author | blanchet |
Tue, 31 Aug 2010 21:01:47 +0200 | |
changeset 38944 | 827c98e8ba8b |
parent 38857 | 97775f3e8722 |
child 39198 | f967a16dfcdd |
permissions | -rw-r--r-- |
10213 | 1 |
(* Title: HOL/Product_Type.thy |
2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
3 |
Copyright 1992 University of Cambridge |
|
11777 | 4 |
*) |
10213 | 5 |
|
11838 | 6 |
header {* Cartesian products *} |
10213 | 7 |
|
15131 | 8 |
theory Product_Type |
33959
2afc55e8ed27
bootstrap datatype_rep_proofs in Datatype.thy (avoids unchecked dynamic name references)
haftmann
parents:
33638
diff
changeset
|
9 |
imports Typedef Inductive Fun |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
10 |
uses |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
11 |
("Tools/split_rule.ML") |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
12 |
("Tools/inductive_codegen.ML") |
31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
13 |
("Tools/inductive_set.ML") |
15131 | 14 |
begin |
11838 | 15 |
|
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
16 |
subsection {* @{typ bool} is a datatype *} |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
17 |
|
27104
791607529f6d
rep_datatype command now takes list of constructors as input arguments
haftmann
parents:
26975
diff
changeset
|
18 |
rep_datatype True False by (auto intro: bool_induct) |
24699
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 |
declare case_split [cases type: bool] |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
21 |
-- "prefer plain propositional version" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
22 |
|
28346
b8390cd56b8f
discontinued special treatment of op = vs. eq_class.eq
haftmann
parents:
28262
diff
changeset
|
23 |
lemma |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
24 |
shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
25 |
and [code]: "HOL.equal True P \<longleftrightarrow> P" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
26 |
and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
27 |
and [code]: "HOL.equal P True \<longleftrightarrow> P" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
28 |
and [code nbe]: "HOL.equal P P \<longleftrightarrow> True" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
29 |
by (simp_all add: equal) |
25534
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset
|
30 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
31 |
code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" |
25534
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset
|
32 |
(Haskell infixl 4 "==") |
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset
|
33 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
34 |
code_instance bool :: equal |
25534
d0b74fdd6067
simplified infrastructure for code generator operational equality
haftmann
parents:
25511
diff
changeset
|
35 |
(Haskell -) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
36 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
37 |
|
37166 | 38 |
subsection {* The @{text unit} type *} |
11838 | 39 |
|
40 |
typedef unit = "{True}" |
|
41 |
proof |
|
20588 | 42 |
show "True : ?unit" .. |
11838 | 43 |
qed |
44 |
||
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
45 |
definition |
11838 | 46 |
Unity :: unit ("'(')") |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
47 |
where |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
48 |
"() = Abs_unit True" |
11838 | 49 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
50 |
lemma unit_eq [no_atp]: "u = ()" |
11838 | 51 |
by (induct u) (simp add: unit_def Unity_def) |
52 |
||
53 |
text {* |
|
54 |
Simplification procedure for @{thm [source] unit_eq}. Cannot use |
|
55 |
this rule directly --- it loops! |
|
56 |
*} |
|
57 |
||
26480 | 58 |
ML {* |
13462 | 59 |
val unit_eq_proc = |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
60 |
let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in |
38715
6513ea67d95d
renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents:
37808
diff
changeset
|
61 |
Simplifier.simproc_global @{theory} "unit_eq" ["x::unit"] |
15531 | 62 |
(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) |
13462 | 63 |
end; |
11838 | 64 |
|
65 |
Addsimprocs [unit_eq_proc]; |
|
66 |
*} |
|
67 |
||
27104
791607529f6d
rep_datatype command now takes list of constructors as input arguments
haftmann
parents:
26975
diff
changeset
|
68 |
rep_datatype "()" by simp |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
69 |
|
11838 | 70 |
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" |
71 |
by simp |
|
72 |
||
73 |
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" |
|
74 |
by (rule triv_forall_equality) |
|
75 |
||
76 |
text {* |
|
77 |
This rewrite counters the effect of @{text unit_eq_proc} on @{term |
|
78 |
[source] "%u::unit. f u"}, replacing it by @{term [source] |
|
79 |
f} rather than by @{term [source] "%u. f ()"}. |
|
80 |
*} |
|
81 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
82 |
lemma unit_abs_eta_conv [simp,no_atp]: "(%u::unit. f ()) = f" |
11838 | 83 |
by (rule ext) simp |
10213 | 84 |
|
30924 | 85 |
instantiation unit :: default |
86 |
begin |
|
87 |
||
88 |
definition "default = ()" |
|
89 |
||
90 |
instance .. |
|
91 |
||
92 |
end |
|
10213 | 93 |
|
28562 | 94 |
lemma [code]: |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
95 |
"HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
96 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
97 |
code_type unit |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
98 |
(SML "unit") |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
99 |
(OCaml "unit") |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
100 |
(Haskell "()") |
34886 | 101 |
(Scala "Unit") |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
102 |
|
37166 | 103 |
code_const Unity |
104 |
(SML "()") |
|
105 |
(OCaml "()") |
|
106 |
(Haskell "()") |
|
107 |
(Scala "()") |
|
108 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
109 |
code_instance unit :: equal |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
110 |
(Haskell -) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
111 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
112 |
code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
113 |
(Haskell infixl 4 "==") |
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 |
|
34886 | 121 |
code_reserved Scala |
122 |
Unit |
|
123 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
124 |
|
37166 | 125 |
subsection {* The product type *} |
10213 | 126 |
|
37166 | 127 |
subsubsection {* Type definition *} |
128 |
||
129 |
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
130 |
"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" |
10213 | 131 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
132 |
typedef ('a, 'b) prod (infixr "*" 20) |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
133 |
= "{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
|
134 |
proof |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
135 |
fix a b show "Pair_Rep a b \<in> ?prod" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
136 |
by rule+ |
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
137 |
qed |
10213 | 138 |
|
35427 | 139 |
type_notation (xsymbols) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
140 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
35427 | 141 |
type_notation (HTML output) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
142 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
10213 | 143 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
144 |
definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
145 |
"Pair a b = Abs_prod (Pair_Rep a b)" |
37166 | 146 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
147 |
