author  blanchet 
Mon, 21 Feb 2011 10:44:19 +0100  
changeset 41792  ff3cb0c418b7 
parent 41505  6d19301074cf 
child 42059  83f3dc509068 
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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*) 
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11838  6 
header {* Cartesian products *} 
10213  7 

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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qualified type "*"; qualified constants Pair, fst, snd, split
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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) 
13462  63 
end; 
11838  64 

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Addsimprocs [unit_eq_proc]; 

66 
*} 

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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 
10213  138 

35427  139 
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 prod.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 
*} 

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nonterminal tuple_args and patterns 
10213  177 

178 
syntax 

179 
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") 

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

181 
"_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  185 

186 
translations 

35115  187 
"(x, y)" == "CONST Pair x y" 
10213  188 
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 
37591  189 
"%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)" 
190 
"%(x, y). b" == "CONST prod_case (%x y. b)" 

35115  191 
"_abs (CONST Pair x y) t" => "%(x, y). t" 
37166  192 
 {* The last rule accommodates tuples in `case C ... (x,y) ... => ...' 
193 
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *} 

10213  194 

35115  195 
(*reconstruct pattern from (nested) splits, avoiding etacontraction of body; 
196 
works best with enclosing "let", if "let" does not avoid etacontraction*) 

14359  197 
print_translation {* 
35115  198 
let 
199 
fun split_tr' [Abs (x, T, t as (Abs abs))] = 

200 
(* split (%x y. t) => %(x,y) t *) 

201 
let 

202 
val (y, t') = atomic_abs_tr' abs; 

203 
val (x', t'') = atomic_abs_tr' (x, T, t'); 

204 
in 

205 
Syntax.const @{syntax_const "_abs"} $ 

206 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

207 
end 

37591  208 
 split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] = 
35115  209 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 
210 
let 

211 
val Const (@{syntax_const "_abs"}, _) $ 

212 
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; 

213 
val (x', t'') = atomic_abs_tr' (x, T, t'); 

214 
in 

215 
Syntax.const @{syntax_const "_abs"} $ 

216 
(Syntax.const @{syntax_const "_pattern"} $ x' $ 

217 
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' 

218 
end 

37591  219 
 split_tr' [Const (@{const_syntax prod_case}, _) $ t] = 
35115  220 
(* split (split (%x y z. t)) => %((x, y), z). t *) 
221 
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) 

222 
 split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = 

223 
(* split (%pttrn z. t) => %(pttrn,z). t *) 

224 
let val (z, t) = atomic_abs_tr' abs in 

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Syntax.const @{syntax_const "_abs"} $ 

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

227 
end 

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 split_tr' _ = raise Match; 

37591  229 
in [(@{const_syntax prod_case}, split_tr')] end 
14359  230 
*} 
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  235 
fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match 
236 
 split_guess_names_tr' _ T [Abs (x, xT, t)] = 

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(case (head_of t) of 
37591  238 
Const (@{const_syntax prod_case}, _) => raise Match 
35115  239 
 _ => 
240 
let 

241 
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

242 
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); 

243 
val (x', t'') = atomic_abs_tr' (x, xT, t'); 

244 
in 

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Syntax.const @{syntax_const "_abs"} $ 

246 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

247 
end) 

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 split_guess_names_tr' _ T [t] = 
35115  249 
(case head_of t of 
37591  250 
Const (@{const_syntax prod_case}, _) => raise Match 
35115  251 
 _ => 
252 
let 

253 
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

254 
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); 

255 
val (x', t'') = atomic_abs_tr' ("x", xT, t'); 

256 
in 

257 
Syntax.const @{syntax_const "_abs"} $ 

258 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

259 
end) 

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260 
 split_guess_names_tr' _ _ _ = raise Match; 
37591  261 
in [(@{const_syntax prod_case}, split_guess_names_tr')] end 
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262 
*} 
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263 

10213  264 

37166  265 
subsubsection {* Code generator setup *} 
266 

37678
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267 
code_type prod 
37166  268 
(SML infix 2 "*") 
269 
(OCaml infix 2 "*") 

270 
(Haskell "!((_),/ (_))") 

271 
(Scala "((_),/ (_))") 

272 

273 
code_const Pair 

274 
(SML "!((_),/ (_))") 

