src/HOL/Gfp.ML
 author wenzelm Thu Jan 23 14:19:16 1997 +0100 (1997-01-23) changeset 2545 d10abc8c11fb parent 2036 62ff902eeffc child 3842 b55686a7b22c permissions -rw-r--r--
```     1 (*  Title:      HOL/gfp
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1993  University of Cambridge
```
```     5
```
```     6 For gfp.thy.  The Knaster-Tarski Theorem for greatest fixed points.
```
```     7 *)
```
```     8
```
```     9 open Gfp;
```
```    10
```
```    11 (*** Proof of Knaster-Tarski Theorem using gfp ***)
```
```    12
```
```    13 (* gfp(f) is the least upper bound of {u. u <= f(u)} *)
```
```    14
```
```    15 val prems = goalw Gfp.thy [gfp_def] "[| X <= f(X) |] ==> X <= gfp(f)";
```
```    16 by (rtac (CollectI RS Union_upper) 1);
```
```    17 by (resolve_tac prems 1);
```
```    18 qed "gfp_upperbound";
```
```    19
```
```    20 val prems = goalw Gfp.thy [gfp_def]
```
```    21     "[| !!u. u <= f(u) ==> u<=X |] ==> gfp(f) <= X";
```
```    22 by (REPEAT (ares_tac ([Union_least]@prems) 1));
```
```    23 by (etac CollectD 1);
```
```    24 qed "gfp_least";
```
```    25
```
```    26 val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) <= f(gfp(f))";
```
```    27 by (EVERY1 [rtac gfp_least, rtac subset_trans, atac,
```
```    28             rtac (mono RS monoD), rtac gfp_upperbound, atac]);
```
```    29 qed "gfp_lemma2";
```
```    30
```
```    31 val [mono] = goal Gfp.thy "mono(f) ==> f(gfp(f)) <= gfp(f)";
```
```    32 by (EVERY1 [rtac gfp_upperbound, rtac (mono RS monoD),
```
```    33             rtac gfp_lemma2, rtac mono]);
```
```    34 qed "gfp_lemma3";
```
```    35
```
```    36 val [mono] = goal Gfp.thy "mono(f) ==> gfp(f) = f(gfp(f))";
```
```    37 by (REPEAT (resolve_tac [equalityI,gfp_lemma2,gfp_lemma3,mono] 1));
```
```    38 qed "gfp_Tarski";
```
```    39
```
```    40 (*** Coinduction rules for greatest fixed points ***)
```
```    41
```
```    42 (*weak version*)
```
```    43 val prems = goal Gfp.thy
```
```    44     "[| a: X;  X <= f(X) |] ==> a : gfp(f)";
```
```    45 by (rtac (gfp_upperbound RS subsetD) 1);
```
```    46 by (REPEAT (ares_tac prems 1));
```
```    47 qed "weak_coinduct";
```
```    48
```
```    49 val [prem,mono] = goal Gfp.thy
```
```    50     "[| X <= f(X Un gfp(f));  mono(f) |] ==>  \
```
```    51 \    X Un gfp(f) <= f(X Un gfp(f))";
```
```    52 by (rtac (prem RS Un_least) 1);
```
```    53 by (rtac (mono RS gfp_lemma2 RS subset_trans) 1);
```
```    54 by (rtac (Un_upper2 RS subset_trans) 1);
```
```    55 by (rtac (mono RS mono_Un) 1);
```
```    56 qed "coinduct_lemma";
```
```    57
```
```    58 (*strong version, thanks to Coen & Frost*)
```
```    59 goal Gfp.thy
```
```    60     "!!X. [| mono(f);  a: X;  X <= f(X Un gfp(f)) |] ==> a : gfp(f)";
```
```    61 by (rtac (coinduct_lemma RSN (2, weak_coinduct)) 1);
```
```    62 by (REPEAT (ares_tac [UnI1, Un_least] 1));
```
```    63 qed "coinduct";
```
```    64
```
```    65 val [mono,prem] = goal Gfp.thy
```
```    66     "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))";
```
```    67 by (rtac (mono RS mono_Un RS subsetD) 1);
```
```    68 by (rtac (mono RS gfp_lemma2 RS subsetD RS UnI2) 1);
```
```    69 by (rtac prem 1);
```
```    70 qed "gfp_fun_UnI2";
```
```    71
```
```    72 (***  Even Stronger version of coinduct  [by Martin Coen]
```
```    73          - instead of the condition  X <= f(X)
```
```    74                            consider  X <= (f(X) Un f(f(X)) ...) Un gfp(X) ***)
