src/HOLCF/Cfun.thy
author huffman
Tue, 23 Mar 2010 10:07:39 -0700
changeset 35933 f135ebcc835c
parent 35914 91a7311177c4
child 36452 d37c6eed8117
permissions -rw-r--r--
remove continuous let-binding function CLet; add cont2cont rule ordinary Let
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
15600
a59f07556a8d fixed filename in header
huffman
parents: 15589
diff changeset
     1
(*  Title:      HOLCF/Cfun.thy
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     2
    Author:     Franz Regensburger
35794
8cd7134275cc use headers consistently
huffman
parents: 35641
diff changeset
     3
    Author:     Brian Huffman
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     4
*)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     5
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     6
header {* The type of continuous functions *}
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     7
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
     8
theory Cfun
29533
7f4a32134447 minimize dependencies
huffman
parents: 29530
diff changeset
     9
imports Pcpodef Ffun Product_Cpo
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    10
begin
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    11
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    12
defaultsort cpo
efb95d0d01f7 converted to new-style 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
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    16
lemma Ex_cont: "\<exists>f. cont f"
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    17
by (rule exI, rule cont_const)
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    18
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    19
lemma adm_cont: "adm cont"
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    20
by (rule admI, rule cont_lub_fun)
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
    21
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
    22
cpodef (CFun)  ('a, 'b) cfun (infixr "->" 0) = "{f::'a => 'b. cont f}"
29063
7619f0561cd7 pcpodef package: state two goals, instead of encoded conjunction;
wenzelm
parents: 29049
diff changeset
    23
by (simp_all add: Ex_cont adm_cont)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    24
35427
ad039d29e01c proper (type_)notation;
wenzelm
parents: 35168
diff changeset
    25
type_notation (xsymbols)
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
    26
  cfun  ("(_ \<rightarrow>/ _)" [1, 0] 0)
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    27
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    28
notation
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    29
  Rep_CFun  ("(_$/_)" [999,1000] 999)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    30
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    31
notation (xsymbols)
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    32
  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    33
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    34
notation (HTML output)
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    35
  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    36
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    37
subsection {* Syntax for continuous lambda abstraction *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    38
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
    39
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
    40
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
    41
parse_translation {*
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    42
(* rewrite (_cabs x t) => (Abs_CFun (%x. t)) *)
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    43
  [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
    44
*}
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    45
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    46
text {* To avoid eta-contraction of body: *}
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    47
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
    48
  let
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    49
    fun cabs_tr' _ _ [Abs abs] = let
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    50
          val (x,t) = atomic_abs_tr' abs
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    51
        in Syntax.const @{syntax_const "_cabs"} $ x $ t end
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    52
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    53
      | cabs_tr' _ T [t] = let
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    54
          val xT = domain_type (domain_type T);
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    55
          val abs' = ("x",xT,(incr_boundvars 1 t)$Bound 0);
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    56
          val (x,t') = atomic_abs_tr' abs';
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    57
        in Syntax.const @{syntax_const "_cabs"} $ x $ t' end;
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    58
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    59
  in [(@{const_syntax Abs_CFun}, cabs_tr')] end;
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    60
*}
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    61
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    62
text {* Syntax for nested abstractions *}
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    63
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    64
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
    65
  "_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
    66
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    67
syntax (xsymbols)
25927
9c544dac6269 add space to binder syntax
huffman
parents: 25921
diff changeset
    68
  "_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
    69
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    70
parse_ast_translation {*
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    71
(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    72
(* 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
    73
  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
    74
    fun Lambda_ast_tr [pats, body] =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    75
          Syntax.fold_ast_p @{syntax_const "_cabs"}
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    76
            (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
    77
      | Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts);
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    78
  in [(@{syntax_const "_Lambda"}, Lambda_ast_tr)] end;
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    79
*}
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    80
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    81
print_ast_translation {*
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    82
(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    83
(* 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
    84
  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
    85
    fun cabs_ast_tr' asts =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    86
      (case Syntax.unfold_ast_p @{syntax_const "_cabs"}
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    87
          (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
    88
        ([], _) => 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
    89
      | (xs, body) => Syntax.Appl
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    90
          [Syntax.Constant @{syntax_const "_Lambda"},
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    91
           Syntax.fold_ast @{syntax_const "_cargs"} xs, body]);
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    92
  in [(@{syntax_const "_cabs"}, cabs_ast_tr')] end
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    93
*}
15641
b389f108c485 added theorems eta_cfun and cont2cont_eta
huffman
parents: 15600
diff changeset
    94
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    95
text {* Dummy patterns for continuous abstraction *}
18079
9d4d70b175fd add translation for wildcard pattern
huffman
parents: 18078
diff changeset
    96
translations
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    97
  "\<Lambda> _. t" => "CONST Abs_CFun (\<lambda> _. t)"
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    98
18079
9d4d70b175fd add translation for wildcard pattern
huffman
parents: 18078
diff changeset
    99
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   100
subsection {* Continuous function space is pointed *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   101
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   102
lemma UU_CFun: "\<bottom> \<in> CFun"
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   103
by (simp add: CFun_def inst_fun_pcpo cont_const)
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   104
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
   105
instance cfun :: (finite_po, finite_po) finite_po
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25786
diff changeset
   106
by (rule typedef_finite_po [OF type_definition_CFun])
