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),
-           (fast_tac upair_cs 1) ]);
+           (Fast_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),
-           (fast_tac upair_cs 1) ]);
+           (Fast_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 (FOL_ss addsimps [doubleton_eq_iff]) 1),
-           (fast_tac FOL_cs 1) ]);
+ (fn _=> [ (simp_tac (!simpset addsimps [doubleton_eq_iff]) 1),
+           (Fast_tac 1) ]);
 
-bind_thm ("Pair_inject", (Pair_iff RS iffD1 RS conjE));
+Addsimps [Pair_iff];
 
-qed_goal "Pair_inject1" ZF.thy "<a,b> = <c,d> ==> a=c"
- (fn [major]=>
-  [ (rtac (major RS Pair_inject) 1), (assume_tac 1) ]);
+bind_thm ("Pair_inject", Pair_iff RS iffD1 RS conjE);
 
-qed_goal "Pair_inject2" ZF.thy "<a,b> = <c,d> ==> b=d"
- (fn [major]=>
-  [ (rtac (major RS Pair_inject) 1), (assume_tac 1) ]);
+AddSEs [Pair_inject];
 
-qed_goalw "Pair_neq_0" ZF.thy [Pair_def] "<a,b>=0 ==> P"
- (fn [major]=>
-  [ (rtac (major RS equalityD1 RS subsetD RS emptyE) 1),
-    (rtac consI1 1) ]);
+bind_thm ("Pair_inject1", Pair_iff RS iffD1 RS conjunct1);
+bind_thm ("Pair_inject2", Pair_iff RS iffD1 RS conjunct2);
 
-qed_goalw "Pair_neq_fst" ZF.thy [Pair_def] "<a,b>=a ==> P"
+qed_goalw "Pair_not_0" thy [Pair_def] "<a,b> ~= 0"
+ (fn _ => [ (fast_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" 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 "SigmaI" ZF.thy [Sigma_def]
-    "[| a:A;  b:B(a) |] ==> <a,b> : Sigma(A,B)"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]);
+qed_goalw "Sigma_iff" thy [Sigma_def] "<a,b>: Sigma(A,B) <-> a:A & b:B(a)"
+ (fn _ => [ Fast_tac 1 ]);
+
+Addsimps [Sigma_iff];
+
+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)) ]);
 
-(** Elimination of <a,b>:A*B -- introduces no eigenvariables **)
-qed_goal "SigmaD1" ZF.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" ZF.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)) ]);
-
-(*Also provable via 
-  rule_by_tactic (REPEAT_FIRST (etac Pair_inject ORELSE' bound_hyp_subst_tac)
-                  THEN prune_params_tac)
-      (read_instantiate [("c","<a,b>")] SigmaE);  *)
-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 (FOL_ss addsimps prems addcongs [RepFun_cong]) 1) ]);
+ (fn prems=> [ (simp_tac (!simpset addsimps prems) 1) ]);
 
-qed_goal "Sigma_empty1" ZF.thy "Sigma(0,B) = 0"
- (fn _ => [ (fast_tac (lemmas_cs addIs [equalityI] addSEs [SigmaE]) 1) ]);
 
-qed_goal "Sigma_empty2" ZF.thy "A*0 = 0"
- (fn _ => [ (fast_tac (lemmas_cs addIs [equalityI] addSEs [SigmaE]) 1) ]);
+(*Sigma_cong, Pi_cong NOT given to Addcongs: they cause
+  flex-flex pairs and the "Check your prover" error.  Most
+  Sigmas and Pis are abbreviated as * or -> *)
 
-val pair_cs = upair_cs 
-    addSIs [SigmaI]
-    addSEs [SigmaE2, SigmaE, Pair_inject, make_elim succ_inject,
-            Pair_neq_0, sym RS Pair_neq_0, succ_neq_0, sym RS succ_neq_0];
+AddSIs [SigmaI];
+AddSEs [SigmaE2, SigmaE];
+
+qed_goal "Sigma_empty1" thy "Sigma(0,B) = 0"
+ (fn _ => [ (fast_tac (!claset addIs [equalityI]) 1) ]);
+
+qed_goal "Sigma_empty2" thy "A*0 = 0"
+ (fn _ => [ (fast_tac (!claset addIs [equalityI]) 1) ]);
+
+Addsimps [Sigma_empty1, Sigma_empty2];
 
 
 (*** Projections: fst, snd ***)
 
-qed_goalw "fst_conv" ZF.thy [fst_def] "fst(<a,b>) = a"
- (fn _=> 
-  [ (fast_tac (pair_cs addIs [the_equality]) 1) ]);
+qed_goalw "fst_conv" thy [fst_def] "fst(<a,b>) = a"
+ (fn _=> [ (fast_tac (!claset addIs [the_equality]) 1) ]);
 