275 
(OCaml "!((_),/ (_))") 

276 
(Haskell "!((_),/ (_))") 

277 
(Scala "!((_),/ (_))") 

278 

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279 
code_instance prod :: equal 
37166  280 
(Haskell ) 
281 

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282 
code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 
39272  283 
(Haskell infix 4 "==") 
37166  284 

285 
types_code 

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286 
"prod" ("(_ */ _)") 
37166  287 
attach (term_of) {* 
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288 
fun term_of_prod aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; 
37166  289 
*} 
290 
attach (test) {* 

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291 
fun gen_prod aG aT bG bT i = 
37166  292 
let 
293 
val (x, t) = aG i; 

294 
val (y, u) = bG i 

295 
in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; 

296 
*} 

297 

298 
consts_code 

299 
"Pair" ("(_,/ _)") 

300 

301 
setup {* 

302 
let 

303 

304 
fun strip_abs_split 0 t = ([], t) 

305 
 strip_abs_split i (Abs (s, T, t)) = 

306 
let 

307 
val s' = Codegen.new_name t s; 

308 
val v = Free (s', T) 

309 
in apfst (cons v) (strip_abs_split (i1) (subst_bound (v, t))) end 

37591  310 
 strip_abs_split i (u as Const (@{const_name prod_case}, _) $ t) = 
37166  311 
(case strip_abs_split (i+1) t of 
312 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 

313 
 _ => ([], u)) 

314 
 strip_abs_split i t = 

315 
strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0)); 

316 

317 
fun let_codegen thy defs dep thyname brack t gr = 

318 
(case strip_comb t of 

319 
(t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) => 

320 
let 

321 
fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) = 

322 
(case strip_abs_split 1 u of 

323 
([p], u') => apfst (cons (p, t)) (dest_let u') 

324 
 _ => ([], l)) 

325 
 dest_let t = ([], t); 

326 
fun mk_code (l, r) gr = 

327 
let 

328 
val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr; 

329 
val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1; 

330 
in ((pl, pr), gr2) end 

331 
in case dest_let (t1 $ t2 $ t3) of 

332 
([], _) => NONE 

333 
 (ps, u) => 

334 
let 

335 
val (qs, gr1) = fold_map mk_code ps gr; 

336 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 

337 
val (pargs, gr3) = fold_map 

338 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 

339 
in 

340 
SOME (Codegen.mk_app brack 

341 
(Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat 

342 
(separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) => 

343 
[Pretty.block [Codegen.str "val ", pl, Codegen.str " =", 

344 
Pretty.brk 1, pr]]) qs))), 

345 
Pretty.brk 1, Codegen.str "in ", pu, 

346 
Pretty.brk 1, Codegen.str "end"])) pargs, gr3) 

347 
end 

348 
end 

349 
 _ => NONE); 

350 

351 
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of 

37591  352 
(t1 as Const (@{const_name prod_case}, _), t2 :: ts) => 
37166  353 
let 
354 
val ([p], u) = strip_abs_split 1 (t1 $ t2); 

355 
val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr; 

356 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 

357 
val (pargs, gr3) = fold_map 

358 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 

359 
in 

360 
SOME (Codegen.mk_app brack 

361 
(Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>", 

362 
Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2) 

363 
end 

364 
 _ => NONE); 

365 

366 
in 

367 

368 
Codegen.add_codegen "let_codegen" let_codegen 

369 
#> Codegen.add_codegen "split_codegen" split_codegen 

370 

371 
end 

372 
*} 

373 

374 

375 
subsubsection {* Fundamental operations and properties *} 

11838  376 

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377 
lemma surj_pair [simp]: "EX x y. p = (x, y)" 
37166  378 
by (cases p) simp 
10213  379 

37389
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380 
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where 
09467cdfa198
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381 
"fst p = (case p of (a, b) \<Rightarrow> a)" 
11838  382 

37389
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383 
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where 
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
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384 
"snd p = (case p of (a, b) \<Rightarrow> b)" 
11838  385 

22886  386 
lemma fst_conv [simp, code]: "fst (a, b) = a" 
37166  387 
unfolding fst_def by simp 
11838  388 

22886  389 
lemma snd_conv [simp, code]: "snd (a, b) = b" 
37166  390 
unfolding snd_def by simp 
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oheimb
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391 

