author | wenzelm |
Fri, 17 Jun 2005 18:33:33 +0200 | |
changeset 16448 | 6c45c5416b79 |
parent 16417 | 9bc16273c2d4 |
child 16514 | 090c6a98c704 |
permissions | -rw-r--r-- |
10213 | 1 |
(* Title: HOL/Transitive_Closure.thy |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 |
Copyright 1992 University of Cambridge |
|
5 |
*) |
|
6 |
||
12691 | 7 |
header {* Reflexive and Transitive closure of a relation *} |
8 |
||
15131 | 9 |
theory Transitive_Closure |
15140 | 10 |
imports Inductive |
16417 | 11 |
uses ("../Provers/trancl.ML") |
15131 | 12 |
begin |
12691 | 13 |
|
14 |
text {* |
|
15 |
@{text rtrancl} is reflexive/transitive closure, |
|
16 |
@{text trancl} is transitive closure, |
|
17 |
@{text reflcl} is reflexive closure. |
|
18 |
||
19 |
These postfix operators have \emph{maximum priority}, forcing their |
|
20 |
operands to be atomic. |
|
21 |
*} |
|
10213 | 22 |
|
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
23 |
consts |
12691 | 24 |
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999) |
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
25 |
|
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
26 |
inductive "r^*" |
12691 | 27 |
intros |
15801 | 28 |
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*" |
29 |
rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*" |
|
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
30 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
31 |
consts |
12691 | 32 |
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999) |
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
33 |
|
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
34 |
inductive "r^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
35 |
intros |
15801 | 36 |
r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+" |
37 |
trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+" |
|
10213 | 38 |
|
39 |
syntax |
|
12691 | 40 |
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) |
10213 | 41 |
translations |
12691 | 42 |
"r^=" == "r \<union> Id" |
10213 | 43 |
|
10827 | 44 |
syntax (xsymbols) |
14361
ad2f5da643b4
* Support for raw latex output in control symbols: \<^raw...>
schirmer
parents:
14337
diff
changeset
|
45 |
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999) |
ad2f5da643b4
* Support for raw latex output in control symbols: \<^raw...>
schirmer
parents:
14337
diff
changeset
|
46 |
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999) |
ad2f5da643b4
* Support for raw latex output in control symbols: \<^raw...>
schirmer
parents:
14337
diff
changeset
|
47 |
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999) |
12691 | 48 |
|
14565 | 49 |
syntax (HTML output) |
50 |
rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999) |
|
51 |
trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999) |
|
52 |
"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999) |
|
53 |
||
12691 | 54 |
|
55 |
subsection {* Reflexive-transitive closure *} |
|
56 |
||
57 |
lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*" |
|
58 |
-- {* @{text rtrancl} of @{text r} contains @{text r} *} |
|
59 |
apply (simp only: split_tupled_all) |
|
60 |
apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl]) |
|
61 |
done |
|
62 |
||
63 |
lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*" |
|
64 |
-- {* monotonicity of @{text rtrancl} *} |
|
65 |
apply (rule subsetI) |
|
66 |
apply (simp only: split_tupled_all) |
|
67 |
apply (erule rtrancl.induct) |
|
14208 | 68 |
apply (rule_tac [2] rtrancl_into_rtrancl, blast+) |
12691 | 69 |
done |
70 |
||
12823 | 71 |
theorem rtrancl_induct [consumes 1, induct set: rtrancl]: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
72 |
assumes a: "(a, b) : r^*" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
73 |
and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
74 |
shows "P b" |
12691 | 75 |
proof - |
76 |
from a have "a = a --> P b" |
|
12823 | 77 |
by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+ |
12691 | 78 |
thus ?thesis by rules |
79 |
qed |
|
80 |
||
14404
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
81 |
lemmas rtrancl_induct2 = |
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
82 |
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), |
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
83 |
consumes 1, case_names refl step] |
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
84 |
|
12691 | 85 |
lemma trans_rtrancl: "trans(r^*)" |
86 |
-- {* transitivity of transitive closure!! -- by induction *} |
|
12823 | 87 |
proof (rule transI) |
88 |
fix x y z |
|
89 |
assume "(x, y) \<in> r\<^sup>*" |
|
90 |
assume "(y, z) \<in> r\<^sup>*" |
|
91 |
thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+ |
|
92 |
qed |
|
12691 | 93 |
|
94 |
lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] |
|
95 |
||
96 |
lemma rtranclE: |
|
97 |
"[| (a::'a,b) : r^*; (a = b) ==> P; |
|
98 |
!!y.[| (a,y) : r^*; (y,b) : r |] ==> P |
|
99 |
|] ==> P" |
|
100 |
-- {* elimination of @{text rtrancl} -- by induction on a special formula *} |
|
101 |
proof - |
|
102 |
assume major: "(a::'a,b) : r^*" |
|
103 |
case rule_context |
|
104 |
show ?thesis |
|
105 |
apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)") |
|
106 |
apply (rule_tac [2] major [THEN rtrancl_induct]) |
|
107 |
prefer 2 apply (blast!) |
|
108 |
prefer 2 apply (blast!) |
|
109 |
apply (erule asm_rl exE disjE conjE prems)+ |
|
110 |
done |
|
111 |
qed |
|
112 |
||
12823 | 113 |
