author  haftmann 
Fri, 28 May 2010 13:37:28 +0200  
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parent 37136  e0c9d3e49e15 
child 37278  307845cc7f51 
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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header {* Cartesian products *} 
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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_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]: "eq_class.eq False P \<longleftrightarrow> \<not> P" 
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and [code]: "eq_class.eq True P \<longleftrightarrow> P" 

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and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 

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and [code]: "eq_class.eq P True \<longleftrightarrow> P" 

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and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True" 
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by (simp_all add: eq) 
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code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" 
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(Haskell infixl 4 "==") 
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code_instance bool :: eq 
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(Haskell ) 
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subsection {* The @{text unit} type *} 
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typedef unit = "{True}" 

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proof 

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show "True : ?unit" .. 
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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: unit_def Unity_def) 
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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 @{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; 
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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" 
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by (rule ext) simp 
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instantiation unit :: default 
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begin 

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definition "default = ()" 

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instance .. 

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end 

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"eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq 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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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 :: eq 
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(Haskell ) 
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code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" 
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(Haskell infixl 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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subsection {* The product type *} 
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37166  126 
subsubsection {* Type definition *} 
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128 
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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global 

132 

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typedef (Prod) 

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('a, 'b) "*" (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  140 

35427  141 
type_notation (xsymbols) 
142 
"*" ("(_ \<times>/ _)" [21, 20] 20) 

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type_notation (HTML output) 

144 
"*" ("(_ \<times>/ _)" [21, 20] 20) 

10213  145 

146 
consts 

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Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" 
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local 
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19535  151 
defs 
152 
Pair_def: "Pair a b == Abs_Prod (Pair_Rep a b)" 

37166  153 

154 
rep_datatype (prod) Pair proof  

155 
fix P :: "'a \<times> 'b \<Rightarrow> bool" and p 

156 
assume "\<And>a b. P (Pair a b)" 

157 
then show "P p" by (cases p) (auto simp add: Prod_def Pair_def Pair_Rep_def) 

158 
next 

159 
fix a c :: 'a and b d :: 'b 

160 
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" 

161 
by (auto simp add: Pair_Rep_def expand_fun_eq) 

162 
moreover have "Pair_Rep a b \<in> Prod" and "Pair_Rep c d \<in> Prod" 

163 
by (auto simp add: Prod_def) 

164 
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" 

165 
by (simp add: Pair_def Abs_Prod_inject) 

166 
qed 

167 

168 

169 
subsubsection {* Tuple syntax *} 

170 

171 
global consts 

172 
split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" 

173 

174 
local defs 

175 
split_prod_case: "split == prod_case" 

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11777  177 
text {* 
178 
Patterns  extends predefined type @{typ pttrn} used in 

179 
abstractions. 

180 
*} 

10213  181 

182 
nonterminals 

183 
tuple_args patterns 

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185 
syntax 

186 
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") 

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

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"_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  192 

193 
translations 

35115  194 
"(x, y)" == "CONST Pair x y" 
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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 
35115  196 
"%(x, y, zs). b" == "CONST split (%x (y, zs). b)" 
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"%(x, y). b" == "CONST split (%x y. b)" 

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

37166  199 
 {* The last rule accommodates tuples in `case C ... (x,y) ... => ...' 
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The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *} 

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(*reconstruct pattern from (nested) splits, avoiding etacontraction of body; 
203 
works best with enclosing "let", if "let" does not avoid etacontraction*) 

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print_translation {* 
35115  205 
let 
206 
fun split_tr' [Abs (x, T, t as (Abs abs))] = 

207 
(* split (%x y. t) => %(x,y) t *) 

208 
let 

209 
val (y, t') = atomic_abs_tr' abs; 

210 
val (x', t'') = atomic_abs_tr' (x, T, t'); 

211 
in 

212 
Syntax.const @{syntax_const "_abs"} $ 

213 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

214 
end 

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 split_tr' [Abs (x, T, (s as Const (@{const_syntax split}, _) $ t))] = 

