author  bulwahn 
Tue, 26 Jul 2011 08:07:00 +0200  
changeset 43973  a907e541b127 
parent 37589  9c33d02656bc 
child 44890  22f665a2e91c 
permissions  rwrr 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

1 
(* Author: Stefan Berghofer, Lukas Bulwahn, TU Muenchen *) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

2 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

3 
header {* A tabled implementation of the reflexive transitive closure *} 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

4 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

5 
theory Transitive_Closure_Table 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

6 
imports Main 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

7 
begin 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

8 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

9 
inductive rtrancl_path :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

10 
for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

11 
where 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

12 
base: "rtrancl_path r x [] x" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

13 
 step: "r x y \<Longrightarrow> rtrancl_path r y ys z \<Longrightarrow> rtrancl_path r x (y # ys) z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

14 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

15 
lemma rtranclp_eq_rtrancl_path: "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

16 
proof 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

17 
assume "r\<^sup>*\<^sup>* x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

18 
then show "\<exists>xs. rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

19 
proof (induct rule: converse_rtranclp_induct) 
34912  20 
case base 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

21 
have "rtrancl_path r y [] y" by (rule rtrancl_path.base) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

22 
then show ?case .. 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

23 
next 
34912  24 
case (step x z) 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

25 
from `\<exists>xs. rtrancl_path r z xs y` 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

26 
obtain xs where "rtrancl_path r z xs y" .. 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

27 
with `r x z` have "rtrancl_path r x (z # xs) y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

28 
by (rule rtrancl_path.step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

29 
then show ?case .. 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

30 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

31 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

32 
assume "\<exists>xs. rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

33 
then obtain xs where "rtrancl_path r x xs y" .. 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

34 
then show "r\<^sup>*\<^sup>* x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

35 
proof induct 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

36 
case (base x) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

37 
show ?case by (rule rtranclp.rtrancl_refl) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

38 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

39 
case (step x y ys z) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

40 
from `r x y` `r\<^sup>*\<^sup>* y z` show ?case 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

41 
by (rule converse_rtranclp_into_rtranclp) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

42 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

43 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

44 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

45 
lemma rtrancl_path_trans: 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

46 
assumes xy: "rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

47 
and yz: "rtrancl_path r y ys z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

48 
shows "rtrancl_path r x (xs @ ys) z" using xy yz 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

49 
proof (induct arbitrary: z) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

50 
case (base x) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

51 
then show ?case by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

52 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

53 
case (step x y xs) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

54 
then have "rtrancl_path r y (xs @ ys) z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

55 
by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

56 
with `r x y` have "rtrancl_path r x (y # (xs @ ys)) z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

57 
by (rule rtrancl_path.step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

58 
then show ?case by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

59 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

60 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

61 
lemma rtrancl_path_appendE: 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

62 
assumes xz: "rtrancl_path r x (xs @ y # ys) z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

63 
obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z" using xz 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

64 
proof (induct xs arbitrary: x) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

65 
case Nil 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

66 
then have "rtrancl_path r x (y # ys) z" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

67 
then obtain xy: "r x y" and yz: "rtrancl_path r y ys z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

68 
by cases auto 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

69 
from xy have "rtrancl_path r x [y] y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

70 
by (rule rtrancl_path.step [OF _ rtrancl_path.base]) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

71 
then have "rtrancl_path r x ([] @ [y]) y" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

72 
then show ?thesis using yz by (rule Nil) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

73 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

74 
case (Cons a as) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

75 
then have "rtrancl_path r x (a # (as @ y # ys)) z" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

76 
then obtain xa: "r x a" and az: "rtrancl_path r a (as @ y # ys) z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

77 
by cases auto 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

78 
show ?thesis 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

79 
proof (rule Cons(1) [OF _ az]) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

80 
assume "rtrancl_path r y ys z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

81 
assume "rtrancl_path r a (as @ [y]) y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

82 
with xa have "rtrancl_path r x (a # (as @ [y])) y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

83 
by (rule rtrancl_path.step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

84 
then have "rtrancl_path r x ((a # as) @ [y]) y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

85 
by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

86 
then show ?thesis using `rtrancl_path r y ys z` 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