lemma converse_rtrancl_into_rtrancl: |
114 |
"(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*" |
|
115 |
by (rule rtrancl_trans) rules+ |
|
12691 | 116 |
|
117 |
text {* |
|
118 |
\medskip More @{term "r^*"} equations and inclusions. |
|
119 |
*} |
|
120 |
||
121 |
lemma rtrancl_idemp [simp]: "(r^*)^* = r^*" |
|
122 |
apply auto |
|
123 |
apply (erule rtrancl_induct) |
|
124 |
apply (rule rtrancl_refl) |
|
125 |
apply (blast intro: rtrancl_trans) |
|
126 |
done |
|
127 |
||
128 |
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" |
|
129 |
apply (rule set_ext) |
|
130 |
apply (simp only: split_tupled_all) |
|
131 |
apply (blast intro: rtrancl_trans) |
|
132 |
done |
|
133 |
||
134 |
lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*" |
|
14208 | 135 |
by (drule rtrancl_mono, simp) |
12691 | 136 |
|
137 |
lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*" |
|
138 |
apply (drule rtrancl_mono) |
|
14398
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14361
diff
changeset
|
139 |
apply (drule rtrancl_mono, simp) |
12691 | 140 |
done |
141 |
||
142 |
lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*" |
|
143 |
by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD]) |
|
144 |
||
145 |
lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*" |
|
146 |
by (blast intro!: rtrancl_subset intro: r_into_rtrancl) |
|
147 |
||
148 |
lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*" |
|
149 |
apply (rule sym) |
|
14208 | 150 |
apply (rule rtrancl_subset, blast, clarify) |
12691 | 151 |
apply (rename_tac a b) |
14208 | 152 |
apply (case_tac "a = b", blast) |
12691 | 153 |
apply (blast intro!: r_into_rtrancl) |
154 |
done |
|
155 |
||
12823 | 156 |
theorem rtrancl_converseD: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
157 |
assumes r: "(x, y) \<in> (r^-1)^*" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
158 |
shows "(y, x) \<in> r^*" |
12823 | 159 |
proof - |
160 |
from r show ?thesis |
|
161 |
by induct (rules intro: rtrancl_trans dest!: converseD)+ |
|
162 |
qed |
|
12691 | 163 |
|
12823 | 164 |
theorem rtrancl_converseI: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
165 |
assumes r: "(y, x) \<in> r^*" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
166 |
shows "(x, y) \<in> (r^-1)^*" |
12823 | 167 |
proof - |
168 |
from r show ?thesis |
|
169 |
by induct (rules intro: rtrancl_trans converseI)+ |
|
170 |
qed |
|
12691 | 171 |
|
172 |
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1" |
|
173 |
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) |
|
174 |
||
14404
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
175 |
theorem converse_rtrancl_induct[consumes 1]: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
176 |
assumes major: "(a, b) : r^*" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
177 |
and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12823
diff
changeset
|
178 |
shows "P a" |
12691 | 179 |
proof - |
12823 | 180 |
from rtrancl_converseI [OF major] |
12691 | 181 |
show ?thesis |
12823 | 182 |
by induct (rules intro: cases dest!: converseD rtrancl_converseD)+ |
12691 | 183 |
qed |
184 |
||
14404
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
185 |
lemmas converse_rtrancl_induct2 = |
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
186 |
converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), |
4952c5a92e04
Transitive_Closure: added consumes and case_names attributes
nipkow
parents:
14398
diff
changeset
|
187 |
consumes 1, case_names refl step] |
12691 | 188 |
|
189 |
lemma converse_rtranclE: |
|
190 |
"[| (x,z):r^*; |
|
191 |
x=z ==> P; |
|
192 |
!!y. [| (x,y):r; (y,z):r^* |] ==> P |
|
193 |
|] ==> P" |
|
194 |
proof - |
|
195 |
assume major: "(x,z):r^*" |
|
196 |
case rule_context |
|
197 |
show ?thesis |
|
198 |
apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)") |
|
199 |
apply (rule_tac [2] major [THEN converse_rtrancl_induct]) |
|
13726 | 200 |
prefer 2 apply rules |
201 |
prefer 2 apply rules |
|
12691 | 202 |
apply (erule asm_rl exE disjE conjE prems)+ |
203 |
done |
|
204 |
qed |
|
205 |
||
206 |
ML_setup {* |
|
207 |
bind_thm ("converse_rtranclE2", split_rule |
|
208 |
(read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE"))); |
|
209 |
*} |
|
210 |
||
211 |
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r" |
|
212 |
by (blast elim: rtranclE converse_rtranclE |
|
213 |
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl) |
|
214 |
||
15551 | 215 |
lemma rtrancl_unfold: "r^* = Id Un (r O r^*)" |
216 |
by (auto intro: rtrancl_into_rtrancl elim: rtranclE) |
|
217 |
||
12691 | 218 |
|
219 |
subsection {* Transitive closure *} |
|
10331 | 220 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
221 |
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
222 |
apply (simp only: split_tupled_all) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
223 |
apply (erule trancl.induct) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
224 |
apply (rules dest: subsetD)+ |
12691 | 225 |
done |
226 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
227 |
lemma r_into_trancl': "!!p. p : r ==> p : r^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
228 |
by (simp only: split_tupled_all) (erule r_into_trancl) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
229 |
|
12691 | 230 |
text {* |
231 |
\medskip Conversions between @{text trancl} and @{text rtrancl}. |
|
232 |
*} |
|
233 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
234 |
lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
235 |
by (erule trancl.induct) rules+ |
12691 | 236 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
237 |
lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
238 |
shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
239 |
by induct rules+ |
12691 | 240 |
|
241 |
lemma rtrancl_into_trancl2: "[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+" |
|
242 |
-- {* intro rule from @{text r} and @{text rtrancl} *} |
|
14208 | 243 |
apply (erule rtranclE, rules) |
12691 | 244 |
apply (rule rtrancl_trans [THEN rtrancl_into_trancl1]) |
245 |
apply (assumption | rule r_into_rtrancl)+ |
|
246 |
done |
|
247 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
248 |
lemma trancl_induct [consumes 1, induct set: trancl]: |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
249 |
assumes a: "(a,b) : r^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
250 |
and cases: "!!y. (a, y) : r ==> P y" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
251 |
"!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
252 |
shows "P b" |
12691 | 253 |
-- {* Nice induction rule for @{text trancl} *} |
254 |
proof - |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
255 |
from a have "a = a --> P b" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
256 |
by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+ |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
257 |
thus ?thesis by rules |
12691 | 258 |
qed |
259 |
||
260 |
lemma trancl_trans_induct: |
|
261 |
"[| (x,y) : r^+; |
|
262 |
!!x y. (x,y) : r ==> P x y; |
|
263 |
!!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z |
|
264 |
|] ==> P x y" |
|
265 |
-- {* Another induction rule for trancl, incorporating transitivity *} |
|
266 |
proof - |
|
267 |
assume major: "(x,y) : r^+" |
|
268 |
case rule_context |
|
269 |
show ?thesis |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
270 |
by (rules intro: r_into_trancl major [THEN trancl_induct] prems) |
12691 | 271 |
qed |
272 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
273 |
inductive_cases tranclE: "(a, b) : r^+" |
10980 | 274 |
|
15551 | 275 |
lemma trancl_unfold: "r^+ = r Un (r O r^+)" |
276 |
by (auto intro: trancl_into_trancl elim: tranclE) |
|
277 |
||
12691 | 278 |
lemma trans_trancl: "trans(r^+)" |
279 |
-- {* Transitivity of @{term "r^+"} *} |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
280 |
proof (rule transI) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
281 |
fix x y z |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
282 |
assume "(x, y) \<in> r^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
283 |
assume "(y, z) \<in> r^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
284 |
thus "(x, z) \<in> r^+" by induct (rules!)+ |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
285 |
qed |
12691 | 286 |
|
287 |
lemmas trancl_trans = trans_trancl [THEN transD, standard] |
|
288 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
289 |
lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
290 |
shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
291 |
by induct (rules intro: trancl_trans)+ |
12691 | 292 |
|
293 |
lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+" |
|
294 |
by (erule transD [OF trans_trancl r_into_trancl]) |
|
295 |
||
296 |
lemma trancl_insert: |
|
297 |
"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}" |
|
298 |
-- {* primitive recursion for @{text trancl} over finite relations *} |
|
299 |
apply (rule equalityI) |
|
300 |
apply (rule subsetI) |
|
301 |
apply (simp only: split_tupled_all) |
|
14208 | 302 |
apply (erule trancl_induct, blast) |
12691 | 303 |
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans) |
304 |
apply (rule subsetI) |
|
305 |
apply (blast intro: trancl_mono rtrancl_mono |
|
306 |
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2) |
|
307 |
done |
|
308 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
309 |
lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
310 |
apply (drule converseD) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
311 |
apply (erule trancl.induct) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
312 |
apply (rules intro: converseI trancl_trans)+ |
12691 | 313 |
done |
314 |
||
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
315 |
lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
316 |
apply (rule converseI) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
317 |
apply (erule trancl.induct) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
318 |
apply (rules dest: converseD intro: trancl_trans)+ |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
319 |
done |
12691 | 320 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
321 |
lemma trancl_converse: "(r^-1)^+ = (r^+)^-1" |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
322 |
by (fastsimp simp add: split_tupled_all |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
323 |
intro!: trancl_converseI trancl_converseD) |
12691 | 324 |
|
325 |
lemma converse_trancl_induct: |
|
326 |
"[| (a,b) : r^+; !!y. (y,b) : r ==> P(y); |
|
327 |
!!y z.[| (y,z) : r; (z,b) : r^+; P(z) |] ==> P(y) |] |
|
328 |
==> P(a)" |
|
329 |
proof - |
|
330 |
assume major: "(a,b) : r^+" |
|
331 |
case rule_context |
|
332 |
show ?thesis |
|
333 |
apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]]) |
|
334 |
apply (rule prems) |
|
335 |
apply (erule converseD) |
|
336 |
apply (blast intro: prems dest!: trancl_converseD) |