216 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 

217 
let 

218 
val Const (@{syntax_const "_abs"}, _) $ 

219 
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; 

220 
val (x', t'') = atomic_abs_tr' (x, T, t'); 

221 
in 

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Syntax.const @{syntax_const "_abs"} $ 

223 
(Syntax.const @{syntax_const "_pattern"} $ x' $ 

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

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end 

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 split_tr' [Const (@{const_syntax split}, _) $ t] = 

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(* 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 *) 

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

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

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

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Syntax.const @{syntax_const "_abs"} $ 

233 
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t 

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end 

235 
 split_tr' _ = raise Match; 

236 
in [(@{const_syntax split}, split_tr')] end 

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*} 
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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 
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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 
35115  245 
Const (@{const_syntax split}, _) => raise Match 
246 
 _ => 

247 
let 

248 
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

249 
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); 

250 
val (x', t'') = atomic_abs_tr' (x, xT, t'); 

251 
in 

252 
Syntax.const @{syntax_const "_abs"} $ 

253 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

254 
end) 

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 split_guess_names_tr' _ T [t] = 
35115  256 
(case head_of t of 
257 
Const (@{const_syntax split}, _) => raise Match 

258 
 _ => 

259 
let 

260 
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

261 
val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); 

262 
val (x', t'') = atomic_abs_tr' ("x", xT, t'); 

263 
in 

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Syntax.const @{syntax_const "_abs"} $ 

265 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

266 
end) 

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 split_guess_names_tr' _ _ _ = raise Match; 
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in [(@{const_syntax split}, split_guess_names_tr')] end 
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*} 
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37166  272 
subsubsection {* Code generator setup *} 
273 

274 
lemma split_case_cert: 

275 
assumes "CASE \<equiv> split f" 

276 
shows "CASE (a, b) \<equiv> f a b" 

277 
using assms by (simp add: split_prod_case) 

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279 
setup {* 

280 
Code.add_case @{thm split_case_cert} 

281 
*} 

282 

283 
code_type * 

284 
(SML infix 2 "*") 

285 
(OCaml infix 2 "*") 

286 
(Haskell "!((_),/ (_))") 

287 
(Scala "((_),/ (_))") 

288 

289 
code_const Pair 

290 
(SML "!((_),/ (_))") 

291 
(OCaml "!((_),/ (_))") 

292 
(Haskell "!((_),/ (_))") 

293 
(Scala "!((_),/ (_))") 

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295 
code_instance * :: eq 

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(Haskell ) 

297 

298 
code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 

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(Haskell infixl 4 "==") 

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301 
types_code 

302 
"*" ("(_ */ _)") 

303 
attach (term_of) {* 

304 
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; 

305 
*} 

306 
attach (test) {* 

307 
fun gen_id_42 aG aT bG bT i = 

308 
let 

309 
val (x, t) = aG i; 

310 
val (y, u) = bG i 

311 
in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; 

312 
*} 

313 

314 
consts_code 

315 
"Pair" ("(_,/ _)") 

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317 
setup {* 

318 
let 

319 

320 
fun strip_abs_split 0 t = ([], t) 

321 
 strip_abs_split i (Abs (s, T, t)) = 

322 
let 

323 
val s' = Codegen.new_name t s; 

324 
val v = Free (s', T) 

325 
in apfst (cons v) (strip_abs_split (i1) (subst_bound (v, t))) end 

326 
 strip_abs_split i (u as Const (@{const_name split}, _) $ t) = 

327 
(case strip_abs_split (i+1) t of 

328 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 

329 
 _ => ([], u)) 

330 
 strip_abs_split i t = 

331 
strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0)); 

332 

333 
fun let_codegen thy defs dep thyname brack t gr = 

334 
(case strip_comb t of 

335 
(t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) => 

336 
let 

337 
fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) = 

338 
(case strip_abs_split 1 u of 

339 
([p], u') => apfst (cons (p, t)) (dest_let u') 

