    ASM_REWRITE_TAC[HAS_SIZE] THEN CONJ_TAC THENL
     [EXPAND_TAC "N" THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN
      SUBGOAL_THEN `(k:form->bool) IN P` ASSUME_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[]; ALL_TAC] THEN
      ASM_MESON_TAC[IN_IMAGE; psatisfies]; ALL_TAC] THEN
    MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `k:form->bool` THEN
    SIMP_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[clause];
    ALL_TAC] THEN
  GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
  DISCH_THEN(X_CHOOSE_THEN `m:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
  DISCH_THEN(X_CHOOSE_THEN `k:form->bool` STRIP_ASSUME_TAC) THEN
  MP_TAC(ASSUME `(k:form->bool) IN N`) THEN EXPAND_TAC "N" THEN
  REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN
  REWRITE_TAC[allpositive; NOT_FORALL_THM; NOT_IMP; positive] THEN
  DISCH_THEN(X_CHOOSE_THEN `r:form` MP_TAC) THEN REWRITE_TAC[negative] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  DISCH_THEN(X_CHOOSE_THEN `l:form` SUBST_ALL_TAC) THEN
  SUBGOAL_THEN `clause k` ASSUME_TAC THENL
   [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[clause]; ALL_TAC] THEN
  SUBGOAL_THEN `atom l` ASSUME_TAC THENL
   [SUBGOAL_THEN `literal(Not l)` MP_TAC THENL
     [ASM_MESON_TAC[clause]; ALL_TAC] THEN
    SIMP_TAC[LEFT_IMP_EXISTS_THM; literal; Not_DEF; form_INJ; atom];
    ALL_TAC] THEN
  SUBGOAL_THEN `v(l:form) = T` ASSUME_TAC THENL
   [UNDISCH_TAC `~pholds v (interp k)` THEN
    ASM_SIMP_TAC[PHOLDS_INTERP; CLAUSE_FINITE; NOT_EXISTS_THM] THEN
    DISCH_THEN(MP_TAC o SPEC `Not l`) THEN ASM_REWRITE_TAC[PHOLDS] THEN
    FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [ATOM]) THEN
    ASM_REWRITE_TAC[PHOLDS]; ALL_TAC] THEN
  SUBGOAL_THEN
   `?j. j IN P /\ l IN j /\ ~(pholds v (interp (j DELETE l)))`
  MP_TAC THENL
   [FIRST_ASSUM(MP_TAC o SPECL
     [`n - 1`; `\p:form. if p = l then F else v(p)`]) THEN
    ANTS_TAC THENL
     [CONJ_TAC THENL
       [MATCH_MP_TAC(ARITH_RULE `~(n = 0) ==> n - 1 < n`) THEN
        DISCH_THEN SUBST_ALL_TAC THEN
        FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_SIZE_0]) THEN
        REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `l:form`) THEN
        REWRITE_TAC[NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[IN];
        ALL_TAC] THEN
      RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
      ASM_REWRITE_TAC[HAS_SIZE] THEN
      SUBGOAL_THEN `(\p:form. if p = l then F else v(p)) = v DELETE l`
      SUBST1_TAC THENL
       [REWRITE_TAC[EXTENSION; IN_DELETE] THEN GEN_TAC THEN
        REWRITE_TAC[IN] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[];
        ALL_TAC] THEN
      ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN
      ASM_REWRITE_TAC[IN];
      ALL_TAC] THEN
    ONCE_REWRITE_TAC[TAUT `~a ==> b <=> ~b ==> a`] THEN
    REWRITE_TAC[NOT_EXISTS_THM] THEN
    REWRITE_TAC[TAUT `~(a /\ b /\ ~c) <=> a /\ b ==> c`] THEN
    DISCH_TAC THEN
    REWRITE_TAC[psatisfies] THEN GEN_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
    SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:form->bool` THEN
    DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN
    FIRST_X_ASSUM(MP_TAC o SPEC `c:form->bool`) THEN ASM_REWRITE_TAC[] THEN
    SUBGOAL_THEN `clause c /\ clause(c DELETE l)` MP_TAC THENL
     [MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[]; ALL_TAC] THEN
      SIMP_TAC[clause; IN_DELETE; FINITE_DELETE]; ALL_TAC] THEN
    SIMP_TAC[clause; PHOLDS_INTERP] THEN
    REWRITE_TAC[GSYM clause] THEN STRIP_TAC THEN
    ASM_CASES_TAC `l:form IN c` THEN ASM_REWRITE_TAC[] THENL
     [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:form` THEN
      SIMP_TAC[IN_DELETE] THEN
      DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
      SUBGOAL_THEN `atom q` MP_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[POSITIVE_LITERAL_ATOM; allpositive; clause];
        ALL_TAC] THEN
      SIMP_TAC[PHOLDS_ATOM] THEN ASM_REWRITE_TAC[];
      UNDISCH_TAC `v psatisfies IMAGE interp P` THEN
      REWRITE_TAC[psatisfies] THEN DISCH_THEN(MP_TAC o SPEC `interp c`) THEN
      REWRITE_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
      DISCH_THEN(MP_TAC o SPEC `c:form->bool`) THEN ASM_REWRITE_TAC[] THEN
      ASM_SIMP_TAC[PHOLDS_INTERP; CLAUSE_FINITE] THEN
      MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:form` THEN STRIP_TAC THEN
      SUBGOAL_THEN `atom q` MP_TAC THENL
       [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
        ASM_MESON_TAC[POSITIVE_LITERAL_ATOM; allpositive; clause];
        ALL_TAC] THEN
      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 `j:form->bool` STRIP_ASSUME_TAC) THEN
  MAP_EVERY EXISTS_TAC [`j:form->bool`; `k:form->bool`; `l:form`] THEN
  REWRITE_TAC[GSYM negative; GSYM positive] THEN
  CONJ_TAC THENL
   [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[allpositive]; ALL_TAC] THEN
  CONJ_TAC THENL
   [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[allpositive]; ALL_TAC] THEN
  CONJ_TAC THENL
   [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[allpositive]; ALL_TAC] THEN
  ASM_REWRITE_TAC[negate] THEN CONJ_TAC THENL
   [COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
    RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    ASM_MESON_TAC[allpositive; positive]; ALL_TAC] THEN
  FIRST_ASSUM(fun th -> MP_TAC(SPEC `m - 1` th) THEN ANTS_TAC) THENL
   [MATCH_MP_TAC(ARITH_RULE `~(n = 0) ==> n - 1 < n`) THEN
    DISCH_THEN SUBST_ALL_TAC THEN
    FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_SIZE_0]) THEN
    REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN
    DISCH_THEN(MP_TAC o SPEC `Not l`) THEN
    ASM_REWRITE_TAC[IN_ELIM_THM; negative] THEN MESON_TAC[];
    ALL_TAC] THEN
  REWRITE_TAC[NOT_EXISTS_THM] THEN
  DISCH_THEN(MP_TAC o SPEC `resolve l j k`) THEN
  ONCE_REWRITE_TAC[TAUT `~a ==> ~b <=> b ==> a`] THEN DISCH_TAC THEN
  SUBGOAL_THEN `~pholds v (interp (resolve l j k))` ASSUME_TAC THENL
   [UNDISCH_TAC `~pholds v (interp k)` THEN
    UNDISCH_TAC `~pholds v (interp (j DELETE l))` THEN
    SUBGOAL_THEN `clause j` ASSUME_TAC THENL
     [RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
      ASM_MESON_TAC[clause]; ALL_TAC] THEN
    ASM_SIMP_TAC[PHOLDS_INTERP; CLAUSE_FINITE; RESOLVE_CLAUSE;
                 FINITE_DELETE] THEN
    REWRITE_TAC[resolve; IN_UNION; IN_DELETE] THEN MESON_TAC[];
    ALL_TAC] THEN
  ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
   [SUBGOAL_THEN `~(resolve l j k IN P)` MP_TAC THENL
     [ASM_MESON_TAC[psatisfies; IN_IMAGE]; ALL_TAC] THEN
    UNDISCH_TAC `resolve l j k IN s` THEN
    MAP_EVERY EXPAND_TAC ["P"; "N"] THEN
    REWRITE_TAC[IN_ELIM_THM] THEN CONV_TAC TAUT; ALL_TAC] THEN
  SUBGOAL_THEN `{p | p IN resolve l j k /\ negative p} =
                {p | p IN k /\ negative p} DELETE (Not l)`
  SUBST1_TAC THENL
   [REWRITE_TAC[EXTENSION; IN_DELETE; IN_ELIM_THM; resolve; IN_UNION] THEN
    SUBGOAL_THEN `~~l = Not l` SUBST1_TAC THENL
     [REWRITE_TAC[negate] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
      RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
      ASM_MESON_TAC[allpositive; positive]; ALL_TAC] THEN
    RULE_ASSUM_TAC(REWRITE_RULE[EXTENSION; IN_ELIM_THM]) THEN
    GEN_TAC THEN MATCH_MP_TAC(TAUT
     `(a ==> ~e) ==> ((a /\ b \/ c /\ d) /\ e <=> (c /\ e) /\ d)`) THEN
    REWRITE_TAC[GSYM positive] THEN
    ASM_MESON_TAC[allpositive]; ALL_TAC] THEN
  SUBGOAL_THEN `FINITE {p | p IN k /\ negative p}` MP_TAC THENL
   [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `k:form->bool` THEN
    ASM_SIMP_TAC[CLAUSE_FINITE; SUBSET; IN_ELIM_THM]; ALL_TAC] THEN
  SIMP_TAC[HAS_SIZE; CARD_DELETE; FINITE_DELETE] THEN
  DISCH_TAC THEN UNDISCH_TAC `{p | p IN k /\ negative p} HAS_SIZE m` THEN
  SIMP_TAC[HAS_SIZE] THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
  COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[negative]);;

