src/FOL/IFOL.ML
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(*  Title: 	FOL/IFOL.ML
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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Tactics and lemmas for IFOL.thy (intuitionistic first-order logic)
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*)
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open IFOL;
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qed_goalw "TrueI" IFOL.thy [True_def] "True"
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 (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
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(*** Sequent-style elimination rules for & --> and ALL ***)
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qed_goal "conjE" IFOL.thy 
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    "[| P&Q; [| P; Q |] ==> R |] ==> R"
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 (fn prems=>
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  [ (REPEAT (resolve_tac prems 1
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      ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
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              resolve_tac prems 1))) ]);
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qed_goal "impE" IFOL.thy 
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    "[| P-->Q;  P;  Q ==> R |] ==> R"
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 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
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qed_goal "allE" IFOL.thy 
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    "[| ALL x.P(x); P(x) ==> R |] ==> R"
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 (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
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(*Duplicates the quantifier; for use with eresolve_tac*)
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qed_goal "all_dupE" IFOL.thy 
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    "[| ALL x.P(x);  [| P(x); ALL x.P(x) |] ==> R \
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\    |] ==> R"
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 (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
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(*** Negation rules, which translate between ~P and P-->False ***)
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qed_goalw "notI" IFOL.thy [not_def] "(P ==> False) ==> ~P"
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 (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
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qed_goalw "notE" IFOL.thy [not_def] "[| ~P;  P |] ==> R"
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 (fn prems=>
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  [ (resolve_tac [mp RS FalseE] 1),
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    (REPEAT (resolve_tac prems 1)) ]);
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(*This is useful with the special implication rules for each kind of P. *)
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qed_goal "not_to_imp" IFOL.thy 
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    "[| ~P;  (P-->False) ==> Q |] ==> Q"
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 (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
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(* For substitution into an assumption P, reduce Q to P-->Q, substitute into
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   this implication, then apply impI to move P back into the assumptions.
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   To specify P use something like
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      eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
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qed_goal "rev_mp" IFOL.thy "[| P;  P --> Q |] ==> Q"
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 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
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(*Contrapositive of an inference rule*)
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qed_goal "contrapos" IFOL.thy "[| ~Q;  P==>Q |] ==> ~P"
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 (fn [major,minor]=> 
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  [ (rtac (major RS notE RS notI) 1), 
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    (etac minor 1) ]);
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(*** Modus Ponens Tactics ***)
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(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
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fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i;
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(*Like mp_tac but instantiates no variables*)
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fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i;
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(*** If-and-only-if ***)
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qed_goalw "iffI" IFOL.thy [iff_def]
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   "[| P ==> Q;  Q ==> P |] ==> P<->Q"
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 (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
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(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
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qed_goalw "iffE" IFOL.thy [iff_def]
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    "[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
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 (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
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(* Destruct rules for <-> similar to Modus Ponens *)
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qed_goalw "iffD1" IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
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 (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
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qed_goalw "iffD2" IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
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 (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
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qed_goal "iff_refl" IFOL.thy "P <-> P"
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 (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
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qed_goal "iff_sym" IFOL.thy "Q <-> P ==> P <-> Q"
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 (fn [major] =>
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  [ (rtac (major RS iffE) 1),
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    (rtac iffI 1),
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    (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
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qed_goal "iff_trans" IFOL.thy
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    "!!P Q R. [| P <-> Q;  Q<-> R |] ==> P <-> R"
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 (fn _ =>
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  [ (rtac iffI 1),
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    (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
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(*** Unique existence.  NOTE THAT the following 2 quantifications
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   EX!x such that [EX!y such that P(x,y)]     (sequential)
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   EX!x,y such that P(x,y)                    (simultaneous)
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 do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
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***)
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qed_goalw "ex1I" IFOL.thy [ex1_def]
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    "[| P(a);  !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
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 (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
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(*Sometimes easier to use: the premises have no shared variables*)