rep_datatype Pair proof - |
37166 | 148 |
fix P :: "'a \<times> 'b \<Rightarrow> bool" and p |
149 |
assume "\<And>a b. P (Pair a b)" |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
150 |
then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
37166 | 151 |
next |
152 |
fix a c :: 'a and b d :: 'b |
|
153 |
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" |
|
154 |
by (auto simp add: Pair_Rep_def expand_fun_eq) |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
155 |
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
156 |
by (auto simp add: prod_def) |
37166 | 157 |
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
158 |
by (simp add: Pair_def Abs_prod_inject) |
37166 | 159 |
qed |
160 |
||
37704
c6161bee8486
adapt Nitpick to "prod_case" and "*" -> "sum" renaming;
blanchet
parents:
37678
diff
changeset
|
161 |
declare prod.simps(2) [nitpick_simp del] |
c6161bee8486
adapt Nitpick to "prod_case" and "*" -> "sum" renaming;
blanchet
parents:
37678
diff
changeset
|
162 |
|
37411
c88c44156083
removed simplifier congruence rule of "prod_case"
haftmann
parents:
37389
diff
changeset
|
163 |
declare weak_case_cong [cong del] |
c88c44156083
removed simplifier congruence rule of "prod_case"
haftmann
parents:
37389
diff
changeset
|
164 |
|
37166 | 165 |
|
166 |
subsubsection {* Tuple syntax *} |
|
167 |
||
37591 | 168 |
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
169 |
"split \<equiv> prod_case" |
|
19535 | 170 |
|
11777 | 171 |
text {* |
172 |
Patterns -- extends pre-defined type @{typ pttrn} used in |
|
173 |
abstractions. |
|
174 |
*} |
|
10213 | 175 |
|
176 |
nonterminals |
|
177 |
tuple_args patterns |
|
178 |
||
179 |
syntax |
|
180 |
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") |
|
181 |
"_tuple_arg" :: "'a => tuple_args" ("_") |
|
182 |
"_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
|
183 |
"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") |
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
184 |
"" :: "pttrn => patterns" ("_") |
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
185 |
"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") |
10213 | 186 |
|
187 |
translations |
|
35115 | 188 |
"(x, y)" == "CONST Pair x y" |
10213 | 189 |
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" |
37591 | 190 |
"%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)" |
191 |
"%(x, y). b" == "CONST prod_case (%x y. b)" |
|
35115 | 192 |
"_abs (CONST Pair x y) t" => "%(x, y). t" |
37166 | 193 |
-- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...' |
194 |
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *} |
|
10213 | 195 |
|
35115 | 196 |
(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body; |
197 |
works best with enclosing "let", if "let" does not avoid eta-contraction*) |
|
14359 | 198 |
print_translation {* |
35115 | 199 |
let |
200 |
fun split_tr' [Abs (x, T, t as (Abs abs))] = |
|
201 |
(* split (%x y. t) => %(x,y) t *) |
|
202 |
let |
|
203 |
val (y, t') = atomic_abs_tr' abs; |
|
204 |
val (x', t'') = atomic_abs_tr' (x, T, t'); |
|
205 |
in |
|
206 |
Syntax.const @{syntax_const "_abs"} $ |
|
207 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
208 |
end |
|
37591 | 209 |
| split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] = |
35115 | 210 |
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) |
211 |
let |
|
212 |
val Const (@{syntax_const "_abs"}, _) $ |
|
213 |
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; |
|
214 |
val (x', t'') = atomic_abs_tr' (x, T, t'); |
|
215 |
in |
|
216 |
Syntax.const @{syntax_const "_abs"} $ |
|
217 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ |
|
218 |
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' |
|
219 |
end |
|
37591 | 220 |
| split_tr' [Const (@{const_syntax prod_case}, _) $ t] = |
35115 | 221 |
(* split (split (%x y z. t)) => %((x, y), z). t *) |
222 |
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) |
|
223 |
| split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = |
|
224 |
(* split (%pttrn z. t) => %(pttrn,z). t *) |
|
225 |
let val (z, t) = atomic_abs_tr' abs in |
|
226 |
Syntax.const @{syntax_const "_abs"} $ |
|
227 |
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t |
|
228 |
end |
|
229 |
| split_tr' _ = raise Match; |
|
37591 | 230 |
in [(@{const_syntax prod_case}, split_tr')] end |
14359 | 231 |
*} |
232 |
||
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
233 |
(* 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
|
234 |
typed_print_translation {* |
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
235 |
let |
35115 | 236 |
fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match |
237 |
| split_guess_names_tr' _ T [Abs (x, xT, t)] = |
|
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
238 |
(case (head_of t) of |
37591 | 239 |
Const (@{const_syntax prod_case}, _) => raise Match |
35115 | 240 |
| _ => |
241 |
let |
|
242 |
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
243 |
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); |
|
244 |
val (x', t'') = atomic_abs_tr' (x, xT, t'); |
|
245 |
in |
|
246 |
Syntax.const @{syntax_const "_abs"} $ |
|
247 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
248 |
end) |
|
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
249 |
| split_guess_names_tr' _ T [t] = |
35115 | 250 |
(case head_of t of |
37591 | 251 |
Const (@{const_syntax prod_case}, _) => raise Match |
35115 | 252 |
| _ => |
253 |
let |
|
254 |
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
255 |
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); |
|
256 |
val (x', t'') = atomic_abs_tr' ("x", xT, t'); |
|
257 |
in |
|
258 |
Syntax.const @{syntax_const "_abs"} $ |
|
259 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
260 |
end) |
|
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
261 |
| split_guess_names_tr' _ _ _ = raise Match; |
37591 | 262 |
in [(@{const_syntax prod_case}, split_guess_names_tr')] end |
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
263 |
*} |
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
264 |
|
10213 | 265 |
|
37166 | 266 |
subsubsection {* Code generator setup *} |
267 |
||
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
268 |
code_type prod |
37166 | 269 |
(SML infix 2 "*") |
270 |
(OCaml infix 2 "*") |
|
271 |
(Haskell "!((_),/ (_))") |
|
272 |
(Scala "((_),/ (_))") |
|
273 |
||
274 |
code_const Pair |
|
275 |
(SML "!((_),/ (_))") |
|
276 |
(OCaml "!((_),/ (_))") |
|
277 |
(Haskell "!((_),/ (_))") |
|
278 |
(Scala "!((_),/ (_))") |
|
279 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
280 |
code_instance prod :: equal |
37166 | 281 |
(Haskell -) |
282 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
283 |
code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" |
37166 | 284 |
(Haskell infixl 4 "==") |
285 |
||
286 |
types_code |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
287 |
"prod" ("(_ */ _)") |
37166 | 288 |
attach (term_of) {* |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
289 |
fun term_of_prod aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; |
37166 | 290 |
*} |
291 |
attach (test) {* |
|
37808
e604e5f9bb6a
correcting function name of generator for products of traditional code generator (introduced in 0040bafffdef)
bulwahn
parents:
37765
diff
changeset
|
292 |
fun gen_prod aG aT bG bT i = |
37166 | 293 |
let |
294 |
val (x, t) = aG i; |
|
295 |
val (y, u) = bG i |
|
296 |
in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; |
|
297 |
*} |
|
298 |
||
299 |
consts_code |
|
300 |
"Pair" ("(_,/ _)") |
|
301 |
||
302 |
setup {* |
|
303 |
let |
|
304 |
||
305 |
fun strip_abs_split 0 t = ([], t) |
|
306 |
| strip_abs_split i (Abs (s, T, t)) = |
|
307 |
let |
|
308 |
val s' = Codegen.new_name t s; |
|
309 |
