src/HOL/HOLCF/Cfun.thy
author wenzelm
Mon, 19 Jan 2015 20:39:01 +0100
changeset 59409 b7cfe12acf2e
parent 58957 c9e744ea8a38
child 59582 0fbed69ff081
permissions -rw-r--r--
tuned;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
42151
4da4fc77664b tuned headers;
wenzelm
parents: 42057
diff changeset
     1
(*  Title:      HOL/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
58880
0baae4311a9f modernized header;
wenzelm
parents: 57954
diff changeset
     6
section {* The type of continuous functions *}
15576
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
40772
c8b52f9e1680 rename Pcpodef.thy to Cpodef.thy;
huffman
parents: 40771
diff changeset
     9
imports Cpodef Fun_Cpo 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
36452
d37c6eed8117 renamed command 'defaultsort' to 'default_sort';
wenzelm
parents: 35933
diff changeset
    12
default_sort cpo
15576
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
45695
b108b3d7c49e prefer cpodef without extra definition;
wenzelm
parents: 45606
diff changeset
    16
definition "cfun = {f::'a => 'b. cont f}"
b108b3d7c49e prefer cpodef without extra definition;
wenzelm
parents: 45606
diff changeset
    17
49759
ecf1d5d87e5e removed support for set constant definitions in HOLCF {cpo,pcpo,domain}def commands;
huffman
parents: 45695
diff changeset
    18
cpodef ('a, 'b) cfun (infixr "->" 0) = "cfun :: ('a => 'b) set"
45695
b108b3d7c49e prefer cpodef without extra definition;
wenzelm
parents: 45606
diff changeset
    19
  unfolding cfun_def by (auto intro: cont_const adm_cont)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    20
35427
ad039d29e01c proper (type_)notation;
wenzelm
parents: 35168
diff changeset
    21
type_notation (xsymbols)
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
    22
  cfun  ("(_ \<rightarrow>/ _)" [1, 0] 0)
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    23
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    24
notation
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    25
  Rep_cfun  ("(_$/_)" [999,1000] 999)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    26
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    27
notation (xsymbols)
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    28
  Rep_cfun  ("(_\<cdot>/_)" [999,1000] 999)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    29
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
    30
notation (HTML output)
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    31
  Rep_cfun  ("(_\<cdot>/_)" [999,1000] 999)
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    32
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    33
subsection {* Syntax for continuous lambda abstraction *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    34
41479
655f583840d0 use proper syntactic types for 'syntax' commands
huffman
parents: 41478
diff changeset
    35
syntax "_cabs" :: "[logic, logic] \<Rightarrow> logic"
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
    36
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
    37
parse_translation {*
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    38
(* rewrite (_cabs x t) => (Abs_cfun (%x. t)) *)
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 42264
diff changeset
    39
  [Syntax_Trans.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
    40
*}
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    41
41478
18500bd1f47b make print translation for Abs_cfun consistent with other binders: prevent eta-contraction, but don't force eta-expansion
huffman
parents: 41430
diff changeset
    42
print_translation {*
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
    43
  [(@{const_syntax Abs_cfun}, fn _ => fn [Abs abs] =>
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 42264
diff changeset
    44
      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
41478
18500bd1f47b make print translation for Abs_cfun consistent with other binders: prevent eta-contraction, but don't force eta-expansion
huffman
parents: 41430
diff changeset
    45
      in Syntax.const @{syntax_const "_cabs"} $ x $ t end)]
18500bd1f47b make print translation for Abs_cfun consistent with other binders: prevent eta-contraction, but don't force eta-expansion
huffman
parents: 41430
diff changeset
    46
*}  -- {* To avoid eta-contraction of body *}
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    47
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    48
text {* Syntax for nested abstractions *}
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    49
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    50
syntax
41479
655f583840d0 use proper syntactic types for 'syntax' commands
huffman
parents: 41478
diff changeset
    51
  "_Lambda" :: "[cargs, logic] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    52
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    53
syntax (xsymbols)
41479
655f583840d0 use proper syntactic types for 'syntax' commands
huffman
parents: 41478
diff changeset
    54
  "_Lambda" :: "[cargs, logic] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    55
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    56
parse_ast_translation {*
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    57
(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    58
(* 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
    59
  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
    60
    fun Lambda_ast_tr [pats, body] =
42224
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    61
          Ast.fold_ast_p @{syntax_const "_cabs"}
42264
b6c1b0c4c511 separate structure Term_Position;
wenzelm
parents: 42224
diff changeset
    62
            (Ast.unfold_ast @{syntax_const "_cargs"} (Ast.strip_positions pats), body)
42224
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    63
      | Lambda_ast_tr asts = raise Ast.AST ("Lambda_ast_tr", asts);
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
    64
  in [(@{syntax_const "_Lambda"}, K Lambda_ast_tr)] end;
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    65
