author  haftmann 
Thu, 18 Nov 2010 17:01:16 +0100  
changeset 40607  30d512bf47a7 
parent 40590  b994d855dbd2 
child 40702  cf26dd7395e4 
permissions  rwrr 
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(* Title: HOL/Product_Type.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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*) 
10213  5 

11838  6 
header {* Cartesian products *} 
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15131  8 
theory Product_Type 
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imports Typedef Inductive Fun 
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uses 
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("Tools/split_rule.ML") 
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("Tools/inductive_codegen.ML") 
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("Tools/inductive_set.ML") 
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begin 
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subsection {* @{typ bool} is a datatype *} 
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rep_datatype True False by (auto intro: bool_induct) 
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declare case_split [cases type: bool] 
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 "prefer plain propositional version" 
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lemma 
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shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P" 
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and [code]: "HOL.equal True P \<longleftrightarrow> P" 
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and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" 
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and [code]: "HOL.equal P True \<longleftrightarrow> P" 
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and [code nbe]: "HOL.equal P P \<longleftrightarrow> True" 
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by (simp_all add: equal) 
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code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" 
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(Haskell infix 4 "==") 
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code_instance bool :: equal 
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(Haskell ) 
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subsection {* The @{text unit} type *} 
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typedef (open) unit = "{True}" 
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proof 
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show "True : ?unit" .. 
11838  43 
qed 
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definition 
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Unity :: unit ("'(')") 
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where 
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"() = Abs_unit True" 
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lemma unit_eq [no_atp]: "u = ()" 
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by (induct u) (simp add: Unity_def) 
11838  52 

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text {* 

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

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this rule directly  it loops! 

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*} 

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ML {* 
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val unit_eq_proc = 
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let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in 
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Simplifier.simproc_global @{theory} "unit_eq" ["x::unit"] 
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(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) 
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end; 
11838  64 

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Addsimprocs [unit_eq_proc]; 

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*} 

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rep_datatype "()" by simp 
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" 
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by simp 

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

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by (rule triv_forall_equality) 

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text {* 

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

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

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

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*} 

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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 
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begin 

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definition "default = ()" 

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instance .. 

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end 

10213  93 

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lemma [code]: 
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"HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ 
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code_type unit 
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(SML "unit") 
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(OCaml "unit") 
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(Haskell "()") 
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(Scala "Unit") 
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37166  103 
code_const Unity 
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(SML "()") 

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(OCaml "()") 

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(Haskell "()") 

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(Scala "()") 

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code_instance unit :: equal 
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(Haskell ) 
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code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" 
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(Haskell infix 4 "==") 
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code_reserved SML 
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unit 
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code_reserved OCaml 
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unit 
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code_reserved Scala 
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Unit 

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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 

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"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" 
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typedef ('a, 'b) prod (infixr "*" 20) 
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= "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" 
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proof 
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fix a b show "Pair_Rep a b \<in> ?prod" 
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by rule+ 
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qed 
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type_notation (xsymbols) 
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"prod" ("(_ \<times>/ _)" [21, 20] 20) 
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type_notation (HTML output) 
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"prod" ("(_ \<times>/ _)" [21, 20] 20) 
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where 
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"Pair a b = Abs_prod (Pair_Rep a b)" 
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rep_datatype Pair proof  
37166  148 
fix P :: "'a \<times> 'b \<Rightarrow> bool" and p 
149 
assume "\<And>a b. P (Pair a b)" 

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then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) 
37166  151 
next 
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fix a c :: 'a and b d :: 'b 

153 
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" 

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by (auto simp add: Pair_Rep_def fun_eq_iff) 
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moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" 
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by (auto simp add: prod_def) 
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ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" 
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by (simp add: Pair_def Abs_prod_inject) 
37166  159 
qed 
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declare prod.simps(2) [nitpick_simp del] 
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declare weak_case_cong [cong del] 
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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 predefined 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" ("_,/ _") 

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"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") 
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"" :: "pttrn => patterns" ("_") 
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"_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 etacontraction of body; 
197 
works best with enclosing "let", if "let" does not avoid etacontraction*) 

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'' 

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end 

37591  220 
 split_tr' [Const (@{const_syntax prod_case}, _) $ t] = 
35115  221 
(* split (split (%x y z. t)) => %((x, y), z). t *) 
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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 

