author  huffman 
Tue, 02 Mar 2010 17:21:10 0800  
changeset 35525  fa231b86cb1e 
parent 35491  92e0028a46f2 
child 35547  991a6af75978 
permissions  rwrr 
15600  1 
(* Title: HOLCF/Ssum.thy 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

2 
Author: Franz Regensburger and Brian Huffman 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

3 
*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

4 

efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

5 
header {* The type of strict sums *} 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

6 

15577  7 
theory Ssum 
31115  8 
imports Tr 
15577  9 
begin 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

10 

16083
fca38c55c8fa
added defaultsort declaration, moved cpair_less to Cprod.thy
huffman
parents:
16070
diff
changeset

11 
defaultsort pcpo 
fca38c55c8fa
added defaultsort declaration, moved cpair_less to Cprod.thy
huffman
parents:
16070
diff
changeset

12 

15593
24d770bbc44a
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

13 
subsection {* Definition of strict sum type *} 
24d770bbc44a
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

14 

35525  15 
pcpodef (Ssum) ('a, 'b) ssum (infixr "++" 10) = 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

16 
"{p :: tr \<times> ('a \<times> 'b). 
31115  17 
(fst p \<sqsubseteq> TT \<longleftrightarrow> snd (snd p) = \<bottom>) \<and> 
18 
(fst p \<sqsubseteq> FF \<longleftrightarrow> fst (snd p) = \<bottom>)}" 

29063
7619f0561cd7
pcpodef package: state two goals, instead of encoded conjunction;
wenzelm
parents:
27310
diff
changeset

19 
by simp_all 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

20 

35525  21 
instance ssum :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po 
25827
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

22 
by (rule typedef_finite_po [OF type_definition_Ssum]) 
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

23 

35525  24 
instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

25 
by (rule typedef_chfin [OF type_definition_Ssum below_Ssum_def]) 
25827
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

26 

15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

27 
syntax (xsymbols) 
35525  28 
ssum :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

29 
syntax (HTML output) 
35525  30 
ssum :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

31 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

32 
subsection {* Definitions of constructors *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

33 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

34 
definition 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

35 
sinl :: "'a \<rightarrow> ('a ++ 'b)" where 
31115  36 
"sinl = (\<Lambda> a. Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>))" 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

37 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

38 
definition 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

39 
sinr :: "'b \<rightarrow> ('a ++ 'b)" where 
31115  40 
"sinr = (\<Lambda> b. Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b))" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

41 

31115  42 
lemma sinl_Ssum: "(strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>) \<in> Ssum" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

43 
by (simp add: Ssum_def strictify_conv_if) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

44 

31115  45 
lemma sinr_Ssum: "(strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b) \<in> Ssum" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

46 
by (simp add: Ssum_def strictify_conv_if) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

47 

31115  48 
lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

49 
by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

50 

31115  51 
lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

52 
by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

53 

31115  54 
lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

55 
by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

56 

31115  57 
lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

58 
by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum) 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

59 

833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

60 
subsection {* Properties of @{term sinl} and @{term sinr} *} 
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

61 

25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

62 
text {* Ordering *} 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

63 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

64 
lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)" 
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

65 
by (simp add: below_Ssum_def Rep_Ssum_sinl strictify_conv_if) 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

66 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

67 
lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)" 
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

68 
by (simp add: below_Ssum_def Rep_Ssum_sinr strictify_conv_if) 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

69 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

70 
lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)" 
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

71 
by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if) 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

72 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

73 
lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)" 
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

74 
by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if) 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

75 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

76 
text {* Equality *} 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

77 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

78 
lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

79 
by (simp add: po_eq_conv) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

80 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

81 
lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

82 
by (simp add: po_eq_conv) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

83 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

84 
lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

85 
by (subst po_eq_conv, simp) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

86 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

87 
lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

88 
by (subst po_eq_conv, simp) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

89 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

90 
lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

91 
by (rule sinl_eq [THEN iffD1]) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

92 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

93 
lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y" 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

94 
by (rule sinr_eq [THEN iffD1]) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

95 

de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

96 
text {* Strictness *} 
17837  97 

16211
faa9691da2bc
changed to use new contI; renamed strict, defined, and inject lemmas
huffman
parents:
16083
diff
changeset

98 
lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>" 
25915  99 
by (simp add: sinl_Abs_Ssum Abs_Ssum_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

100 

16211
faa9691da2bc
changed to use new contI; renamed strict, defined, and inject lemmas
huffman
parents:
16083
diff
changeset

101 
lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>" 
25915  102 
by (simp add: sinr_Abs_Ssum Abs_Ssum_strict) 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

103 

16752
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

104 
lemma sinl_defined_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)" 
17837  105 
by (cut_tac sinl_eq [of "x" "\<bottom>"], simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

106 

16752
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

107 
lemma sinr_defined_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)" 
17837  108 
by (cut_tac sinr_eq [of "x" "\<bottom>"], simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

109 

16752
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

110 
lemma sinl_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>" 
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

111 
by simp 
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

112 

270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

113 
lemma sinr_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>" 
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

