--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOL.ML Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,266 @@
+(* Title: HOL/hol.ML
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1991 University of Cambridge
+
+For hol.thy
+Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68
+*)
+
+open HOL;
+
+
+(** Equality **)
+
+qed_goal "sym" HOL.thy "s=t ==> t=s"
+ (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
+
+(*calling "standard" reduces maxidx to 0*)
+bind_thm ("ssubst", (sym RS subst));
+
+qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
+ (fn prems =>
+ [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
+
+(*Useful with eresolve_tac for proving equalties from known equalities.
+ a = b
+ | |
+ c = d *)
+qed_goal "box_equals" HOL.thy
+ "[| a=b; a=c; b=d |] ==> c=d"
+ (fn prems=>
+ [ (rtac trans 1),
+ (rtac trans 1),
+ (rtac sym 1),
+ (REPEAT (resolve_tac prems 1)) ]);
+
+(** Congruence rules for meta-application **)
+
+(*similar to AP_THM in Gordon's HOL*)
+qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
+ (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
+
+(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
+qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
+ (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
+
+qed_goal "cong" HOL.thy
+ "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
+ (fn [prem1,prem2] =>
+ [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
+
+(** Equality of booleans -- iff **)
+
+qed_goal "iffI" HOL.thy
+ "[| P ==> Q; Q ==> P |] ==> P=Q"
+ (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
+
+qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
+ (fn prems =>
+ [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
+
+val iffD1 = sym RS iffD2;
+
+qed_goal "iffE" HOL.thy
+ "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
+ (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
+
+(** True **)
+
+qed_goalw "TrueI" HOL.thy [True_def] "True"
+ (fn _ => [rtac refl 1]);
+
+qed_goal "eqTrueI " HOL.thy "P ==> P=True"
+ (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
+
+qed_goal "eqTrueE" HOL.thy "P=True ==> P"
+ (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
+
+(** Universal quantifier **)
+
+qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
+ (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
+
+qed_goalw "spec" HOL.thy [All_def] "! x::'a.P(x) ==> P(x)"
+ (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
+
+qed_goal "allE" HOL.thy "[| !x.P(x); P(x) ==> R |] ==> R"
+ (fn major::prems=>
+ [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
+
+qed_goal "all_dupE" HOL.thy
+ "[| ! x.P(x); [| P(x); ! x.P(x) |] ==> R |] ==> R"
+ (fn prems =>
+ [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
+
+
+(** False ** Depends upon spec; it is impossible to do propositional logic
+ before quantifiers! **)
+
+qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
+ (fn [major] => [rtac (major RS spec) 1]);
+
+qed_goal "False_neq_True" HOL.thy "False=True ==> P"
+ (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
+
+
+(** Negation **)
+
+qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
+ (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
+
+qed_goalw "notE" HOL.thy [not_def] "[| ~P; P |] ==> R"
+ (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
+
+(** Implication **)
+
+qed_goal "impE" HOL.thy "[| P-->Q; P; Q ==> R |] ==> R"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
+
+(* Reduces Q to P-->Q, allowing substitution in P. *)
+qed_goal "rev_mp" HOL.thy "[| P; P --> Q |] ==> Q"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
+
+qed_goal "contrapos" HOL.thy "[| ~Q; P==>Q |] ==> ~P"
+ (fn [major,minor]=>
+ [ (rtac (major RS notE RS notI) 1),
+ (etac minor 1) ]);
+
+(* ~(?t = ?s) ==> ~(?s = ?t) *)
+val [not_sym] = compose(sym,2,contrapos);
+
+
+(** Existential quantifier **)
+
+qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)"
+ (fn prems => [rtac selectI 1, resolve_tac prems 1]);
+
+qed_goalw "exE" HOL.thy [Ex_def]
+ "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q"
+ (fn prems => [REPEAT(resolve_tac prems 1)]);
+
+
+(** Conjunction **)
+
+qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
+ (fn prems =>
+ [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
+
+qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
+ (fn prems =>
+ [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
+
+qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
+ (fn prems =>
+ [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
+
+qed_goal "conjE" HOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R"
+ (fn prems =>
+ [cut_facts_tac prems 1, resolve_tac prems 1,
+ etac conjunct1 1, etac conjunct2 1]);
+
+(** Disjunction *)
+
+qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
+ (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
+
+qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
+ (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
+
+qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
+ (fn [a1,a2,a3] =>
+ [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
+ rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
+
+(** CCONTR -- classical logic **)
+
+qed_goalw "classical" HOL.thy [not_def] "(~P ==> P) ==> P"
+ (fn [prem] =>
+ [rtac (True_or_False RS (disjE RS eqTrueE)) 1, assume_tac 1,
+ rtac (impI RS prem RS eqTrueI) 1,
+ etac subst 1, assume_tac 1]);
+
+val ccontr = FalseE RS classical;
+
+(*Double negation law*)
+qed_goal "notnotD" HOL.thy "~~P ==> P"
+ (fn [major]=>
+ [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
+
+
+(** Unique existence **)
+
+qed_goalw "ex1I" HOL.thy [Ex1_def]
+ "[| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
+ (fn prems =>
+ [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
+
+qed_goalw "ex1E" HOL.thy [Ex1_def]
+ "[| ?! x.P(x); !!x. [| P(x); ! y. P(y) --> y=x |] ==> R |] ==> R"
+ (fn major::prems =>
+ [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
+
+
+(** Select: Hilbert's Epsilon-operator **)
+
+(*Easier to apply than selectI: conclusion has only one occurrence of P*)
+qed_goal "selectI2" HOL.thy
+ "[| P(a); !!x. P(x) ==> Q(x) |] ==> Q(@x.P(x))"
+ (fn prems => [ resolve_tac prems 1,
+ rtac selectI 1,
+ resolve_tac prems 1 ]);
+
+qed_goal "select_equality" HOL.thy
+ "[| P(a); !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
+ (fn prems => [ rtac selectI2 1,
+ REPEAT (ares_tac prems 1) ]);
+
+
+(** Classical intro rules for disjunction and existential quantifiers *)
+
+qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
+ (fn prems=>
+ [ (rtac classical 1),
+ (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
+ (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
+
+qed_goal "excluded_middle" HOL.thy "~P | P"
+ (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
+
+(*For disjunctive case analysis*)
+fun excluded_middle_tac sP =
+ res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
+
+(*Classical implies (-->) elimination. *)
+qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
+ (fn major::prems=>
+ [ rtac (excluded_middle RS disjE) 1,
+ REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
+
+(*Classical <-> elimination. *)
+qed_goal "iffCE" HOL.thy
+ "[| P=Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R"
+ (fn major::prems =>
+ [ (rtac (major RS iffE) 1),
+ (REPEAT (DEPTH_SOLVE_1
+ (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
+
+qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
+ (fn prems=>
+ [ (rtac ccontr 1),
+ (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ]);
+
+
+(* case distinction *)
+
+qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
+ (fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1,
+ etac p2 1, etac p1 1]);
+
+fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
+
+(** Standard abbreviations **)
+
+fun stac th = rtac(th RS ssubst);
+fun sstac ths = EVERY' (map stac ths);
+fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);