src/HOL/HOL.ML
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(*  Title:      HOL/HOL.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1991  University of Cambridge
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For HOL.thy
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Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
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*)
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open HOL;
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(** Equality **)
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section "=";
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qed_goal "sym" HOL.thy "s=t ==> t=s"
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 (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
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(*calling "standard" reduces maxidx to 0*)
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bind_thm ("ssubst", (sym RS subst));
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qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
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 (fn prems =>
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        [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
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val prems = goal thy "(A == B) ==> A = B";
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by (rewrite_goals_tac prems);
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by (rtac refl 1);
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qed "def_imp_eq";
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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" HOL.thy
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    "[| a=b;  a=c;  b=d |] ==> c=d"  
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 (fn prems=>
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  [ (rtac trans 1),
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    (rtac trans 1),
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    (rtac sym 1),
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    (REPEAT (resolve_tac prems 1)) ]);
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(** Congruence rules for meta-application **)
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section "Congruence";
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(*similar to AP_THM in Gordon's HOL*)
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qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
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  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
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(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
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qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
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 (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
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qed_goal "cong" HOL.thy
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   "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
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 (fn [prem1,prem2] =>
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   [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
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(** Equality of booleans -- iff **)
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section "iff";
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qed_goal "iffI" HOL.thy
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   "[| P ==> Q;  Q ==> P |] ==> P=Q"
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 (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
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qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
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 (fn prems =>
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        [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
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qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
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 (fn _ => [etac iffD2 1, assume_tac 1]);
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bind_thm ("iffD1", sym RS iffD2);
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bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
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qed_goal "iffE" HOL.thy
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    "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
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 (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
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(** True **)
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section "True";
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qed_goalw "TrueI" HOL.thy [True_def] "True"
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  (fn _ => [rtac refl 1]);
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qed_goal "eqTrueI" HOL.thy "P ==> P=True" 
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 (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
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qed_goal "eqTrueE" HOL.thy "P=True ==> P" 
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 (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
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(** Universal quantifier **)
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section "!";
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qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
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 (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
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qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
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 (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
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qed_goal "allE" HOL.thy "[| !x. P(x);  P(x) ==> R |] ==> R"
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 (fn major::prems=>
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  [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
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qed_goal "all_dupE" HOL.thy 
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    "[| ! x. P(x);  [| P(x); ! x. P(x) |] ==> R |] ==> R"
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 (fn prems =>
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  [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
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(** False ** Depends upon spec; it is impossible to do propositional logic
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             before quantifiers! **)
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section "False";
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qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
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 (fn [major] => [rtac (major RS spec) 1]);
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qed_goal "False_neq_True" HOL.thy "False=True ==> P"
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 (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
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(** Negation **)
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section "~";
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qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
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 (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
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qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
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 (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
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bind_thm ("classical2", notE RS notI);
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qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
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 (fn _ => [REPEAT (ares_tac [notE] 1)]);
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(** Implication **)
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section "-->";
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qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
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 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
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(* Reduces Q to P-->Q, allowing substitution in P. *)
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qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
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 (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
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qed_goal "contrapos" HOL.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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qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
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 (fn [major,minor]=> 
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  [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
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(* ~(?t = ?s) ==> ~(?s = ?t) *)
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bind_thm("not_sym", sym COMP rev_contrapos);
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(** Existential quantifier **)
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section "?";
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qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
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 (fn prems => [rtac selectI 1, resolve_tac prems 1]);
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qed_goalw "exE" HOL.thy [Ex_def]
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  "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
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  (fn prems => [REPEAT(resolve_tac prems 1)]);
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(** Conjunction **)
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section "&";
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qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
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 (fn prems =>
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  [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
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qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
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 (fn prems =>
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   [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
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qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
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 (fn prems =>
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   [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
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qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
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 (fn prems =>
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         [cut_facts_tac prems 1, resolve_tac prems 1,
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          etac conjunct1 1, etac conjunct2 1]);
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(** Disjunction *)
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section "|";
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qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
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 (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
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qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
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 (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
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qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
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 (fn [a1,a2,a3] =>
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        [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
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         rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
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(** CCONTR -- classical logic **)
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section "classical logic";
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qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
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 (fn [prem] =>
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   [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
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    rtac (impI RS prem RS eqTrueI) 1,
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    etac subst 1,  assume_tac 1]);
