1(*  Title:      CCL/Gfp.thy
2    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3    Copyright   1992  University of Cambridge
4*)
5
6section \<open>Greatest fixed points\<close>
7
8theory Gfp
9imports Lfp
10begin
11
12definition
13  gfp :: "['a set\<Rightarrow>'a set] \<Rightarrow> 'a set" where \<comment> \<open>greatest fixed point\<close>
14  "gfp(f) == Union({u. u <= f(u)})"
15
16(* gfp(f) is the least upper bound of {u. u <= f(u)} *)
17
18lemma gfp_upperbound: "A <= f(A) \<Longrightarrow> A <= gfp(f)"
19  unfolding gfp_def by blast
20
21lemma gfp_least: "(\<And>u. u <= f(u) \<Longrightarrow> u <= A) \<Longrightarrow> gfp(f) <= A"
22  unfolding gfp_def by blast
23
24lemma gfp_lemma2: "mono(f) \<Longrightarrow> gfp(f) <= f(gfp(f))"
25  by (rule gfp_least, rule subset_trans, assumption, erule monoD,
26    rule gfp_upperbound, assumption)
27
28lemma gfp_lemma3: "mono(f) \<Longrightarrow> f(gfp(f)) <= gfp(f)"
29  by (rule gfp_upperbound, frule monoD, rule gfp_lemma2, assumption+)
30
31lemma gfp_Tarski: "mono(f) \<Longrightarrow> gfp(f) = f(gfp(f))"
32  by (rule equalityI gfp_lemma2 gfp_lemma3 | assumption)+
33
34
35(*** Coinduction rules for greatest fixed points ***)
36
37(*weak version*)
38lemma coinduct: "\<lbrakk>a: A;  A <= f(A)\<rbrakk> \<Longrightarrow> a : gfp(f)"
39  by (blast dest: gfp_upperbound)
40
41lemma coinduct2_lemma: "\<lbrakk>A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> A Un gfp(f) <= f(A Un gfp(f))"
42  apply (rule subset_trans)
43   prefer 2
44   apply (erule mono_Un)
45  apply (rule subst, erule gfp_Tarski)
46  apply (erule Un_least)
47  apply (rule Un_upper2)
48  done
49
50(*strong version, thanks to Martin Coen*)
51lemma coinduct2: "\<lbrakk>a: A; A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> a : gfp(f)"
52  apply (rule coinduct)
53   prefer 2
54   apply (erule coinduct2_lemma, assumption)
55  apply blast
56  done
57
58(***  Even Stronger version of coinduct  [by Martin Coen]
59         - instead of the condition  A <= f(A)
60                           consider  A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***)
61
62lemma coinduct3_mono_lemma: "mono(f) \<Longrightarrow> mono(\<lambda>x. f(x) Un A Un B)"
63  by (rule monoI) (blast dest: monoD)
64
65lemma coinduct3_lemma:
66  assumes prem: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
67    and mono: "mono(f)"
68  shows "lfp(\<lambda>x. f(x) Un A Un gfp(f)) <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
69  apply (rule subset_trans)
70   apply (rule mono [THEN coinduct3_mono_lemma, THEN lfp_lemma3])
71  apply (rule Un_least [THEN Un_least])
72    apply (rule subset_refl)
73   apply (rule prem)
74  apply (rule mono [THEN gfp_Tarski, THEN equalityD1, THEN subset_trans])
75  apply (rule mono [THEN monoD])
76  apply (subst mono [THEN coinduct3_mono_lemma, THEN lfp_Tarski])
77  apply (rule Un_upper2)
78  done
79
80lemma coinduct3:
81  assumes 1: "a:A"
82    and 2: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
83    and 3: "mono(f)"
84  shows "a : gfp(f)"
85  apply (rule coinduct)
86   prefer 2
87   apply (rule coinduct3_lemma [OF 2 3])
88  apply (subst lfp_Tarski [OF coinduct3_mono_lemma, OF 3])
89  using 1 apply blast
90  done
91
92
93subsection \<open>Definition forms of \<open>gfp_Tarski\<close>, to control unfolding\<close>
94
95lemma def_gfp_Tarski: "\<lbrakk>h == gfp(f); mono(f)\<rbrakk> \<Longrightarrow> h = f(h)"
96  apply unfold
97  apply (erule gfp_Tarski)
98  done
99
100lemma def_coinduct: "\<lbrakk>h == gfp(f); a:A; A <= f(A)\<rbrakk> \<Longrightarrow> a: h"
101  apply unfold
102  apply (erule coinduct)
103  apply assumption
104  done
105
106lemma def_coinduct2: "\<lbrakk>h == gfp(f); a:A; A <= f(A) Un h; mono(f)\<rbrakk> \<Longrightarrow> a: h"
107  apply unfold
108  apply (erule coinduct2)
109   apply assumption
110  apply assumption
111  done
112
113lemma def_coinduct3: "\<lbrakk>h == gfp(f); a:A; A <= f(lfp(\<lambda>x. f(x) Un A Un h)); mono(f)\<rbrakk> \<Longrightarrow> a: h"
114  apply unfold
115  apply (erule coinduct3)
116   apply assumption
117  apply assumption
118  done
119
120(*Monotonicity of gfp!*)
121lemma gfp_mono: "\<lbrakk>mono(f); \<And>Z. f(Z) <= g(Z)\<rbrakk> \<Longrightarrow> gfp(f) <= gfp(g)"
122  apply (rule gfp_upperbound)
123  apply (rule subset_trans)
124   apply (rule gfp_lemma2)
125   apply assumption
126  apply (erule meta_spec)
127  done
128
129end
130