87 
by (rule Cons(2)) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

88 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

89 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

90 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

91 
lemma rtrancl_path_distinct: 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

92 
assumes xy: "rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

93 
obtains xs' where "rtrancl_path r x xs' y" and "distinct (x # xs')" using xy 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

94 
proof (induct xs rule: measure_induct_rule [of length]) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

95 
case (less xs) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

96 
show ?case 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

97 
proof (cases "distinct (x # xs)") 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

98 
case True 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

99 
with `rtrancl_path r x xs y` show ?thesis by (rule less) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

100 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

101 
case False 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

102 
then have "\<exists>as bs cs a. x # xs = as @ [a] @ bs @ [a] @ cs" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

103 
by (rule not_distinct_decomp) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

104 
then obtain as bs cs a where xxs: "x # xs = as @ [a] @ bs @ [a] @ cs" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

105 
by iprover 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

106 
show ?thesis 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

107 
proof (cases as) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

108 
case Nil 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

109 
with xxs have x: "x = a" and xs: "xs = bs @ a # cs" 
35423  110 
by auto 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

111 
from x xs `rtrancl_path r x xs y` have cs: "rtrancl_path r x cs y" 
35423  112 
by (auto elim: rtrancl_path_appendE) 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

113 
from xs have "length cs < length xs" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

114 
then show ?thesis 
35423  115 
by (rule less(1)) (iprover intro: cs less(2))+ 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

116 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

117 
case (Cons d ds) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

118 
with xxs have xs: "xs = ds @ a # (bs @ [a] @ cs)" 
35423  119 
by auto 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

120 
with `rtrancl_path r x xs y` obtain xa: "rtrancl_path r x (ds @ [a]) a" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

121 
and ay: "rtrancl_path r a (bs @ a # cs) y" 
35423  122 
by (auto elim: rtrancl_path_appendE) 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

123 
from ay have "rtrancl_path r a cs y" by (auto elim: rtrancl_path_appendE) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

124 
with xa have xy: "rtrancl_path r x ((ds @ [a]) @ cs) y" 
35423  125 
by (rule rtrancl_path_trans) 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

126 
from xs have "length ((ds @ [a]) @ cs) < length xs" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

127 
then show ?thesis 
35423  128 
by (rule less(1)) (iprover intro: xy less(2))+ 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

129 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

130 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

131 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

132 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

133 
inductive rtrancl_tab :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

134 
for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

135 
where 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

136 
base: "rtrancl_tab r xs x x" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

137 
 step: "x \<notin> set xs \<Longrightarrow> r x y \<Longrightarrow> rtrancl_tab r (x # xs) y z \<Longrightarrow> rtrancl_tab r xs x z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

138 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

139 
lemma rtrancl_path_imp_rtrancl_tab: 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

140 
assumes path: "rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

141 
and x: "distinct (x # xs)" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

142 
and ys: "({x} \<union> set xs) \<inter> set ys = {}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

143 
shows "rtrancl_tab r ys x y" using path x ys 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

144 
proof (induct arbitrary: ys) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

145 
case base 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

146 
show ?case by (rule rtrancl_tab.base) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

147 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

148 
case (step x y zs z) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

149 
then have "x \<notin> set ys" by auto 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

150 
from step have "distinct (y # zs)" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

151 
moreover from step have "({y} \<union> set zs) \<inter> set (x # ys) = {}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

152 
by auto 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

153 
ultimately have "rtrancl_tab r (x # ys) y z" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

154 
by (rule step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

155 
with `x \<notin> set ys` `r x y` 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

156 
show ?case by (rule rtrancl_tab.step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

157 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

158 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

159 
lemma rtrancl_tab_imp_rtrancl_path: 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

160 
assumes tab: "rtrancl_tab r ys x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

161 
obtains xs where "rtrancl_path r x xs y" using tab 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

162 
proof induct 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

163 
case base 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

164 
from rtrancl_path.base show ?case by (rule base) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

165 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

166 
case step show ?case by (iprover intro: step rtrancl_path.step) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

167 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

168 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

169 
lemma rtranclp_eq_rtrancl_tab_nil: "r\<^sup>*\<^sup>* x y = rtrancl_tab r [] x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

170 
proof 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

171 
assume "r\<^sup>*\<^sup>* x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

