src/HOL/Product_Type.thy
 author paulson Mon Dec 13 15:06:59 2004 +0100 (2004-12-13) changeset 15404 a9a762f586b5 parent 15394 a2c34e6ca4f8 child 15422 cbdddc0efadf permissions -rw-r--r--
removal of NatArith.ML and Product_Type.ML
1 (*  Title:      HOL/Product_Type.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1992  University of Cambridge
5 *)
7 header {* Cartesian products *}
9 theory Product_Type
10 imports Fun
11 files ("Tools/split_rule.ML")
12 begin
14 subsection {* Unit *}
16 typedef unit = "{True}"
17 proof
18   show "True : ?unit" by blast
19 qed
21 constdefs
22   Unity :: unit    ("'(')")
23   "() == Abs_unit True"
25 lemma unit_eq: "u = ()"
26   by (induct u) (simp add: unit_def Unity_def)
28 text {*
29   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
30   this rule directly --- it loops!
31 *}
33 ML_setup {*
34   val unit_eq_proc =
35     let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in
36       Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"]
37       (fn _ => fn _ => fn t => if HOLogic.is_unit t then None else Some unit_meta_eq)
38     end;
41 *}
43 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
44   by simp
46 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
47   by (rule triv_forall_equality)
49 lemma unit_induct [induct type: unit]: "P () ==> P x"
50   by simp
52 text {*
53   This rewrite counters the effect of @{text unit_eq_proc} on @{term
54   [source] "%u::unit. f u"}, replacing it by @{term [source]
55   f} rather than by @{term [source] "%u. f ()"}.
56 *}
58 lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
59   by (rule ext) simp
62 subsection {* Pairs *}
64 subsubsection {* Type definition *}
66 constdefs
67   Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
68   "Pair_Rep == (%a b. %x y. x=a & y=b)"
70 global
72 typedef (Prod)
73   ('a, 'b) "*"    (infixr 20)
74     = "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
75 proof
76   fix a b show "Pair_Rep a b : ?Prod"
77     by blast
78 qed
80 syntax (xsymbols)
81   "*"      :: "[type, type] => type"         ("(_ \\<times>/ _)" [21, 20] 20)
82 syntax (HTML output)
83   "*"      :: "[type, type] => type"         ("(_ \\<times>/ _)" [21, 20] 20)
85 local
88 subsubsection {* Abstract constants and syntax *}
90 global
92 consts
93   fst      :: "'a * 'b => 'a"
94   snd      :: "'a * 'b => 'b"
95   split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
96   curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
97   prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
98   Pair     :: "['a, 'b] => 'a * 'b"
99   Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
101 local
103 text {*
104   Patterns -- extends pre-defined type @{typ pttrn} used in
105   abstractions.
