src/HOL/Prod.ML
author nipkow
Mon Oct 21 09:50:50 1996 +0200 (1996-10-21)
changeset 2115 9709f9188549
parent 2089 e2ec077ac90d
child 2637 e9b203f854ae
permissions -rw-r--r--
Added trans_tac (see Provers/nat_transitive.ML)
     1 (*  Title:      HOL/prod
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 For prod.thy.  Ordered Pairs, the Cartesian product type, the unit type
     7 *)
     8 
     9 open Prod;
    10 
    11 (*This counts as a non-emptiness result for admitting 'a * 'b as a type*)
    12 goalw Prod.thy [Prod_def] "Pair_Rep a b : Prod";
    13 by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]);
    14 qed "ProdI";
    15 
    16 val [major] = goalw Prod.thy [Pair_Rep_def]
    17     "Pair_Rep a b = Pair_Rep a' b' ==> a=a' & b=b'";
    18 by (EVERY1 [rtac (major RS fun_cong RS fun_cong RS subst), 
    19             rtac conjI, rtac refl, rtac refl]);
    20 qed "Pair_Rep_inject";
    21 
    22 goal Prod.thy "inj_onto Abs_Prod Prod";
    23 by (rtac inj_onto_inverseI 1);
    24 by (etac Abs_Prod_inverse 1);
    25 qed "inj_onto_Abs_Prod";
    26 
    27 val prems = goalw Prod.thy [Pair_def]
    28     "[| (a, b) = (a',b');  [| a=a';  b=b' |] ==> R |] ==> R";
    29 by (rtac (inj_onto_Abs_Prod RS inj_ontoD RS Pair_Rep_inject RS conjE) 1);
    30 by (REPEAT (ares_tac (prems@[ProdI]) 1));
    31 qed "Pair_inject";
    32 
    33 goal Prod.thy "((a,b) = (a',b')) = (a=a' & b=b')";
    34 by (fast_tac (!claset addIs [Pair_inject]) 1);
    35 qed "Pair_eq";
    36 
    37 goalw Prod.thy [fst_def] "fst((a,b)) = a";
    38 by (fast_tac (!claset addIs [select_equality] addSEs [Pair_inject]) 1);
    39 qed "fst_conv";
    40 
    41 goalw Prod.thy [snd_def] "snd((a,b)) = b";
    42 by (fast_tac (!claset addIs [select_equality] addSEs [Pair_inject]) 1);
    43 qed "snd_conv";
    44 
    45 goalw Prod.thy [Pair_def] "? x y. p = (x,y)";
    46 by (rtac (rewrite_rule [Prod_def] Rep_Prod RS CollectE) 1);
    47 by (EVERY1[etac exE, etac exE, rtac exI, rtac exI,
    48            rtac (Rep_Prod_inverse RS sym RS trans),  etac arg_cong]);
    49 qed "PairE_lemma";
    50 
    51 val [prem] = goal Prod.thy "[| !!x y. p = (x,y) ==> Q |] ==> Q";
    52 by (rtac (PairE_lemma RS exE) 1);
    53 by (REPEAT (eresolve_tac [prem,exE] 1));
    54 qed "PairE";
    55 
    56 (* replace parameters of product type by individual component parameters *)
    57 local
    58 fun is_pair (_,Type("*",_)) = true
    59   | is_pair _ = false;
    60 
    61 fun find_pair_param prem =
    62   let val params = Logic.strip_params prem
    63   in if exists is_pair params
    64      then let val params = rev(rename_wrt_term prem params)
    65                            (*as they are printed*)
    66           in apsome fst (find_first is_pair params) end
    67      else None
    68   end;
    69 
    70 in
    71 
    72 val split_all_tac = REPEAT o SUBGOAL (fn (prem,i) =>
    73   case find_pair_param prem of
    74     None => no_tac
    75   | Some x => EVERY[res_inst_tac[("p",x)] PairE i,
    76                     REPEAT(hyp_subst_tac i), prune_params_tac]);
    77 
    78 end;
    79 
    80 goal Prod.thy "(!x. P x) = (!a b. P(a,b))";
    81 by (fast_tac (!claset addbefore split_all_tac 1) 1);
    82 qed "split_paired_All";
    83 
    84 goalw Prod.thy [split_def] "split c (a,b) = c a b";
    85 by (EVERY1[stac fst_conv, stac snd_conv]);
    86 by (rtac refl 1);