val v = Free (s', T) |
|
310 |
in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end |
|
37591 | 311 |
| strip_abs_split i (u as Const (@{const_name prod_case}, _) $ t) = |
37166 | 312 |
(case strip_abs_split (i+1) t of |
313 |
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) |
|
314 |
| _ => ([], u)) |
|
315 |
| strip_abs_split i t = |
|
316 |
strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0)); |
|
317 |
||
318 |
fun let_codegen thy defs dep thyname brack t gr = |
|
319 |
(case strip_comb t of |
|
320 |
(t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) => |
|
321 |
let |
|
322 |
fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) = |
|
323 |
(case strip_abs_split 1 u of |
|
324 |
([p], u') => apfst (cons (p, t)) (dest_let u') |
|
325 |
| _ => ([], l)) |
|
326 |
| dest_let t = ([], t); |
|
327 |
fun mk_code (l, r) gr = |
|
328 |
let |
|
329 |
val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr; |
|
330 |
val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1; |
|
331 |
in ((pl, pr), gr2) end |
|
332 |
in case dest_let (t1 $ t2 $ t3) of |
|
333 |
([], _) => NONE |
|
334 |
| (ps, u) => |
|
335 |
let |
|
336 |
val (qs, gr1) = fold_map mk_code ps gr; |
|
337 |
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; |
|
338 |
val (pargs, gr3) = fold_map |
|
339 |
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 |
|
340 |
in |
|
341 |
SOME (Codegen.mk_app brack |
|
342 |
(Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat |
|
343 |
(separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) => |
|
344 |
[Pretty.block [Codegen.str "val ", pl, Codegen.str " =", |
|
345 |
Pretty.brk 1, pr]]) qs))), |
|
346 |
Pretty.brk 1, Codegen.str "in ", pu, |
|
347 |
Pretty.brk 1, Codegen.str "end"])) pargs, gr3) |
|
348 |
end |
|
349 |
end |
|
350 |
| _ => NONE); |
|
351 |
||
352 |
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of |
|
37591 | 353 |
(t1 as Const (@{const_name prod_case}, _), t2 :: ts) => |
37166 | 354 |
let |
355 |
val ([p], u) = strip_abs_split 1 (t1 $ t2); |
|
356 |
val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr; |
|
357 |
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; |
|
358 |
val (pargs, gr3) = fold_map |
|
359 |
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 |
|
360 |
in |
|
361 |
SOME (Codegen.mk_app brack |
|
362 |
(Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>", |
|
363 |
Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2) |
|
364 |
end |
|
365 |
| _ => NONE); |
|
366 |
||
367 |
in |
|
368 |
||
369 |
Codegen.add_codegen "let_codegen" let_codegen |
|
370 |
#> Codegen.add_codegen "split_codegen" split_codegen |
|
371 |
||
372 |
end |
|
373 |
*} |
|
374 |
||
375 |
||
376 |
subsubsection {* Fundamental operations and properties *} |
|
11838 | 377 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
378 |
lemma surj_pair [simp]: "EX x y. p = (x, y)" |
37166 | 379 |
by (cases p) simp |
10213 | 380 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
381 |
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
382 |
"fst p = (case p of (a, b) \<Rightarrow> a)" |
11838 | 383 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
384 |
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
385 |
"snd p = (case p of (a, b) \<Rightarrow> b)" |
11838 | 386 |
|
22886 | 387 |
lemma fst_conv [simp, code]: "fst (a, b) = a" |
37166 | 388 |
unfolding fst_def by simp |
11838 | 389 |
|
22886 | 390 |
lemma snd_conv [simp, code]: "snd (a, b) = b" |
37166 | 391 |
unfolding snd_def by simp |
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
392 |
|
37166 | 393 |
code_const fst and snd |
394 |
(Haskell "fst" and "snd") |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
395 |
|
37704
c6161bee8486
adapt Nitpick to "prod_case" and "*" -> "sum" renaming;
blanchet
parents:
37678
diff
changeset
|
396 |
lemma prod_case_unfold [nitpick_def]: "prod_case = (%c p. c (fst p) (snd p))" |
37166 | 397 |
by (simp add: expand_fun_eq split: prod.split) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
398 |
|
11838 | 399 |
lemma fst_eqD: "fst (x, y) = a ==> x = a" |
400 |
by simp |
|
401 |
||
402 |
lemma snd_eqD: "snd (x, y) = a ==> y = a" |
|
403 |
by simp |
|
404 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
405 |
lemma pair_collapse [simp]: "(fst p, snd p) = p" |
11838 | 406 |
by (cases p) simp |
407 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
408 |
lemmas surjective_pairing = pair_collapse [symmetric] |
11838 | 409 |
|
37166 | 410 |
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" |
411 |
by (cases s, cases t) simp |
|
412 |
||
413 |
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" |
|
414 |
by (simp add: Pair_fst_snd_eq) |
|
415 |
||
416 |
lemma split_conv [simp, code]: "split f (a, b) = f a b" |
|
37591 | 417 |
by (fact prod.cases) |
37166 | 418 |
|
419 |
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" |
|
420 |
by (rule split_conv [THEN iffD2]) |
|
421 |
||
422 |
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" |
|
423 |
by (rule split_conv [THEN iffD1]) |
|
424 |
||
425 |
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" |
|
37591 | 426 |
by (simp add: expand_fun_eq split: prod.split) |
37166 | 427 |
|
428 |
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" |
|
429 |
-- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} |
|
37591 | 430 |
by (simp add: expand_fun_eq split: prod.split) |
37166 | 431 |
|
432 |
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" |
|
433 |
by (cases x) simp |
|
434 |
||
435 |
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" |
|
436 |
by (cases p) simp |
|
437 |
||
438 |
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" |
|
37591 | 439 |
by (simp add: prod_case_unfold) |
37166 | 440 |
|
441 |
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" |
|
442 |
-- {* Prevents simplification of @{term c}: much faster *} |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
443 |
by (fact weak_case_cong) |
37166 | 444 |
|
445 |
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" |
|
446 |
by (simp add: split_eta) |
|
447 |
||
11838 | 448 |
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
|
449 |
proof |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
450 |
fix a b |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
451 |
assume "!!x. PROP P x" |
19535 | 452 |
then show "PROP P (a, b)" . |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
453 |
next |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
454 |
fix x |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
455 |
assume "!!a b. PROP P (a, b)" |
19535 | 456 |
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
|
457 |
qed |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
458 |
|
11838 | 459 |
text {* |
460 |
The rule @{thm [source] split_paired_all} does not work with the |
|
461 |
Simplifier because it also affects premises in congrence rules, |
|
462 |
where this can lead to premises of the form @{text "!!a b. ... = |
|
463 |
?P(a, b)"} which cannot be solved by reflexivity. |
|
464 |
*} |
|
465 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
466 |
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
|
467 |
|
26480 | 468 |
ML {* |
11838 | 469 |
(* replace parameters of product type by individual component parameters *) |
470 |
val safe_full_simp_tac = generic_simp_tac true (true, false, false); |
|
471 |
local (* filtering with exists_paired_all is an essential optimization *) |
|
16121 | 472 |
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = |
11838 | 473 |
can HOLogic.dest_prodT T orelse exists_paired_all t |
474 |
| exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u |
|
475 |
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t |
|
476 |
| exists_paired_all _ = false; |