340 
 _ => ([], l)) 

341 
 dest_let t = ([], t); 

342 
fun mk_code (l, r) gr = 

343 
let 

344 
val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr; 

345 
val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1; 

346 
in ((pl, pr), gr2) end 

347 
in case dest_let (t1 $ t2 $ t3) of 

348 
([], _) => NONE 

349 
 (ps, u) => 

350 
let 

351 
val (qs, gr1) = fold_map mk_code ps gr; 

352 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 

353 
val (pargs, gr3) = fold_map 

354 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 

355 
in 

356 
SOME (Codegen.mk_app brack 

357 
(Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat 

358 
(separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) => 

359 
[Pretty.block [Codegen.str "val ", pl, Codegen.str " =", 

360 
Pretty.brk 1, pr]]) qs))), 

361 
Pretty.brk 1, Codegen.str "in ", pu, 

362 
Pretty.brk 1, Codegen.str "end"])) pargs, gr3) 

363 
end 

364 
end 

365 
 _ => NONE); 

366 

367 
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of 

368 
(t1 as Const (@{const_name split}, _), t2 :: ts) => 

369 
let 

370 
val ([p], u) = strip_abs_split 1 (t1 $ t2); 

371 
val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr; 

372 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 

373 
val (pargs, gr3) = fold_map 

374 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 

375 
in 

376 
SOME (Codegen.mk_app brack 

377 
(Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>", 

378 
Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2) 

379 
end 

380 
 _ => NONE); 

381 

382 
in 

383 

384 
Codegen.add_codegen "let_codegen" let_codegen 

385 
#> Codegen.add_codegen "split_codegen" split_codegen 

386 

387 
end 

388 
*} 

389 

390 

391 
subsubsection {* Fundamental operations and properties *} 

11838  392 

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393 
lemma surj_pair [simp]: "EX x y. p = (x, y)" 
37166  394 
by (cases p) simp 
10213  395 

37166  396 
global consts 
397 
fst :: "'a \<times> 'b \<Rightarrow> 'a" 

398 
snd :: "'a \<times> 'b \<Rightarrow> 'b" 

11838  399 

37166  400 
local defs 
401 
fst_def: "fst p == case p of (a, b) \<Rightarrow> a" 

402 
snd_def: "snd p == case p of (a, b) \<Rightarrow> b" 

11838  403 

22886  404 
lemma fst_conv [simp, code]: "fst (a, b) = a" 
37166  405 
unfolding fst_def by simp 
11838  406 

22886  407 
lemma snd_conv [simp, code]: "snd (a, b) = b" 
37166  408 
unfolding snd_def by simp 
11025
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oheimb
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changeset

409 

37166  410 
code_const fst and snd 
411 
(Haskell "fst" and "snd") 

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412 

37166  413 
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))" 
414 
by (simp add: expand_fun_eq split: prod.split) 

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415 

11838  416 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
417 
by simp 

418 

419 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

420 
by simp 

421 

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422 
lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  423 
by (cases p) simp 
424 

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425 
lemmas surjective_pairing = pair_collapse [symmetric] 
11838  426 

37166  427 
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
428 
by (cases s, cases t) simp 

429 

430 
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 

431 
by (simp add: Pair_fst_snd_eq) 

432 

433 
lemma split_conv [simp, code]: "split f (a, b) = f a b" 

434 
by (simp add: split_prod_case) 

435 

436 
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 

437 
by (rule split_conv [THEN iffD2]) 

438 

439 
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 

440 
by (rule split_conv [THEN iffD1]) 

441 

442 
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 

443 
by (simp add: split_prod_case expand_fun_eq split: prod.split) 

444 

445 
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 

446 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} 

447 
by (simp add: split_prod_case expand_fun_eq split: prod.split) 

