author  haftmann 
Thu, 20 Mar 2008 12:04:54 +0100  
changeset 26358  d6a508c16908 
parent 26340  a85fe32e7b2f 
child 26480  544cef16045b 
permissions  rwrr 
10213  1 
(* Title: HOL/Product_Type.thy 
2 
ID: $Id$ 

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 
Copyright 1992 University of Cambridge 

11777  5 
*) 
10213  6 

11838  7 
header {* Cartesian products *} 
10213  8 

15131  9 
theory Product_Type 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

10 
imports Inductive 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

11 
uses 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

12 
("Tools/split_rule.ML") 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

13 
("Tools/inductive_set_package.ML") 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

14 
("Tools/inductive_realizer.ML") 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

15 
("Tools/datatype_realizer.ML") 
15131  16 
begin 
11838  17 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

18 
subsection {* @{typ bool} is a datatype *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

19 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

20 
rep_datatype bool 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

21 
distinct True_not_False False_not_True 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

22 
induction bool_induct 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

23 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

24 
declare case_split [cases type: bool] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

25 
 "prefer plain propositional version" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

26 

25534
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

27 
lemma [code func]: 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

28 
shows "False = P \<longleftrightarrow> \<not> P" 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

29 
and "True = P \<longleftrightarrow> P" 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

30 
and "P = False \<longleftrightarrow> \<not> P" 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

31 
and "P = True \<longleftrightarrow> P" by simp_all 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

32 

d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

33 
code_const "op = \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

34 
(Haskell infixl 4 "==") 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

35 

d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

36 
code_instance bool :: eq 
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset

37 
(Haskell ) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

38 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

39 

11838  40 
subsection {* Unit *} 
41 

42 
typedef unit = "{True}" 

43 
proof 

20588  44 
show "True : ?unit" .. 
11838  45 
qed 
46 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

47 
definition 
11838  48 
Unity :: unit ("'(')") 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

49 
where 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

50 
"() = Abs_unit True" 
11838  51 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

52 
lemma unit_eq [noatp]: "u = ()" 
11838  53 
by (induct u) (simp add: unit_def Unity_def) 
54 

55 
text {* 

56 
Simplification procedure for @{thm [source] unit_eq}. Cannot use 

57 
this rule directly  it loops! 

58 
*} 

59 

60 
ML_setup {* 

13462  61 
val unit_eq_proc = 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

62 
let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

63 
Simplifier.simproc @{theory} "unit_eq" ["x::unit"] 
15531  64 
(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) 
13462  65 
end; 
11838  66 

67 
Addsimprocs [unit_eq_proc]; 

68 
*} 

69 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

70 
lemma unit_induct [noatp,induct type: unit]: "P () ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

71 
by simp 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

72 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

73 
rep_datatype unit 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

74 
induction unit_induct 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

75 

11838  76 
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" 
77 
by simp 

78 

79 
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" 

80 
by (rule triv_forall_equality) 

81 

82 
text {* 

83 
This rewrite counters the effect of @{text unit_eq_proc} on @{term 

84 
[source] "%u::unit. f u"}, replacing it by @{term [source] 

85 
f} rather than by @{term [source] "%u. f ()"}. 

86 
*} 

87 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

88 
lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f" 
11838  89 
by (rule ext) simp 
10213  90 

91 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

92 
text {* code generator setup *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

93 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

94 
instance unit :: eq .. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

95 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

96 
lemma [code func]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

97 
"(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+ 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

98 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

99 
code_type unit 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

100 
(SML "unit") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

101 
(OCaml "unit") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

102 
(Haskell "()") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

103 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

104 
code_instance unit :: eq 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

105 
(Haskell ) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

106 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

107 
code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

108 
(Haskell infixl 4 "==") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

109 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

110 
code_const Unity 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

111 
(SML "()") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

112 
(OCaml "()") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

113 
(Haskell "()") 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

114 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

115 
code_reserved SML 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

116 
unit 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

117 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

118 
code_reserved OCaml 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

119 
unit 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

120 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

121 

11838  122 
subsection {* Pairs *} 
10213  123 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

124 
subsubsection {* Product type, basic operations and concrete syntax *} 
10213  125 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

126 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

127 
Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

128 
where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

129 
"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" 
10213  130 

131 
global 

132 

133 
typedef (Prod) 

22838  134 
('a, 'b) "*" (infixr "*" 20) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

135 
= "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" 
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

136 
proof 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

137 
fix a b show "Pair_Rep a b \<in> ?Prod" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

138 
by rule+ 
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

139 
qed 
10213  140 

12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
11966
diff
changeset

141 
syntax (xsymbols) 
15422
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

142 
"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
10213  143 
syntax (HTML output) 
15422
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

144 
"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
10213  145 

146 
consts 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

147 
Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

148 
fst :: "'a \<times> 'b \<Rightarrow> 'a" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

