enabling parallel execution of testers but removing more informative quickcheck output
(* Title: ZF/ex/misc.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Composition of homomorphisms, Pastre's examples, ...
*)
header{*Miscellaneous ZF Examples*}
theory misc imports Main begin
subsection{*Various Small Problems*}
text{*The singleton problems are much harder in HOL.*}
lemma singleton_example_1:
"\<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
by blast
lemma singleton_example_2:
"\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
-- {*Variant of the problem above. *}
by blast
lemma "\<exists>!x. f (g(x)) = x \<Longrightarrow> \<exists>!y. g (f(y)) = y"
-- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text auto} all fail. *}
apply (erule ex1E, rule ex1I, erule subst_context)
apply (rule subst, assumption, erule allE, rule subst_context, erule mp)
apply (erule subst_context)
done
text{*A weird property of ordered pairs.*}
lemma "b\<noteq>c ==> <a,b> Int <a,c> = <a,a>"
by (simp add: Pair_def Int_cons_left Int_cons_right doubleton_eq_iff, blast)
text{*These two are cited in Benzmueller and Kohlhase's system description of
LEO, CADE-15, 1998 (page 139-143) as theorems LEO could not prove.*}
lemma "(X = Y Un Z) <-> (Y \<subseteq> X & Z \<subseteq> X & (\<forall>V. Y \<subseteq> V & Z \<subseteq> V --> X \<subseteq> V))"
by (blast intro!: equalityI)
text{*the dual of the previous one*}
lemma "(X = Y Int Z) <-> (X \<subseteq> Y & X \<subseteq> Z & (\<forall>V. V \<subseteq> Y & V \<subseteq> Z --> V \<subseteq> X))"
by (blast intro!: equalityI)
text{*trivial example of term synthesis: apparently hard for some provers!*}
schematic_lemma "a \<noteq> b ==> a:?X & b \<notin> ?X"
by blast
text{*Nice blast benchmark. Proved in 0.3s; old tactics can't manage it!*}
lemma "\<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y ==> \<exists>z. S \<subseteq> {z}"
by blast
text{*variant of the benchmark above*}
lemma "\<forall>x \<in> S. Union(S) \<subseteq> x ==> \<exists>z. S \<subseteq> {z}"
by blast
(*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). *)
lemma "(\<forall>F. {x} \<in> F --> {y} \<in> F) --> (\<forall>A. x \<in> A --> y \<in> A)"
by best
text{*A characterization of functions suggested by Tobias Nipkow*}
lemma "r \<in> domain(r)->B <-> r \<subseteq> domain(r)*B & (\<forall>X. r `` (r -`` X) \<subseteq> X)"
by (unfold Pi_def function_def, best)
subsection{*Composition of homomorphisms is a Homomorphism*}
text{*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 *}
text{*collecting the relevant lemmas*}
declare comp_fun [simp] SigmaI [simp] apply_funtype [simp]
(*Force helps prove conditions of rewrites such as comp_fun_apply, since
rewriting does not instantiate Vars.*)
lemma "(\<forall>A f B g. hom(A,f,B,g) =
{H \<in> A->B. f \<in> A*A->A & g \<in> B*B->B &
(\<forall>x \<in> A. \<forall>y \<in> A. H`(f`<x,y>) = g`<H`x,H`y>)}) -->
J \<in> hom(A,f,B,g) & K \<in> hom(B,g,C,h) -->
(K O J) \<in> hom(A,f,C,h)"
by force
text{*Another version, with meta-level rewriting*}
lemma "(!! A f B g. hom(A,f,B,g) ==
{H \<in> A->B. f \<in> A*A->A & g \<in> B*B->B &
(\<forall>x \<in> A. \<forall>y \<in> A. H`(f`<x,y>) = g`<H`x,H`y>)})
==> J \<in> hom(A,f,B,g) & K \<in> hom(B,g,C,h) --> (K O J) \<in> hom(A,f,C,h)"
by force
subsection{*Pastre's Examples*}
text{*D Pastre. Automatic theorem proving in set theory.
Artificial Intelligence, 10:1--27, 1978.
Previously, these were done using ML code, but blast manages fine.*}
lemmas compIs [intro] = comp_surj comp_inj comp_fun [intro]
lemmas compDs [dest] = comp_mem_injD1 comp_mem_surjD1
comp_mem_injD2 comp_mem_surjD2
lemma pastre1:
"[| (h O g O f) \<in> inj(A,A);
(f O h O g) \<in> surj(B,B);
(g O f O h) \<in> surj(C,C);
f \<in> A->B; g \<in> B->C; h \<in> C->A |] ==> h \<in> bij(C,A)"
by (unfold bij_def, blast)
lemma pastre3:
"[| (h O g O f) \<in> surj(A,A);
(f O h O g) \<in> surj(B,B);
(g O f O h) \<in> inj(C,C);
f \<in> A->B; g \<in> B->C; h \<in> C->A |] ==> h \<in> bij(C,A)"
by (unfold bij_def, blast)
lemma pastre4:
"[| (h O g O f) \<in> surj(A,A);
(f O h O g) \<in> inj(B,B);
(g O f O h) \<in> inj(C,C);
f \<in> A->B; g \<in> B->C; h \<in> C->A |] ==> h \<in> bij(C,A)"
by (unfold bij_def, blast)
lemma pastre5:
"[| (h O g O f) \<in> inj(A,A);
(f O h O g) \<in> surj(B,B);
(g O f O h) \<in> inj(C,C);
f \<in> A->B; g \<in> B->C; h \<in> C->A |] ==> h \<in> bij(C,A)"
by (unfold bij_def, blast)
lemma pastre6:
"[| (h O g O f) \<in> inj(A,A);
(f O h O g) \<in> inj(B,B);
(g O f O h) \<in> surj(C,C);
f \<in> A->B; g \<in> B->C; h \<in> C->A |] ==> h \<in> bij(C,A)"
by (unfold bij_def, blast)
end