author  huffman 
Sat, 04 Jun 2005 00:22:08 +0200  
changeset 16221  879400e029bf 
parent 16209  36ee7f6af79f 
child 16314  7102a1aaecfd 
permissions  rwrr 
15600  1 
(* Title: HOLCF/Cfun.thy 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

2 
ID: $Id$ 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

3 
Author: Franz Regensburger 
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 
Definition of the type > of continuous functions. 
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 continuous functions *} 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

9 

15577  10 
theory Cfun 
16094  11 
imports TypedefPcpo 
15577  12 
begin 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

13 

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

14 
defaultsort cpo 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

15 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

16 
subsection {* Definition of continuous function type *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

17 

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

18 
typedef (CFun) ('a, 'b) ">" (infixr 0) = "{f::'a => 'b. cont f}" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

19 
by (rule exI, fast intro: cont_const) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

20 

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

21 
syntax 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

22 
Rep_CFun :: "('a > 'b) => ('a => 'b)" ("_$_" [999,1000] 999) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

23 
(* application *) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

24 
Abs_CFun :: "('a => 'b) => ('a > 'b)" (binder "LAM " 10) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

25 
(* abstraction *) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

26 

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

27 
syntax (xsymbols) 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

28 
">" :: "[type, type] => type" ("(_ \<rightarrow>/ _)" [1,0]0) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

29 
"LAM " :: "[idts, 'a => 'b] => ('a > 'b)" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

30 
("(3\<Lambda>_./ _)" [0, 10] 10) 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

31 
Rep_CFun :: "('a > 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

32 

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

33 
syntax (HTML output) 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

34 
Rep_CFun :: "('a > 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999) 
15641  35 

16098  36 
subsection {* Class instances *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

37 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

38 
instance ">" :: (cpo, cpo) sq_ord .. 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

39 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

40 
defs (overloaded) 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

41 
less_cfun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. Rep_CFun f \<sqsubseteq> Rep_CFun g)" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

42 

16098  43 
lemma adm_CFun: "adm (\<lambda>f. f \<in> CFun)" 
44 
by (simp add: CFun_def, rule admI, rule cont_lub_fun) 

45 

46 
lemma UU_CFun: "\<bottom> \<in> CFun" 

47 
by (simp add: CFun_def inst_fun_pcpo cont_const) 

48 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

49 
instance ">" :: (cpo, cpo) po 
16094  50 
by (rule typedef_po [OF type_definition_CFun less_cfun_def]) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

51 

16098  52 
instance ">" :: (cpo, cpo) cpo 
53 
by (rule typedef_cpo [OF type_definition_CFun less_cfun_def adm_CFun]) 

54 

55 
instance ">" :: (cpo, pcpo) pcpo 

56 
by (rule typedef_pcpo_UU [OF type_definition_CFun less_cfun_def UU_CFun]) 

57 

58 
lemmas cont_Rep_CFun = 

59 
typedef_cont_Rep [OF type_definition_CFun less_cfun_def adm_CFun] 

60 

61 
lemmas cont_Abs_CFun = 

62 
typedef_cont_Abs [OF type_definition_CFun less_cfun_def adm_CFun] 

63 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

64 
lemmas Rep_CFun_strict = 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

65 
typedef_Rep_strict [OF type_definition_CFun less_cfun_def UU_CFun] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

66 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

67 
lemmas Abs_CFun_strict = 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

68 
typedef_Abs_strict [OF type_definition_CFun less_cfun_def UU_CFun] 
16098  69 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

70 
text {* Additional lemma about the isomorphism between 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

71 
@{typ "'a > 'b"} and @{term CFun} *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

72 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

73 
lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

74 
by (simp add: Abs_CFun_inverse CFun_def) 
16098  75 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

76 
text {* Betaequality for continuous functions *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

77 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

78 
lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

79 
by (simp add: Abs_CFun_inverse2) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

80 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

81 
text {* Etaequality for continuous functions *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

82 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

83 
lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

84 
by (rule Rep_CFun_inverse) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

85 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

86 
text {* Extensionality for continuous functions *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

87 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

88 
lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

89 
by (simp add: Rep_CFun_inject [symmetric] ext) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

90 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

91 
text {* lemmas about application of continuous functions *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

92 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

93 
lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

94 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

95 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

96 
lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

97 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

98 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

99 
lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

100 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

101 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

102 
subsection {* Continuity of application *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

103 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

104 
lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

105 
by (rule cont_Rep_CFun [THEN cont2cont_CF1L]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

106 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

107 
lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

108 
apply (rule_tac P = "cont" in CollectD) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