-qed_goalw "snd_conv" ZF.thy [snd_def] "snd(<a,b>) = b"
- (fn _=> 
-  [ (fast_tac (pair_cs addIs [the_equality]) 1) ]);
+qed_goalw "snd_conv" thy [snd_def] "snd(<a,b>) = b"
+ (fn _=> [ (fast_tac (!claset addIs [the_equality]) 1) ]);
 
-val pair_ss = FOL_ss addsimps [fst_conv,snd_conv];
+Addsimps [fst_conv,snd_conv];
 
-qed_goal "fst_type" ZF.thy
-    "!!p. p:Sigma(A,B) ==> fst(p) : A"
- (fn _=> [ (fast_tac (pair_cs addss pair_ss) 1) ]);
+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))"
- (fn _=> [ (fast_tac (pair_cs addss pair_ss) 1) ]);
+qed_goal "snd_type" thy "!!p. p:Sigma(A,B) ==> snd(p) : B(fst(p))"
+ (fn _=> [ Auto_tac() ]);
 
-goal ZF.thy "!!a A B. a: Sigma(A,B) ==> <fst(a),snd(a)> = a";
-by (etac SigmaE 1);
-by (asm_simp_tac pair_ss 1);
-qed "Pair_fst_snd_eq";
+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)"
- (fn _ => [ (simp_tac pair_ss 1),
+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) ]);
 
-qed_goal "split_type" ZF.thy
+Addsimps [split];
+
+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 (pair_ss addsimps (split::prems)) 1) ]);
+    (asm_simp_tac (!simpset addsimps prems) 1) ]);
 
-goalw ZF.thy [split_def]
+goalw thy [split_def]
   "!!u. u: A*B ==>   \
 \       R(split(c,u)) <-> (ALL x:A. ALL y:B. u = <x,y> --> R(c(x,y)))";
-by (etac SigmaE 1);
-by (asm_simp_tac pair_ss 1);
-by (fast_tac pair_cs 1);
+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>)";
-by (asm_simp_tac pair_ss 1);
+goalw thy [split_def] "!!R a b. 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 thy [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 (asm_full_simp_tac (pair_ss addsimps prems) 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)";
-by (asm_full_simp_tac pair_ss 1);
+goalw thy [split_def] "!!R a b. split(R,<a,b>) ==> R(a,b)";
+by (Full_simp_tac 1);
 qed "splitD";
 
 
-(*** Basic simplification for ZF; see simpdata.ML for full version ***)
 
-fun prove_fun s = 
-    (writeln s;  prove_goal ZF.thy s
-       (fn prems => [ (cut_facts_tac prems 1), 
-                      (fast_tac (pair_cs addSIs [equalityI]) 1) ]));
-
-(*INCLUDED IN ZF_ss*)
-val mem_simps = map prove_fun
- [ "a : 0             <-> False",
-   "a : A Un B        <-> a:A | a:B",
-   "a : A Int B       <-> a:A & a:B",
-   "a : A-B           <-> a:A & ~a:B",
-   "<a,b>: Sigma(A,B) <-> a:A & b:B(a)",
-   "a : Collect(A,P)  <-> a:A & P(a)" ];
-
-goal ZF.thy "{x.x:A} = A";
-by (fast_tac (pair_cs addSIs [equalityI]) 1);
-qed "triv_RepFun";
-
-(*INCLUDED: should be?  And what about cons(a,b)?*)
-val bquant_simps = map prove_fun
- [ "(ALL x:0.P(x)) <-> True",
-   "(EX  x:0.P(x)) <-> False",
-   "(ALL x:succ(i).P(x)) <-> P(i) & (ALL x:i.P(x))",
-   "(EX  x:succ(i).P(x)) <-> P(i) | (EX  x:i.P(x))",
-   "(ALL x:cons(a,B).P(x)) <-> P(a) & (ALL x:B.P(x))",
-   "(EX  x:cons(a,B).P(x)) <-> P(a) | (EX  x:B.P(x))" ];
-
-val Collect_simps = map prove_fun
- [ "{x:0. P(x)} = 0",
-   "{x:A. False} = 0",
-   "{x:A. True} = A",
-   "RepFun(0,f) = 0",
-   "RepFun(succ(i),f) = cons(f(i), RepFun(i,f))",
-   "RepFun(cons(a,B),f) = cons(f(a), RepFun(B,f))" ];
-
-val UnInt_simps = map prove_fun
- [ "0 Un A = A",  "A Un 0 = A",
-   "0 Int A = 0", "A Int 0 = 0",
-   "0-A = 0",     "A-0 = A",
-   "Union(0) = 0",
-   "Union(cons(b,A)) = b Un Union(A)",
-   "Inter({b}) = b"];
-