37166  392 
code_const fst and snd 
393 
(Haskell "fst" and "snd") 

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394 

41792
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blanchet
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diff
changeset

395 
lemma prod_case_unfold [nitpick_unfold]: "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
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changeset

396 
by (simp add: fun_eq_iff split: prod.split) 
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397 

11838  398 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
399 
by simp 

400 

401 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

402 
by simp 

403 

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404 
lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  405 
by (cases p) simp 
406 

26358
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changeset

407 
lemmas surjective_pairing = pair_collapse [symmetric] 
11838  408 

37166  409 
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
410 
by (cases s, cases t) simp 

411 

412 
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 

413 
by (simp add: Pair_fst_snd_eq) 

414 

415 
lemma split_conv [simp, code]: "split f (a, b) = f a b" 

37591  416 
by (fact prod.cases) 
37166  417 

418 
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 

419 
by (rule split_conv [THEN iffD2]) 

420 

421 
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 

422 
by (rule split_conv [THEN iffD1]) 

423 

424 
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 

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diff
changeset

425 
by (simp add: fun_eq_iff split: prod.split) 
37166  426 

427 
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 

428 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} 

39302
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nipkow
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39272
diff
changeset

429 
by (simp add: fun_eq_iff split: prod.split) 
37166  430 

431 
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 

432 
by (cases x) simp 

433 

434 
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 

435 
by (cases p) simp 

436 

437 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

37591  438 
by (simp add: prod_case_unfold) 
37166  439 

440 
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" 

441 
 {* Prevents simplification of @{term c}: much faster *} 

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442 
by (fact prod.weak_case_cong) 
37166  443 

444 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

445 
by (simp add: split_eta) 

446 

11838  447 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 
11820
015a82d4ee96
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changeset

448 
proof 
015a82d4ee96
proper proof of split_paired_all (presently unused);
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changeset

449 
fix a b 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
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changeset

450 
assume "!!x. PROP P x" 
19535  451 
then show "PROP P (a, b)" . 
11820
015a82d4ee96
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wenzelm
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diff
changeset

452 
next 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
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diff
changeset

453 
fix x 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
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diff
changeset

454 
assume "!!a b. PROP P (a, b)" 
19535  455 
from `PROP P (fst x, snd x)` show "PROP P x" by simp 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
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11777
diff
changeset

456 
qed 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

457 

11838  458 
text {* 
459 
The rule @{thm [source] split_paired_all} does not work with the 

460 
Simplifier because it also affects premises in congrence rules, 

461 
where this can lead to premises of the form @{text "!!a b. ... = 

462 
?P(a, b)"} which cannot be solved by reflexivity. 

463 
*} 

464 

26358
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changeset

465 
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

466 

26480  467 
ML {* 
11838  468 
(* replace parameters of product type by individual component parameters *) 
469 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

470 
local (* filtering with exists_paired_all is an essential optimization *) 

16121  471 
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = 
11838  472 
can HOLogic.dest_prodT T orelse exists_paired_all t 
473 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

474 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

475 
 exists_paired_all _ = false; 

476 
val ss = HOL_basic_ss 

26340  477 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
11838  478 
addsimprocs [unit_eq_proc]; 
479 
in 

480 
val split_all_tac = SUBGOAL (fn (t, i) => 

481 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

482 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

483 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

484 
fun split_all th = 

26340  485 
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; 
11838  486 
end; 
26340  487 
*} 
11838  488 

26340  489 
declaration {* fn _ => 
490 
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) 

16121  491 
*} 
11838  492 

493 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

494 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

495 
by fast 

496 

26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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diff
changeset

497 
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

498 
by fast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

499 

11838  500 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 
501 
 {* Can't be added to simpset: loops! *} 

26358
d6a508c16908
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haftmann
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26340
diff
changeset

502 
by (simp add: split_eta) 
11838  503 

504 
text {* 

505 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

506 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

507 
and using @{thm [source] cond_split_eta} directly would render some 

508 
existing proofs very inefficient; similarly for @{text 

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

509 
split_beta}. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

510 
*} 
11838  511 

26480  512 
ML {* 
11838  513 
local 
35364  514 
val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta}; 
515 
fun Pair_pat k 0 (Bound m) = (m = k) 