*}
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    66
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    67
print_ast_translation {*
35115
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    68
(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
446c5063e4fd modernized translations;
wenzelm
parents: 31076
diff changeset
    69
(* 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
    70
  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
    71
    fun cabs_ast_tr' asts =
42224
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    72
      (case Ast.unfold_ast_p @{syntax_const "_cabs"}
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    73
          (Ast.Appl (Ast.Constant @{syntax_const "_cabs"} :: asts)) of
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    74
        ([], _) => raise Ast.AST ("cabs_ast_tr'", asts)
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    75
      | (xs, body) => Ast.Appl
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    76
          [Ast.Constant @{syntax_const "_Lambda"},
578a51fae383 discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
wenzelm
parents: 42151
diff changeset
    77
           Ast.fold_ast @{syntax_const "_cargs"} xs, body]);
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
    78
  in [(@{syntax_const "_cabs"}, K cabs_ast_tr')] end
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
    79
*}
15641
b389f108c485 added theorems eta_cfun and cont2cont_eta
huffman
parents: 15600
diff changeset
    80
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    81
text {* Dummy patterns for continuous abstraction *}
18079
9d4d70b175fd add translation for wildcard pattern
huffman
parents: 18078
diff changeset
    82
translations
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    83
  "\<Lambda> _. t" => "CONST Abs_cfun (\<lambda> _. t)"
18087
577d57e51f89 add print translation: Abs_CFun f => LAM x. f x
huffman
parents: 18079
diff changeset
    84
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
    85
subsection {* Continuous function space is pointed *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    86
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
    87
lemma bottom_cfun: "\<bottom> \<in> cfun"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    88
by (simp add: cfun_def inst_fun_pcpo)
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
    89
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
    90
instance cfun :: (cpo, discrete_cpo) discrete_cpo
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    91
by intro_classes (simp add: below_cfun_def Rep_cfun_inject)
26025
ca6876116bb4 instances for class discrete_cpo
huffman
parents: 25927
diff changeset
    92
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 35168
diff changeset
    93
instance cfun :: (cpo, pcpo) pcpo
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
    94
by (rule typedef_pcpo [OF type_definition_cfun below_cfun_def bottom_cfun])
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
    95
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    96
lemmas Rep_cfun_strict =
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
    97
  typedef_Rep_strict [OF type_definition_cfun below_cfun_def bottom_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
    98
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
    99
lemmas Abs_cfun_strict =
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
   100
  typedef_Abs_strict [OF type_definition_cfun below_cfun_def bottom_cfun]
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   101
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   102
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
   103
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   104
lemma Rep_cfun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   105
by (simp add: Rep_cfun_strict)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   106
35641
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   107
lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   108
by (simp add: inst_fun_pcpo [symmetric] Abs_cfun_strict)
35641
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   109
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   110
text {* for compatibility with old HOLCF-Version *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   111
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
35641
a17bc4cec23a add simp rule LAM_strict
huffman
parents: 35547
diff changeset
   112
by simp
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   113
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   114
subsection {* Basic properties of continuous functions *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   115
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   116
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
   117
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   118
lemma Abs_cfun_inverse2: "cont f \<Longrightarrow> Rep_cfun (Abs_cfun f) = f"
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   119
by (simp add: Abs_cfun_inverse cfun_def)
16098
6aef81a6ddd3 use TypedefPcpo for all class instances
huffman
parents: 16094
diff changeset
   120
37083
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   121
lemma beta_cfun: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   122
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
   123
37083
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   124
text {* Beta-reduction simproc *}
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   125
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   126
text {*
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   127
  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
   128
  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
   129
  theorem cannot be completely solved by the cont2cont rules, then
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   130
  the procedure returns the ordinary conditional @{text beta_cfun}
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   131
  rule.
