(* Title: FOL/IFOL_lemmas.ML ID: $Id: IFOL_lemmas.ML,v 1.9 2005/04/07 07:25:34 wenzelm Exp $ 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";