author  haftmann 
Mon, 22 Aug 2011 22:00:36 +0200  
changeset 44414  fb25c131bd73 
parent 44363  53f4f8287606 
child 44415  ce6cd1b2344b 
permissions  rwrr 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

1 
(* Title: HOL/Predicate.thy 
30328  2 
Author: Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

3 
*) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

4 

30328  5 
header {* Predicates as relations and enumerations *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

6 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

7 
theory Predicate 
23708  8 
imports Inductive Relation 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

9 
begin 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

10 

30328  11 
notation 
41082  12 
bot ("\<bottom>") and 
13 
top ("\<top>") and 

30328  14 
inf (infixl "\<sqinter>" 70) and 
15 
sup (infixl "\<squnion>" 65) and 

16 
Inf ("\<Sqinter>_" [900] 900) and 

41082  17 
Sup ("\<Squnion>_" [900] 900) 
30328  18 

41080  19 
syntax (xsymbols) 
41082  20 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
21 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

41080  22 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
23 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

24 

30328  25 

26 
subsection {* Predicates as (complete) lattices *} 

27 

34065
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

28 
text {* 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

29 
Handy introduction and elimination rules for @{text "\<le>"} 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

30 
on unary and binary predicates 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

31 
*} 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

32 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

33 
lemma predicate1I: 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

34 
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

35 
shows "P \<le> Q" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

36 
apply (rule le_funI) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

37 
apply (rule le_boolI) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

38 
apply (rule PQ) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

39 
apply assumption 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

40 
done 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

41 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

42 
lemma predicate1D [Pure.dest?, dest?]: 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

43 
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

44 
apply (erule le_funE) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

45 
apply (erule le_boolE) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

46 
apply assumption+ 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

47 
done 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

48 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

49 
lemma rev_predicate1D: 
44414  50 
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x" 
34065
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

51 
by (rule predicate1D) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

52 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

53 
lemma predicate2I [Pure.intro!, intro!]: 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

54 
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

55 
shows "P \<le> Q" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

56 
apply (rule le_funI)+ 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

57 
apply (rule le_boolI) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

58 
apply (rule PQ) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

59 
apply assumption 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

60 
done 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

61 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

62 
lemma predicate2D [Pure.dest, dest]: 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

63 
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

64 
apply (erule le_funE)+ 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

65 
apply (erule le_boolE) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

66 
apply assumption+ 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

67 
done 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

68 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

69 
lemma rev_predicate2D: 
44414  70 
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y" 
34065
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

71 
by (rule predicate2D) 
6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

72 

6f8f9835e219
moved predicate rules to Predicate.thy; weakened default dest rule predicate1D (is not that reliable wrt. sets)
haftmann
parents:
34007
diff
changeset

73 

32779  74 
subsubsection {* Equality *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

75 

26797
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

76 
lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)" 
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

77 
by (simp add: mem_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

78 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

79 
lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

80 
by (simp add: fun_eq_iff mem_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

81 

32779  82 

83 
subsubsection {* Order relation *} 

84 

44414  85 
lemma pred_subset_eq: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> S)" 
26797
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

86 
by (simp add: mem_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

87 

44414  88 
lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)" 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

89 
by fast 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

90 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

91 

30328  92 
subsubsection {* Top and bottom elements *} 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

93 

44414  94 
lemma bot1E [no_atp, elim!]: "\<bottom> x \<Longrightarrow> P" 
41550  95 
by (simp add: bot_fun_def) 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

96 

44414  97 
lemma bot2E [elim!]: "\<bottom> x y \<Longrightarrow> P" 
41550  98 
by (simp add: bot_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

99 

44414  100 
lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

101 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

102 

44414  103 
lemma bot_empty_eq2: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

104 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

105 

44414  106 
lemma top1I [intro!]: "\<top> x" 
41550  107 
by (simp add: top_fun_def) 
41082  108 

44414  109 
lemma top2I [intro!]: "\<top> x y" 
41550  110 
by (simp add: top_fun_def) 
41082  111 

112 

113 
subsubsection {* Binary intersection *} 

114 

44414  115 
lemma inf1I [intro!]: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x" 
41550  116 
by (simp add: inf_fun_def) 
41082  117 

44414  118 
lemma inf2I [intro!]: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y" 
41550  119 
by (simp add: inf_fun_def) 
41082  120 

44414  121 
lemma inf1E [elim!]: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P" 
41550  122 
by (simp add: inf_fun_def) 
41082  123 

44414  124 
lemma inf2E [elim!]: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P" 
41550  125 
by (simp add: inf_fun_def) 
41082  126 

44414  127 
lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x" 
41550  128 
by (simp add: inf_fun_def) 
41082  129 

44414  130 
lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y" 
41550  131 
by (simp add: inf_fun_def) 
41082  132 

44414  133 
lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x" 
41550  134 
by (simp add: inf_fun_def) 
41082  135 

44414  136 
lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y" 
41550  137 
by (simp add: inf_fun_def) 
41082  138 