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   132
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   133
  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
   134
  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
   135
  simproc is that it can avoid deeply-nested calls to the simplifier
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   136
  that would otherwise be caused by large continuity side conditions.
41322
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   137
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   138
  Update: The simproc now uses rule @{text Abs_cfun_inverse2} instead
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   139
  of @{text beta_cfun}, to avoid problems with eta-contraction.
37083
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
41322
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   142
simproc_setup beta_cfun_proc ("Rep_cfun (Abs_cfun f)") = {*
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 49759
diff changeset
   143
  fn phi => fn ctxt => fn ct =>
37083
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   144
    let
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   145
      val dest = Thm.dest_comb;
41322
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   146
      val f = (snd o dest o snd o dest) ct;
37083
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   147
      val [T, U] = Thm.dest_ctyp (ctyp_of_term f);
41322
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   148
      val tr = instantiate' [SOME T, SOME U] [SOME f]
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   149
          (mk_meta_eq @{thm Abs_cfun_inverse2});
57945
cacb00a569e0 prefer 'named_theorems' over Named_Thms, with subtle change of semantics due to visual order vs. internal reverse order;
wenzelm
parents: 52143
diff changeset
   150
      val rules = Named_Theorems.get ctxt @{named_theorems cont2cont};
58957
c9e744ea8a38 proper context for match_tac etc.;
wenzelm
parents: 58880
diff changeset
   151
      val tac = SOLVED' (REPEAT_ALL_NEW (match_tac ctxt (rev rules)));
37083
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   152
    in SOME (perhaps (SINGLE (tac 1)) tr) end
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
diff changeset
   153
*}
03a70ab79dd9 add beta_cfun simproc, which uses cont2cont rules
huffman
parents: 37079
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
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
   156
36ee7f6af79f removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents: 16098
diff changeset
   157
lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   158
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
   159
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 {* 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
   161
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
   162
lemma cfun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f\<cdot>x = g\<cdot>x)"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   163
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
   164
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
   165
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
   166
by (simp add: cfun_eq_iff)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   167
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   168
text {* Extensionality wrt. ordering for continuous functions *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
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_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   171
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
   172
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
   173
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
   174
by (simp add: cfun_below_iff)
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   175
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   176
text {* Congruence for continuous function application *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   177
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
   178
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
   179
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   180
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
   181
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
   182
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   183
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
   184
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
   185
by simp
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
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
subsection {* Continuity of application *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   188
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   189
lemma cont_Rep_cfun1: "cont (\<lambda>f. f\<cdot>x)"
40834
a1249aeff5b6 change cpodef-generated cont_Rep rules to cont2cont format
huffman
parents: 40794
diff changeset
   190
by (rule cont_Rep_cfun [OF cont_id, THEN cont2cont_fun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   191
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   192
lemma cont_Rep_cfun2: "cont (\<lambda>x. f\<cdot>x)"
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   193
apply (cut_tac x=f in Rep_cfun)
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   194
apply (simp add: cfun_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   195
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   196
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   197
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
   198
45606
b1e1508643b1 eliminated obsolete "standard";
wenzelm
parents: 42284
diff changeset
   199
lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono]
b1e1508643b1 eliminated obsolete "standard";
wenzelm
parents: 42284
diff changeset
   200
lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono]
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
   201
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   202
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
   203
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   204
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   205
by (rule cont_Rep_cfun2 [THEN cont2contlubE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   206
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27274
diff changeset
   207
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   208