(* ------------------------------------------------------------------------- *)
(* Inductive definition of *positive* propositional resolution.              *)
(* ------------------------------------------------------------------------- *)

let pposresproof_RULES,pposresproof_INDUCT,pposresproof_CASES =
 new_inductive_definition
  `(!cl. cl IN hyps ==> pposresproof hyps cl) /\
   (!p cl1 cl2.
       pposresproof hyps cl1 /\ pposresproof hyps cl2 /\
       (allpositive cl1 \/ allpositive cl2) /\
       p IN cl1 /\ ~~p IN cl2
      ==> pposresproof hyps (resolve p cl1 cl2))`;;

(* ------------------------------------------------------------------------- *)
(* Its completeness.                                                         *)
(* ------------------------------------------------------------------------- *)

let POSRESPROOF_FINITE = prove
 (`!hyps. FINITE hyps /\ (!cl. cl IN hyps ==> clause cl)
        ==> FINITE {cl | pposresproof hyps cl}`,
  REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
  EXISTS_TAC `{t | t SUBSET (UNIONS hyps)} :(form->bool)->bool` THEN
  CONJ_TAC THENL
   [MATCH_MP_TAC FINITE_POWERSET THEN RULE_ASSUM_TAC(REWRITE_RULE[clause]) THEN
    ASM_SIMP_TAC[FINITE_UNIONS]; ALL_TAC] THEN
  REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
  MATCH_MP_TAC pposresproof_INDUCT THEN CONJ_TAC THENL
   [MESON_TAC[IN_UNIONS];
    REWRITE_TAC[resolve; IN_UNION; IN_DELETE] THEN MESON_TAC[]]);;

let PPOSRESPROOF_REFUTATION_COMPLETE_FINITE = prove
 (`FINITE hyps /\
   (!cl. cl IN hyps ==> clause cl) /\
   ~(psatisfiable {interp cl | cl IN hyps})
   ==> pposresproof hyps {}`,
  REPEAT STRIP_TAC THEN
  MP_TAC(SPEC `hyps:(form->bool)->bool` POSRESPROOF_FINITE) THEN
  ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
  MP_TAC(SPEC `{cl | pposresproof hyps cl}` PRESOLUTION_LEMMA) THEN
  ASM_REWRITE_TAC[] THEN
  SUBGOAL_THEN `~psatisfiable (IMAGE interp {cl | pposresproof hyps cl})`
  ASSUME_TAC THENL
   [UNDISCH_TAC `~psatisfiable {interp cl | cl IN hyps}` THEN
    REWRITE_TAC[TAUT `~a ==> ~b <=> b ==> a`] THEN
    MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT]
                 PSATISFIABLE_MONO) THEN
    REWRITE_TAC[SUBSET; IN_IMAGE; IN_ELIM_THM] THEN
    MESON_TAC[pposresproof_RULES]; ALL_TAC] THEN
  ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM] THEN
  MATCH_MP_TAC(TAUT `~c /\ a ==> (a /\ ~b ==> c) ==> b`) THEN
  CONJ_TAC THENL [MESON_TAC[pposresproof_RULES]; ALL_TAC] THEN
  MATCH_MP_TAC pposresproof_INDUCT THEN ASM_SIMP_TAC[RESOLVE_CLAUSE]);;

(* ------------------------------------------------------------------------- *)
(* Lift to the non-finite case by compactness.                               *)
(* ------------------------------------------------------------------------- *)

let PPOSRESPROOF_MONO = prove
 (`!hyps1 hyps2 c.
        pposresproof hyps1 c /\ hyps1 SUBSET hyps2 ==> pposresproof 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 pposresproof_INDUCT THEN
  MESON_TAC[pposresproof_RULES; SUBSET]);;

let PPOSRESPROOF_REFUTATION_COMPLETE = prove
 (`(!cl. cl IN hyps ==> clause cl) /\
   ~(psatisfiable {interp cl | cl IN hyps})
   ==> pposresproof hyps {}`,
  REPEAT STRIP_TAC THEN MATCH_MP_TAC PPOSRESPROOF_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 PPOSRESPROOF_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`]);;

(* ------------------------------------------------------------------------- *)
(* Generalization to semantic resolution at the propositional level.         *)
(* ------------------------------------------------------------------------- *)

let psemresproof_RULES,psemresproof_INDUCT,psemresproof_CASES =
 new_inductive_definition
  `(!cl. cl IN hyps ==> psemresproof v hyps cl) /\
   (!p cl1 cl2.
       psemresproof v hyps cl1 /\ psemresproof v hyps cl2 /\
       (~pholds v (interp cl1) \/ ~pholds v (interp cl2)) /\
       p IN cl1 /\ ~~p IN cl2
      ==> psemresproof v hyps (resolve p cl1 cl2))`;;