149 
snd :: "'a \<times> 'b \<Rightarrow> 'b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

150 
split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

151 
curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" 
10213  152 

11777  153 
local 
10213  154 

19535  155 
defs 
156 
Pair_def: "Pair a b == Abs_Prod (Pair_Rep a b)" 

157 
fst_def: "fst p == THE a. EX b. p = Pair a b" 

158 
snd_def: "snd p == THE b. EX a. p = Pair a b" 

24162
8dfd5dd65d82
split off theory Option for benefit of code generator
haftmann
parents:
23247
diff
changeset

159 
split_def: "split == (%c p. c (fst p) (snd p))" 
8dfd5dd65d82
split off theory Option for benefit of code generator
haftmann
parents:
23247
diff
changeset

160 
curry_def: "curry == (%c x y. c (Pair x y))" 
19535  161 

11777  162 
text {* 
163 
Patterns  extends predefined type @{typ pttrn} used in 

164 
abstractions. 

165 
*} 

10213  166 

167 
nonterminals 

168 
tuple_args patterns 

169 

170 
syntax 

171 
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") 

172 
"_tuple_arg" :: "'a => tuple_args" ("_") 

173 
"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") 

11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

174 
"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") 
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

175 
"" :: "pttrn => patterns" ("_") 
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

176 
"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") 
10213  177 

178 
translations 

179 
"(x, y)" == "Pair x y" 

180 
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 

181 
"%(x,y,zs).b" == "split(%x (y,zs).b)" 

182 
"%(x,y).b" == "split(%x y. b)" 

183 
"_abs (Pair x y) t" => "%(x,y).t" 

184 
(* The last rule accommodates tuples in `case C ... (x,y) ... => ...' 

185 
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) 

186 

14359  187 
(* reconstructs pattern from (nested) splits, avoiding etacontraction of body*) 
188 
(* works best with enclosing "let", if "let" does not avoid etacontraction *) 

189 
print_translation {* 

190 
let fun split_tr' [Abs (x,T,t as (Abs abs))] = 

191 
(* split (%x y. t) => %(x,y) t *) 

192 
let val (y,t') = atomic_abs_tr' abs; 

193 
val (x',t'') = atomic_abs_tr' (x,T,t'); 

194 

195 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end 

196 
 split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = 

197 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 

198 
let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; 

199 
val (x',t'') = atomic_abs_tr' (x,T,t'); 

200 
in Syntax.const "_abs"$ 

201 
(Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end 

202 
 split_tr' [Const ("split",_)$t] = 

203 
(* split (split (%x y z. t)) => %((x,y),z). t *) 

204 
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) 

205 
 split_tr' [Const ("_abs",_)$x_y$(Abs abs)] = 

206 
(* split (%pttrn z. t) => %(pttrn,z). t *) 

207 
let val (z,t) = atomic_abs_tr' abs; 

208 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end 

209 
 split_tr' _ = raise Match; 

210 
in [("split", split_tr')] 

211 
end 

212 
*} 

213 

15422
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

214 
(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

215 
typed_print_translation {* 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

216 
let 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

217 
fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

218 
 split_guess_names_tr' _ T [Abs (x,xT,t)] = 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

219 
(case (head_of t) of 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

220 
Const ("split",_) => raise Match 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

221 
 _ => let 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

222 
val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

223 
val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

224 
val (x',t'') = atomic_abs_tr' (x,xT,t'); 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

225 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

226 
 split_guess_names_tr' _ T [t] = 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

227 
(case (head_of t) of 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

228 
Const ("split",_) => raise Match 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

229 
 _ => let 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

230 
val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

231 
val (y,t') = 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

232 
atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

233 
val (x',t'') = atomic_abs_tr' ("x",xT,t'); 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

234 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

235 
 split_guess_names_tr' _ _ _ = raise Match; 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

236 
in [("split", split_guess_names_tr')] 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

237 
end 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

238 
*} 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

239 

10213  240 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

241 
text {* Towards a datatype declaration *} 
11838  242 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

243 
lemma surj_pair [simp]: "EX x y. p = (x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

244 
apply (unfold Pair_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

245 
apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

246 
apply (erule exE, erule exE, rule exI, rule exI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

247 
apply (rule Rep_Prod_inverse [symmetric, THEN trans]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

248 
apply (erule arg_cong) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

249 
done 
11838  250 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

251 
lemma PairE [cases type: *]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

252 
obtains x y where "p = (x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

253 
using surj_pair [of p] by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

254 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

255 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

256 
lemma prod_induct [induct type: *]: "(\<And>a b. P (a, b)) \<Longrightarrow> P x" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

257 
by (cases x) simp 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

258 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

259 
lemma ProdI: "Pair_Rep a b \<in> Prod" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

260 
unfolding Prod_def by rule+ 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

