src/ZF/pair.ML

 Ordered pairs in Zermelo-Fraenkel Set Theory 
 *)
 
 (** Lemmas for showing that <a,b> uniquely determines a and b **)
 
-qed_goal "singleton_eq_iff" ZF.thy
+qed_goal "singleton_eq_iff" thy
     "{a} = {b} <-> a=b"
  (fn _=> [ (resolve_tac [extension RS iff_trans] 1),
            (Blast_tac 1) ]);
 
-qed_goal "doubleton_eq_iff" ZF.thy
+qed_goal "doubleton_eq_iff" thy
     "{a,b} = {c,d} <-> (a=c & b=d) | (a=d & b=c)"
  (fn _=> [ (resolve_tac [extension RS iff_trans] 1),
            (Blast_tac 1) ]);
 
-qed_goalw "Pair_iff" ZF.thy [Pair_def]
+qed_goalw "Pair_iff" thy [Pair_def]
     "<a,b> = <c,d> <-> a=c & b=d"
  (fn _=> [ (simp_tac (simpset() addsimps [doubleton_eq_iff]) 1),
            (Blast_tac 1) ]);
 
 Addsimps [Pair_iff];

 AddSEs [Pair_inject];
 
 bind_thm ("Pair_inject1", Pair_iff RS iffD1 RS conjunct1);
 bind_thm ("Pair_inject2", Pair_iff RS iffD1 RS conjunct2);
 
-qed_goalw "Pair_not_0" ZF.thy [Pair_def] "<a,b> ~= 0"
+qed_goalw "Pair_not_0" thy [Pair_def] "<a,b> ~= 0"
  (fn _ => [ (blast_tac (claset() addEs [equalityE]) 1) ]);
 
 bind_thm ("Pair_neq_0", Pair_not_0 RS notE);
 
 AddSEs [Pair_neq_0, sym RS Pair_neq_0];
 
-qed_goalw "Pair_neq_fst" ZF.thy [Pair_def] "<a,b>=a ==> P"
+qed_goalw "Pair_neq_fst" thy [Pair_def] "<a,b>=a ==> P"
  (fn [major]=>
   [ (rtac (consI1 RS mem_asym RS FalseE) 1),
     (rtac (major RS subst) 1),
     (rtac consI1 1) ]);
 
-qed_goalw "Pair_neq_snd" ZF.thy [Pair_def] "<a,b>=b ==> P"
+qed_goalw "Pair_neq_snd" thy [Pair_def] "<a,b>=b ==> P"
  (fn [major]=>
   [ (rtac (consI1 RS consI2 RS mem_asym RS FalseE) 1),
     (rtac (major RS subst) 1),
     (rtac (consI1 RS consI2) 1) ]);
 
 
 (*** Sigma: Disjoint union of a family of sets
      Generalizes Cartesian product ***)
 
-qed_goalw "Sigma_iff" ZF.thy [Sigma_def] "<a,b>: Sigma(A,B) <-> a:A & b:B(a)"
+qed_goalw "Sigma_iff" thy [Sigma_def] "<a,b>: Sigma(A,B) <-> a:A & b:B(a)"
  (fn _ => [ Blast_tac 1 ]);
 
 Addsimps [Sigma_iff];
 
-qed_goal "SigmaI" ZF.thy
+qed_goal "SigmaI" thy
     "!!a b. [| a:A;  b:B(a) |] ==> <a,b> : Sigma(A,B)"
  (fn _ => [ Asm_simp_tac 1 ]);
 
 bind_thm ("SigmaD1", Sigma_iff RS iffD1 RS conjunct1);
 bind_thm ("SigmaD2", Sigma_iff RS iffD1 RS conjunct2);
 
 (*The general elimination rule*)
-qed_goalw "SigmaE" ZF.thy [Sigma_def]
+qed_goalw "SigmaE" 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)) ]);
 
-qed_goal "SigmaE2" ZF.thy
+qed_goal "SigmaE2" 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) ]);
 
-qed_goalw "Sigma_cong" ZF.thy [Sigma_def]
+qed_goalw "Sigma_cong" thy [Sigma_def]
     "[| A=A';  !!x. x:A' ==> B(x)=B'(x) |] ==> \
 \    Sigma(A,B) = Sigma(A',B')"
  (fn prems=> [ (simp_tac (simpset() addsimps prems) 1) ]);
 
 

   Sigmas and Pis are abbreviated as * or -> *)
 
 AddSIs [SigmaI];
 AddSEs [SigmaE2, SigmaE];
 
