author  haftmann 
Tue, 21 Feb 2012 08:15:42 +0100  
changeset 46557  ae926869a311 
parent 46556  2848e36e0348 
child 46630  3abc964cdc45 
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]: "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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lemma If_case_cert: 
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assumes "CASE \<equiv> (\<lambda>b. If b f g)" 
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shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)" 
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using assms by simp_all 
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setup {* 
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Code.add_case @{thm If_case_cert} 
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*} 
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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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by auto 
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definition Unity :: unit ("'(')") 
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where "() = Abs_unit True" 
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lemma unit_eq [no_atp]: "u = ()" 
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by (induct u) (simp add: 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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simproc_setup unit_eq ("x::unit") = {* 
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fn _ => fn _ => fn ct => 

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if HOLogic.is_unit (term_of ct) then NONE 

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else SOME (mk_meta_eq @{thm unit_eq}) 

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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 simproc @{text unit_eq} 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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lemma UNIV_unit [no_atp]: 
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"UNIV = {()}" by auto 
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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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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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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  128 
subsection {* The product type *} 
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37166  130 
subsubsection {* Type definition *} 
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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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45696  135 
definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" 
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137 
typedef (open) ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set" 

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unfolding prod_def by auto 

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type_notation (xsymbols) 
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"prod" ("(_ \<times>/ _)" [21, 20] 20) 
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type_notation (HTML output) 
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"prod" ("(_ \<times>/ _)" [21, 20] 20) 
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where 
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"Pair a b = Abs_prod (Pair_Rep a b)" 
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rep_datatype Pair proof  
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fix P :: "'a \<times> 'b \<Rightarrow> bool" and p 
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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) 
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next 
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fix a c :: 'a and b d :: 'b 

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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) 
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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  166 

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subsubsection {* Tuple syntax *} 

168 

37591  169 
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 
170 
"split \<equiv> prod_case" 

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11777  172 
text {* 
173 
Patterns  extends predefined type @{typ pttrn} used in 

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abstractions. 

175 
*} 

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nonterminal tuple_args and patterns 
10213  178 

179 
syntax 

180 
"_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" ("_,/ _") 
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187 
translations 

35115  188 
"(x, y)" == "CONST Pair x y" 
10213  189 
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 
37591  190 
"%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)" 
191 
"%(x, y). b" == "CONST prod_case (%x y. b)" 

35115  192 
"_abs (CONST Pair x y) t" => "%(x, y). t" 
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 {* 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 *} 

10213  195 

35115  196 
(*reconstruct pattern from (nested) splits, avoiding etacontraction of body; 
197 
works best with enclosing "let", if "let" does not avoid etacontraction*) 

14359  198 
print_translation {* 
35115  199 
let 
200 
fun split_tr' [Abs (x, T, t as (Abs abs))] = 

201 
(* split (%x y. t) => %(x,y) t *) 

202 
let 

42284  203 
val (y, t') = Syntax_Trans.atomic_abs_tr' abs; 
204 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); 

35115  205 
in 
206 
Syntax.const @{syntax_const "_abs"} $ 

207 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

208 
end 

37591  209 
 split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] = 
35115  210 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 
211 
let 

212 
val Const (@{syntax_const "_abs"}, _) $ 

213 
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; 

42284  214 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); 
35115  215 
in 
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Syntax.const @{syntax_const "_abs"} $ 

217 
(Syntax.const @{syntax_const "_pattern"} $ x' $ 

218 
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' 

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end 

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 split_tr' [Const (@{const_syntax prod_case}, _) $ 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 *) 

223 
 split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = 

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

42284  225 
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in 
35115  226 
Syntax.const @{syntax_const "_abs"} $ 
227 
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t 

228 
end 

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 split_tr' _ = raise Match; 

37591  230 
in [(@{const_syntax prod_case}, split_tr')] end 
14359  231 
*} 
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {* 
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let 
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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 
37591  239 
Const (@{const_syntax prod_case}, _) => raise Match 
35115  240 
 _ => 
241 
let 