```
```    75
```
```    76 val [prem] = goal Gfp.thy "mono(f) ==> mono(%x.f(x) Un X Un B)";
```
```    77 by (REPEAT (ares_tac [subset_refl, monoI, Un_mono, prem RS monoD] 1));
```
```    78 qed "coinduct3_mono_lemma";
```
```    79
```
```    80 val [prem,mono] = goal Gfp.thy
```
```    81     "[| X <= f(lfp(%x.f(x) Un X Un gfp(f)));  mono(f) |] ==> \
```
```    82 \    lfp(%x.f(x) Un X Un gfp(f)) <= f(lfp(%x.f(x) Un X Un gfp(f)))";
```
```    83 by (rtac subset_trans 1);
```
```    84 by (rtac (mono RS coinduct3_mono_lemma RS lfp_lemma3) 1);
```
```    85 by (rtac (Un_least RS Un_least) 1);
```
```    86 by (rtac subset_refl 1);
```
```    87 by (rtac prem 1);
```
```    88 by (rtac (mono RS gfp_Tarski RS equalityD1 RS subset_trans) 1);
```
```    89 by (rtac (mono RS monoD) 1);
```
```    90 by (stac (mono RS coinduct3_mono_lemma RS lfp_Tarski) 1);
```
```    91 by (rtac Un_upper2 1);
```
```    92 qed "coinduct3_lemma";
```
```    93
```
```    94 val prems = goal Gfp.thy
```
```    95     "[| mono(f);  a:X;  X <= f(lfp(%x.f(x) Un X Un gfp(f))) |] ==> a : gfp(f)";
```
```    96 by (rtac (coinduct3_lemma RSN (2,weak_coinduct)) 1);
```
```    97 by (resolve_tac (prems RL [coinduct3_mono_lemma RS lfp_Tarski RS ssubst]) 1);
```
```    98 by (rtac (UnI2 RS UnI1) 1);
```
```    99 by (REPEAT (resolve_tac prems 1));
```
```   100 qed "coinduct3";
```
```   101
```
```   102
```
```   103 (** Definition forms of gfp_Tarski and coinduct, to control unfolding **)
```
```   104
```
```   105 val [rew,mono] = goal Gfp.thy "[| A==gfp(f);  mono(f) |] ==> A = f(A)";
```
```   106 by (rewtac rew);
```
```   107 by (rtac (mono RS gfp_Tarski) 1);
```
```   108 qed "def_gfp_Tarski";
```
```   109
```
```   110 val rew::prems = goal Gfp.thy
```
```   111     "[| A==gfp(f);  mono(f);  a:X;  X <= f(X Un A) |] ==> a: A";
```
```   112 by (rewtac rew);
```
```   113 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct]) 1));
```
```   114 qed "def_coinduct";
```
```   115
```
```   116 (*The version used in the induction/coinduction package*)
```
```   117 val prems = goal Gfp.thy
```
```   118     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));  \
```
```   119 \       a: X;  !!z. z: X ==> P (X Un A) z |] ==> \
```
```   120 \    a : A";
```
```   121 by (rtac def_coinduct 1);
```
```   122 by (REPEAT (ares_tac (prems @ [subsetI,CollectI]) 1));
```
```   123 qed "def_Collect_coinduct";
```
```   124
```
```   125 val rew::prems = goal Gfp.thy
```
```   126     "[| A==gfp(f); mono(f);  a:X;  X <= f(lfp(%x.f(x) Un X Un A)) |] ==> a: A";
```
```   127 by (rewtac rew);
```
```   128 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct3]) 1));
```
```   129 qed "def_coinduct3";
```
```   130
```
```   131 (*Monotonicity of gfp!*)
```
```   132 val prems = goal Gfp.thy
```
```   133     "[| mono(f);  !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
```
```   134 by (rtac gfp_upperbound 1);
```
```   135 by (rtac subset_trans 1);
```
```   136 by (rtac gfp_lemma2 1);
```
```   137 by (resolve_tac prems 1);
```
```   138 by (resolve_tac prems 1);
```
```   139 val gfp_mono = result();
```
```   140
```
```   141 (*Monotonicity of gfp!*)
```
```   142 val [prem] = goal Gfp.thy "[| !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
```
```   143 by (rtac (gfp_upperbound RS gfp_least) 1);
```
```   144 by (etac (prem RSN (2,subset_trans)) 1);
```
```   145 qed "gfp_mono";
```