516 
 Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = 

517 
i > 0 andalso m = k + i andalso Pair_pat k (i  1) t 

518 
 Pair_pat _ _ _ = false; 

519 
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t 

520 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

521 
 no_args k i (Bound m) = m < k orelse m > k + i 

522 
 no_args _ _ _ = true; 

523 
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE 

37591  524 
 split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t 
35364  525 
 split_pat tp i _ = NONE; 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
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19656
diff
changeset

526 
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] 
35364  527 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) 
18328  528 
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); 
11838  529 

35364  530 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t 
531 
 beta_term_pat k i (t $ u) = 

532 
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) 

533 
 beta_term_pat k i t = no_args k i t; 

534 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

535 
 eta_term_pat _ _ _ = false; 

11838  536 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 
35364  537 
 subst arg k i (t $ u) = 
538 
if Pair_pat k i (t $ u) then incr_boundvars k arg 

539 
else (subst arg k i t $ subst arg k i u) 

540 
 subst arg k i t = t; 

37591  541 
fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) = 
11838  542 
(case split_pat beta_term_pat 1 t of 
35364  543 
SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f)) 
15531  544 
 NONE => NONE) 
35364  545 
 beta_proc _ _ = NONE; 
37591  546 
fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = 
11838  547 
(case split_pat eta_term_pat 1 t of 
35364  548 
SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) 
15531  549 
 NONE => NONE) 
35364  550 
 eta_proc _ _ = NONE; 
11838  551 
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

552 
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

553 
val split_eta_proc = Simplifier.simproc_global @{theory} "split_eta" ["split f"] (K eta_proc); 
11838  554 
end; 
555 

556 
Addsimprocs [split_beta_proc, split_eta_proc]; 

557 
*} 

558 

26798
a9134a089106
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berghofe
parents:
26588
diff
changeset

559 
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" 
11838  560 
by (subst surjective_pairing, rule split_conv) 
561 

35828
46cfc4b8112e
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blanchet
parents:
35427
diff
changeset

562 
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) > R(c x y))" 
11838  563 
 {* For use with @{text split} and the Simplifier. *} 
15481  564 
by (insert surj_pair [of p], clarify, simp) 
11838  565 

566 
text {* 

567 
@{thm [source] split_split} could be declared as @{text "[split]"} 

568 
done after the Splitter has been speeded up significantly; 

569 
precompute the constants involved and don't do anything unless the 

570 
current goal contains one of those constants. 

571 
*} 

572 

35828
46cfc4b8112e
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blanchet
parents:
35427
diff
changeset

573 
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 
14208  574 
by (subst split_split, simp) 
11838  575 

576 
text {* 

577 
\medskip @{term split} used as a logical connective or set former. 

578 

579 
\medskip These rules are for use with @{text blast}; could instead 

40929
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40702
diff
changeset

580 
call @{text simp} using @{thm [source] prod.split} as rewrite. *} 
11838  581 

582 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

583 
apply (simp only: split_tupled_all) 

584 
apply (simp (no_asm_simp)) 

585 
done 

586 

587 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

588 
apply (simp only: split_tupled_all) 

589 
apply (simp (no_asm_simp)) 

590 
done 

591 

592 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

37591  593 
by (induct p) auto 
11838  594 

595 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

37591  596 
by (induct p) auto 
11838  597 

598 
lemma splitE2: 

599 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

600 
proof  

601 
assume q: "Q (split P z)" 

602 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

603 
show R 

604 
apply (rule r surjective_pairing)+ 

605 
apply (rule split_beta [THEN subst], rule q) 

606 
done 

607 
qed 

608 

609 
lemma splitD': "split R (a,b) c ==> R a b c" 

610 
by simp 

611 

612 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

613 
by simp 

614 

615 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  616 
by (simp only: split_tupled_all, simp) 
11838  617 

18372  618 
lemma mem_splitE: 
37166  619 
assumes major: "z \<in> split c p" 
620 
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" 

18372  621 
shows Q 
37591  622 
by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+ 
11838  623 

624 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

625 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

626 

26340  627 
ML {* 
11838  628 
local (* filtering with exists_p_split is an essential optimization *) 
37591  629 
fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true 
11838  630 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
631 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

632 
 exists_p_split _ = false; 

35364  633 
val ss = HOL_basic_ss addsimps @{thms split_conv}; 
11838  634 
in 
635 
val split_conv_tac = SUBGOAL (fn (t, i) => 