448 

449 
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 

450 
by (cases x) simp 

451 

452 
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 

453 
by (cases p) simp 

454 

455 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

456 
by (simp add: split_prod_case prod_case_unfold) 

457 

458 
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" 

459 
 {* Prevents simplification of @{term c}: much faster *} 

460 
by (erule arg_cong) 

461 

462 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

463 
by (simp add: split_eta) 

464 

11838  465 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 
11820
015a82d4ee96
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wenzelm
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changeset

466 
proof 
015a82d4ee96
proper proof of split_paired_all (presently unused);
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parents:
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diff
changeset

467 
fix a b 
015a82d4ee96
proper proof of split_paired_all (presently unused);
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diff
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468 
assume "!!x. PROP P x" 
19535  469 
then show "PROP P (a, b)" . 
11820
015a82d4ee96
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parents:
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changeset

470 
next 
015a82d4ee96
proper proof of split_paired_all (presently unused);
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changeset

471 
fix x 
015a82d4ee96
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472 
assume "!!a b. PROP P (a, b)" 
19535  473 
from `PROP P (fst x, snd x)` show "PROP P x" by simp 
11820
015a82d4ee96
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wenzelm
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changeset

474 
qed 
015a82d4ee96
proper proof of split_paired_all (presently unused);
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475 

11838  476 
text {* 
477 
The rule @{thm [source] split_paired_all} does not work with the 

478 
Simplifier because it also affects premises in congrence rules, 

479 
where this can lead to premises of the form @{text "!!a b. ... = 

480 
?P(a, b)"} which cannot be solved by reflexivity. 

481 
*} 

482 

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483 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
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484 

26480  485 
ML {* 
11838  486 
(* replace parameters of product type by individual component parameters *) 
487 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

488 
local (* filtering with exists_paired_all is an essential optimization *) 

16121  489 
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = 
11838  490 
can HOLogic.dest_prodT T orelse exists_paired_all t 
491 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

492 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

493 
 exists_paired_all _ = false; 

494 
val ss = HOL_basic_ss 

26340  495 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
11838  496 
addsimprocs [unit_eq_proc]; 
497 
in 

498 
val split_all_tac = SUBGOAL (fn (t, i) => 

499 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

500 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

501 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

502 
fun split_all th = 

26340  503 
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; 
11838  504 
end; 
26340  505 
*} 
11838  506 

26340  507 
declaration {* fn _ => 
508 
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) 

16121  509 
*} 
11838  510 

511 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

512 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

513 
by fast 

514 

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515 
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 
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516 
by fast 
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changeset

517 

11838  518 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 
519 
 {* Can't be added to simpset: loops! *} 

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diff
changeset

520 
by (simp add: split_eta) 
11838  521 

522 
text {* 

523 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

524 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

525 
and using @{thm [source] cond_split_eta} directly would render some 

526 
existing proofs very inefficient; similarly for @{text 

26358
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diff
changeset

527 
split_beta}. 
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diff
changeset

528 
*} 
11838  529 

26480  530 
ML {* 
11838  531 
local 
35364  532 
val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta}; 
533 
fun Pair_pat k 0 (Bound m) = (m = k) 

534 
 Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = 

535 
i > 0 andalso m = k + i andalso Pair_pat k (i  1) t 

536 
 Pair_pat _ _ _ = false; 

537 
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t 

538 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

539 
 no_args k i (Bound m) = m < k orelse m > k + i 

540 
 no_args _ _ _ = true; 

541 
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE 

542 
 split_pat tp i (Const (@{const_name split}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t 

543 
 split_pat tp i _ = NONE; 

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

544 
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] 
35364  545 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) 
18328  546 
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); 
11838  547 

35364  548 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t 
549 
 beta_term_pat k i (t $ u) = 

550 
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) 

551 
 beta_term_pat k i t = no_args k i t; 

552 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

553 
 eta_term_pat _ _ _ = false; 