172 
then obtain xs where "rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

173 
by (auto simp add: rtranclp_eq_rtrancl_path) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

174 
then obtain xs' where xs': "rtrancl_path r x xs' y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

175 
and distinct: "distinct (x # xs')" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

176 
by (rule rtrancl_path_distinct) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

177 
have "({x} \<union> set xs') \<inter> set [] = {}" by simp 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

178 
with xs' distinct show "rtrancl_tab r [] x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

179 
by (rule rtrancl_path_imp_rtrancl_tab) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

180 
next 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

181 
assume "rtrancl_tab r [] x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

182 
then obtain xs where "rtrancl_path r x xs y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

183 
by (rule rtrancl_tab_imp_rtrancl_path) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

184 
then show "r\<^sup>*\<^sup>* x y" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

185 
by (auto simp add: rtranclp_eq_rtrancl_path) 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

186 
qed 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

187 

33870
5b0d23d2c08f
improving the setup for the tabled transitive closure thanks to usage of Andreas Lochbihler
bulwahn
parents:
33649
diff
changeset

188 
declare rtranclp_eq_rtrancl_tab_nil [code_unfold, code_inline del] 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

189 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

190 
declare rtranclp_eq_rtrancl_tab_nil[THEN iffD2, code_pred_intro] 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

191 

37589  192 
code_pred rtranclp using rtranclp_eq_rtrancl_tab_nil [THEN iffD1] by fastsimp 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

193 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

194 
subsection {* A simple example *} 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

195 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

196 
datatype ty = A  B  C 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

197 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

198 
inductive test :: "ty \<Rightarrow> ty \<Rightarrow> bool" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

199 
where 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

200 
"test A B" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

201 
 "test B A" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

202 
 "test B C" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

203 

43973
a907e541b127
adding remarks after static inspection of the invocation of the SML code generator
bulwahn
parents:
37589
diff
changeset

204 
subsubsection {* Invoking with the (legacy) SML code generator *} 
a907e541b127
adding remarks after static inspection of the invocation of the SML code generator
bulwahn
parents:
37589
diff
changeset

205 

a907e541b127
adding remarks after static inspection of the invocation of the SML code generator
bulwahn
parents:
37589
diff
changeset

206 
text {* this test can be removed once the SML code generator is deactivated *} 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

207 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

208 
code_module Test 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

209 
contains 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

210 
test1 = "test\<^sup>*\<^sup>* A C" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

211 
test2 = "test\<^sup>*\<^sup>* C A" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

212 
test3 = "test\<^sup>*\<^sup>* A _" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

213 
test4 = "test\<^sup>*\<^sup>* _ C" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

214 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

215 
ML "Test.test1" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

216 
ML "Test.test2" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

217 
ML "DSeq.list_of Test.test3" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

218 
ML "DSeq.list_of Test.test4" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

219 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

220 
subsubsection {* Invoking with the predicate compiler and the generic code generator *} 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

221 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

222 
code_pred test . 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

223 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

224 
values "{x. test\<^sup>*\<^sup>* A C}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

225 
values "{x. test\<^sup>*\<^sup>* C A}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

226 
values "{x. test\<^sup>*\<^sup>* A x}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

227 
values "{x. test\<^sup>*\<^sup>* x C}" 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

228 

33870
5b0d23d2c08f
improving the setup for the tabled transitive closure thanks to usage of Andreas Lochbihler
bulwahn
parents:
33649
diff
changeset

229 
value "test\<^sup>*\<^sup>* A C" 
5b0d23d2c08f
improving the setup for the tabled transitive closure thanks to usage of Andreas Lochbihler
bulwahn
parents:
33649
diff
changeset

230 
value "test\<^sup>*\<^sup>* C A" 
5b0d23d2c08f
improving the setup for the tabled transitive closure thanks to usage of Andreas Lochbihler
bulwahn
parents:
33649
diff
changeset

231 

36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
35423
diff
changeset

232 
hide_type ty 
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
35423
diff
changeset

233 
hide_const test A B C 
33649
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

234 

854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

235 
end 
854173fcd21c
added a tabled implementation of the reflexive transitive closure
bulwahn
parents:
diff
changeset

236 