636 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

637 
end; 

26340  638 
*} 
639 

11838  640 
(* This prevents applications of splitE for already splitted arguments leading 
641 
to quite timeconsuming computations (in particular for nested tuples) *) 

26340  642 
declaration {* fn _ => 
643 
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) 

16121  644 
*} 
11838  645 

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

646 
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  647 
by (rule ext) fast 
11838  648 

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

649 
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  650 
by (rule ext) fast 
11838  651 

652 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

653 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

18372  654 
by (rule ext) blast 
11838  655 

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

656 
(* 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

657 
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

658 
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

659 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

660 
lemma split_comp_eq: 
20415  661 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
662 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  663 
by (rule ext) auto 
14101  664 

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

665 
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

666 
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

667 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

668 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

669 

11838  670 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
671 
by blast 

672 

673 
(* 

674 
the following would be slightly more general, 

675 
but cannot be used as rewrite rule: 

676 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

677 
### ?y = .x 

678 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  679 
by (rtac some_equality 1) 
680 
by ( Simp_tac 1) 

681 
by (split_all_tac 1) 

682 
by (Asm_full_simp_tac 1) 

11838  683 
qed "The_split_eq"; 
684 
*) 

685 

686 
text {* 

687 
Setup of internal @{text split_rule}. 

688 
*} 

689 

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

690 
lemmas prod_caseI = prod.cases [THEN iffD2, standard] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

691 

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

692 
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

693 
by (fact splitI2) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

694 

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

695 
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

696 
by (fact splitI2') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

697 

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

698 
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

699 
by (fact splitE) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

700 

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

701 
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

702 
by (fact splitE') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

703 

37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

704 
declare prod_caseI [intro!] 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

705 

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

706 
lemma prod_case_beta: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

707 
"prod_case f p = f (fst p) (snd p)" 
37591  708 
by (fact split_beta) 
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

709 

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

710 
lemma prod_cases3 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

711 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

712 
by (cases y, case_tac b) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

713 

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

714 
lemma prod_induct3 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

715 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

716 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

717 

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

718 
lemma prod_cases4 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

719 
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

720 
by (cases y, case_tac c) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

721 

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

722 
lemma prod_induct4 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

723 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

724 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

725 

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

726 
lemma prod_cases5 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

727 
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

728 
by (cases y, case_tac d) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

729 

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

730 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

731 
"(!!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

732 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

733 

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

734 
lemma prod_cases6 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

735 
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

736 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

737 

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

738 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

739 
"(!!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

740 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

741 

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

742 
lemma prod_cases7 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

743 
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

744 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

745 

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

746 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

747 
"(!!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

748 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

749 

37166  750 
lemma split_def: 
751 
"split = (\<lambda>c p. c (fst p) (snd p))" 

37591  752 
by (fact prod_case_unfold) 
37166  753 

754 
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 

755 
"internal_split == split" 

756 

757 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

758 
by (simp only: internal_split_def split_conv) 

759 

760 
use "Tools/split_rule.ML" 

761 
setup Split_Rule.setup 

762 

763 
hide_const internal_split 

764 

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

765 

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

766 
subsubsection {* Derived operations *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

767 

37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset

768 
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

769 
"curry = (\<lambda>c x y. c (x, y))" 
37166  770 

771 
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" 

772 
by (simp add: curry_def) 

773 

774 
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 

775 
by (simp add: curry_def) 

776 

777 
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 

778 
by (simp add: curry_def) 

779 

780 
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 

781 
by (simp add: curry_def) 

782 

783 
lemma curry_split [simp]: "curry (split f) = f" 

784 
by (simp add: curry_def split_def) 

785 

786 
lemma split_curry [simp]: "split (curry f) = f" 

787 
by (simp add: curry_def split_def) 

788 

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

789 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

790 
The compositionuncurry combinator. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

792 

37751  793 
notation fcomp (infixl "\<circ>>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

794 

37751  795 
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where 
796 
"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

797 

37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

798 
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

799 
by (simp add: fun_eq_iff scomp_def prod_case_unfold) 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

800 

37751  801 
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)" 
802 
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