11838  554 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 
35364  555 
 subst arg k i (t $ u) = 
556 
if Pair_pat k i (t $ u) then incr_boundvars k arg 

557 
else (subst arg k i t $ subst arg k i u) 

558 
 subst arg k i t = t; 

559 
fun beta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t) $ arg) = 

11838  560 
(case split_pat beta_term_pat 1 t of 
35364  561 
SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f)) 
15531  562 
 NONE => NONE) 
35364  563 
 beta_proc _ _ = NONE; 
564 
fun eta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t)) = 

11838  565 
(case split_pat eta_term_pat 1 t of 
35364  566 
SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) 
15531  567 
 NONE => NONE) 
35364  568 
 eta_proc _ _ = NONE; 
11838  569 
in 
32010  570 
val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc); 
571 
val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc); 

11838  572 
end; 
573 

574 
Addsimprocs [split_beta_proc, split_eta_proc]; 

575 
*} 

576 

26798
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split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents:
26588
diff
changeset

577 
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" 
11838  578 
by (subst surjective_pairing, rule split_conv) 
579 

35828
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blanchet
parents:
35427
diff
changeset

580 
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) > R(c x y))" 
11838  581 
 {* For use with @{text split} and the Simplifier. *} 
15481  582 
by (insert surj_pair [of p], clarify, simp) 
11838  583 

584 
text {* 

585 
@{thm [source] split_split} could be declared as @{text "[split]"} 

586 
done after the Splitter has been speeded up significantly; 

587 
precompute the constants involved and don't do anything unless the 

588 
current goal contains one of those constants. 

589 
*} 

590 

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diff
changeset

591 
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 
14208  592 
by (subst split_split, simp) 
11838  593 

594 
text {* 

595 
\medskip @{term split} used as a logical connective or set former. 

596 

597 
\medskip These rules are for use with @{text blast}; could instead 

598 
call @{text simp} using @{thm [source] split} as rewrite. *} 

599 

600 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

601 
apply (simp only: split_tupled_all) 

602 
apply (simp (no_asm_simp)) 

603 
done 

604 

605 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

606 
apply (simp only: split_tupled_all) 

607 
apply (simp (no_asm_simp)) 

608 
done 

609 

610 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

37166  611 
by (induct p) (auto simp add: split_prod_case) 
11838  612 

613 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

37166  614 
by (induct p) (auto simp add: split_prod_case) 
11838  615 

616 
lemma splitE2: 

617 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

618 
proof  

619 
assume q: "Q (split P z)" 

620 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

621 
show R 

622 
apply (rule r surjective_pairing)+ 

623 
apply (rule split_beta [THEN subst], rule q) 

624 
done 

625 
qed 

626 

627 
lemma splitD': "split R (a,b) c ==> R a b c" 

628 
by simp 

629 

630 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

631 
by simp 

632 

633 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  634 
by (simp only: split_tupled_all, simp) 
11838  635 

18372  636 
lemma mem_splitE: 
37166  637 
assumes major: "z \<in> split c p" 
638 
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" 

18372  639 
shows Q 
37166  640 
by (rule major [unfolded split_prod_case prod_case_unfold] cases surjective_pairing)+ 
11838  641 

642 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

643 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

644 

26340  645 
ML {* 
11838  646 
local (* filtering with exists_p_split is an essential optimization *) 
35364  647 
fun exists_p_split (Const (@{const_name split},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true 
11838  648 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
649 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

650 
 exists_p_split _ = false; 

35364  651 
val ss = HOL_basic_ss addsimps @{thms split_conv}; 
11838  652 
in 
653 
val split_conv_tac = SUBGOAL (fn (t, i) => 

654 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

655 
end; 

26340  656 
*} 
657 

11838  658 
(* This prevents applications of splitE for already splitted arguments leading 
659 
to quite timeconsuming computations (in particular for nested tuples) *) 

