src/HOL/Library/Continuity.thy
author wenzelm
Thu, 16 Feb 2012 22:53:56 +0100
changeset 46508 ec9630fe9ca7
parent 44928 7ef6505bde7f
child 54257 5c7a3b6b05a9
permissions -rw-r--r--
tuned imports;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
     1
(*  Title:      HOL/Library/Continuity.thy
11355
wenzelm
parents: 11351
diff changeset
     2
    Author:     David von Oheimb, TU Muenchen
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
     3
*)
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
     4
14706
71590b7733b7 tuned document;
wenzelm
parents: 11461
diff changeset
     5
header {* Continuity and iterations (of set transformers) *}
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14981
diff changeset
     7
theory Continuity
46508
ec9630fe9ca7 tuned imports;
wenzelm
parents: 44928
diff changeset
     8
imports Main
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14981
diff changeset
     9
begin
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    10
22367
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    11
subsection {* Continuity for complete lattices *}
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    12
22367
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    13
definition
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22431
diff changeset
    14
  chain :: "(nat \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
22367
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    15
  "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    16
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    17
definition
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22431
diff changeset
    18
  continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
22367
6860f09242bf tuned document;
wenzelm
parents: 21404
diff changeset
    19
  "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    20
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    21
lemma SUP_nat_conv:
22431
28344ccffc35 Adapted to changes in definition of SUP.
berghofe
parents: 22422
diff changeset
    22
  "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    23
apply(rule order_antisym)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 32960
diff changeset
    24
 apply(rule SUP_least)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    25
 apply(case_tac n)
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    26
  apply simp
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 32960
diff changeset
    27
 apply (fast intro:SUP_upper le_supI2)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    28
apply(simp)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 32960
diff changeset
    29
apply (blast intro:SUP_least SUP_upper)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    30
done
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    31
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22431
diff changeset
    32
lemma continuous_mono: fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    33
  assumes "continuous F" shows "mono F"
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    34
proof
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    35
  fix A B :: "'a" assume "A <= B"
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    36
  let ?C = "%i::nat. if i=0 then A else B"
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    37
  have "chain ?C" using `A <= B` by(simp add:chain_def)
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22367
diff changeset
    38
  have "F B = sup (F A) (F B)"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    39
  proof -
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22367
diff changeset
    40
    have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
22431
28344ccffc35 Adapted to changes in definition of SUP.
berghofe
parents: 22422
diff changeset
    41
    hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    42
    also have "\<dots> = (SUP i. F(?C i))"
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    43
      using `chain ?C` `continuous F` by(simp add:continuous_def)
22431
28344ccffc35 Adapted to changes in definition of SUP.
berghofe
parents: 22422
diff changeset
    44
    also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    45
    finally show ?thesis .
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    46
  qed
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22367
diff changeset
    47
  thus "F A \<le> F B" by(subst le_iff_sup, simp)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    48
qed
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    49
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    50
lemma continuous_lfp:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    51
 assumes "continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    52
proof -
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    53
  note mono = continuous_mono[OF `continuous F`]
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    54
  { fix i have "(F ^^ i) bot \<le> lfp F"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    55
    proof (induct i)
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    56
      show "(F ^^ 0) bot \<le> lfp F" by simp
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    57
    next
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    58
      case (Suc i)
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    59
      have "(F ^^ Suc i) bot = F((F ^^ i) bot)" by simp
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    60
      also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    61
      also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    62
      finally show ?case .
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    63
    qed }
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 32960
diff changeset
    64
  hence "(SUP i. (F ^^ i) bot) \<le> lfp F" by (blast intro!:SUP_least)
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    65
  moreover have "lfp F \<le> (SUP i. (F ^^ i) bot)" (is "_ \<le> ?U")
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    66
  proof (rule lfp_lowerbound)
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    67
    have "chain(%i. (F ^^ i) bot)"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    68
    proof -
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    69
      { fix i have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
    70
        proof (induct i)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
    71
          case 0 show ?case by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
    72
        next
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
    73
          case Suc thus ?case using monoD[OF mono Suc] by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
    74
        qed }
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    75
      thus ?thesis by(auto simp add:chain_def)
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    76
    qed
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
    77
    hence "F ?U = (SUP i. (F ^^ (i+1)) bot)" using `continuous F` by (simp add:continuous_def)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 32960
diff changeset
    78
    also have "\<dots> \<le> ?U" by(fast intro:SUP_least SUP_upper)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    79
    finally show "F ?U \<le> ?U" .
