--- a/src/FOL/IFOL_lemmas.ML Sun Nov 26 23:09:25 2006 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,451 +0,0 @@
-(* Title: FOL/IFOL_lemmas.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1992 University of Cambridge
-
-Tactics and lemmas for theory IFOL (intuitionistic first-order logic).
-*)
-
-(* ML bindings *)
-
-val refl = thm "refl";
-val subst = thm "subst";
-val conjI = thm "conjI";
-val conjunct1 = thm "conjunct1";
-val conjunct2 = thm "conjunct2";
-val disjI1 = thm "disjI1";
-val disjI2 = thm "disjI2";
-val disjE = thm "disjE";
-val impI = thm "impI";
-val mp = thm "mp";
-val FalseE = thm "FalseE";
-val True_def = thm "True_def";
-val not_def = thm "not_def";
-val iff_def = thm "iff_def";
-val ex1_def = thm "ex1_def";
-val allI = thm "allI";
-val spec = thm "spec";
-val exI = thm "exI";
-val exE = thm "exE";
-val eq_reflection = thm "eq_reflection";
-val iff_reflection = thm "iff_reflection";
-
-structure IFOL =
-struct
- val thy = the_context ();
- val refl = refl;
- val subst = subst;
- val conjI = conjI;
- val conjunct1 = conjunct1;
- val conjunct2 = conjunct2;
- val disjI1 = disjI1;
- val disjI2 = disjI2;
- val disjE = disjE;
- val impI = impI;
- val mp = mp;
- val FalseE = FalseE;
- val True_def = True_def;
- val not_def = not_def;
- val iff_def = iff_def;
- val ex1_def = ex1_def;
- val allI = allI;
- val spec = spec;
- val exI = exI;
- val exE = exE;
- val eq_reflection = eq_reflection;
- val iff_reflection = iff_reflection;
-end;
-
-
-Goalw [True_def] "True";
-by (REPEAT (ares_tac [impI] 1)) ;
-qed "TrueI";
-
-(*** Sequent-style elimination rules for & --> and ALL ***)
-
-val major::prems = Goal
- "[| P&Q; [| P; Q |] ==> R |] ==> R";
-by (resolve_tac prems 1);
-by (rtac (major RS conjunct1) 1);
-by (rtac (major RS conjunct2) 1);
-qed "conjE";
-
-val major::prems = Goal
- "[| P-->Q; P; Q ==> R |] ==> R";
-by (resolve_tac prems 1);
-by (rtac (major RS mp) 1);
-by (resolve_tac prems 1);
-qed "impE";
-
-val major::prems = Goal
- "[| ALL x. P(x); P(x) ==> R |] ==> R";
-by (resolve_tac prems 1);
-by (rtac (major RS spec) 1);
-qed "allE";
-
-(*Duplicates the quantifier; for use with eresolve_tac*)
-val major::prems = Goal
- "[| ALL x. P(x); [| P(x); ALL x. P(x) |] ==> R \
-\ |] ==> R";
-by (resolve_tac prems 1);
-by (rtac (major RS spec) 1);
-by (rtac major 1);
-qed "all_dupE";
-
-
-(*** Negation rules, which translate between ~P and P-->False ***)
-
-val prems = Goalw [not_def] "(P ==> False) ==> ~P";
-by (REPEAT (ares_tac (prems@[impI]) 1)) ;
-qed "notI";
-
-Goalw [not_def] "[| ~P; P |] ==> R";
-by (etac (mp RS FalseE) 1);
-by (assume_tac 1);
-qed "notE";
-
-Goal "[| P; ~P |] ==> R";
-by (etac notE 1);
-by (assume_tac 1);
-qed "rev_notE";
-
-(*This is useful with the special implication rules for each kind of P. *)
-val prems = Goal
- "[| ~P; (P-->False) ==> Q |] ==> Q";
-by (REPEAT (ares_tac (prems@[impI,notE]) 1)) ;
-qed "not_to_imp";
-
-(* For substitution into 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 |] ==> Q";
-by (etac mp 1);
-by (assume_tac 1);
-qed "rev_mp";
-
-(*Contrapositive of an inference rule*)
-val [major,minor]= Goal "[| ~Q; P==>Q |] ==> ~P";
-by (rtac (major RS notE RS notI) 1);
-by (etac minor 1) ;
-qed "contrapos";
-
-
-(*** Modus Ponens Tactics ***)
-
-(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
-fun mp_tac i = eresolve_tac [notE,impE] i THEN assume_tac i;
-