106 *}
108 nonterminals
109   tuple_args patterns
111 syntax
112   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
113   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
114   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
115   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
116   ""            :: "pttrn => patterns"                  ("_")
117   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
118   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
119   "@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)
121 translations
122   "(x, y)"       == "Pair x y"
123   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
124   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
125   "%(x,y).b"     == "split(%x y. b)"
126   "_abs (Pair x y) t" => "%(x,y).t"
127   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
128      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
130   "SIGMA x:A. B" => "Sigma A (%x. B)"
131   "A <*> B"      => "Sigma A (_K B)"
133 (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
134 (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
135 print_translation {*
136 let fun split_tr' [Abs (x,T,t as (Abs abs))] =
137       (* split (%x y. t) => %(x,y) t *)
138       let val (y,t') = atomic_abs_tr' abs;
139           val (x',t'') = atomic_abs_tr' (x,T,t');
141       in Syntax.const "_abs" \$ (Syntax.const "_pattern" \$x'\$y) \$ t'' end
142     | split_tr' [Abs (x,T,(s as Const ("split",_)\$t))] =
143        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
144        let val (Const ("_abs",_)\$(Const ("_pattern",_)\$y\$z)\$t') = split_tr' [t];
145            val (x',t'') = atomic_abs_tr' (x,T,t');
146        in Syntax.const "_abs"\$
147            (Syntax.const "_pattern"\$x'\$(Syntax.const "_patterns"\$y\$z))\$t'' end
148     | split_tr' [Const ("split",_)\$t] =
149        (* split (split (%x y z. t)) => %((x,y),z). t *)
150        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
151     | split_tr' [Const ("_abs",_)\$x_y\$(Abs abs)] =
152        (* split (%pttrn z. t) => %(pttrn,z). t *)
153        let val (z,t) = atomic_abs_tr' abs;
154        in Syntax.const "_abs" \$ (Syntax.const "_pattern" \$x_y\$z) \$ t end
155     | split_tr' _ =  raise Match;
156 in [("split", split_tr')]
157 end
158 *}
160 text{*Deleted x-symbol and html support using @{text"\\<Sigma>"} (Sigma) because of the danger of confusion with Sum.*}
161 syntax (xsymbols)
162   "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \\<times> _" [81, 80] 80)
164 syntax (HTML output)
165   "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \\<times> _" [81, 80] 80)
167 print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}
170 subsubsection {* Definitions *}
172 defs
173   Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
174   fst_def:      "fst p == THE a. EX b. p = (a, b)"
175   snd_def:      "snd p == THE b. EX a. p = (a, b)"
176   split_def:    "split == (%c p. c (fst p) (snd p))"
177   curry_def:    "curry == (%c x y. c (x,y))"
178   prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
179   Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
182 subsubsection {* Lemmas and proof tool setup *}
184 lemma ProdI: "Pair_Rep a b : Prod"
185   by (unfold Prod_def) blast
187 lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
188   apply (unfold Pair_Rep_def)
189   apply (drule fun_cong [THEN fun_cong], blast)
190   done
192 lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
193   apply (rule inj_on_inverseI)
194   apply (erule Abs_Prod_inverse)
195   done
197 lemma Pair_inject:
198   "(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R"
199 proof -
200   case rule_context [unfolded Pair_def]
201   show ?thesis
202     apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
203     apply (rule rule_context ProdI)+
204     .
205 qed
207 lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
208   by (blast elim!: Pair_inject)
210 lemma fst_conv [simp]: "fst (a, b) = a"
211   by (unfold fst_def) blast
213 lemma snd_conv [simp]: "snd (a, b) = b"
214   by (unfold snd_def) blast
216 lemma fst_eqD: "fst (x, y) = a ==> x = a"
217   by simp
219 lemma snd_eqD: "snd (x, y) = a ==> y = a"
220   by simp
222 lemma PairE_lemma: "EX x y. p = (x, y)"
223   apply (unfold Pair_def)
224   apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
225   apply (erule exE, erule exE, rule exI, rule exI)
226   apply (rule Rep_Prod_inverse [symmetric, THEN trans])
227   apply (erule arg_cong)
228   done
230 lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
231   by (insert PairE_lemma [of p]) blast
233 ML_setup {*
234   local val PairE = thm "PairE" in
235     fun pair_tac s =
236       EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
237   end;
238 *}
240 lemma surjective_pairing: "p = (fst p, snd p)"
241   -- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
242   by (cases p) simp
244 declare surjective_pairing [symmetric, simp]
246 lemma surj_pair [simp]: "EX x y. z = (x, y)"
247   apply (rule exI)
248   apply (rule exI)
249   apply (rule surjective_pairing)
250   done
252 lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
253 proof
254   fix a b
255   assume "!!x. PROP P x"
256   thus "PROP P (a, b)" .
257 next
258   fix x
259   assume "!!a b. PROP P (a, b)"
260   hence "PROP P (fst x, snd x)" .