242 
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

42284  243 
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); 
244 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); 

35115  245 
in 
246 
Syntax.const @{syntax_const "_abs"} $ 

247 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

248 
end) 

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 split_guess_names_tr' T [t] = 
35115  250 
(case head_of t of 
37591  251 
Const (@{const_syntax prod_case}, _) => raise Match 
35115  252 
 _ => 
253 
let 

254 
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

42284  255 
val (y, t') = 
256 
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); 

257 
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); 

35115  258 
in 
259 
Syntax.const @{syntax_const "_abs"} $ 

260 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

261 
end) 

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262 
 split_guess_names_tr' _ _ = raise Match; 
37591  263 
in [(@{const_syntax prod_case}, split_guess_names_tr')] end 
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264 
*} 
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265 

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266 
(* Force etacontraction for terms of the form "Q A (%p. prod_case P p)" 
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267 
where Q is some bounded quantifier or set operator. 
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268 
Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y" 
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269 
whereas we want "Q (x,y):A. P x y". 
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270 
Otherwise prevent etacontraction. 
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271 
*) 
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272 
print_translation {* 
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273 
let 
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274 
fun contract Q f ts = 
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275 
case ts of 
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276 
[A, Abs(_, _, (s as Const (@{const_syntax prod_case},_) $ t) $ Bound 0)] 
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277 
=> if Term.is_dependent t then f ts else Syntax.const Q $ A $ s 
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278 
 _ => f ts; 
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279 
fun contract2 (Q,f) = (Q, contract Q f); 
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280 
val pairs = 
42284  281 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"}, 
282 
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}, 

283 
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, 

284 
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] 

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285 
in map contract2 pairs end 
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286 
*} 
10213  287 

37166  288 
subsubsection {* Code generator setup *} 
289 

37678
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290 
code_type prod 
37166  291 
(SML infix 2 "*") 
292 
(OCaml infix 2 "*") 

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

294 
(Scala "((_),/ (_))") 

295 

296 
code_const Pair 

297 
(SML "!((_),/ (_))") 

298 
(OCaml "!((_),/ (_))") 

299 
(Haskell "!((_),/ (_))") 

300 
(Scala "!((_),/ (_))") 

301 

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302 
code_instance prod :: equal 
37166  303 
(Haskell ) 
304 

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305 
code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 
39272  306 
(Haskell infix 4 "==") 
37166  307 

308 

309 
subsubsection {* Fundamental operations and properties *} 

11838  310 

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311 
lemma surj_pair [simp]: "EX x y. p = (x, y)" 
37166  312 
by (cases p) simp 
10213  313 

37389
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314 
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where 
09467cdfa198
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315 
"fst p = (case p of (a, b) \<Rightarrow> a)" 
11838  316 

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317 
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where 
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318 
"snd p = (case p of (a, b) \<Rightarrow> b)" 
11838  319 

22886  320 
lemma fst_conv [simp, code]: "fst (a, b) = a" 
37166  321 
unfolding fst_def by simp 
11838  322 

22886  323 
lemma snd_conv [simp, code]: "snd (a, b) = b" 
37166  324 
unfolding snd_def by simp 
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325 

37166  326 
code_const fst and snd 
327 
(Haskell "fst" and "snd") 

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328 

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329 
lemma prod_case_unfold [nitpick_unfold]: "prod_case = (%c p. c (fst p) (snd p))" 
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330 
by (simp add: fun_eq_iff split: prod.split) 
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331 

11838  332 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
333 
by simp 

334 

335 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

336 
by simp 

337 

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338 
lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  339 
by (cases p) simp 
340 

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341 
lemmas surjective_pairing = pair_collapse [symmetric] 
11838  342 

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343 
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
37166  344 
by (cases s, cases t) simp 
345 

346 
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 

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347 
by (simp add: prod_eq_iff) 
37166  348 

