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permissions  rwrr 
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(* Title: HOL/Product_Type.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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*) 
10213  6 

11838  7 
header {* Cartesian products *} 
10213  8 

15131  9 
theory Product_Type 
15140  10 
imports Fun 
15131  11 
files ("Tools/split_rule.ML") 
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begin 

11838  13 

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subsection {* Unit *} 

15 

16 
typedef unit = "{True}" 

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proof 

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show "True : ?unit" by blast 

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qed 

20 

21 
constdefs 

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Unity :: unit ("'(')") 

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"() == Abs_unit True" 

24 

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lemma unit_eq: "u = ()" 

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by (induct u) (simp add: unit_def Unity_def) 

27 

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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! 

31 
*} 

32 

33 
ML_setup {* 

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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.sign_of (the_context ())) "unit_eq" ["x::unit"] 

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(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) 
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end; 
11838  39 

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Addsimprocs [unit_eq_proc]; 

41 
*} 

42 

43 
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" 

44 
by simp 

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46 
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" 

47 
by (rule triv_forall_equality) 

48 

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lemma unit_induct [induct type: unit]: "P () ==> P x" 

50 
by simp 

51 

52 
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 ()"}. 

56 
*} 

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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f" 

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by (rule ext) simp 

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61 

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subsection {* Pairs *} 
10213  63 

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subsubsection {* Type definition *} 
10213  65 

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constdefs 

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Pair_Rep :: "['a, 'b] => ['a, 'b] => bool" 
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"Pair_Rep == (%a b. %x y. x=a & y=b)" 
10213  69 

70 
global 

71 

72 
typedef (Prod) 

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('a, 'b) "*" (infixr 20) 
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= "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}" 
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proof 
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fix a b show "Pair_Rep a b : ?Prod" 
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by blast 
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qed 
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syntax (xsymbols) 
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
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syntax (HTML output) 
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
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local 
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11777  87 

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subsubsection {* Abstract constants and syntax *} 

89 

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global 

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consts 

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fst :: "'a * 'b => 'a" 
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snd :: "'a * 'b => 'b" 
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split :: "[['a, 'b] => 'c, 'a * 'b] => 'c" 
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curry :: "['a * 'b => 'c, 'a, 'b] => 'c" 
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prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd" 
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Pair :: "['a, 'b] => 'a * 'b" 
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Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set" 
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local 
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text {* 
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Patterns  extends predefined type @{typ pttrn} used in 

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abstractions. 

106 
*} 

10213  107 

108 
nonterminals 

109 
tuple_args patterns 

110 

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syntax 

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"_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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"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10) 
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"@Times" ::"['a set, 'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80) 
10213  120 

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translations 

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"(x, y)" == "Pair x y" 

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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 

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"%(x,y,zs).b" == "split(%x (y,zs).b)" 

125 
"%(x,y).b" == "split(%x y. b)" 

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"_abs (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 *) 

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"SIGMA x:A. B" => "Sigma A (%x. B)" 

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"A <*> B" => "Sigma A (_K B)" 

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

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print_translation {* 

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let fun split_tr' [Abs (x,T,t as (Abs abs))] = 

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

138 
let val (y,t') = atomic_abs_tr' abs; 

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

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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end 

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 split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = 

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

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let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; 

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

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in Syntax.const "_abs"$ 

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(Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end 

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 split_tr' [Const ("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 ("_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; 

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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end 

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 split_tr' _ = raise Match; 

156 
in [("split", split_tr')] 

157 
end 

158 
*} 

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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 
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Const ("split",_) => raise Match 
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 _ => let 
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val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
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val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
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val (x',t'') = atomic_abs_tr' (x,xT,t'); 
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
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 split_guess_names_tr' _ T [t] = 
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(case (head_of t) of 
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Const ("split",_) => raise Match 
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 _ => let 
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val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
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val (y,t') = 
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atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
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val (x',t'') = atomic_abs_tr' ("x",xT,t'); 
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
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 split_guess_names_tr' _ _ _ = raise Match; 
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in [("split", split_guess_names_tr')] 
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end 
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*} 
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text{*Deleted xsymbol and html support using @{text"\<Sigma>"} (Sigma) because of the danger of confusion with Sum.*} 
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syntax (xsymbols) 
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \<times> _" [81, 80] 80) 
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14565  192 
syntax (HTML output) 
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \<times> _" [81, 80] 80) 
14565  194 

