--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/ex/misc.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,213 @@
+(* Title: ZF/ex/misc
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+Miscellaneous examples for Zermelo-Fraenkel Set Theory
+Cantor's Theorem; Schroeder-Bernstein Theorem; Composition of homomorphisms...
+*)
+
+writeln"ZF/ex/misc";
+
+
+(*Example 12 (credited to Peter Andrews) from
+ W. Bledsoe. A Maximal Method for Set Variables in Automatic Theorem-proving.
+ In: J. Hayes and D. Michie and L. Mikulich, eds. Machine Intelligence 9.
+ Ellis Horwood, 53-100 (1979). *)
+goal ZF.thy "(ALL F. {x}: F --> {y}:F) --> (ALL A. x:A --> y:A)";
+by (best_tac ZF_cs 1);
+result();
+
+
+(*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
+
+val cantor_cs = FOL_cs (*precisely the rules needed for the proof*)
+ addSIs [ballI, CollectI, PowI, subsetI] addIs [bexI]
+ addSEs [CollectE, equalityCE];
+
+(*The search is undirected and similar proof attempts fail*)
+goal ZF.thy "ALL f: A->Pow(A). EX S: Pow(A). ALL x:A. ~ f`x = S";
+by (best_tac cantor_cs 1);
+result();
+
+(*This form displays the diagonal term, {x: A . ~ x: f`x} *)
+val [prem] = goal ZF.thy
+ "f: A->Pow(A) ==> (ALL x:A. ~ f`x = ?S) & ?S: Pow(A)";
+by (best_tac cantor_cs 1);
+result();
+
+(*yet another version...*)
+goalw Perm.thy [surj_def] "~ f : surj(A,Pow(A))";
+by (safe_tac ZF_cs);
+by (etac ballE 1);
+by (best_tac (cantor_cs addSEs [bexE]) 1);
+by (fast_tac ZF_cs 1);
+result();
+
+
+(**** The Schroeder-Bernstein Theorem -- see Davey & Priestly, page 106 ****)
+
+val SB_thy = merge_theories (Fixedpt.thy, Perm.thy);
+
+(** Lemma: Banach's Decomposition Theorem **)
+
+goal SB_thy "bnd_mono(X, %W. X - g``(Y - f``W))";
+by (rtac bnd_monoI 1);
+by (REPEAT (ares_tac [Diff_subset, subset_refl, Diff_mono, image_mono] 1));
+val decomp_bnd_mono = result();
+
+val [gfun] = goal SB_thy
+ "g: Y->X ==> \
+\ g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) = \
+\ X - lfp(X, %W. X - g``(Y - f``W)) ";
+by (res_inst_tac [("P", "%u. ?v = X-u")]
+ (decomp_bnd_mono RS lfp_Tarski RS ssubst) 1);
+by (SIMP_TAC (ZF_ss addrews [subset_refl, double_complement, Diff_subset,
+ gfun RS fun_is_rel RS image_subset]) 1);
+val Banach_last_equation = result();
+
+val prems = goal SB_thy
+ "[| f: X->Y; g: Y->X |] ==> \
+\ EX XA XB YA YB. (XA Int XB = 0) & (XA Un XB = X) & \
+\ (YA Int YB = 0) & (YA Un YB = Y) & \
+\ f``XA=YA & g``YB=XB";
+by (REPEAT
+ (FIRSTGOAL
+ (resolve_tac [refl, exI, conjI, Diff_disjoint, Diff_partition])));
+by (rtac Banach_last_equation 3);
+by (REPEAT (resolve_tac (prems@[fun_is_rel, image_subset, lfp_subset]) 1));
+val decomposition = result();
+
+val prems = goal SB_thy
+ "[| f: inj(X,Y); g: inj(Y,X) |] ==> EX h. h: bij(X,Y)";
+by (cut_facts_tac prems 1);
+by (cut_facts_tac [(prems RL [inj_is_fun]) MRS decomposition] 1);
+by (fast_tac (ZF_cs addSIs [restrict_bij,bij_disjoint_Un]
+ addIs [bij_converse_bij]) 1);
+(* The instantiation of exI to "restrict(f,XA) Un converse(restrict(g,YB))"
+ is forced by the context!! *)
+val schroeder_bernstein = result();
+
+
+(*** Composition of homomorphisms is a homomorphism ***)
+
+(*Given as a challenge problem in
+ R. Boyer et al.,
+ Set Theory in First-Order Logic: Clauses for G\"odel's Axioms,
+ JAR 2 (1986), 287-327
+*)
+
+val hom_ss = (*collecting the relevant lemmas*)
+ ZF_ss addrews [comp_func,comp_func_apply,SigmaI,apply_type]