261 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

262 
lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

263 
unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast 
10213  264 

11838  265 
lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" 
266 
apply (rule inj_on_inverseI) 

267 
apply (erule Abs_Prod_inverse) 

268 
done 

269 

270 
lemma Pair_inject: 

18372  271 
assumes "(a, b) = (a', b')" 
272 
and "a = a' ==> b = b' ==> R" 

273 
shows R 

274 
apply (insert prems [unfolded Pair_def]) 

275 
apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) 

276 
apply (assumption  rule ProdI)+ 

277 
done 

10213  278 

11838  279 
lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')" 
280 
by (blast elim!: Pair_inject) 

281 

22886  282 
lemma fst_conv [simp, code]: "fst (a, b) = a" 
19535  283 
unfolding fst_def by blast 
11838  284 

22886  285 
lemma snd_conv [simp, code]: "snd (a, b) = b" 
19535  286 
unfolding snd_def by blast 
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

287 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

288 
rep_datatype prod 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

289 
inject Pair_eq 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

290 
induction prod_induct 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

291 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

292 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

293 
subsubsection {* Basic rules and proof tools *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

294 

11838  295 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
296 
by simp 

297 

298 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

299 
by simp 

300 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

301 
lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  302 
by (cases p) simp 
303 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

304 
lemmas surjective_pairing = pair_collapse [symmetric] 
11838  305 

306 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 

11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

307 
proof 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

308 
fix a b 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

309 
assume "!!x. PROP P x" 
19535  310 
then show "PROP P (a, b)" . 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

311 
next 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

312 
fix x 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

313 
assume "!!a b. PROP P (a, b)" 
19535  314 
from `PROP P (fst x, snd x)` show "PROP P x" by simp 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

315 
qed 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

316 

11838  317 
text {* 
318 
The rule @{thm [source] split_paired_all} does not work with the 

319 
Simplifier because it also affects premises in congrence rules, 

320 
where this can lead to premises of the form @{text "!!a b. ... = 

321 
?P(a, b)"} which cannot be solved by reflexivity. 

322 
*} 

323 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

324 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

325 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

326 
ML_setup {* 
11838  327 
(* replace parameters of product type by individual component parameters *) 
328 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

329 
local (* filtering with exists_paired_all is an essential optimization *) 

16121  330 
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = 
11838  331 
can HOLogic.dest_prodT T orelse exists_paired_all t 
332 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

333 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

334 
 exists_paired_all _ = false; 

335 
val ss = HOL_basic_ss 

26340  336 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
11838  337 
addsimprocs [unit_eq_proc]; 
338 
in 

339 
val split_all_tac = SUBGOAL (fn (t, i) => 

340 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

341 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

342 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

343 
fun split_all th = 

26340  344 
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; 
11838  345 
end; 
26340  346 
*} 
11838  347 

26340  348 
declaration {* fn _ => 
349 
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) 

16121  350 
*} 
11838  351 

352 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

353 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

354 
by fast 

355 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

356 
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

357 
by fast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

358 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

359 
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

360 
by (cases s, cases t) simp 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

361 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

362 
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

363 
by (simp add: Pair_fst_snd_eq) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

364 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

365 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

366 
subsubsection {* @{text split} and @{text curry} *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

367 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

368 
lemma split_conv [simp, code func]: "split f (a, b) = f a b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

369 
by (simp add: split_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

370 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

371 
lemma curry_conv [simp, code func]: "curry f a b = f (a, b)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

372 
by (simp add: curry_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

373 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

374 
lemmas split = split_conv  {* for backwards compatibility *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

375 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

376 
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

377 
by (rule split_conv [THEN iffD2]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

378 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

379 
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

380 
by (rule split_conv [THEN iffD1]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

381 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

382 
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

383 
by (simp add: curry_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

384 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

385 
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

386 
by (simp add: curry_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

387 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

388 
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

389 
by (simp add: curry_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

390 

14189  391 
lemma curry_split [simp]: "curry (split f) = f" 
392 
by (simp add: curry_def split_def) 

393 

394 
lemma split_curry [simp]: "split (curry f) = f" 

395 
by (simp add: curry_def split_def) 

396 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

397 
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

398 
by (simp add: split_def id_def) 
11838  399 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

400 
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

401 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

402 
by (rule ext) auto 
11838  403 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

404 
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

405 
by (cases x) simp 
11838  406 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

407 
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

408 
unfolding split_def .. 
11838  409 

410 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 

411 
 {* Can't be added to simpset: loops! *} 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

412 
by (simp add: split_eta) 
11838  413 

414 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

415 
by (simp add: split_def) 

416 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

417 
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" 
11838  418 
 {* Prevents simplification of @{term c}: much faster *} 
419 
by (erule arg_cong) 

420 

421 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

422 
by (simp add: split_eta) 