26340  660 
declaration {* fn _ => 
661 
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) 

16121  662 
*} 
11838  663 

35828
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blanchet
parents:
35427
diff
changeset

664 
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  665 
by (rule ext) fast 
11838  666 

35828
46cfc4b8112e
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blanchet
parents:
35427
diff
changeset

667 
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  668 
by (rule ext) fast 
11838  669 

670 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

671 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

18372  672 
by (rule ext) blast 
11838  673 

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

674 
(* 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

675 
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

676 
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

677 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

678 
lemma split_comp_eq: 
20415  679 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
680 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  681 
by (rule ext) auto 
14101  682 

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

683 
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

684 
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

685 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

686 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

687 

11838  688 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
689 
by blast 

690 

691 
(* 

692 
the following would be slightly more general, 

693 
but cannot be used as rewrite rule: 

694 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

695 
### ?y = .x 

696 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  697 
by (rtac some_equality 1) 
698 
by ( Simp_tac 1) 

699 
by (split_all_tac 1) 

700 
by (Asm_full_simp_tac 1) 

11838  701 
qed "The_split_eq"; 
702 
*) 

703 

704 
text {* 

705 
Setup of internal @{text split_rule}. 

706 
*} 

707 

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

708 
lemmas prod_caseI = prod.cases [THEN iffD2, standard] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

709 

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

710 
lemma prod_caseI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> prod_case c p" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

711 
by auto 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

712 

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

713 
lemma prod_caseI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> prod_case c p x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

714 
by (auto simp: split_tupled_all) 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

715 

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

716 
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

717 
by (induct p) auto 
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_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

720 
by (induct p) auto 
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 
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

723 
declare prod_caseE' [elim!] prod_caseE [elim!] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

724 

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

725 
lemma prod_case_split: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

726 
"prod_case = split" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

727 
by (auto simp add: expand_fun_eq) 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

728 

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

729 
lemma prod_case_beta: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

730 
"prod_case f p = f (fst p) (snd p)" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

731 
unfolding prod_case_split split_beta .. 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

732 

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

733 
lemma prod_cases3 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

734 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

735 
by (cases y, case_tac b) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

736 

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

737 
lemma prod_induct3 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

738 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

739 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

740 

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

741 
lemma prod_cases4 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

742 
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

743 
by (cases y, case_tac c) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

744 

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

745 
lemma prod_induct4 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

746 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

747 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

748 

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

749 
lemma prod_cases5 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

750 
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

751 
by (cases y, case_tac d) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

752 

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

753 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

754 
"(!!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

755 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

756 

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

757 
lemma prod_cases6 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

758 
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

759 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

760 

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

761 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

762 
"(!!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

763 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

764 

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

765 
lemma prod_cases7 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

766 
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

767 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

768 

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

769 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

770 
"(!!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

771 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

772 

37166  773 
lemma split_def: 
774 
"split = (\<lambda>c p. c (fst p) (snd p))" 

775 
unfolding split_prod_case prod_case_unfold .. 

776 

777 
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 

778 
"internal_split == split" 

779 

780 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

781 
by (simp only: internal_split_def split_conv) 

782 

783 
use "Tools/split_rule.ML" 

784 
setup Split_Rule.setup 

785 

786 
hide_const internal_split 

787 

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

788 

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

789 
subsubsection {* Derived operations *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

790 

37166  791 
global consts 
792 
curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" 

793 

794 
local defs 

795 
curry_def: "curry == (%c x y. c (Pair x y))" 

796 

797 
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" 

798 
by (simp add: curry_def) 

799 

800 
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 

801 
by (simp add: curry_def) 

802 

803 
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 

804 
by (simp add: curry_def) 

805 

806 
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 

807 
by (simp add: curry_def) 

808 

809 
lemma curry_split [simp]: "curry (split f) = f" 

810 
by (simp add: curry_def split_def) 

811 

812 
lemma split_curry [simp]: "split (curry f) = f" 