    87 qed "split";
    88 
    89 Addsimps [fst_conv, snd_conv, split_paired_All, split, Pair_eq];
    90 
    91 goal Prod.thy "(s=t) = (fst(s)=fst(t) & snd(s)=snd(t))";
    92 by (res_inst_tac[("p","s")] PairE 1);
    93 by (res_inst_tac[("p","t")] PairE 1);
    94 by (Asm_simp_tac 1);
    95 qed "Pair_fst_snd_eq";
    96 
    97 (*Prevents simplification of c: much faster*)
    98 qed_goal "split_weak_cong" Prod.thy
    99   "p=q ==> split c p = split c q"
   100   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   101 
   102 (* Do not add as rewrite rule: invalidates some proofs in IMP *)
   103 goal Prod.thy "p = (fst(p),snd(p))";
   104 by (res_inst_tac [("p","p")] PairE 1);
   105 by (Asm_simp_tac 1);
   106 qed "surjective_pairing";
   107 
   108 goal Prod.thy "p = split (%x y.(x,y)) p";
   109 by (res_inst_tac [("p","p")] PairE 1);
   110 by (Asm_simp_tac 1);
   111 qed "surjective_pairing2";
   112 
   113 qed_goal "split_eta" Prod.thy "(%(x,y). f(x,y)) = f"
   114   (fn _ => [rtac ext 1, split_all_tac 1, rtac split 1]);
   115 
   116 (*For use with split_tac and the simplifier*)
   117 goal Prod.thy "R(split c p) = (! x y. p = (x,y) --> R(c x y))";
   118 by (stac surjective_pairing 1);
   119 by (stac split 1);
   120 by (fast_tac (!claset addSEs [Pair_inject]) 1);
   121 qed "expand_split";
   122 
   123 (** split used as a logical connective or set former **)
   124 
   125 (*These rules are for use with fast_tac.
   126   Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*)
   127 
   128 goal Prod.thy "!!p. [| !!a b. p=(a,b) ==> c a b |] ==> split c p";
   129 by (split_all_tac 1);
   130 by (Asm_simp_tac 1);
   131 qed "splitI2";
   132 
   133 goal Prod.thy "!!a b c. c a b ==> split c (a,b)";
   134 by (Asm_simp_tac 1);
   135 qed "splitI";
   136 
   137 val prems = goalw Prod.thy [split_def]
   138     "[| split c p;  !!x y. [| p = (x,y);  c x y |] ==> Q |] ==> Q";
   139 by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
   140 qed "splitE";
   141 
   142 goal Prod.thy "!!R a b. split R (a,b) ==> R a b";
   143 by (etac (split RS iffD1) 1);
   144 qed "splitD";
   145 
   146 goal Prod.thy "!!a b c. z: c a b ==> z: split c (a,b)";
   147 by (Asm_simp_tac 1);
   148 qed "mem_splitI";
   149 
   150 goal Prod.thy "!!p. [| !!a b. p=(a,b) ==> z: c a b |] ==> z: split c p";
   151 by (split_all_tac 1);
   152 by (Asm_simp_tac 1);
   153 qed "mem_splitI2";
   154 
   155 val prems = goalw Prod.thy [split_def]
   156     "[| z: split c p;  !!x y. [| p = (x,y);  z: c x y |] ==> Q |] ==> Q";
   157 by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
   158 qed "mem_splitE";
   159 
   160 (*** prod_fun -- action of the product functor upon functions ***)
   161 
   162 goalw Prod.thy [prod_fun_def] "prod_fun f g (a,b) = (f(a),g(b))";
   163 by (rtac split 1);
   164 qed "prod_fun";
   165 
   166 goal Prod.thy 
   167     "prod_fun (f1 o f2) (g1 o g2) = ((prod_fun f1 g1) o (prod_fun f2 g2))";
   168 by (rtac ext 1);
   169 by (res_inst_tac [("p","x")] PairE 1);
   170 by (asm_simp_tac (!simpset addsimps [prod_fun,o_def]) 1);
   171 qed "prod_fun_compose";
   172 
   173 goal Prod.thy "prod_fun (%x.x) (%y.y) = (%z.z)";
   174 by (rtac ext 1);
   175 by (res_inst_tac [("p","z")] PairE 1);
   176 by (asm_simp_tac (!simpset addsimps [prod_fun]) 1);
   177 qed "prod_fun_ident";
   178 
   179 val prems = goal Prod.thy "(a,b):r ==> (f(a),g(b)) : (prod_fun f g)``r";