349 
lemma split_conv [simp, code]: "split f (a, b) = f a b" 

37591  350 
by (fact prod.cases) 
37166  351 

352 
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 

353 
by (rule split_conv [THEN iffD2]) 

354 

355 
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 

356 
by (rule split_conv [THEN iffD1]) 

357 

358 
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 

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359 
by (simp add: fun_eq_iff split: prod.split) 
37166  360 

361 
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 

362 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} 

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363 
by (simp add: fun_eq_iff split: prod.split) 
37166  364 

365 
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 

366 
by (cases x) simp 

367 

368 
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 

369 
by (cases p) simp 

370 

371 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

37591  372 
by (simp add: prod_case_unfold) 
37166  373 

374 
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" 

375 
 {* Prevents simplification of @{term c}: much faster *} 

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376 
by (fact prod.weak_case_cong) 
37166  377 

378 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

379 
by (simp add: split_eta) 

380 

11838  381 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 
11820
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382 
proof 
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383 
fix a b 
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384 
assume "!!x. PROP P x" 
19535  385 
then show "PROP P (a, b)" . 
11820
015a82d4ee96
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386 
next 
015a82d4ee96
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387 
fix x 
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388 
assume "!!a b. PROP P (a, b)" 
19535  389 
from `PROP P (fst x, snd x)` show "PROP P x" by simp 
11820
015a82d4ee96
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390 
qed 
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391 

11838  392 
text {* 
393 
The rule @{thm [source] split_paired_all} does not work with the 

394 
Simplifier because it also affects premises in congrence rules, 

395 
where this can lead to premises of the form @{text "!!a b. ... = 

396 
?P(a, b)"} which cannot be solved by reflexivity. 

397 
*} 

398 

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399 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
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400 

26480  401 
ML {* 
11838  402 
(* replace parameters of product type by individual component parameters *) 
403 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

404 
local (* filtering with exists_paired_all is an essential optimization *) 

16121  405 
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = 
11838  406 
can HOLogic.dest_prodT T orelse exists_paired_all t 
407 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

408 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

409 
 exists_paired_all _ = false; 

410 
val ss = HOL_basic_ss 

26340  411 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
43594  412 
addsimprocs [@{simproc unit_eq}]; 
11838  413 
in 
414 
val split_all_tac = SUBGOAL (fn (t, i) => 

415 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

416 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

417 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

418 
fun split_all th = 

26340  419 
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; 
11838  420 
end; 
26340  421 
*} 
11838  422 

26340  423 
declaration {* fn _ => 
424 
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) 

16121  425 
*} 
11838  426 

427 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

428 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

429 
by fast 

430 

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431 
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 
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432 
by fast 
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433 

11838  434 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 
435 
 {* Can't be added to simpset: loops! *} 

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436 
by (simp add: split_eta) 
11838  437 

438 
text {* 

439 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

440 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

441 
and using @{thm [source] cond_split_eta} directly would render some 

442 
existing proofs very inefficient; similarly for @{text 

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443 
split_beta}. 
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444 
*} 
11838  445 

26480  446 
ML {* 
11838  447 
local 
35364  448 
val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta}; 
449 
fun Pair_pat k 0 (Bound m) = (m = k) 

450 
 Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = 

451 
i > 0 andalso m = k + i andalso Pair_pat k (i  1) t 

452 
 Pair_pat _ _ _ = false; 

453 
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t 

454 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

455 
 no_args k i (Bound m) = m < k orelse m > k + i 

456 
 no_args _ _ _ = true; 

457 
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE 

37591  458 
 split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t 
35364  459 
 split_pat tp i _ = NONE; 
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460 
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] 
35364  461 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) 
18328  462 
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); 
11838  463 

35364  464 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t 
465 
 beta_term_pat k i (t $ u) = 

466 
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) 

467 
 beta_term_pat k i t = no_args k i t; 

468 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

469 
 eta_term_pat _ _ _ = false; 

