src/HOLCF/Cprod.thy
author huffman
Thu, 10 Mar 2005 20:22:45 +0100
changeset 15600 a59f07556a8d
parent 15593 24d770bbc44a
child 15609 d12c459e2325
permissions -rw-r--r--
fixed filename in header
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
15600
a59f07556a8d fixed filename in header
huffman
parents: 15593
diff changeset
     1
(*  Title:      HOLCF/Cprod.thy
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     2
    ID:         $Id$
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     3
    Author:     Franz Regensburger
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     4
    License:    GPL (GNU GENERAL PUBLIC LICENSE)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     5
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     6
Partial ordering for cartesian product of HOL theory prod.thy
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
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
     9
header {* The cpo of cartesian products *}
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    10
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    11
theory Cprod
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    12
imports Cfun
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
    13
begin
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    14
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    15
defaultsort cpo
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    16
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    17
subsection {* Ordering on @{typ "'a * 'b"} *}
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    18
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    19
instance "*" :: (sq_ord, sq_ord) sq_ord ..
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    20
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    21
defs (overloaded)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    22
  less_cprod_def: "p1 << p2 == (fst p1<<fst p2 & snd p1 << snd p2)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    23
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    24
subsection {* Type @{typ "'a * 'b"} is a partial order *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    25
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    26
lemma refl_less_cprod: "(p::'a*'b) << p"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    27
apply (unfold less_cprod_def)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    28
apply simp
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    29
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    30
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    31
lemma antisym_less_cprod: "[|(p1::'a * 'b) << p2;p2 << p1|] ==> p1=p2"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    32
apply (unfold less_cprod_def)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    33
apply (rule injective_fst_snd)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    34
apply (fast intro: antisym_less)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    35
apply (fast intro: antisym_less)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    36
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    37
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    38
lemma trans_less_cprod: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    39
        "[|(p1::'a*'b) << p2;p2 << p3|] ==> p1 << p3"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    40
apply (unfold less_cprod_def)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    41
apply (rule conjI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    42
apply (fast intro: trans_less)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    43
apply (fast intro: trans_less)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    44
done
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
defaultsort pcpo
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    47
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    48
instance "*" :: (cpo, cpo) po
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    49
by intro_classes
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    50
  (assumption | rule refl_less_cprod antisym_less_cprod trans_less_cprod)+
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    51
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    52
text {* for compatibility with old HOLCF-Version *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    53
lemma inst_cprod_po: "(op <<)=(%x y. fst x<<fst y & snd x<<snd y)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    54
apply (fold less_cprod_def)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    55
apply (rule refl)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    56
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    57
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    58
lemma less_cprod4c: "(x1,y1) << (x2,y2) ==> x1 << x2 & y1 << y2"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    59
apply (simp add: inst_cprod_po)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    60
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    61
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    62
subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    63
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    64
text {* Pair @{text "(_,_)"}  is monotone in both arguments *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    65
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    66
lemma monofun_pair1: "monofun Pair"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    67
by (simp add: monofun less_fun inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    68
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    69
lemma monofun_pair2: "monofun(Pair x)"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    70
by (simp add: monofun inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    71
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    72
lemma monofun_pair: "[|x1<<x2; y1<<y2|] ==> (x1::'a::cpo,y1::'b::cpo)<<(x2,y2)"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    73
by (simp add: inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    74
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    75
text {* @{term fst} and @{term snd} are monotone *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    76
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    77
lemma monofun_fst: "monofun fst"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    78
by (simp add: monofun inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    79
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    80
lemma monofun_snd: "monofun snd"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    81
by (simp add: monofun inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    82
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
    83
subsection {* Type @{typ "'a * 'b"} is a cpo *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    84
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    85
lemma lub_cprod: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    86
"chain S ==> range S<<|(lub(range(%i. fst(S i))),lub(range(%i. snd(S i))))"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    87
