author  huffman 
Mon, 06 Dec 2010 08:30:00 0800  
changeset 41027  c599955d9806 
parent 40834  a1249aeff5b6 
child 41030  ff7d177128ef 
permissions  rwrr 
15600  1 
(* Title: HOLCF/Cfun.thy 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

2 
Author: Franz Regensburger 
35794  3 
Author: 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 

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

6 
header {* The type of continuous functions *} 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

7 

15577  8 
theory Cfun 
40772  9 
imports Cpodef Fun_Cpo Product_Cpo 
15577  10 
begin 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

11 

36452  12 
default_sort cpo 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

13 

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

14 
subsection {* Definition of continuous function type *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

15 

40327  16 
cpodef ('a, 'b) cfun (infixr ">" 0) = "{f::'a => 'b. cont f}" 
40011
b974cf829099
cleaned up Fun_Cpo.thy; deprecated a few theorem names
huffman
parents:
40008
diff
changeset

17 
by (auto intro: cont_const adm_cont) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

18 

35427  19 
type_notation (xsymbols) 
35525  20 
cfun ("(_ \<rightarrow>/ _)" [1, 0] 0) 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

21 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

22 
notation 
40327  23 
Rep_cfun ("(_$/_)" [999,1000] 999) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

24 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

25 
notation (xsymbols) 
40327  26 
Rep_cfun ("(_\<cdot>/_)" [999,1000] 999) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

27 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

28 
notation (HTML output) 
40327  29 
Rep_cfun ("(_\<cdot>/_)" [999,1000] 999) 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

30 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

31 
subsection {* Syntax for continuous lambda abstraction *} 
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

32 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

33 
syntax "_cabs" :: "'a" 
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

34 

20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

35 
parse_translation {* 
40327  36 
(* rewrite (_cabs x t) => (Abs_cfun (%x. t)) *) 
37 
[mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_cfun})]; 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

38 
*} 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

39 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

40 
text {* To avoid etacontraction of body: *} 
18087  41 
typed_print_translation {* 
18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

42 
let 
18087  43 
fun cabs_tr' _ _ [Abs abs] = let 
44 
val (x,t) = atomic_abs_tr' abs 

35115  45 
in Syntax.const @{syntax_const "_cabs"} $ x $ t end 
18087  46 

47 
 cabs_tr' _ T [t] = let 

48 
val xT = domain_type (domain_type T); 

49 
val abs' = ("x",xT,(incr_boundvars 1 t)$Bound 0); 

50 
val (x,t') = atomic_abs_tr' abs'; 

35115  51 
in Syntax.const @{syntax_const "_cabs"} $ x $ t' end; 
18087  52 

40327  53 
in [(@{const_syntax Abs_cfun}, cabs_tr')] end; 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

54 
*} 
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

55 

18087  56 
text {* Syntax for nested abstractions *} 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

57 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

58 
syntax 
18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

59 
"_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3LAM _./ _)" [1000, 10] 10) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

60 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

61 
syntax (xsymbols) 
25927  62 
"_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

63 

17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

64 
parse_ast_translation {* 
35115  65 
(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *) 
66 
(* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *) 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

67 
let 
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

68 
fun Lambda_ast_tr [pats, body] = 
35115  69 
Syntax.fold_ast_p @{syntax_const "_cabs"} 
70 
(Syntax.unfold_ast @{syntax_const "_cargs"} pats, body) 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

71 
 Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts); 
35115  72 
in [(@{syntax_const "_Lambda"}, Lambda_ast_tr)] end; 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

73 
*} 
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

74 

9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

75 
print_ast_translation {* 
35115  76 
(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *) 
77 
(* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *) 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

78 
let 
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

79 
fun cabs_ast_tr' asts = 
35115  80 
(case Syntax.unfold_ast_p @{syntax_const "_cabs"} 
81 
(Syntax.Appl (Syntax.Constant @{syntax_const "_cabs"} :: asts)) of 

18078
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

82 
([], _) => raise Syntax.AST ("cabs_ast_tr'", asts) 
20e5a6440790
change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents:
18076
diff
changeset

83 
 (xs, body) => Syntax.Appl 
35115  84 
[Syntax.Constant @{syntax_const "_Lambda"}, 
85 
Syntax.fold_ast @{syntax_const "_cargs"} xs, body]); 