44414  139 
lemma inf_Int_eq: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)" 
41550  140 
by (simp add: inf_fun_def mem_def) 
41082  141 

44414  142 
lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)" 
41550  143 
by (simp add: inf_fun_def mem_def) 
41082  144 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

145 

30328  146 
subsubsection {* Binary union *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

147 

44414  148 
lemma sup1I1 [elim?]: "A x \<Longrightarrow> (A \<squnion> B) x" 
41550  149 
by (simp add: sup_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

150 

44414  151 
lemma sup2I1 [elim?]: "A x y \<Longrightarrow> (A \<squnion> B) x y" 
41550  152 
by (simp add: sup_fun_def) 
32883
7cbd93dacef3
inf/sup1/2_iff are mere duplicates of underlying definitions: dropped
haftmann
parents:
32782
diff
changeset

153 

44414  154 
lemma sup1I2 [elim?]: "B x \<Longrightarrow> (A \<squnion> B) x" 
41550  155 
by (simp add: sup_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

156 

44414  157 
lemma sup2I2 [elim?]: "B x y \<Longrightarrow> (A \<squnion> B) x y" 
41550  158 
by (simp add: sup_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

159 

44414  160 
lemma sup1E [elim!]: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P" 
41550  161 
by (simp add: sup_fun_def) iprover 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

162 

44414  163 
lemma sup2E [elim!]: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P" 
41550  164 
by (simp add: sup_fun_def) iprover 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

165 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

166 
text {* 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

167 
\medskip Classical introduction rule: no commitment to @{text A} vs 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

168 
@{text B}. 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

169 
*} 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

170 

44414  171 
lemma sup1CI [intro!]: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x" 
41550  172 
by (auto simp add: sup_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

173 

44414  174 
lemma sup2CI [intro!]: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y" 
41550  175 
by (auto simp add: sup_fun_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

176 

44414  177 
lemma sup_Un_eq: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)" 
41550  178 
by (simp add: sup_fun_def mem_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

179 

44414  180 
lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)" 
41550  181 
by (simp add: sup_fun_def mem_def) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

182 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

183 

30328  184 
subsubsection {* Intersections of families *} 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

185 

44414  186 
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)" 
41080  187 
by (simp add: INFI_apply) 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

188 

44414  189 
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)" 
41080  190 
by (simp add: INFI_apply) 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

191 

44414  192 
lemma INF1_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" 
41080  193 
by (auto simp add: INFI_apply) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

194 

44414  195 
lemma INF2_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" 
41080  196 
by (auto simp add: INFI_apply) 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

197 

44414  198 
lemma INF1_D [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" 
41080  199 
by (auto simp add: INFI_apply) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

200 

44414  201 
lemma INF2_D [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" 
41080  202 
by (auto simp add: INFI_apply) 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

203 

44414  204 
lemma INF1_E [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" 
41080  205 
by (auto simp add: INFI_apply) 
22430
6a56bf1b3a64
Generalized version of SUP and INF (with index set).
berghofe
parents:
22422
diff
changeset

206 

44414  207 
lemma INF2_E [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" 
41080  208 
by (auto simp add: INFI_apply) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

209 

44414  210 
lemma INF_INT_eq: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Sqinter>i. r i))" 
41080  211 
by (simp add: INFI_apply fun_eq_iff) 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

212 

44414  213 
lemma INF_INT_eq2: "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Sqinter>i. r i))" 
41080  214 
by (simp add: INFI_apply fun_eq_iff) 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

215 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

216 

41082  217 
subsubsection {* Unions of families *} 
218 

44414  219 
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)" 
41082  220 
by (simp add: SUPR_apply) 
221 

44414  222 
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)" 
41082  223 
by (simp add: SUPR_apply) 
224 

44414  225 
lemma SUP1_I [intro]: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" 
41082  226 
by (auto simp add: SUPR_apply) 
227 

44414  228 
lemma SUP2_I [intro]: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" 
41082  229 
by (auto simp add: SUPR_apply) 
230 

44414  231 
lemma SUP1_E [elim!]: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R" 
41082  232 
by (auto simp add: SUPR_apply) 
233 

44414  234 
lemma SUP2_E [elim!]: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R" 
41082  235 
by (auto simp add: SUPR_apply) 
236 

44414  237 
lemma SUP_UN_eq: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))" 
41082  238 
by (simp add: SUPR_apply fun_eq_iff) 
239 

44414  240 
lemma SUP_UN_eq2: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))" 
41082  241 
by (simp add: SUPR_apply fun_eq_iff) 
242 

243 

30328  244 
subsection {* Predicates as relations *} 
245 

246 
subsubsection {* Composition *} 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

247 

44414  248 
inductive pred_comp :: "['a \<Rightarrow> 'b \<Rightarrow> bool, 'b \<Rightarrow> 'c \<Rightarrow> bool] \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75) 
249 
for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where 

250 
pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

251 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

252 
inductive_cases pred_compE [elim!]: "(r OO s) a c" 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