261   thus "PROP P x" by simp
262 qed
264 lemmas split_tupled_all = split_paired_all unit_all_eq2
266 text {*
267   The rule @{thm [source] split_paired_all} does not work with the
268   Simplifier because it also affects premises in congrence rules,
269   where this can lead to premises of the form @{text "!!a b. ... =
270   ?P(a, b)"} which cannot be solved by reflexivity.
271 *}
273 ML_setup "
274   (* replace parameters of product type by individual component parameters *)
275   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
276   local (* filtering with exists_paired_all is an essential optimization *)
277     fun exists_paired_all (Const (\"all\", _) \$ Abs (_, T, t)) =
278           can HOLogic.dest_prodT T orelse exists_paired_all t
279       | exists_paired_all (t \$ u) = exists_paired_all t orelse exists_paired_all u
280       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
281       | exists_paired_all _ = false;
282     val ss = HOL_basic_ss
283       addsimps [thm \"split_paired_all\", thm \"unit_all_eq2\", thm \"unit_abs_eta_conv\"]
285   in
286     val split_all_tac = SUBGOAL (fn (t, i) =>
287       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
288     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
289       if exists_paired_all t then full_simp_tac ss i else no_tac);
290     fun split_all th =
291    if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
292   end;
294 claset_ref() := claset() addSbefore (\"split_all_tac\", split_all_tac);
295 "
297 lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
298   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
299   by fast
301 lemma curry_split [simp]: "curry (split f) = f"
302   by (simp add: curry_def split_def)
304 lemma split_curry [simp]: "split (curry f) = f"
305   by (simp add: curry_def split_def)
307 lemma curryI [intro!]: "f (a,b) ==> curry f a b"
310 lemma curryD [dest!]: "curry f a b ==> f (a,b)"
313 lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
316 lemma curry_conv [simp]: "curry f a b = f (a,b)"
319 lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
320   by fast
322 lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
323   by fast
325 lemma split_conv [simp]: "split c (a, b) = c a b"
328 lemmas split = split_conv  -- {* for backwards compatibility *}
330 lemmas splitI = split_conv [THEN iffD2, standard]
331 lemmas splitD = split_conv [THEN iffD1, standard]
333 lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
334   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
335   apply (rule ext)
336   apply (tactic {* pair_tac "x" 1 *}, simp)
337   done
339 lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
340   -- {* Can't be added to simpset: loops! *}
343 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
346 lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
347 by (simp only: split_tupled_all, simp)
349 lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
352 lemma split_weak_cong: "p = q ==> split c p = split c q"
353   -- {* Prevents simplification of @{term c}: much faster *}
354   by (erule arg_cong)
356 lemma split_eta: "(%(x, y). f (x, y)) = f"
357   apply (rule ext)
358   apply (simp only: split_tupled_all)
359   apply (rule split_conv)
360   done
362 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
365 text {*
366   Simplification procedure for @{thm [source] cond_split_eta}.  Using
367   @{thm [source] split_eta} as a rewrite rule is not general enough,
368   and using @{thm [source] cond_split_eta} directly would render some
369   existing proofs very inefficient; similarly for @{text
370   split_beta}. *}
372 ML_setup {*
374 local
375   val cond_split_eta = thm "cond_split_eta";
376   fun  Pair_pat k 0 (Bound m) = (m = k)
377   |    Pair_pat k i (Const ("Pair",  _) \$ Bound m \$ t) = i > 0 andalso
378                         m = k+i andalso Pair_pat k (i-1) t
379   |    Pair_pat _ _ _ = false;
380   fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
381   |   no_args k i (t \$ u) = no_args k i t andalso no_args k i u
382   |   no_args k i (Bound m) = m < k orelse m > k+i
383   |   no_args _ _ _ = true;
384   fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then Some (i,t) else None
385   |   split_pat tp i (Const ("split", _) \$ Abs (_, _, t)) = split_pat tp (i+1) t
386   |   split_pat tp i _ = None;
387   fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] []
388         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
389         (K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1)));
391   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
392   |   beta_term_pat k i (t \$ u) = Pair_pat k i (t \$ u) orelse
393                         (beta_term_pat k i t andalso beta_term_pat k i u)
394   |   beta_term_pat k i t = no_args k i t;