|
477 |
val ss = HOL_basic_ss |
|
26340 | 478 |
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] |
11838 | 479 |
addsimprocs [unit_eq_proc]; |
480 |
in |
|
481 |
val split_all_tac = SUBGOAL (fn (t, i) => |
|
482 |
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); |
|
483 |
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => |
|
484 |
if exists_paired_all t then full_simp_tac ss i else no_tac); |
|
485 |
fun split_all th = |
|
26340 | 486 |
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; |
11838 | 487 |
end; |
26340 | 488 |
*} |
11838 | 489 |
|
26340 | 490 |
declaration {* fn _ => |
491 |
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) |
|
16121 | 492 |
*} |
11838 | 493 |
|
494 |
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" |
|
495 |
-- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} |
|
496 |
by fast |
|
497 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
498 |
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
|
499 |
by fast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
500 |
|
11838 | 501 |
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" |
502 |
-- {* Can't be added to simpset: loops! *} |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
503 |
by (simp add: split_eta) |
11838 | 504 |
|
505 |
text {* |
|
506 |
Simplification procedure for @{thm [source] cond_split_eta}. Using |
|
507 |
@{thm [source] split_eta} as a rewrite rule is not general enough, |
|
508 |
and using @{thm [source] cond_split_eta} directly would render some |
|
509 |
existing proofs very inefficient; similarly for @{text |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
510 |
split_beta}. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
511 |
*} |
11838 | 512 |
|
26480 | 513 |
ML {* |
11838 | 514 |
local |
35364 | 515 |
val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta}; |
516 |
fun Pair_pat k 0 (Bound m) = (m = k) |
|
517 |
| Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = |
|
518 |
i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t |
|
519 |
| Pair_pat _ _ _ = false; |
|
520 |
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t |
|
521 |
| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
|
522 |
| no_args k i (Bound m) = m < k orelse m > k + i |
|
523 |
| no_args _ _ _ = true; |
|
524 |
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE |
|
37591 | 525 |
| split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t |
35364 | 526 |
| split_pat tp i _ = NONE; |
20044
92cc2f4c7335
simprocs: no theory argument -- use simpset context instead;
wenzelm
parents:
19656
diff
changeset
|
527 |
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] |
35364 | 528 |
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) |
18328 | 529 |
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); |
11838 | 530 |
|
35364 | 531 |
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t |
532 |
| beta_term_pat k i (t $ u) = |
|
533 |
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) |
|
534 |
| beta_term_pat k i t = no_args k i t; |
|
535 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
|
536 |
| eta_term_pat _ _ _ = false; |
|
11838 | 537 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
35364 | 538 |
| subst arg k i (t $ u) = |
539 |
if Pair_pat k i (t $ u) then incr_boundvars k arg |
|
540 |
else (subst arg k i t $ subst arg k i u) |
|
541 |
| subst arg k i t = t; |
|
37591 | 542 |
fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) = |
11838 | 543 |
(case split_pat beta_term_pat 1 t of |
35364 | 544 |
SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f)) |
15531 | 545 |
| NONE => NONE) |
35364 | 546 |
| beta_proc _ _ = NONE; |
37591 | 547 |
fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = |
11838 | 548 |
(case split_pat eta_term_pat 1 t of |
35364 | 549 |
SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) |
15531 | 550 |
| NONE => NONE) |
35364 | 551 |
| eta_proc _ _ = NONE; |
11838 | 552 |
in |
38715
6513ea67d95d
renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents:
37808
diff
changeset
|
553 |
val split_beta_proc = Simplifier.simproc_global @{theory} "split_beta" ["split f z"] (K beta_proc); |
6513ea67d95d
renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents:
37808
diff
changeset
|
554 |
val split_eta_proc = Simplifier.simproc_global @{theory} "split_eta" ["split f"] (K eta_proc); |
11838 | 555 |
end; |
556 |
||
557 |
Addsimprocs [split_beta_proc, split_eta_proc]; |
|
558 |
*} |
|
559 |
||
26798
a9134a089106
split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents:
26588
diff
changeset
|
560 |
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" |
11838 | 561 |
by (subst surjective_pairing, rule split_conv) |
562 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
563 |
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))" |
11838 | 564 |
-- {* For use with @{text split} and the Simplifier. *} |
15481 | 565 |
by (insert surj_pair [of p], clarify, simp) |
11838 | 566 |
|
567 |
text {* |
|
568 |
@{thm [source] split_split} could be declared as @{text "[split]"} |
|
569 |
done after the Splitter has been speeded up significantly; |
|
570 |
precompute the constants involved and don't do anything unless the |
|
571 |
current goal contains one of those constants. |
|
572 |
*} |
|
573 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
574 |
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" |
14208 | 575 |
by (subst split_split, simp) |
11838 | 576 |
|
577 |
text {* |
|
578 |
\medskip @{term split} used as a logical connective or set former. |
|
579 |
||
580 |
\medskip These rules are for use with @{text blast}; could instead |
|
581 |
call @{text simp} using @{thm [source] split} as rewrite. *} |
|
582 |
||
583 |
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p" |
|
584 |
apply (simp only: split_tupled_all) |
|
585 |
apply (simp (no_asm_simp)) |
|
586 |
done |
|
587 |
||
588 |
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x" |
|
589 |
apply (simp only: split_tupled_all) |
|
590 |
apply (simp (no_asm_simp)) |
|
591 |
done |
|
592 |
||
593 |
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
|
37591 | 594 |
by (induct p) auto |
11838 | 595 |
|
596 |
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
|
37591 | 597 |
by (induct p) auto |
11838 | 598 |
|
599 |
lemma splitE2: |
|
600 |
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R" |
|
601 |
proof - |
|
602 |
assume q: "Q (split P z)" |
|
603 |
assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R" |
|
604 |
show R |
|
605 |
apply (rule r surjective_pairing)+ |
|
606 |
apply (rule split_beta [THEN subst], rule q) |
|
607 |
done |
|
608 |
qed |
|
609 |
||
610 |
lemma splitD': "split R (a,b) c ==> R a b c" |
|
611 |
by simp |
|
612 |
||
613 |
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" |
|
614 |
by simp |
|
615 |
||
616 |
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p" |
|
14208 | 617 |
by (simp only: split_tupled_all, simp) |
11838 | 618 |
|
18372 | 619 |
lemma mem_splitE: |
37166 | 620 |
assumes major: "z \<in> split c p" |
621 |
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" |
|
18372 | 622 |
shows Q |
37591 | 623 |
by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+ |
11838 | 624 |
|
625 |
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] |
|
626 |
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] |
|
627 |
||
26340 | 628 |
ML {* |
11838 | 629 |
local (* filtering with exists_p_split is an essential optimization *) |
37591 | 630 |
fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true |
11838 | 631 |
| exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u |
632 |
| exists_p_split (Abs (_, _, t)) = exists_p_split t |
|
633 |
| exists_p_split _ = false; |
|
35364 | 634 |