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25786
diff changeset
   107
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
   108
instance cfun :: (finite_po, chfin) chfin
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
   109
by (rule typedef_chfin [OF type_definition_CFun below_CFun_def])
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25786
diff changeset
   110
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
   111
instance cfun :: (cpo, discrete_cpo) discrete_cpo
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
   112
by intro_classes (simp add: below_CFun_def Rep_CFun_inject)
26025
ca6876116bb4 instances for class discrete_cpo
huffman
parents: 25927
diff changeset
   113
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
   114
instance cfun :: (cpo, pcpo) pcpo
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
   115
by (rule typedef_pcpo [OF type_definition_CFun below_CFun_def UU_CFun])
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   116
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
   117
lemmas Rep_CFun_strict =
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
   118
  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
   119
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
lemmas Abs_CFun_strict =
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
   121
  typedef_Abs_strict [OF type_definition_CFun below_CFun_def UU_CFun]
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   122
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   123
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
   124
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   125
lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   126
by (simp add: Rep_CFun_strict)
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   127
35641
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   128
lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>"
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   129
by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   130
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   131
text {* for compatibility with old HOLCF-Version *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   132
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
35641
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   133
by simp
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   134
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   135
subsection {* Basic properties of continuous functions *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   136
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   137
text {* Beta-equality 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
   138
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
   139
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
   140
by (simp add: Abs_CFun_inverse CFun_def)
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   141
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
   142
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
   143
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
   144
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
text {* Eta-equality 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
   146
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
   147
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
   148
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
   149
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
   150
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
   151
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   152
lemma expand_cfun_eq: "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   153
by (simp add: Rep_CFun_inject [symmetric] expand_fun_eq)
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   154
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
   155
lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   156
by (simp add: expand_cfun_eq)
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   157
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   158
text {* Extensionality wrt. ordering for continuous functions *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   159
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
   160
lemma expand_cfun_below: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 31041
diff changeset
   161
by (simp add: below_CFun_def expand_fun_below)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   162
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
   163
lemma below_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 31041
diff changeset
   164
by (simp add: expand_cfun_below)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   165
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   166
text {* Congruence for continuous function application *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   167
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
   168
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
   169
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
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 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
   172
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   173
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
   174
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
   175
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   176
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
   177
subsection {* Continuity of application *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   178
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
   179
lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   180
by (rule cont_Rep_CFun [THEN cont2cont_fun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   181
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
   182
lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   183
apply (cut_tac x=f in Rep_CFun)
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   184
apply (simp add: CFun_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   185
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   186
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
   187
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
   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
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
   190
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
   191
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
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
   193
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   194
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
35914
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   195
by (rule cont_Rep_CFun2 [THEN cont2contlubE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   196
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   197
lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(\<Squnion>i. 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
   198
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
   199
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   200
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
35914
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   201
by (rule cont_Rep_CFun1 [THEN cont2contlubE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   202
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   203
lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| (\<Squnion>i. F i)\<cdot>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
   204
by (rule cont_Rep_CFun1 [THEN contE])
15576
efb95d0d01f7 converted to new-style 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
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
   207
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
lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
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
   209
by (simp add: expand_cfun_below)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   210
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
   211
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
   212
by (rule monofun_Rep_CFun2 [THEN monofunE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   213
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
   214
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
   215
by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   216
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
   217
text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   218
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
   219
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
   220
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
   221
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
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
   223
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   224
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
   225
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
   226
by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   227
18076
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   228