86 
in [(@{syntax_const "_cabs"}, cabs_ast_tr')] end 

17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

87 
*} 
15641  88 

18087  89 
text {* Dummy patterns for continuous abstraction *} 
18079  90 
translations 
40327  91 
"\<Lambda> _. t" => "CONST Abs_cfun (\<lambda> _. t)" 
18087  92 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

93 
subsection {* Continuous function space is pointed *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

94 

40327  95 
lemma UU_cfun: "\<bottom> \<in> cfun" 
96 
by (simp add: cfun_def inst_fun_pcpo) 

16098  97 

35525  98 
instance cfun :: (cpo, discrete_cpo) discrete_cpo 
40327  99 
by intro_classes (simp add: below_cfun_def Rep_cfun_inject) 
26025  100 

35525  101 
instance cfun :: (cpo, pcpo) pcpo 
40327  102 
by (rule typedef_pcpo [OF type_definition_cfun below_cfun_def UU_cfun]) 
16098  103 

40327  104 
lemmas Rep_cfun_strict = 
105 
typedef_Rep_strict [OF type_definition_cfun below_cfun_def UU_cfun] 

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

106 

40327  107 
lemmas Abs_cfun_strict = 
108 
typedef_Abs_strict [OF type_definition_cfun below_cfun_def UU_cfun] 

16098  109 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

110 
text {* function application is strict in its first argument *} 
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

111 

40327  112 
lemma Rep_cfun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>" 
113 
by (simp add: Rep_cfun_strict) 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

114 

35641  115 
lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>" 
40327  116 
by (simp add: inst_fun_pcpo [symmetric] Abs_cfun_strict) 
35641  117 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

118 
text {* for compatibility with old HOLCFVersion *} 
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

119 
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)" 
35641  120 
by simp 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

121 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

122 
subsection {* Basic properties of continuous functions *} 
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

123 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

124 
text {* Betaequality for continuous functions *} 
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

125 

40327  126 
lemma Abs_cfun_inverse2: "cont f \<Longrightarrow> Rep_cfun (Abs_cfun f) = f" 
127 
by (simp add: Abs_cfun_inverse cfun_def) 

16098  128 

37083
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

129 
lemma beta_cfun: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u" 
40327  130 
by (simp add: Abs_cfun_inverse2) 
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

131 

37083
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

132 
text {* Betareduction simproc *} 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

133 

03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

134 
text {* 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

135 
Given the term @{term "(\<Lambda> x. f x)\<cdot>y"}, the procedure tries to 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

136 
construct the theorem @{term "(\<Lambda> x. f x)\<cdot>y == f y"}. If this 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

137 
theorem cannot be completely solved by the cont2cont rules, then 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

138 
the procedure returns the ordinary conditional @{text beta_cfun} 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

139 
rule. 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

140 

03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

141 
The simproc does not solve any more goals that would be solved by 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

142 
using @{text beta_cfun} as a simp rule. The advantage of the 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

143 
simproc is that it can avoid deeplynested calls to the simplifier 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

144 
that would otherwise be caused by large continuity side conditions. 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

145 
*} 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

146 

40327  147 
simproc_setup beta_cfun_proc ("Abs_cfun f\<cdot>x") = {* 
37083
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

148 
fn phi => fn ss => fn ct => 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

149 
let 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

150 
val dest = Thm.dest_comb; 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

151 
val (f, x) = (apfst (snd o dest o snd o dest) o dest) ct; 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

152 
val [T, U] = Thm.dest_ctyp (ctyp_of_term f); 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

153 
val tr = instantiate' [SOME T, SOME U] [SOME f, SOME x] 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

154 
(mk_meta_eq @{thm beta_cfun}); 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

155 
val rules = Cont2ContData.get (Simplifier.the_context ss); 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

156 
val tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules)); 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

157 
in SOME (perhaps (SINGLE (tac 1)) tr) end 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

158 
*} 
03a70ab79dd9
add beta_cfun simproc, which uses cont2cont rules
huffman
parents:
37079
diff
changeset

159 

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

160 
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

161 

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 eta_cfun: "(\<Lambda> x. f\<cdot>x) = f" 
40327  163 
by (rule Rep_cfun_inverse) 
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 

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 
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

166 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

167 
lemma cfun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f\<cdot>x = g\<cdot>x)" 
40327  168 
by (simp add: Rep_cfun_inject [symmetric] fun_eq_iff) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

169 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

170 
lemma cfun_eqI: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g" 
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

171 
by (simp add: cfun_eq_iff) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