|
337 |
done |
|
338 |
qed |
|
339 |
||
340 |
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*" |
|
14208 | 341 |
apply (erule converse_trancl_induct, auto) |
12691 | 342 |
apply (blast intro: rtrancl_trans) |
343 |
done |
|
344 |
||
13867 | 345 |
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" |
346 |
by(blast elim: tranclE dest: trancl_into_rtrancl) |
|
12691 | 347 |
|
348 |
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" |
|
349 |
by (blast dest: r_into_trancl) |
|
350 |
||
351 |
lemma trancl_subset_Sigma_aux: |
|
352 |
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" |
|
14208 | 353 |
apply (erule rtrancl_induct, auto) |
12691 | 354 |
done |
355 |
||
356 |
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A" |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
357 |
apply (rule subsetI) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
358 |
apply (simp only: split_tupled_all) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
359 |
apply (erule tranclE) |
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset
|
360 |
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ |
12691 | 361 |
done |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
362 |
|
11090 | 363 |
lemma reflcl_trancl [simp]: "(r^+)^= = r^*" |
11084 | 364 |
apply safe |
12691 | 365 |
apply (erule trancl_into_rtrancl) |
11084 | 366 |
apply (blast elim: rtranclE dest: rtrancl_into_trancl1) |
367 |
done |
|
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
368 |
|
11090 | 369 |
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" |
11084 | 370 |
apply safe |
14208 | 371 |
apply (drule trancl_into_rtrancl, simp) |
372 |
apply (erule rtranclE, safe) |
|
373 |
apply (rule r_into_trancl, simp) |
|
11084 | 374 |
apply (rule rtrancl_into_trancl1) |
14208 | 375 |
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast) |
11084 | 376 |
done |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
377 |
|
11090 | 378 |
lemma trancl_empty [simp]: "{}^+ = {}" |
11084 | 379 |
by (auto elim: trancl_induct) |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
380 |
|
11090 | 381 |
lemma rtrancl_empty [simp]: "{}^* = Id" |
11084 | 382 |
by (rule subst [OF reflcl_trancl]) simp |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
383 |
|
11090 | 384 |
lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+" |
11084 | 385 |
by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl) |
386 |
||
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
387 |
|
12691 | 388 |
text {* @{text Domain} and @{text Range} *} |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
389 |
|
11090 | 390 |
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV" |
11084 | 391 |
by blast |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
392 |
|
11090 | 393 |
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV" |
11084 | 394 |
by blast |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
395 |
|
11090 | 396 |
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*" |
11084 | 397 |
by (rule rtrancl_Un_rtrancl [THEN subst]) fast |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
398 |
|
11090 | 399 |
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*" |
11084 | 400 |
by (blast intro: subsetD [OF rtrancl_Un_subset]) |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
401 |
|
11090 | 402 |
lemma trancl_domain [simp]: "Domain (r^+) = Domain r" |
11084 | 403 |
by (unfold Domain_def) (blast dest: tranclD) |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
404 |
|
11090 | 405 |
lemma trancl_range [simp]: "Range (r^+) = Range r" |
11084 | 406 |
by (simp add: Range_def trancl_converse [symmetric]) |
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset
|
407 |
|
11115 | 408 |
lemma Not_Domain_rtrancl: |
12691 | 409 |
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)" |
410 |
apply auto |
|
411 |
by (erule rev_mp, erule rtrancl_induct, auto) |
|
412 |
||
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
413 |
|
12691 | 414 |
text {* More about converse @{text rtrancl} and @{text trancl}, should |
415 |
be merged with main body. *} |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
416 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
417 |
lemma single_valued_confluent: |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
418 |
"\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk> |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
419 |
\<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*" |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
420 |
apply(erule rtrancl_induct) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
421 |
apply simp |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
422 |
apply(erule disjE) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
423 |
apply(blast elim:converse_rtranclE dest:single_valuedD) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
424 |
apply(blast intro:rtrancl_trans) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
425 |
done |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
426 |
|
12691 | 427 |
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" |
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
428 |
by (fast intro: trancl_trans) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
429 |
|
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
430 |
lemma trancl_into_trancl [rule_format]: |
12691 | 431 |
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+" |