lemma ch2ch_Rep_CFun [simp]:
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   229
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
25884
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   230
by (simp add: chain_def monofun_cfun)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   231
25884
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   232
lemma ch2ch_LAM [simp]:
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   233
  "\<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)"
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
   234
by (simp add: chain_def expand_cfun_below)
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   235
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
   236
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   237
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 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
   239
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
18076
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   240
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   241
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
   242
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
   243
  "\<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
   244
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
   245
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
   246
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
   247
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
   248
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   249
lemma contlub_LAM:
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   250
  "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   251
    \<Longrightarrow> (\<Lambda> x. \<Squnion>i. F i x) = (\<Squnion>i. \<Lambda> x. F i x)"
25884
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   252
apply (simp add: thelub_CFun)
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   253
apply (simp add: Abs_CFun_inverse2)
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   254
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
   255
done
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   256
25901
bb178c8251e0 added lemmas lub_distribs
huffman
parents: 25884
diff changeset
   257
lemmas lub_distribs = 
bb178c8251e0 added lemmas lub_distribs
huffman
parents: 25884
diff changeset
   258
  contlub_cfun [symmetric]
bb178c8251e0 added lemmas lub_distribs
huffman
parents: 25884
diff changeset
   259
  contlub_LAM [symmetric]
bb178c8251e0 added lemmas lub_distribs
huffman
parents: 25884
diff changeset
   260
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
   261
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
   262
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
   263
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
   264
apply (rule UU_I)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   265
apply (erule subst)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   266
apply (rule minimal [THEN monofun_cfun_arg])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   267
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   268
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
   269
text {* the lub of a chain of continous functions is monotone *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   270
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
   271
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
   272
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
   273
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
   274
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
   275
apply (simp add: monofun_Rep_CFun2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   276
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   277
16386
c6f5ade29608 moved continuity simproc to a separate file
huffman
parents: 16314
diff changeset
   278
text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   279
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
   280
lemma ex_lub_cfun:
24b494ff8f0f use new pcpodef package
huffman
parents: 16417
diff changeset
   281
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>j. \<Squnion>i. F j\<cdot>(Y i)) = (\<Squnion>i. \<Squnion>j. F j\<cdot>(Y i))"
18076
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   282
by (simp add: diag_lub)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   283
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   284
text {* the lub of a chain of cont. functions is continuous *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   285
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
   286
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
   287
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
   288
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
   289
apply (rule cont_Rep_CFun2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   290
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   291
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   292
text {* type @{typ "'a -> 'b"} is chain complete *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   293
16920
ded12c9e88c2 cleaned up
huffman
parents: 16699
diff changeset
   294
lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
ded12c9e88c2 cleaned up
huffman
parents: 16699
diff changeset
   295
by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   296
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   297
lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
16920
ded12c9e88c2 cleaned up
huffman
parents: 16699
diff changeset
   298
by (rule lub_cfun [THEN thelubI])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   299
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   300
subsection {* Continuity simplification procedure *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   301
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   302
text {* cont2cont lemma for @{term Rep_CFun} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   303
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   304
lemma cont2cont_Rep_CFun [cont2cont]:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   305
  assumes f: "cont (\<lambda>x. f x)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   306
  assumes t: "cont (\<lambda>x. t x)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   307
  shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   308
proof -
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   309
  have "cont (\<lambda>x. Rep_CFun (f x))"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   310
    using cont_Rep_CFun f by (rule cont2cont_app3)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   311
  thus "cont (\<lambda>x. (f x)\<cdot>(t x))"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   312
    using cont_Rep_CFun2 t by (rule cont2cont_app2)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   313
qed
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   314
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   315
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   316
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   317
lemma cont2mono_LAM:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   318
  "\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   319
    \<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x 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
   320
  unfolding monofun_def expand_cfun_below by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   321
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   322
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   323
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   324
text {*
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   325
  Not suitable as a cont2cont rule, because on nested lambdas
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   326
  it causes exponential blow-up in the number of subgoals.
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   327
*}
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   328
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   329
lemma cont2cont_LAM:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   330
  assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   331
  assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   332
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   333
proof (rule cont_Abs_CFun)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   334
  fix x
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   335
  from f1 show "f x \<in> CFun" by (simp add: CFun_def)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   336
  from f2 show "cont f" by (rule cont2cont_lambda)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   337
qed
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   338
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   339
text {*
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   340
  This version does work as a cont2cont rule, since it
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   341
  has only a single subgoal.