val ss = HOL_basic_ss addsimps @{thms split_conv}; |
11838 | 635 |
in |
636 |
val split_conv_tac = SUBGOAL (fn (t, i) => |
|
637 |
if exists_p_split t then safe_full_simp_tac ss i else no_tac); |
|
638 |
end; |
|
26340 | 639 |
*} |
640 |
||
11838 | 641 |
(* This prevents applications of splitE for already splitted arguments leading |
642 |
to quite time-consuming computations (in particular for nested tuples) *) |
|
26340 | 643 |
declaration {* fn _ => |
644 |
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) |
|
16121 | 645 |
*} |
11838 | 646 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
647 |
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" |
18372 | 648 |
by (rule ext) fast |
11838 | 649 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
650 |
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" |
18372 | 651 |
by (rule ext) fast |
11838 | 652 |
|
653 |
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" |
|
654 |
-- {* Allows simplifications of nested splits in case of independent predicates. *} |
|
18372 | 655 |
by (rule ext) blast |
11838 | 656 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
657 |
(* 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
|
658 |
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
|
659 |
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
|
660 |
*) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
661 |
lemma split_comp_eq: |
20415 | 662 |
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" |
663 |
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" |
|
18372 | 664 |
by (rule ext) auto |
14101 | 665 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
666 |
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
|
667 |
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
|
668 |
apply auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
669 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
670 |
|
11838 | 671 |
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" |
672 |
by blast |
|
673 |
||
674 |
(* |
|
675 |
the following would be slightly more general, |
|
676 |
but cannot be used as rewrite rule: |
|
677 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
678 |
### ?y = .x |
|
679 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" |
|
14208 | 680 |
by (rtac some_equality 1) |
681 |
by ( Simp_tac 1) |
|
682 |
by (split_all_tac 1) |
|
683 |
by (Asm_full_simp_tac 1) |
|
11838 | 684 |
qed "The_split_eq"; |
685 |
*) |
|
686 |
||
687 |
text {* |
|
688 |
Setup of internal @{text split_rule}. |
|
689 |
*} |
|
690 |
||
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
691 |
lemmas prod_caseI = prod.cases [THEN iffD2, standard] |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
692 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
693 |
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
694 |
by (fact splitI2) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
695 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
696 |
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
697 |
by (fact splitI2') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
698 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
699 |
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
700 |
by (fact splitE) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
701 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
702 |
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
703 |
by (fact splitE') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
704 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
705 |
declare prod_caseI [intro!] |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
706 |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
707 |
lemma prod_case_beta: |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
708 |
"prod_case f p = f (fst p) (snd p)" |
37591 | 709 |
by (fact split_beta) |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
710 |
|
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
711 |
lemma prod_cases3 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
712 |
obtains (fields) a b c where "y = (a, b, c)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
713 |
by (cases y, case_tac b) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
714 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
715 |
lemma prod_induct3 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
716 |
"(!!a b c. P (a, b, c)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
717 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
718 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
719 |
lemma prod_cases4 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
720 |
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
|
721 |
by (cases y, case_tac c) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
722 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
723 |
lemma prod_induct4 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
724 |
"(!!a b c d. P (a, b, c, d)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
725 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
726 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
727 |
lemma prod_cases5 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
728 |
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
|
729 |
by (cases y, case_tac d) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
730 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
731 |
lemma prod_induct5 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
732 |
"(!!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
|
733 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
734 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
735 |
lemma prod_cases6 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
736 |
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
|
737 |
by (cases y, case_tac e) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
738 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
739 |
lemma prod_induct6 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
740 |
"(!!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
|
741 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
742 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
743 |
lemma prod_cases7 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
744 |
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
|
745 |
by (cases y, case_tac f) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
746 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
747 |
lemma prod_induct7 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
748 |
"(!!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
|
749 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
750 |
|
37166 | 751 |
lemma split_def: |
752 |
"split = (\<lambda>c p. c (fst p) (snd p))" |
|
37591 | 753 |
by (fact prod_case_unfold) |
37166 | 754 |
|
755 |
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
|
756 |
"internal_split == split" |
|
757 |
||
758 |
lemma internal_split_conv: "internal_split c (a, b) = c a b" |
|
759 |
by (simp only: internal_split_def split_conv) |
|
760 |
||
761 |
use "Tools/split_rule.ML" |
|
762 |
setup Split_Rule.setup |
|
763 |
||
764 |
hide_const internal_split |
|
765 |
||
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
766 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
767 |
subsubsection {* Derived operations *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
768 |
|
37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
769 |
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where |
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
770 |
"curry = (\<lambda>c x y. c (x, y))" |
37166 | 771 |
|
772 |
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" |
|
773 |
