src/ZF/ex/counit.ML
 author lcp Tue Aug 16 18:58:42 1994 +0200 (1994-08-16) changeset 532 851df239ac8b parent 173 85071e6ad295 permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
```     1 (*  Title: 	ZF/ex/counit.ML
```
```     2     ID:         \$Id\$
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```     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1993  University of Cambridge
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```     5
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```     6 Trivial codatatype definitions, one of which goes wrong!
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```     7
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```     8 Need to find sufficient conditions for codatatypes to work correctly!
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```     9 *)
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```    10
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```    11 (*This degenerate definition does not work well because the one constructor's
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```    12   definition is trivial!  The same thing occurs with Aczel's Special Final
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```    13   Coalgebra Theorem
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```    14 *)
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```    15 structure CoUnit = CoDatatype_Fun
```
```    16  (val thy = QUniv.thy;
```
```    17   val rec_specs =
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```    18       [("counit", "quniv(0)",
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```    19 	  [(["Con"],	"i=>i")])];
```
```    20   val rec_styp = "i";
```
```    21   val ext = None
```
```    22   val sintrs = ["x: counit ==> Con(x) : counit"];
```
```    23   val monos = [];
```
```    24   val type_intrs = codatatype_intrs
```
```    25   val type_elims = codatatype_elims);
```
```    26
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```    27 val [ConI] = CoUnit.intrs;
```
```    28
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```    29 (*USELESS because folding on Con(?xa) == ?xa fails*)
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```    30 val ConE = CoUnit.mk_cases CoUnit.con_defs "Con(x) : counit";
```
```    31
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```    32 (*Proving freeness results*)
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```    33 val Con_iff = CoUnit.mk_free "Con(x)=Con(y) <-> x=y";
```
```    34
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```    35 (*Should be a singleton, not everything!*)
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```    36 goal CoUnit.thy "counit = quniv(0)";
```
```    37 by (rtac (CoUnit.dom_subset RS equalityI) 1);
```
```    38 by (rtac subsetI 1);
```
```    39 by (etac CoUnit.coinduct 1);
```
```    40 by (rtac subset_refl 1);
```
```    41 by (rewrite_goals_tac CoUnit.con_defs);
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```    42 by (fast_tac ZF_cs 1);
```
```    43 val counit_eq_univ = result();
```
```    44
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```    45
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```    46 (*****************************************************************)
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```    47
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```    48 (*A similar example, but the constructor is non-degenerate and it works!
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```    49   The resulting set is a singleton.
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```    50 *)
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```    51
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```    52 structure CoUnit2 = CoDatatype_Fun
```
```    53  (val thy = QUniv.thy;
```
```    54   val rec_specs =
```
```    55       [("counit2", "quniv(0)",
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```    56 	  [(["Con2"],	"[i,i]=>i")])];
```
```    57   val rec_styp = "i";
```
```    58   val ext = None
```
```    59   val sintrs = ["[| x: counit2;  y: counit2 |] ==> Con2(x,y) : counit2"];
```
```    60   val monos = [];
```
```    61   val type_intrs = codatatype_intrs
```
```    62   val type_elims = codatatype_elims);
```
```    63
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```    64 val [Con2I] = CoUnit2.intrs;
```
```    65
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```    66 val Con2E = CoUnit2.mk_cases CoUnit2.con_defs "Con2(x,y) : counit2";
```
```    67
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```    68 (*Proving freeness results*)
```
```    69 val Con2_iff = CoUnit2.mk_free "Con2(x,y)=Con2(x',y') <-> x=x' & y=y'";
```
```    70
```
```    71 goalw CoUnit2.thy CoUnit2.con_defs "bnd_mono(univ(0), %x. Con2(x,x))";
```
```    72 by (rtac bnd_monoI 1);
```
```    73 by (REPEAT (ares_tac [subset_refl, QPair_subset_univ, QPair_mono] 1));
```
```    74 val Con2_bnd_mono = result();
```
```    75
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```    76 goal CoUnit2.thy "lfp(univ(0), %x. Con2(x,x)) : counit2";
```
```    77 by (rtac (singletonI RS CoUnit2.coinduct) 1);
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```    78 by (rtac (qunivI RS singleton_subsetI) 1);
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```    79 by (rtac ([lfp_subset, empty_subsetI RS univ_mono] MRS subset_trans) 1);
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```    80 by (fast_tac (ZF_cs addSIs [Con2_bnd_mono RS lfp_Tarski]) 1);
```
```    81 val lfp_Con2_in_counit2 = result();
```
```    82
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```    83 (*Lemma for proving finality.  Borrowed from ex/llist_eq.ML!*)
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```    84 goal CoUnit2.thy
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```    85     "!!i. Ord(i) ==> ALL x y. x: counit2 & y: counit2 --> x Int Vset(i) <= y";
```
```    86 by (etac trans_induct 1);
```
```    87 by (safe_tac subset_cs);
```
```    88 by (etac CoUnit2.elim 1);
```
```    89 by (etac CoUnit2.elim 1);
```
```    90 by (rewrite_goals_tac CoUnit2.con_defs);
```
```    91 by (fast_tac lleq_cs 1);
```
```    92 val counit2_Int_Vset_subset_lemma = result();
```
```    93
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```    94 val counit2_Int_Vset_subset = standard
```
```    95 	(counit2_Int_Vset_subset_lemma RS spec RS spec RS mp);
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```    96
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```    97 goal CoUnit2.thy "!!x y. [| x: counit2;  y: counit2 |] ==> x=y";
```
```    98 by (rtac equalityI 1);
```
```    99 by (REPEAT (ares_tac [conjI, counit2_Int_Vset_subset RS Int_Vset_subset] 1));
```
```   100 val counit2_implies_equal = result();
```
```   101
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```   102 goal CoUnit2.thy "counit2 = {lfp(univ(0), %x. Con2(x,x))}";
```
```   103 by (rtac equalityI 1);
```
```   104 by (rtac (lfp_Con2_in_counit2 RS singleton_subsetI) 2);
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```   105 by (rtac subsetI 1);
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```   106 by (dtac (lfp_Con2_in_counit2 RS counit2_implies_equal) 1);
```
```   107 by (etac subst 1);
```
```   108 by (rtac singletonI 1);
```
```   109 val counit2_eq_univ = result();
```