813 
by (simp add: curry_def split_def) 

814 

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

815 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

816 
The compositionuncurry combinator. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

817 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

818 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

819 
notation fcomp (infixl "o>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

820 

37166  821 
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60) where 
26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

822 
"f o\<rightarrow> g = (\<lambda>x. split g (f x))" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

823 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

824 
lemma scomp_apply: "(f o\<rightarrow> g) x = split g (f x)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

825 
by (simp add: scomp_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

826 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

827 
lemma Pair_scomp: "Pair x o\<rightarrow> f = f x" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

828 
by (simp add: expand_fun_eq scomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

829 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

830 
lemma scomp_Pair: "x o\<rightarrow> Pair = x" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

831 
by (simp add: expand_fun_eq scomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

832 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

833 
lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

834 
by (simp add: expand_fun_eq split_twice scomp_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

835 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

836 
lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

837 
by (simp add: expand_fun_eq scomp_apply fcomp_def split_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

838 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

839 
lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

840 
by (simp add: expand_fun_eq scomp_apply fcomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

841 

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

842 
code_const scomp 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

843 
(Eval infixl 3 "#>") 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

844 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

845 
no_notation fcomp (infixl "o>" 60) 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

846 
no_notation scomp (infixl "o\<rightarrow>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

847 

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

848 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

849 
@{term prod_fun}  action of the product functor upon 
36664
6302f9ad7047
repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents:
36622
diff
changeset

850 
functions. 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

851 
*} 
21195  852 

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

853 
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
28562  854 
[code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

855 

28562  856 
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

857 
by (simp add: prod_fun_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

858 

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

859 
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

860 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

861 

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

862 
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

863 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

864 

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

865 
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

866 
apply (rule image_eqI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

867 
apply (rule prod_fun [symmetric], assumption) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

868 
done 
21195  869 

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

870 
lemma prod_fun_imageE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

871 
assumes major: "c: (prod_fun f g)`r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

872 
and cases: "!!x y. [ c=(f(x),g(y)); (x,y):r ] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

873 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

874 
apply (rule major [THEN imageE]) 
37166  875 
apply (case_tac x) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

877 
apply (blast intro: prod_fun) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

878 
apply blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

879 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

880 

37166  881 
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where 
882 
"apfst f = prod_fun f id" 

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

883 

37166  884 
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where 
885 
"apsnd f = prod_fun id f" 

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

886 

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

887 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

888 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

889 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

890 

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

891 
lemma apsnd_conv [simp, code]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

892 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

893 
by (simp add: apsnd_def) 
21195  894 

33594  895 
lemma fst_apfst [simp]: 
896 
"fst (apfst f x) = f (fst x)" 

897 
by (cases x) simp 

898 

899 
lemma fst_apsnd [simp]: 

900 
"fst (apsnd f x) = fst x" 

901 
by (cases x) simp 

902 

903 
lemma snd_apfst [simp]: 

904 
"snd (apfst f x) = snd x" 

905 
by (cases x) simp 

906 

907 
lemma snd_apsnd [simp]: 

908 
"snd (apsnd f x) = f (snd x)" 

909 
by (cases x) simp 

910 

911 
lemma apfst_compose: 

912 
"apfst f (apfst g x) = apfst (f \<circ> g) x" 

913 
by (cases x) simp 

914 

915 
lemma apsnd_compose: 

916 
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" 

917 
by (cases x) simp 

918 

919 
lemma apfst_apsnd [simp]: 

920 
"apfst f (apsnd g x) = (f (fst x), g (snd x))" 

921 
by (cases x) simp 

922 

923 
lemma apsnd_apfst [simp]: 

924 
"apsnd f (apfst g x) = (g (fst x), f (snd x))" 

925 
by (cases x) simp 

926 

927 
lemma apfst_id [simp] : 

928 
"apfst id = id" 