(* ------------------------------------------------------------------------- *)
(* Proof by propositional variable flipping.                                 *)
(* ------------------------------------------------------------------------- *)

let propflip = new_definition
 `propflip w p = if (negative p = pholds w p) then p else ~~p`;;

let PHOLDS_LITERAL_PROPFLIP = prove
 (`!p w. literal(p) ==> (pholds w p <=> pholds (\x. F) (propflip w p))`,
  REWRITE_TAC[literal; ATOM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[propflip] THEN REWRITE_TAC[PHOLDS] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM] THEN
  COND_CASES_TAC THEN ASM_REWRITE_TAC[PHOLDS_NEGATE; PHOLDS]);;

let PROPFLIP_INVOLUTE = prove
 (`!w p. literal p ==> (propflip w (propflip w p) = p)`,
  REWRITE_TAC[literal; ATOM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[propflip] THEN REWRITE_TAC[PHOLDS] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM] THENL
   [ASM_CASES_TAC `w(Atom q l):bool` THEN
    ASM_REWRITE_TAC[negate; NOT_NEGATIVE_ATOM; NEGATIVE_NOT; PHOLDS] THEN
    REWRITE_TAC[Not_DEF; form_INJ; SELECT_REFL];
    ASM_CASES_TAC `w(Atom q' l):bool` THEN
    ASM_REWRITE_TAC[negate; NOT_NEGATIVE_ATOM; NEGATIVE_NOT; PHOLDS] THEN
    REWRITE_TAC[Not_DEF; form_INJ; SELECT_REFL] THEN
    ASM_REWRITE_TAC[NOT_NEGATIVE_ATOM; PHOLDS]]);;

let PROPFLIP_INJ = prove
 (`!w p q. literal p /\ literal q /\ (propflip w p = propflip w q)
           ==> (p = q)`,
  MESON_TAC[PROPFLIP_INVOLUTE]);;

let PROPFLIP_NEGATE = prove
 (`!w p. literal p ==> (propflip w (~~p) = ~~(propflip w p))`,
  REWRITE_TAC[literal; ATOM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[propflip] THEN REWRITE_TAC[PHOLDS] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM; NEGATE_NEG] THEN
  SIMP_TAC[NEGATE_ATOM; atom] THEN REWRITE_TAC[PHOLDS; NEGATIVE_NOT] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM; NEGATE_NEG] THEN
  SIMP_TAC[NEGATE_ATOM; atom] THEN
  COND_CASES_TAC THEN SIMP_TAC[NEGATE_ATOM; atom; NEGATE_NEG]);;

let PROPFLIP_RESOLVE = prove
 (`!cl1 cl2 p w.
     clause cl1 /\ clause cl2 /\ p IN cl1
     ==> (IMAGE (propflip w) (resolve p cl1 cl2) =
          resolve (propflip w p)
                  (IMAGE (propflip w) cl1) (IMAGE (propflip w) cl2))`,
  REPEAT STRIP_TAC THEN REWRITE_TAC[resolve; IMAGE_UNION] THEN BINOP_TAC THEN
  (REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DELETE] THEN
   X_GEN_TAC `q:form` THEN EQ_TAC THENL
    [ALL_TAC; ASM_MESON_TAC[PROPFLIP_NEGATE; clause]] THEN
   REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
   MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:form` THEN
   STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
   ASM_MESON_TAC[PROPFLIP_INJ; clause; PROPFLIP_NEGATE; NEGATE_LITERAL]));;

let PPOSRESPROOF_CLAUSE = prove
 (`!hyps. (!c. c IN hyps ==> clause c)
          ==> !c. pposresproof hyps c ==> clause c`,
  GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC pposresproof_INDUCT THEN
  ASM_SIMP_TAC[RESOLVE_CLAUSE]);;

let PSEMRESPROOF_CLAUSE = prove
 (`!hyps w. (!c. c IN hyps ==> clause c)
            ==> !c. psemresproof w hyps c ==> clause c`,
  REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC psemresproof_INDUCT THEN
  ASM_SIMP_TAC[RESOLVE_CLAUSE]);;

let LITERAL_PROPFLIP = prove
 (`!p w. literal p ==> literal (propflip w p)`,
  REPEAT GEN_TAC THEN REWRITE_TAC[propflip] THEN
  COND_CASES_TAC THEN SIMP_TAC[NEGATE_LITERAL]);;

let CLAUSE_IMAGE_PROPFLIP = prove
 (`!cl w. clause cl ==> clause (IMAGE (propflip w) cl)`,
  SIMP_TAC[clause; FINITE_IMAGE] THEN
  MESON_TAC[LITERAL_PROPFLIP; IN_IMAGE]);;

let PHOLDS_LITERAL_PROPFLIP_SAME = prove
 (`!p w. literal(p) ==> (pholds w (propflip w p) <=> ~(positive p))`,
  REWRITE_TAC[literal; ATOM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[propflip] THEN REWRITE_TAC[PHOLDS] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM; positive] THEN
  COND_CASES_TAC THEN ASM_REWRITE_TAC[PHOLDS_NEGATE; PHOLDS]);;

let PHOLDS_IMAGE_PROPFLIP_SAME = prove
 (`!v cl. clause cl
          ==> (pholds v (interp (IMAGE (propflip v) cl)) <=> ~(allpositive cl))`,
  SIMP_TAC[clause; PHOLDS_INTERP; FINITE_IMAGE; allpositive] THEN
  ONCE_REWRITE_TAC[TAUT `a /\ b <=> ~(a ==> ~b)`] THEN
  SIMP_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
  REWRITE_TAC[NOT_IMP; NOT_FORALL_THM] THEN
  MESON_TAC[PHOLDS_LITERAL_PROPFLIP_SAME]);;

let PPOSRESPROOF_PSEMRESPROOF = prove
 (`!hyps. (!c. c IN hyps ==> clause c)
          ==> !w cl. pposresproof hyps cl
                     ==> psemresproof w (IMAGE (IMAGE (propflip w)) hyps)
                                        (IMAGE (propflip w) cl)`,
  GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
  SUBGOAL_THEN
   `!cl. pposresproof hyps cl
         ==> clause cl /\ psemresproof w (IMAGE (IMAGE (propflip w)) hyps)
                                         (IMAGE (propflip w) cl)`
   (fun th -> SIMP_TAC[th]) THEN
  MATCH_MP_TAC pposresproof_INDUCT THEN
  ASM_SIMP_TAC[RESOLVE_CLAUSE] THEN CONJ_TAC THENL
   [ASM_MESON_TAC[psemresproof_RULES; IN_IMAGE]; ALL_TAC] THEN
  ASM_SIMP_TAC[PROPFLIP_RESOLVE] THEN
  REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
  MATCH_MP_TAC(CONJUNCT2(SPEC_ALL psemresproof_RULES)) THEN
  ASM_SIMP_TAC[PHOLDS_IMAGE_PROPFLIP_SAME] THEN
  ASM_MESON_TAC[PROPFLIP_NEGATE; clause; NEGATE_LITERAL; IN_IMAGE]);;

(* ------------------------------------------------------------------------- *)
(* Hence refutation completeness.                                            *)
(* ------------------------------------------------------------------------- *)

let PHOLDS_ATOM_PROPFLIP_DIFF = prove
 (`!p v w. atom(p) ==> (pholds v (propflip w p) <=> ~(v p = w p))`,
  SIMP_TAC[ATOM; LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN
  REWRITE_TAC[propflip; NOT_NEGATIVE_ATOM; positive; negate; PHOLDS] THEN
  COND_CASES_TAC THEN ASM_REWRITE_TAC[PHOLDS]);;

let PHOLDS_LITERAL_PROPFLIP_DIFF = prove
 (`!p v w. literal(p)
           ==> (pholds v (propflip w p) <=> pholds (\x. ~(v x = w x)) p)`,
  REWRITE_TAC[literal; ATOM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[propflip] THEN REWRITE_TAC[PHOLDS] THEN
  REWRITE_TAC[NEGATIVE_NOT; NOT_NEGATIVE_ATOM; positive] THEN
  COND_CASES_TAC THEN ASM_REWRITE_TAC[PHOLDS_NEGATE; PHOLDS]);;

let PHOLDS_INTERP_IMAGE_PROPFLIP_DIFF = prove
 (`!v cl. clause cl
          ==> (pholds v (interp (IMAGE (propflip w) cl)) <=>
               pholds (\x. ~(v x = w x)) (interp cl))`,
  SIMP_TAC[clause; PHOLDS_INTERP; FINITE_IMAGE] THEN
  REPEAT STRIP_TAC THEN
  REWRITE_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM] THEN
  ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
  REWRITE_TAC[UNWIND_THM2; GSYM CONJ_ASSOC] THEN
  AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
  GEN_TAC THEN REWRITE_TAC[] THEN
  MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
  ASM_SIMP_TAC[PHOLDS_LITERAL_PROPFLIP_DIFF]);;