11838  470 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 
35364  471 
 subst arg k i (t $ u) = 
472 
if Pair_pat k i (t $ u) then incr_boundvars k arg 

473 
else (subst arg k i t $ subst arg k i u) 

474 
 subst arg k i t = t; 

43595  475 
in 
37591  476 
fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) = 
11838  477 
(case split_pat beta_term_pat 1 t of 
35364  478 
SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f)) 
15531  479 
 NONE => NONE) 
35364  480 
 beta_proc _ _ = NONE; 
37591  481 
fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = 
11838  482 
(case split_pat eta_term_pat 1 t of 
35364  483 
SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) 
15531  484 
 NONE => NONE) 
35364  485 
 eta_proc _ _ = NONE; 
11838  486 
end; 
487 
*} 

43595  488 
simproc_setup split_beta ("split f z") = {* fn _ => fn ss => fn ct => beta_proc ss (term_of ct) *} 
489 
simproc_setup split_eta ("split f") = {* fn _ => fn ss => fn ct => eta_proc ss (term_of ct) *} 

11838  490 

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491 
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" 
11838  492 
by (subst surjective_pairing, rule split_conv) 
493 

35828
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494 
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) > R(c x y))" 
11838  495 
 {* For use with @{text split} and the Simplifier. *} 
15481  496 
by (insert surj_pair [of p], clarify, simp) 
11838  497 

498 
text {* 

499 
@{thm [source] split_split} could be declared as @{text "[split]"} 

500 
done after the Splitter has been speeded up significantly; 

501 
precompute the constants involved and don't do anything unless the 

502 
current goal contains one of those constants. 

503 
*} 

504 

35828
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blanchet
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changeset

505 
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 
14208  506 
by (subst split_split, simp) 
11838  507 

508 
text {* 

509 
\medskip @{term split} used as a logical connective or set former. 

510 

511 
\medskip These rules are for use with @{text blast}; could instead 

40929
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512 
call @{text simp} using @{thm [source] prod.split} as rewrite. *} 
11838  513 

514 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

515 
apply (simp only: split_tupled_all) 

516 
apply (simp (no_asm_simp)) 

517 
done 

518 

519 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

520 
apply (simp only: split_tupled_all) 

521 
apply (simp (no_asm_simp)) 

522 
done 

523 

524 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

37591  525 
by (induct p) auto 
11838  526 

527 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

37591  528 
by (induct p) auto 
11838  529 

530 
lemma splitE2: 

531 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

532 
proof  

533 
assume q: "Q (split P z)" 

534 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

535 
show R 

536 
apply (rule r surjective_pairing)+ 

537 
apply (rule split_beta [THEN subst], rule q) 

538 
done 

539 
qed 

540 

541 
lemma splitD': "split R (a,b) c ==> R a b c" 

542 
by simp 

543 

544 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

545 
by simp 

546 

547 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  548 
by (simp only: split_tupled_all, simp) 
11838  549 

18372  550 
lemma mem_splitE: 
37166  551 
assumes major: "z \<in> split c p" 
552 
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" 

18372  553 
shows Q 
37591  554 
by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+ 
11838  555 

556 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

557 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

558 

26340  559 
ML {* 
11838  560 
local (* filtering with exists_p_split is an essential optimization *) 
37591  561 
fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true 
11838  562 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
563 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

564 
 exists_p_split _ = false; 

35364  565 
val ss = HOL_basic_ss addsimps @{thms split_conv}; 
11838  566 
in 
567 
val split_conv_tac = SUBGOAL (fn (t, i) => 

568 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

569 
end; 

26340  570 
*} 
571 

11838  572 
(* This prevents applications of splitE for already splitted arguments leading 
573 
to quite timeconsuming computations (in particular for nested tuples) *) 

26340  574 
declaration {* fn _ => 
575 
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) 

16121  576 
*} 
11838  577 

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

578 
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  579 
by (rule ext) fast 
11838  580 

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

581 
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  582 
by (rule ext) fast 
11838  583 