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    80
  qed
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    81
  ultimately show ?thesis by (blast intro:order_antisym)
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    82
qed
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    83
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    84
text{* The following development is just for sets but presents an up
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    85
and a down version of chains and continuity and covers @{const gfp}. *}
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    86
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 19736
diff changeset
    87
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    88
subsection "Chains"
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    89
19736
wenzelm
parents: 15140
diff changeset
    90
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
    91
  up_chain :: "(nat => 'a set) => bool" where
19736
wenzelm
parents: 15140
diff changeset
    92
  "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    93
11355
wenzelm
parents: 11351
diff changeset
    94
lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
wenzelm
parents: 11351
diff changeset
    95
  by (simp add: up_chain_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    96
11355
wenzelm
parents: 11351
diff changeset
    97
lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
wenzelm
parents: 11351
diff changeset
    98
  by (simp add: up_chain_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
    99
19736
wenzelm
parents: 15140
diff changeset
   100
lemma up_chain_less_mono:
wenzelm
parents: 15140
diff changeset
   101
    "up_chain F ==> x < y ==> F x \<subseteq> F y"
wenzelm
parents: 15140
diff changeset
   102
  apply (induct y)
wenzelm
parents: 15140
diff changeset
   103
   apply (blast dest: up_chainD elim: less_SucE)+
11355
wenzelm
parents: 11351
diff changeset
   104
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   105
11355
wenzelm
parents: 11351
diff changeset
   106
lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
wenzelm
parents: 11351
diff changeset
   107
  apply (drule le_imp_less_or_eq)
wenzelm
parents: 11351
diff changeset
   108
  apply (blast dest: up_chain_less_mono)
wenzelm
parents: 11351
diff changeset
   109
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   110
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   111
19736
wenzelm
parents: 15140
diff changeset
   112
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
   113
  down_chain :: "(nat => 'a set) => bool" where
19736
wenzelm
parents: 15140
diff changeset
   114
  "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   115
11355
wenzelm
parents: 11351
diff changeset
   116
lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
wenzelm
parents: 11351
diff changeset
   117
  by (simp add: down_chain_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   118
11355
wenzelm
parents: 11351
diff changeset
   119
lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
wenzelm
parents: 11351
diff changeset
   120
  by (simp add: down_chain_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   121
19736
wenzelm
parents: 15140
diff changeset
   122
lemma down_chain_less_mono:
wenzelm
parents: 15140
diff changeset
   123
    "down_chain F ==> x < y ==> F y \<subseteq> F x"
wenzelm
parents: 15140
diff changeset
   124
  apply (induct y)
wenzelm
parents: 15140
diff changeset
   125
   apply (blast dest: down_chainD elim: less_SucE)+
11355
wenzelm
parents: 11351
diff changeset
   126
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   127
11355
wenzelm
parents: 11351
diff changeset
   128
lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
wenzelm
parents: 11351
diff changeset
   129
  apply (drule le_imp_less_or_eq)
wenzelm
parents: 11351
diff changeset
   130
  apply (blast dest: down_chain_less_mono)
wenzelm
parents: 11351
diff changeset
   131
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   132
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   133
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   134
subsection "Continuity"
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   135
19736
wenzelm
parents: 15140
diff changeset
   136
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
   137
  up_cont :: "('a set => 'a set) => bool" where
19736
wenzelm
parents: 15140
diff changeset
   138
  "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   139
11355
wenzelm
parents: 11351
diff changeset
   140
lemma up_contI:
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   141
  "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   142
apply (unfold up_cont_def)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   143
apply blast
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   144
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   145
11355
wenzelm
parents: 11351
diff changeset
   146
lemma up_contD:
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   147
  "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   148
apply (unfold up_cont_def)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   149
apply auto
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   150
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   151
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   152
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   153