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   342
*}
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   343
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   344
lemma cont2cont_LAM' [cont2cont]:
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   345
  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   346
  assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   347
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   348
proof (rule cont2cont_LAM)
31041
85b4843d9939 replace cont2cont_apply with cont_apply; add new cont2cont lemmas
huffman
parents: 29533
diff changeset
   349
  fix x :: 'a show "cont (\<lambda>y. f x y)"
85b4843d9939 replace cont2cont_apply with cont_apply; add new cont2cont lemmas
huffman
parents: 29533
diff changeset
   350
    using f by (rule cont_fst_snd_D2)
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   351
next
31041
85b4843d9939 replace cont2cont_apply with cont_apply; add new cont2cont lemmas
huffman
parents: 29533
diff changeset
   352
  fix y :: 'b show "cont (\<lambda>x. f x y)"
85b4843d9939 replace cont2cont_apply with cont_apply; add new cont2cont lemmas
huffman
parents: 29533
diff changeset
   353
    using f by (rule cont_fst_snd_D1)
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   354
qed
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   355
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   356
lemma cont2cont_LAM_discrete [cont2cont]:
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   357
  "(\<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
   358
by (simp add: cont2cont_LAM)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   359
16055
58186c507750 moved continuity simproc to Cfun.thy
huffman
parents: 15641
diff changeset
   360
lemmas cont_lemmas1 =
58186c507750 moved continuity simproc to Cfun.thy
huffman
parents: 15641
diff changeset
   361
  cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
58186c507750 moved continuity simproc to Cfun.thy
huffman
parents: 15641
diff changeset
   362
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   363
subsection {* Miscellaneous *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   364
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   365
text {* Monotonicity of @{term Abs_CFun} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   366
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   367
lemma semi_monofun_Abs_CFun:
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   368
  "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
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
   369
by (simp add: below_CFun_def Abs_CFun_inverse2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   370
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   371
text {* some lemmata for functions with flat/chfin domain/range types *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   372
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   373
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   374
      ==> !s. ? n. (LUB i. Y i)$s = Y n$s"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   375
apply (rule allI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   376
apply (subst contlub_cfun_fun)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   377
apply assumption
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   378
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   379
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   380
18089
35c091a9841a moved adm_chfindom from Fix.thy to Cfun.thy
huffman
parents: 18087
diff changeset
   381
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))"
35c091a9841a moved adm_chfindom from Fix.thy to Cfun.thy
huffman
parents: 18087
diff changeset
   382
by (rule adm_subst, simp, rule adm_chfin)
35c091a9841a moved adm_chfindom from Fix.thy to Cfun.thy
huffman
parents: 18087
diff changeset
   383
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   384
subsection {* Continuous injection-retraction pairs *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   385
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   386
text {* Continuous retractions are strict. *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   387
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   388
lemma retraction_strict:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   389
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   390
apply (rule UU_I)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   391
apply (drule_tac x="\<bottom>" in spec)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   392
apply (erule subst)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   393
apply (rule monofun_cfun_arg)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   394
apply (rule minimal)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   395
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   396
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   397
lemma injection_eq:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   398
  "\<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
   399
apply (rule iffI)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   400
apply (drule_tac f=f in cfun_arg_cong)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   401
apply simp
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   402
apply simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   403
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   404
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
   405
lemma injection_below:
16314
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   406
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   407
apply (rule iffI)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   408
apply (drule_tac f=f in monofun_cfun_arg)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   409
apply simp
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   410
apply (erule monofun_cfun_arg)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   411
done
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   412
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   413
lemma injection_defined_rev:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   414
  "\<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
   415
apply (drule_tac f=f in cfun_arg_cong)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   416
apply (simp add: retraction_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   417
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   418
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   419
lemma injection_defined:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   420
  "\<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
   421
by (erule contrapos_nn, rule injection_defined_rev)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   422
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   423
text {* propagation of flatness and chain-finiteness by retractions *}
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   424
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   425
lemma chfin2chfin:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   426
  "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   427
    \<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
   428
apply clarify
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   429
apply (drule_tac f=g in chain_monofun)
25921
0ca392ab7f37 change class axiom chfin to rule_format
huffman
parents: 25920
diff changeset
   430
apply (drule chfin)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   431
apply (unfold max_in_chain_def)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   432
apply (simp add: injection_eq)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   433
done
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   434
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   435
lemma flat2flat:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   436
  "\<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
   437
    \<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
   438
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
   439
apply (drule_tac f=g in monofun_cfun_arg)
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 25901
diff changeset
   440
apply (drule ax_flat)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   441
apply (erule disjE)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   442
apply (simp add: injection_defined_rev)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   443
apply (simp add: injection_eq)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   444
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   445
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   446
text {* a result about functions with flat codomain *}
15576
efb95d0d01f7 converted to new-style 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 flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 25901
diff changeset
   449
by (drule ax_flat, simp)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   450
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   451
lemma flat_codom:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   452
  "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