584 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

585 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

18372  586 
by (rule ext) blast 
11838  587 

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

588 
(* Do NOT make this a simp rule as it 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
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parents:
14208
diff
changeset

589 
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

590 
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

591 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
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diff
changeset

592 
lemma split_comp_eq: 
20415  593 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
594 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  595 
by (rule ext) auto 
14101  596 

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

597 
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

598 
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

599 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

600 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

601 

11838  602 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
603 
by blast 

604 

605 
(* 

606 
the following would be slightly more general, 

607 
but cannot be used as rewrite rule: 

608 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

609 
### ?y = .x 

610 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  611 
by (rtac some_equality 1) 
612 
by ( Simp_tac 1) 

613 
by (split_all_tac 1) 

614 
by (Asm_full_simp_tac 1) 

11838  615 
qed "The_split_eq"; 
616 
*) 

617 

618 
text {* 

619 
Setup of internal @{text split_rule}. 

620 
*} 

621 

45607  622 
lemmas prod_caseI = prod.cases [THEN iffD2] 
24699
c6674504103f
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haftmann
parents:
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diff
changeset

623 

c6674504103f
datatype interpretators for size and datatype_realizer
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changeset

624 
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

625 
by (fact splitI2) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

626 

c6674504103f
datatype interpretators for size and datatype_realizer
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parents:
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diff
changeset

627 
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

628 
by (fact splitI2') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

629 

c6674504103f
datatype interpretators for size and datatype_realizer
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parents:
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diff
changeset

630 
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

631 
by (fact splitE) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

632 

c6674504103f
datatype interpretators for size and datatype_realizer
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parents:
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diff
changeset

633 
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

634 
by (fact splitE') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

635 

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

636 
declare prod_caseI [intro!] 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

637 

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

638 
lemma prod_case_beta: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

639 
"prod_case f p = f (fst p) (snd p)" 
37591  640 
by (fact split_beta) 
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

641 

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

642 
lemma prod_cases3 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

643 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

644 
by (cases y, case_tac b) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

645 

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

646 
lemma prod_induct3 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

647 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

648 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

649 

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

650 
lemma prod_cases4 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

651 
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

652 
by (cases y, case_tac c) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

653 

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

654 
lemma prod_induct4 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

655 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

656 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

657 

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

658 
lemma prod_cases5 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

659 
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

660 
by (cases y, case_tac d) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

661 

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

662 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

663 
"(!!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

664 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

665 

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

666 
lemma prod_cases6 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

667 
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

668 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

669 

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

670 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

671 
"(!!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

672 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

673 

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

674 
lemma prod_cases7 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

675 
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

676 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

677 

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

678 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

679 
"(!!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

680 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

681 

37166  682 
lemma split_def: 
683 
"split = (\<lambda>c p. c (fst p) (snd p))" 

37591  684 
by (fact prod_case_unfold) 
37166  685 

686 
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 

687 
"internal_split == split" 

688 

689 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

690 
by (simp only: internal_split_def split_conv) 

691 

692 
use "Tools/split_rule.ML" 

693 
setup Split_Rule.setup 

694 

695 
hide_const internal_split 

696 

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

697 

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

698 
subsubsection {* Derived operations *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

699 

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

700 
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

701 
"curry = (\<lambda>c x y. c (x, y))" 
37166  702 

703 
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" 

704 
by (simp add: curry_def) 

705 

706 
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 

707 
by (simp add: curry_def) 

708 

709 
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 

710 
by (simp add: curry_def) 

711 

712 
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 

713 
by (simp add: curry_def) 

714 

715 
lemma curry_split [simp]: "curry (split f) = f" 

716 
by (simp add: curry_def split_def) 

717 

718 
lemma split_curry [simp]: "split (curry f) = f" 

719 
by (simp add: curry_def split_def) 

720 

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

721 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

722 
The compositionuncurry combinator. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

723 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

724 

37751  725 
notation fcomp (infixl "\<circ>>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