11032  195 
print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *} 
10213  196 

197 

11777  198 
subsubsection {* Definitions *} 
10213  199 

200 
defs 

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Pair_def: "Pair a b == Abs_Prod(Pair_Rep a b)" 
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fst_def: "fst p == THE a. EX b. p = (a, b)" 
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snd_def: "snd p == THE b. EX a. p = (a, b)" 
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split_def: "split == (%c p. c (fst p) (snd p))" 
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curry_def: "curry == (%c x y. c (x,y))" 
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prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))" 
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Sigma_def: "Sigma A B == UN x:A. UN y:B(x). {(x, y)}" 
10213  208 

209 

11966  210 
subsubsection {* Lemmas and proof tool setup *} 
11838  211 

212 
lemma ProdI: "Pair_Rep a b : Prod" 

213 
by (unfold Prod_def) blast 

214 

215 
lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'" 

216 
apply (unfold Pair_Rep_def) 

14208  217 
apply (drule fun_cong [THEN fun_cong], blast) 
11838  218 
done 
10213  219 

11838  220 
lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" 
221 
apply (rule inj_on_inverseI) 

222 
apply (erule Abs_Prod_inverse) 

223 
done 

224 

225 
lemma Pair_inject: 

226 
"(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R" 

227 
proof  

228 
case rule_context [unfolded Pair_def] 

229 
show ?thesis 

230 
apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) 

231 
apply (rule rule_context ProdI)+ 

232 
. 

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qed 
10213  234 

11838  235 
lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')" 
236 
by (blast elim!: Pair_inject) 

237 

238 
lemma fst_conv [simp]: "fst (a, b) = a" 

239 
by (unfold fst_def) blast 

240 

241 
lemma snd_conv [simp]: "snd (a, b) = b" 

242 
by (unfold snd_def) blast 

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11838  244 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
245 
by simp 

246 

247 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

248 
by simp 

249 

250 
lemma PairE_lemma: "EX x y. p = (x, y)" 

251 
apply (unfold Pair_def) 

252 
apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) 

253 
apply (erule exE, erule exE, rule exI, rule exI) 

254 
apply (rule Rep_Prod_inverse [symmetric, THEN trans]) 

255 
apply (erule arg_cong) 

256 
done 

11032  257 

11838  258 
lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q" 
259 
by (insert PairE_lemma [of p]) blast 

260 

261 
ML_setup {* 

262 
local val PairE = thm "PairE" in 

263 
fun pair_tac s = 

264 
EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac]; 

265 
end; 

266 
*} 

11032  267 

11838  268 
lemma surjective_pairing: "p = (fst p, snd p)" 
269 
 {* Do not add as rewrite rule: invalidates some proofs in IMP *} 

270 
by (cases p) simp 

271 

272 
declare surjective_pairing [symmetric, simp] 

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11838  274 
lemma surj_pair [simp]: "EX x y. z = (x, y)" 
275 
apply (rule exI) 

276 
apply (rule exI) 

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apply (rule surjective_pairing) 

278 
done 

279 

280 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 

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proof 
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fix a b 
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assume "!!x. PROP P x" 
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thus "PROP P (a, b)" . 
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next 
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fix x 
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assume "!!a b. PROP P (a, b)" 
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hence "PROP P (fst x, snd x)" . 
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thus "PROP P x" by simp 
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changeset

290 
qed 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset

291 

11838  292 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
293 

294 
text {* 

295 
The rule @{thm [source] split_paired_all} does not work with the 

296 
Simplifier because it also affects premises in congrence rules, 

297 
where this can lead to premises of the form @{text "!!a b. ... = 

298 
?P(a, b)"} which cannot be solved by reflexivity. 