109 
apply (fold CFun_def) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

110 
apply (rule Rep_CFun) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

112 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

113 
lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

114 
lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub] 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

115 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

116 
lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

117 
lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

118 
lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

119 
lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard] 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

120 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

121 
text {* contlub, cont properties of @{term Rep_CFun} in each argument *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

122 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

123 
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(lub (range Y)) = (\<Squnion>i. f\<cdot>(Y i))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

124 
by (rule contlub_Rep_CFun2 [THEN contlubE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

125 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

126 
lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) << f\<cdot>(lub (range Y))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

127 
by (rule cont_Rep_CFun2 [THEN contE]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

128 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

129 
lemma contlub_cfun_fun: "chain F \<Longrightarrow> lub (range F)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

130 
by (rule contlub_Rep_CFun1 [THEN contlubE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

131 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

132 
lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) << lub (range F)\<cdot>x" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

133 
by (rule cont_Rep_CFun1 [THEN contE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

134 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

135 
text {* Extensionality wrt. @{term "op <<"} in @{typ "'a > 'b"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

136 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

137 
lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

138 
by (simp add: less_cfun_def less_fun_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

139 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

140 
text {* monotonicity of application *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

141 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

142 
lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

143 
by (simp add: less_cfun_def less_fun_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

144 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

145 
lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

146 
by (rule monofun_Rep_CFun2 [THEN monofunE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

147 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

148 
lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

149 
by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

150 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

151 
text {* ch2ch  rules for the type @{typ "'a > 'b"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

152 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

153 
lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

154 
by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

155 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

156 
lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

157 
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

158 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

159 
lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

160 
by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

161 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

162 
lemma ch2ch_Rep_CFun: "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

163 
apply (rule chainI) 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

164 
apply (rule monofun_cfun) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

165 
apply (erule chainE) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

166 
apply (erule chainE) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

167 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

168 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

169 
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

170 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

171 
lemma contlub_cfun: 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

172 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

173 
apply (simp only: contlub_cfun_fun) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

174 
apply (simp only: contlub_cfun_arg) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

175 
apply (rule diag_lub) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

176 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

177 
apply (erule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

178 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

179 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

180 
lemma cont_cfun: 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

181 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) << (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

182 
apply (rule thelubE) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

183 
apply (simp only: ch2ch_Rep_CFun) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

184 
apply (simp only: contlub_cfun) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

185 
done 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

186 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

187 
text {* strictness *} 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

188 

36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

189 
lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

190 
apply (rule UU_I) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

191 
apply (erule subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

192 
apply (rule minimal [THEN monofun_cfun_arg]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

193 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

194 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

195 
text {* the lub of a chain of continous functions is monotone *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

196 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

197 
lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

198 
apply (drule ch2ch_monofun [OF monofun_Rep_CFun]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

199 
apply (simp add: thelub_fun [symmetric]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

200 
apply (erule monofun_lub_fun) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

201 
apply (simp add: monofun_Rep_CFun2) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

202 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

203 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

204 
text {* a lemma about the exchange of lubs for type @{typ "'a > 'b"}: uses MF2 lemmas from Cont.thy *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

205 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

206 
lemma ex_lub_cfun: "[ chain(F); chain(Y) ] ==> 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

207 
lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

208 
lub(range(%i. lub(range(%j. F(j)$(Y i)))))" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

209 
by (simp add: diag_lub ch2ch_Rep_CFunL ch2ch_Rep_CFunR) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

210 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

211 
text {* the lub of a chain of cont. functions is continuous *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

212 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

213 
lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

214 
apply (rule cont2cont_lub) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

215 
apply (erule monofun_Rep_CFun [THEN ch2ch_monofun]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

216 
apply (rule cont_Rep_CFun2) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

217 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

218 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

219 
text {* type @{typ "'a > 'b"} is chain complete *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

220 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

221 
lemma lub_cfun: "chain F \<Longrightarrow> range F << (LAM x. LUB i. F i$x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

222 
apply (subst thelub_fun [symmetric]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

223 
apply (erule monofun_Rep_CFun [THEN ch2ch_monofun]) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

224 
apply (erule typedef_is_lub [OF type_definition_CFun less_cfun_def adm_CFun]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

225 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

226 

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

227 
lemmas thelub_cfun = lub_cfun [THEN thelubI, standard] 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

228 
 {* @{thm thelub_cfun} *} (* chain F \<Longrightarrow> lub (range F) = (\<Lambda>x. \<Squnion>i. F i\<cdot>x) *) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

229 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

230 
subsection {* Miscellaneous *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