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Syntax.const @{syntax_const "_abs"} $ 

227 
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t 

228 
end 

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 split_tr' _ = raise Match; 

37591  230 
in [(@{const_syntax prod_case}, split_tr')] end 
14359  231 
*} 
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {* 
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let 
35115  236 
fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match 
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 split_guess_names_tr' _ T [Abs (x, xT, t)] = 

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(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) 

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 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) 

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261 
 split_guess_names_tr' _ _ _ = raise Match; 
37591  262 
in [(@{const_syntax prod_case}, split_guess_names_tr')] end 
15422
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263 
*} 
cbdddc0efadf
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264 

10213  265 

37166  266 
subsubsection {* Code generator setup *} 
267 

37678
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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
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280 
code_instance prod :: equal 
37166  281 
(Haskell ) 
282 

38857
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283 
code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 
39272  284 
(Haskell infix 4 "==") 
37166  285 

286 
types_code 

37678
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287 
"prod" ("(_ */ _)") 
37166  288 
attach (term_of) {* 
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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) {* 

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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 (i1) (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 

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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
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diff
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381 
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where 
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
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changeset

382 
"fst p = (case p of (a, b) \<Rightarrow> a)" 
11838  383 

37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
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changeset

384 
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where 
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
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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
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haftmann
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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))" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

397 
by (simp add: fun_eq_iff split: prod.split) 
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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
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diff
changeset

405 
lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  406 
by (cases p) simp 
407 

26358
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haftmann
parents:
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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" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

426 
by (simp add: fun_eq_iff 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. *} 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

430 
by (simp add: fun_eq_iff 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 timeconsuming 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 compositionuncurry 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)))" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

800 
by (simp add: fun_eq_iff 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" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

806 
by (simp add: fun_eq_iff 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" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

809 
by (simp add: fun_eq_iff 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)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

812 
by (simp add: fun_eq_iff 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)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

815 
by (simp add: fun_eq_iff 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)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

818 
by (simp add: fun_eq_iff 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 {* 
40607  827 
@{term map_pair}  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 

40607  831 
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
832 
"map_pair 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 

40607  834 
lemma map_pair_simp [simp, code]: 
835 
"map_pair f g (a, b) = (f a, g b)" 

836 
by (simp add: map_pair_def) 

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

837 

40607  838 
type_mapper map_pair 
839 
by (simp_all add: split_paired_all) 

37278  840 

40607  841 
lemma fst_map_pair [simp]: 
842 
"fst (map_pair f g x) = f (fst x)" 

843 
by (cases x) simp_all 

37278  844 

40607  845 
lemma snd_prod_fun [simp]: 
846 
"snd (map_pair f g x) = g (snd x)" 

847 
by (cases x) simp_all 

37278  848 

40607  849 
lemma fst_comp_map_pair [simp]: 
850 
"fst \<circ> map_pair f g = f \<circ> fst" 

851 
by (rule ext) simp_all 

37278  852 

40607  853 
lemma snd_comp_map_pair [simp]: 
854 
"snd \<circ> map_pair f g = g \<circ> snd" 

855 
by (rule ext) simp_all 

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

856 

40607  857 
lemma map_pair_compose: 
858 
"map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)" 

859 
by (rule ext) (simp add: map_pair.compositionality comp_def) 

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

860 

40607  861 
lemma map_pair_ident [simp]: 
862 
"map_pair (%x. x) (%y. y) = (%z. z)" 

863 
by (rule ext) (simp add: map_pair.identity) 

864 

865 
lemma map_pair_imageI [intro]: 

866 
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R" 

867 
by (rule image_eqI) simp_all 

21195  868 

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

869 
lemma prod_fun_imageE [elim!]: 
40607  870 
assumes major: "c \<in> map_pair f g ` R" 
871 
and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P" 

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

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

873 
apply (rule major [THEN imageE]) 
37166  874 
apply (case_tac x) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

875 
apply (rule cases) 
40607  876 
apply simp_all 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

878 

37166  879 
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where 
40607  880 
"apfst f = map_pair f id" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

881 

37166  882 
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where 
40607  883 
"apsnd f = map_pair id f" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

884 

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

885 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

886 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

887 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

888 

33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

889 
lemma apsnd_conv [simp, code]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