by (simp add: curry_def) |
|
774 |
||
775 |
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" |
|
776 |
by (simp add: curry_def) |
|
777 |
||
778 |
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" |
|
779 |
by (simp add: curry_def) |
|
780 |
||
781 |
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" |
|
782 |
by (simp add: curry_def) |
|
783 |
||
784 |
lemma curry_split [simp]: "curry (split f) = f" |
|
785 |
by (simp add: curry_def split_def) |
|
786 |
||
787 |
lemma split_curry [simp]: "split (curry f) = f" |
|
788 |
by (simp add: curry_def split_def) |
|
789 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
790 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
791 |
The composition-uncurry combinator. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
792 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
793 |
|
37751 | 794 |
notation fcomp (infixl "\<circ>>" 60) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
795 |
|
37751 | 796 |
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where |
797 |
"f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
798 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
799 |
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))" |
37751 | 800 |
by (simp add: expand_fun_eq scomp_def prod_case_unfold) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
801 |
|
37751 | 802 |
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)" |
803 |
by (simp add: scomp_unfold prod_case_unfold) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
804 |
|
37751 | 805 |
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" |
26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset
|
806 |
by (simp add: expand_fun_eq scomp_apply) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
807 |
|
37751 | 808 |
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" |
26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset
|
809 |
by (simp add: expand_fun_eq scomp_apply) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
810 |
|
37751 | 811 |
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
812 |
by (simp add: expand_fun_eq scomp_unfold) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
813 |
|
37751 | 814 |
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
815 |
by (simp add: expand_fun_eq scomp_unfold fcomp_def) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
816 |
|
37751 | 817 |
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
818 |
by (simp add: expand_fun_eq scomp_unfold fcomp_apply) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
819 |
|
31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
820 |
code_const scomp |
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
821 |
(Eval infixl 3 "#->") |
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
822 |
|
37751 | 823 |
no_notation fcomp (infixl "\<circ>>" 60) |
824 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
26358
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 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
827 |
@{term prod_fun} --- action of the product functor upon |
36664
6302f9ad7047
repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents:
36622
diff
changeset
|
828 |
functions. |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
829 |
*} |
21195 | 830 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
831 |
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where |
37765 | 832 |
"prod_fun f g = (\<lambda>(x, y). (f x, g y))" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
833 |
|
28562 | 834 |
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
835 |
by (simp add: prod_fun_def) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
836 |
|
37278 | 837 |
lemma fst_prod_fun[simp]: "fst (prod_fun f g x) = f (fst x)" |
838 |
by (cases x, auto) |
|
839 |
||
840 |
lemma snd_prod_fun[simp]: "snd (prod_fun f g x) = g (snd x)" |
|
841 |
by (cases x, auto) |
|
842 |
||
843 |
lemma fst_comp_prod_fun[simp]: "fst \<circ> prod_fun f g = f \<circ> fst" |
|
844 |
by (rule ext) auto |
|
845 |
||
846 |
lemma snd_comp_prod_fun[simp]: "snd \<circ> prod_fun f g = g \<circ> snd" |
|
847 |
by (rule ext) auto |
|
848 |
||
849 |
||
850 |
lemma prod_fun_compose: |
|
851 |
"prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" |
|
852 |
by (rule ext) auto |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
853 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
854 |
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
|
855 |
by (rule ext) auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
856 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
857 |
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
|
858 |
apply (rule image_eqI) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
859 |
apply (rule prod_fun [symmetric], assumption) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
860 |
done |
21195 | 861 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
862 |
lemma prod_fun_imageE [elim!]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
863 |
assumes major: "c: (prod_fun f g)`r" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
864 |
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
|
865 |
shows P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
866 |
apply (rule major [THEN imageE]) |
37166 | 867 |
apply (case_tac x) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
868 |
apply (rule cases) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
869 |
apply (blast intro: prod_fun) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
870 |
apply blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
871 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
872 |
|
37278 | 873 |
|
37166 | 874 |
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where |
875 |
"apfst f = prod_fun f id" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
876 |
|
37166 | 877 |
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where |
878 |
"apsnd f = prod_fun id f" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
879 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
880 |
lemma apfst_conv [simp, code]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
881 |
"apfst f (x, y) = (f x, y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
882 |
by (simp add: apfst_def) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
883 |
|
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
884 |
lemma apsnd_conv [simp, code]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
885 |
"apsnd f (x, y) = (x, f y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
886 |
by (simp add: apsnd_def) |
21195 | 887 |
|
33594 | 888 |
lemma fst_apfst [simp]: |
889 |
"fst (apfst f x) = f (fst x)" |
|
890 |
by (cases x) simp |
|
891 |
||
892 |
lemma fst_apsnd [simp]: |
|
893 |
"fst (apsnd f x) = fst x" |
|
894 |
by (cases x) simp |
|
895 |
||
896 |
lemma snd_apfst [simp]: |
|
897 |
"snd (apfst f x) = snd x" |
|
898 |
by (cases x) simp |
|
899 |
||
900 |
lemma snd_apsnd [simp]: |
|
901 |
"snd (apsnd f x) = f (snd x)" |
|
902 |
by (cases x) simp |
|
903 |
||
904 |
lemma apfst_compose: |
|
905 |
"apfst f (apfst g x) = apfst (f \<circ> g) x" |
|
906 |
by (cases x) simp |
|
907 |
||
908 |
lemma apsnd_compose: |
|
909 |
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" |
|
910 |
by (cases x) simp |
|
911 |
||
912 |
lemma apfst_apsnd [simp]: |
|
913 |
"apfst f (apsnd g x) = (f (fst x), g (snd x))" |
|
914 |
by (cases x) simp |
|
915 |
||
916 |
lemma apsnd_apfst [simp]: |
|
917 |
"apsnd f (apfst g x) = (g (fst x), f (snd x))" |
|
918 |
by (cases x) simp |
|
919 |
||
920 |
lemma apfst_id [simp] : |
|
921 |
"apfst id = id" |
|
922 |
by (simp add: expand_fun_eq) |
|
923 |
||
924 |
lemma apsnd_id [simp] : |
|
925 |
"apsnd id = id" |
|
926 |
by (simp add: expand_fun_eq) |