395   fun  eta_term_pat k i (f \$ arg) = no_args k i f andalso Pair_pat k i arg
396   |    eta_term_pat _ _ _ = false;
397   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
398   |   subst arg k i (t \$ u) = if Pair_pat k i (t \$ u) then incr_boundvars k arg
399                               else (subst arg k i t \$ subst arg k i u)
400   |   subst arg k i t = t;
401   fun beta_proc sg _ (s as Const ("split", _) \$ Abs (_, _, t) \$ arg) =
402         (case split_pat beta_term_pat 1 t of
403         Some (i,f) => Some (metaeq sg s (subst arg 0 i f))
404         | None => None)
405   |   beta_proc _ _ _ = None;
406   fun eta_proc sg _ (s as Const ("split", _) \$ Abs (_, _, t)) =
407         (case split_pat eta_term_pat 1 t of
408           Some (_,ft) => Some (metaeq sg s (let val (f \$ arg) = ft in f end))
409         | None => None)
410   |   eta_proc _ _ _ = None;
411 in
412   val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
413     "split_beta" ["split f z"] beta_proc;
414   val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
415     "split_eta" ["split f"] eta_proc;
416 end;
419 *}
421 lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
422   by (subst surjective_pairing, rule split_conv)
424 lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))"
425   -- {* For use with @{text split} and the Simplifier. *}
426   apply (subst surjective_pairing)
427   apply (subst split_conv, blast)
428   done
430 text {*
431   @{thm [source] split_split} could be declared as @{text "[split]"}
432   done after the Splitter has been speeded up significantly;
433   precompute the constants involved and don't do anything unless the
434   current goal contains one of those constants.
435 *}
437 lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
438 by (subst split_split, simp)
441 text {*
442   \medskip @{term split} used as a logical connective or set former.
444   \medskip These rules are for use with @{text blast}; could instead
445   call @{text simp} using @{thm [source] split} as rewrite. *}
447 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
448   apply (simp only: split_tupled_all)
449   apply (simp (no_asm_simp))
450   done
452 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
453   apply (simp only: split_tupled_all)
454   apply (simp (no_asm_simp))
455   done
457 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
458   by (induct p) (auto simp add: split_def)
460 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
461   by (induct p) (auto simp add: split_def)
463 lemma splitE2:
464   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
465 proof -
466   assume q: "Q (split P z)"
467   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
468   show R
469     apply (rule r surjective_pairing)+
470     apply (rule split_beta [THEN subst], rule q)
471     done
472 qed
474 lemma splitD': "split R (a,b) c ==> R a b c"
475   by simp
477 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
478   by simp
480 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
481 by (simp only: split_tupled_all, simp)
483 lemma mem_splitE: "[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"
484 proof -
485   case rule_context [unfolded split_def]
486   show ?thesis by (rule rule_context surjective_pairing)+
487 qed
489 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
490 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
492 ML_setup "
493 local (* filtering with exists_p_split is an essential optimization *)
494   fun exists_p_split (Const (\"split\",_) \$ _ \$ (Const (\"Pair\",_)\$_\$_)) = true
495     | exists_p_split (t \$ u) = exists_p_split t orelse exists_p_split u
496     | exists_p_split (Abs (_, _, t)) = exists_p_split t
497     | exists_p_split _ = false;
498   val ss = HOL_basic_ss addsimps [thm \"split_conv\"];
499 in
500 val split_conv_tac = SUBGOAL (fn (t, i) =>
501     if exists_p_split t then safe_full_simp_tac ss i else no_tac);
502 end;
503 (* This prevents applications of splitE for already splitted arguments leading
504    to quite time-consuming computations (in particular for nested tuples) *)
505 claset_ref() := claset() addSbefore (\"split_conv_tac\", split_conv_tac);
506 "
508 lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
509 by (rule ext, fast)
511 lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
512 by (rule ext, fast)
514 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
515   -- {* Allows simplifications of nested splits in case of independent predicates. *}
516   apply (rule ext, blast)
517   done
519 (* Do NOT make this a simp rule as it
520    a) only helps in special situations
521    b) can lead to nontermination in the presence of split_def
522 *)
523 lemma split_comp_eq:
524 "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
