src/ZF/Ordinal.ML

--- a/src/ZF/Ordinal.ML	Mon Jun 22 15:53:24 1998 +0200
+++ b/src/ZF/Ordinal.ML	Mon Jun 22 17:12:27 1998 +0200
@@ -12,17 +12,17 @@
 
 (** Two neat characterisations of Transset **)
 
-goalw Ordinal.thy [Transset_def] "Transset(A) <-> A<=Pow(A)";
+Goalw [Transset_def] "Transset(A) <-> A<=Pow(A)";
 by (Blast_tac 1);
 qed "Transset_iff_Pow";
 
-goalw Ordinal.thy [Transset_def] "Transset(A) <-> Union(succ(A)) = A";
+Goalw [Transset_def] "Transset(A) <-> Union(succ(A)) = A";
 by (blast_tac (claset() addSEs [equalityE]) 1);
 qed "Transset_iff_Union_succ";
 
 (** Consequences of downwards closure **)
 
-goalw Ordinal.thy [Transset_def]
+Goalw [Transset_def]
     "!!C a b. [| Transset(C); {a,b}: C |] ==> a:C & b: C";
 by (Blast_tac 1);
 qed "Transset_doubleton_D";
@@ -47,29 +47,29 @@
 
 (** Closure properties **)
 
-goalw Ordinal.thy [Transset_def] "Transset(0)";
+Goalw [Transset_def] "Transset(0)";
 by (Blast_tac 1);
 qed "Transset_0";
 
-goalw Ordinal.thy [Transset_def]
+Goalw [Transset_def]
     "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Un j)";
 by (Blast_tac 1);
 qed "Transset_Un";
 
-goalw Ordinal.thy [Transset_def]
+Goalw [Transset_def]
     "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Int j)";
 by (Blast_tac 1);
 qed "Transset_Int";
 
-goalw Ordinal.thy [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))";
+Goalw [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))";
 by (Blast_tac 1);
 qed "Transset_succ";
 
-goalw Ordinal.thy [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))";
+Goalw [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))";
 by (Blast_tac 1);
 qed "Transset_Pow";
 
-goalw Ordinal.thy [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))";
+Goalw [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))";
 by (Blast_tac 1);
 qed "Transset_Union";
 
@@ -103,7 +103,7 @@
 
 (*** Lemmas for ordinals ***)
 
-goalw Ordinal.thy [Ord_def,Transset_def] "!!i j.[| Ord(i);  j:i |] ==> Ord(j)";
+Goalw [Ord_def,Transset_def] "!!i j.[| Ord(i);  j:i |] ==> Ord(j)";
 by (Blast_tac 1);
 qed "Ord_in_Ord";
 
@@ -112,31 +112,31 @@
 
 AddSDs [Ord_succD];
 
-goal Ordinal.thy "!!i j. [| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)";
+Goal "!!i j. [| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)";
 by (REPEAT (ares_tac [OrdI] 1
      ORELSE eresolve_tac [Ord_contains_Transset, subsetD] 1));
 qed "Ord_subset_Ord";
 
-goalw Ordinal.thy [Ord_def,Transset_def] "!!i j. [| j:i;  Ord(i) |] ==> j<=i";
+Goalw [Ord_def,Transset_def] "!!i j. [| j:i;  Ord(i) |] ==> j<=i";
 by (Blast_tac 1);
 qed "OrdmemD";
 
-goal Ordinal.thy "!!i j k. [| i:j;  j:k;  Ord(k) |] ==> i:k";
+Goal "!!i j k. [| i:j;  j:k;  Ord(k) |] ==> i:k";
 by (REPEAT (ares_tac [OrdmemD RS subsetD] 1));
 qed "Ord_trans";
 
-goal Ordinal.thy "!!i j. [| i:j;  Ord(j) |] ==> succ(i) <= j";
+Goal "!!i j. [| i:j;  Ord(j) |] ==> succ(i) <= j";
 by (REPEAT (ares_tac [OrdmemD RSN (2,succ_subsetI)] 1));
 qed "Ord_succ_subsetI";
 