   180 by (rtac image_eqI 1);
   181 by (rtac (prod_fun RS sym) 1);
   182 by (resolve_tac prems 1);
   183 qed "prod_fun_imageI";
   184 
   185 val major::prems = goal Prod.thy
   186     "[| c: (prod_fun f g)``r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P  \
   187 \    |] ==> P";
   188 by (rtac (major RS imageE) 1);
   189 by (res_inst_tac [("p","x")] PairE 1);
   190 by (resolve_tac prems 1);
   191 by (Fast_tac 2);
   192 by (fast_tac (!claset addIs [prod_fun]) 1);
   193 qed "prod_fun_imageE";
   194 
   195 (*** Disjoint union of a family of sets - Sigma ***)
   196 
   197 qed_goalw "SigmaI" Prod.thy [Sigma_def]
   198     "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
   199  (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]);
   200 
   201 (*The general elimination rule*)
   202 qed_goalw "SigmaE" Prod.thy [Sigma_def]
   203     "[| c: Sigma A B;  \
   204 \       !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P \
   205 \    |] ==> P"
   206  (fn major::prems=>
   207   [ (cut_facts_tac [major] 1),
   208     (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]);
   209 
   210 (** Elimination of (a,b):A*B -- introduces no eigenvariables **)
   211 qed_goal "SigmaD1" Prod.thy "(a,b) : Sigma A B ==> a : A"
   212  (fn [major]=>
   213   [ (rtac (major RS SigmaE) 1),
   214     (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
   215 
   216 qed_goal "SigmaD2" Prod.thy "(a,b) : Sigma A B ==> b : B(a)"
   217  (fn [major]=>
   218   [ (rtac (major RS SigmaE) 1),
   219     (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
   220 
   221 qed_goal "SigmaE2" Prod.thy
   222     "[| (a,b) : Sigma A B;    \
   223 \       [| a:A;  b:B(a) |] ==> P   \
   224 \    |] ==> P"
   225  (fn [major,minor]=>
   226   [ (rtac minor 1),
   227     (rtac (major RS SigmaD1) 1),
   228     (rtac (major RS SigmaD2) 1) ]);
   229 
   230 val prems = goal Prod.thy
   231     "[| A<=C;  !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D";
   232 by (cut_facts_tac prems 1);
   233 by (fast_tac (!claset addIs (prems RL [subsetD]) 
   234                      addSIs [SigmaI] 
   235                      addSEs [SigmaE]) 1);
   236 qed "Sigma_mono";
   237 
   238 qed_goal "Sigma_empty1" Prod.thy "Sigma {} B = {}"
   239  (fn _ => [ (fast_tac (!claset addSEs [SigmaE]) 1) ]);
   240 
   241 qed_goal "Sigma_empty2" Prod.thy "A Times {} = {}"
   242  (fn _ => [ (fast_tac (!claset addSEs [SigmaE]) 1) ]);
   243 
   244 Addsimps [Sigma_empty1,Sigma_empty2]; 
   245 
   246 goal Prod.thy "((a,b): Sigma A B) = (a:A & b:B(a))";
   247 by (fast_tac (!claset addSIs [SigmaI] addSEs [SigmaE, Pair_inject]) 1);
   248 qed "mem_Sigma_iff";
   249 Addsimps [mem_Sigma_iff]; 
   250 
   251 
   252 (*** Domain of a relation ***)
   253 
   254 val prems = goalw Prod.thy [image_def] "(a,b) : r ==> a : fst``r";
   255 by (rtac CollectI 1);
   256 by (rtac bexI 1);
   257 by (rtac (fst_conv RS sym) 1);
   258 by (resolve_tac prems 1);
   259 qed "fst_imageI";
   260 
   261 val major::prems = goal Prod.thy
   262     "[| a : fst``r;  !!y.[| (a,y) : r |] ==> P |] ==> P"; 
   263 by (rtac (major RS imageE) 1);
   264 by (resolve_tac prems 1);
   265 by (etac ssubst 1);
   266 by (rtac (surjective_pairing RS subst) 1);
   267 by (assume_tac 1);
   268 qed "fst_imageE";
   269 
   270 (*** Range of a relation ***)
   271 
