author  huffman 
Wed, 12 Oct 2005 01:43:37 +0200  
changeset 17837  2922be3544f8 
parent 17817  405fb812e738 
child 18078  20e5a6440790 
permissions  rwrr 
15600  1 
(* Title: HOLCF/Sprod.thy 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

2 
ID: $Id$ 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

3 
Author: Franz Regensburger and Brian Huffman 
15576
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 
Strict product with typedef. 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

6 
*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

7 

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

8 
header {* The type of strict products *} 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

9 

15577  10 
theory Sprod 
16699  11 
imports Cprod 
15577  12 
begin 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

13 

16082  14 
defaultsort pcpo 
15 

15591
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

16 
subsection {* Definition of strict product type *} 
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

17 

17817  18 
pcpodef (Sprod) ('a, 'b) "**" (infixr "**" 20) = 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

19 
"{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<bottom>)}" 
16699  20 
by simp 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

21 

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

22 
syntax (xsymbols) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

23 
"**" :: "[type, type] => type" ("(_ \<otimes>/ _)" [21,20] 20) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

24 
syntax (HTML output) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

25 
"**" :: "[type, type] => type" ("(_ \<otimes>/ _)" [21,20] 20) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

26 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

27 
lemma spair_lemma: 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

28 
"<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a> \<in> Sprod" 
16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

29 
by (simp add: Sprod_def strictify_conv_if cpair_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

30 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

31 
subsection {* Definitions of constants *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

32 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

33 
consts 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

34 
sfst :: "('a ** 'b) \<rightarrow> 'a" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

35 
ssnd :: "('a ** 'b) \<rightarrow> 'b" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

36 
spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

37 
ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

38 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

39 
defs 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

40 
sfst_def: "sfst \<equiv> \<Lambda> p. cfst\<cdot>(Rep_Sprod p)" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

41 
ssnd_def: "ssnd \<equiv> \<Lambda> p. csnd\<cdot>(Rep_Sprod p)" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

42 
spair_def: "spair \<equiv> \<Lambda> a b. Abs_Sprod 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

43 
<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

44 
ssplit_def: "ssplit \<equiv> \<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p))" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

45 

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

46 
syntax 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

47 
"@stuple" :: "['a, args] => 'a ** 'b" ("(1'(:_,/ _:'))") 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

48 

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

49 
translations 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

50 
"(:x, y, z:)" == "(:x, (:y, z:):)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

51 
"(:x, y:)" == "spair$x$y" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

52 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

53 
subsection {* Case analysis *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

54 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

55 
lemma spair_Abs_Sprod: 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

56 
"(:a, b:) = Abs_Sprod <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

57 
apply (unfold spair_def) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

58 
apply (simp add: cont_Abs_Sprod spair_lemma) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

59 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

60 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

61 
lemma Exh_Sprod2: 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

62 
"z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

63 
apply (rule_tac x=z in Abs_Sprod_cases) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

64 
apply (simp add: Sprod_def) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

65 
apply (erule disjE) 
16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

66 
apply (simp add: Abs_Sprod_strict) 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

67 
apply (rule disjI2) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

68 
apply (rule_tac x="cfst\<cdot>y" in exI) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

69 
apply (rule_tac x="csnd\<cdot>y" in exI) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

70 
apply (simp add: spair_Abs_Sprod Abs_Sprod_inject spair_lemma) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

71 
apply (simp add: surjective_pairing_Cprod2) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

72 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

73 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

74 
lemma sprodE: 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

75 
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

76 
by (cut_tac z=p in Exh_Sprod2, auto) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

77 

dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

78 
subsection {* Properties of @{term spair} *} 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

79 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

80 
lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>" 
16920  81 
by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

82 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

83 
lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>" 
16920  84 
by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

85 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

86 
lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

87 
by auto 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

88 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

89 
lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

90 
by (erule contrapos_np, auto) 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

91 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

92 
lemma spair_defined [simp]: 
16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

93 
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>" 
16920  94 
by (simp add: spair_Abs_Sprod Abs_Sprod_defined cpair_defined_iff Sprod_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

95 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

96 
lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

97 
by (erule contrapos_pp, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

98 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

99 
lemma spair_eq: 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

100 
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ((:x, y:) = (:a, b:)) = (x = a \<and> y = b)" 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

101 
apply (simp add: spair_Abs_Sprod) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

102 
apply (simp add: Abs_Sprod_inject [OF _ spair_lemma] Sprod_def) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

103 
apply (simp add: strictify_conv_if) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

104 
done 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

105 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

106 
lemma spair_inject: 
16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

107 
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>; (:x, y:) = (:a, b:)\<rbrakk> \<Longrightarrow> x = a \<and> y = b" 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

108 
by (rule spair_eq [THEN iffD1]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

109 

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

110 
lemma inst_sprod_pcpo2: "UU = (:UU,UU:)" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

111 
by simp 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

112 

17837  113 
lemma Rep_Sprod_spair: 
114 
"Rep_Sprod (:a, b:) = <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>" 

115 
apply (unfold spair_def) 

116 
apply (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma) 

117 
done 

118 

119 
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)" 

120 
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if) 

121 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

122 
subsection {* Properties of @{term sfst} and @{term ssnd} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

123 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

124 
lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>" 
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

125 
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

126 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

127 
lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>" 
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

128 
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

129 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

130 
lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

131 
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

132 

16212
422f836f6b39
renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents:
16082
diff
changeset

133 
lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

134 
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

135 

16777
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

136 
lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)" 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

137 
by (rule_tac p=p in sprodE, simp_all) 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

138 

555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

139 
lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

140 
by (rule_tac p=p in sprodE, simp_all) 
16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

141 

16777
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

142 
lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>" 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

143 
by simp 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

144 

555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

145 
lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>" 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

146 
by simp 
555c8951f05c
added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents:
16751
diff
changeset

147 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

148 
lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p" 
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

149 
by (rule_tac p=p in sprodE, simp_all) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

150 

16751  151 
lemma less_sprod: "x \<sqsubseteq> y = (sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y)" 
16699  152 
apply (simp add: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod) 
16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

153 
apply (rule less_cprod) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

154 
done 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

155 

16751  156 
lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)" 
157 
by (auto simp add: po_eq_conv less_sprod) 

158 

16317
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

159 
lemma spair_less: 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

160 
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> b)" 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

161 
apply (case_tac "a = \<bottom>") 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

162 
apply (simp add: eq_UU_iff [symmetric]) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

163 
apply (case_tac "b = \<bottom>") 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

164 
apply (simp add: eq_UU_iff [symmetric]) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

165 
apply (simp add: less_sprod) 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

166 
done 
868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

167 

868eddbcaf6e
added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents:
16212
diff
changeset

168 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

169 
subsection {* Properties of @{term ssplit} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

170 

16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

171 
lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>" 
15591
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

172 
by (simp add: ssplit_def) 
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

173 

16920  174 
lemma ssplit2 [simp]: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ssplit\<cdot>f\<cdot>(:x, y:) = f\<cdot>x\<cdot>y" 
15591
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

175 
by (simp add: ssplit_def) 
50c3384ca6c4
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

176 

16553  177 
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z" 
16059
dab0d004732f
Simplified version of strict product theory, using TypedefPcpo
huffman
parents:
15930
diff
changeset

178 
by (rule_tac p=z in sprodE, simp_all) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

179 

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

180 
end 