 
 (*** The construction of ordinals: 0, succ, Union ***)
 
-goal Ordinal.thy "Ord(0)";
+Goal "Ord(0)";
 by (REPEAT (ares_tac [OrdI,Transset_0] 1 ORELSE etac emptyE 1));
 qed "Ord_0";
 
-goal Ordinal.thy "!!i. Ord(i) ==> Ord(succ(i))";
+Goal "!!i. Ord(i) ==> Ord(succ(i))";
 by (REPEAT (ares_tac [OrdI,Transset_succ] 1
      ORELSE eresolve_tac [succE,ssubst,Ord_is_Transset,
                           Ord_contains_Transset] 1));
@@ -144,18 +144,18 @@
 
 bind_thm ("Ord_1", Ord_0 RS Ord_succ);
 
-goal Ordinal.thy "Ord(succ(i)) <-> Ord(i)";
+Goal "Ord(succ(i)) <-> Ord(i)";
 by (blast_tac (claset() addIs [Ord_succ] addDs [Ord_succD]) 1);
 qed "Ord_succ_iff";
 
 Addsimps [Ord_0, Ord_succ_iff];
 AddSIs   [Ord_0, Ord_succ];
 
-goalw Ordinal.thy [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Un j)";
+Goalw [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Un j)";
 by (blast_tac (claset() addSIs [Transset_Un]) 1);
 qed "Ord_Un";
 
-goalw Ordinal.thy [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Int j)";
+Goalw [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Int j)";
 by (blast_tac (claset() addSIs [Transset_Int]) 1);
 qed "Ord_Int";
 
@@ -176,7 +176,7 @@
 qed "Ord_INT";
 
 (*There is no set of all ordinals, for then it would contain itself*)
-goal Ordinal.thy "~ (ALL i. i:X <-> Ord(i))";
+Goal "~ (ALL i. i:X <-> Ord(i))";
 by (rtac notI 1);
 by (forw_inst_tac [("x", "X")] spec 1);
 by (safe_tac (claset() addSEs [mem_irrefl]));
@@ -190,7 +190,7 @@
 
 (*** < is 'less than' for ordinals ***)
 
-goalw Ordinal.thy [lt_def] "!!i j. [| i:j;  Ord(j) |] ==> i<j";
+Goalw [lt_def] "!!i j. [| i:j;  Ord(j) |] ==> i<j";
 by (REPEAT (ares_tac [conjI] 1));
 qed "ltI";
 
@@ -200,22 +200,22 @@
 by (REPEAT (ares_tac (prems@[Ord_in_Ord]) 1));
 qed "ltE";
 
-goal Ordinal.thy "!!i j. i<j ==> i:j";
+Goal "!!i j. i<j ==> i:j";
 by (etac ltE 1);
 by (assume_tac 1);
 qed "ltD";
 
-goalw Ordinal.thy [lt_def] "~ i<0";
+Goalw [lt_def] "~ i<0";
 by (Blast_tac 1);
 qed "not_lt0";
 
 Addsimps [not_lt0];
 
-goal Ordinal.thy "!!i j. j<i ==> Ord(j)";
+Goal "!!i j. j<i ==> Ord(j)";
 by (etac ltE 1 THEN assume_tac 1);
 qed "lt_Ord";
 
-goal Ordinal.thy "!!i j. j<i ==> Ord(i)";
+Goal "!!i j. j<i ==> Ord(i)";
 by (etac ltE 1 THEN assume_tac 1);
 qed "lt_Ord2";
 
@@ -225,11 +225,11 @@
 (* i<0 ==> R *)
 bind_thm ("lt0E", not_lt0 RS notE);
 
-goal Ordinal.thy "!!i j k. [| i<j;  j<k |] ==> i<k";
+Goal "!!i j k. [| i<j;  j<k |] ==> i<k";
 by (blast_tac (claset() addSIs [ltI] addSEs [ltE] addIs [Ord_trans]) 1);
 qed "lt_trans";
 