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qed_goal "ex_ex1I" IFOL.thy
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    "[| EX x.P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"
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 (fn [ex,eq] => [ (rtac (ex RS exE) 1),
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		  (REPEAT (ares_tac [ex1I,eq] 1)) ]);
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qed_goalw "ex1E" IFOL.thy [ex1_def]
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    "[| EX! x.P(x);  !!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R |] ==> R"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
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(*** <-> congruence rules for simplification ***)
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(*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
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fun iff_tac prems i =
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    resolve_tac (prems RL [iffE]) i THEN
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    REPEAT1 (eresolve_tac [asm_rl,mp] i);
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qed_goal "conj_cong" IFOL.thy 
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    "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT  (ares_tac [iffI,conjI] 1
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      ORELSE  eresolve_tac [iffE,conjE,mp] 1
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      ORELSE  iff_tac prems 1)) ]);
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(*Reversed congruence rule!   Used in ZF/Order*)
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qed_goal "conj_cong2" IFOL.thy 
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    "[| P <-> P';  P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT  (ares_tac [iffI,conjI] 1
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      ORELSE  eresolve_tac [iffE,conjE,mp] 1
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      ORELSE  iff_tac prems 1)) ]);
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qed_goal "disj_cong" IFOL.thy 
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    "[| P <-> P';  Q <-> Q' |] ==> (P|Q) <-> (P'|Q')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT  (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
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      ORELSE  ares_tac [iffI] 1
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      ORELSE  mp_tac 1)) ]);
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qed_goal "imp_cong" IFOL.thy 
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    "[| P <-> P';  P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT   (ares_tac [iffI,impI] 1
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      ORELSE  eresolve_tac [iffE] 1
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      ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);
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qed_goal "iff_cong" IFOL.thy 
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    "[| P <-> P';  Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT   (eresolve_tac [iffE] 1
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      ORELSE  ares_tac [iffI] 1
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      ORELSE  mp_tac 1)) ]);
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qed_goal "not_cong" IFOL.thy 
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    "P <-> P' ==> ~P <-> ~P'"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (REPEAT   (ares_tac [iffI,notI] 1
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      ORELSE  mp_tac 1
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      ORELSE  eresolve_tac [iffE,notE] 1)) ]);
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qed_goal "all_cong" IFOL.thy 
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    "(!!x.P(x) <-> Q(x)) ==> (ALL x.P(x)) <-> (ALL x.Q(x))"
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 (fn prems =>
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  [ (REPEAT   (ares_tac [iffI,allI] 1
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      ORELSE   mp_tac 1
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      ORELSE   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
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qed_goal "ex_cong" IFOL.thy 
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    "(!!x.P(x) <-> Q(x)) ==> (EX x.P(x)) <-> (EX x.Q(x))"
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 (fn prems =>
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  [ (REPEAT   (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
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      ORELSE   mp_tac 1
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      ORELSE   iff_tac prems 1)) ]);
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qed_goal "ex1_cong" IFOL.thy 
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    "(!!x.P(x) <-> Q(x)) ==> (EX! x.P(x)) <-> (EX! x.Q(x))"
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 (fn prems =>
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  [ (REPEAT   (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
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      ORELSE   mp_tac 1
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      ORELSE   iff_tac prems 1)) ]);
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(*** Equality rules ***)
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qed_goal "sym" IFOL.thy "a=b ==> b=a"
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 (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
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qed_goal "trans" IFOL.thy "[| a=b;  b=c |] ==> a=c"
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 (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
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(** ~ b=a ==> ~ a=b **)
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val [not_sym] = compose(sym,2,contrapos);
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(*calling "standard" reduces maxidx to 0*)
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bind_thm ("ssubst", (sym RS subst));
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(*A special case of ex1E that would otherwise need quantifier expansion*)
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qed_goal "ex1_equalsE" IFOL.thy
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    "[| EX! x.P(x);  P(a);  P(b) |] ==> a=b"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (etac ex1E 1),
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    (rtac trans 1),
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    (rtac sym 2),
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    (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
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(** Polymorphic congruence rules **)
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qed_goal "subst_context" IFOL.thy 
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   "[| a=b |]  ==>  t(a)=t(b)"
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 (fn prems=>
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  [ (resolve_tac (prems RL [ssubst]) 1),
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    (resolve_tac [refl] 1) ]);
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qed_goal "subst_context2" IFOL.thy 
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   "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
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 (fn prems=>
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  [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
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qed_goal "subst_context3" IFOL.thy 
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   "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
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 (fn prems=>
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  [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
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(*Useful with eresolve_tac for proving equalties from known equalities.