231 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

232 
text {* Monotonicity of @{term Abs_CFun} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

233 

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

234 
lemma semi_monofun_Abs_CFun: "[ cont(f); cont(g); f<<g] ==> Abs_CFun(f)<<Abs_CFun(g)" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

235 
by (simp add: less_cfun_def Abs_CFun_inverse2) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

236 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

237 
text {* for compatibility with old HOLCFVersion *} 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

238 
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

239 
by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

240 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

241 
subsection {* Continuity of application *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

242 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

243 
text {* cont2cont lemma for @{term Rep_CFun} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

244 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

245 
lemma cont2cont_Rep_CFun: 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

246 
"\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

247 
by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

248 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

249 
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

250 

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

251 
lemma cont2mono_LAM: 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

252 
assumes p1: "!!x. cont(c1 x)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

253 
assumes p2: "!!y. monofun(%x. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

254 
shows "monofun(%x. LAM y. c1 x y)" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

255 
apply (rule monofunI) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

256 
apply (rule less_cfun_ext) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

257 
apply (simp add: p1) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

258 
apply (erule p2 [THEN monofunE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

259 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

260 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

261 
text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

262 

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

263 
lemma cont2cont_LAM: 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

264 
assumes p1: "!!x. cont(c1 x)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

265 
assumes p2: "!!y. cont(%x. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

266 
shows "cont(%x. LAM y. c1 x y)" 
16098  267 
apply (rule cont_Abs_CFun) 
268 
apply (simp add: p1 CFun_def) 

269 
apply (simp add: p2 cont2cont_CF1L_rev) 

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

270 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

271 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

272 
text {* cont2cont tactic *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

273 

16055  274 
lemmas cont_lemmas1 = 
275 
cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM 

276 

277 
text {* 

278 
Continuity simproc by Brian Huffman. 

279 
Given the term @{term "cont f"}, the procedure tries to 

280 
construct the theorem @{prop "cont f == True"}. If this 

281 
theorem cannot be completely solved by the introduction 

282 
rules, then the procedure returns a conditional rewrite 

283 
rule with the unsolved subgoals as premises. 

284 
*} 

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

285 

16055  286 
ML_setup {* 
287 
local 

288 
val rules = thms "cont_lemmas1"; 

289 
fun solve_cont sg _ t = 

290 
let val tr = instantiate' [] [SOME (cterm_of sg t)] Eq_TrueI; 

291 
val tac = REPEAT_ALL_NEW (resolve_tac rules) 1; 

292 
in Option.map fst (Seq.pull (tac tr)) end; 

293 
in 

294 
val cont_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 

295 
"continuity" ["cont f"] solve_cont; 

296 
end; 

297 
Addsimprocs [cont_proc]; 

298 
*} 

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

299 

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

300 
(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

301 
(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

302 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

303 
text {* function application is strict in its first argument *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

304 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

305 
lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

306 
by (simp add: Rep_CFun_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

307 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

308 
text {* some lemmata for functions with flat/chfin domain/range types *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

309 

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

310 
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

311 
==> !s. ? n. lub(range(Y))$s = Y n$s" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

312 
apply (rule allI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

313 
apply (subst contlub_cfun_fun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

314 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

315 
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

316 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

317 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

318 
subsection {* Continuous injectionretraction pairs *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

319 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

320 
text {* Continuous retractions are strict. *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

321 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

322 
lemma retraction_strict: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

323 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

324 
apply (rule UU_I) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

325 
apply (drule_tac x="\<bottom>" in spec) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

326 
apply (erule subst) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

327 
apply (rule monofun_cfun_arg) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

328 
apply (rule minimal) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

329 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

330 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

331 
lemma injection_eq: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

332 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

333 
apply (rule iffI) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

334 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

335 
apply simp 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

336 
apply simp 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

337 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

338 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

339 
lemma injection_defined_rev: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

340 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

341 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

342 
apply (simp add: retraction_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

343 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

344 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

345 
lemma injection_defined: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

346 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

347 
by (erule contrapos_nn, rule injection_defined_rev) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

348 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

349 
text {* propagation of flatness and chainfiniteness by retractions *} 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

350 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

351 
lemma chfin2chfin: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

352 
"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

353 
\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

354 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

355 
apply (drule_tac f=g in chain_monofun) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

356 
apply (drule chfin [rule_format]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

357 
apply (unfold max_in_chain_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

358 
apply (simp add: injection_eq) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

359 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

360 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

361 
lemma flat2flat: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

362 
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

363 
\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

364 
apply clarify 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

365 
apply (drule_tac f=g in monofun_cfun_arg) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