-goalw Ordinal.thy [lt_def] "!!i j. [| i<j;  j<i |] ==> P";
+Goalw [lt_def] "!!i j. [| i<j;  j<i |] ==> P";
 by (REPEAT (eresolve_tac [asm_rl, conjE, mem_asym] 1));
 qed "lt_asym";
 
@@ -243,16 +243,16 @@
 
 (** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
 
-goalw Ordinal.thy [lt_def] "i le j <-> i<j | (i=j & Ord(j))";
+Goalw [lt_def] "i le j <-> i<j | (i=j & Ord(j))";
 by (blast_tac (claset() addSIs [Ord_succ] addSDs [Ord_succD]) 1);
 qed "le_iff";
 
 (*Equivalently, i<j ==> i < succ(j)*)
-goal Ordinal.thy "!!i j. i<j ==> i le j";
+Goal "!!i j. i<j ==> i le j";
 by (asm_simp_tac (simpset() addsimps [le_iff]) 1);
 qed "leI";
 
-goal Ordinal.thy "!!i. [| i=j;  Ord(j) |] ==> i le j";
+Goal "!!i. [| i=j;  Ord(j) |] ==> i le j";
 by (asm_simp_tac (simpset() addsimps [le_iff]) 1);
 qed "le_eqI";
 
@@ -269,12 +269,12 @@
 by (DEPTH_SOLVE (ares_tac prems 1 ORELSE etac conjE 1));
 qed "leE";
 
-goal Ordinal.thy "!!i j. [| i le j;  j le i |] ==> i=j";
+Goal "!!i j. [| i le j;  j le i |] ==> i=j";
 by (asm_full_simp_tac (simpset() addsimps [le_iff]) 1);
 by (blast_tac (claset() addEs [lt_asym]) 1);
 qed "le_anti_sym";
 
-goal Ordinal.thy "i le 0 <-> i=0";
+Goal "i le 0 <-> i=0";
 by (blast_tac (claset() addSIs [Ord_0 RS le_refl] addSEs [leE]) 1);
 qed "le0_iff";
 
@@ -290,11 +290,11 @@
 
 (*** Natural Deduction rules for Memrel ***)
 
-goalw Ordinal.thy [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A";
+Goalw [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A";
 by (Blast_tac 1);
 qed "Memrel_iff";
 
-goal Ordinal.thy "!!A. [| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)";
+Goal "!!A. [| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)";
 by (REPEAT (ares_tac [conjI, Memrel_iff RS iffD2] 1));
 qed "MemrelI";
 
@@ -311,19 +311,19 @@
 AddSIs [MemrelI];
 AddSEs [MemrelE];
 
-goalw Ordinal.thy [Memrel_def] "Memrel(A) <= A*A";
+Goalw [Memrel_def] "Memrel(A) <= A*A";
 by (Blast_tac 1);
 qed "Memrel_type";
 
-goalw Ordinal.thy [Memrel_def] "!!A B. A<=B ==> Memrel(A) <= Memrel(B)";
+Goalw [Memrel_def] "!!A B. A<=B ==> Memrel(A) <= Memrel(B)";
 by (Blast_tac 1);
 qed "Memrel_mono";
 
-goalw Ordinal.thy [Memrel_def] "Memrel(0) = 0";
+Goalw [Memrel_def] "Memrel(0) = 0";
 by (Blast_tac 1);
 qed "Memrel_0";
 
-goalw Ordinal.thy [Memrel_def] "Memrel(1) = 0";
+Goalw [Memrel_def] "Memrel(1) = 0";
 by (Blast_tac 1);
 qed "Memrel_1";
 
@@ -331,7 +331,7 @@
 
 (*The membership relation (as a set) is well-founded.
   Proof idea: show A<=B by applying the foundation axiom to A-B *)
-goalw Ordinal.thy [wf_def] "wf(Memrel(A))";
+Goalw [wf_def] "wf(Memrel(A))";
 by (EVERY1 [rtac (foundation RS disjE RS allI),
             etac disjI1,
             etac bexE, 
@@ -342,13 +342,13 @@
 qed "wf_Memrel";
 