let PSATISFIABLE_CLAUSES_PROPFLIP = prove
 (`!w s. (!c. c IN s ==> clause c)
         ==> (psatisfiable (IMAGE (interp o IMAGE (propflip w)) s) <=>
              psatisfiable (IMAGE interp s))`,
  REPEAT STRIP_TAC THEN REWRITE_TAC[psatisfiable; IMAGE_o] THEN EQ_TAC THEN
  DISCH_THEN(X_CHOOSE_THEN `v:form->bool` STRIP_ASSUME_TAC) THEN
  EXISTS_TAC `\p:form. ~(v(p):bool = w(p))` THEN
  ASM_SIMP_TAC[IN_IMAGE; LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM] THENL
   [ASM_SIMP_TAC[GSYM PHOLDS_INTERP_IMAGE_PROPFLIP_DIFF];
    ASM_SIMP_TAC[PHOLDS_INTERP_IMAGE_PROPFLIP_DIFF] THEN
    REWRITE_TAC[TAUT `~(~(a <=> b) <=> b) <=> a`] THEN
    CONV_TAC(ONCE_DEPTH_CONV ETA_CONV)] THEN
  ASM_MESON_TAC[IN_IMAGE]);;

let PSEMRESPROOF_MONO = prove
 (`!w hyps1 hyps2 c.
        psemresproof w hyps1 c /\ hyps1 SUBSET hyps2
        ==> psemresproof w hyps2 c`,
  GEN_TAC THEN GEN_TAC THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN
  REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
  MATCH_MP_TAC psemresproof_INDUCT THEN
  MESON_TAC[psemresproof_RULES; SUBSET]);;

let PROPFLIP_INVOLUTE_CLAUSE = prove
 (`!w cl. clause cl ==> (IMAGE (propflip w) (IMAGE (propflip w) cl) = cl)`,
  REWRITE_TAC[clause] THEN REPEAT STRIP_TAC THEN
  GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_IMAGE] THEN
  ASM_MESON_TAC[PROPFLIP_INVOLUTE]);;

let PSEMRESPROOF_REFUTATION_COMPLETE = prove
 (`!hyps w. (!cl. cl IN hyps ==> clause cl) /\
            ~(psatisfiable {interp cl | cl IN hyps})
            ==> psemresproof w hyps {}`,
  let lemma = prove
   (`{interp cl | cl IN hyps} = IMAGE interp hyps`,
    REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[]) in
  REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  REWRITE_TAC[lemma] THEN
  ASM_SIMP_TAC[GSYM PSATISFIABLE_CLAUSES_PROPFLIP] THEN
  REWRITE_TAC[IMAGE_o; GSYM lemma] THEN
  SUBGOAL_THEN `!cl. cl IN IMAGE (IMAGE (propflip w)) hyps ==> clause cl`
  MP_TAC THENL
   [ASM_SIMP_TAC[CLAUSE_IMAGE_PROPFLIP; IN_IMAGE; LEFT_IMP_EXISTS_THM];
    ALL_TAC] THEN
  ONCE_REWRITE_TAC[TAUT `a ==> b ==> c <=> a /\ b ==> a ==> c`] THEN
  DISCH_THEN(MP_TAC o MATCH_MP PPOSRESPROOF_REFUTATION_COMPLETE) THEN
  ONCE_REWRITE_TAC[TAUT `b ==> a ==> c <=> a /\ b ==> c`] THEN
  DISCH_THEN(MP_TAC o MATCH_MP
    (REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]
                  PPOSRESPROOF_PSEMRESPROOF)) THEN
  DISCH_THEN(MP_TAC o SPEC `w:form->bool`) THEN REWRITE_TAC[IMAGE_CLAUSES] THEN
  MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT]
                         PSEMRESPROOF_MONO) THEN
  SIMP_TAC[SUBSET; IN_IMAGE; LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
  ASM_MESON_TAC[PROPFLIP_INVOLUTE_CLAUSE]);;

(* ------------------------------------------------------------------------- *)
(* Lifting positive resolution to first order level.                         *)
(* ------------------------------------------------------------------------- *)

let posresproof_RULES,posresproof_INDUCT,posresproof_CASES =
  new_inductive_definition
  `(!cl. cl IN hyps ==> posresproof hyps cl) /\
   (!cl1 cl2 cl2' ps1 ps2 i.
        posresproof hyps cl1 /\ posresproof hyps cl2 /\
        (allpositive cl1 \/ allpositive 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)
        ==> posresproof hyps
               (IMAGE (formsubst i) ((cl1 DIFF ps1) UNION (cl2' DIFF ps2))))`;;

let POSRESPROOF_CLAUSE = prove
 (`(!cl. cl IN hyps ==> clause cl)
   ==> !cl. posresproof hyps cl ==> clause cl`,
  let lemma = prove (`s DIFF t SUBSET s`,SET_TAC[]) in
  DISCH_TAC THEN MATCH_MP_TAC posresproof_INDUCT THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[clause; IMAGE_UNION; FINITE_UNION] THEN
  REPEAT GEN_TAC THEN
  DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_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 ALLPOSITIVE_INSTANCE_OF = prove
 (`!cl1 cl2. cl1 instance_of cl2 /\ allpositive cl1 ==> allpositive cl2`,
  REWRITE_TAC[allpositive; instance_of] THEN REPEAT GEN_TAC THEN
  DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
  ASM_REWRITE_TAC[] THEN MESON_TAC[positive; NEGATIVE_FORMSUBST; IN_IMAGE]);;

let POSRESOLUTION_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))
   ==> posresproof 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 | v,cl | 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]
                PPOSRESPROOF_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. pposresproof {IMAGE (formsubst v) cl | v,cl | cl IN hyps} cl0
           ==> ?cl. posresproof 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 pposresproof_INDUCT THEN CONJ_TAC THENL
   [REWRITE_TAC[IN_IMAGE; instance_of; IN_ELIM_THM] THEN
    MESON_TAC[posresproof_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)
               (REPEAT_TCL CONJUNCTS_THEN 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; POSRESPROOF_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[POSRESPROOF_CLAUSE];
    ASM_MESON_TAC[IMAGE_FORMSUBST_CLAUSE; POSRESPROOF_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[POSRESPROOF_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[POSRESPROOF_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[POSRESPROOF_CLAUSE; clause; FINITE_SUBSET; FINITE_IMAGE];
      REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN
      ASM_MESON_TAC[POSRESPROOF_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 posresproof_RULES)) THEN
  EXISTS_TAC `B:form->bool` THEN ASM_REWRITE_TAC[] THEN
  ASM_MESON_TAC[ALLPOSITIVE_INSTANCE_OF]);;

(* ------------------------------------------------------------------------- *)
(* Lift semantic resolution to first order level as well.                    *)
(* ------------------------------------------------------------------------- *)

let semresproof_RULES,semresproof_INDUCT,semresproof_CASES =
  new_inductive_definition
  `(!cl. cl IN hyps ==> semresproof M hyps cl) /\
   (!cl1 cl2 cl2' ps1 ps2 i.
        semresproof M hyps cl1 /\ semresproof M hyps cl2 /\
        (~(!v:num->A. holds M v (interp cl1)) \/
         ~(!v:num->A. holds M v (interp 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)
        ==> semresproof M hyps
               (IMAGE (formsubst i) ((cl1 DIFF ps1) UNION (cl2' DIFF ps2))))`;;

let SEMRESPROOF_CLAUSE = prove
 (`(!c. c IN hyps ==> clause c) ==> (!c. semresproof M hyps c ==> clause c)`,
  let lemma = prove (`s DIFF t SUBSET s`,SET_TAC[]) in
  DISCH_TAC THEN MATCH_MP_TAC semresproof_INDUCT THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[clause; IMAGE_UNION; FINITE_UNION] THEN
  REPEAT GEN_TAC THEN
  DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_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 QFREE_HOLDS_PHOLDS = prove
 (`!p. qfree(p) ==> (holds M v p <=> pholds (holds M v) p)`,
  MATCH_MP_TAC form_INDUCTION THEN SIMP_TAC[HOLDS; PHOLDS; qfree]);;