253 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

254 
lemma pred_comp_rel_comp_eq [pred_set_conv]: 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

255 
"((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)" 
41550  256 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

257 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

258 

30328  259 
subsubsection {* Converse *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

260 

44414  261 
inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^1)" [1000] 1000) 
262 
for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

263 
conversepI: "r a b \<Longrightarrow> r^1 b a" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

264 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

265 
notation (xsymbols) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

266 
conversep ("(_\<inverse>\<inverse>)" [1000] 1000) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

267 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

268 
lemma conversepD: 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

269 
assumes ab: "r^1 a b" 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

270 
shows "r b a" using ab 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

271 
by cases simp 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

272 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

273 
lemma conversep_iff [iff]: "r^1 a b = r b a" 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

274 
by (iprover intro: conversepI dest: conversepD) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

275 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

276 
lemma conversep_converse_eq [pred_set_conv]: 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

277 
"(\<lambda>x y. (x, y) \<in> r)^1 = (\<lambda>x y. (x, y) \<in> r^1)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

278 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

279 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

280 
lemma conversep_conversep [simp]: "(r^1)^1 = r" 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

281 
by (iprover intro: order_antisym conversepI dest: conversepD) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

282 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

283 
lemma converse_pred_comp: "(r OO s)^1 = s^1 OO r^1" 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

284 
by (iprover intro: order_antisym conversepI pred_compI 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

285 
elim: pred_compE dest: conversepD) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

286 

44414  287 
lemma converse_meet: "(r \<sqinter> s)^1 = r^1 \<sqinter> s^1" 
41550  288 
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

289 

44414  290 
lemma converse_join: "(r \<squnion> s)^1 = r^1 \<squnion> s^1" 
41550  291 
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

292 

44414  293 
lemma conversep_noteq [simp]: "(op \<noteq>)^1 = op \<noteq>" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

294 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

295 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

296 
lemma conversep_eq [simp]: "(op =)^1 = op =" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

297 
by (auto simp add: fun_eq_iff) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

298 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

299 

30328  300 
subsubsection {* Domain *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

301 

44414  302 
inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" 
303 
for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

304 
DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

305 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

306 
inductive_cases DomainPE [elim!]: "DomainP r a" 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

307 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

308 
lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)" 
26797
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

309 
by (blast intro!: Orderings.order_antisym predicate1I) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

310 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

311 

30328  312 
subsubsection {* Range *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

313 

44414  314 
inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" 
315 
for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

316 
RangePI [intro]: "r a b \<Longrightarrow> RangeP r b" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

317 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

318 
inductive_cases RangePE [elim!]: "RangeP r b" 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

319 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

320 
lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)" 
26797
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

321 
by (blast intro!: Orderings.order_antisym predicate1I) 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

322 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

323 

30328  324 
subsubsection {* Inverse image *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

325 

44414  326 
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where 
327 
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

328 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

329 
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

330 
by (simp add: inv_image_def inv_imagep_def) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

331 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

332 
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)" 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

333 
by (simp add: inv_imagep_def) 
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

334 

476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

335 

30328  336 
subsubsection {* Powerset *} 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

337 

1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

338 
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where 
44414  339 
"Powp A = (\<lambda>B. \<forall>x \<in> B. A x)" 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

340 

1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

341 
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

342 
by (auto simp add: Powp_def fun_eq_iff) 
23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

343 

26797
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

344 
lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq] 
a6cb51c314f2
 Added mem_def and predicate1I in some of the proofs
berghofe
parents:
24345
diff
changeset

345 

23741
1801a921df13
 Moved infrastructure for converting between sets and predicates
berghofe
parents:
23708
diff
changeset

346 

30328  347 
subsubsection {* Properties of relations *} 
22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

348 

44414  349 
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
350 
"antisymP r \<equiv> antisym {(x, y). r x y}" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

351 

44414  352 
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
353 
"transP r \<equiv> trans {(x, y). r x y}" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

354 

44414  355 
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
356 
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

357 

40813
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

358 
(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

359 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

360 
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

361 
"reflp r \<longleftrightarrow> refl {(x, y). r x y}" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

362 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

363 
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

364 
"symp r \<longleftrightarrow> sym {(x, y). r x y}" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

365 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

366 
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

367 
"transp r \<longleftrightarrow> trans {(x, y). r x y}" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

368 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

369 
lemma reflpI: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

370 
"(\<And>x. r x x) \<Longrightarrow> reflp r" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

371 
by (auto intro: refl_onI simp add: reflp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

372 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

373 
lemma reflpE: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

374 
assumes "reflp r" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

375 
obtains "r x x" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

376 
using assms by (auto dest: refl_onD simp add: reflp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

377 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

378 
lemma sympI: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

379 
"(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

380 
by (auto intro: symI simp add: symp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

381 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

382 
lemma sympE: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

383 
assumes "symp r" and "r x y" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

384 
obtains "r y x" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

385 
using assms by (auto dest: symD simp add: symp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

