src/HOLCF/Sprod.thy
author huffman
Wed, 12 Oct 2005 01:43:37 +0200
changeset 17837 2922be3544f8
parent 17817 405fb812e738
child 18078 20e5a6440790
permissions -rw-r--r--
added compactness lemmas; cleaned up
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
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     2
    ID:         $Id$
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
     3
    Author:     Franz Regensburger and 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
Strict product with typedef.
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     6
*)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     7
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     8
header {* The type of strict products *}
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     9
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    10
theory Sprod
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16553
diff changeset
    11
imports Cprod
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    12
begin
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    13
16082
ebb53ebfd4e2 added defaultsort declaration
huffman
parents: 16070
diff changeset
    14
defaultsort pcpo
ebb53ebfd4e2 added defaultsort declaration
huffman
parents: 16070
diff changeset
    15
15591
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    16
subsection {* Definition of strict product type *}
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    17
17817
405fb812e738 add names to infix declarations
huffman
parents: 16920
diff changeset
    18
pcpodef (Sprod)  ('a, 'b) "**" (infixr "**" 20) =
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    19
        "{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<bottom>)}"
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16553
diff changeset
    20
by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    21
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    22
syntax (xsymbols)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    23
  "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    24
syntax (HTML output)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    25
  "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    26
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    27
lemma spair_lemma:
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    28
  "<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a> \<in> Sprod"
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
    29
by (simp add: Sprod_def strictify_conv_if cpair_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    30
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    31
subsection {* Definitions of constants *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    32
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    33
consts
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    34
  sfst :: "('a ** 'b) \<rightarrow> 'a"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    35
  ssnd :: "('a ** 'b) \<rightarrow> 'b"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    36
  spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    37
  ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    38
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    39
defs
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    40
  sfst_def: "sfst \<equiv> \<Lambda> p. cfst\<cdot>(Rep_Sprod p)"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    41
  ssnd_def: "ssnd \<equiv> \<Lambda> p. csnd\<cdot>(Rep_Sprod p)"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    42
  spair_def: "spair \<equiv> \<Lambda> a b. Abs_Sprod
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    43
                <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    44
  ssplit_def: "ssplit \<equiv> \<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p))"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    45
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    46
syntax  
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    47
  "@stuple"	:: "['a, args] => 'a ** 'b"	("(1'(:_,/ _:'))")
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    48
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    49
translations
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    50
        "(:x, y, z:)"   == "(:x, (:y, z:):)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    51
        "(:x, y:)"      == "spair$x$y"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    52
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    53
subsection {* Case analysis *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    54
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    55
lemma spair_Abs_Sprod:
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    56
  "(:a, b:) = Abs_Sprod <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    57
apply (unfold spair_def)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    58
apply (simp add: cont_Abs_Sprod spair_lemma)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    59
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    60
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    61
lemma Exh_Sprod2:
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    62
  "z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    63
apply (rule_tac x=z in Abs_Sprod_cases)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    64
apply (simp add: Sprod_def)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    65
apply (erule disjE)
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
    66
apply (simp add: Abs_Sprod_strict)
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    67
apply (rule disjI2)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    68
apply (rule_tac x="cfst\<cdot>y" in exI)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    69
apply (rule_tac x="csnd\<cdot>y" in exI)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    70
apply (simp add: spair_Abs_Sprod Abs_Sprod_inject spair_lemma)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    71
apply (simp add: surjective_pairing_Cprod2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    72
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    73
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    74
lemma sprodE:
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    75
  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    76
by (cut_tac z=p in Exh_Sprod2, auto)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    77
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    78
subsection {* Properties of @{term spair} *}
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    79
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    80
lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"
16920
ded12c9e88c2 cleaned up
huffman
parents: 16777
diff changeset
    81
by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    82
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    83
lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
16920
ded12c9e88c2 cleaned up
huffman
parents: 16777
diff changeset
    84
by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    85
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    86
lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    87
by auto
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    88
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
    89
lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    90
by (erule contrapos_np, auto)
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    91
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
    92
lemma spair_defined [simp]: 
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    93
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>"
16920
ded12c9e88c2 cleaned up
huffman
parents: 16777
diff changeset
    94
by (simp add: spair_Abs_Sprod Abs_Sprod_defined cpair_defined_iff Sprod_def)
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_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
    97
by (erule contrapos_pp, simp)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    98
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
    99
lemma spair_eq:
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   100
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ((:x, y:) = (:a, b:)) = (x = a \<and> y = b)"
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   101
apply (simp add: spair_Abs_Sprod)
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   102
apply (simp add: Abs_Sprod_inject [OF _ spair_lemma] Sprod_def)
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   103
apply (simp add: strictify_conv_if)
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   104
done
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   105
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   106
lemma spair_inject:
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   107
  "\<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
   108
by (rule spair_eq [THEN iffD1])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   109
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   110
lemma inst_sprod_pcpo2: "UU = (:UU,UU:)"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   111
by simp
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   112
17837
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   113
lemma Rep_Sprod_spair:
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   114
  "Rep_Sprod (:a, b:) = <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   115
apply (unfold spair_def)
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   116
apply (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   117
done
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   118
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   119
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   120
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)
2922be3544f8 added compactness lemmas; cleaned up
huffman
parents: 17817
diff changeset
   121
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   122
subsection {* Properties of @{term sfst} and @{term ssnd} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   123
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   124
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
   125
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
   126
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   127
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
   128
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
   129
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   130
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
   131
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
   132
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
huffman
parents: 16082
diff changeset
   133
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
   134
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
   135
16777
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   136
lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)"
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   137
by (rule_tac p=p in sprodE, simp_all)
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   138
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   139
lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   140
by (rule_tac p=p in sprodE, simp_all)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   141
16777
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   142
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
   143
by simp
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   144
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   145
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
   146
by simp
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
huffman
parents: 16751
diff changeset
   147
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   148
lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p"
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   149
by (rule_tac p=p in sprodE, simp_all)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   150
16751
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   151
lemma less_sprod: "x \<sqsubseteq> y = (sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y)"
16699
24b494ff8f0f use new pcpodef package
huffman
parents: 16553
diff changeset
   152
apply (simp add: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   153
apply (rule less_cprod)
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   154
done
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   155
16751
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   156
lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)"
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   157
by (auto simp add: po_eq_conv less_sprod)
7af6723ad741 add lemma eq_sprod
huffman
parents: 16699
diff changeset
   158
16317
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   159
lemma spair_less:
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   160
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> b)"
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   161
apply (case_tac "a = \<bottom>")
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   162
apply (simp add: eq_UU_iff [symmetric])
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   163
apply (case_tac "b = \<bottom>")
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   164
apply (simp add: eq_UU_iff [symmetric])
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   165
apply (simp add: less_sprod)
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
868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
huffman
parents: 16212
diff changeset
   168
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   169
subsection {* Properties of @{term ssplit} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   170
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   171
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
   172
by (simp add: ssplit_def)
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   173
16920
ded12c9e88c2 cleaned up
huffman
parents: 16777
diff changeset
   174
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
   175
by (simp add: ssplit_def)
50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   176
16553
aa36d41e4263 add csplit3, ssplit3, fup3 as simp rules
huffman
parents: 16317
diff changeset
   177
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
huffman
parents: 15930
diff changeset
   178
by (rule_tac p=z in sprodE, simp_all)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   179
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   180
end