366 
apply (drule ax_flat [rule_format]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

367 
apply (erule disjE) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

368 
apply (simp add: injection_defined_rev) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

369 
apply (simp add: injection_eq) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

370 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

371 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

372 
text {* a result about functions with flat codomain *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

373 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

374 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

375 
by (drule ax_flat [rule_format], simp) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

376 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

377 
lemma flat_codom: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

378 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

379 
apply (case_tac "f\<cdot>x = \<bottom>") 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

380 
apply (rule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

381 
apply (rule UU_I) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

382 
apply (erule_tac t="\<bottom>" in subst) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

383 
apply (rule minimal [THEN monofun_cfun_arg]) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

384 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

385 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

386 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

387 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

388 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

389 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

390 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

391 
subsection {* Identity and composition *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

392 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

393 
consts 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

394 
ID :: "'a \<rightarrow> 'a" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

395 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

396 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

397 
syntax "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

398 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

399 
translations "f1 oo f2" == "cfcomp$f1$f2" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

400 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

401 
defs 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

402 
ID_def: "ID \<equiv> (\<Lambda> x. x)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

403 
oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

404 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

405 
lemma ID1 [simp]: "ID\<cdot>x = x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

406 
by (simp add: ID_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

407 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

408 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

409 
by (simp add: oo_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

410 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

411 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

412 
by (simp add: cfcomp1) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

413 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

414 
text {* 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

415 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

416 
The class of objects is interpretation of syntactical class pcpo. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

417 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

418 
The identity arrow is interpretation of @{term ID}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

419 
The composition of f and g is interpretation of @{text "oo"}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

420 
*} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

421 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

422 
lemma ID2 [simp]: "f oo ID = f" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

423 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

424 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

425 
lemma ID3 [simp]: "ID oo f = f" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

426 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

427 

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

428 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

429 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

430 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

431 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

432 
subsection {* Strictified functions *} 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

433 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

434 
defaultsort pcpo 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

435 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

436 
consts 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

437 
Istrictify :: "('a \<rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

438 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

439 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

440 
defs 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

441 
Istrictify_def: "Istrictify f x \<equiv> if x = \<bottom> then \<bottom> else f\<cdot>x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

442 
strictify_def: "strictify \<equiv> (\<Lambda> f x. Istrictify f x)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

443 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

444 
text {* results about strictify *} 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

445 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

446 
lemma Istrictify1: "Istrictify f \<bottom> = \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

447 
by (simp add: Istrictify_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

448 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

449 
lemma Istrictify2: "x \<noteq> \<bottom> \<Longrightarrow> Istrictify f x = f\<cdot>x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

450 
by (simp add: Istrictify_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

451 

16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

452 
lemma cont_Istrictify1: "cont (\<lambda>f. Istrictify f x)" 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

453 
apply (case_tac "x = \<bottom>") 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

454 
apply (simp add: Istrictify1) 
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

455 
apply (simp add: Istrictify2) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

456 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

457 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

458 
lemma monofun_Istrictify2: "monofun (\<lambda>x. Istrictify f x)" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

459 
apply (rule monofunI) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

460 
apply (simp add: Istrictify_def monofun_cfun_arg) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

461 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

462 
apply (simp add: eq_UU_iff) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

463 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

464 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

465 
lemma contlub_Istrictify2: "contlub (\<lambda>x. Istrictify f x)" 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

466 
apply (rule contlubI) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

467 
apply (case_tac "lub (range Y) = \<bottom>") 
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset

468 
apply (simp add: Istrictify1 chain_UU_I thelub_const) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

469 
apply (simp add: Istrictify2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

470 
apply (simp add: contlub_cfun_arg) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

471 
apply (rule lub_equal2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

472 
apply (rule chain_mono2 [THEN exE]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

473 
apply (erule chain_UU_I_inverse2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

474 
apply (assumption) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

475 
apply (blast intro: Istrictify2 [symmetric]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

476 
apply (erule chain_monofun) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

477 
apply (erule monofun_Istrictify2 [THEN ch2ch_monofun]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

478 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

479 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

480 
lemmas cont_Istrictify2 = 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

481 
monocontlub2cont [OF monofun_Istrictify2 contlub_Istrictify2, standard] 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

482 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

483 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

484 
apply (unfold strictify_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

485 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

486 
apply (rule Istrictify1) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

487 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

488 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

489 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

490 
apply (unfold strictify_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

491 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

492 
apply (erule Istrictify2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

493 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

494 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

495 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

496 
by simp 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

497 

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

498 
end 