423 

424 
text {* 

425 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

426 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

427 
and using @{thm [source] cond_split_eta} directly would render some 

428 
existing proofs very inefficient; similarly for @{text 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

429 
split_beta}. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

430 
*} 
11838  431 

432 
ML_setup {* 

433 

434 
local 

18328  435 
val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"] 
11838  436 
fun Pair_pat k 0 (Bound m) = (m = k) 
437 
 Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso 

438 
m = k+i andalso Pair_pat k (i1) t 

439 
 Pair_pat _ _ _ = false; 

440 
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t 

441 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

442 
 no_args k i (Bound m) = m < k orelse m > k+i 

443 
 no_args _ _ _ = true; 

15531  444 
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE 
11838  445 
 split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t 
15531  446 
 split_pat tp i _ = NONE; 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

447 
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] 
13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

448 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) 
18328  449 
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); 
11838  450 

451 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t 

452 
 beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse 

453 
(beta_term_pat k i t andalso beta_term_pat k i u) 

454 
 beta_term_pat k i t = no_args k i t; 

455 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

456 
 eta_term_pat _ _ _ = false; 

457 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 

458 
 subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg 

459 
else (subst arg k i t $ subst arg k i u) 

460 
 subst arg k i t = t; 

20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

461 
fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) = 
11838  462 
(case split_pat beta_term_pat 1 t of 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

463 
SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f)) 
15531  464 
 NONE => NONE) 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

465 
 beta_proc _ _ = NONE; 
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

466 
fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) = 
11838  467 
(case split_pat eta_term_pat 1 t of 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

468 
SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) 
15531  469 
 NONE => NONE) 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19656
diff
changeset

470 
 eta_proc _ _ = NONE; 
11838  471 
in 
22577  472 
val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc); 
473 
val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc); 

11838  474 
end; 
475 

476 
Addsimprocs [split_beta_proc, split_eta_proc]; 

477 
*} 

478 

479 
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)" 

480 
by (subst surjective_pairing, rule split_conv) 

481 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

482 
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) > R(c x y))" 
11838  483 
 {* For use with @{text split} and the Simplifier. *} 
15481  484 
by (insert surj_pair [of p], clarify, simp) 
11838  485 

486 
text {* 

487 
@{thm [source] split_split} could be declared as @{text "[split]"} 

488 
done after the Splitter has been speeded up significantly; 

489 
precompute the constants involved and don't do anything unless the 

490 
current goal contains one of those constants. 

491 
*} 

492 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

493 
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 
14208  494 
by (subst split_split, simp) 
11838  495 

496 

497 
text {* 

498 
\medskip @{term split} used as a logical connective or set former. 

499 

500 
\medskip These rules are for use with @{text blast}; could instead 

501 
call @{text simp} using @{thm [source] split} as rewrite. *} 

502 

503 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

504 
apply (simp only: split_tupled_all) 

505 
apply (simp (no_asm_simp)) 

506 
done 

507 

508 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

509 
apply (simp only: split_tupled_all) 

510 
apply (simp (no_asm_simp)) 

511 
done 

512 

513 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

514 
by (induct p) (auto simp add: split_def) 

515 

516 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

517 
by (induct p) (auto simp add: split_def) 

518 

519 
lemma splitE2: 

520 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

521 
proof  

522 
assume q: "Q (split P z)" 

523 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

524 
show R 

525 
apply (rule r surjective_pairing)+ 

526 
apply (rule split_beta [THEN subst], rule q) 

527 
done 

528 
qed 

529 

530 
lemma splitD': "split R (a,b) c ==> R a b c" 

531 
by simp 

532 

533 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

534 
by simp 

535 

536 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  537 
by (simp only: split_tupled_all, simp) 
11838  538 

18372  539 
lemma mem_splitE: 
540 
assumes major: "z: split c p" 

541 
and cases: "!!x y. [ p = (x,y); z: c x y ] ==> Q" 

542 
shows Q 

543 
by (rule major [unfolded split_def] cases surjective_pairing)+ 

11838  544 

545 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

546 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

547 

26340  548 
ML {* 
11838  549 
local (* filtering with exists_p_split is an essential optimization *) 
16121  550 
fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true 
11838  551 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
552 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

553 
 exists_p_split _ = false; 

16121  554 
val ss = HOL_basic_ss addsimps [thm "split_conv"]; 
11838  555 
in 
556 
val split_conv_tac = SUBGOAL (fn (t, i) => 

557 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

558 
end; 

26340  559 
*} 
560 

11838  561 
(* This prevents applications of splitE for already splitted arguments leading 
562 
to quite timeconsuming computations (in particular for nested tuples) *) 

26340  563 
declaration {* fn _ => 
564 
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) 

16121  565 
*} 
11838  566 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

567 
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  568 
by (rule ext) fast 
11838  569 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24162
diff
changeset

570 
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  571 
by (rule ext) fast 
11838  572 