by (rule cont_Rep_cfun1 [THEN cont2contlubE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   209
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
   210
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
   211
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
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
   213
by (simp add: cfun_below_iff)
15576
efb95d0d01f7 converted to new-style 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
lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   216
by (rule monofun_Rep_cfun2 [THEN monofunE])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   217
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
   218
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
   219
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
   220
16209
36ee7f6af79f removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents: 16098
diff changeset
   221
text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
15576
efb95d0d01f7 converted to new-style 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 chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   224
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
   225
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   226
lemma ch2ch_Rep_cfunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   227
by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   228
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   229
lemma ch2ch_Rep_cfunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   230
by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   231
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   232
lemma ch2ch_Rep_cfun [simp]:
18076
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   233
  "\<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
   234
by (simp add: chain_def monofun_cfun)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   235
25884
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   236
lemma ch2ch_LAM [simp]:
a69e665b7df8 declare ch2ch_LAM [simp]
huffman
parents: 25827
diff changeset
   237
  "\<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
   238
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
   239
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   240
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
   241
41027
c599955d9806 add lemmas lub_APP, lub_LAM
huffman
parents: 40834
diff changeset
   242
lemma lub_APP:
c599955d9806 add lemmas lub_APP, lub_LAM
huffman
parents: 40834
diff changeset
   243
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i\<cdot>(Y i)) = (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
18076
e2e626b673b3 cleaned up; ch2ch_Rep_CFun is a simp rule
huffman
parents: 17832
diff changeset
   244
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
   245
41027
c599955d9806 add lemmas lub_APP, lub_LAM
huffman
parents: 40834
diff changeset
   246
lemma lub_LAM:
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   247
  "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
41027
c599955d9806 add lemmas lub_APP, lub_LAM
huffman
parents: 40834
diff changeset
   248
    \<Longrightarrow> (\<Squnion>i. \<Lambda> x. F i x) = (\<Lambda> x. \<Squnion>i. F i x)"
41322
43a5b9a0ee8a beta-reduction simproc uses lemma Abs_cfun_inverse2 instead of beta_cfun, to be more robust against eta-contraction
huffman
parents: 41031
diff changeset
   249
by (simp add: lub_cfun lub_fun ch2ch_lambda)
18092
2c5d5da79a1e renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents: 18091
diff changeset
   250
41027
c599955d9806 add lemmas lub_APP, lub_LAM
huffman
parents: 40834
diff changeset
   251
lemmas lub_distribs = lub_APP lub_LAM
25901
bb178c8251e0 added lemmas lub_distribs
huffman
parents: 25884
diff changeset
   252
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
   253
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
   254
36ee7f6af79f removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents: 16098
diff changeset
   255
lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
   256
apply (rule bottomI)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   257
apply (erule subst)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   258
apply (rule minimal [THEN monofun_cfun_arg])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   259
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   260
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   261
text {* type @{typ "'a -> 'b"} is chain complete *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   262
41031
9883d1417ce1 remove lemma cont_cfun;
huffman
parents: 41030
diff changeset
   263
lemma lub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
9883d1417ce1 remove lemma cont_cfun;
huffman
parents: 41030
diff changeset
   264
by (simp add: lub_cfun lub_fun ch2ch_lambda)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   265
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   266
subsection {* Continuity simplification procedure *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   267
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   268
text {* cont2cont lemma for @{term Rep_cfun} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   269
40326
73d45866dbda renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents: 40093
diff changeset
   270
lemma cont2cont_APP [simp, cont2cont]:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   271
  assumes f: "cont (\<lambda>x. f x)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   272
  assumes t: "cont (\<lambda>x. t x)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   273
  shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   274
proof -
40006
116e94f9543b remove unneeded lemmas from Fun_Cpo.thy
huffman
parents: 40005
diff changeset
   275
  have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   276
    using cont_Rep_cfun1 f by (rule cont_compose)
40006
116e94f9543b remove unneeded lemmas from Fun_Cpo.thy
huffman
parents: 40005
diff changeset
   277
  show "cont (\<lambda>x. (f x)\<cdot>(t x))"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   278
    using t cont_Rep_cfun2 1 by (rule cont_apply)
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   279
qed
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   280
40008
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   281
text {*
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   282
  Two specific lemmas for the combination of LCF and HOL terms.