|
927 |
||
928 |
lemma apfst_eq_conv [simp]: |
|
929 |
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" |
|
930 |
by (cases x) simp |
|
931 |
||
932 |
lemma apsnd_eq_conv [simp]: |
|
933 |
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" |
|
934 |
by (cases x) simp |
|
935 |
||
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
936 |
lemma apsnd_apfst_commute: |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
937 |
"apsnd f (apfst g p) = apfst g (apsnd f p)" |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
938 |
by simp |
21195 | 939 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
940 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
941 |
Disjoint union of a family of sets -- Sigma. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
942 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
943 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
944 |
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
|
945 |
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
|
946 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
947 |
abbreviation |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
948 |
Times :: "['a set, 'b set] => ('a * 'b) set" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
949 |
(infixr "<*>" 80) where |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
950 |
"A <*> B == Sigma A (%_. B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
951 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
952 |
notation (xsymbols) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
953 |
Times (infixr "\<times>" 80) |
15394 | 954 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
955 |
notation (HTML output) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
956 |
Times (infixr "\<times>" 80) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
957 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
958 |
syntax |
35115 | 959 |
"_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
960 |
translations |
35115 | 961 |
"SIGMA x:A. B" == "CONST Sigma A (%x. B)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
962 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
963 |
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
|
964 |
by (unfold Sigma_def) blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
965 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
966 |
lemma SigmaE [elim!]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
967 |
"[| c: Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
968 |
!!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
|
969 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
970 |
-- {* The general elimination rule. *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
971 |
by (unfold Sigma_def) blast |
20588 | 972 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
973 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
974 |
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
|
975 |
eigenvariables. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
976 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
977 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
978 |
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
|
979 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
980 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
981 |
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
|
982 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
983 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
984 |
lemma SigmaE2: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
985 |
"[| (a, b) : Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
986 |
[| a:A; b:B(a) |] ==> P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
987 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
988 |
by blast |
20588 | 989 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
990 |
lemma Sigma_cong: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
991 |
"\<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
|
992 |
\<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
|
993 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
994 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
995 |
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
|
996 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
997 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
998 |
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
999 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1000 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1001 |
lemma Sigma_empty2 [simp]: "A <*> {} = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1002 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1003 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1004 |
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1005 |
by auto |
21908 | 1006 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1007 |
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
|
1008 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1009 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1010 |
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
|
1011 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1012 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1013 |
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
|
1014 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1015 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1016 |
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
|
1017 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1018 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1019 |
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
|
1020 |
by (blast elim: equalityE) |
20588 | 1021 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1022 |
lemma SetCompr_Sigma_eq: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1023 |
"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
|
1024 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1025 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1026 |
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
|
1027 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1028 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1029 |
lemma UN_Times_distrib: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1030 |
"(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
|
1031 |
-- {* Suggested by Pierre Chartier *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1032 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1033 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
1034 |
lemma split_paired_Ball_Sigma [simp,no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1035 |
"(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
|
1036 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1037 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
1038 |
lemma split_paired_Bex_Sigma [simp,no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1039 |
"(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
|
1040 |
by blast |
21908 | 1041 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1042 |
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
|
1043 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1044 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1045 |
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
|
1046 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1047 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1048 |
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
|
1049 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1050 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1051 |
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
|