573 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

574 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

18372  575 
by (rule ext) blast 
11838  576 

14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

577 
(* Do NOT make this a simp rule as it 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

578 
a) only helps in special situations 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

579 
b) can lead to nontermination in the presence of split_def 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

580 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

581 
lemma split_comp_eq: 
20415  582 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
583 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  584 
by (rule ext) auto 
14101  585 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

586 
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

587 
apply (rule_tac x = "(a, b)" in image_eqI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

588 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

589 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

590 

11838  591 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
592 
by blast 

593 

594 
(* 

595 
the following would be slightly more general, 

596 
but cannot be used as rewrite rule: 

597 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

598 
### ?y = .x 

599 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  600 
by (rtac some_equality 1) 
601 
by ( Simp_tac 1) 

602 
by (split_all_tac 1) 

603 
by (Asm_full_simp_tac 1) 

11838  604 
qed "The_split_eq"; 
605 
*) 

606 

607 
text {* 

608 
Setup of internal @{text split_rule}. 

609 
*} 

610 

25511  611 
definition 
612 
internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" 

613 
where 

11032  614 
"internal_split == split" 
615 

616 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

617 
by (simp only: internal_split_def split_conv) 

618 

619 
hide const internal_split 

620 

11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

621 
use "Tools/split_rule.ML" 
11032  622 
setup SplitRule.setup 
10213  623 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

624 
lemmas prod_caseI = prod.cases [THEN iffD2, standard] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

625 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

626 
lemma prod_caseI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> prod_case c p" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

627 
by auto 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

628 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

629 
lemma prod_caseI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> prod_case c p x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

630 
by (auto simp: split_tupled_all) 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

631 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

632 
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

633 
by (induct p) auto 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

634 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

635 
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

636 
by (induct p) auto 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

637 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

638 
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

639 
by (simp add: expand_fun_eq) 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

640 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

641 
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

642 
declare prod_caseE' [elim!] prod_caseE [elim!] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

643 

24844
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

644 
lemma prod_case_split: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

645 
"prod_case = split" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

646 
by (auto simp add: expand_fun_eq) 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

647 

26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

648 
lemma prod_case_beta: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

649 
"prod_case f p = f (fst p) (snd p)" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

650 
unfolding prod_case_split split_beta .. 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

651 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

652 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

653 
subsection {* Further cases/induct rules for tuples *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

654 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

655 
lemma prod_cases3 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

656 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

657 
by (cases y, case_tac b) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

658 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

659 
lemma prod_induct3 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

660 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

661 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

662 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

663 
lemma prod_cases4 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

664 
obtains (fields) a b c d where "y = (a, b, c, d)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

665 
by (cases y, case_tac c) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

666 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

667 
lemma prod_induct4 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

668 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

669 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

670 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

671 
lemma prod_cases5 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

672 
obtains (fields) a b c d e where "y = (a, b, c, d, e)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

673 
by (cases y, case_tac d) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

674 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

675 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

676 
"(!!a b c d e. P (a, b, c, d, e)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

677 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

678 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

679 
lemma prod_cases6 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

680 
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

681 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

682 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

683 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

684 
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

685 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

686 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

687 
lemma prod_cases7 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

688 
obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

689 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

690 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

691 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

692 
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

693 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

694 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

695 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

696 
subsubsection {* Derived operations *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

697 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

698 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

699 
The compositionuncurry combinator. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

700 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

701 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

702 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

703 
mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o>" 60) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

704 
where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

705 
"f o> g = (\<lambda>x. split g (f x))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

706 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

707 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

708 
mbind (infixl "\<circ>\<rightarrow>" 60) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

709 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

710 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

711 
mbind (infixl "\<circ>\<rightarrow>" 60) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

712 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

713 
lemma mbind_apply: "(f \<circ>\<rightarrow> g) x = split g (f x)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

714 
by (simp add: mbind_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

715 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

716 
lemma Pair_mbind: "Pair x \<circ>\<rightarrow> f = f x" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

717 
by (simp add: expand_fun_eq mbind_apply) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

718 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

719 
lemma mbind_Pair: "x \<circ>\<rightarrow> Pair = x" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

720 
by (simp add: expand_fun_eq mbind_apply) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

721 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

722 
lemma mbind_mbind: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

723 
by (simp add: expand_fun_eq split_twice mbind_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

724 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

725 
lemma mbind_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

726 
by (simp add: expand_fun_eq mbind_apply fcomp_def split_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

727 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

728 
lemma fcomp_mbind: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

729 
by (simp add: expand_fun_eq mbind_apply fcomp_apply) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

730 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

731 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

732 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

733 
@{term prod_fun}  action of the product functor upon 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

734 
functions. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

735 
*} 
21195  736 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

737 
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

738 
[code func del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

739 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

740 
lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

741 
by (simp add: prod_fun_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