525 by (rule ext, auto)
527 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
528   by blast
530 (*
531 the following  would be slightly more general,
532 but cannot be used as rewrite rule:
533 ### Cannot add premise as rewrite rule because it contains (type) unknowns:
534 ### ?y = .x
535 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
536 by (rtac some_equality 1)
537 by ( Simp_tac 1)
538 by (split_all_tac 1)
539 by (Asm_full_simp_tac 1)
540 qed "The_split_eq";
541 *)
543 lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
544   by auto
547 text {*
548   \bigskip @{term prod_fun} --- action of the product functor upon
549   functions.
550 *}
552 lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)"
555 lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
556   apply (rule ext)
557   apply (tactic {* pair_tac "x" 1 *}, simp)
558   done
560 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
561   apply (rule ext)
562   apply (tactic {* pair_tac "z" 1 *}, simp)
563   done
565 lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
566   apply (rule image_eqI)
567   apply (rule prod_fun [symmetric], assumption)
568   done
570 lemma prod_fun_imageE [elim!]:
571   "[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P
572     |] ==> P"
573 proof -
574   case rule_context
575   assume major: "c: (prod_fun f g)`r"
576   show ?thesis
577     apply (rule major [THEN imageE])
578     apply (rule_tac p = x in PairE)
579     apply (rule rule_context)
580      prefer 2
581      apply blast
582     apply (blast intro: prod_fun)
583     done
584 qed
587 constdefs
588   upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b"
589  "upd_fst f == prod_fun f id"
591   upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c"
592  "upd_snd f == prod_fun id f"
594 lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)"
597 lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)"
600 text {*
601   \bigskip Disjoint union of a family of sets -- Sigma.
602 *}
604 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
605   by (unfold Sigma_def) blast
607 lemma SigmaE [elim!]:
608     "[| c: Sigma A B;
609         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
610      |] ==> P"
611   -- {* The general elimination rule. *}
612   by (unfold Sigma_def) blast
614 text {*
615   Elimination of @{term "(a, b) : A \\<times> B"} -- introduces no
616   eigenvariables.
617 *}
619 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
620 by blast
622 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
623 by blast
625 lemma SigmaE2:
626     "[| (a, b) : Sigma A B;
627         [| a:A;  b:B(a) |] ==> P
628      |] ==> P"
629   by blast
631 lemma Sigma_cong:
632      "\\<lbrakk>A = B; !!x. x \\<in> B \\<Longrightarrow> C x = D x\\<rbrakk>
633       \\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
634 by auto
636 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
637   by blast
639 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
640   by blast
642 lemma Sigma_empty2 [simp]: "A <*> {} = {}"
643   by blast
645 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
646   by auto
648 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
649   by auto
651 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
652   by auto
654 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
655   by blast
657 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
658   by blast
660 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
661   by (blast elim: equalityE)
663 lemma SetCompr_Sigma_eq:
664     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
665   by blast
667 text {*
668   \bigskip Complex rules for Sigma.
669 *}
671 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
672   by blast
674 lemma UN_Times_distrib:
675   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
676   -- {* Suggested by Pierre Chartier *}
677   by blast
679 lemma split_paired_Ball_Sigma [simp]:
680     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
681   by blast
683 lemma split_paired_Bex_Sigma [simp]:
684     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
685   by blast
687 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
688   by blast
690 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
691   by blast
693 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
694   by blast
696 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
697   by blast
699 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
700   by blast
702 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
703   by blast
705 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
706   by blast
708 text {*
709   Non-dependent versions are needed to avoid the need for higher-order
710   matching, especially when the rules are re-oriented.
