src/ZF/ex/misc.ML
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(*  Title: 	ZF/ex/misc
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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Miscellaneous examples for Zermelo-Fraenkel Set Theory 
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Cantor's Theorem; Schroeder-Bernstein Theorem; Composition of homomorphisms...
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*)
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writeln"ZF/ex/misc";
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(*Example 12 (credited to Peter Andrews) from
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 W. Bledsoe.  A Maximal Method for Set Variables in Automatic Theorem-proving.
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 In: J. Hayes and D. Michie and L. Mikulich, eds.  Machine Intelligence 9.
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 Ellis Horwood, 53-100 (1979). *)
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goal ZF.thy "(ALL F. {x}: F --> {y}:F) --> (ALL A. x:A --> y:A)";
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by (best_tac ZF_cs 1);
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result();
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(*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
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val cantor_cs = FOL_cs   (*precisely the rules needed for the proof*)
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  addSIs [ballI, CollectI, PowI, subsetI] addIs [bexI]
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  addSEs [CollectE, equalityCE];
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(*The search is undirected and similar proof attempts fail*)
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goal ZF.thy "ALL f: A->Pow(A). EX S: Pow(A). ALL x:A. f`x ~= S";
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by (best_tac cantor_cs 1);
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result();
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(*This form displays the diagonal term, {x: A . x ~: f`x} *)
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val [prem] = goal ZF.thy
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    "f: A->Pow(A) ==> (ALL x:A. f`x ~= ?S) & ?S: Pow(A)";
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by (best_tac cantor_cs 1);
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result();
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(*yet another version...*)
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goalw Perm.thy [surj_def] "f ~: surj(A,Pow(A))";
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by (safe_tac ZF_cs);
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by (etac ballE 1);
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by (best_tac (cantor_cs addSEs [bexE]) 1);
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by (fast_tac ZF_cs 1);
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result();
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(*** Composition of homomorphisms is a homomorphism ***)
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(*Given as a challenge problem in
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  R. Boyer et al.,
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  Set Theory in First-Order Logic: Clauses for G\"odel's Axioms,
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  JAR 2 (1986), 287-327 
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*)
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(*collecting the relevant lemmas*)
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val hom_ss = ZF_ss addsimps [comp_fun,comp_fun_apply,SigmaI,apply_funtype];
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(*The problem below is proving conditions of rewrites such as comp_fun_apply;
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  rewriting does not instantiate Vars; we must prove the conditions
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  explicitly.*)
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fun hom_tac hyps =
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    resolve_tac (TrueI::refl::iff_refl::hyps) ORELSE' 
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    (cut_facts_tac hyps THEN' fast_tac prop_cs);
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(*This version uses a super application of simp_tac*)
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goal Perm.thy
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    "(ALL A f B g. hom(A,f,B,g) = \
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\          {H: A->B. f:A*A->A & g:B*B->B & \
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\                    (ALL x:A. ALL y:A. H`(f`<x,y>) = g`<H`x,H`y>)}) --> \
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\    J : hom(A,f,B,g) & K : hom(B,g,C,h) -->  \
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\    (K O J) : hom(A,f,C,h)";
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by (simp_tac (hom_ss setsolver hom_tac) 1);
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(*Also works but slower:
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    by (asm_simp_tac (hom_ss setloop (K (safe_tac FOL_cs))) 1); *)
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val comp_homs = result();
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(*This version uses meta-level rewriting, safe_tac and asm_simp_tac*)
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val [hom_def] = goal Perm.thy
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    "(!! A f B g. hom(A,f,B,g) == \
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\          {H: A->B. f:A*A->A & g:B*B->B & \
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\                    (ALL x:A. ALL y:A. H`(f`<x,y>) = g`<H`x,H`y>)}) ==> \
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\    J : hom(A,f,B,g) & K : hom(B,g,C,h) -->  \
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\    (K O J) : hom(A,f,C,h)";
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by (rewtac hom_def);
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by (safe_tac ZF_cs);
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by (asm_simp_tac hom_ss 1);
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by (asm_simp_tac hom_ss 1);
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val comp_homs = result();
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(** A characterization of functions, suggested by Tobias Nipkow **)
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goalw ZF.thy [Pi_def]
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    "r: domain(r)->B  <->  r <= domain(r)*B & (ALL X. r `` (r -`` X) <= X)";
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by (safe_tac ZF_cs);
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by (fast_tac (ZF_cs addSDs [bspec RS ex1_equalsE]) 1);
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by (eres_inst_tac [("x", "{y}")] allE 1);
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by (fast_tac ZF_cs 1);
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result();
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(**** From D Pastre.  Automatic theorem proving in set theory. 