803 

37751  804 
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

805 
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

806 

37751  807 
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

808 
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

809 

37751  810 
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

811 
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

812 

37751  813 
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

814 
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

815 

37751  816 
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

817 
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

818 

31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

819 
code_const scomp 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

820 
(Eval infixl 3 "#>") 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

821 

37751  822 
no_notation fcomp (infixl "\<circ>>" 60) 
823 
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) 

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

824 

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

825 
text {* 
40607  826 
@{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

827 
functions. 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

828 
*} 
21195  829 

40607  830 
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
831 
"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

832 

40607  833 
lemma map_pair_simp [simp, code]: 
834 
"map_pair f g (a, b) = (f a, g b)" 

835 
by (simp add: map_pair_def) 

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

836 

41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41372
diff
changeset

837 
enriched_type map_pair: map_pair 
41372  838 
by (auto simp add: split_paired_all intro: ext) 
37278  839 

40607  840 
lemma fst_map_pair [simp]: 
841 
"fst (map_pair f g x) = f (fst x)" 

842 
by (cases x) simp_all 

37278  843 

40607  844 
lemma snd_prod_fun [simp]: 
845 
"snd (map_pair f g x) = g (snd x)" 

846 
by (cases x) simp_all 

37278  847 

40607  848 
lemma fst_comp_map_pair [simp]: 
849 
"fst \<circ> map_pair f g = f \<circ> fst" 

850 
by (rule ext) simp_all 

37278  851 

40607  852 
lemma snd_comp_map_pair [simp]: 
853 
"snd \<circ> map_pair f g = g \<circ> snd" 

854 
by (rule ext) simp_all 

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

855 

40607  856 
lemma map_pair_compose: 
857 
"map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)" 

858 
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

859 

40607  860 
lemma map_pair_ident [simp]: 
861 
"map_pair (%x. x) (%y. y) = (%z. z)" 

862 
by (rule ext) (simp add: map_pair.identity) 

863 

864 
lemma map_pair_imageI [intro]: 

865 
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R" 

866 
by (rule image_eqI) simp_all 

21195  867 

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

868 
lemma prod_fun_imageE [elim!]: 
40607  869 
assumes major: "c \<in> map_pair f g ` R" 
870 
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

871 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

872 
apply (rule major [THEN imageE]) 
37166  873 
apply (case_tac x) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

874 
apply (rule cases) 
40607  875 
apply simp_all 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

876 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

877 

37166  878 
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where 
40607  879 
"apfst f = map_pair f id" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

880 

37166  881 
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where 
40607  882 
"apsnd f = map_pair id f" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

883 

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

884 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

885 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

886 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

887 

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

888 
lemma apsnd_conv [simp, code]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

889 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

890 
by (simp add: apsnd_def) 
21195  891 

33594  892 
lemma fst_apfst [simp]: 
893 
"fst (apfst f x) = f (fst x)" 

894 
by (cases x) simp 

895 

896 
lemma fst_apsnd [simp]: 

897 
"fst (apsnd f x) = fst x" 

898 
by (cases x) simp 

899 

900 
lemma snd_apfst [simp]: 

901 
"snd (apfst f x) = snd x" 

902 
by (cases x) simp 

903 

904 
lemma snd_apsnd [simp]: 

905 
"snd (apsnd f x) = f (snd x)" 

906 
by (cases x) simp 

907 

908 
lemma apfst_compose: 

909 
"apfst f (apfst g x) = apfst (f \<circ> g) x" 

910 
by (cases x) simp 

911 

912 
lemma apsnd_compose: 

913 
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" 

914 
by (cases x) simp 

915 

916 
lemma apfst_apsnd [simp]: 

917 
"apfst f (apsnd g x) = (f (fst x), g (snd x))" 

918 
by (cases x) simp 

919 

920 
lemma apsnd_apfst [simp]: 

921 
"apsnd f (apfst g x) = (g (fst x), f (snd x))" 

922 
by (cases x) simp 

923 

924 
lemma apfst_id [simp] : 

925 
"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

926 
by (simp add: fun_eq_iff) 
33594  927 

928 
lemma apsnd_id [simp] : 

929 
"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

930 
by (simp add: fun_eq_iff) 
33594  931 

932 
lemma apfst_eq_conv [simp]: 

933 
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" 

934 
by (cases x) simp 

935 

936 
lemma apsnd_eq_conv [simp]: 

