1(*  Title:      CCL/Trancl.thy
2    Author:     Martin Coen, Cambridge University Computer Laboratory
3    Copyright   1993  University of Cambridge
4*)
5
6section \<open>Transitive closure of a relation\<close>
7
8theory Trancl
9imports CCL
10begin
11
12definition trans :: "i set \<Rightarrow> o"  (*transitivity predicate*)
13  where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"
14
15definition id :: "i set"  (*the identity relation*)
16  where "id == {p. EX x. p = <x,x>}"
17
18definition relcomp :: "[i set,i set] \<Rightarrow> i set"  (infixr "O" 60)  (*composition of relations*)
19  where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"
20
21definition rtrancl :: "i set \<Rightarrow> i set"  ("(_^*)" [100] 100)
22  where "r^* == lfp(\<lambda>s. id Un (r O s))"
23
24definition trancl :: "i set \<Rightarrow> i set"  ("(_^+)" [100] 100)
25  where "r^+ == r O rtrancl(r)"
26
27
28subsection \<open>Natural deduction for \<open>trans(r)\<close>\<close>
29
30lemma transI: "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)"
31  unfolding trans_def by blast
32
33lemma transD: "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r"
34  unfolding trans_def by blast
35
36
37subsection \<open>Identity relation\<close>
38
39lemma idI: "<a,a> : id"
40  apply (unfold id_def)
41  apply (rule CollectI)
42  apply (rule exI)
43  apply (rule refl)
44  done
45
46lemma idE: "\<lbrakk>p: id;  \<And>x. p = <x,x> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
47  apply (unfold id_def)
48  apply (erule CollectE)
49  apply blast
50  done
51
52
53subsection \<open>Composition of two relations\<close>
54
55lemma compI: "\<lbrakk><a,b>:s; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c> : r O s"
56  unfolding relcomp_def by blast
57
58(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
59lemma compE: "\<lbrakk>xz : r O s; \<And>x y z. \<lbrakk>xz = <x,z>; <x,y>:s; <y,z>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
60  unfolding relcomp_def by blast
61
62lemma compEpair: "\<lbrakk><a,c> : r O s; \<And>y. \<lbrakk><a,y>:s; <y,c>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
63  apply (erule compE)
64  apply (simp add: pair_inject)
65  done
66
67lemmas [intro] = compI idI
68  and [elim] = compE idE
69  and [elim!] = pair_inject
70
71lemma comp_mono: "\<lbrakk>r'<=r; s'<=s\<rbrakk> \<Longrightarrow> (r' O s') <= (r O s)"
72  by blast
73
74
75subsection \<open>The relation rtrancl\<close>
76
77lemma rtrancl_fun_mono: "mono(\<lambda>s. id Un (r O s))"
78  apply (rule monoI)
79  apply (rule monoI subset_refl comp_mono Un_mono)+
80  apply assumption
81  done
82
83lemma rtrancl_unfold: "r^* = id Un (r O r^*)"
84  by (rule rtrancl_fun_mono [THEN rtrancl_def [THEN def_lfp_Tarski]])
85
86(*Reflexivity of rtrancl*)
87lemma rtrancl_refl: "<a,a> : r^*"
88  apply (subst rtrancl_unfold)
89  apply blast
90  done
91
92(*Closure under composition with r*)
93lemma rtrancl_into_rtrancl: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^*"
94  apply (subst rtrancl_unfold)
95  apply blast
96  done
97
98(*rtrancl of r contains r*)
99lemma r_into_rtrancl: "<a,b> : r \<Longrightarrow> <a,b> : r^*"
100  apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])
101  apply assumption
102  done
103
104
105subsection \<open>standard induction rule\<close>
106
107lemma rtrancl_full_induct:
108  "\<lbrakk><a,b> : r^*;
109      \<And>x. P(<x,x>);
110      \<And>x y z. \<lbrakk>P(<x,y>); <x,y>: r^*; <y,z>: r\<rbrakk>  \<Longrightarrow> P(<x,z>)\<rbrakk>
