# File IFOLP.ML

    (*  Title:      FOLP/IFOLP.ML
        ID:         $Id: IFOLP.ML,v 1.7 2005/09/18 12:25:48 wenzelm Exp $
        Author:     Martin D Coen, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    (*** Sequent-style elimination rules for & --> and ALL ***)

    val prems= Goal
        "[| p:P&Q; !!x y.[| x:P; y:Q |] ==> f(x,y):R |] ==> ?a:R";
    by (REPEAT (resolve_tac prems 1
       ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN resolve_tac prems 1))) ;
    qed "conjE";

    val prems= Goal
        "[| p:P-->Q;  q:P;  !!x. x:Q ==> r(x):R |] ==> ?p:R";
    by (REPEAT (resolve_tac (prems@[mp]) 1)) ;
    qed "impE";

    val prems= Goal
        "[| p:ALL x. P(x); !!y. y:P(x) ==> q(y):R |] ==> ?p:R";
    by (REPEAT (resolve_tac (prems@[spec]) 1)) ;
    qed "allE";

    (*Duplicates the quantifier; for use with eresolve_tac*)
    val prems= Goal
        "[| p:ALL x. P(x);  !!y z.[| y:P(x); z:ALL x. P(x) |] ==> q(y,z):R \
    \    |] ==> ?p:R";
    by (REPEAT (resolve_tac (prems@[spec]) 1)) ;
    qed "all_dupE";

    (*** Negation rules, which translate between ~P and P-->False ***)

    val notI = prove_goalw (the_context ()) [not_def]  "(!!x. x:P ==> q(x):False) ==> ?p:~P"
     (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);

    val notE = prove_goalw (the_context ()) [not_def] "[| p:~P;  q:P |] ==> ?p:R"
     (fn prems=>
      [ (resolve_tac [mp RS FalseE] 1),
        (REPEAT (resolve_tac prems 1)) ]);

    (*This is useful with the special implication rules for each kind of P. *)
    val prems= Goal
        "[| p:~P;  !!x. x:(P-->False) ==> q(x):Q |] ==> ?p:Q";
    by (REPEAT (ares_tac (prems@[impI,notE]) 1)) ;
    qed "not_to_imp";

    (* For substitution int an assumption P, reduce Q to P-->Q, substitute into
       this implication, then apply impI to move P back into the assumptions.
       To specify P use something like
          eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
    Goal "[| p:P;  q:P --> Q |] ==> ?p:Q";
    by (REPEAT (ares_tac [mp] 1)) ;
    qed "rev_mp";

    (*Contrapositive of an inference rule*)
    val [major,minor]= Goal "[| p:~Q;  !!y. y:P==>q(y):Q |] ==> ?a:~P";
    by (rtac (major RS notE RS notI) 1);
    by (etac minor 1) ;
    qed "contrapos";

    (** Unique assumption tactic.
        Ignores proof objects.
        Fails unless one assumption is equal and exactly one is unifiable
    **)

    local
        fun discard_proof (Const("Proof",_) $ P $ _) = P;
    in
    val uniq_assume_tac =
      SUBGOAL
        (fn (prem,i) =>
          let val hyps = map discard_proof (Logic.strip_assums_hyp prem)
              and concl = discard_proof (Logic.strip_assums_concl prem)
          in
              if exists (fn hyp => hyp aconv concl) hyps
              then case distinct (List.filter (fn hyp=> could_unify(hyp,concl)) hyps) of
                       [_] => assume_tac i
                     |  _  => no_tac
              else no_tac
          end);
    end;

    (*** Modus Ponens Tactics ***)

    (*Finds P-->Q and P in the assumptions, replaces implication by Q *)
    fun mp_tac i = eresolve_tac [notE,make_elim mp] i  THEN  assume_tac i;

    (*Like mp_tac but instantiates no variables*)
    fun int_uniq_mp_tac i = eresolve_tac [notE,impE] i  THEN  uniq_assume_tac i;

    (*** If-and-only-if ***)

    val iffI = prove_goalw (the_context ()) [iff_def]
       "[| !!x. x:P ==> q(x):Q;  !!x. x:Q ==> r(x):P |] ==> ?p:P<->Q"
     (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);

    (*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
    val iffE = prove_goalw (the_context ()) [iff_def]
        "[| p:P <-> Q;  !!x y.[| x:P-->Q; y:Q-->P |] ==> q(x,y):R |] ==> ?p:R"
     (fn prems => [ (rtac conjE 1), (REPEAT (ares_tac prems 1)) ]);

    (* Destruct rules for <-> similar to Modus Ponens *)

    val iffD1 = prove_goalw (the_context ()) [iff_def] "[| p:P <-> Q;  q:P |] ==> ?p:Q"
     (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

    val iffD2 = prove_goalw (the_context ()) [iff_def] "[| p:P <-> Q;  q:Q |] ==> ?p:P"
     (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

    Goal "?p:P <-> P";
    by (REPEAT (ares_tac [iffI] 1)) ;
    qed "iff_refl";