172 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

173 
text {* Extensionality wrt. ordering for continuous functions *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

174 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

175 
lemma cfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
40327  176 
by (simp add: below_cfun_def fun_below_iff) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

177 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

178 
lemma cfun_belowI: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g" 
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

179 
by (simp add: cfun_below_iff) 
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

180 

e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

181 
text {* Congruence for continuous function application *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

182 

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

183 
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

184 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

185 

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

186 
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

187 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

188 

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

189 
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

190 
by simp 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

191 

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

192 
subsection {* Continuity of application *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

193 

40327  194 
lemma cont_Rep_cfun1: "cont (\<lambda>f. f\<cdot>x)" 
40834
a1249aeff5b6
change cpodefgenerated cont_Rep rules to cont2cont format
huffman
parents:
40794
diff
changeset

195 
by (rule cont_Rep_cfun [OF cont_id, THEN cont2cont_fun]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

196 

40327  197 
lemma cont_Rep_cfun2: "cont (\<lambda>x. f\<cdot>x)" 
198 
apply (cut_tac x=f in Rep_cfun) 

199 
apply (simp add: cfun_def) 

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

200 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

201 

40327  202 
lemmas monofun_Rep_cfun = cont_Rep_cfun [THEN cont2mono] 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

203 

40327  204 
lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono, standard] 
205 
lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono, 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

206 

40327  207 
text {* contlub, cont properties of @{term Rep_cfun} in each argument *} 
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

208 

27413  209 
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))" 
40327  210 
by (rule cont_Rep_cfun2 [THEN cont2contlubE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

211 

27413  212 
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" 
40327  213 
by (rule cont_Rep_cfun1 [THEN cont2contlubE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

214 

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

215 
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

216 

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

217 
lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x" 
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

218 
by (simp add: cfun_below_iff) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

219 

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

220 
lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y" 
40327  221 
by (rule monofun_Rep_cfun2 [THEN monofunE]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

222 

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

223 
lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y" 
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
31041
diff
changeset

224 
by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

225 

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

226 
text {* ch2ch  rules for the type @{typ "'a > 'b"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

227 

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 
lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
40327  229 
by (erule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 
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

230 

40327  231 
lemma ch2ch_Rep_cfunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
232 
by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 

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

233 

40327  234 
lemma ch2ch_Rep_cfunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" 
235 
by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun]) 

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

236 

40327  237 
lemma ch2ch_Rep_cfun [simp]: 
18076  238 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 
25884  239 
by (simp add: chain_def monofun_cfun) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

240 

25884  241 
lemma ch2ch_LAM [simp]: 
242 
"\<lbrakk>\<And>x. chain (\<lambda>i. S i x); \<And>i. cont (\<lambda>x. S i x)\<rbrakk> \<Longrightarrow> chain (\<lambda>i. \<Lambda> x. S i x)" 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

243 
by (simp add: chain_def cfun_below_iff) 
18092
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
18091
diff
changeset

244 

40327  245 
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

246 

41027  247 
lemma lub_APP: 
248 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i\<cdot>(Y i)) = (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)" 

18076  249 
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

250 

41027  251 
lemma cont_cfun: 
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

252 
"\<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

253 
apply (rule thelubE) 
40327  254 
apply (simp only: ch2ch_Rep_cfun) 
41027  255 
apply (simp only: lub_APP) 
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

256 
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

257 

41027  258 
lemma lub_LAM: 
18092
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
18091
diff
changeset

259 
"\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk> 
41027  260 
\<Longrightarrow> (\<Squnion>i. \<Lambda> x. F i x) = (\<Lambda> x. \<Squnion>i. F i x)" 
40770
6023808b38d4
rename cpodef theorems: lub_foo > is_lub_foo, thelub_foo > lub_foo
huffman
parents:
40767
diff
changeset

261 
apply (simp add: lub_cfun) 
40327  262 
apply (simp add: Abs_cfun_inverse2) 
18092
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
18091
diff
changeset

263 
apply (simp add: thelub_fun ch2ch_lambda) 
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
18091
diff
changeset

264 
done 
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
18091
diff
changeset

265 

41027  266 
lemmas lub_distribs = lub_APP lub_LAM 
25901  267 

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

268 
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

269 

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

270 
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

271 
apply (rule UU_I) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

272 
apply (erule subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

273 
apply (rule minimal [THEN monofun_cfun_arg]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