-qed_goal "Sigma_empty1" ZF.thy "Sigma(0,B) = 0"
+qed_goal "Sigma_empty1" thy "Sigma(0,B) = 0"
  (fn _ => [ (Blast_tac 1) ]);
 
-qed_goal "Sigma_empty2" ZF.thy "A*0 = 0"
+qed_goal "Sigma_empty2" thy "A*0 = 0"
  (fn _ => [ (Blast_tac 1) ]);
 
 Addsimps [Sigma_empty1, Sigma_empty2];
 
 Goal "A*B=0 <-> A=0 | B=0";

 qed "Sigma_empty_iff";
 
 
 (*** Projections: fst, snd ***)
 
-qed_goalw "fst_conv" ZF.thy [fst_def] "fst(<a,b>) = a"
+qed_goalw "fst_conv" thy [fst_def] "fst(<a,b>) = a"
  (fn _=> [ (blast_tac (claset() addIs [the_equality]) 1) ]);
 
-qed_goalw "snd_conv" ZF.thy [snd_def] "snd(<a,b>) = b"
+qed_goalw "snd_conv" thy [snd_def] "snd(<a,b>) = b"
  (fn _=> [ (blast_tac (claset() addIs [the_equality]) 1) ]);
 
 Addsimps [fst_conv,snd_conv];
 
-qed_goal "fst_type" ZF.thy "!!p. p:Sigma(A,B) ==> fst(p) : A"
+qed_goal "fst_type" thy "!!p. p:Sigma(A,B) ==> fst(p) : A"
  (fn _=> [ Auto_tac ]);
 
-qed_goal "snd_type" ZF.thy "!!p. p:Sigma(A,B) ==> snd(p) : B(fst(p))"
+qed_goal "snd_type" thy "!!p. p:Sigma(A,B) ==> snd(p) : B(fst(p))"
  (fn _=> [ Auto_tac ]);
 
-qed_goal "Pair_fst_snd_eq" ZF.thy
+qed_goal "Pair_fst_snd_eq" thy
     "!!a A B. a: Sigma(A,B) ==> <fst(a),snd(a)> = a"
  (fn _=> [ Auto_tac ]);
 
 
 (*** Eliminator - split ***)
 
 (*A META-equality, so that it applies to higher types as well...*)
-qed_goalw "split" ZF.thy [split_def] "split(%x y. c(x,y), <a,b>) == c(a,b)"
+qed_goalw "split" thy [split_def] "split(%x y. c(x,y), <a,b>) == c(a,b)"
  (fn _ => [ (Simp_tac 1),
             (rtac reflexive_thm 1) ]);
 
 Addsimps [split];
 
-qed_goal "split_type" ZF.thy
+qed_goal "split_type" thy
     "[|  p:Sigma(A,B);   \
 \        !!x y.[| x:A; y:B(x) |] ==> c(x,y):C(<x,y>) \
 \    |] ==> split(%x y. c(x,y), p) : C(p)"
  (fn major::prems=>
   [ (rtac (major RS SigmaE) 1),
     (asm_simp_tac (simpset() addsimps prems) 1) ]);
 
-goalw ZF.thy [split_def]
-  "!!u. u: A*B ==>   \
+Goalw [split_def]
+  "u: A*B ==>   \
 \       R(split(c,u)) <-> (ALL x:A. ALL y:B. u = <x,y> --> R(c(x,y)))";
 by Auto_tac;
 qed "expand_split";
 
 
 (*** split for predicates: result type o ***)
 
-goalw ZF.thy [split_def] "!!R a b. R(a,b) ==> split(R, <a,b>)";
+Goalw [split_def] "R(a,b) ==> split(R, <a,b>)";
 by (Asm_simp_tac 1);
 qed "splitI";
 
-val major::sigma::prems = goalw ZF.thy [split_def]
+val major::sigma::prems = Goalw [split_def]
     "[| split(R,z);  z:Sigma(A,B);                      \
 \       !!x y. [| z = <x,y>;  R(x,y) |] ==> P           \
 \    |] ==> P";
 by (rtac (sigma RS SigmaE) 1);
 by (cut_facts_tac [major] 1);
 by (REPEAT (ares_tac prems 1));
 by (Asm_full_simp_tac 1);
 qed "splitE";
 
-goalw ZF.thy [split_def] "!!R a b. split(R,<a,b>) ==> R(a,b)";
+Goalw [split_def] "split(R,<a,b>) ==> R(a,b)";
 by (Full_simp_tac 1);
 qed "splitD";