let LIFTING_FALSIFY = prove
 (`!p M w. qfree(p) /\ (!v. holds M v p)
           ==> pholds (holds M w) (formsubst i p)`,
  SIMP_TAC[GSYM QFREE_HOLDS_PHOLDS; QFREE_FORMSUBST; HOLDS_FORMSUBST]);;

let LIFTING_FALSITY_CLAUSE = prove
 (`clause A /\ (!v:num->A. holds M v (interp A)) /\ A' instance_of A
   ==> pholds (holds M w) (interp A')`,
  REPEAT STRIP_TAC THEN
  FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [instance_of]) THEN
  DISCH_THEN(X_CHOOSE_THEN `i:num->term` SUBST1_TAC) THEN
  SUBGOAL_THEN `pholds (holds M (w:num->A)) (formsubst i (interp A))`
  MP_TAC THENL [ASM_MESON_TAC[LIFTING_FALSIFY; QFREE_INTERP]; ALL_TAC] THEN
  ASM_SIMP_TAC[PHOLDS_INTERP; IMAGE_FORMSUBST_CLAUSE; FINITE_IMAGE;
               CLAUSE_FINITE; PHOLDS_FORMSUBST; QFREE_INTERP] THEN
  ASM_MESON_TAC[IN_IMAGE; clause; QFREE_LITERAL; PHOLDS_FORMSUBST]);;

let SEMRESOLUTION_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))
   ==> !M:(A->bool)#(num->A list->A)#(num->A list->bool).
           semresproof M 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 | v,cl |
                cl IN hyps /\
                (!x. v(x) IN herbase (functions (IMAGE interp 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]
                PSEMRESPROOF_REFUTATION_COMPLETE)) THEN
  REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL
   [SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
    ASM_SIMP_TAC[IMAGE_FORMSUBST_CLAUSE]; ALL_TAC] THEN
  DISCH_THEN(MP_TAC o SPEC `holds M (@x:num->A. T)`) THEN
  ABBREV_TAC `w = @x:num->A. T` THEN
  ABBREV_TAC
    `ghyps = {IMAGE(formsubst v) cl | v,cl |
              cl IN hyps /\
              (!x. v(x) IN herbase (functions (IMAGE interp hyps)))}` THEN
  SUBGOAL_THEN
    `!cl0. psemresproof (holds M (w:num->A)) ghyps cl0
           ==> ?cl. semresproof M 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 psemresproof_INDUCT THEN CONJ_TAC THENL
   [EXPAND_TAC "ghyps" THEN
    REWRITE_TAC[IN_IMAGE; instance_of; IN_ELIM_THM] THEN
    MESON_TAC[semresproof_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)
               (REPEAT_TCL CONJUNCTS_THEN 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; SEMRESPROOF_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[SEMRESPROOF_CLAUSE];
    ASM_MESON_TAC[IMAGE_FORMSUBST_CLAUSE; SEMRESPROOF_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[SEMRESPROOF_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[SEMRESPROOF_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[SEMRESPROOF_CLAUSE; clause; FINITE_SUBSET; FINITE_IMAGE];
      REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN
      ASM_MESON_TAC[SEMRESPROOF_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 semresproof_RULES)) THEN
  EXISTS_TAC `B:form->bool` THEN ASM_REWRITE_TAC[] THEN
  ASM_MESON_TAC[SEMRESPROOF_CLAUSE; LIFTING_FALSITY_CLAUSE]);;

(* ------------------------------------------------------------------------- *)
(* More refined variant based on genuine models and valuations.              *)
(* ------------------------------------------------------------------------- *)

let semresproof2_RULES,semresproof2_INDUCT,semresproof2_CASES =
  new_inductive_definition
  `(!cl. cl IN hyps ==> semresproof2 M hyps cl) /\
   (!cl1 cl2 cl2' ps1 ps2 i.
        semresproof2 M hyps cl1 /\ semresproof2 M hyps cl2 /\
        (~(!v:num->A. valuation M v ==> holds M v (interp cl1)) \/
         ~(!v:num->A. valuation M v ==> holds M v (interp 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)
        ==> semresproof2 M hyps
               (IMAGE (formsubst i) ((cl1 DIFF ps1) UNION (cl2' DIFF ps2))))`;;

let SEMRESPROOF2_CLAUSE = prove
 (`(!c. c IN hyps ==> clause c) ==> (!c. semresproof2 M hyps c ==> clause c)`,
  let lemma = prove (`s DIFF t SUBSET s`,SET_TAC[]) in
  DISCH_TAC THEN MATCH_MP_TAC semresproof2_INDUCT THEN ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[clause; IMAGE_UNION; FINITE_UNION] THEN
  REPEAT GEN_TAC THEN
  DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_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 QFREE_HOLDS_PHOLDS = prove
 (`!p. qfree(p) ==> (holds M v p <=> pholds (holds M v) p)`,
  MATCH_MP_TAC form_INDUCTION THEN SIMP_TAC[HOLDS; PHOLDS; qfree]);;

let LIFTING_FALSIFY = prove
 (`!p M w. qfree(p) /\
           (!v. valuation M v ==> holds M v p) /\
           (!x f l. (f,LENGTH l) IN functions_term(i x) /\
                    ALL (\a. a IN Dom(M)) l
                    ==> Fun M f l IN Dom(M))
           ==> !w. valuation M w ==> pholds (holds M w) (formsubst i p)`,
  SIMP_TAC[GSYM QFREE_HOLDS_PHOLDS; QFREE_FORMSUBST; HOLDS_FORMSUBST] THEN
  REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
  REWRITE_TAC[valuation; o_THM] THEN X_GEN_TAC `v:num` THEN
  MATCH_MP_TAC INTERPRETATION_TERMVAL THEN ASM_REWRITE_TAC[] THEN
  ASM_REWRITE_TAC[interpretation]);;

let LIFTING_FALSITY_CLAUSE = prove
 (`clause A /\ (A' = IMAGE (formsubst i) A) /\
   (!v:num->A. valuation M v ==> holds M v (interp A)) /\
   (!x f l. (f,LENGTH l) IN functions_term(i x) /\
            ALL (\a. a IN Dom(M)) l
            ==> Fun M f l IN Dom(M))
   ==> !w. valuation M w ==> pholds (holds M w) (interp A')`,
  REPEAT STRIP_TAC THEN
  SUBGOAL_THEN `pholds (holds M (w:num->A)) (formsubst i (interp A))`
  MP_TAC THENL
   [UNDISCH_TAC `valuation M (w:num->A)` THEN
    SPEC_TAC(`w:num->A`,`w:num->A`) THEN
    MATCH_MP_TAC LIFTING_FALSIFY THEN ASM_REWRITE_TAC[] THEN
    ASM_SIMP_TAC[QFREE_INTERP]; ALL_TAC] THEN
  ASM_SIMP_TAC[PHOLDS_INTERP; IMAGE_FORMSUBST_CLAUSE; FINITE_IMAGE;
               CLAUSE_FINITE; PHOLDS_FORMSUBST; QFREE_INTERP] THEN
  ASM_MESON_TAC[IN_IMAGE; clause; QFREE_LITERAL; PHOLDS_FORMSUBST]);;

let FUNCTIONS_FORM_INTERP = prove
 (`!s. FINITE s ==> (functions_form(interp s) = functions s)`,
  REWRITE_TAC[interp] THEN
  SUBGOAL_THEN
   `!l. functions_form(ITLIST (||)  l False) = functions(set_of_list l)`
   (fun th -> MESON_TAC[SET_OF_LIST_OF_SET; th]) THEN
  LIST_INDUCT_TAC THEN
  REWRITE_TAC[ITLIST; And_DEF; Or_DEF; Not_DEF;
              functions_form; set_of_list] THENL
   [REWRITE_TAC[functions; NOT_IN_EMPTY; EXTENSION; IN_ELIM_THM; IN_UNIONS];
    ALL_TAC] THEN
  ASM_REWRITE_TAC[] THEN
  REWRITE_TAC[functions; IN_INSERT; EXTENSION; IN_ELIM_THM; IN_UNIONS;
              IN_UNION] THEN
  MESON_TAC[]);;