274 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

275 

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

276 
text {* type @{typ "'a > 'b"} is chain complete *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

277 

16920  278 
lemma lub_cfun: "chain F \<Longrightarrow> range F << (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" 
41027  279 
by (simp only: contlub_cfun_fun [symmetric] eta_cfun cpo_lubI) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

280 

27413  281 
lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" 
40771  282 
by (rule lub_cfun [THEN lub_eqI]) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

283 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

284 
subsection {* Continuity simplification procedure *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

285 

40327  286 
text {* cont2cont lemma for @{term Rep_cfun} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

287 

40326
73d45866dbda
renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents:
40093
diff
changeset

288 
lemma cont2cont_APP [simp, cont2cont]: 
29049  289 
assumes f: "cont (\<lambda>x. f x)" 
290 
assumes t: "cont (\<lambda>x. t x)" 

291 
shows "cont (\<lambda>x. (f x)\<cdot>(t x))" 

292 
proof  

40006  293 
have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)" 
40327  294 
using cont_Rep_cfun1 f by (rule cont_compose) 
40006  295 
show "cont (\<lambda>x. (f x)\<cdot>(t x))" 
40327  296 
using t cont_Rep_cfun2 1 by (rule cont_apply) 
29049  297 
qed 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

298 

40008  299 
text {* 
300 
Two specific lemmas for the combination of LCF and HOL terms. 

301 
These lemmas are needed in theories that use types like @{typ "'a \<rightarrow> 'b \<Rightarrow> 'c"}. 

302 
*} 

303 

40326
73d45866dbda
renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents:
40093
diff
changeset

304 
lemma cont_APP_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s)" 
73d45866dbda
renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents:
40093
diff
changeset

305 
by (rule cont2cont_APP [THEN cont2cont_fun]) 
40008  306 

40326
73d45866dbda
renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents:
40093
diff
changeset

307 
lemma cont_APP_app_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s t)" 
73d45866dbda
renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents:
40093
diff
changeset

308 
by (rule cont_APP_app [THEN cont2cont_fun]) 
40008  309 

310 

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

311 
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

312 

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

313 
lemma cont2mono_LAM: 
29049  314 
"\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk> 
315 
\<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)" 

40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

316 
unfolding monofun_def cfun_below_iff by simp 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

317 

29049  318 
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

319 

29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

320 
text {* 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

321 
Not suitable as a cont2cont rule, because on nested lambdas 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

322 
it causes exponential blowup in the number of subgoals. 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

323 
*} 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

324 

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

325 
lemma cont2cont_LAM: 
29049  326 
assumes f1: "\<And>x. cont (\<lambda>y. f x y)" 
327 
assumes f2: "\<And>y. cont (\<lambda>x. f x y)" 

328 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 

40327  329 
proof (rule cont_Abs_cfun) 
29049  330 
fix x 
40327  331 
from f1 show "f x \<in> cfun" by (simp add: cfun_def) 
29049  332 
from f2 show "cont f" by (rule cont2cont_lambda) 
333 
qed 

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

334 

29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

335 
text {* 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

336 
This version does work as a cont2cont rule, since it 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

337 
has only a single subgoal. 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

338 
*} 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

339 

37079
0cd15d8c90a0
remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents:
36452
diff
changeset

340 
lemma cont2cont_LAM' [simp, cont2cont]: 
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

341 
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

342 
assumes f: "cont (\<lambda>p. f (fst p) (snd p))" 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

343 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 
39808
1410c84013b9
rename cont2cont_split to cont2cont_prod_case; add lemmas prod_contI, prod_cont_iff; simplify some proofs
huffman
parents:
39302
diff
changeset

344 
using assms by (simp add: cont2cont_LAM prod_cont_iff) 
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

345 

37079
0cd15d8c90a0
remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents:
36452
diff
changeset

346 
lemma cont2cont_LAM_discrete [simp, cont2cont]: 
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

347 
"(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)" 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset

348 
by (simp add: cont2cont_LAM) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

349 

17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

350 
subsection {* Miscellaneous *} 
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset

351 

40327  352 
text {* Monotonicity of @{term Abs_cfun} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

353 

40433
3128c2a54785
remove some unnecessary lemmas; move monofun_LAM to Cfun.thy
huffman
parents:
40327
diff
changeset

354 
lemma monofun_LAM: 
3128c2a54785
remove some unnecessary lemmas; move monofun_LAM to Cfun.thy
huffman
parents:
40327
diff
changeset