   272 val prems = goalw Prod.thy [image_def] "(a,b) : r ==> b : snd``r";
   273 by (rtac CollectI 1);
   274 by (rtac bexI 1);
   275 by (rtac (snd_conv RS sym) 1);
   276 by (resolve_tac prems 1);
   277 qed "snd_imageI";
   278 
   279 val major::prems = goal Prod.thy
   280     "[| a : snd``r;  !!y.[| (y,a) : r |] ==> P |] ==> P"; 
   281 by (rtac (major RS imageE) 1);
   282 by (resolve_tac prems 1);
   283 by (etac ssubst 1);
   284 by (rtac (surjective_pairing RS subst) 1);
   285 by (assume_tac 1);
   286 qed "snd_imageE";
   287 
   288 (** Exhaustion rule for unit -- a degenerate form of induction **)
   289 
   290 goalw Prod.thy [Unity_def]
   291     "u = ()";
   292 by (stac (rewrite_rule [Unit_def] Rep_Unit RS CollectD RS sym) 1);
   293 by (rtac (Rep_Unit_inverse RS sym) 1);
   294 qed "unit_eq";
   295  
   296 AddSIs [SigmaI, splitI, splitI2, mem_splitI, mem_splitI2];
   297 AddIs  [fst_imageI, snd_imageI, prod_fun_imageI];
   298 AddSEs [SigmaE2, SigmaE, splitE, mem_splitE, 
   299         fst_imageE, snd_imageE, prod_fun_imageE,
   300         Pair_inject];
   301 
   302 structure Prod_Syntax =
   303 struct
   304 
   305 val unitT = Type("unit",[]);
   306 
   307 fun mk_prod (T1,T2) = Type("*", [T1,T2]);
   308 
   309 (*Maps the type T1*...*Tn to [T1,...,Tn], however nested*)
   310 fun factors (Type("*", [T1,T2])) = factors T1 @ factors T2
   311   | factors T                    = [T];
   312 
   313 (*Make a correctly typed ordered pair*)
   314 fun mk_Pair (t1,t2) = 
   315   let val T1 = fastype_of t1
   316       and T2 = fastype_of t2
   317   in  Const("Pair", [T1, T2] ---> mk_prod(T1,T2)) $ t1 $ t2  end;
   318    
   319 fun split_const(Ta,Tb,Tc) = 
   320     Const("split", [[Ta,Tb]--->Tc, mk_prod(Ta,Tb)] ---> Tc);
   321 
   322 (*In ap_split S T u, term u expects separate arguments for the factors of S,
   323   with result type T.  The call creates a new term expecting one argument
   324   of type S.*)
   325 fun ap_split (Type("*", [T1,T2])) T3 u = 
   326       split_const(T1,T2,T3) $ 
   327       Abs("v", T1, 
   328           ap_split T2 T3
   329              ((ap_split T1 (factors T2 ---> T3) (incr_boundvars 1 u)) $ 
   330               Bound 0))
   331   | ap_split T T3 u = u;
   332 
   333 (*Makes a nested tuple from a list, following the product type structure*)
   334 fun mk_tuple (Type("*", [T1,T2])) tms = 
   335         mk_Pair (mk_tuple T1 tms, 
   336                  mk_tuple T2 (drop (length (factors T1), tms)))
   337   | mk_tuple T (t::_) = t;
   338 
   339 (*Attempts to remove occurrences of split, and pair-valued parameters*)
   340 val remove_split = rewrite_rule [split RS eq_reflection]  o  
   341                    rule_by_tactic (ALLGOALS split_all_tac);
   342 
   343 (*Uncurries any Var of function type in the rule*)
   344 fun split_rule_var (t as Var(v, Type("fun",[T1,T2])), rl) =
   345       let val T' = factors T1 ---> T2
   346           val newt = ap_split T1 T2 (Var(v,T'))
   347           val cterm = Thm.cterm_of (#sign(rep_thm rl))
   348       in
   349           remove_split (instantiate ([], [(cterm t, cterm newt)]) rl)
   350       end
   351   | split_rule_var (t,rl) = rl;
   352 
   353 (*Uncurries ALL function variables occurring in a rule's conclusion*)
   354 fun split_rule rl = foldr split_rule_var (term_vars (concl_of rl), rl)
   355                     |> standard;
   356 
   357 end;