lemma up_cont_mono: "up_cont f ==> mono f"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   154
apply (rule monoI)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 24331
diff changeset
   155
apply (drule_tac F = "\<lambda>i. if i = 0 then x else y" in up_contD)
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   156
 apply (rule up_chainI)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   157
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   158
apply (drule Un_absorb1)
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 30971
diff changeset
   159
apply (auto split:split_if_asm)
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   160
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   161
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   162
19736
wenzelm
parents: 15140
diff changeset
   163
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
   164
  down_cont :: "('a set => 'a set) => bool" where
19736
wenzelm
parents: 15140
diff changeset
   165
  "down_cont f =
wenzelm
parents: 15140
diff changeset
   166
    (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   167
11355
wenzelm
parents: 11351
diff changeset
   168
lemma down_contI:
wenzelm
parents: 11351
diff changeset
   169
  "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
wenzelm
parents: 11351
diff changeset
   170
    down_cont f"
wenzelm
parents: 11351
diff changeset
   171
  apply (unfold down_cont_def)
wenzelm
parents: 11351
diff changeset
   172
  apply blast
wenzelm
parents: 11351
diff changeset
   173
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   174
11355
wenzelm
parents: 11351
diff changeset
   175
lemma down_contD: "down_cont f ==> down_chain F ==>
wenzelm
parents: 11351
diff changeset
   176
    f (Inter (range F)) = Inter (f ` range F)"
wenzelm
parents: 11351
diff changeset
   177
  apply (unfold down_cont_def)
wenzelm
parents: 11351
diff changeset
   178
  apply auto
wenzelm
parents: 11351
diff changeset
   179
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   180
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   181
lemma down_cont_mono: "down_cont f ==> mono f"
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   182
apply (rule monoI)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 24331
diff changeset
   183
apply (drule_tac F = "\<lambda>i. if i = 0 then y else x" in down_contD)
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   184
 apply (rule down_chainI)
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   185
 apply simp
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   186
apply (drule Int_absorb1)
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 30971
diff changeset
   187
apply (auto split:split_if_asm)
24331
76f7a8c6e842 Made UN_Un simp
nipkow
parents: 23752
diff changeset
   188
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   189
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   190
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   191
subsection "Iteration"
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   192
19736
wenzelm
parents: 15140
diff changeset
   193
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
   194
  up_iterate :: "('a set => 'a set) => nat => 'a set" where
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   195
  "up_iterate f n = (f ^^ n) {}"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   196
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   197
lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
11355
wenzelm
parents: 11351
diff changeset
   198
  by (simp add: up_iterate_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   199
11355
wenzelm
parents: 11351
diff changeset
   200
lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
wenzelm
parents: 11351
diff changeset
   201
  by (simp add: up_iterate_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   202
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   203
lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
11355
wenzelm
parents: 11351
diff changeset
   204
  apply (rule up_chainI)
wenzelm
parents: 11351
diff changeset
   205
  apply (induct_tac i)
wenzelm
parents: 11351
diff changeset
   206
   apply simp+
wenzelm
parents: 11351
diff changeset
   207
  apply (erule (1) monoD)
wenzelm
parents: 11351
diff changeset
   208
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   209
11355
wenzelm
parents: 11351
diff changeset
   210
lemma UNION_up_iterate_is_fp:
wenzelm
parents: 11351
diff changeset
   211
  "up_cont F ==>
wenzelm
parents: 11351
diff changeset
   212
    F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
wenzelm
parents: 11351
diff changeset
   213
  apply (frule up_cont_mono [THEN up_iterate_chain])
wenzelm
parents: 11351
diff changeset
   214
  apply (drule (1) up_contD)
wenzelm
parents: 11351
diff changeset
   215
  apply simp
wenzelm
parents: 11351
diff changeset
   216
  apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
wenzelm
parents: 11351
diff changeset
   217
  apply (case_tac xa)
wenzelm
parents: 11351
diff changeset
   218
   apply auto
wenzelm
parents: 11351
diff changeset
   219
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   220
11355
wenzelm
parents: 11351
diff changeset
   221
lemma UNION_up_iterate_lowerbound:
wenzelm
parents: 11351
diff changeset
   222
    "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
wenzelm
parents: 11351
diff changeset
   223
  apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
wenzelm
parents: 11351