432 |
apply (erule trancl_induct) |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
433 |
apply (fast intro: r_r_into_trancl) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
434 |
apply (fast intro: r_r_into_trancl trancl_trans) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
435 |
done |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
436 |
|
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
437 |
lemma trancl_rtrancl_trancl: |
12691 | 438 |
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+" |
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
439 |
apply (drule tranclD) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
440 |
apply (erule exE, erule conjE) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
441 |
apply (drule rtrancl_trans, assumption) |
14208 | 442 |
apply (drule rtrancl_into_trancl2, assumption, assumption) |
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
443 |
done |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
444 |
|
12691 | 445 |
lemmas transitive_closure_trans [trans] = |
446 |
r_r_into_trancl trancl_trans rtrancl_trans |
|
447 |
trancl_into_trancl trancl_into_trancl2 |
|
448 |
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl |
|
449 |
rtrancl_trancl_trancl trancl_rtrancl_trancl |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
450 |
|
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
451 |
declare trancl_into_rtrancl [elim] |
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
452 |
|
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
453 |
declare rtranclE [cases set: rtrancl] |
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
454 |
declare tranclE [cases set: trancl] |
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
455 |
|
15551 | 456 |
|
457 |
||
458 |
||
459 |
||
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
460 |
subsection {* Setup of transitivity reasoner *} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
461 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
462 |
use "../Provers/trancl.ML"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
463 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
464 |
ML_setup {* |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
465 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
466 |
structure Trancl_Tac = Trancl_Tac_Fun ( |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
467 |
struct |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
468 |
val r_into_trancl = thm "r_into_trancl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
469 |
val trancl_trans = thm "trancl_trans"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
470 |
val rtrancl_refl = thm "rtrancl_refl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
471 |
val r_into_rtrancl = thm "r_into_rtrancl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
472 |
val trancl_into_rtrancl = thm "trancl_into_rtrancl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
473 |
val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
474 |
val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl"; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
475 |
val rtrancl_trans = thm "rtrancl_trans"; |
15096 | 476 |
|
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
477 |
fun decomp (Trueprop $ t) = |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
478 |
let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
479 |
let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*") |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
480 |
| decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+") |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
481 |
| decr r = (r,"r"); |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
482 |
val (rel,r) = decr rel; |
15531 | 483 |
in SOME (a,b,rel,r) end |
484 |
| dec _ = NONE |
|
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
485 |
in dec t end; |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
486 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
487 |
end); (* struct *) |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
488 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
489 |
simpset_ref() := simpset () |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
490 |
addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac)) |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
491 |
addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac)); |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
492 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
493 |
*} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
494 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
495 |
(* Optional methods |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
496 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
497 |
method_setup trancl = |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
498 |
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
499 |
{* simple transitivity reasoner *} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
500 |
method_setup rtrancl = |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
501 |
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
502 |
{* simple transitivity reasoner *} |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
503 |
|
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
504 |
*) |
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14565
diff
changeset
|
505 |
|
10213 | 506 |
end |