src/HOL/Gfp.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 15386 06757406d8cf permissions -rw-r--r--
Constant "If" is now local
```     1 (*  ID:         \$Id\$
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```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     3     Copyright   1994  University of Cambridge
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```     4
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```     5 *)
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```     6
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```     7 header{*Greatest Fixed Points and the Knaster-Tarski Theorem*}
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```     8
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```     9 theory Gfp
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```    10 imports Lfp
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```    11 begin
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```    12
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```    13 constdefs
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```    14   gfp :: "['a set=>'a set] => 'a set"
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```    15     "gfp(f) == Union({u. u \<subseteq> f(u)})"
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```    16
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```    17
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```    18
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```    19 subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*}
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```    20
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```    21
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```    22 text{*@{term "gfp f"} is the greatest lower bound of
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```    23       the set @{term "{u. u \<subseteq> f(u)}"} *}
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```    24
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```    25 lemma gfp_upperbound: "[| X \<subseteq> f(X) |] ==> X \<subseteq> gfp(f)"
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```    26 by (auto simp add: gfp_def)
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```    27
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```    28 lemma gfp_least: "[| !!u. u \<subseteq> f(u) ==> u\<subseteq>X |] ==> gfp(f) \<subseteq> X"
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```    29 by (auto simp add: gfp_def)
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```    30
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```    31 lemma gfp_lemma2: "mono(f) ==> gfp(f) \<subseteq> f(gfp(f))"
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```    32 by (rules intro: gfp_least subset_trans monoD gfp_upperbound)
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```    33
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```    34 lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) \<subseteq> gfp(f)"
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```    35 by (rules intro: gfp_lemma2 monoD gfp_upperbound)
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```    36
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```    37 lemma gfp_unfold: "mono(f) ==> gfp(f) = f(gfp(f))"
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```    38 by (rules intro: equalityI gfp_lemma2 gfp_lemma3)
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```    39
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```    40 subsection{*Coinduction rules for greatest fixed points*}
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```    41
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```    42 text{*weak version*}
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```    43 lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
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```    44 by (rule gfp_upperbound [THEN subsetD], auto)
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```    45
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```    46 lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
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```    47 apply (erule gfp_upperbound [THEN subsetD])
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```    48 apply (erule imageI)
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```    49 done
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```    50
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```    51 lemma coinduct_lemma:
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```    52      "[| X \<subseteq> f(X Un gfp(f));  mono(f) |] ==> X Un gfp(f) \<subseteq> f(X Un gfp(f))"
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```    53 by (blast dest: gfp_lemma2 mono_Un)
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```    54
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```    55 text{*strong version, thanks to Coen and Frost*}
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```    56 lemma coinduct: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
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```    57 by (blast intro: weak_coinduct [OF _ coinduct_lemma])
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```    58
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```    59 lemma gfp_fun_UnI2: "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))"
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```    60 by (blast dest: gfp_lemma2 mono_Un)
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```    61
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```    62 subsection{*Even Stronger Coinduction Rule, by Martin Coen*}
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```    63
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```    64 text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
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```    65   @{term lfp} and @{term gfp}*}
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```    66
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```    67 lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
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```    68 by (rules intro: subset_refl monoI Un_mono monoD)
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```    69
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```    70 lemma coinduct3_lemma:
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```    71      "[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
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```    72       ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
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```    73 apply (rule subset_trans)
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```    74 apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
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```    75 apply (rule Un_least [THEN Un_least])
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```    76 apply (rule subset_refl, assumption)
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```    77 apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
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```    78 apply (rule monoD, assumption)
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```    79 apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
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```    80 done
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```    81
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```    82 lemma coinduct3:
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```    83   "[| mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
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```    84 apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
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```    85 apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto)
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```    86 done
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```    87
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```    88
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```    89 text{*Definition forms of @{text gfp_unfold} and @{text coinduct},
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```    90     to control unfolding*}
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```    91
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```    92 lemma def_gfp_unfold: "[| A==gfp(f);  mono(f) |] ==> A = f(A)"
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```    93 by (auto intro!: gfp_unfold)
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```    94
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```    95 lemma def_coinduct:
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```    96      "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
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```    97 by (auto intro!: coinduct)
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```    98
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```    99 (*The version used in the induction/coinduction package*)
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```   100 lemma def_Collect_coinduct:
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```   101     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));
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```   102         a: X;  !!z. z: X ==> P (X Un A) z |] ==>
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```   103      a : A"
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```   104 apply (erule def_coinduct, auto)
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```   105 done
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```   106
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```   107 lemma def_coinduct3:
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```   108     "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
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```   109 by (auto intro!: coinduct3)
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```   110
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```   111 text{*Monotonicity of @{term gfp}!*}
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```   112 lemma gfp_mono: "[| !!Z. f(Z)\<subseteq>g(Z) |] ==> gfp(f) \<subseteq> gfp(g)"
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```   113 by (rule gfp_upperbound [THEN gfp_least], blast)
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```   114
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```   115
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```   116 ML
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```   117 {*
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```   118 val gfp_def = thm "gfp_def";
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```   119 val gfp_upperbound = thm "gfp_upperbound";
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```   120 val gfp_least = thm "gfp_least";
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```   121 val gfp_unfold = thm "gfp_unfold";
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```   122 val weak_coinduct = thm "weak_coinduct";
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```   123 val weak_coinduct_image = thm "weak_coinduct_image";
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```   124 val coinduct = thm "coinduct";
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```   125 val gfp_fun_UnI2 = thm "gfp_fun_UnI2";
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```   126 val coinduct3 = thm "coinduct3";
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```   127 val def_gfp_unfold = thm "def_gfp_unfold";
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```   128 val def_coinduct = thm "def_coinduct";
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```   129 val def_Collect_coinduct = thm "def_Collect_coinduct";
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```   130 val def_coinduct3 = thm "def_coinduct3";
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```   131 val gfp_mono = thm "gfp_mono";
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```   132 *}
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```   133
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```   134
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```   135 end
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