711 *}
713 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
714   by blast
716 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
717   by blast
719 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
720   by blast
723 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
724   apply (rule_tac x = "(a, b)" in image_eqI)
725    apply auto
726   done
729 text {*
730   Setup of internal @{text split_rule}.
731 *}
733 constdefs
734   internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
735   "internal_split == split"
737 lemma internal_split_conv: "internal_split c (a, b) = c a b"
738   by (simp only: internal_split_def split_conv)
740 hide const internal_split
742 use "Tools/split_rule.ML"
743 setup SplitRule.setup
746 subsection {* Code generator setup *}
748 types_code
749   "*"     ("(_ */ _)")
751 consts_code
752   "Pair"    ("(_,/ _)")
753   "fst"     ("fst")
754   "snd"     ("snd")
756 ML {*
757 fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U \$ f x \$ g y;
759 fun gen_id_42 aG bG i = (aG i, bG i);
761 local
763 fun strip_abs 0 t = ([], t)
764   | strip_abs i (Abs (s, T, t)) =
765     let
766       val s' = Codegen.new_name t s;
767       val v = Free (s', T)
768     in apfst (cons v) (strip_abs (i-1) (subst_bound (v, t))) end
769   | strip_abs i (u as Const ("split", _) \$ t) = (case strip_abs (i+1) t of
770         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
771       | _ => ([], u))
772   | strip_abs i t = ([], t);
774 fun let_codegen thy gr dep brack (t as Const ("Let", _) \$ _ \$ _) =
775     let
776       fun dest_let (l as Const ("Let", _) \$ t \$ u) =
777           (case strip_abs 1 u of
778              ([p], u') => apfst (cons (p, t)) (dest_let u')
779            | _ => ([], l))
780         | dest_let t = ([], t);
781       fun mk_code (gr, (l, r)) =
782         let
783           val (gr1, pl) = Codegen.invoke_codegen thy dep false (gr, l);
784           val (gr2, pr) = Codegen.invoke_codegen thy dep false (gr1, r);
785         in (gr2, (pl, pr)) end
786     in case dest_let t of
787         ([], _) => None
788       | (ps, u) =>
789           let
790             val (gr1, qs) = foldl_map mk_code (gr, ps);
791             val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u)
792           in
793             Some (gr2, Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, flat
794                 (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
795                   [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
796                      Pretty.brk 1, pr]]) qs))),
797               Pretty.brk 1, Pretty.str "in ", pu,
798               Pretty.brk 1, Pretty.str "end"]))
799           end
800     end
801   | let_codegen thy gr dep brack t = None;
803 fun split_codegen thy gr dep brack (t as Const ("split", _) \$ _) =
804     (case strip_abs 1 t of
805        ([p], u) =>
806          let
807            val (gr1, q) = Codegen.invoke_codegen thy dep false (gr, p);
808            val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u)
809          in
810            Some (gr2, Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
811              Pretty.brk 1, pu, Pretty.str ")"])
812          end
813      | _ => None)
814   | split_codegen thy gr dep brack t = None;
816 in
818 val prod_codegen_setup =
822 end;
823 *}
825 setup prod_codegen_setup
827 ML
828 {*
829 val Collect_split = thm "Collect_split";
830 val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
831 val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
832 val PairE = thm "PairE";
833 val PairE_lemma = thm "PairE_lemma";
834 val Pair_Rep_inject = thm "Pair_Rep_inject";
835 val Pair_def = thm "Pair_def";
836 val Pair_eq = thm "Pair_eq";