386 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

387 
lemma transpI: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

388 
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

389 
by (auto intro: transI simp add: transp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

390 

f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

391 
lemma transpE: 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

392 
assumes "transp r" and "r x y" and "r y z" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

393 
obtains "r x z" 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

394 
using assms by (auto dest: transD simp add: transp_def) 
f1fc2a1547eb
moved generic definitions about relations from Quotient.thy to Predicate;
haftmann
parents:
40674
diff
changeset

395 

30328  396 

397 
subsection {* Predicates as enumerations *} 

398 

399 
subsubsection {* The type of predicate enumerations (a monad) *} 

400 

401 
datatype 'a pred = Pred "'a \<Rightarrow> bool" 

402 

403 
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where 

404 
eval_pred: "eval (Pred f) = f" 

405 

406 
lemma Pred_eval [simp]: 

407 
"Pred (eval x) = x" 

408 
by (cases x) simp 

409 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

410 
lemma pred_eqI: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

411 
"(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

412 
by (cases P, cases Q) (auto simp add: fun_eq_iff) 
30328  413 

44033  414 
instantiation pred :: (type) complete_lattice 
30328  415 
begin 
416 

417 
definition 

418 
"P \<le> Q \<longleftrightarrow> eval P \<le> eval Q" 

419 

420 
definition 

421 
"P < Q \<longleftrightarrow> eval P < eval Q" 

422 

423 
definition 

424 
"\<bottom> = Pred \<bottom>" 

425 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

426 
lemma eval_bot [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

427 
"eval \<bottom> = \<bottom>" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

428 
by (simp add: bot_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

429 

30328  430 
definition 
431 
"\<top> = Pred \<top>" 

432 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

433 
lemma eval_top [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

434 
"eval \<top> = \<top>" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

435 
by (simp add: top_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

436 

30328  437 
definition 
438 
"P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)" 

439 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

440 
lemma eval_inf [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

441 
"eval (P \<sqinter> Q) = eval P \<sqinter> eval Q" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

442 
by (simp add: inf_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

443 

30328  444 
definition 
445 
"P \<squnion> Q = Pred (eval P \<squnion> eval Q)" 

446 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

447 
lemma eval_sup [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

448 
"eval (P \<squnion> Q) = eval P \<squnion> eval Q" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

449 
by (simp add: sup_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

450 

30328  451 
definition 
37767  452 
"\<Sqinter>A = Pred (INFI A eval)" 
30328  453 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

454 
lemma eval_Inf [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

455 
"eval (\<Sqinter>A) = INFI A eval" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

456 
by (simp add: Inf_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

457 

30328  458 
definition 
37767  459 
"\<Squnion>A = Pred (SUPR A eval)" 
30328  460 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

461 
lemma eval_Sup [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

462 
"eval (\<Squnion>A) = SUPR A eval" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

463 
by (simp add: Sup_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

464 

44033  465 
instance proof 
466 
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def) 

467 

468 
end 

469 

470 
lemma eval_INFI [simp]: 

471 
"eval (INFI A f) = INFI A (eval \<circ> f)" 

472 
by (unfold INFI_def) simp 

473 

474 
lemma eval_SUPR [simp]: 

475 
"eval (SUPR A f) = SUPR A (eval \<circ> f)" 

476 
by (unfold SUPR_def) simp 

477 

478 
instantiation pred :: (type) complete_boolean_algebra 

479 
begin 

480 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

481 
definition 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

482 
" P = Pred ( eval P)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

483 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

484 
lemma eval_compl [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

485 
"eval ( P) =  eval P" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

486 
by (simp add: uminus_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

487 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

488 
definition 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

489 
"P  Q = Pred (eval P  eval Q)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

490 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

491 
lemma eval_minus [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

492 
"eval (P  Q) = eval P  eval Q" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

493 
by (simp add: minus_pred_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

494 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

495 
instance proof 
44033  496 
qed (auto intro!: pred_eqI simp add: uminus_apply minus_apply) 
30328  497 

22259
476604be7d88
New theory for converting between predicates and sets.
berghofe
parents:
diff
changeset

498 
end 
30328  499 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

500 
definition single :: "'a \<Rightarrow> 'a pred" where 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

501 
"single x = Pred ((op =) x)" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

502 

c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

503 
lemma eval_single [simp]: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

504 
"eval (single x) = (op =) x" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

505 
by (simp add: single_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

506 

c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

507 
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where 
41080  508 
"P \<guillemotright>= f = (SUPR {x. eval P x} f)" 
40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

509 

c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

510 
lemma eval_bind [simp]: 
41080  511 
"eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)" 
40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

512 
by (simp add: bind_def) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

513 

30328  514 
lemma bind_bind: 
515 
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)" 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

516 
by (rule pred_eqI) auto 
30328  517 

518 
lemma bind_single: 

519 
"P \<guillemotright>= single = P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

520 
by (rule pred_eqI) auto 
30328  521 

522 
lemma single_bind: 

523 
"single x \<guillemotright>= P = P x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