let FUNCTIONS_IMAGE_INTERP = prove
 (`!s. (!c. c IN s ==> FINITE(c))
       ==> (functions (IMAGE interp s) = UNIONS {functions p | p IN s})`,
  REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
  REWRITE_TAC[functions_form; functions; IN_UNIONS;
              IN_ELIM_THM; IN_IMAGE] THEN
  GEN_REWRITE_TAC I [FORALL_PAIR_THM] THEN
  ONCE_REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
  ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
  ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c <=> b /\ a /\ c`] THEN
  REWRITE_TAC[UNWIND_THM2] THEN
  REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
  ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
  REWRITE_TAC[UNWIND_THM2; GSYM CONJ_ASSOC] THEN
  REWRITE_TAC[GSYM functions] THEN
  ASM_MESON_TAC[FUNCTIONS_FORM_INTERP]);;

let FUNCTIONS_RESOLVE = prove
 (`functions(resolve p cl1 cl2) SUBSET (functions cl1 UNION functions cl2)`,
  REWRITE_TAC[SUBSET; functions; IN_UNION; resolve; IN_DIFF; IN_UNION;
              IN_UNIONS; IN_ELIM_THM; IN_DELETE] THEN
  MESON_TAC[]);;

let PSEMRESPROOF_FUNCTIONS = prove
 (`(!c. c IN hyps ==> clause c)
   ==> !c. psemresproof M hyps c
           ==> functions c SUBSET functions(IMAGE interp hyps)`,
  DISCH_TAC THEN
  MATCH_MP_TAC psemresproof_INDUCT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
   [ASM_SIMP_TAC[FUNCTIONS_IMAGE_INTERP;
                 PSEMRESPROOF_CLAUSE; CLAUSE_FINITE] THEN
    REWRITE_TAC[SUBSET; IN_UNIONS; IN_ELIM_THM] THEN MESON_TAC[];
    ALL_TAC] THEN
  REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
  EXISTS_TAC `functions cl1 UNION functions cl2` THEN
  REWRITE_TAC[FUNCTIONS_RESOLVE] THEN ASM_MESON_TAC[SUBSET; IN_UNION]);;

let FUNCTIONS_TERM_NOCONSTANTS = prove
 (`!t. ~(?c. c,0 IN functions_term t) ==> ~(FVT t = {})`,
  MATCH_MP_TAC term_INDUCT THEN
  REWRITE_TAC[functions_term; NOT_IN_EMPTY; FVT] THEN CONJ_TAC THENL
   [REWRITE_TAC[EXTENSION; IN_SING; NOT_IN_EMPTY] THEN MESON_TAC[];
    ALL_TAC] THEN
  GEN_TAC THEN LIST_INDUCT_TAC THEN
  REWRITE_TAC[ALL; LENGTH; IN_INSERT; MAP; LIST_UNION] THENL
   [MESON_TAC[]; ALL_TAC] THEN
  REWRITE_TAC[IN_UNION; EMPTY_UNION] THEN MESON_TAC[]);;

let HERBASE = prove
 (`!t. t IN herbase fns <=>
        functions_term t SUBSET fns /\
        (FVT(t) = if ?c. c,0 IN fns then {} else {0})`,
  GEN_TAC THEN EQ_TAC THEN SPEC_TAC(`t:term`,`t:term`) THENL
   [GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV) [IN] THEN
    MATCH_MP_TAC herbase_INDUCT THEN
    SIMP_TAC[FVT; functions_term; EMPTY_SUBSET] THEN
    REWRITE_TAC[GSYM ALL_MEM] THEN
    MAP_EVERY X_GEN_TAC [`f:num`; `tms:term list`] THEN STRIP_TAC THEN
    CONJ_TAC THENL
     [REWRITE_TAC[SUBSET; IN_INSERT; IN_LIST_UNION; GSYM EX_MEM; MEM_MAP] THEN
      ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
    GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:num` THEN
    REWRITE_TAC[IN_LIST_UNION; GSYM EX_MEM; MEM_MAP] THEN
    REWRITE_TAC[RIGHT_AND_EXISTS_THM; GSYM CONJ_ASSOC] THEN
    ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> c /\ a /\ b`] THEN
    ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
    GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
     [TAUT `a /\ b <=> ~(b ==> ~a)`] THEN
    ASM_SIMP_TAC[] THEN REWRITE_TAC[NOT_IMP] THEN
    COND_CASES_TAC THEN REWRITE_TAC[NOT_IN_EMPTY] THEN
    SUBGOAL_THEN `~(tms:term list = [])`
     (fun th -> ASM_MESON_TAC[th; list_CASES; MEM; LENGTH_EQ_NIL]) THEN
    ASM_MESON_TAC[LENGTH];
    ALL_TAC] THEN
  MATCH_MP_TAC term_INDUCT THEN CONJ_TAC THENL
   [REWRITE_TAC[functions_term; EMPTY_SUBSET; FVT] THEN
    COND_CASES_TAC THEN REWRITE_TAC[EXTENSION; IN_SING; NOT_IN_EMPTY] THEN
    ASM_MESON_TAC[IN; herbase_RULES]; ALL_TAC] THEN
  MAP_EVERY X_GEN_TAC [`f:num`; `tms:term list`] THEN
  REWRITE_TAC[GSYM ALL_MEM] THEN REPEAT STRIP_TAC THEN
  REWRITE_TAC[IN] THEN MATCH_MP_TAC(CONJUNCT2(SPEC_ALL herbase_RULES)) THEN
  UNDISCH_TAC `functions_term (Fn f tms) SUBSET fns` THEN
  REWRITE_TAC[SUBSET; functions_term; IN_INSERT; IN_LIST_UNION] THEN
  SIMP_TAC[TAUT `(a \/ b ==> c) <=> (a ==> c) /\ (b ==> c)`;
           FORALL_AND_THM] THEN
  MATCH_MP_TAC(TAUT `(a ==> a') /\ (b ==> b') ==> a /\ b ==> a' /\ b'`) THEN
  CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
  REWRITE_TAC[GSYM ALL_MEM; GSYM EX_MEM; MEM_MAP] THEN
  SIMP_TAC[LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
  GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
  GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
  MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:term` THEN
  ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN
  SIMP_TAC[] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
  REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
  DISCH_THEN(MP_TAC o SPEC `functions_term t`) THEN REWRITE_TAC[] THEN
  REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [GSYM IN] THEN
  FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[TAUT
   `a ==> b ==> c <=> a /\ b ==> c`]) THEN
  ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
   [ASM_MESON_TAC[SUBSET; IN]; ALL_TAC] THEN
  UNDISCH_TAC `FVT(Fn f tms) = (if ?c:num. c,0 IN fns then {} else {0})` THEN
  REWRITE_TAC[FVT] THEN COND_CASES_TAC THENL
   [REWRITE_TAC[EXTENSION; IN_LIST_UNION; MEM_MAP; NOT_IN_EMPTY;
                GSYM EX_MEM] THEN ASM_MESON_TAC[];
    ALL_TAC] THEN
  SUBGOAL_THEN `~(FVT t = {})` MP_TAC THENL
   [ASM_MESON_TAC[FUNCTIONS_TERM_NOCONSTANTS]; ALL_TAC] THEN
  REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_SING] THEN
  REWRITE_TAC[IN_LIST_UNION; MEM_MAP; NOT_IN_EMPTY;
                GSYM EX_MEM] THEN ASM_MESON_TAC[]);;