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   283
  These lemmas are needed in theories that use types like @{typ "'a \<rightarrow> 'b \<Rightarrow> 'c"}.
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   284
*}
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   285
40326
73d45866dbda renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents: 40093
diff changeset
   286
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
   287
by (rule cont2cont_APP [THEN cont2cont_fun])
40008
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   288
40326
73d45866dbda renamed lemma cont2cont_Rep_CFun to cont2cont_APP
huffman
parents: 40093
diff changeset
   289
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
   290
by (rule cont_APP_app [THEN cont2cont_fun])
40008
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   291
58ead6f77f8e move lemmas from Lift.thy to Cfun.thy
huffman
parents: 40006
diff changeset
   292
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   293
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
   294
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   295
lemma cont2mono_LAM:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   296
  "\<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
   297
    \<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
   298
  unfolding monofun_def cfun_below_iff by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   299
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   300
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
   301
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   302
text {*
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   303
  Not suitable as a cont2cont rule, because on nested lambdas
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   304
  it causes exponential blow-up in the number of subgoals.
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   305
*}
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   306
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   307
lemma cont2cont_LAM:
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   308
  assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   309
  assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   310
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   311
proof (rule cont_Abs_cfun)
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   312
  fix x
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   313
  from f1 show "f x \<in> cfun" by (simp add: cfun_def)
29049
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   314
  from f2 show "cont f" by (rule cont2cont_lambda)
4e5b9e508e1e cleaned up some proofs in Cfun.thy
huffman
parents: 27413
diff changeset
   315
qed
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   316
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   317
text {*
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   318
  This version does work as a cont2cont rule, since it
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   319
  has only a single subgoal.
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   320
*}
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   321
37079
0cd15d8c90a0 remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents: 36452
diff changeset
   322
lemma cont2cont_LAM' [simp, cont2cont]:
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   323
  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   324
  assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   325
  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
   326
using assms by (simp add: cont2cont_LAM prod_cont_iff)
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   327
37079
0cd15d8c90a0 remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents: 36452
diff changeset
   328
lemma cont2cont_LAM_discrete [simp, cont2cont]:
29530
9905b660612b change to simpler, more extensible continuity simproc
huffman
parents: 29138
diff changeset
   329
  "(\<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
   330
by (simp add: cont2cont_LAM)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   331
17832
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   332
subsection {* Miscellaneous *}
e18fc1a9a0e0 rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents: 17817
diff changeset
   333
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   334
text {* Monotonicity of @{term Abs_cfun} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   335
40433
3128c2a54785 remove some unnecessary lemmas; move monofun_LAM to Cfun.thy
huffman
parents: 40327
diff changeset
   336
lemma monofun_LAM:
3128c2a54785 remove some unnecessary lemmas; move monofun_LAM to Cfun.thy
huffman
parents: 40327
diff changeset
   337
  "\<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
   338
by (simp add: cfun_below_iff)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   339
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   340
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
   341
40327
1dfdbd66093a renamed {Rep,Abs}_CFun to {Rep,Abs}_cfun
huffman
parents: 40326
diff changeset
   342
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
   343
      ==> !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
   344
apply (rule allI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   345
apply (subst contlub_cfun_fun)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   346
apply assumption
40771
1c6f7d4b110e renamed several HOLCF theorems (listed in NEWS)
huffman
parents: 40770
diff changeset
   347
apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   348
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   349
18089
35c091a9841a moved adm_chfindom from Fix.thy to Cfun.thy
huffman
parents: 18087
diff changeset
   350
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
   351
by (rule adm_subst, simp, rule adm_chfin)
35c091a9841a moved adm_chfindom from Fix.thy to Cfun.thy
huffman
parents: 18087
diff changeset
   352
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   353
subsection {* Continuous injection-retraction pairs *}
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   354
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   355
text {* Continuous retractions are strict. *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   356
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   357
