src/HOL/Lfp.ML
 author wenzelm Thu Jan 23 14:19:16 1997 +0100 (1997-01-23) changeset 2545 d10abc8c11fb parent 1873 b07ee188f061 child 3842 b55686a7b22c permissions -rw-r--r--
```     1 (*  Title:      HOL/lfp.ML
```
```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1992  University of Cambridge
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```     5
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```     6 For lfp.thy.  The Knaster-Tarski Theorem
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```     7 *)
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```     8
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```     9 open Lfp;
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```    10
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```    11 (*** Proof of Knaster-Tarski Theorem ***)
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```    12
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```    13 (* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
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```    14
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```    15 val prems = goalw Lfp.thy [lfp_def] "[| f(A) <= A |] ==> lfp(f) <= A";
```
```    16 by (rtac (CollectI RS Inter_lower) 1);
```
```    17 by (resolve_tac prems 1);
```
```    18 qed "lfp_lowerbound";
```
```    19
```
```    20 val prems = goalw Lfp.thy [lfp_def]
```
```    21     "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)";
```
```    22 by (REPEAT (ares_tac ([Inter_greatest]@prems) 1));
```
```    23 by (etac CollectD 1);
```
```    24 qed "lfp_greatest";
```
```    25
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```    26 val [mono] = goal Lfp.thy "mono(f) ==> f(lfp(f)) <= lfp(f)";
```
```    27 by (EVERY1 [rtac lfp_greatest, rtac subset_trans,
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```    28             rtac (mono RS monoD), rtac lfp_lowerbound, atac, atac]);
```
```    29 qed "lfp_lemma2";
```
```    30
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```    31 val [mono] = goal Lfp.thy "mono(f) ==> lfp(f) <= f(lfp(f))";
```
```    32 by (EVERY1 [rtac lfp_lowerbound, rtac (mono RS monoD),
```
```    33             rtac lfp_lemma2, rtac mono]);
```
```    34 qed "lfp_lemma3";
```
```    35
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```    36 val [mono] = goal Lfp.thy "mono(f) ==> lfp(f) = f(lfp(f))";
```
```    37 by (REPEAT (resolve_tac [equalityI,lfp_lemma2,lfp_lemma3,mono] 1));
```
```    38 qed "lfp_Tarski";
```
```    39
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```    40 (*** General induction rule for least fixed points ***)
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```    41
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```    42 val [lfp,mono,indhyp] = goal Lfp.thy
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```    43     "[| a: lfp(f);  mono(f);                            \
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```    44 \       !!x. [| x: f(lfp(f) Int {x.P(x)}) |] ==> P(x)   \
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```    45 \    |] ==> P(a)";
```
```    46 by (res_inst_tac [("a","a")] (Int_lower2 RS subsetD RS CollectD) 1);
```
```    47 by (rtac (lfp RSN (2, lfp_lowerbound RS subsetD)) 1);
```
```    48 by (EVERY1 [rtac Int_greatest, rtac subset_trans,
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```    49             rtac (Int_lower1 RS (mono RS monoD)),
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```    50             rtac (mono RS lfp_lemma2),
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```    51             rtac (CollectI RS subsetI), rtac indhyp, atac]);
```
```    52 qed "induct";
```
```    53
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```    54 bind_thm
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```    55   ("induct2",
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```    56    Prod_Syntax.split_rule
```
```    57      (read_instantiate [("a","(a,b)")] induct));
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```    58
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```    59
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```    60 (** Definition forms of lfp_Tarski and induct, to control unfolding **)
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```    61
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```    62 val [rew,mono] = goal Lfp.thy "[| h==lfp(f);  mono(f) |] ==> h = f(h)";
```
```    63 by (rewtac rew);
```
```    64 by (rtac (mono RS lfp_Tarski) 1);
```
```    65 qed "def_lfp_Tarski";
```
```    66
```
```    67 val rew::prems = goal Lfp.thy
```
```    68     "[| A == lfp(f);  mono(f);   a:A;                   \
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```    69 \       !!x. [| x: f(A Int {x.P(x)}) |] ==> P(x)        \
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```    70 \    |] ==> P(a)";
```
```    71 by (EVERY1 [rtac induct,        (*backtracking to force correct induction*)
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```    72             REPEAT1 o (ares_tac (map (rewrite_rule [rew]) prems))]);
```
```    73 qed "def_induct";
```
```    74
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```    75 (*Monotonicity of lfp!*)
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```    76 val [prem] = goal Lfp.thy "[| !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)";
```
```    77 by (rtac (lfp_lowerbound RS lfp_greatest) 1);
```
```    78 by (etac (prem RS subset_trans) 1);
```
```    79 qed "lfp_mono";
```