524 
by (rule pred_eqI) auto 
30328  525 

526 
lemma bottom_bind: 

527 
"\<bottom> \<guillemotright>= P = \<bottom>" 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

528 
by (rule pred_eqI) auto 
30328  529 

530 
lemma sup_bind: 

531 
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R" 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

532 
by (rule pred_eqI) auto 
30328  533 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

534 
lemma Sup_bind: 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

535 
"(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)" 
40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

536 
by (rule pred_eqI) auto 
30328  537 

538 
lemma pred_iffI: 

539 
assumes "\<And>x. eval A x \<Longrightarrow> eval B x" 

540 
and "\<And>x. eval B x \<Longrightarrow> eval A x" 

541 
shows "A = B" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

542 
using assms by (auto intro: pred_eqI) 
30328  543 

544 
lemma singleI: "eval (single x) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

545 
by simp 
30328  546 

547 
lemma singleI_unit: "eval (single ()) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

548 
by simp 
30328  549 

550 
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

551 
by simp 
30328  552 

553 
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

554 
by simp 
30328  555 

556 
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

557 
by auto 
30328  558 

559 
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

560 
by auto 
30328  561 

562 
lemma botE: "eval \<bottom> x \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

563 
by auto 
30328  564 

565 
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

566 
by auto 
30328  567 

568 
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

569 
by auto 
30328  570 

571 
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

572 
by auto 
30328  573 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

574 
lemma single_not_bot [simp]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

575 
"single x \<noteq> \<bottom>" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

576 
by (auto simp add: single_def bot_pred_def fun_eq_iff) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

577 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

578 
lemma not_bot: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

579 
assumes "A \<noteq> \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

580 
obtains x where "eval A x" 
40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

581 
using assms by (cases A) 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

582 
(auto simp add: bot_pred_def, auto simp add: mem_def) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

583 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

584 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

585 
subsubsection {* Emptiness check and definite choice *} 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

586 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

587 
definition is_empty :: "'a pred \<Rightarrow> bool" where 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

588 
"is_empty A \<longleftrightarrow> A = \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

589 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

590 
lemma is_empty_bot: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

591 
"is_empty \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

592 
by (simp add: is_empty_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

593 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

594 
lemma not_is_empty_single: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

595 
"\<not> is_empty (single x)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

596 
by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

597 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

598 
lemma is_empty_sup: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

599 
"is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B" 
36008  600 
by (auto simp add: is_empty_def) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

601 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

602 
definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where 
33111  603 
"singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

604 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

605 
lemma singleton_eqI: 
33110  606 
"\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

607 
by (auto simp add: singleton_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

608 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

609 
lemma eval_singletonI: 
33110  610 
"\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

611 
proof  
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

612 
assume assm: "\<exists>!x. eval A x" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

613 
then obtain x where "eval A x" .. 
33110  614 
moreover with assm have "singleton dfault A = x" by (rule singleton_eqI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

615 
ultimately show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

616 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

617 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

618 
lemma single_singleton: 
33110  619 
"\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

620 
proof  
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

621 
assume assm: "\<exists>!x. eval A x" 
33110  622 
then have "eval A (singleton dfault A)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

623 
by (rule eval_singletonI) 
33110  624 
moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

625 
by (rule singleton_eqI) 
33110  626 
ultimately have "eval (single (singleton dfault A)) = eval A" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

627 
by (simp (no_asm_use) add: single_def fun_eq_iff) blast 
40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

628 
then have "\<And>x. eval (single (singleton dfault A)) x = eval A x" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

629 
by simp 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

630 
then show ?thesis by (rule pred_eqI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

631 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

632 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

633 
lemma singleton_undefinedI: 
33111  634 
"\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

635 
by (simp add: singleton_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

636 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

637 
lemma singleton_bot: 
33111  638 
"singleton dfault \<bottom> = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

639 
by (auto simp add: bot_pred_def intro: singleton_undefinedI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

640 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

641 
lemma singleton_single: 
33110  642 
"singleton dfault (single x) = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

643 
by (auto simp add: intro: singleton_eqI singleI elim: singleE) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

644 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

645 
lemma singleton_sup_single_single: 
33111  646 
"singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

647 
proof (cases "x = y") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

648 
case True then show ?thesis by (simp add: singleton_single) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

649 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

650 
case False 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

651 
have "eval (single x \<squnion> single y) x" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

652 
and "eval (single x \<squnion> single y) y" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

653 
by (auto intro: supI1 supI2 singleI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

654 
with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

655 
by blast 
33111  656 
then have "singleton dfault (single x \<squnion> single y) = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

657 
by (rule singleton_undefinedI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

658 
with False show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

659 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

660 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

661 
lemma singleton_sup_aux: 
33110  662 
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B 
663 
else if B = \<bottom> then singleton dfault A 

664 
else singleton dfault 

665 
(single (singleton dfault A) \<squnion> single (singleton dfault B)))" 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

666 
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

667 
case True then show ?thesis by (simp add: single_singleton) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

