src/ZF/arith.ML

 by (rtac nat_case_0 1);
 val rec_0 = result();
 
 goal Arith.thy "rec(succ(m),a,b) = b(m, rec(m,a,b))";
 val rec_ss = ZF_ss 
-      addcongs (mk_typed_congs Arith.thy [("b", "[i,i]=>i")])
-      addrews [nat_case_succ, nat_succI];
+      addsimps [nat_case_succ, nat_succI];
 by (rtac rec_trans 1);
-by (SIMP_TAC rec_ss 1);
+by (simp_tac rec_ss 1);
 val rec_succ = result();
 
 val major::prems = goal Arith.thy
     "[| n: nat;  \
 \       a: C(0);  \
 \       !!m z. [| m: nat;  z: C(m) |] ==> b(m,z): C(succ(m))  \
 \    |] ==> rec(n,a,b) : C(n)";
 by (rtac (major RS nat_induct) 1);
 by (ALLGOALS
-    (ASM_SIMP_TAC (ZF_ss addrews (prems@[rec_0,rec_succ]))));
+    (asm_simp_tac (ZF_ss addsimps (prems@[rec_0,rec_succ]))));
 val rec_type = result();
 
-val prems = goalw Arith.thy [rec_def]
-    "[| n=n';  a=a';  !!m z. b(m,z)=b'(m,z)  \
-\    |] ==> rec(n,a,b)=rec(n',a',b')";
-by (SIMP_TAC (ZF_ss addcongs [transrec_cong,nat_case_cong] 
-		    addrews (prems RL [sym])) 1);
-val rec_cong = result();
-
 val nat_typechecks = [rec_type,nat_0I,nat_1I,nat_succI,Ord_nat];
 
-val nat_ss = ZF_ss addcongs [nat_case_cong,rec_cong]
-	       	   addrews ([rec_0,rec_succ] @ nat_typechecks);
+val nat_ss = ZF_ss addsimps ([rec_0,rec_succ] @ nat_typechecks);
 
 
 (** Addition **)
 
 val add_type = prove_goalw Arith.thy [add_def]

 
 val diff_0_eq_0 = prove_goalw Arith.thy [diff_def]
     "n:nat ==> 0 #- n = 0"
  (fn [prem]=>
   [ (rtac (prem RS nat_induct) 1),
-    (ALLGOALS (ASM_SIMP_TAC nat_ss)) ]);
+    (ALLGOALS (asm_simp_tac nat_ss)) ]);
 
 (*Must simplify BEFORE the induction!!  (Else we get a critical pair)
   succ(m) #- succ(n)   rewrites to   pred(succ(m) #- n)  *)
 val diff_succ_succ = prove_goalw Arith.thy [diff_def]
     "[| m:nat;  n:nat |] ==> succ(m) #- succ(n) = m #- n"
  (fn prems=>
-  [ (ASM_SIMP_TAC (nat_ss addrews prems) 1),
+  [ (asm_simp_tac (nat_ss addsimps prems) 1),
     (nat_ind_tac "n" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (nat_ss addrews prems))) ]);
+    (ALLGOALS (asm_simp_tac (nat_ss addsimps prems))) ]);
 
 val prems = goal Arith.thy 
     "[| m:nat;  n:nat |] ==> m #- n : succ(m)";
 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
 by (resolve_tac prems 1);
 by (resolve_tac prems 1);
 by (etac succE 3);
 by (ALLGOALS
-    (ASM_SIMP_TAC
-     (nat_ss addrews (prems@[diff_0,diff_0_eq_0,diff_succ_succ]))));
+    (asm_simp_tac
+     (nat_ss addsimps (prems@[diff_0,diff_0_eq_0,diff_succ_succ]))));
 val diff_leq = result();
 
 (*** Simplification over add, mult, diff ***)
 
 val arith_typechecks = [add_type, mult_type, diff_type];
 val arith_rews = [add_0, add_succ,
 		  mult_0, mult_succ,
 		  diff_0, diff_0_eq_0, diff_succ_succ];
 
-val arith_congs = mk_congs Arith.thy ["op #+", "op #-", "op #*"];
-
-val arith_ss = nat_ss addcongs arith_congs
-                      addrews  (arith_rews@arith_typechecks);
+val arith_ss = nat_ss addsimps (arith_rews@arith_typechecks);
 
 (*** Addition ***)
 