    Goal "p:Q <-> P ==> ?p:P <-> Q";
    by (etac iffE 1);
    by (rtac iffI 1);
    by (REPEAT (eresolve_tac [asm_rl,mp] 1)) ;
    qed "iff_sym";

    Goal "[| p:P <-> Q; q:Q<-> R |] ==> ?p:P <-> R";
    by (rtac iffI 1);
    by (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ;
    qed "iff_trans";

    (*** Unique existence.  NOTE THAT the following 2 quantifications
       EX!x such that [EX!y such that P(x,y)]     (sequential)
       EX!x,y such that P(x,y)                    (simultaneous)
     do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
    ***)

    val prems = goalw (the_context ()) [ex1_def]
        "[| p:P(a);  !!x u. u:P(x) ==> f(u) : x=a |] ==> ?p:EX! x. P(x)";
    by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ;
    qed "ex1I";

    val prems = goalw (the_context ()) [ex1_def]
        "[| p:EX! x. P(x);  \
    \       !!x u v. [| u:P(x);  v:ALL y. P(y) --> y=x |] ==> f(x,u,v):R |] ==>\
    \    ?a : R";
    by (cut_facts_tac prems 1);
    by (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ;
    qed "ex1E";

    (*** <-> congruence rules for simplification ***)

    (*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
    fun iff_tac prems i =
        resolve_tac (prems RL [iffE]) i THEN
        REPEAT1 (eresolve_tac [asm_rl,mp] i);

    val conj_cong = prove_goal (the_context ())
        "[| p:P <-> P';  !!x. x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P&Q) <-> (P'&Q')"
     (fn prems =>
      [ (cut_facts_tac prems 1),
        (REPEAT  (ares_tac [iffI,conjI] 1
          ORELSE  eresolve_tac [iffE,conjE,mp] 1
          ORELSE  iff_tac prems 1)) ]);

    val disj_cong = prove_goal (the_context ())
        "[| p:P <-> P';  q:Q <-> Q' |] ==> ?p:(P|Q) <-> (P'|Q')"
     (fn prems =>
      [ (cut_facts_tac prems 1),
        (REPEAT  (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
          ORELSE  ares_tac [iffI] 1
          ORELSE  mp_tac 1)) ]);

    val imp_cong = prove_goal (the_context ())
        "[| p:P <-> P';  !!x. x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P-->Q) <-> (P'-->Q')"
     (fn prems =>
      [ (cut_facts_tac prems 1),
        (REPEAT   (ares_tac [iffI,impI] 1
          ORELSE  etac iffE 1
          ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);

    val iff_cong = prove_goal (the_context ())
        "[| p:P <-> P';  q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')"
     (fn prems =>
      [ (cut_facts_tac prems 1),
        (REPEAT   (etac iffE 1
          ORELSE  ares_tac [iffI] 1
          ORELSE  mp_tac 1)) ]);

    val not_cong = prove_goal (the_context ())
        "p:P <-> P' ==> ?p:~P <-> ~P'"
     (fn prems =>
      [ (cut_facts_tac prems 1),
        (REPEAT   (ares_tac [iffI,notI] 1
          ORELSE  mp_tac 1
          ORELSE  eresolve_tac [iffE,notE] 1)) ]);

    val all_cong = prove_goal (the_context ())
        "(!!x. f(x):P(x) <-> Q(x)) ==> ?p:(ALL x. P(x)) <-> (ALL x. Q(x))"
     (fn prems =>
      [ (REPEAT   (ares_tac [iffI,allI] 1
          ORELSE   mp_tac 1
          ORELSE   etac allE 1 ORELSE iff_tac prems 1)) ]);

    val ex_cong = prove_goal (the_context ())
        "(!!x. f(x):P(x) <-> Q(x)) ==> ?p:(EX x. P(x)) <-> (EX x. Q(x))"
     (fn prems =>
      [ (REPEAT   (etac exE 1 ORELSE ares_tac [iffI,exI] 1
          ORELSE   mp_tac 1
          ORELSE   iff_tac prems 1)) ]);

    (*NOT PROVED
    val ex1_cong = prove_goal (the_context ())
        "(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX! x.P(x)) <-> (EX! x.Q(x))"
     (fn prems =>
      [ (REPEAT   (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
          ORELSE   mp_tac 1
          ORELSE   iff_tac prems 1)) ]);
    *)

    (*** Equality rules ***)

    val refl = ieqI;

    val subst = prove_goal (the_context ()) "[| p:a=b;  q:P(a) |] ==> ?p : P(b)"
     (fn [prem1,prem2] => [ rtac (prem2 RS rev_mp) 1, (rtac (prem1 RS ieqE) 1),
                            rtac impI 1, atac 1 ]);

    Goal "q:a=b ==> ?c:b=a";
    by (etac subst 1);
    by (rtac refl 1) ;
    qed "sym";

    Goal "[| p:a=b;  q:b=c |] ==> ?d:a=c";
    by (etac subst 1 THEN assume_tac 1);
    qed "trans";