let HERBASE_LEMMA = prove
 (`functions_form q SUBSET fns /\
   (!v. i(v) IN herbase fns) /\
   ~(j(x) IN herbase fns) /\
   x IN FV(p)
   ==> ~(formsubst j p = formsubst i q)`,
  REWRITE_TAC[HERBASE] THEN
  REWRITE_TAC[DE_MORGAN_THM] THEN REPEAT STRIP_TAC THENL
   [SUBGOAL_THEN `functions_form(formsubst i q) SUBSET fns /\
                  ~(functions_form(formsubst j p) SUBSET fns)`
     (fun th -> ASM_MESON_TAC[th]) THEN
    REWRITE_TAC[FORMSUBST_FUNCTIONS_FORM] THEN CONJ_TAC THENL
     [REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN
      ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
    UNDISCH_TAC `~(functions_term (j(x:num)) SUBSET fns)` THEN
    REWRITE_TAC[SUBSET] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
    MATCH_MP_TAC MONO_EXISTS THEN
    X_GEN_TAC `fn:num#num` THEN STRIP_TAC THEN
    ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[];
    ALL_TAC] THEN
  SUBGOAL_THEN `?y. y IN FVT(j(x:num)) /\ !z:num. ~(y IN FVT(i z))`
  MP_TAC THENL
   [ALL_TAC;
    DISCH_THEN(X_CHOOSE_THEN `y:num` STRIP_ASSUME_TAC) THEN
    SUBGOAL_THEN `~(FV(formsubst j p) = FV(formsubst i q))`
     (fun th -> ASM_MESON_TAC[th]) THEN
    REWRITE_TAC[EXTENSION; NOT_FORALL_THM; IN_ELIM_THM; FORMSUBST_FV] THEN
    ASM_MESON_TAC[]] THEN
  ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
   [UNDISCH_TAC
      `~(FVT(j(x:num)) = (if ?c:num. c,0 IN fns then {} else {0}))` THEN
    ASM_REWRITE_TAC[EXTENSION; IN_SING; NOT_IN_EMPTY] THEN MESON_TAC[];
    ALL_TAC] THEN
  UNDISCH_TAC
   `~(FVT(j(x:num)) = (if ?c:num. c,0 IN fns then {} else {0}))` THEN
  ASM_REWRITE_TAC[] THEN
  SUBGOAL_THEN `~(FVT(j(x:num)) = {})` MP_TAC THENL
   [ALL_TAC;
    REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_SING] THEN MESON_TAC[]] THEN
  MATCH_MP_TAC FUNCTIONS_TERM_NOCONSTANTS THEN
  SUBGOAL_THEN `functions_term(j(x:num)) SUBSET fns`
   (fun th -> ASM_MESON_TAC[th; SUBSET]) THEN
  MATCH_MP_TAC SUBSET_TRANS THEN
  EXISTS_TAC `functions_form(formsubst j p)` THEN CONJ_TAC THENL
   [REWRITE_TAC[FORMSUBST_FUNCTIONS_FORM] THEN
    REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[];
    ALL_TAC] THEN
  ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORMSUBST_FUNCTIONS_FORM; SUBSET] THEN
  REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN ASM_MESON_TAC[SUBSET]);;

