src/ZF/ex/CoUnit.ML
 author lcp Mon, 15 Aug 1994 18:31:29 +0200 changeset 527 35c70ab82940 parent 515 abcc438e7c27 child 760 f0200e91b272 permissions -rw-r--r--
ZF/ex/CoUnit/lleq_cs: copied from LList.ML
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(*  Title: 	ZF/ex/CoUnit.ML
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

Trivial codatatype definitions, one of which goes wrong!

See discussion in
L C Paulson.  A Concrete Final Coalgebra Theorem for ZF Set Theory.
Report 334,  Cambridge University Computer Laboratory.  1994.
*)

open CoUnit;

(*USELESS because folding on Con(?xa) == ?xa fails*)
val ConE = counit.mk_cases counit.con_defs "Con(x) : counit";

(*Proving freeness results*)
val Con_iff = counit.mk_free "Con(x)=Con(y) <-> x=y";

(*Should be a singleton, not everything!*)
goal CoUnit.thy "counit = quniv(0)";
by (rtac (counit.dom_subset RS equalityI) 1);
by (rtac subsetI 1);
by (etac counit.coinduct 1);
by (rtac subset_refl 1);
by (rewrite_goals_tac counit.con_defs);
by (fast_tac ZF_cs 1);
val counit_eq_univ = result();

(*A similar example, but the constructor is non-degenerate and it works!
The resulting set is a singleton.
*)

val Con2E = counit2.mk_cases counit2.con_defs "Con2(x,y) : counit2";

(*Proving freeness results*)
val Con2_iff = counit2.mk_free "Con2(x,y)=Con2(x',y') <-> x=x' & y=y'";

goalw CoUnit.thy counit2.con_defs "bnd_mono(univ(0), %x. Con2(x,x))";
by (rtac bnd_monoI 1);
by (REPEAT (ares_tac [subset_refl, QPair_subset_univ, QPair_mono] 1));
val Con2_bnd_mono = result();

goal CoUnit.thy "lfp(univ(0), %x. Con2(x,x)) : counit2";
by (rtac (singletonI RS counit2.coinduct) 1);
by (rtac (qunivI RS singleton_subsetI) 1);
by (rtac ([lfp_subset, empty_subsetI RS univ_mono] MRS subset_trans) 1);
by (fast_tac (ZF_cs addSIs [Con2_bnd_mono RS lfp_Tarski]) 1);
val lfp_Con2_in_counit2 = result();

(*Lemma for proving finality.  Borrowed from ex/llist_eq.ML!*)
goal CoUnit.thy
"!!i. Ord(i) ==> ALL x y. x: counit2 & y: counit2 --> x Int Vset(i) <= y";
by (etac trans_induct 1);
by (safe_tac subset_cs);
by (etac counit2.elim 1);
by (etac counit2.elim 1);
by (rewrite_goals_tac counit2.con_defs);
val lleq_cs = subset_cs
by (fast_tac lleq_cs 1);
val counit2_Int_Vset_subset_lemma = result();

val counit2_Int_Vset_subset = standard
(counit2_Int_Vset_subset_lemma RS spec RS spec RS mp);

goal CoUnit.thy "!!x y. [| x: counit2;  y: counit2 |] ==> x=y";
by (rtac equalityI 1);
by (REPEAT (ares_tac [conjI, counit2_Int_Vset_subset RS Int_Vset_subset] 1));
val counit2_implies_equal = result();

goal CoUnit.thy "counit2 = {lfp(univ(0), %x. Con2(x,x))}";
by (rtac equalityI 1);
by (rtac (lfp_Con2_in_counit2 RS singleton_subsetI) 2);
by (rtac subsetI 1);
by (dtac (lfp_Con2_in_counit2 RS counit2_implies_equal) 1);
by (etac subst 1);
by (rtac singletonI 1);
val counit2_eq_univ = result();
```