ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
ex/prop-log/hyps_thms_if: split up the fast_tac call for more speed
called expandshort
(* 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 addsimps [subset_refl, double_complement,
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
*)
(*collecting the relevant lemmas*)
val hom_ss = ZF_ss addsimps [comp_func,comp_func_apply,SigmaI,apply_funtype];
(*The problem below is proving conditions of rewrites such as comp_func_apply;
rewriting does not instantiate Vars; we must prove the conditions
explicitly.*)
fun hom_tac hyps =
resolve_tac (TrueI::refl::iff_refl::hyps) ORELSE'
(cut_facts_tac hyps THEN' fast_tac prop_cs);
(*This version uses a super application of simp_tac*)
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 setsolver hom_tac) 1);
(*Also works but slower:
by (asm_simp_tac (hom_ss setloop (K (safe_tac FOL_cs))) 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();
(** Yet another example... **)
goalw (merge_theories(Sum.thy,Perm.thy)) [bij_def,inj_def,surj_def]
"(lam Z:Pow(A+B). <{x:A. Inl(x):Z}, {y:B. Inr(y):Z}>) \
\ : bij(Pow(A+B), Pow(A)*Pow(B))";
by (DO_GOAL
[rtac IntI,
DO_GOAL
[rtac CollectI,
fast_tac (ZF_cs addSIs [lam_type]),
simp_tac ZF_ss,
fast_tac (eq_cs addSEs [sumE]
addEs [equalityD1 RS subsetD RS CollectD2,
equalityD2 RS subsetD RS CollectD2])],
DO_GOAL
[rtac CollectI,
fast_tac (ZF_cs addSIs [lam_type]),
simp_tac ZF_ss,
K(safe_tac ZF_cs),
res_inst_tac [("x", "{Inl(u). u: ?U} Un {Inr(v). v: ?V}")] bexI,
DO_GOAL
[res_inst_tac [("t", "Pair")] subst_context2,
fast_tac (sum_cs addSIs [equalityI]),
fast_tac (sum_cs addSIs [equalityI])],
DO_GOAL [fast_tac sum_cs]]] 1);
val Pow_bij = result();
writeln"Reached end of file.";