668 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

669 
case False 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

670 
from False have A_or_B: 
33111  671 
"singleton dfault A = dfault () \<or> singleton dfault B = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

672 
by (auto intro!: singleton_undefinedI) 
33110  673 
then have rhs: "singleton dfault 
33111  674 
(single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

675 
by (auto simp add: singleton_sup_single_single singleton_single) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

676 
from False have not_unique: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

677 
"\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

678 
show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

679 
case True 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

680 
then obtain a b where a: "eval A a" and b: "eval B b" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

681 
by (blast elim: not_bot) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

682 
with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

683 
by (auto simp add: sup_pred_def bot_pred_def) 
33111  684 
then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

685 
with True rhs show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

686 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

687 
case False then show ?thesis by auto 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

688 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

689 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

690 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

691 
lemma singleton_sup: 
33110  692 
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B 
693 
else if B = \<bottom> then singleton dfault A 

33111  694 
else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())" 
33110  695 
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

696 

30328  697 

698 
subsubsection {* Derived operations *} 

699 

700 
definition if_pred :: "bool \<Rightarrow> unit pred" where 

701 
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)" 

702 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

703 
definition holds :: "unit pred \<Rightarrow> bool" where 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

704 
holds_eq: "holds P = eval P ()" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

705 

30328  706 
definition not_pred :: "unit pred \<Rightarrow> unit pred" where 
707 
not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())" 

708 

709 
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()" 

710 
unfolding if_pred_eq by (auto intro: singleI) 

711 

712 
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P" 

713 
unfolding if_pred_eq by (cases b) (auto elim: botE) 

714 

715 
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()" 

716 
unfolding not_pred_eq eval_pred by (auto intro: singleI) 

717 

718 
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()" 

719 
unfolding not_pred_eq by (auto intro: singleI) 

720 

721 
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

722 
unfolding not_pred_eq 

723 
by (auto split: split_if_asm elim: botE) 

724 

725 
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

726 
unfolding not_pred_eq 

727 
by (auto split: split_if_asm elim: botE) 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

728 
lemma "f () = False \<or> f () = True" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

729 
by simp 
30328  730 

37549  731 
lemma closure_of_bool_cases [no_atp]: 
44007  732 
fixes f :: "unit \<Rightarrow> bool" 
733 
assumes "f = (\<lambda>u. False) \<Longrightarrow> P f" 

734 
assumes "f = (\<lambda>u. True) \<Longrightarrow> P f" 

735 
shows "P f" 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

736 
proof  
44007  737 
have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)" 
33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

738 
apply (cases "f ()") 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

739 
apply (rule disjI2) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

740 
apply (rule ext) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

741 
apply (simp add: unit_eq) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

742 
apply (rule disjI1) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

743 
apply (rule ext) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

744 
apply (simp add: unit_eq) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

745 
done 
41550  746 
from this assms show ?thesis by blast 
33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

747 
qed 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

748 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

749 
lemma unit_pred_cases: 
44007  750 
assumes "P \<bottom>" 
751 
assumes "P (single ())" 

752 
shows "P Q" 

753 
using assms unfolding bot_pred_def Collect_def empty_def single_def proof (cases Q) 

754 
fix f 

755 
assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))" 

756 
then have "P (Pred f)" 

757 
by (cases _ f rule: closure_of_bool_cases) simp_all 

758 
moreover assume "Q = Pred f" 

759 
ultimately show "P Q" by simp 

760 
qed 

761 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

762 
lemma holds_if_pred: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

763 
"holds (if_pred b) = b" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

764 
unfolding if_pred_eq holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

765 
by (cases b) (auto intro: singleI elim: botE) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

766 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

767 
lemma if_pred_holds: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

768 
"if_pred (holds P) = P" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

769 
unfolding if_pred_eq holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

770 
by (rule unit_pred_cases) (auto intro: singleI elim: botE) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

771 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

772 
lemma is_empty_holds: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

773 
"is_empty P \<longleftrightarrow> \<not> holds P" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

774 
unfolding is_empty_def holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

775 
by (rule unit_pred_cases) (auto elim: botE intro: singleI) 
30328  776 

41311  777 
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where 
778 
"map f P = P \<guillemotright>= (single o f)" 

779 

780 
lemma eval_map [simp]: 

44363  781 
"eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))" 
41311  782 
by (auto simp add: map_def) 
783 

41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41372
diff
changeset

784 
enriched_type map: map 
44363  785 
by (rule ext, rule pred_eqI, auto)+ 
41311  786 

787 

30328  788 
subsubsection {* Implementation *} 
789 

790 
datatype 'a seq = Empty  Insert "'a" "'a pred"  Join "'a pred" "'a seq" 

791 

792 
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where 

44414  793 
"pred_of_seq Empty = \<bottom>" 
794 
 "pred_of_seq (Insert x P) = single x \<squnion> P" 

795 
 "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq" 

30328  796 

797 
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where 

798 
"Seq f = pred_of_seq (f ())" 