-(*Like mp_tac but instantiates no variables*)
-fun eq_mp_tac i = eresolve_tac [notE,impE] i THEN eq_assume_tac i;
-
-
-(*** If-and-only-if ***)
-
-val prems = Goalw [iff_def]
- "[| P ==> Q; Q ==> P |] ==> P<->Q";
-by (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ;
-qed "iffI";
-
-
-(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
-val prems = Goalw [iff_def]
- "[| P <-> Q; [| P-->Q; Q-->P |] ==> R |] ==> R";
-by (rtac conjE 1);
-by (REPEAT (ares_tac prems 1)) ;
-qed "iffE";
-
-(* Destruct rules for <-> similar to Modus Ponens *)
-
-Goalw [iff_def] "[| P <-> Q; P |] ==> Q";
-by (etac (conjunct1 RS mp) 1);
-by (assume_tac 1);
-qed "iffD1";
-
-val prems = Goalw [iff_def] "[| P <-> Q; Q |] ==> P";
-by (etac (conjunct2 RS mp) 1);
-by (assume_tac 1);
-qed "iffD2";
-
-Goal "[| P; P <-> Q |] ==> Q";
-by (etac iffD1 1);
-by (assume_tac 1);
-qed "rev_iffD1";
-
-Goal "[| Q; P <-> Q |] ==> P";
-by (etac iffD2 1);
-by (assume_tac 1);
-qed "rev_iffD2";
-
-Goal "P <-> P";
-by (REPEAT (ares_tac [iffI] 1)) ;
-qed "iff_refl";
-
-Goal "Q <-> P ==> P <-> Q";
-by (etac iffE 1);
-by (rtac iffI 1);
-by (REPEAT (eresolve_tac [asm_rl,mp] 1)) ;
-qed "iff_sym";
-
-Goal "[| P <-> Q; Q<-> R |] ==> 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 [ex1_def]
- "[| P(a); !!x. P(x) ==> x=a |] ==> EX! x. P(x)";
-by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ;
-qed "ex1I";
-
-(*Sometimes easier to use: the premises have no shared variables. Safe!*)
-val [ex,eq] = Goal
- "[| EX x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)";
-by (rtac (ex RS exE) 1);
-by (REPEAT (ares_tac [ex1I,eq] 1)) ;
-qed "ex_ex1I";
-
-val prems = Goalw [ex1_def]
- "[| EX! x. P(x); !!x. [| P(x); ALL y. P(y) --> y=x |] ==> R |] ==> 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 prems = Goal
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')";
-by (cut_facts_tac prems 1);
-by (REPEAT (ares_tac [iffI,conjI] 1
- ORELSE eresolve_tac [iffE,conjE,mp] 1
- ORELSE iff_tac prems 1)) ;
-qed "conj_cong";
-
-(*Reversed congruence rule! Used in ZF/Order*)
-val prems = Goal
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')";
-by (cut_facts_tac prems 1);
-by (REPEAT (ares_tac [iffI,conjI] 1
- ORELSE eresolve_tac [iffE,conjE,mp] 1 ORELSE iff_tac prems 1)) ;
-qed "conj_cong2";
-
-val prems = Goal
- "[| P <-> P'; Q <-> Q' |] ==> (P|Q) <-> (P'|Q')";
-by (cut_facts_tac prems 1);
-by (REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
- ORELSE ares_tac [iffI] 1 ORELSE mp_tac 1)) ;
-qed "disj_cong";
-
-val prems = Goal
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')";
-by (cut_facts_tac prems 1);
-by (REPEAT (ares_tac [iffI,impI] 1
- ORELSE etac iffE 1 ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ;
-qed "imp_cong";
-
-val prems = Goal
- "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')";
-by (cut_facts_tac prems 1);
-by (REPEAT (etac iffE 1 ORELSE ares_tac [iffI] 1 ORELSE mp_tac 1)) ;
-qed "iff_cong";
-
-val prems = Goal
- "P <-> P' ==> ~P <-> ~P'";
-by (cut_facts_tac prems 1);
-by (REPEAT (ares_tac [iffI,notI] 1
- ORELSE mp_tac 1 ORELSE eresolve_tac [iffE,notE] 1)) ;
-qed "not_cong";
-
-val prems = Goal
- "(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))";
-by (REPEAT (ares_tac [iffI,allI] 1
- ORELSE mp_tac 1 ORELSE etac allE 1 ORELSE iff_tac prems 1)) ;
-qed "all_cong";
-
-val prems = Goal
- "(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))";