299 
*} 

300 

301 
ML_setup " 

302 
(* replace parameters of product type by individual component parameters *) 

303 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

304 
local (* filtering with exists_paired_all is an essential optimization *) 

305 
fun exists_paired_all (Const (\"all\", _) $ Abs (_, T, t)) = 

306 
can HOLogic.dest_prodT T orelse exists_paired_all t 

307 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

308 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

309 
 exists_paired_all _ = false; 

310 
val ss = HOL_basic_ss 

311 
addsimps [thm \"split_paired_all\", thm \"unit_all_eq2\", thm \"unit_abs_eta_conv\"] 

312 
addsimprocs [unit_eq_proc]; 

313 
in 

314 
val split_all_tac = SUBGOAL (fn (t, i) => 

315 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

316 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

317 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

318 
fun split_all th = 

319 
if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th; 

320 
end; 

321 

322 
claset_ref() := claset() addSbefore (\"split_all_tac\", split_all_tac); 

323 
" 

324 

325 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

326 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

327 
by fast 

328 

14189  329 
lemma curry_split [simp]: "curry (split f) = f" 
330 
by (simp add: curry_def split_def) 

331 

332 
lemma split_curry [simp]: "split (curry f) = f" 

333 
by (simp add: curry_def split_def) 

334 

335 
lemma curryI [intro!]: "f (a,b) ==> curry f a b" 

336 
by (simp add: curry_def) 

337 

14190
609c072edf90
Fixed blunder in the setup of the classical reasoner wrt. the constant
skalberg
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14189
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changeset

338 
lemma curryD [dest!]: "curry f a b ==> f (a,b)" 
14189  339 
by (simp add: curry_def) 
340 

14190
609c072edf90
Fixed blunder in the setup of the classical reasoner wrt. the constant
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changeset

341 
lemma curryE: "[ curry f a b ; f (a,b) ==> Q ] ==> Q" 
14189  342 
by (simp add: curry_def) 
343 

344 
lemma curry_conv [simp]: "curry f a b = f (a,b)" 

345 
by (simp add: curry_def) 

346 

11838  347 
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x" 
348 
by fast 

349 

350 
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 

351 
by fast 

352 

353 
lemma split_conv [simp]: "split c (a, b) = c a b" 

354 
by (simp add: split_def) 

355 

356 
lemmas split = split_conv  {* for backwards compatibility *} 

357 

358 
lemmas splitI = split_conv [THEN iffD2, standard] 

359 
lemmas splitD = split_conv [THEN iffD1, standard] 

360 

361 
lemma split_Pair_apply: "split (%x y. f (x, y)) = f" 

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

363 
apply (rule ext) 

14208  364 
apply (tactic {* pair_tac "x" 1 *}, simp) 
11838  365 
done 
366 

367 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 

368 
 {* Can't be added to simpset: loops! *} 

369 
by (simp add: split_Pair_apply) 

370 

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

372 
by (simp add: split_def) 

373 

374 
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)" 

14208  375 
by (simp only: split_tupled_all, simp) 
11838  376 

377 
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q" 

378 
by (simp add: Pair_fst_snd_eq) 

379 

380 
lemma split_weak_cong: "p = q ==> split c p = split c q" 

381 
 {* Prevents simplification of @{term c}: much faster *} 

382 
by (erule arg_cong) 

383 

384 
lemma split_eta: "(%(x, y). f (x, y)) = f" 

385 
apply (rule ext) 

386 
apply (simp only: split_tupled_all) 

387 
apply (rule split_conv) 

388 
done 

389 

390 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

391 
by (simp add: split_eta) 

392 

393 
text {* 

394 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

395 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

396 
and using @{thm [source] cond_split_eta} directly would render some 

397 
existing proofs very inefficient; similarly for @{text 

398 
split_beta}. *} 

399 

400 
ML_setup {* 

401 

402 
local 

403 
val cond_split_eta = thm "cond_split_eta"; 

404 
fun Pair_pat k 0 (Bound m) = (m = k) 