apply (rule is_lubI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    88
apply (rule ub_rangeI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    89
apply (rule_tac t = "S i" in surjective_pairing [THEN ssubst])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    90
apply (rule monofun_pair)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    91
apply (rule is_ub_thelub)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    92
apply (erule monofun_fst [THEN ch2ch_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    93
apply (rule is_ub_thelub)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    94
apply (erule monofun_snd [THEN ch2ch_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    95
apply (rule_tac t = "u" in surjective_pairing [THEN ssubst])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    96
apply (rule monofun_pair)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    97
apply (rule is_lub_thelub)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    98
apply (erule monofun_fst [THEN ch2ch_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
    99
apply (erule monofun_fst [THEN ub2ub_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   100
apply (rule is_lub_thelub)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   101
apply (erule monofun_snd [THEN ch2ch_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   102
apply (erule monofun_snd [THEN ub2ub_monofun])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   103
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   104
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   105
lemmas thelub_cprod = lub_cprod [THEN thelubI, standard]
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   106
(*
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   107
"chain ?S1 ==>
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   108
 lub (range ?S1) =
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   109
 (lub (range (%i. fst (?S1 i))), lub (range (%i. snd (?S1 i))))" : thm
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   110
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   111
*)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   112
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   113
lemma cpo_cprod: "chain(S::nat=>'a::cpo*'b::cpo)==>EX x. range S<<| x"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   114
by (rule exI, erule lub_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   115
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   116
instance "*" :: (cpo,cpo)cpo
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   117
by intro_classes (rule cpo_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   118
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   119
subsection {* Type @{typ "'a * 'b"} is pointed *}
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   120
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   121
lemma minimal_cprod: "(UU,UU)<<p"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   122
by (simp add: inst_cprod_po)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   123
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   124
lemmas UU_cprod_def = minimal_cprod [THEN minimal2UU, symmetric, standard]
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   125
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   126
lemma least_cprod: "EX x::'a*'b. ALL y. x<<y"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   127
apply (rule_tac x = " (UU,UU) " in exI)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   128
apply (rule minimal_cprod [THEN allI])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   129
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   130
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   131
instance "*" :: (pcpo,pcpo)pcpo
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   132
by intro_classes (rule least_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   133
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   134
text {* for compatibility with old HOLCF-Version *}
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   135
lemma inst_cprod_pcpo: "UU = (UU,UU)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   136
apply (simp add: UU_cprod_def[folded UU_def])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   137
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   138
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   139
subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   140
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   141
lemma contlub_pair1: "contlub(Pair)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   142
apply (rule contlubI [rule_format])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   143
apply (rule ext)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   144
apply (subst lub_fun [THEN thelubI])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   145
apply (erule monofun_pair1 [THEN ch2ch_monofun])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   146
apply (subst thelub_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   147
apply (rule ch2ch_fun)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   148
apply (erule monofun_pair1 [THEN ch2ch_monofun])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   149
apply (simp add: lub_const [THEN thelubI])
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   150
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   151
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   152
lemma contlub_pair2: "contlub(Pair(x))"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   153
apply (rule contlubI [rule_format])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   154
apply (subst thelub_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   155
apply (erule monofun_pair2 [THEN ch2ch_monofun])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   156
apply (simp add: lub_const [THEN thelubI])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   157
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   158
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   159
lemma cont_pair1: "cont(Pair)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   160
apply (rule monocontlub2cont)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   161
apply (rule monofun_pair1)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   162
apply (rule contlub_pair1)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   163
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   164
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   165
lemma cont_pair2: "cont(Pair(x))"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   166
apply (rule monocontlub2cont)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   167
apply (rule monofun_pair2)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   168
apply (rule contlub_pair2)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   169
done
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   170
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   171
lemma contlub_fst: "contlub(fst)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   172