diff changeset
   224
   apply fast
wenzelm
parents: 11351
diff changeset
   225
  apply (induct_tac i)
wenzelm
parents: 11351
diff changeset
   226
  prefer 2 apply (drule (1) monoD)
wenzelm
parents: 11351
diff changeset
   227
   apply auto
wenzelm
parents: 11351
diff changeset
   228
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   229
11355
wenzelm
parents: 11351
diff changeset
   230
lemma UNION_up_iterate_is_lfp:
wenzelm
parents: 11351
diff changeset
   231
    "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
wenzelm
parents: 11351
diff changeset
   232
  apply (rule set_eq_subset [THEN iffD2])
wenzelm
parents: 11351
diff changeset
   233
  apply (rule conjI)
wenzelm
parents: 11351
diff changeset
   234
   prefer 2
wenzelm
parents: 11351
diff changeset
   235
   apply (drule up_cont_mono)
wenzelm
parents: 11351
diff changeset
   236
   apply (rule UNION_up_iterate_lowerbound)
wenzelm
parents: 11351
diff changeset
   237
    apply assumption
wenzelm
parents: 11351
diff changeset
   238
   apply (erule lfp_unfold [symmetric])
wenzelm
parents: 11351
diff changeset
   239
  apply (rule lfp_lowerbound)
wenzelm
parents: 11351
diff changeset
   240
  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
wenzelm
parents: 11351
diff changeset
   241
  apply (erule UNION_up_iterate_is_fp [symmetric])
wenzelm
parents: 11351
diff changeset
   242
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   243
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   244
19736
wenzelm
parents: 15140
diff changeset
   245
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21312
diff changeset
   246
  down_iterate :: "('a set => 'a set) => nat => 'a set" where
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   247
  "down_iterate f n = (f ^^ n) UNIV"
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   248
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   249
lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
11355
wenzelm
parents: 11351
diff changeset
   250
  by (simp add: down_iterate_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   251
11355
wenzelm
parents: 11351
diff changeset
   252
lemma down_iterate_Suc [simp]:
wenzelm
parents: 11351
diff changeset
   253
    "down_iterate f (Suc i) = f (down_iterate f i)"
wenzelm
parents: 11351
diff changeset
   254
  by (simp add: down_iterate_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   255
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   256
lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
11355
wenzelm
parents: 11351
diff changeset
   257
  apply (rule down_chainI)
wenzelm
parents: 11351
diff changeset
   258
  apply (induct_tac i)
wenzelm
parents: 11351
diff changeset
   259
   apply simp+
wenzelm
parents: 11351
diff changeset
   260
  apply (erule (1) monoD)
wenzelm
parents: 11351
diff changeset
   261
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   262
11355
wenzelm
parents: 11351
diff changeset
   263
lemma INTER_down_iterate_is_fp:
wenzelm
parents: 11351
diff changeset
   264
  "down_cont F ==>
wenzelm
parents: 11351
diff changeset
   265
    F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
wenzelm
parents: 11351
diff changeset
   266
  apply (frule down_cont_mono [THEN down_iterate_chain])
wenzelm
parents: 11351
diff changeset
   267
  apply (drule (1) down_contD)
wenzelm
parents: 11351
diff changeset
   268
  apply simp
wenzelm
parents: 11351
diff changeset
   269
  apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
wenzelm
parents: 11351
diff changeset
   270
  apply (case_tac xa)
wenzelm
parents: 11351
diff changeset
   271
   apply auto
wenzelm
parents: 11351
diff changeset
   272
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   273
11355
wenzelm
parents: 11351
diff changeset
   274
lemma INTER_down_iterate_upperbound:
wenzelm
parents: 11351
diff changeset
   275
    "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
wenzelm
parents: 11351
diff changeset
   276
  apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
wenzelm
parents: 11351
diff changeset
   277
   apply fast
wenzelm
parents: 11351
diff changeset
   278
  apply (induct_tac i)
wenzelm
parents: 11351
diff changeset
   279
  prefer 2 apply (drule (1) monoD)
wenzelm
parents: 11351
diff changeset
   280
   apply auto
wenzelm
parents: 11351
diff changeset
   281
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   282
11355
wenzelm
parents: 11351
diff changeset
   283
lemma INTER_down_iterate_is_gfp:
wenzelm
parents: 11351
diff changeset
   284
    "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
wenzelm
parents: 11351
diff changeset
   285
  apply (rule set_eq_subset [THEN iffD2])
wenzelm
parents: 11351
diff changeset
   286
  apply (rule conjI)
wenzelm
parents: 11351
diff changeset
   287
   apply (drule down_cont_mono)
wenzelm
parents: 11351
diff changeset
   288
   apply (rule INTER_down_iterate_upperbound)
wenzelm
parents: 11351
diff changeset
   289
    apply assumption
wenzelm
parents: 11351
diff changeset
   290
   apply (erule gfp_unfold [symmetric])
wenzelm
parents: 11351
diff changeset
   291
  apply (rule gfp_upperbound)
wenzelm
parents: 11351
diff changeset
   292
  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
wenzelm
parents: 11351
diff changeset
   293
  apply (erule INTER_down_iterate_is_fp)
wenzelm
parents: 11351
diff changeset
   294
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   295
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   296
end