src/HOLCF/Sprod.thy
author huffman
Fri, 08 May 2009 16:19:51 -0700
changeset 31076 99fe356cbbc2
parent 29138 661a8db7e647
child 31114 2e9cc546e5b0
permissions -rw-r--r--
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
15600
a59f07556a8d fixed filename in header
huffman
parents: 15591
diff changeset
     1
(*  Title:      HOLCF/Sprod.thy
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
     2
    Author:     Franz Regensburger and Brian Huffman
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     3
*)
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
header {* The type of strict products *}
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     6
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
     7
theory Sprod
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16553
diff changeset
     8
imports Cprod
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
     9
begin
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    10
16082
ebb53ebfd4e2 added defaultsort declaration
huffman
parents: 16070
diff changeset
    11
defaultsort pcpo
ebb53ebfd4e2 added defaultsort declaration
huffman
parents: 16070
diff changeset
    12
15591
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    13
subsection {* Definition of strict product type *}
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    14
17817
405fb812e738 add names to infix declarations
huffman
parents: 16920
diff changeset
    15
pcpodef (Sprod)  ('a, 'b) "**" (infixr "**" 20) =
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    16
        "{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<bottom>)}"
29063
7619f0561cd7 pcpodef package: state two goals, instead of encoded conjunction;
wenzelm
parents: 27310
diff changeset
    17
by simp_all
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    18
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
    19
instance "**" :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
    20
by (rule typedef_finite_po [OF type_definition_Sprod])
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
    21
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
    22
instance "**" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
    23
by (rule typedef_chfin [OF type_definition_Sprod below_Sprod_def])
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
    24
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    25
syntax (xsymbols)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    26
  "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    27
syntax (HTML output)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    28
  "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    29
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    30
lemma spair_lemma:
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    31
  "<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a> \<in> Sprod"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    32
by (simp add: Sprod_def strictify_conv_if)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    33
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    34
subsection {* Definitions of constants *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    35
25135
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    36
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    37
  sfst :: "('a ** 'b) \<rightarrow> 'a" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    38
  "sfst = (\<Lambda> p. cfst\<cdot>(Rep_Sprod p))"
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    39
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    40
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    41
  ssnd :: "('a ** 'b) \<rightarrow> 'b" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    42
  "ssnd = (\<Lambda> p. csnd\<cdot>(Rep_Sprod p))"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    43
25135
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    44
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    45
  spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    46
  "spair = (\<Lambda> a b. Abs_Sprod
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    47
             <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>)"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    48
25135
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    49
definition
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    50
  ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c" where
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    51
  "ssplit = (\<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    52
4f8176c940cf modernized specifications ('definition', 'axiomatization');
wenzelm
parents: 25131
diff changeset
    53
syntax
18078
20e5a6440790 change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents: 17837
diff changeset
    54
  "@stuple" :: "['a, args] => 'a ** 'b"  ("(1'(:_,/ _:'))")
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    55
translations
18078
20e5a6440790 change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents: 17837
diff changeset
    56
  "(:x, y, z:)" == "(:x, (:y, z:):)"
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 18078
diff changeset
    57
  "(:x, y:)"    == "CONST spair\<cdot>x\<cdot>y"
18078
20e5a6440790 change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents: 17837
diff changeset
    58
20e5a6440790 change syntax for LAM to use expressions as patterns; define LAM pattern syntax for cpair, spair, sinl, sinr, up
huffman
parents: 17837
diff changeset
    59
translations
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 18078
diff changeset
    60
  "\<Lambda>(CONST spair\<cdot>x\<cdot>y). t" == "CONST ssplit\<cdot>(\<Lambda> x y. t)"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    61
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    62
subsection {* Case analysis *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    63
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    64
lemma Rep_Sprod_spair:
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    65
  "Rep_Sprod (:a, b:) = <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    66
unfolding spair_def
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    67
by (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    68
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    69
lemmas Rep_Sprod_simps =
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
    70
  Rep_Sprod_inject [symmetric] below_Sprod_def
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    71
  Rep_Sprod_strict Rep_Sprod_spair
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    72
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
    73
lemma Exh_Sprod:
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    74
  "z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    75
apply (insert Rep_Sprod [of z])
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    76
apply (simp add: Rep_Sprod_simps eq_cprod)
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    77
apply (simp add: Sprod_def)
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    78
apply (erule disjE, simp)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    79
apply (simp add: strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    80
apply fast
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    81
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    82
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
    83
lemma sprodE [cases type: **]:
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    84
  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
    85
by (cut_tac z=p in Exh_Sprod, auto)
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    86
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
    87
lemma sprod_induct [induct type: **]:
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
    88
  "\<lbrakk>P \<bottom>; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P (:x, y:)\<rbrakk> \<Longrightarrow> P x"
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
    89
by (cases x, simp_all)
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
    90
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    91
subsection {* Properties of @{term spair} *}
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    92
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    93
lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    94
by (simp add: Rep_Sprod_simps strictify_conv_if)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    95
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    96
lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    97
by (simp add: Rep_Sprod_simps strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    98
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
    99
lemma spair_strict_iff [simp]: "((:x, y:) = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   100
by (simp add: Rep_Sprod_simps strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   101
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   102
lemma spair_below_iff:
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   103
  "((:a, b:) \<sqsubseteq> (:c, d:)) = (a = \<bottom> \<or> b = \<bottom> \<or> (a \<sqsubseteq> c \<and> b \<sqsubseteq> d))"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   104
by (simp add: Rep_Sprod_simps strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   105
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   106
lemma spair_eq_iff:
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   107
  "((:a, b:) = (:c, d:)) =
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   108
    (a = c \<and> b = d \<or> (a = \<bottom> \<or> b = \<bottom>) \<and> (c = \<bottom> \<or> d = \<bottom>))"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   109
by (simp add: Rep_Sprod_simps strictify_conv_if)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   110
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   111
lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   112
by simp
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   113
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   114
lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   115
by simp
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   116
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   117
lemma spair_defined: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   118
by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   119
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   120
lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   121
by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   122
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   123
lemma spair_eq:
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   124
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ((:x, y:) = (:a, b:)) = (x = a \<and> y = b)"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   125
by (simp add: spair_eq_iff)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   126
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   127
lemma spair_inject:
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   128
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>; (:x, y:) = (:a, b:)\<rbrakk> \<Longrightarrow> x = a \<and> y = b"
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   129
by (rule spair_eq [THEN iffD1])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   130
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   131
lemma inst_sprod_pcpo2: "UU = (:UU,UU:)"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   132
by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   133
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   134
subsection {* Properties of @{term sfst} and @{term ssnd} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   135
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   136
lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>"
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   137
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   138
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   139
lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>"
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   140
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   141
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   142
lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   143
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   144
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   145
lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   146
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   147
16777
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   148
lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)"
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   149
by (cases p, simp_all)
16777
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   150
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   151
lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)"
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   152
by (cases p, simp_all)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   153
16777
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   154
lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>"
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   155
by simp
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   156
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   157
lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>"
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   158
by simp
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   159
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   160
lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p"
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   161
by (cases p, simp_all)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   162
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   163
lemma below_sprod: "x \<sqsubseteq> y = (sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y)"
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   164
apply (simp add: below_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   165
apply (rule below_cprod)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   166
done
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   167
16751
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   168
lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)"
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   169
by (auto simp add: po_eq_conv below_sprod)
16751
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   170
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   171
lemma spair_below:
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   172
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> b)"
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   173
apply (cases "a = \<bottom>", simp)
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   174
apply (cases "b = \<bottom>", simp)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   175
apply (simp add: below_sprod)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   176
done
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   177
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   178
lemma sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)"
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   179
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   180
apply (simp add: below_sprod)
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   181
done
