diff -r f04b33ce250f -r a4dc62a46ee4 Prod.ML
--- a/Prod.ML	Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,237 +0,0 @@
-(*  Title: 	HOL/prod
-    ID:         $Id$
-    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1991  University of Cambridge
-
-For prod.thy.  Ordered Pairs, the Cartesian product type, the unit type
-*)
-
-open Prod;
-
-(*This counts as a non-emptiness result for admitting 'a * 'b as a type*)
-goalw Prod.thy [Prod_def] "Pair_Rep(a,b) : Prod";
-by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]);
-qed "ProdI";
-
-val [major] = goalw Prod.thy [Pair_Rep_def]
-    "Pair_Rep(a, b) = Pair_Rep(a',b') ==> a=a' & b=b'";
-by (EVERY1 [rtac (major RS fun_cong RS fun_cong RS subst), 
-	    rtac conjI, rtac refl, rtac refl]);
-qed "Pair_Rep_inject";
-
-goal Prod.thy "inj_onto(Abs_Prod,Prod)";
-by (rtac inj_onto_inverseI 1);
-by (etac Abs_Prod_inverse 1);
-qed "inj_onto_Abs_Prod";
-
-val prems = goalw Prod.thy [Pair_def]
-    "[| <a, b> = <a',b'>;  [| a=a';  b=b' |] ==> R |] ==> R";
-by (rtac (inj_onto_Abs_Prod RS inj_ontoD RS Pair_Rep_inject RS conjE) 1);
-by (REPEAT (ares_tac (prems@[ProdI]) 1));
-qed "Pair_inject";
-
-goal Prod.thy "(<a,b> = <a',b'>) = (a=a' & b=b')";
-by (fast_tac (set_cs addIs [Pair_inject]) 1);
-qed "Pair_eq";
-
-goalw Prod.thy [fst_def] "fst(<a,b>) = a";
-by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1);
-qed "fst_conv";
-
-goalw Prod.thy [snd_def] "snd(<a,b>) = b";
-by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1);
-qed "snd_conv";
-
-goalw Prod.thy [Pair_def] "? x y. p = <x,y>";
-by (rtac (rewrite_rule [Prod_def] Rep_Prod RS CollectE) 1);
-by (EVERY1[etac exE, etac exE, rtac exI, rtac exI,
-	   rtac (Rep_Prod_inverse RS sym RS trans),  etac arg_cong]);
-qed "PairE_lemma";
-
-val [prem] = goal Prod.thy "[| !!x y. p = <x,y> ==> Q |] ==> Q";
-by (rtac (PairE_lemma RS exE) 1);
-by (REPEAT (eresolve_tac [prem,exE] 1));
-qed "PairE";
-
-goalw Prod.thy [split_def] "split(c, <a,b>) = c(a,b)";
-by (sstac [fst_conv, snd_conv] 1);
-by (rtac refl 1);
-qed "split";
-
-val prod_ss = set_ss addsimps [fst_conv, snd_conv, split, Pair_eq];
-
-goal Prod.thy "(s=t) = (fst(s)=fst(t) & snd(s)=snd(t))";
-by (res_inst_tac[("p","s")] PairE 1);
-by (res_inst_tac[("p","t")] PairE 1);
-by (asm_simp_tac prod_ss 1);
-qed "Pair_fst_snd_eq";
-
-(*Prevents simplification of c: much faster*)
-qed_goal "split_weak_cong" Prod.thy
-  "p=q ==> split(c,p) = split(c,q)"
-  (fn [prem] => [rtac (prem RS arg_cong) 1]);
-
-(* Do not add as rewrite rule: invalidates some proofs in IMP *)
-goal Prod.thy "p = <fst(p),snd(p)>";
-by (res_inst_tac [("p","p")] PairE 1);
-by (asm_simp_tac prod_ss 1);
-qed "surjective_pairing";
-
-goal Prod.thy "p = split(%x y.<x,y>, p)";
-by (res_inst_tac [("p","p")] PairE 1);
-by (asm_simp_tac prod_ss 1);
-qed "surjective_pairing2";
-
-(*For use with split_tac and the simplifier*)
-goal Prod.thy "R(split(c,p)) = (! x y. p = <x,y> --> R(c(x,y)))";
-by (stac surjective_pairing 1);
-by (stac split 1);
-by (fast_tac (HOL_cs addSEs [Pair_inject]) 1);
-qed "expand_split";
-
-(** split used as a logical connective or set former **)
-
-(*These rules are for use with fast_tac.