lemma retraction_strict:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   358
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
   359
apply (rule bottomI)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   360
apply (drule_tac x="\<bottom>" in spec)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   361
apply (erule subst)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   362
apply (rule monofun_cfun_arg)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   363
apply (rule minimal)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   364
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   365
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   366
lemma injection_eq:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   367
  "\<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
   368
apply (rule iffI)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   369
apply (drule_tac f=f in cfun_arg_cong)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   370
apply simp
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   371
apply simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   372
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   373
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
   374
lemma injection_below:
16314
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   375
  "\<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
   376
apply (rule iffI)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   377
apply (drule_tac f=f in monofun_cfun_arg)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   378
apply simp
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   379
apply (erule monofun_cfun_arg)
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   380
done
7102a1aaecfd added theorem injection_less
huffman
parents: 16209
diff changeset
   381
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   382
lemma injection_defined_rev:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   383
  "\<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
   384
apply (drule_tac f=f in cfun_arg_cong)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   385
apply (simp add: retraction_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   386
done
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 injection_defined:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   389
  "\<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
   390
by (erule contrapos_nn, rule injection_defined_rev)
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   391
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   392
text {* a result about functions with flat codomain *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   393
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   394
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
   395
by (drule ax_flat, simp)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   396
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   397
lemma flat_codom:
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   398
  "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
   399
apply (case_tac "f\<cdot>x = \<bottom>")
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   400
apply (rule disjI1)
41430
1aa23e9f2c87 change some lemma names containing 'UU' to 'bottom'
huffman
parents: 41400
diff changeset
   401
apply (rule bottomI)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   402
apply (erule_tac t="\<bottom>" in subst)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   403
apply (rule minimal [THEN monofun_cfun_arg])
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   404
apply clarify
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   405
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   406
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   407
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
   408
done
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   409
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   410
subsection {* Identity and composition *}
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   411
25135
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   412
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   413
  ID :: "'a \<rightarrow> 'a" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   414
  "ID = (\<Lambda> x. x)"
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   415
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   416
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   417
  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
   418
  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
   419
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   420
abbreviation
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   421
  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
   422
  "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
   423
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   424
lemma ID1 [simp]: "ID\<cdot>x = x"
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   425
by (simp add: ID_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   426
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   427
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
   428
by (simp add: oo_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   429
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   430
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
   431
by (simp add: cfcomp1)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   432
27274
1c97c471db82 add lemma cfcomp_LAM
huffman
parents: 26025
diff changeset
   433
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
   434
by (simp add: cfcomp1)
1c97c471db82 add lemma cfcomp_LAM
huffman
parents: 26025
diff changeset
   435
19709
78cd5f6af8e8 add theorem cfcomp_strict
huffman
parents: 18092
diff changeset
   436
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
   437
by (simp add: cfun_eq_iff)
19709
78cd5f6af8e8 add theorem cfcomp_strict
huffman
parents: 18092
diff changeset
   438
15589
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   439
text {*
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   440
  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
   441
  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
   442
  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
   443
  The identity arrow is interpretation of @{term ID}.