111   \<Longrightarrow>  P(<a,b>)"
112  apply (erule def_induct [OF rtrancl_def])
113   apply (rule rtrancl_fun_mono)
114  apply blast
115  done
116
117(*nice induction rule*)
118lemma rtrancl_induct:
119  "\<lbrakk><a,b> : r^*;
120      P(a);
121      \<And>y z. \<lbrakk><a,y> : r^*; <y,z> : r;  P(y)\<rbrakk> \<Longrightarrow> P(z) \<rbrakk>
122    \<Longrightarrow> P(b)"
123(*by induction on this formula*)
124  apply (subgoal_tac "ALL y. <a,b> = <a,y> \<longrightarrow> P(y)")
125(*now solve first subgoal: this formula is sufficient*)
126  apply blast
127(*now do the induction*)
128  apply (erule rtrancl_full_induct)
129   apply blast
130  apply blast
131  done
132
133(*transitivity of transitive closure!! -- by induction.*)
134lemma trans_rtrancl: "trans(r^*)"
135  apply (rule transI)
136  apply (rule_tac b = z in rtrancl_induct)
137    apply (fast elim: rtrancl_into_rtrancl)+
138  done
139
140(*elimination of rtrancl -- by induction on a special formula*)
141lemma rtranclE:
142  "\<lbrakk><a,b> : r^*; a = b \<Longrightarrow> P; \<And>y. \<lbrakk><a,y> : r^*; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
143  apply (subgoal_tac "a = b | (EX y. <a,y> : r^* \<and> <y,b> : r)")
144   prefer 2
145   apply (erule rtrancl_induct)
146    apply blast
147   apply blast
148  apply blast
149  done
150
151
152subsection \<open>The relation trancl\<close>
153
154subsubsection \<open>Conversions between trancl and rtrancl\<close>
155
156lemma trancl_into_rtrancl: "<a,b> : r^+ \<Longrightarrow> <a,b> : r^*"
157  apply (unfold trancl_def)
158  apply (erule compEpair)
159  apply (erule rtrancl_into_rtrancl)
160  apply assumption
161  done
162
163(*r^+ contains r*)
164lemma r_into_trancl: "<a,b> : r \<Longrightarrow> <a,b> : r^+"
165  unfolding trancl_def by (blast intro: rtrancl_refl)
166
167(*intro rule by definition: from rtrancl and r*)
168lemma rtrancl_into_trancl1: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^+"
169  unfolding trancl_def by blast
170
171(*intro rule from r and rtrancl*)
172lemma rtrancl_into_trancl2: "\<lbrakk><a,b> : r; <b,c> : r^*\<rbrakk> \<Longrightarrow> <a,c> : r^+"
173  apply (erule rtranclE)
174   apply (erule subst)
175   apply (erule r_into_trancl)
176  apply (rule trans_rtrancl [THEN transD, THEN rtrancl_into_trancl1])
177    apply (assumption | rule r_into_rtrancl)+
178  done
179
180(*elimination of r^+ -- NOT an induction rule*)
181lemma tranclE:
182  "\<lbrakk><a,b> : r^+;
183    <a,b> : r \<Longrightarrow> P;
184    \<And>y. \<lbrakk><a,y> : r^+; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
185  apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ \<and> <y,b> : r)")
186   apply blast
187  apply (unfold trancl_def)
188  apply (erule compEpair)
189  apply (erule rtranclE)
190   apply blast
191  apply (blast intro!: rtrancl_into_trancl1)
192  done
193
194(*Transitivity of r^+.
195  Proved by unfolding since it uses transitivity of rtrancl. *)
196lemma trans_trancl: "trans(r^+)"
197  apply (unfold trancl_def)
198  apply (rule transI)
199  apply (erule compEpair)+
200  apply (erule rtrancl_into_rtrancl [THEN trans_rtrancl [THEN transD, THEN compI]])
201    apply assumption+
202  done
203
204lemma trancl_into_trancl2: "\<lbrakk><a,b> : r; <b,c> : r^+\<rbrakk> \<Longrightarrow> <a,c> : r^+"
205  apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])
206   apply assumption+
207  done
208
209end
210