+ addcongs (mk_congs Perm.thy ["op O"]);
+
+(*This version uses a super application of SIMP_TAC; it is SLOW
+ Expressing the goal by --> instead of ==> would make it slower still*)
+val [hom_eq] = goal Perm.thy
+ "(ALL A f B g. hom(A,f,B,g) = \
+\ {H: A->B. f:A*A->A & g:B*B->B & \
+\ (ALL x:A. ALL y:A. H`(f`<x,y>) = g`<H`x,H`y>)}) ==> \
+\ J : hom(A,f,B,g) & K : hom(B,g,C,h) --> \
+\ (K O J) : hom(A,f,C,h)";
+by (SIMP_TAC (hom_ss setauto K(fast_tac prop_cs) addrews [hom_eq]) 1);
+val comp_homs = result();
+
+(*This version uses meta-level rewriting, safe_tac and ASM_SIMP_TAC*)
+val [hom_def] = goal Perm.thy
+ "(!! A f B g. hom(A,f,B,g) == \
+\ {H: A->B. f:A*A->A & g:B*B->B & \
+\ (ALL x:A. ALL y:A. H`(f`<x,y>) = g`<H`x,H`y>)}) ==> \
+\ J : hom(A,f,B,g) & K : hom(B,g,C,h) --> \
+\ (K O J) : hom(A,f,C,h)";
+by (rewtac hom_def);
+by (safe_tac ZF_cs);
+by (ASM_SIMP_TAC hom_ss 1);
+by (ASM_SIMP_TAC hom_ss 1);
+val comp_homs = result();
+
+
+(** A characterization of functions, suggested by Tobias Nipkow **)
+
+goalw ZF.thy [Pi_def]
+ "r: domain(r)->B <-> r <= domain(r)*B & (ALL X. r `` (r -`` X) <= X)";
+by (safe_tac ZF_cs);
+by (fast_tac (ZF_cs addSDs [bspec RS ex1_equalsE]) 1);
+by (eres_inst_tac [("x", "{y}")] allE 1);
+by (fast_tac ZF_cs 1);
+result();
+
+
+(**** From D Pastre. Automatic theorem proving in set theory.
+ Artificial Intelligence, 10:1--27, 1978.
+ These examples require forward reasoning! ****)
+
+(*reduce the clauses to units by type checking -- beware of nontermination*)
+fun forw_typechk tyrls [] = []
+ | forw_typechk tyrls clauses =
+ let val (units, others) = partition (has_fewer_prems 1) clauses
+ in gen_union eq_thm (units, forw_typechk tyrls (tyrls RL others))
+ end;
+
+(*A crude form of forward reasoning*)
+fun forw_iterate tyrls rls facts 0 = facts
+ | forw_iterate tyrls rls facts n =
+ let val facts' =
+ gen_union eq_thm (forw_typechk (tyrls@facts) (facts RL rls), facts);
+ in forw_iterate tyrls rls facts' (n-1) end;
+
+val pastre_rls =
+ [comp_mem_injD1, comp_mem_surjD1, comp_mem_injD2, comp_mem_surjD2];
+
+fun pastre_facts (fact1::fact2::fact3::prems) =
+ forw_iterate (prems @ [comp_surj, comp_inj, comp_func])
+ pastre_rls [fact1,fact2,fact3] 4;
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): inj(A,A); \
+\ (f O h O g): surj(B,B); \
+\ (g O f O h): surj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre1 = result();
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): surj(A,A); \
+\ (f O h O g): inj(B,B); \
+\ (g O f O h): surj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre2 = result();
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): surj(A,A); \
+\ (f O h O g): surj(B,B); \
+\ (g O f O h): inj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre3 = result();
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): surj(A,A); \
+\ (f O h O g): inj(B,B); \
+\ (g O f O h): inj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre4 = result();
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): inj(A,A); \
+\ (f O h O g): surj(B,B); \
+\ (g O f O h): inj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre5 = result();
+
+val prems = goalw Perm.thy [bij_def]
+ "[| (h O g O f): inj(A,A); \
+\ (f O h O g): inj(B,B); \
+\ (g O f O h): surj(C,C); \
+\ f: A->B; g: B->C; h: C->A |] ==> h: bij(C,A)";
+by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
+val pastre6 = result();
+
+writeln"Reached end of file.";