799 

800 
code_datatype Seq 

801 

802 
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool" where 

803 
"member Empty x \<longleftrightarrow> False" 

44414  804 
 "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x" 
805 
 "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x" 

30328  806 

807 
lemma eval_member: 

808 
"member xq = eval (pred_of_seq xq)" 

809 
proof (induct xq) 

810 
case Empty show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

811 
by (auto simp add: fun_eq_iff elim: botE) 
30328  812 
next 
813 
case Insert show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

814 
by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI) 
30328  815 
next 
816 
case Join then show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

817 
by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2) 
30328  818 
qed 
819 

820 
lemma eval_code [code]: "eval (Seq f) = member (f ())" 

821 
unfolding Seq_def by (rule sym, rule eval_member) 

822 

823 
lemma single_code [code]: 

824 
"single x = Seq (\<lambda>u. Insert x \<bottom>)" 

825 
unfolding Seq_def by simp 

826 

41080  827 
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where 
30328  828 
"apply f Empty = Empty" 
829 
 "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)" 

830 
 "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)" 

831 

832 
lemma apply_bind: 

833 
"pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f" 

834 
proof (induct xq) 

835 
case Empty show ?case 

836 
by (simp add: bottom_bind) 

837 
next 

838 
case Insert show ?case 

839 
by (simp add: single_bind sup_bind) 

840 
next 

841 
case Join then show ?case 

842 
by (simp add: sup_bind) 

843 
qed 

844 

845 
lemma bind_code [code]: 

846 
"Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))" 

847 
unfolding Seq_def by (rule sym, rule apply_bind) 

848 

849 
lemma bot_set_code [code]: 

850 
"\<bottom> = Seq (\<lambda>u. Empty)" 

851 
unfolding Seq_def by simp 

852 

30376  853 
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where 
854 
"adjunct P Empty = Join P Empty" 

855 
 "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)" 

856 
 "adjunct P (Join Q xq) = Join Q (adjunct P xq)" 

857 

858 
lemma adjunct_sup: 

859 
"pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq" 

860 
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute) 

861 

30328  862 
lemma sup_code [code]: 
863 
"Seq f \<squnion> Seq g = Seq (\<lambda>u. case f () 

864 
of Empty \<Rightarrow> g () 

865 
 Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g) 

30376  866 
 Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))" 
30328  867 
proof (cases "f ()") 
868 
case Empty 

869 
thus ?thesis 

34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
33988
diff
changeset

870 
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"]) 
30328  871 
next 
872 
case Insert 

873 
thus ?thesis 

874 
unfolding Seq_def by (simp add: sup_assoc) 

875 
next 

876 
case Join 

877 
thus ?thesis 

30376  878 
unfolding Seq_def 
879 
by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute) 

30328  880 
qed 
881 

30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

882 
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

883 
"contained Empty Q \<longleftrightarrow> True" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

884 
 "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

885 
 "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

886 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

887 
lemma single_less_eq_eval: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

888 
"single x \<le> P \<longleftrightarrow> eval P x" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

889 
by (auto simp add: single_def less_eq_pred_def mem_def) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

890 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

891 
lemma contained_less_eq: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

892 
"contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

893 
by (induct xq) (simp_all add: single_less_eq_eval) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

894 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

895 
lemma less_eq_pred_code [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

896 
"Seq f \<le> Q = (case f () 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

897 
of Empty \<Rightarrow> True 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

898 
 Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

899 
 Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

900 
by (cases "f ()") 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

901 
(simp_all add: Seq_def single_less_eq_eval contained_less_eq) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

902 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

903 
lemma eq_pred_code [code]: 
31133  904 
fixes P Q :: "'a pred" 
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

905 
shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P" 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

906 
by (auto simp add: equal) 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

907 

97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

908 
lemma [code nbe]: 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

909 
"HOL.equal (x :: 'a pred) x \<longleftrightarrow> True" 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

910 
by (fact equal_refl) 
30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

911 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

912 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

913 
"pred_case f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

914 
by (cases P) simp 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

915 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

916 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

917 
"pred_rec f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

918 
by (cases P) simp 
30328  919 

31105
95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

920 
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x" 
95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

921 

95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

922 
lemma eq_is_eq: "eq x y \<equiv> (x = y)" 
31108  923 
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases) 
30948  924 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

925 
primrec null :: "'a seq \<Rightarrow> bool" where 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

926 
"null Empty \<longleftrightarrow> True" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

927 
 "null (Insert x P) \<longleftrightarrow> False" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

928 
 "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

929 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

930 
lemma null_is_empty: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

931 
"null xq \<longleftrightarrow> is_empty (pred_of_seq xq)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

932 
by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

933 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

934 
lemma is_empty_code [code]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

935 
"is_empty (Seq f) \<longleftrightarrow> null (f ())" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

936 
by (simp add: null_is_empty Seq_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

937 

33111  938 
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where 
939 
[code del]: "the_only dfault Empty = dfault ()" 