1052 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1053 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1054 |
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
|
1055 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1056 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1057 |
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
|
1058 |
by blast |
21908 | 1059 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1060 |
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
|
1061 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1062 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1063 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1064 |
Non-dependent versions are needed to avoid the need for higher-order |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1065 |
matching, especially when the rules are re-oriented. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1066 |
*} |
21908 | 1067 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1068 |
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" |
28719 | 1069 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1070 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1071 |
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" |
28719 | 1072 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1073 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1074 |
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" |
28719 | 1075 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1076 |
|
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1077 |
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1078 |
by auto |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1079 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1080 |
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1081 |
by (auto intro!: image_eqI) |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1082 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1083 |
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1084 |
by (auto intro!: image_eqI) |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1085 |
|
28719 | 1086 |
lemma insert_times_insert[simp]: |
1087 |
"insert a A \<times> insert b B = |
|
1088 |
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" |
|
1089 |
by blast |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1090 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1091 |
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)" |
37166 | 1092 |
by (auto, case_tac "f x", auto) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1093 |
|
37278 | 1094 |
text{* The following @{const prod_fun} lemmas are due to Joachim Breitner: *} |
1095 |
||
1096 |
lemma prod_fun_inj_on: |
|
1097 |
assumes "inj_on f A" and "inj_on g B" |
|
1098 |
shows "inj_on (prod_fun f g) (A \<times> B)" |
|
1099 |
proof (rule inj_onI) |
|
1100 |
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" |
|
1101 |
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto |
|
1102 |
assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto |
|
1103 |
assume "prod_fun f g x = prod_fun f g y" |
|
1104 |
hence "fst (prod_fun f g x) = fst (prod_fun f g y)" by (auto) |
|
1105 |
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) |
|
1106 |
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` |
|
1107 |
have "fst x = fst y" by (auto dest:dest:inj_onD) |
|
1108 |
moreover from `prod_fun f g x = prod_fun f g y` |
|
1109 |
have "snd (prod_fun f g x) = snd (prod_fun f g y)" by (auto) |
|
1110 |
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) |
|
1111 |
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` |
|
1112 |
have "snd x = snd y" by (auto dest:dest:inj_onD) |
|
1113 |
ultimately show "x = y" by(rule prod_eqI) |
|
1114 |
qed |
|
1115 |
||
1116 |
lemma prod_fun_surj: |
|
1117 |
assumes "surj f" and "surj g" |
|
1118 |
shows "surj (prod_fun f g)" |
|
1119 |
unfolding surj_def |
|
1120 |
proof |
|
1121 |
fix y :: "'b \<times> 'd" |
|
1122 |
from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) |
|
1123 |
moreover |
|
1124 |
from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) |
|
1125 |
ultimately have "(fst y, snd y) = prod_fun f g (a,b)" by auto |
|
1126 |
thus "\<exists>x. y = prod_fun f g x" by auto |
|
1127 |
qed |
|
1128 |
||
1129 |
lemma prod_fun_surj_on: |
|
1130 |
assumes "f ` A = A'" and "g ` B = B'" |
|
1131 |
shows "prod_fun f g ` (A \<times> B) = A' \<times> B'" |
|
1132 |
unfolding image_def |
|
1133 |
proof(rule set_ext,rule iffI) |
|
1134 |
fix x :: "'a \<times> 'c" |
|
1135 |
assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = prod_fun f g x}" |
|
1136 |
then obtain y where "y \<in> A \<times> B" and "x = prod_fun f g y" by blast |
|
1137 |
from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto |
|
1138 |
moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto |
|
1139 |
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto |
|
1140 |
with `x = prod_fun f g y` show "x \<in> A' \<times> B'" by (cases y, auto) |
|
1141 |
next |
|
1142 |
fix x :: "'a \<times> 'c" |
|
1143 |
assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto |
|
1144 |
from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto |
|
1145 |
then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE) |
|
1146 |
moreover from `image g B = B'` and `snd x \<in> B'` |
|
1147 |
obtain b where "b \<in> B" and "snd x = g b" by auto |
|
1148 |
ultimately have "(fst x, snd x) = prod_fun f g (a,b)" by auto |
|
1149 |
moreover from `a \<in> A` and `b \<in> B` have "(a , b) \<in> A \<times> B" by auto |
|
1150 |
ultimately have "\<exists>y \<in> A \<times> B. x = prod_fun f g y" by auto |
|
1151 |
thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = prod_fun f g y}" by auto |
|
1152 |
qed |
|
1153 |
||
35822 | 1154 |
lemma swap_inj_on: |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1155 |
"inj_on (\<lambda>(i, j). (j, i)) A" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1156 |
by (auto intro!: inj_onI) |
35822 | 1157 |
|
1158 |
lemma swap_product: |
|
1159 |
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
|
1160 |
by (simp add: split_def image_def) blast |
|
1161 |
||
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1162 |
lemma image_split_eq_Sigma: |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1163 |
"(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1164 |
proof (safe intro!: imageI vimageI) |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1165 |
fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1166 |
show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1167 |
using * eq[symmetric] by auto |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1168 |
qed simp_all |
35822 | 1169 |
|
21908 | 1170 |
|
37166 | 1171 |
subsection {* Inductively defined sets *} |
15394 | 1172 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
1173 |
use "Tools/inductive_codegen.ML" |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
1174 |
setup Inductive_Codegen.setup |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
1175 |
|
31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
1176 |
use "Tools/inductive_set.ML" |
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
1177 |
setup Inductive_Set.setup |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
1178 |
|
37166 | 1179 |
|
1180 |
subsection {* Legacy theorem bindings and duplicates *} |
|
1181 |
||
1182 |
lemma PairE: |
|
1183 |
obtains x y where "p = (x, y)" |
|
1184 |
by (fact prod.exhaust) |
|
1185 |
||
1186 |
lemma Pair_inject: |
|
1187 |
assumes "(a, b) = (a', b')" |
|
1188 |
and "a = a' ==> b = b' ==> R" |
|
1189 |
shows R |
|
1190 |
using assms by simp |
|
1191 |
||
1192 |
lemmas Pair_eq = prod.inject |
|
1193 |
||
1194 |
lemmas split = split_conv -- {* for backwards compatibility *} |
|
1195 |
||
10213 | 1196 |
end |