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	a = b
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	|   |
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	c = d	*)
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qed_goal "box_equals" IFOL.thy
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    "[| a=b;  a=c;  b=d |] ==> c=d"  
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 (fn prems=>
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  [ (resolve_tac [trans] 1),
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    (resolve_tac [trans] 1),
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    (resolve_tac [sym] 1),
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    (REPEAT (resolve_tac prems 1)) ]);
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(*Dual of box_equals: for proving equalities backwards*)
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qed_goal "simp_equals" IFOL.thy
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    "[| a=c;  b=d;  c=d |] ==> a=b"  
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 (fn prems=>
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  [ (resolve_tac [trans] 1),
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    (resolve_tac [trans] 1),
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    (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
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(** Congruence rules for predicate letters **)
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qed_goal "pred1_cong" IFOL.thy
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    "a=a' ==> P(a) <-> P(a')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (rtac iffI 1),
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    (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
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qed_goal "pred2_cong" IFOL.thy
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    "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (rtac iffI 1),
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    (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
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qed_goal "pred3_cong" IFOL.thy
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    "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
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 (fn prems =>
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  [ (cut_facts_tac prems 1),
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    (rtac iffI 1),
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    (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
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(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
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val pred_congs = 
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    flat (map (fn c => 
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	       map (fn th => read_instantiate [("P",c)] th)
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		   [pred1_cong,pred2_cong,pred3_cong])
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	       (explode"PQRS"));
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(*special case for the equality predicate!*)
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val eq_cong = read_instantiate [("P","op =")] pred2_cong;
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(*** Simplifications of assumed implications.
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     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
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     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
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     intuitionistic propositional logic.  See
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   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
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    (preprint, University of St Andrews, 1991)  ***)
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qed_goal "conj_impE" IFOL.thy 
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    "[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
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qed_goal "disj_impE" IFOL.thy 
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    "[| (P|Q)-->S;  [| P-->S; Q-->S |] ==> R |] ==> R"
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 (fn major::prems=>
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  [ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
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(*Simplifies the implication.  Classical version is stronger. 
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  Still UNSAFE since Q must be provable -- backtracking needed.  *)
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qed_goal "imp_impE" IFOL.thy 
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    "[| (P-->Q)-->S;  [| P; Q-->S |] ==> Q;  S ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
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(*Simplifies the implication.  Classical version is stronger. 
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  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
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qed_goal "not_impE" IFOL.thy
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    "[| ~P --> S;  P ==> False;  S ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
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(*Simplifies the implication.   UNSAFE.  *)
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qed_goal "iff_impE" IFOL.thy 
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    "[| (P<->Q)-->S;  [| P; Q-->S |] ==> Q;  [| Q; P-->S |] ==> P;  \
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\       S ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
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(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
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qed_goal "all_impE" IFOL.thy 
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    "[| (ALL x.P(x))-->S;  !!x.P(x);  S ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
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(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
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qed_goal "ex_impE" IFOL.thy 
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    "[| (EX x.P(x))-->S;  P(x)-->S ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
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(*Courtesy Krzysztof Grabczewski*)
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val major::prems = goal IFOL.thy "[| P|Q;  P==>R;  Q==>S |] ==> R|S";
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br (major RS disjE) 1;
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by (REPEAT (eresolve_tac (prems RL [disjI1, disjI2]) 1));
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qed "disj_imp_disj";