726 

37751  727 
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where 
728 
"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

729 

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

730 
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

731 
by (simp add: fun_eq_iff scomp_def prod_case_unfold) 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

732 

37751  733 
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)" 
734 
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

735 

37751  736 
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" 
44921  737 
by (simp add: fun_eq_iff) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

738 

37751  739 
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" 
44921  740 
by (simp add: fun_eq_iff) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

741 

37751  742 
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

743 
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

744 

37751  745 
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

746 
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

747 

37751  748 
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" 
44921  749 
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

750 

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

751 
code_const scomp 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

752 
(Eval infixl 3 "#>") 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

753 

37751  754 
no_notation fcomp (infixl "\<circ>>" 60) 
755 
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) 

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

756 

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

757 
text {* 
40607  758 
@{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

759 
functions. 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

760 
*} 
21195  761 

40607  762 
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
763 
"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

764 

40607  765 
lemma map_pair_simp [simp, code]: 
766 
"map_pair f g (a, b) = (f a, g b)" 

767 
by (simp add: map_pair_def) 

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

768 

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

769 
enriched_type map_pair: map_pair 
44921  770 
by (auto simp add: split_paired_all) 
37278  771 

40607  772 
lemma fst_map_pair [simp]: 
773 
"fst (map_pair f g x) = f (fst x)" 

774 
by (cases x) simp_all 

37278  775 

40607  776 
lemma snd_prod_fun [simp]: 
777 
"snd (map_pair f g x) = g (snd x)" 

778 
by (cases x) simp_all 

37278  779 

40607  780 
lemma fst_comp_map_pair [simp]: 
781 
"fst \<circ> map_pair f g = f \<circ> fst" 

782 
by (rule ext) simp_all 

37278  783 

40607  784 
lemma snd_comp_map_pair [simp]: 
785 
"snd \<circ> map_pair f g = g \<circ> snd" 

786 
by (rule ext) simp_all 

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

787 

40607  788 
lemma map_pair_compose: 
789 
"map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)" 

790 
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

791 

40607  792 
lemma map_pair_ident [simp]: 
793 
"map_pair (%x. x) (%y. y) = (%z. z)" 

794 
by (rule ext) (simp add: map_pair.identity) 

795 

796 
lemma map_pair_imageI [intro]: 

797 
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R" 

798 
by (rule image_eqI) simp_all 

21195  799 

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

800 
lemma prod_fun_imageE [elim!]: 
40607  801 
assumes major: "c \<in> map_pair f g ` R" 
802 
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

803 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

804 
apply (rule major [THEN imageE]) 
37166  805 
apply (case_tac x) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

806 
apply (rule cases) 
40607  807 
apply simp_all 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

808 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

809 

37166  810 
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where 
40607  811 
"apfst f = map_pair f id" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

812 

37166  813 
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where 
40607  814 
"apsnd f = map_pair id f" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

815 

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

816 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

817 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

818 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

819 

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

820 
lemma apsnd_conv [simp, code]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

821 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

822 
by (simp add: apsnd_def) 
21195  823 

33594  824 
lemma fst_apfst [simp]: 
825 
"fst (apfst f x) = f (fst x)" 

826 
by (cases x) simp 

827 

828 
lemma fst_apsnd [simp]: 

829 
"fst (apsnd f x) = fst x" 

830 
by (cases x) simp 

831 

832 
lemma snd_apfst [simp]: 

833 
"snd (apfst f x) = snd x" 

834 
by (cases x) simp 

835 

836 
lemma snd_apsnd [simp]: 

837 
"snd (apsnd f x) = f (snd x)" 

838 
by (cases x) simp 

839 

840 
lemma apfst_compose: 

841 
"apfst f (apfst g x) = apfst (f \<circ> g) x" 

842 
by (cases x) simp 

843 

844 
lemma apsnd_compose: 

845 
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" 

846 
by (cases x) simp 

847 

848 
lemma apfst_apsnd [simp]: 

