# Theory misc

theory misc
imports ZF
```(*  Title:      ZF/ex/misc.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Composition of homomorphisms, Pastre's examples, ...
*)

section‹Miscellaneous ZF Examples›

theory misc imports ZF begin

subsection‹Various Small Problems›

text‹The singleton problems are much harder in HOL.›
lemma singleton_example_1:
"∀x ∈ S. ∀y ∈ S. x ⊆ y ⟹ ∃z. S ⊆ {z}"
by blast

lemma singleton_example_2:
"∀x ∈ S. ⋃S ⊆ x ⟹ ∃z. S ⊆ {z}"
― ‹Variant of the problem above.›
by blast

lemma "∃!x. f (g(x)) = x ⟹ ∃!y. g (f(y)) = y"
― ‹A unique fixpoint theorem --- ‹fast›/‹best›/‹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≠c ==> <a,b> ∩ <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 ∪ Z) ⟷ (Y ⊆ X & Z ⊆ X & (∀V. Y ⊆ V & Z ⊆ V ⟶ X ⊆ V))"
by (blast intro!: equalityI)

text‹the dual of the previous one›
lemma "(X = Y ∩ Z) ⟷ (X ⊆ Y & X ⊆ Z & (∀V. V ⊆ Y & V ⊆ Z ⟶ V ⊆ X))"
by (blast intro!: equalityI)

text‹trivial example of term synthesis: apparently hard for some provers!›
schematic_goal "a ≠ b ==> a:?X & b ∉ ?X"
by blast

text‹Nice blast benchmark.  Proved in 0.3s; old tactics can't manage it!›
lemma "∀x ∈ S. ∀y ∈ S. x ⊆ y ==> ∃z. S ⊆ {z}"
by blast

text‹variant of the benchmark above›
lemma "∀x ∈ S. ⋃(S) ⊆ x ==> ∃z. S ⊆ {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 "(∀F. {x} ∈ F ⟶ {y} ∈ F) ⟶ (∀A. x ∈ A ⟶ y ∈ A)"
by best

text‹A characterization of functions suggested by Tobias Nipkow›
lemma "r ∈ domain(r)->B  ⟷  r ⊆ domain(r)*B & (∀X. r `` (r -`` X) ⊆ 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 "(∀A f B g. hom(A,f,B,g) =
{H ∈ A->B. f ∈ A*A->A & g ∈ B*B->B &
(∀x ∈ A. ∀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 force

text‹Another version, with meta-level rewriting›
lemma "(!! A f B g. hom(A,f,B,g) ==
{H ∈ A->B. f ∈ A*A->A & g ∈ B*B->B &
(∀x ∈ A. ∀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 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) ∈ 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 (unfold bij_def, blast)

lemma pastre3:
"[| (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 (unfold bij_def, blast)

lemma pastre4:
"[| (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 (unfold bij_def, blast)

lemma pastre5:
"[| (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 (unfold bij_def, blast)

lemma pastre6:
"[| (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 (unfold bij_def, blast)

end

```