src/HOL/Gfp.ML
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 14169 0590de71a016 permissions -rw-r--r--
import -> imports
```     1 (*  Title:      HOL/Gfp.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1993  University of Cambridge
```
```     5
```
```     6 The Knaster-Tarski Theorem for greatest fixed points.
```
```     7 *)
```
```     8
```
```     9 (*** Proof of Knaster-Tarski Theorem using gfp ***)
```
```    10
```
```    11 val gfp_def = thm "gfp_def";
```
```    12
```
```    13 (* gfp(f) is the least upper bound of {u. u <= f(u)} *)
```
```    14
```
```    15 Goalw [gfp_def] "[| X <= f(X) |] ==> X <= gfp(f)";
```
```    16 by (etac (CollectI RS Union_upper) 1);
```
```    17 qed "gfp_upperbound";
```
```    18
```
```    19 val prems = Goalw [gfp_def]
```
```    20     "[| !!u. u <= f(u) ==> u<=X |] ==> gfp(f) <= X";
```
```    21 by (REPEAT (ares_tac ([Union_least]@prems) 1));
```
```    22 by (etac CollectD 1);
```
```    23 qed "gfp_least";
```
```    24
```
```    25 Goal "mono(f) ==> gfp(f) <= f(gfp(f))";
```
```    26 by (EVERY1 [rtac gfp_least, rtac subset_trans, atac,
```
```    27             etac monoD, rtac gfp_upperbound, atac]);
```
```    28 qed "gfp_lemma2";
```
```    29
```
```    30 Goal "mono(f) ==> f(gfp(f)) <= gfp(f)";
```
```    31 by (EVERY1 [rtac gfp_upperbound, rtac monoD, assume_tac,
```
```    32             etac gfp_lemma2]);
```
```    33 qed "gfp_lemma3";
```
```    34
```
```    35 Goal "mono(f) ==> gfp(f) = f(gfp(f))";
```
```    36 by (REPEAT (ares_tac [equalityI,gfp_lemma2,gfp_lemma3] 1));
```
```    37 qed "gfp_unfold";
```
```    38
```
```    39 (*** Coinduction rules for greatest fixed points ***)
```
```    40
```
```    41 (*weak version*)
```
```    42 Goal "[| a: X;  X <= f(X) |] ==> a : gfp(f)";
```
```    43 by (rtac (gfp_upperbound RS subsetD) 1);
```
```    44 by Auto_tac;
```
```    45 qed "weak_coinduct";
```
```    46
```
```    47 Goal "!!X. [| a : X; g`X <= f (g`X) |] ==> g a : gfp f";
```
```    48 by (etac (gfp_upperbound RS subsetD) 1);
```
```    49 by (etac imageI 1);
```
```    50 qed "weak_coinduct_image";
```
```    51
```
```    52 Goal "[| X <= f(X Un gfp(f));  mono(f) |] ==>  \
```
```    53 \    X Un gfp(f) <= f(X Un gfp(f))";
```
```    54 by (blast_tac (claset() addDs [gfp_lemma2, mono_Un]) 1);
```
```    55 qed "coinduct_lemma";
```
```    56
```
```    57 (*strong version, thanks to Coen & Frost*)
```
```    58 Goal "[| mono(f);  a: X;  X <= f(X Un gfp(f)) |] ==> a : gfp(f)";
```
```    59 by (rtac (coinduct_lemma RSN (2, weak_coinduct)) 1);
```
```    60 by (REPEAT (ares_tac [UnI1, Un_least] 1));
```
```    61 qed "coinduct";
```
```    62
```
```    63 Goal "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))";
```
```    64 by (blast_tac (claset() addDs [gfp_lemma2, mono_Un]) 1);
```
```    65 qed "gfp_fun_UnI2";
```
```    66
```
```    67 (***  Even Stronger version of coinduct  [by Martin Coen]
```
```    68          - instead of the condition  X <= f(X)
```
```    69                            consider  X <= (f(X) Un f(f(X)) ...) Un gfp(X) ***)
```
```    70
```
```    71 Goal "mono(f) ==> mono(%x. f(x) Un X Un B)";
```
```    72 by (REPEAT (ares_tac [subset_refl, monoI, Un_mono] 1 ORELSE etac monoD 1));
```
```    73 qed "coinduct3_mono_lemma";
```
```    74
```
```    75 Goal "[| X <= f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |] ==> \
```
```    76 \    lfp(%x. f(x) Un X Un gfp(f)) <= f(lfp(%x. f(x) Un X Un gfp(f)))";
```
```    77 by (rtac subset_trans 1);
```
```    78 by (etac (coinduct3_mono_lemma RS lfp_lemma3) 1);
```
```    79 by (rtac (Un_least RS Un_least) 1);
```
```    80 by (rtac subset_refl 1);
```
```    81 by (assume_tac 1);
```
```    82 by (rtac (gfp_unfold RS equalityD1 RS subset_trans) 1);
```
```    83 by (assume_tac 1);
```
```    84 by (rtac monoD 1 THEN assume_tac 1);
```
```    85 by (stac (coinduct3_mono_lemma RS lfp_unfold) 1);
```
```    86 by Auto_tac;
```
```    87 qed "coinduct3_lemma";
```
```    88
```
```    89 Goal
```
```    90   "[| mono(f);  a:X;  X <= f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)";
```
```    91 by (rtac (coinduct3_lemma RSN (2,weak_coinduct)) 1);
```
```    92 by (resolve_tac [coinduct3_mono_lemma RS lfp_unfold RS ssubst] 1);
```
```    93 by Auto_tac;
```
```    94 qed "coinduct3";
```
```    95
```
```    96
```
```    97 (** Definition forms of gfp_unfold and coinduct, to control unfolding **)
```
```    98
```
```    99 Goal "[| A==gfp(f);  mono(f) |] ==> A = f(A)";
```
```   100 by (auto_tac (claset() addSIs [gfp_unfold], simpset()));
```
```   101 qed "def_gfp_unfold";
```
```   102
```
```   103 Goal "[| A==gfp(f);  mono(f);  a:X;  X <= f(X Un A) |] ==> a: A";
```
```   104 by (auto_tac (claset() addSIs [coinduct], simpset()));
```
```   105 qed "def_coinduct";
```
```   106
```
```   107 (*The version used in the induction/coinduction package*)
```
```   108 val prems = Goal
```
```   109     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));  \
```
```   110 \       a: X;  !!z. z: X ==> P (X Un A) z |] ==> \
```
```   111 \    a : A";
```
```   112 by (rtac def_coinduct 1);
```
```   113 by (REPEAT (ares_tac (prems @ [subsetI,CollectI]) 1));
```
```   114 qed "def_Collect_coinduct";
```
```   115
```
```   116 Goal "[| A==gfp(f); mono(f);  a:X;  X <= f(lfp(%x. f(x) Un X Un A)) |] \
```
```   117 \     ==> a: A";
```
```   118 by (auto_tac (claset() addSIs [coinduct3], simpset()));
```
```   119 qed "def_coinduct3";
```
```   120
```
```   121 (*Monotonicity of gfp!*)
```
```   122 val [prem] = Goal "[| !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
```
```   123 by (rtac (gfp_upperbound RS gfp_least) 1);
```
```   124 by (etac (prem RSN (2,subset_trans)) 1);
```
```   125 qed "gfp_mono";
```