890 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

891 
by (simp add: apsnd_def) 
21195  892 

33594  893 
lemma fst_apfst [simp]: 
894 
"fst (apfst f x) = f (fst x)" 

895 
by (cases x) simp 

896 

897 
lemma fst_apsnd [simp]: 

898 
"fst (apsnd f x) = fst x" 

899 
by (cases x) simp 

900 

901 
lemma snd_apfst [simp]: 

902 
"snd (apfst f x) = snd x" 

903 
by (cases x) simp 

904 

905 
lemma snd_apsnd [simp]: 

906 
"snd (apsnd f x) = f (snd x)" 

907 
by (cases x) simp 

908 

909 
lemma apfst_compose: 

910 
"apfst f (apfst g x) = apfst (f \<circ> g) x" 

911 
by (cases x) simp 

912 

913 
lemma apsnd_compose: 

914 
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" 

915 
by (cases x) simp 

916 

917 
lemma apfst_apsnd [simp]: 

918 
"apfst f (apsnd g x) = (f (fst x), g (snd x))" 

919 
by (cases x) simp 

920 

921 
lemma apsnd_apfst [simp]: 

922 
"apsnd f (apfst g x) = (g (fst x), f (snd x))" 

923 
by (cases x) simp 

924 

925 
lemma apfst_id [simp] : 

926 
"apfst id = id" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

927 
by (simp add: fun_eq_iff) 
33594  928 

929 
lemma apsnd_id [simp] : 

930 
"apsnd id = id" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

931 
by (simp add: fun_eq_iff) 
33594  932 

933 
lemma apfst_eq_conv [simp]: 

934 
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" 

935 
by (cases x) simp 

936 

937 
lemma apsnd_eq_conv [simp]: 

938 
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" 

939 
by (cases x) simp 

940 

33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

941 
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

942 
"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

943 
by simp 
21195  944 

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

945 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

946 
Disjoint union of a family of sets  Sigma. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

947 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

948 

40607  949 
definition Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

950 
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

951 

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

952 
abbreviation 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

953 
Times :: "['a set, 'b set] => ('a * 'b) set" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

954 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

955 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

956 

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

957 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

958 
Times (infixr "\<times>" 80) 
15394  959 

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

960 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

961 
Times (infixr "\<times>" 80) 
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 
syntax 
35115  964 
"_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

965 
translations 
35115  966 
"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

967 

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

968 
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

969 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

970 

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

971 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

972 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

973 
!!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

974 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

975 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

976 
by (unfold Sigma_def) blast 
20588  977 

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

978 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

979 
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

980 
eigenvariables. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

981 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

982 

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

983 
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

984 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

985 

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

986 
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

987 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

988 

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

989 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

990 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

991 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

992 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

993 
by blast 
20588  994 

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

995 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

996 
"\<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

997 
\<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

998 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

999 

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

1000 
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

1001 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1002 

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

1003 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1004 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1005 

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

1006 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1007 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1008 

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

1009 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1010 
by auto 
21908  1011 

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

1012 
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

1013 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1014 

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

1015 
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

1016 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1017 

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

1018 
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

1019 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1020 

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

1021 
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

1022 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1023 

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

1024 
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

1025 
by (blast elim: equalityE) 
20588  1026 

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

1027 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1028 
"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

1029 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1030 

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

1031 
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

1032 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1033 

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

1034 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1035 
"(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

1036 
 {* Suggested by Pierre Chartier *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1037 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1038 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset

1039 
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

1040 
"(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

1041 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1042 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset

1043 
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

1044 
"(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

1045 
by blast 
21908  1046 

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

1047 
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

1048 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1049 

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

1050 
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

1051 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1052 

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

1053 
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

1054 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1055 

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

1056 
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

1057 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1058 

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

1059 
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

1060 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1061 

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

1062 
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

1063 
by blast 
21908  1064 

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

1065 
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

1066 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1067 

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

1068 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1069 
Nondependent versions are needed to avoid the need for higherorder 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1070 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1071 
*} 
21908  1072 

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

1073 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
28719  1074 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1075 

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

1076 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
28719  1077 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1078 

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

1079 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
28719  1080 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1081 

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

1082 
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

1083 
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

1084 

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 
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