 (*Transset(i) does not suffice, though ALL j:i.Transset(j) does*)
-goalw Ordinal.thy [Ord_def, Transset_def, trans_def]
+Goalw [Ord_def, Transset_def, trans_def]
     "!!i. Ord(i) ==> trans(Memrel(i))";
 by (Blast_tac 1);
 qed "trans_Memrel";
 
 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
-goalw Ordinal.thy [Transset_def]
+Goalw [Transset_def]
     "!!A. Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A";
 by (Blast_tac 1);
 qed "Transset_Memrel_iff";
@@ -425,11 +425,11 @@
 by (DEPTH_SOLVE (ares_tac ([leI,le_eqI] @ prems) 1));
 qed "Ord_linear_le";
 
-goal Ordinal.thy "!!i j. j le i ==> ~ i<j";
+Goal "!!i j. j le i ==> ~ i<j";
 by (blast_tac le_cs 1);
 qed "le_imp_not_lt";
 
-goal Ordinal.thy "!!i j. [| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i";
+Goal "!!i j. [| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i";
 by (res_inst_tac [("i","i"),("j","j")] Ord_linear2 1);
 by (REPEAT (SOMEGOAL assume_tac));
 by (blast_tac le_cs 1);
@@ -437,25 +437,25 @@
 
 (** Some rewrite rules for <, le **)
 
-goalw Ordinal.thy [lt_def] "!!i j. Ord(j) ==> i:j <-> i<j";
+Goalw [lt_def] "!!i j. Ord(j) ==> i:j <-> i<j";
 by (Blast_tac 1);
 qed "Ord_mem_iff_lt";
 
-goal Ordinal.thy "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i";
+Goal "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i";
 by (REPEAT (ares_tac [iffI, le_imp_not_lt, not_lt_imp_le] 1));
 qed "not_lt_iff_le";
 
-goal Ordinal.thy "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i";
+Goal "!!i j. [| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i";
 by (asm_simp_tac (simpset() addsimps [not_lt_iff_le RS iff_sym]) 1);
 qed "not_le_iff_lt";
 
 (*This is identical to 0<succ(i) *)
-goal Ordinal.thy "!!i. Ord(i) ==> 0 le i";
+Goal "!!i. Ord(i) ==> 0 le i";
 by (etac (not_lt_iff_le RS iffD1) 1);
 by (REPEAT (resolve_tac [Ord_0, not_lt0] 1));
 qed "Ord_0_le";
 
-goal Ordinal.thy "!!i. [| Ord(i);  i~=0 |] ==> 0<i";
+Goal "!!i. [| Ord(i);  i~=0 |] ==> 0<i";
 by (etac (not_le_iff_lt RS iffD1) 1);
 by (rtac Ord_0 1);
 by (Blast_tac 1);
@@ -465,25 +465,25 @@
 
 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
 
-goal Ordinal.thy "!!i j. [| j<=i;  Ord(i);  Ord(j) |] ==> j le i";
+Goal "!!i j. [| j<=i;  Ord(i);  Ord(j) |] ==> j le i";
 by (rtac (not_lt_iff_le RS iffD1) 1);
 by (assume_tac 1);
 by (assume_tac 1);
 by (blast_tac (claset() addEs [ltE, mem_irrefl]) 1);
 qed "subset_imp_le";
 
-goal Ordinal.thy "!!i j. i le j ==> i<=j";
+Goal "!!i j. i le j ==> i<=j";
 by (etac leE 1);
 by (Blast_tac 2);
 by (blast_tac (subset_cs addIs [OrdmemD] addEs [ltE]) 1);
 qed "le_imp_subset";
 