742 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

743 
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

744 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

745 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

746 
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

747 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

748 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

749 
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

750 
apply (rule image_eqI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

751 
apply (rule prod_fun [symmetric], assumption) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

752 
done 
21195  753 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

754 
lemma prod_fun_imageE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

755 
assumes major: "c: (prod_fun f g)`r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

756 
and cases: "!!x y. [ c=(f(x),g(y)); (x,y):r ] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

757 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

758 
apply (rule major [THEN imageE]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

759 
apply (rule_tac p = x in PairE) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

760 
apply (rule cases) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

761 
apply (blast intro: prod_fun) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

762 
apply blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

763 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

764 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

765 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

766 
apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

767 
where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

768 
[code func del]: "apfst f = prod_fun f id" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

769 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

770 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

771 
apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

772 
where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

773 
[code func del]: "apsnd f = prod_fun id f" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

774 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

775 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

776 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

777 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

778 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

779 
lemma upd_snd_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

780 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

781 
by (simp add: apsnd_def) 
21195  782 

783 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

784 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

785 
Disjoint union of a family of sets  Sigma. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

786 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

787 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

788 
definition Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

789 
Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

790 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

791 
abbreviation 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

792 
Times :: "['a set, 'b set] => ('a * 'b) set" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

793 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

794 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

795 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

796 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

797 
Times (infixr "\<times>" 80) 
15394  798 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

799 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

800 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

801 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

802 
syntax 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

803 
"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

804 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

805 
translations 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

806 
"SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

807 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

808 
lemma SigmaI [intro!]: "[ a:A; b:B(a) ] ==> (a,b) : Sigma A B" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

809 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

810 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

811 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

812 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

813 
!!x y.[ x:A; y:B(x); c=(x,y) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

814 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

815 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

816 
by (unfold Sigma_def) blast 
20588  817 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

818 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

819 
Elimination of @{term "(a, b) : A \<times> B"}  introduces no 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

820 
eigenvariables. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

821 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

822 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

823 
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

824 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

825 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

826 
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

827 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

828 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

829 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

830 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

831 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

832 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

833 
by blast 
20588  834 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

835 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

836 
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

837 
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

838 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

839 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

840 
lemma Sigma_mono: "[ A <= C; !!x. x:A ==> B x <= D x ] ==> Sigma A B <= Sigma C D" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

841 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

842 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

843 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

844 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

845 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

846 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

847 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

848 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

849 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

850 
by auto 
21908  851 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

852 
lemma Compl_Times_UNIV1 [simp]: " (UNIV <*> A) = UNIV <*> (A)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

853 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

854 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

855 
lemma Compl_Times_UNIV2 [simp]: " (A <*> UNIV) = (A) <*> UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

856 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

857 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

858 
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

859 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

860 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

861 
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

862 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

863 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

864 
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

865 
by (blast elim: equalityE) 
20588  866 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

867 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

868 
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

869 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

870 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

871 
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

872 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

873 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

874 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

875 
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

876 
 {* Suggested by Pierre Chartier *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

877 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

878 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

879 
lemma split_paired_Ball_Sigma [simp,noatp]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

880 
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

881 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

882 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

883 
lemma split_paired_Bex_Sigma [simp,noatp]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

884 
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

885 
by blast 
21908  886 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

887 
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

888 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

889 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

890 
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

891 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

892 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

893 
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

894 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

895 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

896 
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

897 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

898 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

899 
lemma Sigma_Diff_distrib1: "(SIGMA i:I  J. C(i)) = (SIGMA i:I. C(i))  (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

900 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

901 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

902 
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i)  B(i)) = (SIGMA i:I. A(i))  (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

903 
by blast 
21908  904 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

905 
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

906 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

907 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

908 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

909 
Nondependent versions are needed to avoid the need for higherorder 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

910 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

911 
*} 
21908  912 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

913 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

914 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

915 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

916 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

917 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

918 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

919 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

920 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

921 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

922 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

923 
subsubsection {* Code generator setup *} 
21908  924 

20588  925 
instance * :: (eq, eq) eq .. 
926 

927 
lemma [code func]: 

21454  928 
"(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto 
20588  929 

24844
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

930 
lemma split_case_cert: 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

931 
assumes "CASE \<equiv> split f" 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

932 
shows "CASE (a, b) \<equiv> f a b" 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

933 
using assms by simp 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

934 

98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

935 
setup {* 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

936 
Code.add_case @{thm split_case_cert} 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

937 
*} 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

938 

21908  939 
code_type * 
940 
(SML infix 2 "*") 

941 
(OCaml infix 2 "*") 

942 
(Haskell "!((_),/ (_))") 

943 

20588  944 
code_instance * :: eq 
945 
(Haskell ) 

946 

21908  947 
code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 
20588  948 
(Haskell infixl 4 "==") 
949 