   453
apply (case_tac "f\<cdot>x = \<bottom>")
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   454
apply (rule disjI1)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   455
apply (rule UU_I)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   456
apply (erule_tac t="\<bottom>" in subst)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   457
apply (rule minimal [THEN monofun_cfun_arg])
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   458
apply clarify
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   459
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   460
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   461
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
   462
done
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   463
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   464
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   465
subsection {* Identity and composition *}
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   466
25135
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   467
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   468
  ID :: "'a \<rightarrow> 'a" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   469
  "ID = (\<Lambda> x. x)"
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   470
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   471
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   472
  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   473
  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
   474
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   475
abbreviation
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   476
  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
   477
  "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
   478
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   479
lemma ID1 [simp]: "ID\<cdot>x = x"
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   480
by (simp add: ID_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   481
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   482
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
   483
by (simp add: oo_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   484
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   485
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
   486
by (simp add: cfcomp1)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   487
27274
1c97c471db82 add lemma cfcomp_LAM
huffman
parents: 26025
diff changeset
   488
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
1c97c471db82 add lemma cfcomp_LAM
huffman
parents: 26025
diff changeset
   489
by (simp add: cfcomp1)
1c97c471db82 add lemma cfcomp_LAM
huffman
parents: 26025
diff changeset
   490
19709
78cd5f6af8e8 add theorem cfcomp_strict
huffman
parents: 18092
diff changeset
   491
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>"
78cd5f6af8e8 add theorem cfcomp_strict
huffman
parents: 18092
diff changeset
   492
by (simp add: expand_cfun_eq)
78cd5f6af8e8 add theorem cfcomp_strict
huffman
parents: 18092
diff changeset
   493
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   494
text {*
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   495
  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
   496
  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
   497
  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
   498
  The identity arrow is interpretation of @{term ID}.
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   499
  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
   500
*}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   501
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   502
lemma ID2 [simp]: "f oo ID = f"
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   503
by (rule ext_cfun, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   504
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   505
lemma ID3 [simp]: "ID oo f = f"
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   506
by (rule ext_cfun, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   507
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   508
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
   509
by (rule ext_cfun, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   510
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
subsection {* Strictified functions *}
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   513
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   514
defaultsort pcpo
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   515
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   516
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   517
  strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   518
  "strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   519
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   520
text {* results about strictify *}
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   521
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   522
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
35168
07b3112e464b fix warnings about duplicate simp rules
huffman
parents: 35115
diff changeset
   523
by simp
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   524
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   525
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   526
apply (rule monofunI)
25786
6b3c79acac1f move lemmas from Cont.thy to Ffun.thy;
huffman
parents: 25723
diff changeset
   527
apply (auto simp add: monofun_cfun_arg)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   528
done
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   529
35914
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   530
lemma cont_strictify2: "cont (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   531
apply (rule contI2)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   532
apply (rule monofun_strictify2)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   533
apply (case_tac "(\<Squnion>i. Y i) = \<bottom>", simp)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   534
apply (simp add: contlub_cfun_arg del: if_image_distrib)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   535
apply (drule chain_UU_I_inverse2, clarify, rename_tac j)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   536
apply (rule lub_mono2, rule_tac x=j in exI, simp_all)
91a7311177c4 remove contlub predicate
huffman
parents: 35794
diff changeset
   537
apply (auto dest!: chain_mono_less)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   538
done
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   539
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   540
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   541
  unfolding strictify_def
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   542
  by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   543
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   544
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   545
by (simp add: strictify_conv_if)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   546
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   547
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   548
by (simp add: strictify_conv_if)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   549
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   550
subsection {* Continuity of let-bindings *}
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
   551
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   552
lemma cont2cont_Let:
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   553
  assumes f: "cont (\<lambda>x. f x)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   554
  assumes g1: "\<And>y. cont (\<lambda>x. g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   555
  assumes g2: "\<And>x. cont (\<lambda>y. g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   556
  shows "cont (\<lambda>x. let y = f x in g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   557
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
   558
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   559
lemma cont2cont_Let' [cont2cont]:
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   560
  assumes f: "cont (\<lambda>x. f x)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   561
  assumes g: "cont (\<lambda>p. g (fst p) (snd p))"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   562
  shows "cont (\<lambda>x. let y = f x in g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   563
using f
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   564
proof (rule cont2cont_Let)
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   565
  fix x show "cont (\<lambda>y. g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   566
    using g by (rule cont_fst_snd_D2)
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   567
next
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   568
  fix y show "cont (\<lambda>x. g x y)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   569
    using g by (rule cont_fst_snd_D1)
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   570
qed
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
   571
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   572
end