849 
"apfst f (apsnd g x) = (f (fst x), g (snd x))" 

850 
by (cases x) simp 

851 

852 
lemma apsnd_apfst [simp]: 

853 
"apsnd f (apfst g x) = (g (fst x), f (snd x))" 

854 
by (cases x) simp 

855 

856 
lemma apfst_id [simp] : 

857 
"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

858 
by (simp add: fun_eq_iff) 
33594  859 

860 
lemma apsnd_id [simp] : 

861 
"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

862 
by (simp add: fun_eq_iff) 
33594  863 

864 
lemma apfst_eq_conv [simp]: 

865 
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" 

866 
by (cases x) simp 

867 

868 
lemma apsnd_eq_conv [simp]: 

869 
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" 

870 
by (cases x) simp 

871 

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

872 
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

873 
"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

874 
by simp 
21195  875 

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

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

877 
Disjoint union of a family of sets  Sigma. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

879 

45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset

880 
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

881 
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

882 

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

883 
abbreviation 
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset

884 
Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

885 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

886 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

887 

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

888 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

889 
Times (infixr "\<times>" 80) 
15394  890 

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

891 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

892 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

893 

45662
4f7c05990420
Hide Product_Type.Times  too precious an identifier
nipkow
parents:
45607
diff
changeset

894 
hide_const (open) Times 
4f7c05990420
Hide Product_Type.Times  too precious an identifier
nipkow
parents:
45607
diff
changeset

895 

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

896 
syntax 
35115  897 
"_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

898 
translations 
35115  899 
"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

900 

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

901 
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

902 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

903 

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

904 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

905 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

906 
!!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

907 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

908 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

909 
by (unfold Sigma_def) blast 
20588  910 

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

911 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

912 
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

913 
eigenvariables. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

915 

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

916 
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

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

918 

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

919 
lemma 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

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

921 

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

922 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

923 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

924 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

925 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

926 
by blast 
20588  927 

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

928 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

929 
"\<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

930 
\<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

931 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

932 

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

933 
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

934 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

935 

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

936 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

937 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

938 

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

939 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

940 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

941 

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

942 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

943 
by auto 
21908  944 

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

945 
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

946 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

947 

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

948 
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

949 
by auto 
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 
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

952 
by blast 
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 
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

955 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

956 

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

957 
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

958 
by (blast elim: equalityE) 
20588  959 

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

960 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

961 
"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

962 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

963 

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

964 
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

965 
by blast 
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 UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

968 
"(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

969 
 {* Suggested by Pierre Chartier *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

970 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

971 

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

972 
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

973 
"(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

974 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

975 

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

976 
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

977 
"(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

978 
by blast 
21908  979 

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

980 
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

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

982 

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

983 
lemma 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

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

985 

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

986 
lemma 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

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

988 

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

989 
lemma 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

990 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

991 

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

992 
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

993 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

994 

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

995 
lemma Sigma_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

996 
by blast 
21908  997 

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

998 
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

999 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1000 

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

1001 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1002 
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

1003 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1004 
*} 
21908  1005 

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

1006 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
28719  1007 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1008 

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

1009 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
28719  1010 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1011 

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

1012 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
28719  1013 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1014 

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

1015 
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

1016 
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

1017 

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

1018 
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" 
44921  1019 
by force 
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

1020 

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

1021 
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" 
44921  1022 
by force 
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

1023 

28719  1024 
lemma insert_times_insert[simp]: 
1025 
"insert a A \<times> insert b B = 

1026 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

1027 
by blast 

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

1028 

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

1029 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
37166  1030 
by (auto, case_tac "f x", auto) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1031 

35822  1032 
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

1033 
"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

1034 
by (auto intro!: inj_onI) 
35822  1035 

1036 
lemma swap_product: 

1037 
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" 

1038 
by (simp add: split_def image_def) blast 

1039 

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

1040 
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

1041 
"(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f ` {x} \<inter> A))" 
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1042 
proof (safe intro!: imageI) 
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