1086 
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

1087 

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

1088 
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

1089 
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

1090 

28719  1091 
lemma insert_times_insert[simp]: 
1092 
"insert a A \<times> insert b B = 

1093 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

1094 
by blast 

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

1095 

33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1096 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
37166  1097 
by (auto, case_tac "f x", auto) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1098 

35822  1099 
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

1100 
"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

1101 
by (auto intro!: inj_onI) 
35822  1102 

1103 
lemma swap_product: 

1104 
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" 

1105 
by (simp add: split_def image_def) blast 

1106 

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

1107 
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

1108 
"(\<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

1109 
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

1110 
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

1111 
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

1112 
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

1113 
qed simp_all 
35822  1114 

40607  1115 
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *} 
1116 

1117 
lemma map_pair_inj_on: 

1118 
assumes "inj_on f A" and "inj_on g B" 

1119 
shows "inj_on (map_pair f g) (A \<times> B)" 

1120 
proof (rule inj_onI) 

1121 
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" 

1122 
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto 

1123 
assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto 

1124 
assume "map_pair f g x = map_pair f g y" 

1125 
hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto) 

1126 
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) 

1127 
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` 

1128 
have "fst x = fst y" by (auto dest:dest:inj_onD) 

1129 
moreover from `map_pair f g x = map_pair f g y` 

1130 
have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto) 

1131 
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) 

1132 
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` 

1133 
have "snd x = snd y" by (auto dest:dest:inj_onD) 

1134 
ultimately show "x = y" by(rule prod_eqI) 

1135 
qed 

1136 

1137 
lemma map_pair_surj: 

1138 
assumes "surj f" and "surj g" 

1139 
shows "surj (map_pair f g)" 

1140 
unfolding surj_def 

1141 
proof 

1142 
fix y :: "'b \<times> 'd" 

1143 
from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) 

1144 
moreover 

1145 
from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) 

1146 
ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto 

1147 
thus "\<exists>x. y = map_pair f g x" by auto 

1148 
qed 

1149 

1150 
lemma map_pair_surj_on: 

1151 
assumes "f ` A = A'" and "g ` B = B'" 

1152 
shows "map_pair f g ` (A \<times> B) = A' \<times> B'" 

1153 
unfolding image_def 

1154 
proof(rule set_eqI,rule iffI) 

1155 
fix x :: "'a \<times> 'c" 

1156 
assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}" 

1157 
then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast 

1158 
from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto 

1159 
moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto 

1160 
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto 

1161 
with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto) 

1162 
next 

1163 
fix x :: "'a \<times> 'c" 

1164 
assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto 

1165 
from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto 

1166 
then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE) 

1167 
moreover from `image g B = B'` and `snd x \<in> B'` 

1168 
obtain b where "b \<in> B" and "snd x = g b" by auto 

1169 
ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto 

1170 
moreover from `a \<in> A` and `b \<in> B` have "(a , b) \<in> A \<times> B" by auto 

1171 
ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto 

1172 
thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto 

1173 
qed 

1174 

21908  1175 

37166  1176 
subsection {* Inductively defined sets *} 
15394  1177 

37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset

1178 
use "Tools/inductive_codegen.ML" 
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
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parents:
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1179 
setup Inductive_Codegen.setup 
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qualified type "*"; qualified constants Pair, fst, snd, split
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parents:
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diff
changeset

1180 

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discontinued ancient tradition to suffix certain ML module names with "_package"
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parents:
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changeset

1181 
use "Tools/inductive_set.ML" 
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discontinued ancient tradition to suffix certain ML module names with "_package"
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parents:
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diff
changeset

1182 
setup Inductive_Set.setup 
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datatype interpretators for size and datatype_realizer
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1183 

37166  1184 

1185 
subsection {* Legacy theorem bindings and duplicates *} 

1186 

1187 
lemma PairE: 

1188 
obtains x y where "p = (x, y)" 

1189 
by (fact prod.exhaust) 

1190 

1191 
lemma Pair_inject: 

1192 
assumes "(a, b) = (a', b')" 

1193 
and "a = a' ==> b = b' ==> R" 

1194 
shows R 

1195 
using assms by simp 

1196 

1197 
lemmas Pair_eq = prod.inject 

1198 

1199 
lemmas split = split_conv  {* for backwards compatibility *} 

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10213  1201 
end 