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   182
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   183
lemma ssnd_below_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)"
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   184
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   185
apply (simp add: below_sprod)
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   186
done
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   187
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   188
subsection {* Compactness *}
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   189
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   190
lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)"
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   191
by (rule compactI, simp add: sfst_below_iff)
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   192
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   193
lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)"
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   194
by (rule compactI, simp add: ssnd_below_iff)
25881
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   195
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   196
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   197
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   198
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   199
lemma compact_spair_iff:
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   200
  "compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))"
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   201
apply (safe elim!: compact_spair)
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   202
apply (drule compact_sfst, simp)
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   203
apply (drule compact_ssnd, simp)
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   204
apply simp
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   205
apply simp
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   206
done
d80bd899ea95 Compactness subsection with new lemmas
huffman
parents: 25827
diff changeset
   207
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   208
subsection {* Properties of @{term ssplit} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   209
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   210
lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>"
15591
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   211
by (simp add: ssplit_def)
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   212
16920
ded12c9e88c2 cleaned up
huffman
parents: 16777
diff changeset
   213
lemma ssplit2 [simp]: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ssplit\<cdot>f\<cdot>(:x, y:) = f\<cdot>x\<cdot>y"
15591
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   214
by (simp add: ssplit_def)
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   215
16553
aa36d41e4263 add csplit3, ssplit3, fup3 as simp rules
huffman
parents: 16317
diff changeset
   216
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"
25757
5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
huffman
parents: 25135
diff changeset
   217
by (cases z, simp_all)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   218
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
   219
subsection {* Strict product preserves flatness *}
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
   220
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
   221
instance "**" :: (flat, flat) flat
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   222
proof
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   223
  fix x y :: "'a \<otimes> 'b"
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   224
  assume "x \<sqsubseteq> y" thus "x = \<bottom> \<or> x = y"
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   225
    apply (induct x, simp)
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   226
    apply (induct y, simp)
31076
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 29138
diff changeset
   227
    apply (simp add: spair_below_iff flat_below_iff)
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   228
    done
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   229
qed
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
huffman
parents: 25757
diff changeset
   230
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   231
subsection {* Strict product is a bifinite domain *}
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   232
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   233
instantiation "**" :: (bifinite, bifinite) bifinite
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   234
begin
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   235
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   236
definition
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   237
  approx_sprod_def:
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   238
    "approx = (\<lambda>n. \<Lambda>(:x, y:). (:approx n\<cdot>x, approx n\<cdot>y:))"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   239
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   240
instance proof
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   241
  fix i :: nat and x :: "'a \<otimes> 'b"
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   242
  show "chain (approx :: nat \<Rightarrow> 'a \<otimes> 'b \<rightarrow> 'a \<otimes> 'b)"
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   243
    unfolding approx_sprod_def by simp
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   244
  show "(\<Squnion>i. approx i\<cdot>x) = x"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   245
    unfolding approx_sprod_def
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   246
    by (simp add: lub_distribs eta_cfun)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   247
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   248
    unfolding approx_sprod_def
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   249
    by (simp add: ssplit_def strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   250
  have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   251
    unfolding approx_sprod_def
27310
d0229bc6c461 simplify profinite class axioms
huffman
parents: 26962
diff changeset
   252
    apply (clarify, case_tac x)
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   253
     apply (simp add: Rep_Sprod_strict)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   254
    apply (simp add: Rep_Sprod_spair spair_eq_iff)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   255
    done
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   256
  hence "finite (Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x})"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   257
    using finite_fixes_approx by (rule finite_subset)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   258
  thus "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   259
    by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   260
qed
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   261
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   262
end
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 25914
diff changeset
   263
25914
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   264
lemma approx_spair [simp]:
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   265
  "approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)"
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   266
unfolding approx_sprod_def
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   267
by (simp add: ssplit_def strictify_conv_if)
ff835e25ae87 clean up some proofs;
huffman
parents: 25881
diff changeset
   268
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   269
end