405 
 Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso 

406 
m = k+i andalso Pair_pat k (i1) t 

407 
 Pair_pat _ _ _ = false; 

408 
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t 

409 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

410 
 no_args k i (Bound m) = m < k orelse m > k+i 

411 
 no_args _ _ _ = true; 

15531  412 
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE 
11838  413 
 split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t 
15531  414 
 split_pat tp i _ = NONE; 
13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

415 
fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] [] 
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

416 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) 
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

417 
(K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1))); 
11838  418 

419 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t 

420 
 beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse 

421 
(beta_term_pat k i t andalso beta_term_pat k i u) 

422 
 beta_term_pat k i t = no_args k i t; 

423 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

424 
 eta_term_pat _ _ _ = false; 

425 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 

426 
 subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg 

427 
else (subst arg k i t $ subst arg k i u) 

428 
 subst arg k i t = t; 

429 
fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) = 

430 
(case split_pat beta_term_pat 1 t of 

15531  431 
SOME (i,f) => SOME (metaeq sg s (subst arg 0 i f)) 
432 
 NONE => NONE) 

433 
 beta_proc _ _ _ = NONE; 

11838  434 
fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) = 
435 
(case split_pat eta_term_pat 1 t of 

15531  436 
SOME (_,ft) => SOME (metaeq sg s (let val (f $ arg) = ft in f end)) 
437 
 NONE => NONE) 

438 
 eta_proc _ _ _ = NONE; 

11838  439 
in 
13462  440 
val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 
441 
"split_beta" ["split f z"] beta_proc; 

442 
val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 

443 
"split_eta" ["split f"] eta_proc; 

11838  444 
end; 
445 

446 
Addsimprocs [split_beta_proc, split_eta_proc]; 

447 
*} 

448 

449 
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)" 

450 
by (subst surjective_pairing, rule split_conv) 

451 

452 
lemma split_split: "R (split c p) = (ALL x y. p = (x, y) > R (c x y))" 

453 
 {* For use with @{text split} and the Simplifier. *} 

15481  454 
by (insert surj_pair [of p], clarify, simp) 
11838  455 

456 
text {* 

457 
@{thm [source] split_split} could be declared as @{text "[split]"} 

458 
done after the Splitter has been speeded up significantly; 

459 
precompute the constants involved and don't do anything unless the 

460 
current goal contains one of those constants. 

461 
*} 

462 

463 
lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 

14208  464 
by (subst split_split, simp) 
11838  465 

466 

467 
text {* 

468 
\medskip @{term split} used as a logical connective or set former. 

469 

470 
\medskip These rules are for use with @{text blast}; could instead 

471 
call @{text simp} using @{thm [source] split} as rewrite. *} 

472 

473 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

474 
apply (simp only: split_tupled_all) 

475 
apply (simp (no_asm_simp)) 

476 
done 

477 

478 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

479 
apply (simp only: split_tupled_all) 

480 
apply (simp (no_asm_simp)) 

481 
done 

482 

483 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

484 
by (induct p) (auto simp add: split_def) 

485 

486 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

487 
by (induct p) (auto simp add: split_def) 

488 

489 
lemma splitE2: 

490 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

491 
proof  

492 
assume q: "Q (split P z)" 

493 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

494 
show R 

495 
apply (rule r surjective_pairing)+ 

496 
apply (rule split_beta [THEN subst], rule q) 

497 
done 

498 
qed 

499 

500 
lemma splitD': "split R (a,b) c ==> R a b c" 

501 
by simp 

502 

503 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

504 
by simp 

505 

506 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  507 
by (simp only: split_tupled_all, simp) 
11838  508 

509 
lemma mem_splitE: "[ z: split c p; !!x y. [ p = (x,y); z: c x y ] ==> Q ] ==> Q" 

510 
proof  

511 
case rule_context [unfolded split_def] 

512 
show ?thesis by (rule rule_context surjective_pairing)+ 

513 
qed 

514 

515 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

516 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

517 

518 
ML_setup " 

519 
local (* filtering with exists_p_split is an essential optimization *) 