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val ccontr = FalseE RS classical;
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(*Double negation law*)
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qed_goal "notnotD" HOL.thy "~~P ==> P"
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 (fn [major]=>
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  [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
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qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
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        rtac classical 1,
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        dtac p2 1,
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        etac notE 1,
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        rtac p1 1]);
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qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
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        rtac notI 1,
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        dtac p2 1,
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        etac notE 1,
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        rtac p1 1]);
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(** Unique existence **)
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section "?!";
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qed_goalw "ex1I" HOL.thy [Ex1_def]
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            "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
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 (fn prems =>
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  [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
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(*Sometimes easier to use: the premises have no shared variables.  Safe!*)
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qed_goal "ex_ex1I" HOL.thy
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    "[| ? x. P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! 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" HOL.thy [Ex1_def]
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    "[| ?! x. P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
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 (fn major::prems =>
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  [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
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Goal "?! x. P x ==> ? x. P x";
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by (etac ex1E 1);
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by (rtac exI 1);
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by (assume_tac 1);
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qed "ex1_implies_ex";
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(** Select: Hilbert's Epsilon-operator **)
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section "@";
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(*Easier to apply than selectI: conclusion has only one occurrence of P*)
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qed_goal "selectI2" HOL.thy
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    "[| P a;  !!x. P x ==> Q x |] ==> Q (@x. P x)"
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 (fn prems => [ resolve_tac prems 1, 
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                rtac selectI 1, 
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                resolve_tac prems 1 ]);
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(*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
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qed_goal "selectI2EX" HOL.thy
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  "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
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(fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
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qed_goal "select_equality" HOL.thy
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    "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a"
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 (fn prems => [ rtac selectI2 1, 
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                REPEAT (ares_tac prems 1) ]);
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qed_goalw "select1_equality" HOL.thy [Ex1_def]
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  "!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [
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	  rtac select_equality 1, atac 1,
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          etac exE 1, etac conjE 1,
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          rtac allE 1, atac 1,
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          etac impE 1, atac 1, etac ssubst 1,
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          etac allE 1, etac impE 1, atac 1, etac ssubst 1,
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          rtac refl 1]);
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qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (K [
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        rtac iffI 1,
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        etac exI 1,
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        etac exE 1,
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        etac selectI 1]);
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qed_goal "Eps_eq" HOL.thy "(Eps (op = x)) = x" (K [
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	rtac select_equality 1,
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	rtac refl 1,
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	etac sym 1]);
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(** Classical intro rules for disjunction and existential quantifiers *)
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section "classical intro rules";
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qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
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 (fn prems=>
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  [ (rtac classical 1),
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    (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
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    (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
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qed_goal "excluded_middle" HOL.thy "~P | P"
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 (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
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(*For disjunctive case analysis*)
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fun excluded_middle_tac sP =
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    res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
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(*Classical implies (-->) elimination. *)
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qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" 
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 (fn major::prems=>
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  [ rtac (excluded_middle RS disjE) 1,
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    REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
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(*This version of --> elimination works on Q before P.  It works best for
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  those cases in which P holds "almost everywhere".  Can't install as
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  default: would break old proofs.*)
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qed_goal "impCE'" thy 
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    "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R"
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 (fn major::prems=>
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  [ (resolve_tac [excluded_middle RS disjE] 1),
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    (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
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(*Classical <-> elimination. *)
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qed_goal "iffCE" HOL.thy
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    "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
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 (fn major::prems =>
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  [ (rtac (major RS iffE) 1),
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    (REPEAT (DEPTH_SOLVE_1 
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        (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
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qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)"
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 (fn prems=>
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  [ (rtac ccontr 1),
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    (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
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(* case distinction *)
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qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
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  (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
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                  etac p2 1, etac p1 1]);
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fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
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(** Standard abbreviations **)
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(*Apply an equality or definition ONCE.
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  Fails unless the substitution has an effect*)
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fun stac th = 
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  let val th' = th RS def_imp_eq handle _ => th
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  in  CHANGED_GOAL (rtac (th' RS ssubst))
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  end;
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fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 
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(** strip ! and --> from proved goal while preserving !-bound var names **)
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local
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(* Use XXX to avoid forall_intr failing because of duplicate variable name *)
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val myspec = read_instantiate [("P","?XXX")] spec;
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val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
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val cvx = cterm_of (#sign(rep_thm myspec)) vx;
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val aspec = forall_intr cvx myspec;
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in
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fun RSspec th =
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  (case concl_of th of
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     _ $ (Const("All",_) $ Abs(a,_,_)) =>
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         let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
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         in th RS forall_elim ca aspec end
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  | _ => raise THM("RSspec",0,[th]));
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fun RSmp th =
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  (case concl_of th of
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     _ $ (Const("op -->",_)$_$_) => th RS mp
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  | _ => raise THM("RSmp",0,[th]));
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fun normalize_thm funs =
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let fun trans [] th = th
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      | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
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in trans funs end;
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fun qed_spec_mp name =
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  let val thm = normalize_thm [RSspec,RSmp] (result())
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  in bind_thm(name, thm) end;
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fun qed_goal_spec_mp name thy s p = 
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	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
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fun qed_goalw_spec_mp name thy defs s p = 
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	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
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end;