-  Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*)
-
-goal Prod.thy "!!a b c. c(a,b) ==> split(c, <a,b>)";
-by (asm_simp_tac prod_ss 1);
-qed "splitI";
-
-val prems = goalw Prod.thy [split_def]
-    "[| split(c,p);  !!x y. [| p = <x,y>;  c(x,y) |] ==> Q |] ==> Q";
-by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
-qed "splitE";
-
-goal Prod.thy "!!R a b. split(R,<a,b>) ==> R(a,b)";
-by (etac (split RS iffD1) 1);
-qed "splitD";
-
-goal Prod.thy "!!a b c. z: c(a,b) ==> z: split(c, <a,b>)";
-by (asm_simp_tac prod_ss 1);
-qed "mem_splitI";
-
-val prems = goalw Prod.thy [split_def]
-    "[| z: split(c,p);  !!x y. [| p = <x,y>;  z: c(x,y) |] ==> Q |] ==> Q";
-by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
-qed "mem_splitE";
-
-(*** prod_fun -- action of the product functor upon functions ***)
-
-goalw Prod.thy [prod_fun_def] "prod_fun(f,g,<a,b>) = <f(a),g(b)>";
-by (rtac split 1);
-qed "prod_fun";
-
-goal Prod.thy 
-    "prod_fun(f1 o f2, g1 o g2) = (prod_fun(f1,g1) o prod_fun(f2,g2))";
-by (rtac ext 1);
-by (res_inst_tac [("p","x")] PairE 1);
-by (asm_simp_tac (prod_ss addsimps [prod_fun,o_def]) 1);
-qed "prod_fun_compose";
-
-goal Prod.thy "prod_fun(%x.x, %y.y) = (%z.z)";
-by (rtac ext 1);
-by (res_inst_tac [("p","z")] PairE 1);
-by (asm_simp_tac (prod_ss addsimps [prod_fun]) 1);
-qed "prod_fun_ident";
-
-val prems = goal Prod.thy "<a,b>:r ==> <f(a),g(b)> : prod_fun(f,g)``r";
-by (rtac image_eqI 1);
-by (rtac (prod_fun RS sym) 1);
-by (resolve_tac prems 1);
-qed "prod_fun_imageI";
-
-val major::prems = goal Prod.thy
-    "[| c: prod_fun(f,g)``r;  !!x y. [| c=<f(x),g(y)>;  <x,y>:r |] ==> P  \
-\    |] ==> P";
-by (rtac (major RS imageE) 1);
-by (res_inst_tac [("p","x")] PairE 1);
-by (resolve_tac prems 1);
-by (fast_tac HOL_cs 2);
-by (fast_tac (HOL_cs addIs [prod_fun]) 1);
-qed "prod_fun_imageE";
-
-(*** Disjoint union of a family of sets - Sigma ***)
-
-qed_goalw "SigmaI" Prod.thy [Sigma_def]
-    "[| a:A;  b:B(a) |] ==> <a,b> : Sigma(A,B)"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]);
-
-(*The general elimination rule*)
-qed_goalw "SigmaE" Prod.thy [Sigma_def]
-    "[| c: Sigma(A,B);  \
-\       !!x y.[| x:A;  y:B(x);  c=<x,y> |] ==> P \
-\    |] ==> P"
- (fn major::prems=>
-  [ (cut_facts_tac [major] 1),
-    (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]);
-
-(** Elimination of <a,b>:A*B -- introduces no eigenvariables **)
-qed_goal "SigmaD1" Prod.thy "<a,b> : Sigma(A,B) ==> a : A"
- (fn [major]=>
-  [ (rtac (major RS SigmaE) 1),
-    (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
-
-qed_goal "SigmaD2" Prod.thy "<a,b> : Sigma(A,B) ==> b : B(a)"
- (fn [major]=>
-  [ (rtac (major RS SigmaE) 1),
-    (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
-
-qed_goal "SigmaE2" Prod.thy
-    "[| <a,b> : Sigma(A,B);    \
-\       [| a:A;  b:B(a) |] ==> P   \
-\    |] ==> P"
- (fn [major,minor]=>
-  [ (rtac minor 1),
-    (rtac (major RS SigmaD1) 1),
-    (rtac (major RS SigmaD2) 1) ]);
-
-(*** Domain of a relation ***)
-
-val prems = goalw Prod.thy [image_def] "<a,b> : r ==> a : fst``r";
-by (rtac CollectI 1);
-by (rtac bexI 1);
-by (rtac (fst_conv RS sym) 1);
-by (resolve_tac prems 1);
-qed "fst_imageI";
-
-val major::prems = goal Prod.thy
-    "[| a : fst``r;  !!y.[| <a,y> : r |] ==> P |] ==> P"; 
-by (rtac (major RS imageE) 1);
-by (resolve_tac prems 1);
-by (etac ssubst 1);
-by (rtac (surjective_pairing RS subst) 1);
-by (assume_tac 1);
-qed "fst_imageE";
-
-(*** Range of a relation ***)
-
-val prems = goalw Prod.thy [image_def] "<a,b> : r ==> b : snd``r";
-by (rtac CollectI 1);
-by (rtac bexI 1);
-by (rtac (snd_conv RS sym) 1);
-by (resolve_tac prems 1);
-qed "snd_imageI";
-
-val major::prems = goal Prod.thy
-    "[| a : snd``r;  !!y.[| <y,a> : r |] ==> P |] ==> P"; 
-by (rtac (major RS imageE) 1);
-by (resolve_tac prems 1);
-by (etac ssubst 1);
-by (rtac (surjective_pairing RS subst) 1);
-by (assume_tac 1);
-qed "snd_imageE";
-
-(** Exhaustion rule for unit -- a degenerate form of induction **)
-
-goalw Prod.thy [Unity_def]
-    "u = Unity";
-by (stac (rewrite_rule [Unit_def] Rep_Unit RS CollectD RS sym) 1);
-by (rtac (Rep_Unit_inverse RS sym) 1);
-qed "unit_eq";
-
-val prod_cs = set_cs addSIs [SigmaI, mem_splitI] 
-                     addIs  [fst_imageI, snd_imageI, prod_fun_imageI]
-                     addSEs [SigmaE2, SigmaE, mem_splitE, 
-			     fst_imageE, snd_imageE, prod_fun_imageE,
-			     Pair_inject];