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         Artificial Intelligence, 10:1--27, 1978.
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             These examples require forward reasoning! ****)
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(*reduce the clauses to units by type checking -- beware of nontermination*)
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fun forw_typechk tyrls [] = []
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  | forw_typechk tyrls clauses =
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    let val (units, others) = partition (has_fewer_prems 1) clauses
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    in  gen_union eq_thm (units, forw_typechk tyrls (tyrls RL others))
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    end;
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(*A crude form of forward reasoning*)
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fun forw_iterate tyrls rls facts 0 = facts
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  | forw_iterate tyrls rls facts n =
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      let val facts' = 
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	  gen_union eq_thm (forw_typechk (tyrls@facts) (facts RL rls), facts);
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      in  forw_iterate tyrls rls facts' (n-1)  end;
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val pastre_rls =
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    [comp_mem_injD1, comp_mem_surjD1, comp_mem_injD2, comp_mem_surjD2];
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fun pastre_facts (fact1::fact2::fact3::prems) = 
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    forw_iterate (prems @ [comp_surj, comp_inj, comp_fun])
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               pastre_rls [fact1,fact2,fact3] 4;
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): inj(A,A);		\
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\       (f O h O g): surj(B,B); 	\
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\       (g O f O h): surj(C,C); 	\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre1 = result();
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): surj(A,A);		\
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\       (f O h O g): inj(B,B); 		\
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\       (g O f O h): surj(C,C); 	\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre2 = result();
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): surj(A,A);		\
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\       (f O h O g): surj(B,B); 	\
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\       (g O f O h): inj(C,C); 		\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre3 = result();
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): surj(A,A);		\
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\       (f O h O g): inj(B,B); 		\
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\       (g O f O h): inj(C,C); 		\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre4 = result();
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): inj(A,A);		\
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\       (f O h O g): surj(B,B); 	\
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\       (g O f O h): inj(C,C); 		\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre5 = result();
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val prems = goalw Perm.thy [bij_def]
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    "[| (h O g O f): inj(A,A);		\
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\       (f O h O g): inj(B,B); 		\
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\       (g O f O h): surj(C,C); 	\
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\       f: A->B;  g: B->C;  h: C->A |] ==> h: bij(C,A)";
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by (REPEAT (resolve_tac (IntI :: pastre_facts prems) 1));
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val pastre6 = result();
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7
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(** Yet another example... **)
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goalw (merge_theories(Sum.thy,Perm.thy)) [bij_def,inj_def,surj_def]
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    "(lam Z:Pow(A+B). <{x:A. Inl(x):Z}, {y:B. Inr(y):Z}>) \
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\    : bij(Pow(A+B), Pow(A)*Pow(B))";
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by (DO_GOAL
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      [rtac IntI,
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       DO_GOAL
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	 [rtac CollectI,
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	  fast_tac (ZF_cs addSIs [lam_type]),
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	  simp_tac ZF_ss,
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	  fast_tac (eq_cs addSEs [sumE]
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			  addEs  [equalityD1 RS subsetD RS CollectD2,
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				  equalityD2 RS subsetD RS CollectD2])],
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       DO_GOAL
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	 [rtac CollectI,
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	  fast_tac (ZF_cs addSIs [lam_type]),
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	  simp_tac ZF_ss,
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	  K(safe_tac ZF_cs),
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	  res_inst_tac [("x", "{Inl(u). u: ?U} Un {Inr(v). v: ?V}")] bexI,
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	  DO_GOAL
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	    [res_inst_tac [("t", "Pair")] subst_context2,
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	    fast_tac (sum_cs addSIs [equalityI]),
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	    fast_tac (sum_cs addSIs [equalityI])],
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	  DO_GOAL [fast_tac sum_cs]]] 1);
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val Pow_bij = result();
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0
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writeln"Reached end of file.";