114 
by simp 
270ec60cc9e8
added lemmas sinl_defined_iff sinr_defined_iff, sinl_eq_sinr, sinr_eq_sinl; added more simp rules; cleaned up
huffman
parents:
16742
diff
changeset

115 

25882
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

116 
text {* Compactness *} 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

117 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

118 
lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

119 
by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

120 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

121 
lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

122 
by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

123 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

124 
lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

125 
unfolding compact_def 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

126 
by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinl]], simp) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

127 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

128 
lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

129 
unfolding compact_def 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

130 
by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinr]], simp) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

131 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

132 
lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

133 
by (safe elim!: compact_sinl compact_sinlD) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

134 

c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

135 
lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x" 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

136 
by (safe elim!: compact_sinr compact_sinrD) 
c58e380d9f7d
new compactness lemmas; removed duplicated flat_less_iff
huffman
parents:
25827
diff
changeset

137 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

138 
subsection {* Case analysis *} 
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

139 

16921  140 
lemma Exh_Ssum: 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

141 
"z = \<bottom> \<or> (\<exists>a. z = sinl\<cdot>a \<and> a \<noteq> \<bottom>) \<or> (\<exists>b. z = sinr\<cdot>b \<and> b \<noteq> \<bottom>)" 
31115  142 
apply (induct z rule: Abs_Ssum_induct) 
143 
apply (case_tac y, rename_tac t a b) 

144 
apply (case_tac t rule: trE) 

25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

145 
apply (rule disjI1) 
31115  146 
apply (simp add: Ssum_def Abs_Ssum_strict) 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

147 
apply (rule disjI2, rule disjI1, rule_tac x=a in exI) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

148 
apply (simp add: sinl_Abs_Ssum Ssum_def) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

149 
apply (rule disjI2, rule disjI2, rule_tac x=b in exI) 
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

150 
apply (simp add: sinr_Abs_Ssum Ssum_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

151 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

152 

35525  153 
lemma ssumE [cases type: ssum]: 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

154 
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; 
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

155 
\<And>x. \<lbrakk>p = sinl\<cdot>x; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q; 
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

156 
\<And>y. \<lbrakk>p = sinr\<cdot>y; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
16921  157 
by (cut_tac z=p in Exh_Ssum, auto) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

158 

35525  159 
lemma ssum_induct [induct type: ssum]: 
25756  160 
"\<lbrakk>P \<bottom>; 
161 
\<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x); 

162 
\<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x" 

163 
by (cases x, simp_all) 

164 

15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

165 
lemma ssumE2: 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

166 
"\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

167 
by (cases p, simp only: sinl_strict [symmetric], simp, simp) 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

168 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

169 
lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

170 
by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

171 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
29530
diff
changeset

172 
lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

173 
by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

174 

25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

175 
subsection {* Case analysis combinator *} 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

176 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

177 
definition 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
19440
diff
changeset

178 
sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where 
31115  179 
"sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s))" 
16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

180 

833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

181 
translations 
26046  182 
"case s of XCONST sinl\<cdot>x \<Rightarrow> t1  XCONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s" 
18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
17837
diff
changeset

183 

20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
17837
diff
changeset

184 
translations 
26046  185 
"\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>" 
186 
"\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)" 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

187 

25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

188 
lemma beta_sscase: 
31115  189 
"sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s)" 
190 
unfolding sscase_def by (simp add: cont_Rep_Ssum [THEN cont_compose]) 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

191 

833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

192 
lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

193 
unfolding beta_sscase by (simp add: Rep_Ssum_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

194 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

195 
lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

196 
unfolding beta_sscase by (simp add: Rep_Ssum_sinl) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

197 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

198 
lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y" 
25740
de65baf89106
changed type definition to make Iwhen and reasoning about chains unnecessary;
huffman
parents:
25131
diff
changeset

199 
unfolding beta_sscase by (simp add: Rep_Ssum_sinr) 
15593
24d770bbc44a
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

200 

16060
833be7f71ecd
Simplified version of strict sum theory, using TypedefPcpo
huffman
parents:
15606
diff
changeset

201 
lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z" 
25756  202 
by (cases z, simp_all) 
15593
24d770bbc44a
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

203 

25827
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

204 
subsection {* Strict sum preserves flatness *} 
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

205 

35525  206 
instance ssum :: (flat, flat) flat 
25827
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

207 
apply (intro_classes, clarify) 
31115  208 
apply (case_tac x, simp) 
209 
apply (case_tac y, simp_all add: flat_below_iff) 

210 
apply (case_tac y, simp_all add: flat_below_iff) 

25827
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

211 
done 
c2adeb1bae5c
new instance proofs for classes finite_po, chfin, flat
huffman
parents:
25756
diff
changeset

212 

33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

213 
subsection {* Map function for strict sums *} 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

214 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

215 
definition 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

216 
ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

217 
where 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

218 
"ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

219 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

220 
lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

221 
unfolding ssum_map_def by simp 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

222 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

223 
lemma ssum_map_sinl [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

224 
unfolding ssum_map_def by simp 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