69bea57212ef reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   444
  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
   445
*}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   446
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   447
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
   448
by (rule cfun_eqI, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   449
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   450
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
   451
by (rule cfun_eqI, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   452
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   453
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
   454
by (rule cfun_eqI, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   455
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   456
subsection {* Strictified functions *}
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   457
36452
d37c6eed8117 renamed command 'defaultsort' to 'default_sort';
wenzelm
parents: 35933
diff changeset
   458
default_sort pcpo
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   459
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   460
definition
40767
a3e505b236e7 rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents: 40502
diff changeset
   461
  seq :: "'a \<rightarrow> 'b \<rightarrow> 'b" where
a3e505b236e7 rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents: 40502
diff changeset
   462
  "seq = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)"
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   463
40794
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   464
lemma cont2cont_if_bottom [cont2cont, simp]:
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   465
  assumes f: "cont (\<lambda>x. f x)" and g: "cont (\<lambda>x. g x)"
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   466
  shows "cont (\<lambda>x. if f x = \<bottom> then \<bottom> else g x)"
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   467
proof (rule cont_apply [OF f])
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   468
  show "\<And>x. cont (\<lambda>y. if y = \<bottom> then \<bottom> else g x)"
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   469
    unfolding cont_def is_lub_def is_ub_def ball_simps
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   470
    by (simp add: lub_eq_bottom_iff)
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   471
  show "\<And>y. cont (\<lambda>x. if y = \<bottom> then \<bottom> else g x)"
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   472
    by (simp add: g)
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   473
qed
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   474
40767
a3e505b236e7 rename function 'strict' to 'seq', which is its name in Haskell
huffman
parents: 40502
diff changeset
   475
lemma seq_conv_if: "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
40794
d28d41ee4cef add lemma cont2cont_if_bottom
huffman
parents: 40774
diff changeset
   476
unfolding seq_def by simp
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   477
41400
1a7557cc686a replaced separate lemmas seq{1,2,3} with seq_simps
huffman
parents: 41322
diff changeset
   478
lemma seq_simps [simp]:
1a7557cc686a replaced separate lemmas seq{1,2,3} with seq_simps
huffman
parents: 41322
diff changeset
   479
  "seq\<cdot>\<bottom> = \<bottom>"
1a7557cc686a replaced separate lemmas seq{1,2,3} with seq_simps
huffman
parents: 41322
diff changeset
   480
  "seq\<cdot>x\<cdot>\<bottom> = \<bottom>"
1a7557cc686a replaced separate lemmas seq{1,2,3} with seq_simps
huffman
parents: 41322
diff changeset
   481
  "x \<noteq> \<bottom> \<Longrightarrow> seq\<cdot>x = ID"
1a7557cc686a replaced separate lemmas seq{1,2,3} with seq_simps
huffman
parents: 41322
diff changeset
   482
by (simp_all add: seq_conv_if)
40093
c2d36bc4cd14 add lemma strict3
huffman
parents: 40091
diff changeset
   483
c2d36bc4cd14 add lemma strict3
huffman
parents: 40091
diff changeset
   484
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
   485
  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
   486
  "strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))"
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   487
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   488
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
   489
unfolding strictify_def by simp
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   490
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   491
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
17815
ccf54e3cabfa removed Istrictify; simplified some proofs
huffman
parents: 16920
diff changeset
   492
by (simp add: strictify_conv_if)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   493
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   494
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
   495
by (simp add: strictify_conv_if)
16085
c004b9bc970e rewrote continuous isomorphism section, cleaned up
huffman
parents: 16070
diff changeset
   496
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   497
subsection {* Continuity of let-bindings *}
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
   498
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   499
lemma cont2cont_Let:
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   500
  assumes f: "cont (\<lambda>x. f x)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   501
  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
   502
  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
   503
  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
   504
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
   505
37079
0cd15d8c90a0 remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents: 36452
diff changeset
   506
lemma cont2cont_Let' [simp, cont2cont]:
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   507
  assumes f: "cont (\<lambda>x. f x)"
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   508
  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
   509
  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
   510
using f
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   511
proof (rule cont2cont_Let)
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   512
  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
   513
    using g by (simp add: prod_cont_iff)
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   514
next
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   515
  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
   516
    using g by (simp add: prod_cont_iff)
35933
f135ebcc835c remove continuous let-binding function CLet; add cont2cont rule ordinary Let
huffman
parents: 35914
diff changeset
   517
qed
17816
9942c5ed866a new syntax translations for continuous lambda abstraction
huffman
parents: 17815
diff changeset
   518
39145
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   519
text {* The simple version (suggested by Joachim Breitner) is needed if
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   520
  the type of the defined term is not a cpo. *}
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   521
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   522
lemma cont2cont_Let_simple [simp, cont2cont]:
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   523
  assumes "\<And>y. cont (\<lambda>x. g x y)"
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   524
  shows "cont (\<lambda>x. let y = t in g x y)"
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   525
unfolding Let_def using assms .
154fd9c06c63 add lemma cont2cont_Let_simple
huffman
parents: 37083
diff changeset
   526
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   527
end