520 
fun exists_p_split (Const (\"split\",_) $ _ $ (Const (\"Pair\",_)$_$_)) = true 

521 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 

522 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

523 
 exists_p_split _ = false; 

524 
val ss = HOL_basic_ss addsimps [thm \"split_conv\"]; 

525 
in 

526 
val split_conv_tac = SUBGOAL (fn (t, i) => 

527 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

528 
end; 

529 
(* This prevents applications of splitE for already splitted arguments leading 

530 
to quite timeconsuming computations (in particular for nested tuples) *) 

531 
claset_ref() := claset() addSbefore (\"split_conv_tac\", split_conv_tac); 

532 
" 

533 

534 
lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 

14208  535 
by (rule ext, fast) 
11838  536 

537 
lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 

14208  538 
by (rule ext, fast) 
11838  539 

540 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

541 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

14208  542 
apply (rule ext, blast) 
11838  543 
done 
544 

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

545 
(* 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

546 
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

547 
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

548 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

549 
lemma split_comp_eq: 
14101  550 
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 
551 
by (rule ext, auto) 

552 

11838  553 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
554 
by blast 

555 

556 
(* 

557 
the following would be slightly more general, 

558 
but cannot be used as rewrite rule: 

559 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

560 
### ?y = .x 

561 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  562 
by (rtac some_equality 1) 
563 
by ( Simp_tac 1) 

564 
by (split_all_tac 1) 

565 
by (Asm_full_simp_tac 1) 

11838  566 
qed "The_split_eq"; 
567 
*) 

568 

569 
lemma injective_fst_snd: "!!x y. [fst x = fst y; snd x = snd y] ==> x = y" 

570 
by auto 

571 

572 

573 
text {* 

574 
\bigskip @{term prod_fun}  action of the product functor upon 

575 
functions. 

576 
*} 

577 

578 
lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)" 

579 
by (simp add: prod_fun_def) 

580 

581 
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" 

582 
apply (rule ext) 

14208  583 
apply (tactic {* pair_tac "x" 1 *}, simp) 
11838  584 
done 
585 

586 
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" 

587 
apply (rule ext) 

14208  588 
apply (tactic {* pair_tac "z" 1 *}, simp) 
11838  589 
done 
590 

591 
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" 

592 
apply (rule image_eqI) 

14208  593 
apply (rule prod_fun [symmetric], assumption) 
11838  594 
done 
595 

596 
lemma prod_fun_imageE [elim!]: 

597 
"[ c: (prod_fun f g)`r; !!x y. [ c=(f(x),g(y)); (x,y):r ] ==> P 

598 
] ==> P" 

599 
proof  

600 
case rule_context 

601 
assume major: "c: (prod_fun f g)`r" 

602 
show ?thesis 

603 
apply (rule major [THEN imageE]) 

604 
apply (rule_tac p = x in PairE) 

605 
apply (rule rule_context) 

606 
prefer 2 

607 
apply blast 

608 
apply (blast intro: prod_fun) 

609 
done 

610 
qed 

611 

612 

14101  613 
constdefs 
614 
upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b" 

615 
"upd_fst f == prod_fun f id" 

616 

617 
upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c" 

618 
"upd_snd f == prod_fun id f" 

619 

620 
lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)" 

621 
by (simp add: upd_fst_def) 

622 

623 
lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)" 

624 
by (simp add: upd_snd_def) 

625 

11838  626 
text {* 
627 
\bigskip Disjoint union of a family of sets  Sigma. 