let SEMRESOLUTION_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))
   ==> !M:(A->bool)#(num->A list->A)#(num->A list->bool).
           interpretation (language(IMAGE interp hyps)) M /\ ~(Dom M = {})
           ==> semresproof2 M 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 | v,cl |
                cl IN hyps /\
                (!x. v(x) IN herbase (functions (IMAGE interp 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]
                PSEMRESPROOF_REFUTATION_COMPLETE)) THEN
  REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL
   [SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
    ASM_SIMP_TAC[IMAGE_FORMSUBST_CLAUSE]; ALL_TAC] THEN
  FIRST_ASSUM(X_CHOOSE_TAC `w:num->A` o MATCH_MP VALUATION_EXISTS) THEN
  DISCH_THEN(MP_TAC o SPEC `holds M (w:num->A)`) THEN
  ABBREV_TAC
    `ghyps = {IMAGE(formsubst v) cl | v,cl |
              cl IN hyps /\
              (!x. v(x) IN herbase (functions (IMAGE interp hyps)))}` THEN
  SUBGOAL_THEN
    `!cl0. psemresproof (holds M (w:num->A)) ghyps cl0
           ==> ?cl. semresproof2 M hyps cl /\
                    ?i. (!x. i(x) IN herbase(functions(IMAGE interp hyps))) /\
                        (cl0 = IMAGE (formsubst i) 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; instance_of]] THEN
  ONCE_REWRITE_TAC[TAUT `a ==> b <=> a ==> a /\ b`] THEN
  MATCH_MP_TAC psemresproof_INDUCT THEN CONJ_TAC THENL
   [SIMP_TAC[CONJUNCT1(SPEC_ALL psemresproof_RULES)] THEN
    EXPAND_TAC "ghyps" THEN
    REWRITE_TAC[IN_IMAGE; instance_of; IN_ELIM_THM] THEN
    MESON_TAC[semresproof2_RULES]; ALL_TAC] THEN
  MAP_EVERY X_GEN_TAC [`p:form`; `A':form->bool`; `B':form->bool`] THEN
  DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC
     (X_CHOOSE_THEN `A:form->bool` MP_TAC)) MP_TAC) THEN
  DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC
     (X_CHOOSE_THEN `B:form->bool` MP_TAC)) MP_TAC) THEN
  DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  DISCH_THEN(X_CHOOSE_THEN `k1:num->term` (STRIP_ASSUME_TAC o GSYM)) THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  DISCH_THEN(X_CHOOSE_THEN `k2:num->term` (STRIP_ASSUME_TAC o GSYM)) THEN
  CONJ_TAC THENL [ASM_SIMP_TAC[psemresproof_RULES]; ALL_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; SEMRESPROOF2_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[SEMRESPROOF2_CLAUSE];
    ASM_MESON_TAC[IMAGE_FORMSUBST_CLAUSE; SEMRESPROOF2_CLAUSE];
    ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REWRITE_TAC[];
    ASM_MESON_TAC[instance_of];
    SUBGOAL_THEN `B' instance_of B` MP_TAC THENL
     [ASM_MESON_TAC[instance_of]; ALL_TAC] 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[SEMRESPROOF2_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[SEMRESPROOF2_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[SEMRESPROOF2_CLAUSE; clause; FINITE_SUBSET; FINITE_IMAGE];
      REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN
      ASM_MESON_TAC[SEMRESPROOF2_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 CONJ_TAC THENL
   [MATCH_MP_TAC(CONJUNCT2(SPEC_ALL semresproof2_RULES)) THEN
    EXISTS_TAC `B:form->bool` THEN ASM_REWRITE_TAC[] THEN
    FIRST_ASSUM(UNDISCH_TAC o check is_disj o concl) THEN
    MAP_EVERY EXPAND_TAC ["A'"; "B'"] THEN
    UNDISCH_TAC `valuation M (w:num->A)` THEN
    MATCH_MP_TAC(TAUT
     `(d ==> a ==> b) /\ (e ==> a ==> c)
      ==> a ==> ~b \/ ~c ==> ~d \/ ~e`) THEN
    CONJ_TAC THEN DISCH_TAC THEN SPEC_TAC(`w:num->A`,`w:num->A`) THEN
    MATCH_MP_TAC(GEN_ALL LIFTING_FALSITY_CLAUSE) THENL
     [MAP_EVERY EXISTS_TAC [`A:form->bool`; `k2:num->term`];
      MAP_EVERY EXISTS_TAC [`B:form->bool`; `k1:num->term`]] THEN
    (ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
      [ASM_MESON_TAC[SEMRESPROOF2_CLAUSE]; ALL_TAC] THEN
     REPEAT STRIP_TAC THEN
     FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [interpretation] o
                   REWRITE_RULE[language]) THEN
     ASM_REWRITE_TAC[])
    THENL
     [UNDISCH_TAC `f,LENGTH(l:A list) IN functions_term (k2(x:num))`;
      UNDISCH_TAC `f,LENGTH(l:A list) IN functions_term (k1(x:num))`] THEN
    SPEC_TAC(`f:num,LENGTH(l:A list)`,`fn:num#num`) THEN
    REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC HERBASE_FUNCTIONS THEN
    ASM_REWRITE_TAC[]; ALL_TAC] THEN
  UNDISCH_TAC
   `resolve p A' B' instance_of
     IMAGE (formsubst (mgu (A1 UNION {~~ l | l IN B1})))
     (A DIFF A1 UNION C DIFF B1)` THEN
  REWRITE_TAC[instance_of] THEN
  DISCH_THEN(X_CHOOSE_TAC `i:num->term`) THEN
  ABBREV_TAC `D = IMAGE (formsubst (mgu (A1 UNION {~~ l | l IN B1})))
                        (A DIFF A1 UNION C DIFF B1)` THEN
  ABBREV_TAC
   `i' = \x:num. if i(x) IN herbase (functions (IMAGE interp hyps))
                 then i(x)
                 else @x. x IN herbase (functions (IMAGE interp hyps))` THEN
  EXISTS_TAC `i':num->term` THEN CONJ_TAC THENL
   [GEN_TAC THEN EXPAND_TAC "i'" THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN
    ASM_REWRITE_TAC[] THEN CONV_TAC SELECT_CONV THEN
    REWRITE_TAC[HERBASE_NONEMPTY]; ALL_TAC] THEN
  SUBGOAL_THEN
   `!p x. p IN D /\ x IN FV(p) ==> (i'(x):term = i(x))`
  MP_TAC THENL
   [ALL_TAC;
    ASM_REWRITE_TAC[] THEN REWRITE_TAC[EXTENSION; IN_IMAGE] THEN
    MESON_TAC[FORMSUBST_VALUATION]] THEN
  SUBGOAL_THEN `!p x. p IN D /\ x IN FV(p)
                      ==> i(x) IN herbase(functions (IMAGE interp hyps))`
  MP_TAC THENL
   [ALL_TAC;
    EXPAND_TAC "i'" THEN SIMP_TAC[] THEN
    REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
    ASM_MESON_TAC[]] THEN
  SUBGOAL_THEN
   `!p. p IN D ==> ?v q. (!x. v x IN herbase(functions(IMAGE interp hyps))) /\
                         functions_form q SUBSET
                            functions(IMAGE interp hyps) /\
                         (formsubst i p = formsubst v q)`
   (fun th -> ASM_MESON_TAC[th; HERBASE_LEMMA]) THEN
  SUBGOAL_THEN
   `!p. p IN D ==> functions_form(formsubst i p) SUBSET
                      functions(IMAGE interp ghyps) /\
                   ?v q. (!x. v x IN herbase(functions(IMAGE interp hyps))) /\
                         (formsubst i p = formsubst v q)`
  MP_TAC THENL
   [X_GEN_TAC `q:form` THEN DISCH_TAC THEN
    SUBGOAL_THEN `(formsubst i q) IN resolve p A' B'` ASSUME_TAC THENL
     [ASM_MESON_TAC[EXTENSION; IN_IMAGE]; ALL_TAC] THEN
    CONJ_TAC THENL
     [ALL_TAC;
      UNDISCH_TAC `(formsubst i q) IN resolve p A' B'` THEN
      REWRITE_TAC[resolve; IN_UNION; IN_DELETE] THEN
      MAP_EVERY EXPAND_TAC ["A'"; "B'"] THEN ASM_MESON_TAC[IN_IMAGE]] THEN
    MATCH_MP_TAC SUBSET_TRANS THEN
    EXISTS_TAC `functions(resolve p A' B')` THEN CONJ_TAC THENL
     [REWRITE_TAC[functions; SUBSET; IN_UNIONS; IN_ELIM_THM] THEN
      ASM_MESON_TAC[]; ALL_TAC] THEN
    SUBGOAL_THEN `psemresproof (holds M (w:num->A)) ghyps (resolve p A' B')`
    MP_TAC THENL
     [MATCH_MP_TAC(CONJUNCT2(SPEC_ALL psemresproof_RULES)) THEN
      ASM_REWRITE_TAC[]; ALL_TAC] THEN
    SPEC_TAC(`resolve p A' B'`,`cl:form->bool`) THEN
    MATCH_MP_TAC PSEMRESPROOF_FUNCTIONS THEN
    EXPAND_TAC "ghyps" THEN REWRITE_TAC[IN_ELIM_THM] THEN
    ASM_MESON_TAC[IMAGE_FORMSUBST_CLAUSE]; ALL_TAC] THEN
  MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `q:form` THEN
  DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
  DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_THEN `ii:num->term` MP_TAC)) THEN
  DISCH_THEN(X_CHOOSE_THEN `r:form` STRIP_ASSUME_TAC) THEN DISCH_TAC THEN
  MAP_EVERY EXISTS_TAC [`ii:num->term`; `r:form`] THEN ASM_REWRITE_TAC[] THEN
  MATCH_MP_TAC SUBSET_TRANS THEN
  EXISTS_TAC `functions_form(formsubst i q)` THEN CONJ_TAC THENL
   [ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORMSUBST_FUNCTIONS_FORM] THEN
    SIMP_TAC[SUBSET; IN_UNION]; ALL_TAC] THEN
  MATCH_MP_TAC SUBSET_TRANS THEN
  EXISTS_TAC `functions(IMAGE interp ghyps)` THEN ASM_REWRITE_TAC[] THEN
  SUBGOAL_THEN
   `functions(IMAGE interp ghyps) = UNIONS {functions p | p IN ghyps}`
  SUBST1_TAC THENL
   [MATCH_MP_TAC FUNCTIONS_IMAGE_INTERP THEN
    ASM_SIMP_TAC[CLAUSE_FINITE] THEN EXPAND_TAC "ghyps" THEN
    REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
    ASM_SIMP_TAC[FINITE_IMAGE; CLAUSE_FINITE]; ALL_TAC] THEN
  REWRITE_TAC[SUBSET; IN_UNIONS; IN_ELIM_THM] THEN
  X_GEN_TAC `fn:num#num` THEN
  DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
  DISCH_THEN(X_CHOOSE_THEN `cl:form->bool`
   (CONJUNCTS_THEN2 MP_TAC SUBST_ALL_TAC)) THEN
  EXPAND_TAC "ghyps" THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE] THEN
  DISCH_THEN(X_CHOOSE_THEN `vv:num->term` MP_TAC) THEN
  DISCH_THEN(X_CHOOSE_THEN `c:form->bool` MP_TAC) THEN
  DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC SUBST_ALL_TAC) THEN
  UNDISCH_TAC `fn IN functions (IMAGE (formsubst vv) c)` THEN
  GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [functions] THEN
  REWRITE_TAC[IN_UNIONS; IN_ELIM_THM; IN_IMAGE] THEN
  DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
  DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 MP_TAC SUBST_ALL_TAC)) THEN
  DISCH_THEN(X_CHOOSE_THEN `s:form`
   (CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN
  UNDISCH_TAC `fn IN functions_form (formsubst vv s)` THEN
  REWRITE_TAC[FORMSUBST_FUNCTIONS_FORM] THEN
  REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN
  DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL
   [ASM_SIMP_TAC[FUNCTIONS_IMAGE_INTERP; CLAUSE_FINITE] THEN
    REWRITE_TAC[IN_UNIONS; functions; IN_ELIM_THM] THEN
    EXISTS_TAC `UNIONS {functions_form f | f IN c}` THEN
    CONJ_TAC THENL
     [EXISTS_TAC `c:form->bool` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
    REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN ASM_MESON_TAC[];
    ALL_TAC] THEN
  ASM_MESON_TAC[HERBASE_FUNCTIONS; SUBSET]);;