355 
"\<lbrakk>cont f; cont g; \<And>x. f x \<sqsubseteq> g x\<rbrakk> \<Longrightarrow> (\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)" 
3128c2a54785
remove some unnecessary lemmas; move monofun_LAM to Cfun.thy
huffman
parents:
40327
diff
changeset

356 
by (simp add: cfun_below_iff) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

357 

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

358 
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

359 

40327  360 
lemma chfin_Rep_cfunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
27413  361 
==> !s. ? n. (LUB i. Y i)$s = Y n$s" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

362 
apply (rule allI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

363 
apply (subst contlub_cfun_fun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

364 
apply assumption 
40771  365 
apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

366 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

367 

18089  368 
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))" 
369 
by (rule adm_subst, simp, rule adm_chfin) 

370 

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

371 
subsection {* Continuous injectionretraction pairs *} 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

372 

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

373 
text {* Continuous retractions are strict. *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

374 

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

375 
lemma retraction_strict: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

376 
"\<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

377 
apply (rule UU_I) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

378 
apply (drule_tac x="\<bottom>" in spec) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

379 
apply (erule subst) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

380 
apply (rule monofun_cfun_arg) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

381 
apply (rule minimal) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

382 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

383 

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

384 
lemma injection_eq: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

385 
"\<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

386 
apply (rule iffI) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

387 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

388 
apply simp 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

389 
apply simp 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

390 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

391 

31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less>below in many lemma names
huffman
parents:
31041
diff
changeset

392 
lemma injection_below: 
16314  393 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)" 
394 
apply (rule iffI) 

395 
apply (drule_tac f=f in monofun_cfun_arg) 

396 
apply simp 

397 
apply (erule monofun_cfun_arg) 

398 
done 

399 

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

400 
lemma injection_defined_rev: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

401 
"\<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

402 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

403 
apply (simp add: retraction_strict) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

404 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

405 

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

406 
lemma injection_defined: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

407 
"\<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

408 
by (erule contrapos_nn, rule injection_defined_rev) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

409 

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

410 
text {* a result about functions with flat codomain *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

411 

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

412 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
25920  413 
by (drule ax_flat, simp) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

414 

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

415 
lemma flat_codom: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

416 
"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

417 
apply (case_tac "f\<cdot>x = \<bottom>") 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

418 
apply (rule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

419 
apply (rule UU_I) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

420 
apply (erule_tac t="\<bottom>" in subst) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

421 
apply (rule minimal [THEN monofun_cfun_arg]) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

422 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

423 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

424 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

425 
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

426 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

427 

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

428 
subsection {* Identity and composition *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

429 

25135
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

430 
definition 
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

431 
ID :: "'a \<rightarrow> 'a" where 
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

432 
"ID = (\<Lambda> x. x)" 
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

433 

4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

434 
definition 
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

435 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where 
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset

436 
oo_def: "cfcomp = (\<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

437 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

438 
abbreviation 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

439 
cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) where 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

440 
"f oo g == cfcomp\<cdot>f\<cdot>g" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

441 

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

442 
lemma ID1 [simp]: "ID\<cdot>x = x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

443 
by (simp add: ID_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

444 

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

445 
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

446 
by (simp add: oo_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

447 

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

448 
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

449 
by (simp add: cfcomp1) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

450 

27274  451 
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))" 
452 
by (simp add: cfcomp1) 

453 

19709  454 
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>" 
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

455 
by (simp add: cfun_eq_iff) 
19709  456 

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

457 
text {* 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

458 
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

459 
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

460 
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

461 
The identity arrow is interpretation of @{term ID}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

462 
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

463 
*} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

464 

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

465 
lemma ID2 [simp]: "f oo ID = f" 
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

466 
by (rule cfun_eqI, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

467 

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

468 
lemma ID3 [simp]: "ID oo f = f" 
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

469 
by (rule cfun_eqI, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

470 

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

471 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset

472 
by (rule cfun_eqI, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

473 

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

474 
subsection {* Strictified functions *} 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

475 

36452  476 
default_sort pcpo 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

477 

25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset

478 
definition 
40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

479 
seq :: "'a \<rightarrow> 'b \<rightarrow> 'b" where 
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

480 
"seq = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)" 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

481 

40794  482 
lemma cont2cont_if_bottom [cont2cont, simp]: 
483 
assumes f: "cont (\<lambda>x. f x)" and g: "cont (\<lambda>x. g x)" 