628 
*} 

629 

630 
lemma SigmaI [intro!]: "[ a:A; b:B(a) ] ==> (a,b) : Sigma A B" 

631 
by (unfold Sigma_def) blast 

632 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

633 
lemma SigmaE [elim!]: 
11838  634 
"[ c: Sigma A B; 
635 
!!x y.[ x:A; y:B(x); c=(x,y) ] ==> P 

636 
] ==> P" 

637 
 {* The general elimination rule. *} 

638 
by (unfold Sigma_def) blast 

639 

640 
text {* 

15422
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

641 
Elimination of @{term "(a, b) : A \<times> B"}  introduces no 
11838  642 
eigenvariables. 
643 
*} 

644 

645 
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

646 
by blast 
11838  647 

648 
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

649 
by blast 
11838  650 

651 
lemma SigmaE2: 

652 
"[ (a, b) : Sigma A B; 

653 
[ a:A; b:B(a) ] ==> P 

654 
] ==> P" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

655 
by blast 
11838  656 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

657 
lemma Sigma_cong: 
15422
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

658 
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> 
cbdddc0efadf
added print translation for split: split f > %(x,y). f x y
schirmer
parents:
15404
diff
changeset

659 
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" 
14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

660 
by auto 
11838  661 

662 
lemma Sigma_mono: "[ A <= C; !!x. x:A ==> B x <= D x ] ==> Sigma A B <= Sigma C D" 

663 
by blast 

664 

665 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 

666 
by blast 

667 

668 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 

669 
by blast 

670 

671 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 

672 
by auto 

673 

674 
lemma Compl_Times_UNIV1 [simp]: " (UNIV <*> A) = UNIV <*> (A)" 

675 
by auto 

676 

677 
lemma Compl_Times_UNIV2 [simp]: " (A <*> UNIV) = (A) <*> UNIV" 

678 
by auto 

679 

680 
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" 

681 
by blast 

682 

683 
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" 

684 
by blast 

685 

686 
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" 

687 
by (blast elim: equalityE) 

688 

689 
lemma SetCompr_Sigma_eq: 

690 
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" 

691 
by blast 

692 

693 
text {* 

694 
\bigskip Complex rules for Sigma. 

695 
*} 

696 

697 
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" 

698 
by blast 

699 

700 
lemma UN_Times_distrib: 

701 
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" 

702 
 {* Suggested by Pierre Chartier *} 

703 
by blast 

704 

705 
lemma split_paired_Ball_Sigma [simp]: 

706 
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" 

707 
by blast 

708 

709 
lemma split_paired_Bex_Sigma [simp]: 

710 
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" 

711 
by blast 

712 

713 
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" 

714 
by blast 

715 

716 
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" 

717 
by blast 

718 

719 
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" 

720 
by blast 

721 

722 
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" 

723 
by blast 

724 

725 
lemma Sigma_Diff_distrib1: "(SIGMA i:I  J. C(i)) = (SIGMA i:I. C(i))  (SIGMA j:J. C(j))" 

726 
by blast 

727 

728 
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i)  B(i)) = (SIGMA i:I. A(i))  (SIGMA i:I. B(i))" 

729 
by blast 

730 

731 
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" 

732 
by blast 

733 

734 
text {* 

735 
Nondependent versions are needed to avoid the need for higherorder 

736 
matching, especially when the rules are reoriented. 

737 
*} 

738 

739 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 

740 
by blast 

741 

742 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 

743 
by blast 

744 

745 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 

746 
by blast 

747 

748 

11493  749 
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" 
11777  750 
apply (rule_tac x = "(a, b)" in image_eqI) 
751 
apply auto 

752 
done 

753 

11493  754 

11838  755 
text {* 
756 
Setup of internal @{text split_rule}. 

757 
*} 

758 

11032  759 
constdefs 
11425  760 
internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c" 
11032  761 
"internal_split == split" 
762 

763 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

764 
by (simp only: internal_split_def split_conv) 

765 

766 
hide const internal_split 

767 

11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

768 
use "Tools/split_rule.ML" 
11032  769 
setup SplitRule.setup 
10213  770 

15394  771 

772 
subsection {* Code generator setup *} 

773 

774 
types_code 

775 
"*" ("(_ */ _)") 

776 

777 
consts_code 

778 
"Pair" ("(_,/ _)") 

779 
"fst" ("fst") 

780 
"snd" ("snd") 

781 

782 
ML {* 

783 
fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y; 

784 

785 
fun gen_id_42 aG bG i = (aG i, bG i); 