940 
 "the_only dfault (Insert x P) = (if is_empty P then x else let y = singleton dfault P in if x = y then x else dfault ())" 

33110  941 
 "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P 
942 
else let x = singleton dfault P; y = the_only dfault xq in 

33111  943 
if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

944 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

945 
lemma the_only_singleton: 
33110  946 
"the_only dfault xq = singleton dfault (pred_of_seq xq)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

947 
by (induct xq) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

948 
(auto simp add: singleton_bot singleton_single is_empty_def 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

949 
null_is_empty Let_def singleton_sup) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

950 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

951 
lemma singleton_code [code]: 
33110  952 
"singleton dfault (Seq f) = (case f () 
33111  953 
of Empty \<Rightarrow> dfault () 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

954 
 Insert x P \<Rightarrow> if is_empty P then x 
33110  955 
else let y = singleton dfault P in 
33111  956 
if x = y then x else dfault () 
33110  957 
 Join P xq \<Rightarrow> if is_empty P then the_only dfault xq 
958 
else if null xq then singleton dfault P 

959 
else let x = singleton dfault P; y = the_only dfault xq in 

33111  960 
if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

961 
by (cases "f ()") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

962 
(auto simp add: Seq_def the_only_singleton is_empty_def 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

963 
null_is_empty singleton_bot singleton_single singleton_sup Let_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

964 

44414  965 
definition the :: "'a pred \<Rightarrow> 'a" where 
37767  966 
"the A = (THE x. eval A x)" 
33111  967 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

968 
lemma the_eqI: 
41080  969 
"(THE x. eval P x) = x \<Longrightarrow> the P = x" 
40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

970 
by (simp add: the_def) 
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

971 

44414  972 
definition not_unique :: "'a pred \<Rightarrow> 'a" where 
973 
[code del]: "not_unique A = (THE x. eval A x)" 

974 

975 
code_abort not_unique 

976 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

977 
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A" 
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

978 
by (rule the_eqI) (simp add: singleton_def not_unique_def) 
33110  979 

36531
19f6e3b0d9b6
code_reflect: specify module name directly after keyword
haftmann
parents:
36513
diff
changeset

980 
code_reflect Predicate 
36513  981 
datatypes pred = Seq and seq = Empty  Insert  Join 
982 
functions map 

983 

30948  984 
ML {* 
985 
signature PREDICATE = 

986 
sig 

987 
datatype 'a pred = Seq of (unit > 'a seq) 

988 
and 'a seq = Empty  Insert of 'a * 'a pred  Join of 'a pred * 'a seq 

30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

989 
val yield: 'a pred > ('a * 'a pred) option 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

990 
val yieldn: int > 'a pred > 'a list * 'a pred 
31222  991 
val map: ('a > 'b) > 'a pred > 'b pred 
30948  992 
end; 
993 

994 
structure Predicate : PREDICATE = 

995 
struct 

996 

36513  997 
datatype pred = datatype Predicate.pred 
998 
datatype seq = datatype Predicate.seq 

999 

1000 
fun map f = Predicate.map f; 

30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1001 

36513  1002 
fun yield (Seq f) = next (f ()) 
1003 
and next Empty = NONE 

1004 
 next (Insert (x, P)) = SOME (x, P) 

1005 
 next (Join (P, xq)) = (case yield P 

30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1006 
of NONE => next xq 
36513  1007 
 SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq)))); 
30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1008 

458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1009 
fun anamorph f k x = (if k = 0 then ([], x) 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1010 
else case f x 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1011 
of NONE => ([], x) 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1012 
 SOME (v, y) => let 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1013 
val (vs, z) = anamorph f (k  1) y 
33607  1014 
in (v :: vs, z) end); 
30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1015 

458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

1016 
fun yieldn P = anamorph yield P; 
30948  1017 

1018 
end; 

1019 
*} 

1020 

44363  1021 
lemma eval_mem [simp]: 
1022 
"x \<in> eval P \<longleftrightarrow> eval P x" 

1023 
by (simp add: mem_def) 

1024 

1025 
lemma eq_mem [simp]: 

1026 
"x \<in> (op =) y \<longleftrightarrow> x = y" 

1027 
by (auto simp add: mem_def) 

1028 

30328  1029 
no_notation 
41082  1030 
bot ("\<bottom>") and 
1031 
top ("\<top>") and 

30328  1032 
inf (infixl "\<sqinter>" 70) and 
1033 
sup (infixl "\<squnion>" 65) and 

1034 
Inf ("\<Sqinter>_" [900] 900) and 

1035 
Sup ("\<Squnion>_" [900] 900) and 

1036 
bind (infixl "\<guillemotright>=" 70) 

1037 

41080  1038 
no_syntax (xsymbols) 
41082  1039 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
1040 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

41080  1041 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
1042 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

1043 

36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
36008
diff
changeset

1044 
hide_type (open) pred seq 
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
36008
diff
changeset

1045 
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds 
33111  1046 
Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the 
30328  1047 

1048 
end 