937 
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" 

938 
by (cases x) simp 

939 

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

940 
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

941 
"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

942 
by simp 
21195  943 

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

944 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

945 
Disjoint union of a family of sets  Sigma. 
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 

40607  948 
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

949 
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

950 

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

951 
abbreviation 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

952 
Times :: "['a set, 'b set] => ('a * 'b) set" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

953 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

954 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

955 

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

956 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

957 
Times (infixr "\<times>" 80) 
15394  958 

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

959 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

960 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

961 

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

962 
syntax 
35115  963 
"_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

964 
translations 
35115  965 
"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

966 

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

967 
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

968 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

969 

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

970 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

971 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

972 
!!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

973 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

974 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

975 
by (unfold Sigma_def) blast 
20588  976 

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

977 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

978 
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

979 
eigenvariables. 
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 

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

982 
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

983 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

984 

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

985 
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

986 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

987 

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

988 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

989 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

990 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

991 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

992 
by blast 
20588  993 

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

994 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

995 
"\<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

996 
\<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

997 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

998 

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

999 
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

1000 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1001 

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

1002 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1003 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1004 

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

1005 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1006 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1007 

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

1008 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1009 
by auto 
21908  1010 

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

1011 
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

1012 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1013 

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

1014 
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

1015 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1016 

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

1017 
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

1018 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1019 

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

1020 
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

1021 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1022 

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

1023 
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

1024 
by (blast elim: equalityE) 
20588  1025 

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

1026 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1027 
"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

1028 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1029 

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

1030 
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

1031 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1032 

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

1033 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1034 
"(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

1035 
 {* Suggested by Pierre Chartier *} 
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_Ball_Sigma [simp,no_atp]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1039 
"(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

1040 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1041 

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

1042 
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

1043 
"(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

1044 
by blast 
21908  1045 

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

1046 
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

1047 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1048 

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

1049 
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

1050 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1051 

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

1052 
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

1053 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1054 

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

1055 
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

1056 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1057 

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

1058 
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

1059 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1060 

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

1061 
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

1062 
by blast 
21908  1063 

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

1064 
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

1065 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1066 

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

1067 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1068 
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

1069 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1070 
*} 
21908  1071 

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

1072 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
28719  1073 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1074 

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

1075 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
28719  1076 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1077 

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

1078 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
28719  1079 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1080 

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

1081 
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

1082 
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

1083 

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 
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

1085 
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

1086 

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 
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

1088 
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

1089 

28719  1090 
lemma insert_times_insert[simp]: 
1091 
"insert a A \<times> insert b B = 

1092 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

1093 
by blast 

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

1094 

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

1095 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
37166  1096 
by (auto, case_tac "f x", auto) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1097 

35822  1098 
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

1099 
"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

1100 
by (auto intro!: inj_onI) 
35822  1101 

1102 
lemma swap_product: 

1103 
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" 

1104 
by (simp add: split_def image_def) blast 

1105 

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

1106 
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

1107 
"(\<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

1108 
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

1109 
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

1110 
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

1111 
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

1112 
qed simp_all 
35822  1113 

40607  1114 
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *} 
1115 

1116 
lemma map_pair_inj_on: 

1117 
assumes "inj_on f A" and "inj_on g B" 

1118 
shows "inj_on (map_pair f g) (A \<times> B)" 

1119 
proof (rule inj_onI) 

1120 
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" 

1121 
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto 

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

1123 
assume "map_pair f g x = map_pair f g y" 

1124 
hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto) 

1125 
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) 

1126 
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` 

1127 
have "fst x = fst y" by (auto dest:dest:inj_onD) 

1128 
moreover from `map_pair f g x = map_pair f g y` 

1129 
have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto) 

1130 
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) 

1131 
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` 

1132 
have "snd x = snd y" by (auto dest:dest:inj_onD) 

1133 
ultimately show "x = y" by(rule prod_eqI) 

1134 
qed 

1135 

1136 
lemma map_pair_surj: 

40702  1137 
fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd" 
40607  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
haftmann
parents:
37387
diff
changeset

1179 
setup Inductive_Codegen.setup 
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset

1180 

31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset

1181 
use "Tools/inductive_set.ML" 
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset

1182 
setup Inductive_Set.setup 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

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 *} 

1200 

10213  1201 
end 