    (** ~ b=a ==> ~ a=b **)
    Goal "p:~ b=a ==> ?q:~ a=b";
    by (etac contrapos 1);
    by (etac sym 1) ;
    qed "not_sym";

    (*calling "standard" reduces maxidx to 0*)
    val ssubst = standard (sym RS subst);

    (*A special case of ex1E that would otherwise need quantifier expansion*)
    Goal "[| p:EX! x. P(x);  q:P(a);  r:P(b) |] ==> ?d:a=b";
    by (etac ex1E 1);
    by (rtac trans 1);
    by (rtac sym 2);
    by (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ;
    qed "ex1_equalsE";

    (** Polymorphic congruence rules **)

    Goal "[| p:a=b |]  ==>  ?d:t(a)=t(b)";
    by (etac ssubst 1);
    by (rtac refl 1) ;
    qed "subst_context";

    Goal "[| p:a=b;  q:c=d |]  ==>  ?p:t(a,c)=t(b,d)";
    by (REPEAT (etac ssubst 1));
    by (rtac refl 1) ;
    qed "subst_context2";

    Goal "[| p:a=b;  q:c=d;  r:e=f |]  ==>  ?p:t(a,c,e)=t(b,d,f)";
    by (REPEAT (etac ssubst 1));
    by (rtac refl 1) ;
    qed "subst_context3";

    (*Useful with eresolve_tac for proving equalties from known equalities.
            a = b
            |   |
            c = d   *)
    Goal "[| p:a=b;  q:a=c;  r:b=d |] ==> ?p:c=d";
    by (rtac trans 1);
    by (rtac trans 1);
    by (rtac sym 1);
    by (REPEAT (assume_tac 1)) ;
    qed "box_equals";

    (*Dual of box_equals: for proving equalities backwards*)
    Goal "[| p:a=c;  q:b=d;  r:c=d |] ==> ?p:a=b";
    by (rtac trans 1);
    by (rtac trans 1);
    by (REPEAT (eresolve_tac [asm_rl, sym] 1)) ;
    qed "simp_equals";

    (** Congruence rules for predicate letters **)

    Goal "p:a=a' ==> ?p:P(a) <-> P(a')";
    by (rtac iffI 1);
    by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
    qed "pred1_cong";

    Goal "[| p:a=a';  q:b=b' |] ==> ?p:P(a,b) <-> P(a',b')";
    by (rtac iffI 1);
    by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
    qed "pred2_cong";

    Goal "[| p:a=a';  q:b=b';  r:c=c' |] ==> ?p:P(a,b,c) <-> P(a',b',c')";
    by (rtac iffI 1);
    by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
    qed "pred3_cong";

    (*special cases for free variables P, Q, R, S -- up to 3 arguments*)

    val pred_congs =
        List.concat (map (fn c =>
                   map (fn th => read_instantiate [("P",c)] th)
                       [pred1_cong,pred2_cong,pred3_cong])
                   (explode"PQRS"));

    (*special case for the equality predicate!*)
    val eq_cong = read_instantiate [("P","op =")] pred2_cong;

    (*** Simplifications of assumed implications.
         Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
         used with mp_tac (restricted to atomic formulae) is COMPLETE for
         intuitionistic propositional logic.  See
       R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
        (preprint, University of St Andrews, 1991)  ***)

    val major::prems= Goal
        "[| p:(P&Q)-->S;  !!x. x:P-->(Q-->S) ==> q(x):R |] ==> ?p:R";
    by (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ;
    qed "conj_impE";

    val major::prems= Goal
        "[| p:(P|Q)-->S;  !!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R |] ==> ?p:R";
    by (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ;
    qed "disj_impE";

    (*Simplifies the implication.  Classical version is stronger.
      Still UNSAFE since Q must be provable -- backtracking needed.  *)
    val major::prems= Goal
        "[| p:(P-->Q)-->S;  !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q;  !!x. x:S ==> r(x):R |] ==> \
    \    ?p:R";
    by (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ;
    qed "imp_impE";

    (*Simplifies the implication.  Classical version is stronger.
      Still UNSAFE since ~P must be provable -- backtracking needed.  *)
    val major::prems= Goal
        "[| p:~P --> S;  !!y. y:P ==> q(y):False;  !!y. y:S ==> r(y):R |] ==> ?p:R";
    by (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ;
    qed "not_impE";

    (*Simplifies the implication.   UNSAFE.  *)
    val major::prems= Goal
        "[| p:(P<->Q)-->S;  !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q;  \
    \       !!x y.[| x:Q; y:P-->S |] ==> r(x,y):P;  !!x. x:S ==> s(x):R |] ==> ?p:R";
    by (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ;
    qed "iff_impE";

    (*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
    val major::prems= Goal
        "[| p:(ALL x. P(x))-->S;  !!x. q:P(x);  !!y. y:S ==> r(y):R |] ==> ?p:R";
    by (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ;
    qed "all_impE";

    (*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
    val major::prems= Goal
        "[| p:(EX x. P(x))-->S;  !!y. y:P(a)-->S ==> q(y):R |] ==> ?p:R";
    by (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ;
    qed "ex_impE";