apply (rule contlubI [rule_format])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   173
apply (simp add: lub_cprod [THEN thelubI])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   174
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   175
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   176
lemma contlub_snd: "contlub(snd)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   177
apply (rule contlubI [rule_format])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   178
apply (simp add: lub_cprod [THEN thelubI])
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   179
done
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   180
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   181
lemma cont_fst: "cont(fst)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   182
apply (rule monocontlub2cont)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   183
apply (rule monofun_fst)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   184
apply (rule contlub_fst)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   185
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   186
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   187
lemma cont_snd: "cont(snd)"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   188
apply (rule monocontlub2cont)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   189
apply (rule monofun_snd)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   190
apply (rule contlub_snd)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   191
done
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   192
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   193
subsection {* Continuous versions of constants *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   194
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   195
consts
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   196
        cpair        :: "'a::cpo -> 'b::cpo -> ('a*'b)" (* continuous pairing *)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   197
        cfst         :: "('a::cpo*'b::cpo)->'a"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   198
        csnd         :: "('a::cpo*'b::cpo)->'b"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   199
        csplit       :: "('a::cpo->'b::cpo->'c::cpo)->('a*'b)->'c"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   200
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   201
syntax
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   202
        "@ctuple"    :: "['a, args] => 'a * 'b"         ("(1<_,/ _>)")
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   203
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   204
translations
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   205
        "<x, y, z>"   == "<x, <y, z>>"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   206
        "<x, y>"      == "cpair$x$y"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   207
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   208
defs
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   209
cpair_def:       "cpair  == (LAM x y.(x,y))"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   210
cfst_def:        "cfst   == (LAM p. fst(p))"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   211
csnd_def:        "csnd   == (LAM p. snd(p))"      
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   212
csplit_def:      "csplit == (LAM f p. f$(cfst$p)$(csnd$p))"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   213
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   214
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   215
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   216
(* introduce syntax for
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   217
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   218
   Let <x,y> = e1; z = E2 in E3
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   219
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   220
   and
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   221
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   222
   LAM <x,y,z>.e
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   223
*)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   224
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   225
constdefs
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   226
  CLet           :: "'a -> ('a -> 'b) -> 'b"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   227
  "CLet == LAM s f. f$s"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   228
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   229
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   230
(* syntax for Let *)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   231
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   232
nonterminals
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   233
  Cletbinds  Cletbind
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   234
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   235
syntax
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   236
  "_Cbind"  :: "[pttrn, 'a] => Cletbind"             ("(2_ =/ _)" 10)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   237
  ""        :: "Cletbind => Cletbinds"               ("_")
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   238
  "_Cbinds" :: "[Cletbind, Cletbinds] => Cletbinds"  ("_;/ _")
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   239
  "_CLet"   :: "[Cletbinds, 'a] => 'a"               ("(Let (_)/ in (_))" 10)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   240
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   241
translations
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   242
  "_CLet (_Cbinds b bs) e"  == "_CLet b (_CLet bs e)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   243
  "Let x = a in e"          == "CLet$a$(LAM x. e)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   244
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   245
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   246
(* syntax for LAM <x,y,z>.e *)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   247
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   248
syntax
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   249
  "_LAM"    :: "[patterns, 'a => 'b] => ('a -> 'b)"  ("(3LAM <_>./ _)" [0, 10] 10)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   250
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   251
translations
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   252
  "LAM <x,y,zs>.b"        == "csplit$(LAM x. LAM <y,zs>.b)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   253
  "LAM <x,y>. LAM zs. b"  <= "csplit$(LAM x y zs. b)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   254
  "LAM <x,y>.b"           == "csplit$(LAM x y. b)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   255
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   256
syntax (xsymbols)
15577
e16da3068ad6 fix headers
huffman
parents: 15576
diff changeset
   257
  "_LAM"    :: "[patterns, 'a => 'b] => ('a -> 'b)"  ("(3\<Lambda>()<_>./ _)" [0, 10] 10)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   258