929 
by (simp add: expand_fun_eq) 

930 

931 
lemma apsnd_id [simp] : 

932 
"apsnd id = id" 

933 
by (simp add: expand_fun_eq) 

934 

935 
lemma apfst_eq_conv [simp]: 

936 
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" 

937 
by (cases x) simp 

938 

939 
lemma apsnd_eq_conv [simp]: 

940 
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" 

941 
by (cases x) simp 

942 

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

943 
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

944 
"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

945 
by simp 
21195  946 

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

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

948 
Disjoint union of a family of sets  Sigma. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

952 
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

953 

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

954 
abbreviation 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

955 
Times :: "['a set, 'b set] => ('a * 'b) set" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

956 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

957 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

958 

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

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

960 
Times (infixr "\<times>" 80) 
15394  961 

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

962 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

963 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

964 

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

965 
syntax 
35115  966 
"_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

967 
translations 
35115  968 
"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

969 

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

970 
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

971 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

972 

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

973 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

974 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

975 
!!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

976 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

977 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

978 
by (unfold Sigma_def) blast 
20588  979 

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

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

981 
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

982 
eigenvariables. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

984 

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

985 
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

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 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

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

990 

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

991 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

992 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

993 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

994 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

995 
by blast 
20588  996 

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

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

998 
"\<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

999 
\<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

1000 
by auto 
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_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

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_empty1 [simp]: "Sigma {} B = {}" 
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 Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1009 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1010 

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

1011 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1012 
by auto 
21908  1013 

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

1014 
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

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

1018 
by auto 
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 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

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_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

1024 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1025 

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

1026 
lemma 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

1027 
by (blast elim: equalityE) 
20588  1028 

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

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

1030 
"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

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 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

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

1035 

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

1036 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1037 
"(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

1038 
 {* Suggested by Pierre Chartier *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1039 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1040 

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

1041 
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

1042 
"(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

1043 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1044 

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

1045 
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

1046 
"(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

1047 
by blast 
21908  1048 

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

1049 
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

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_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

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_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

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_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

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_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

1062 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1063 

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

1064 
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

1065 
by blast 
21908  1066 

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

1067 
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

1068 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1069 

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

1070 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1071 
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

1072 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1073 
*} 
21908  1074 

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

1075 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (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_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
28719  1079 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1080 

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

1081 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
28719  1082 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1083 

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

1084 
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

1085 
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

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 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

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 

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 
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

1091 
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

1092 

28719  1093 
lemma insert_times_insert[simp]: 
1094 
"insert a A \<times> insert b B = 

1095 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

1096 
by blast 

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

1097 

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

1098 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
37166  1099 
by (auto, case_tac "f x", auto) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1100 

35822  1101 
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

1102 
"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

1103 
by (auto intro!: inj_onI) 
35822  1104 

1105 
lemma swap_product: 

1106 
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" 

1107 
by (simp add: split_def image_def) blast 

1108 

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

1109 
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

1110 
"(\<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

1111 
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

1112 
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

1113 
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

1114 
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

1115 
qed simp_all 
35822  1116 

21908  1117 

37166  1118 
subsection {* Inductively defined sets *} 
15394  1119 

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

1120 
use "Tools/inductive_set.ML" 
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset

1121 
setup Inductive_Set.setup 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1122 

37166  1123 

1124 
subsection {* Legacy theorem bindings and duplicates *} 

1125 

1126 
lemma PairE: 

1127 
obtains x y where "p = (x, y)" 

1128 
by (fact prod.exhaust) 

1129 

1130 
lemma Pair_inject: 

1131 
assumes "(a, b) = (a', b')" 

1132 
and "a = a' ==> b = b' ==> R" 

1133 
shows R 

1134 
using assms by simp 

1135 

1136 
lemmas Pair_eq = prod.inject 

1137 

1138 
lemmas split = split_conv  {* for backwards compatibility *} 

1139 

10213  1140 
end 