21908  950 
code_const Pair 
951 
(SML "!((_),/ (_))") 

952 
(OCaml "!((_),/ (_))") 

953 
(Haskell "!((_),/ (_))") 

20588  954 

22389  955 
code_const fst and snd 
956 
(Haskell "fst" and "snd") 

957 

15394  958 
types_code 
959 
"*" ("(_ */ _)") 

16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

960 
attach (term_of) {* 
25885  961 
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; 
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

962 
*} 
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

963 
attach (test) {* 
25885  964 
fun gen_id_42 aG aT bG bT i = 
965 
let 

966 
val (x, t) = aG i; 

967 
val (y, u) = bG i 

968 
in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; 

16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

969 
*} 
15394  970 

18706
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

971 
consts_code 
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

972 
"Pair" ("(_,/ _)") 
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

973 

21908  974 
setup {* 
975 

976 
let 

18013  977 

19039  978 
fun strip_abs_split 0 t = ([], t) 
979 
 strip_abs_split i (Abs (s, T, t)) = 

18013  980 
let 
981 
val s' = Codegen.new_name t s; 

982 
val v = Free (s', T) 

19039  983 
in apfst (cons v) (strip_abs_split (i1) (subst_bound (v, t))) end 
984 
 strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of 

15394  985 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 
986 
 _ => ([], u)) 

19039  987 
 strip_abs_split i t = ([], t); 
18013  988 

16634  989 
fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of 
990 
(t1 as Const ("Let", _), t2 :: t3 :: ts) => 

15394  991 
let 
992 
fun dest_let (l as Const ("Let", _) $ t $ u) = 

19039  993 
(case strip_abs_split 1 u of 
15394  994 
([p], u') => apfst (cons (p, t)) (dest_let u') 
995 
 _ => ([], l)) 

996 
 dest_let t = ([], t); 

997 
fun mk_code (gr, (l, r)) = 

998 
let 

16634  999 
val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l); 
1000 
val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r); 

15394  1001 
in (gr2, (pl, pr)) end 
16634  1002 
in case dest_let (t1 $ t2 $ t3) of 
15531  1003 
([], _) => NONE 
15394  1004 
 (ps, u) => 
1005 
let 

1006 
val (gr1, qs) = foldl_map mk_code (gr, ps); 

16634  1007 
val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); 
1008 
val (gr3, pargs) = foldl_map 

17021
1c361a3de73d
Fixed bug in code generator for let and split leading to illformed code.
berghofe
parents:
17002
diff
changeset

1009 
(Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) 
15394  1010 
in 
16634  1011 
SOME (gr3, Codegen.mk_app brack 
1012 
(Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat 

1013 
(separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) => 

1014 
[Pretty.block [Pretty.str "val ", pl, Pretty.str " =", 

1015 
Pretty.brk 1, pr]]) qs))), 

1016 
Pretty.brk 1, Pretty.str "in ", pu, 

1017 
Pretty.brk 1, Pretty.str "end"])) pargs) 

15394  1018 
end 
1019 
end 

16634  1020 
 _ => NONE); 
15394  1021 

16634  1022 
fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of 
1023 
(t1 as Const ("split", _), t2 :: ts) => 

19039  1024 
(case strip_abs_split 1 (t1 $ t2) of 
16634  1025 
([p], u) => 
1026 
let 

1027 
val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p); 

1028 
val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); 

1029 
val (gr3, pargs) = foldl_map 

17021
1c361a3de73d
Fixed bug in code generator for let and split leading to illformed code.
berghofe
parents:
17002
diff
changeset

1030 
(Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) 
16634  1031 
in 
1032 
SOME (gr2, Codegen.mk_app brack 

1033 
(Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>", 

1034 
Pretty.brk 1, pu, Pretty.str ")"]) pargs) 

1035 
end 

1036 
 _ => NONE) 

1037 
 _ => NONE); 

15394  1038 

21908  1039 
in 
1040 

20105  1041 
Codegen.add_codegen "let_codegen" let_codegen 
1042 
#> Codegen.add_codegen "split_codegen" split_codegen 

15394  1043 

21908  1044 
end 
15394  1045 
*} 
1046 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1047 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1048 
subsection {* Legacy bindings *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1049 

21908  1050 
ML {* 
15404  1051 
val Collect_split = thm "Collect_split"; 
1052 
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; 

1053 
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; 

1054 
val PairE = thm "PairE"; 

1055 
val Pair_Rep_inject = thm "Pair_Rep_inject"; 

1056 
val Pair_def = thm "Pair_def"; 

1057 
val Pair_eq = thm "Pair_eq"; 

1058 
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; 

1059 
val ProdI = thm "ProdI"; 

1060 
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; 

1061 
val SigmaD1 = thm "SigmaD1"; 

1062 
val SigmaD2 = thm "SigmaD2"; 