-by (REPEAT (etac exE 1 ORELSE ares_tac [iffI,exI] 1
- ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ;
-qed "ex_cong";
-
-val prems = Goal
- "(!!x. P(x) <-> Q(x)) ==> (EX! x. P(x)) <-> (EX! x. Q(x))";
-by (REPEAT (eresolve_tac [ex1E, spec RS mp] 1
- ORELSE ares_tac [iffI,ex1I] 1 ORELSE mp_tac 1
- ORELSE iff_tac prems 1)) ;
-qed "ex1_cong";
-
-(*** Equality rules ***)
-
-Goal "a=b ==> b=a";
-by (etac subst 1);
-by (rtac refl 1) ;
-qed "sym";
-
-Goal "[| a=b; b=c |] ==> a=c";
-by (etac subst 1 THEN assume_tac 1) ;
-qed "trans";
-
-(** ~ b=a ==> ~ a=b **)
-bind_thm ("not_sym", hd (compose(sym,2,contrapos)));
-
-
-(* Two theorms for rewriting only one instance of a definition:
- the first for definitions of formulae and the second for terms *)
-
-val prems = goal (the_context()) "(A == B) ==> A <-> B";
-by (rewrite_goals_tac prems);
-by (rtac iff_refl 1);
-qed "def_imp_iff";
-
-val prems = goal (the_context()) "(A == B) ==> A = B";
-by (rewrite_goals_tac prems);
-by (rtac refl 1);
-qed "meta_eq_to_obj_eq";
-
-(*substitution*)
-bind_thm ("ssubst", sym RS subst);
-
-(*A special case of ex1E that would otherwise need quantifier expansion*)
-val prems = Goal
- "[| EX! x. P(x); P(a); P(b) |] ==> a=b";
-by (cut_facts_tac prems 1);
-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 "[| a=b |] ==> t(a)=t(b)";
-by (etac ssubst 1);
-by (rtac refl 1) ;
-qed "subst_context";
-
-Goal "[| a=b; c=d |] ==> t(a,c)=t(b,d)";
-by (REPEAT (etac ssubst 1));
-by (rtac refl 1) ;
-qed "subst_context2";
-
-Goal "[| a=b; c=d; e=f |] ==> 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 "[| a=b; a=c; b=d |] ==> 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 "[| a=c; b=d; c=d |] ==> a=b";
-by (rtac trans 1);
-by (rtac trans 1);
-by (REPEAT (assume_tac 1));
-by (etac sym 1);
-qed "simp_equals";
-
-(** Congruence rules for predicate letters **)
-
-Goal "a=a' ==> P(a) <-> P(a')";
-by (rtac iffI 1);
-by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
-qed "pred1_cong";
-
-Goal "[| a=a'; b=b' |] ==> P(a,b) <-> P(a',b')";
-by (rtac iffI 1);
-by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
-qed "pred2_cong";
-
-Goal "[| a=a'; b=b'; c=c' |] ==> 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!*)
-bind_thm ("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&Q)-->S; P-->(Q-->S) ==> R |] ==> R";
-by (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ;
-qed "conj_impE";
-
-val major::prems= Goal
- "[| (P|Q)-->S; [| P-->S; Q-->S |] ==> R |] ==> 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-->Q)-->S; [| P; Q-->S |] ==> Q; S ==> R |] ==> 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 --> S; P ==> False; S ==> R |] ==> R";
-by (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ;
-qed "not_impE";
-
-(*Simplifies the implication. UNSAFE. *)
-val major::prems= Goal
- "[| (P<->Q)-->S; [| P; Q-->S |] ==> Q; [| Q; P-->S |] ==> P; \
-\ S ==> R |] ==> 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
- "[| (ALL x. P(x))-->S; !!x. P(x); S ==> R |] ==> 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
- "[| (EX x. P(x))-->S; P(x)-->S ==> R |] ==> R";
-by (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ;
-qed "ex_impE";
-
-(*** Courtesy of Krzysztof Grabczewski ***)
-
-val major::prems = Goal "[| P|Q; P==>R; Q==>S |] ==> R|S";
-by (rtac (major RS disjE) 1);
-by (REPEAT (eresolve_tac (prems RL [disjI1, disjI2]) 1));
-qed "disj_imp_disj";