786 

787 
local 

788 

789 
fun strip_abs 0 t = ([], t) 

790 
 strip_abs i (Abs (s, T, t)) = 

791 
let 

792 
val s' = Codegen.new_name t s; 

793 
val v = Free (s', T) 

794 
in apfst (cons v) (strip_abs (i1) (subst_bound (v, t))) end 

795 
 strip_abs i (u as Const ("split", _) $ t) = (case strip_abs (i+1) t of 

796 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 

797 
 _ => ([], u)) 

798 
 strip_abs i t = ([], t); 

799 

800 
fun let_codegen thy gr dep brack (t as Const ("Let", _) $ _ $ _) = 

801 
let 

802 
fun dest_let (l as Const ("Let", _) $ t $ u) = 

803 
(case strip_abs 1 u of 

804 
([p], u') => apfst (cons (p, t)) (dest_let u') 

805 
 _ => ([], l)) 

806 
 dest_let t = ([], t); 

807 
fun mk_code (gr, (l, r)) = 

808 
let 

809 
val (gr1, pl) = Codegen.invoke_codegen thy dep false (gr, l); 

810 
val (gr2, pr) = Codegen.invoke_codegen thy dep false (gr1, r); 

811 
in (gr2, (pl, pr)) end 

812 
in case dest_let t of 

15531  813 
([], _) => NONE 
15394  814 
 (ps, u) => 
815 
let 

816 
val (gr1, qs) = foldl_map mk_code (gr, ps); 

817 
val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u) 

818 
in 

15531  819 
SOME (gr2, Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, flat 
15394  820 
(separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) => 
821 
[Pretty.block [Pretty.str "val ", pl, Pretty.str " =", 

822 
Pretty.brk 1, pr]]) qs))), 

823 
Pretty.brk 1, Pretty.str "in ", pu, 

824 
Pretty.brk 1, Pretty.str "end"])) 

825 
end 

826 
end 

15531  827 
 let_codegen thy gr dep brack t = NONE; 
15394  828 

829 
fun split_codegen thy gr dep brack (t as Const ("split", _) $ _) = 

830 
(case strip_abs 1 t of 

831 
([p], u) => 

832 
let 

833 
val (gr1, q) = Codegen.invoke_codegen thy dep false (gr, p); 

834 
val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u) 

835 
in 

15531  836 
SOME (gr2, Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>", 
15394  837 
Pretty.brk 1, pu, Pretty.str ")"]) 
838 
end 

15531  839 
 _ => NONE) 
840 
 split_codegen thy gr dep brack t = NONE; 

15394  841 

842 
in 

843 

844 
val prod_codegen_setup = 

845 
[Codegen.add_codegen "let_codegen" let_codegen, 

846 
Codegen.add_codegen "split_codegen" split_codegen]; 

847 

848 
end; 

849 
*} 

850 

851 
setup prod_codegen_setup 

852 

15404  853 
ML 
854 
{* 

855 
val Collect_split = thm "Collect_split"; 

856 
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; 

857 
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; 

858 
val PairE = thm "PairE"; 

859 
val PairE_lemma = thm "PairE_lemma"; 

860 
val Pair_Rep_inject = thm "Pair_Rep_inject"; 

861 
val Pair_def = thm "Pair_def"; 

862 
val Pair_eq = thm "Pair_eq"; 

863 
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; 

864 
val Pair_inject = thm "Pair_inject"; 

865 
val ProdI = thm "ProdI"; 

866 
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; 

867 
val SigmaD1 = thm "SigmaD1"; 

868 
val SigmaD2 = thm "SigmaD2"; 

869 
val SigmaE = thm "SigmaE"; 

870 
val SigmaE2 = thm "SigmaE2"; 

871 
val SigmaI = thm "SigmaI"; 

872 
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; 

873 
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; 

874 
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; 

875 
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; 

876 
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; 

877 
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; 

878 
val Sigma_Union = thm "Sigma_Union"; 

879 
val Sigma_def = thm "Sigma_def"; 