1063 
val SigmaE = thm "SigmaE"; 

1064 
val SigmaE2 = thm "SigmaE2"; 

1065 
val SigmaI = thm "SigmaI"; 

1066 
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; 

1067 
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; 

1068 
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; 

1069 
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; 

1070 
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; 

1071 
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; 

1072 
val Sigma_Union = thm "Sigma_Union"; 

1073 
val Sigma_def = thm "Sigma_def"; 

1074 
val Sigma_empty1 = thm "Sigma_empty1"; 

1075 
val Sigma_empty2 = thm "Sigma_empty2"; 

1076 
val Sigma_mono = thm "Sigma_mono"; 

1077 
val The_split = thm "The_split"; 

1078 
val The_split_eq = thm "The_split_eq"; 

1079 
val The_split_eq = thm "The_split_eq"; 

1080 
val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; 

1081 
val Times_Int_distrib1 = thm "Times_Int_distrib1"; 

1082 
val Times_Un_distrib1 = thm "Times_Un_distrib1"; 

1083 
val Times_eq_cancel2 = thm "Times_eq_cancel2"; 

1084 
val Times_subset_cancel2 = thm "Times_subset_cancel2"; 

1085 
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; 

1086 
val UN_Times_distrib = thm "UN_Times_distrib"; 

1087 
val Unity_def = thm "Unity_def"; 

1088 
val cond_split_eta = thm "cond_split_eta"; 

1089 
val fst_conv = thm "fst_conv"; 

1090 
val fst_def = thm "fst_def"; 

1091 
val fst_eqD = thm "fst_eqD"; 

1092 
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; 

1093 
val mem_Sigma_iff = thm "mem_Sigma_iff"; 

1094 
val mem_splitE = thm "mem_splitE"; 

1095 
val mem_splitI = thm "mem_splitI"; 

1096 
val mem_splitI2 = thm "mem_splitI2"; 

1097 
val prod_eqI = thm "prod_eqI"; 

1098 
val prod_fun = thm "prod_fun"; 

1099 
val prod_fun_compose = thm "prod_fun_compose"; 

1100 
val prod_fun_def = thm "prod_fun_def"; 

1101 
val prod_fun_ident = thm "prod_fun_ident"; 

1102 
val prod_fun_imageE = thm "prod_fun_imageE"; 

1103 
val prod_fun_imageI = thm "prod_fun_imageI"; 

1104 
val prod_induct = thm "prod_induct"; 

1105 
val snd_conv = thm "snd_conv"; 

1106 
val snd_def = thm "snd_def"; 

1107 
val snd_eqD = thm "snd_eqD"; 

1108 
val split = thm "split"; 

1109 
val splitD = thm "splitD"; 

1110 
val splitD' = thm "splitD'"; 

1111 
val splitE = thm "splitE"; 

1112 
val splitE' = thm "splitE'"; 

1113 
val splitE2 = thm "splitE2"; 

1114 
val splitI = thm "splitI"; 

1115 
val splitI2 = thm "splitI2"; 

1116 
val splitI2' = thm "splitI2'"; 

1117 
val split_beta = thm "split_beta"; 

1118 
val split_conv = thm "split_conv"; 

1119 
val split_def = thm "split_def"; 

1120 
val split_eta = thm "split_eta"; 

1121 
val split_eta_SetCompr = thm "split_eta_SetCompr"; 

1122 
val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; 

1123 
val split_paired_All = thm "split_paired_All"; 

1124 
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; 

1125 
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; 

1126 
val split_paired_Ex = thm "split_paired_Ex"; 

1127 
val split_paired_The = thm "split_paired_The"; 

1128 
val split_paired_all = thm "split_paired_all"; 

1129 
val split_part = thm "split_part"; 

1130 
val split_split = thm "split_split"; 

1131 
val split_split_asm = thm "split_split_asm"; 

1132 
val split_tupled_all = thms "split_tupled_all"; 

1133 
val split_weak_cong = thm "split_weak_cong"; 

1134 
val surj_pair = thm "surj_pair"; 

1135 
val surjective_pairing = thm "surjective_pairing"; 

1136 
val unit_abs_eta_conv = thm "unit_abs_eta_conv"; 

1137 
val unit_all_eq1 = thm "unit_all_eq1"; 

1138 
val unit_all_eq2 = thm "unit_all_eq2"; 

1139 
val unit_eq = thm "unit_eq"; 

1140 
*} 

1141 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1142 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1143 
subsection {* Further inductive packages *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1144 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1145 
use "Tools/inductive_realizer.ML" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1146 
setup InductiveRealizer.setup 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1147 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1148 
use "Tools/inductive_set_package.ML" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1149 
setup InductiveSetPackage.setup 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1150 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1151 
use "Tools/datatype_realizer.ML" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1152 
setup DatatypeRealizer.setup 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1153 

10213  1154 
end 