1043 
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

1044 
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

1045 
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

1046 
qed simp_all 
35822  1047 

46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1048 
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1049 
[code_abbrev]: "product A B = A \<times> B" 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1050 

53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1051 
hide_const (open) product 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1052 

53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1053 
lemma member_product: 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1054 
"x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B" 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1055 
by (simp add: product_def) 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1056 

40607  1057 
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *} 
1058 

1059 
lemma map_pair_inj_on: 

1060 
assumes "inj_on f A" and "inj_on g B" 

1061 
shows "inj_on (map_pair f g) (A \<times> B)" 

1062 
proof (rule inj_onI) 

1063 
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" 

1064 
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto 

1065 
assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto 

1066 
assume "map_pair f g x = map_pair f g y" 

1067 
hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto) 

1068 
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) 

1069 
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` 

1070 
have "fst x = fst y" by (auto dest:dest:inj_onD) 

1071 
moreover from `map_pair f g x = map_pair f g y` 

1072 
have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto) 

1073 
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) 

1074 
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` 

1075 
have "snd x = snd y" by (auto dest:dest:inj_onD) 

1076 
ultimately show "x = y" by(rule prod_eqI) 

1077 
qed 

1078 

1079 
lemma map_pair_surj: 

40702  1080 
fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd" 
40607  1081 
assumes "surj f" and "surj g" 
1082 
shows "surj (map_pair f g)" 

1083 
unfolding surj_def 

1084 
proof 

1085 
fix y :: "'b \<times> 'd" 

1086 
from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) 

1087 
moreover 

1088 
from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) 

1089 
ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto 

1090 
thus "\<exists>x. y = map_pair f g x" by auto 

1091 
qed 

1092 

1093 
lemma map_pair_surj_on: 

1094 
assumes "f ` A = A'" and "g ` B = B'" 

1095 
shows "map_pair f g ` (A \<times> B) = A' \<times> B'" 

1096 
unfolding image_def 

1097 
proof(rule set_eqI,rule iffI) 

1098 
fix x :: "'a \<times> 'c" 

1099 
assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}" 

1100 
then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast 

1101 
from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto 

1102 
moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto 

1103 
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto 

1104 
with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto) 

1105 
next 

1106 
fix x :: "'a \<times> 'c" 

1107 
assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto 

1108 
from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto 

1109 
then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE) 

1110 
moreover from `image g B = B'` and `snd x \<in> B'` 

1111 
obtain b where "b \<in> B" and "snd x = g b" by auto 

1112 
ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto 

1113 
moreover from `a \<in> A` and `b \<in> B` have "(a , b) \<in> A \<times> B" by auto 

1114 
ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto 

1115 
thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto 

1116 
qed 

1117 

21908  1118 

37166  1119 
subsection {* Inductively defined sets *} 
15394  1120 

31723
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1121 
use "Tools/inductive_set.ML" 
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diff
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1122 
setup Inductive_Set.setup 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
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24286
diff
changeset

1123 

37166  1124 

1125 
subsection {* Legacy theorem bindings and duplicates *} 

1126 

1127 
lemma PairE: 

1128 
obtains x y where "p = (x, y)" 

1129 
by (fact prod.exhaust) 

1130 

1131 
lemma Pair_inject: 

1132 
assumes "(a, b) = (a', b')" 

1133 
and "a = a' ==> b = b' ==> R" 

1134 
shows R 

1135 
using assms by simp 

1136 

1137 
lemmas Pair_eq = prod.inject 

1138 

1139 
lemmas split = split_conv  {* for backwards compatibility *} 

1140 

44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
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parents:
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diff
changeset

1141 
lemmas Pair_fst_snd_eq = prod_eq_iff 
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset

1142 

45204
5e4a1270c000
hide typedefgenerated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset

1143 
hide_const (open) prod 
5e4a1270c000
hide typedefgenerated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset

1144 

10213  1145 
end 