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   259
subsection {* Convert all lemmas to the continuous versions *}
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   260
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   261
lemma beta_cfun_cprod: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   262
        "(LAM x y.(x,y))$a$b = (a,b)"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   263
apply (subst beta_cfun)
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   264
apply (simp add: cont_pair1 cont_pair2 cont2cont_CF1L)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   265
apply (subst beta_cfun)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   266
apply (rule cont_pair2)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   267
apply (rule refl)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   268
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   269
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   270
lemma inject_cpair: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   271
        "<a,b> = <aa,ba>  ==> a=aa & b=ba"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   272
by (simp add: cpair_def beta_cfun_cprod)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   273
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   274
lemma inst_cprod_pcpo2: "UU = <UU,UU>"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   275
by (simp add: cpair_def beta_cfun_cprod inst_cprod_pcpo)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   276
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   277
lemma defined_cpair_rev: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   278
 "<a,b> = UU ==> a = UU & b = UU"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   279
apply (drule inst_cprod_pcpo2 [THEN subst])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   280
apply (erule inject_cpair)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   281
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   282
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   283
lemma Exh_Cprod2: "? a b. z=<a,b>"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   284
apply (unfold cpair_def)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   285
apply (rule PairE)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   286
apply (rule exI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   287
apply (rule exI)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   288
apply (erule beta_cfun_cprod [THEN ssubst])
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   289
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   290
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   291
lemma cprodE:
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   292
assumes prems: "!!x y. [| p = <x,y> |] ==> Q"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   293
shows "Q"
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   294
apply (rule PairE)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   295
apply (rule prems)
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   296
apply (simp add: cpair_def beta_cfun_cprod)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   297
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   298
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   299
lemma cfst2 [simp]: "cfst$<x,y> = x"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   300
by (simp add: cpair_def cfst_def beta_cfun_cprod cont_fst)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   301
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   302
lemma csnd2 [simp]: "csnd$<x,y> = y"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   303
by (simp add: cpair_def csnd_def beta_cfun_cprod cont_snd)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   304
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   305
lemma cfst_strict: "cfst$UU = UU"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   306
by (simp add: inst_cprod_pcpo2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   307
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   308
lemma csnd_strict: "csnd$UU = UU"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   309
by (simp add: inst_cprod_pcpo2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   310
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   311
lemma surjective_pairing_Cprod2: "<cfst$p, csnd$p> = p"
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   312
apply (unfold cfst_def csnd_def cpair_def)
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   313
apply (simp add: cont_fst cont_snd beta_cfun_cprod)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   314
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   315
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   316
lemma less_cprod5c: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   317
 "<xa,ya> << <x,y> ==> xa<<x & ya << y"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   318
by (simp add: cpair_def beta_cfun_cprod inst_cprod_po)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   319
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   320
lemma lub_cprod2: 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   321
"[|chain(S)|] ==> range(S) <<|  
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   322
  <(lub(range(%i. cfst$(S i)))) , lub(range(%i. csnd$(S i)))>"
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   323
apply (simp add: cpair_def beta_cfun_cprod)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   324
apply (simp add: cfst_def csnd_def cont_fst cont_snd)
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   325
apply (erule lub_cprod)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   326
done
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   327
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   328
lemmas thelub_cprod2 = lub_cprod2 [THEN thelubI, standard]
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   329
(*
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   330
chain ?S1 ==>
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   331
 lub (range ?S1) =
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   332
 <lub (range (%i. cfst$(?S1 i))), lub (range (%i. csnd$(?S1 i)))>" 
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   333
*)
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   334
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   335
lemma csplit2 [simp]: "csplit$f$<x,y> = f$x$y"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   336
by (simp add: csplit_def)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   337
15593
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   338
lemma csplit3: "csplit$cpair$z=z"
24d770bbc44a reordered and arranged for document generation, cleaned up some proofs
huffman
parents: 15577
diff changeset
   339
by (simp add: csplit_def surjective_pairing_Cprod2)
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   340
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   341
lemmas Cprod_rews = cfst2 csnd2 csplit2
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   342
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
parents:
diff changeset
   343
end