837 val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
838 val Pair_inject = thm "Pair_inject";
839 val ProdI = thm "ProdI";
840 val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
843 val SigmaE = thm "SigmaE";
844 val SigmaE2 = thm "SigmaE2";
845 val SigmaI = thm "SigmaI";
846 val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
847 val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
848 val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
849 val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
850 val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
851 val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
852 val Sigma_Union = thm "Sigma_Union";
853 val Sigma_def = thm "Sigma_def";
854 val Sigma_empty1 = thm "Sigma_empty1";
855 val Sigma_empty2 = thm "Sigma_empty2";
856 val Sigma_mono = thm "Sigma_mono";
857 val The_split = thm "The_split";
858 val The_split_eq = thm "The_split_eq";
859 val The_split_eq = thm "The_split_eq";
860 val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
861 val Times_Int_distrib1 = thm "Times_Int_distrib1";
862 val Times_Un_distrib1 = thm "Times_Un_distrib1";
863 val Times_eq_cancel2 = thm "Times_eq_cancel2";
864 val Times_subset_cancel2 = thm "Times_subset_cancel2";
865 val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
866 val UN_Times_distrib = thm "UN_Times_distrib";
867 val Unity_def = thm "Unity_def";
868 val cond_split_eta = thm "cond_split_eta";
869 val fst_conv = thm "fst_conv";
870 val fst_def = thm "fst_def";
871 val fst_eqD = thm "fst_eqD";
872 val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
873 val injective_fst_snd = thm "injective_fst_snd";
874 val mem_Sigma_iff = thm "mem_Sigma_iff";
875 val mem_splitE = thm "mem_splitE";
876 val mem_splitI = thm "mem_splitI";
877 val mem_splitI2 = thm "mem_splitI2";
878 val prod_eqI = thm "prod_eqI";
879 val prod_fun = thm "prod_fun";
880 val prod_fun_compose = thm "prod_fun_compose";
881 val prod_fun_def = thm "prod_fun_def";
882 val prod_fun_ident = thm "prod_fun_ident";
883 val prod_fun_imageE = thm "prod_fun_imageE";
884 val prod_fun_imageI = thm "prod_fun_imageI";
885 val prod_induct = thm "prod_induct";
886 val snd_conv = thm "snd_conv";
887 val snd_def = thm "snd_def";
888 val snd_eqD = thm "snd_eqD";
889 val split = thm "split";
890 val splitD = thm "splitD";
891 val splitD' = thm "splitD'";
892 val splitE = thm "splitE";
893 val splitE' = thm "splitE'";
894 val splitE2 = thm "splitE2";
895 val splitI = thm "splitI";
896 val splitI2 = thm "splitI2";
897 val splitI2' = thm "splitI2'";
898 val split_Pair_apply = thm "split_Pair_apply";
899 val split_beta = thm "split_beta";
900 val split_conv = thm "split_conv";
901 val split_def = thm "split_def";
902 val split_eta = thm "split_eta";
903 val split_eta_SetCompr = thm "split_eta_SetCompr";
904 val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
905 val split_paired_All = thm "split_paired_All";
906 val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
907 val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
908 val split_paired_Ex = thm "split_paired_Ex";
909 val split_paired_The = thm "split_paired_The";
910 val split_paired_all = thm "split_paired_all";
911 val split_part = thm "split_part";
912 val split_split = thm "split_split";
913 val split_split_asm = thm "split_split_asm";
914 val split_tupled_all = thms "split_tupled_all";
915 val split_weak_cong = thm "split_weak_cong";
916 val surj_pair = thm "surj_pair";
917 val surjective_pairing = thm "surjective_pairing";
918 val unit_abs_eta_conv = thm "unit_abs_eta_conv";
919 val unit_all_eq1 = thm "unit_all_eq1";
920 val unit_all_eq2 = thm "unit_all_eq2";
921 val unit_eq = thm "unit_eq";
922 val unit_induct = thm "unit_induct";
923 *}
925 end