484 
shows "cont (\<lambda>x. if f x = \<bottom> then \<bottom> else g x)" 

485 
proof (rule cont_apply [OF f]) 

486 
show "\<And>x. cont (\<lambda>y. if y = \<bottom> then \<bottom> else g x)" 

487 
unfolding cont_def is_lub_def is_ub_def ball_simps 

488 
by (simp add: lub_eq_bottom_iff) 

489 
show "\<And>y. cont (\<lambda>x. if y = \<bottom> then \<bottom> else g x)" 

490 
by (simp add: g) 

491 
qed 

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

492 

40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

493 
lemma seq_conv_if: "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)" 
40794  494 
unfolding seq_def by simp 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

495 

40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

496 
lemma seq1 [simp]: "seq\<cdot>\<bottom> = \<bottom>" 
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

497 
by (simp add: seq_conv_if) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

498 

40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

499 
lemma seq2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> seq\<cdot>x = ID" 
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

500 
by (simp add: seq_conv_if) 
40046
ba2e41c8b725
introduce function strict :: 'a > 'b > 'b, which works like Haskell's seq; use strict instead of strictify in various definitions
huffman
parents:
40011
diff
changeset

501 

40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

502 
lemma seq3 [simp]: "seq\<cdot>x\<cdot>\<bottom> = \<bottom>" 
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

503 
by (simp add: seq_conv_if) 
40093  504 

505 
definition 

40046
ba2e41c8b725
introduce function strict :: 'a > 'b > 'b, which works like Haskell's seq; use strict instead of strictify in various definitions
huffman
parents:
40011
diff
changeset

506 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where 
40767
a3e505b236e7
rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents:
40502
diff
changeset

507 
"strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))" 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

508 

17815  509 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
40046
ba2e41c8b725
introduce function strict :: 'a > 'b > 'b, which works like Haskell's seq; use strict instead of strictify in various definitions
huffman
parents:
40011
diff
changeset

510 
unfolding strictify_def by simp 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

511 

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

512 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
17815  513 
by (simp add: strictify_conv_if) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

514 

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

515 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
17815  516 
by (simp add: strictify_conv_if) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

517 

35933
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

518 
subsection {* Continuity of letbindings *} 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

519 

35933
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

520 
lemma cont2cont_Let: 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

521 
assumes f: "cont (\<lambda>x. f x)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

522 
assumes g1: "\<And>y. cont (\<lambda>x. g x y)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

523 
assumes g2: "\<And>x. cont (\<lambda>y. g x y)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

524 
shows "cont (\<lambda>x. let y = f x in g x y)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

525 
unfolding Let_def using f g2 g1 by (rule cont_apply) 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

526 

37079
0cd15d8c90a0
remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents:
36452
diff
changeset

527 
lemma cont2cont_Let' [simp, cont2cont]: 
35933
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

528 
assumes f: "cont (\<lambda>x. f x)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

529 
assumes g: "cont (\<lambda>p. g (fst p) (snd p))" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

530 
shows "cont (\<lambda>x. let y = f x in g x y)" 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

531 
using f 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

532 
proof (rule cont2cont_Let) 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

533 
fix x show "cont (\<lambda>y. g x y)" 
40003
427106657e04
remove unused lemmas cont_fst_snd_D1, cont_fst_snd_D2
huffman
parents:
40002
diff
changeset

534 
using g by (simp add: prod_cont_iff) 
35933
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

535 
next 
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

536 
fix y show "cont (\<lambda>x. g x y)" 
40003
427106657e04
remove unused lemmas cont_fst_snd_D1, cont_fst_snd_D2
huffman
parents:
40002
diff
changeset

537 
using g by (simp add: prod_cont_iff) 
35933
f135ebcc835c
remove continuous letbinding function CLet; add cont2cont rule ordinary Let
huffman
parents:
35914
diff
changeset

538 
qed 
17816
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

539 

39145  540 
text {* The simple version (suggested by Joachim Breitner) is needed if 
541 
the type of the defined term is not a cpo. *} 

542 

543 
lemma cont2cont_Let_simple [simp, cont2cont]: 

544 
assumes "\<And>y. cont (\<lambda>x. g x y)" 

545 
shows "cont (\<lambda>x. let y = t in g x y)" 

546 
unfolding Let_def using assms . 

547 

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

548 
end 