225 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

226 
lemma ssum_map_sinr [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

227 
unfolding ssum_map_def by simp 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

228 

35491  229 
lemma ssum_map_sinl': "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)" 
230 
by (cases "x = \<bottom>") simp_all 

231 

232 
lemma ssum_map_sinr': "g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)" 

233 
by (cases "x = \<bottom>") simp_all 

234 

33808  235 
lemma ssum_map_ID: "ssum_map\<cdot>ID\<cdot>ID = ID" 
236 
unfolding ssum_map_def by (simp add: expand_cfun_eq eta_cfun) 

237 

33587  238 
lemma ssum_map_map: 
239 
"\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow> 

240 
ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) = 

241 
ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p" 

242 
apply (induct p, simp) 

243 
apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp) 

244 
apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp) 

245 
done 

246 

33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

247 
lemma ep_pair_ssum_map: 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

248 
assumes "ep_pair e1 p1" and "ep_pair e2 p2" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

249 
shows "ep_pair (ssum_map\<cdot>e1\<cdot>e2) (ssum_map\<cdot>p1\<cdot>p2)" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

250 
proof 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

251 
interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

252 
interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

253 
fix x show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

254 
by (induct x) simp_all 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

255 
fix y show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

256 
apply (induct y, simp) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

257 
apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

258 
apply (case_tac "p2\<cdot>y = \<bottom>", simp, simp add: e2p2.e_p_below) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

259 
done 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

260 
qed 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

261 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

262 
lemma deflation_ssum_map: 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

263 
assumes "deflation d1" and "deflation d2" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

264 
shows "deflation (ssum_map\<cdot>d1\<cdot>d2)" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

265 
proof 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

266 
interpret d1: deflation d1 by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

267 
interpret d2: deflation d2 by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

268 
fix x 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

269 
show "ssum_map\<cdot>d1\<cdot>d2\<cdot>(ssum_map\<cdot>d1\<cdot>d2\<cdot>x) = ssum_map\<cdot>d1\<cdot>d2\<cdot>x" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

270 
apply (induct x, simp) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

271 
apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.idem) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

272 
apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.idem) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

273 
done 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

274 
show "ssum_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

275 
apply (induct x, simp) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

276 
apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.below) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

277 
apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.below) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

278 
done 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

279 
qed 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

280 

b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

281 
lemma finite_deflation_ssum_map: 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

282 
assumes "finite_deflation d1" and "finite_deflation d2" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

283 
shows "finite_deflation (ssum_map\<cdot>d1\<cdot>d2)" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

284 
proof (intro finite_deflation.intro finite_deflation_axioms.intro) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

285 
interpret d1: finite_deflation d1 by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

286 
interpret d2: finite_deflation d2 by fact 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

287 
have "deflation d1" and "deflation d2" by fact+ 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

288 
thus "deflation (ssum_map\<cdot>d1\<cdot>d2)" by (rule deflation_ssum_map) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

289 
have "{x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq> 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

290 
(\<lambda>x. sinl\<cdot>x) ` {x. d1\<cdot>x = x} \<union> 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

291 
(\<lambda>x. sinr\<cdot>x) ` {x. d2\<cdot>x = x} \<union> {\<bottom>}" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

292 
by (rule subsetI, case_tac x, simp_all) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

293 
thus "finite {x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x}" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

294 
by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes) 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

295 
qed 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

296 

25915  297 
subsection {* Strict sum is a bifinite domain *} 
298 

35525  299 
instantiation ssum :: (bifinite, bifinite) bifinite 
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26046
diff
changeset

300 
begin 
25915  301 

26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26046
diff
changeset

302 
definition 
25915  303 
approx_ssum_def: 
33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

304 
"approx = (\<lambda>n. ssum_map\<cdot>(approx n)\<cdot>(approx n))" 
25915  305 

306 
lemma approx_sinl [simp]: "approx i\<cdot>(sinl\<cdot>x) = sinl\<cdot>(approx i\<cdot>x)" 

307 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

308 

309 
lemma approx_sinr [simp]: "approx i\<cdot>(sinr\<cdot>x) = sinr\<cdot>(approx i\<cdot>x)" 

310 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

311 

26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26046
diff
changeset

312 
instance proof 
25915  313 
fix i :: nat and x :: "'a \<oplus> 'b" 
27310  314 
show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)" 
25915  315 
unfolding approx_ssum_def by simp 
316 
show "(\<Squnion>i. approx i\<cdot>x) = x" 

317 
unfolding approx_ssum_def 

33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

318 
by (cases x, simp_all add: lub_distribs) 
25915  319 
show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" 
320 
by (cases x, simp add: approx_ssum_def, simp, simp) 

33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

321 
show "finite {x::'a \<oplus> 'b. approx i\<cdot>x = x}" 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

322 
unfolding approx_ssum_def 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

323 
by (intro finite_deflation.finite_fixes 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

324 
finite_deflation_ssum_map 
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
32960
diff
changeset

325 
finite_deflation_approx) 
25915  326 
qed 
327 

15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

328 
end 
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26046
diff
changeset

329 

c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26046
diff
changeset

330 
end 