880 
val Sigma_empty1 = thm "Sigma_empty1"; 

881 
val Sigma_empty2 = thm "Sigma_empty2"; 

882 
val Sigma_mono = thm "Sigma_mono"; 

883 
val The_split = thm "The_split"; 

884 
val The_split_eq = thm "The_split_eq"; 

885 
val The_split_eq = thm "The_split_eq"; 

886 
val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; 

887 
val Times_Int_distrib1 = thm "Times_Int_distrib1"; 

888 
val Times_Un_distrib1 = thm "Times_Un_distrib1"; 

889 
val Times_eq_cancel2 = thm "Times_eq_cancel2"; 

890 
val Times_subset_cancel2 = thm "Times_subset_cancel2"; 

891 
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; 

892 
val UN_Times_distrib = thm "UN_Times_distrib"; 

893 
val Unity_def = thm "Unity_def"; 

894 
val cond_split_eta = thm "cond_split_eta"; 

895 
val fst_conv = thm "fst_conv"; 

896 
val fst_def = thm "fst_def"; 

897 
val fst_eqD = thm "fst_eqD"; 

898 
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; 

899 
val injective_fst_snd = thm "injective_fst_snd"; 

900 
val mem_Sigma_iff = thm "mem_Sigma_iff"; 

901 
val mem_splitE = thm "mem_splitE"; 

902 
val mem_splitI = thm "mem_splitI"; 

903 
val mem_splitI2 = thm "mem_splitI2"; 

904 
val prod_eqI = thm "prod_eqI"; 

905 
val prod_fun = thm "prod_fun"; 

906 
val prod_fun_compose = thm "prod_fun_compose"; 

907 
val prod_fun_def = thm "prod_fun_def"; 

908 
val prod_fun_ident = thm "prod_fun_ident"; 

909 
val prod_fun_imageE = thm "prod_fun_imageE"; 

910 
val prod_fun_imageI = thm "prod_fun_imageI"; 

911 
val prod_induct = thm "prod_induct"; 

912 
val snd_conv = thm "snd_conv"; 

913 
val snd_def = thm "snd_def"; 

914 
val snd_eqD = thm "snd_eqD"; 

915 
val split = thm "split"; 

916 
val splitD = thm "splitD"; 

917 
val splitD' = thm "splitD'"; 

918 
val splitE = thm "splitE"; 

919 
val splitE' = thm "splitE'"; 

920 
val splitE2 = thm "splitE2"; 

921 
val splitI = thm "splitI"; 

922 
val splitI2 = thm "splitI2"; 

923 
val splitI2' = thm "splitI2'"; 

924 
val split_Pair_apply = thm "split_Pair_apply"; 

925 
val split_beta = thm "split_beta"; 

926 
val split_conv = thm "split_conv"; 

927 
val split_def = thm "split_def"; 

928 
val split_eta = thm "split_eta"; 

929 
val split_eta_SetCompr = thm "split_eta_SetCompr"; 

930 
val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; 

931 
val split_paired_All = thm "split_paired_All"; 

932 
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; 

933 
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; 

934 
val split_paired_Ex = thm "split_paired_Ex"; 

935 
val split_paired_The = thm "split_paired_The"; 

936 
val split_paired_all = thm "split_paired_all"; 

937 
val split_part = thm "split_part"; 

938 
val split_split = thm "split_split"; 

939 
val split_split_asm = thm "split_split_asm"; 

940 
val split_tupled_all = thms "split_tupled_all"; 

941 
val split_weak_cong = thm "split_weak_cong"; 

942 
val surj_pair = thm "surj_pair"; 

943 
val surjective_pairing = thm "surjective_pairing"; 

944 
val unit_abs_eta_conv = thm "unit_abs_eta_conv"; 

945 
val unit_all_eq1 = thm "unit_all_eq1"; 

946 
val unit_all_eq2 = thm "unit_all_eq2"; 

947 
val unit_eq = thm "unit_eq"; 

948 
val unit_induct = thm "unit_induct"; 

949 
*} 

950 

10213  951 
end 