-goal Ordinal.thy "j le i <-> j<=i & Ord(i) & Ord(j)";
+Goal "j le i <-> j<=i & Ord(i) & Ord(j)";
 by (blast_tac (claset() addDs [Ord_succD, subset_imp_le, le_imp_subset]
                        addEs [ltE]) 1);
 qed "le_subset_iff";
 
-goal Ordinal.thy "i le succ(j) <-> i le j | i=succ(j) & Ord(i)";
+Goal "i le succ(j) <-> i le j | i=succ(j) & Ord(i)";
 by (simp_tac (simpset() addsimps [le_iff]) 1);
 by (blast_tac (claset() addIs [Ord_succ] addDs [Ord_succD]) 1);
 qed "le_succ_iff";
@@ -497,55 +497,55 @@
 
 (** Transitive laws **)
 
-goal Ordinal.thy "!!i j. [| i le j;  j<k |] ==> i<k";
+Goal "!!i j. [| i le j;  j<k |] ==> i<k";
 by (blast_tac (claset() addSEs [leE] addIs [lt_trans]) 1);
 qed "lt_trans1";
 
-goal Ordinal.thy "!!i j. [| i<j;  j le k |] ==> i<k";
+Goal "!!i j. [| i<j;  j le k |] ==> i<k";
 by (blast_tac (claset() addSEs [leE] addIs [lt_trans]) 1);
 qed "lt_trans2";
 
-goal Ordinal.thy "!!i j. [| i le j;  j le k |] ==> i le k";
+Goal "!!i j. [| i le j;  j le k |] ==> i le k";
 by (REPEAT (ares_tac [lt_trans1] 1));
 qed "le_trans";
 
-goal Ordinal.thy "!!i j. i<j ==> succ(i) le j";
+Goal "!!i j. i<j ==> succ(i) le j";
 by (rtac (not_lt_iff_le RS iffD1) 1);
 by (blast_tac le_cs 3);
 by (ALLGOALS (blast_tac (claset() addEs [ltE])));
 qed "succ_leI";
 
 (*Identical to  succ(i) < succ(j) ==> i<j  *)
-goal Ordinal.thy "!!i j. succ(i) le j ==> i<j";
+Goal "!!i j. succ(i) le j ==> i<j";
 by (rtac (not_le_iff_lt RS iffD1) 1);
 by (blast_tac le_cs 3);
 by (ALLGOALS (blast_tac (claset() addEs [ltE, make_elim Ord_succD])));
 qed "succ_leE";
 
-goal Ordinal.thy "succ(i) le j <-> i<j";
+Goal "succ(i) le j <-> i<j";
 by (REPEAT (ares_tac [iffI,succ_leI,succ_leE] 1));
 qed "succ_le_iff";
 
 Addsimps [succ_le_iff];
 
-goal Ordinal.thy "!!i j. succ(i) le succ(j) ==> i le j";
+Goal "!!i j. succ(i) le succ(j) ==> i le j";
 by (blast_tac (claset() addSDs [succ_leE]) 1);
 qed "succ_le_imp_le";
 
 (** Union and Intersection **)
 
-goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> i le i Un j";
+Goal "!!i j. [| Ord(i); Ord(j) |] ==> i le i Un j";
 by (rtac (Un_upper1 RS subset_imp_le) 1);
 by (REPEAT (ares_tac [Ord_Un] 1));
 qed "Un_upper1_le";
 
-goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> j le i Un j";
+Goal "!!i j. [| Ord(i); Ord(j) |] ==> j le i Un j";
 by (rtac (Un_upper2 RS subset_imp_le) 1);
 by (REPEAT (ares_tac [Ord_Un] 1));
 qed "Un_upper2_le";
 
 (*Replacing k by succ(k') yields the similar rule for le!*)
-goal Ordinal.thy "!!i j k. [| i<k;  j<k |] ==> i Un j < k";
+Goal "!!i j k. [| i<k;  j<k |] ==> i Un j < k";
 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1);
 by (stac Un_commute 4);
 by (asm_full_simp_tac (simpset() addsimps [le_subset_iff, subset_Un_iff]) 4);
@@ -553,7 +553,7 @@
 by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
 qed "Un_least_lt";
 
-goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k";
+Goal "!!i j. [| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k";
 by (safe_tac (claset() addSIs [Un_least_lt]));
 by (rtac (Un_upper2_le RS lt_trans1) 2);
 by (rtac (Un_upper1_le RS lt_trans1) 1);
@@ -568,7 +568,7 @@
 qed "Un_least_mem_iff";
 
 (*Replacing k by succ(k') yields the similar rule for le!*)
-goal Ordinal.thy "!!i j k. [| i<k;  j<k |] ==> i Int j < k";
+Goal "!!i j k. [| i<k;  j<k |] ==> i Int j < k";
 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1);
 by (stac Int_commute 4);
 by (asm_full_simp_tac (simpset() addsimps [le_subset_iff, subset_Int_iff]) 4);
@@ -625,35 +625,35 @@
      ORELSE dtac Ord_succD 1));
 qed "le_implies_UN_le_UN";
 
-goal Ordinal.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
+Goal "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
 by (blast_tac (claset() addIs [Ord_trans]) 1);
 qed "Ord_equality";
 
 (*Holds for all transitive sets, not just ordinals*)
-goal Ordinal.thy "!!i. Ord(i) ==> Union(i) <= i";
+Goal "!!i. Ord(i) ==> Union(i) <= i";
 by (blast_tac (claset() addIs [Ord_trans]) 1);
 qed "Ord_Union_subset";
 
 
 (*** Limit ordinals -- general properties ***)
 
-goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> Union(i) = i";
+Goalw [Limit_def] "!!i. Limit(i) ==> Union(i) = i";
 by (fast_tac (claset() addSIs [ltI] addSEs [ltE] addEs [Ord_trans]) 1);
 qed "Limit_Union_eq";
 
-goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> Ord(i)";
+Goalw [Limit_def] "!!i. Limit(i) ==> Ord(i)";
 by (etac conjunct1 1);
 qed "Limit_is_Ord";
 
-goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> 0 < i";
+Goalw [Limit_def] "!!i. Limit(i) ==> 0 < i";
 by (etac (conjunct2 RS conjunct1) 1);
 qed "Limit_has_0";
 
-goalw Ordinal.thy [Limit_def] "!!i. [| Limit(i);  j<i |] ==> succ(j) < i";
+Goalw [Limit_def] "!!i. [| Limit(i);  j<i |] ==> succ(j) < i";
 by (Blast_tac 1);
 qed "Limit_has_succ";
 
-goalw Ordinal.thy [Limit_def]
+Goalw [Limit_def]
     "!!i. [| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)";
 by (safe_tac subset_cs);
 by (rtac (not_le_iff_lt RS iffD1) 2);
@@ -661,20 +661,20 @@
 by (REPEAT (eresolve_tac [asm_rl, ltE, Ord_succ] 1));
 qed "non_succ_LimitI";
 
-goal Ordinal.thy "!!i. Limit(succ(i)) ==> P";
+Goal "!!i. Limit(succ(i)) ==> P";
 by (rtac lt_irrefl 1);
 by (rtac Limit_has_succ 1);
 by (assume_tac 1);
 by (etac (Limit_is_Ord RS Ord_succD RS le_refl) 1);
 qed "succ_LimitE";
 
-goal Ordinal.thy "!!i. [| Limit(i);  i le succ(j) |] ==> i le j";
+Goal "!!i. [| Limit(i);  i le succ(j) |] ==> i le j";
 by (safe_tac (claset() addSEs [succ_LimitE, leE]));
 qed "Limit_le_succD";
 
 (** Traditional 3-way case analysis on ordinals **)
 
-goal Ordinal.thy "!!i. Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)";
+Goal "!!i. Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)";
 by (blast_tac (claset() addSIs [non_succ_LimitI, Ord_0_lt]
                       addSDs [Ord_succD]) 1);
 qed "Ord_cases_disj";