 (*Associative law for addition*)
 val add_assoc = prove_goal Arith.thy 
     "m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]);
 
 (*The following two lemmas are used for add_commute and sometimes
   elsewhere, since they are safe for rewriting.*)
 val add_0_right = prove_goal Arith.thy
     "m:nat ==> m #+ 0 = m"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); 
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
 
 val add_succ_right = prove_goal Arith.thy
     "m:nat ==> m #+ succ(n) = succ(m #+ n)"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); 
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
 
 (*Commutative law for addition*)  
 val add_commute = prove_goal Arith.thy 
     "[| m:nat;  n:nat |] ==> m #+ n = n #+ m"
  (fn prems=>
   [ (nat_ind_tac "n" prems 1),
     (ALLGOALS
-     (ASM_SIMP_TAC
-      (arith_ss addrews (prems@[add_0_right, add_succ_right])))) ]);
+     (asm_simp_tac
+      (arith_ss addsimps (prems@[add_0_right, add_succ_right])))) ]);
 
 (*Cancellation law on the left*)
 val [knat,eqn] = goal Arith.thy 
     "[| k:nat;  k #+ m = k #+ n |] ==> m=n";
 by (rtac (eqn RS rev_mp) 1);
 by (nat_ind_tac "k" [knat] 1);
-by (ALLGOALS (SIMP_TAC arith_ss));
+by (ALLGOALS (simp_tac arith_ss));
 by (fast_tac ZF_cs 1);
 val add_left_cancel = result();
 
 (*** Multiplication ***)
 
 (*right annihilation in product*)
 val mult_0_right = prove_goal Arith.thy 
     "m:nat ==> m #* 0 = 0"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems)))  ]);
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps prems)))  ]);
 
 (*right successor law for multiplication*)
 val mult_succ_right = prove_goal Arith.thy 
-    "[| m:nat;  n:nat |] ==> m #* succ(n) = m #+ (m #* n)"
- (fn prems=>
-  [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews ([add_assoc RS sym]@prems)))),
+    "!!m n. [| m:nat;  n:nat |] ==> m #* succ(n) = m #+ (m #* n)"
+ (fn _=>
+  [ (nat_ind_tac "m" [] 1),
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]))),
        (*The final goal requires the commutative law for addition*)
-    (REPEAT (ares_tac (prems@[refl,add_commute]@ZF_congs@arith_congs) 1))  ]);
+    (rtac (add_commute RS subst_context) 1),
+    (REPEAT (assume_tac 1))  ]);
 
 (*Commutative law for multiplication*)
 val mult_commute = prove_goal Arith.thy 
     "[| m:nat;  n:nat |] ==> m #* n = n #* m"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC
-	       (arith_ss addrews (prems@[mult_0_right, mult_succ_right])))) ]);
+    (ALLGOALS (asm_simp_tac
+	     (arith_ss addsimps (prems@[mult_0_right, mult_succ_right])))) ]);
 
 (*addition distributes over multiplication*)
 val add_mult_distrib = prove_goal Arith.thy 
     "[| m:nat;  k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews ([add_assoc RS sym]@prems)))) ]);
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps ([add_assoc RS sym]@prems)))) ]);
 
 (*Distributive law on the left; requires an extra typing premise*)
 val add_mult_distrib_left = prove_goal Arith.thy 
     "[| m:nat;  n:nat;  k:nat |] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)"
  (fn prems=>
       let val mult_commute' = read_instantiate [("m","k")] mult_commute
-          val ss = arith_ss addrews ([mult_commute',add_mult_distrib]@prems)
-      in [ (SIMP_TAC ss 1) ]
+          val ss = arith_ss addsimps ([mult_commute',add_mult_distrib]@prems)
+      in [ (simp_tac ss 1) ]
       end);
 
 (*Associative law for multiplication*)
 val mult_assoc = prove_goal Arith.thy 
     "[| m:nat;  n:nat;  k:nat |] ==> (m #* n) #* k = m #* (n #* k)"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews (prems@[add_mult_distrib])))) ]);
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps (prems@[add_mult_distrib])))) ]);
 
 
 (*** Difference ***)
 
 val diff_self_eq_0 = prove_goal Arith.thy 
     "m:nat ==> m #- m = 0"
  (fn prems=>
   [ (nat_ind_tac "m" prems 1),
-    (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]);
+    (ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]);
 
 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
 val notless::prems = goal Arith.thy
     "[| ~m:n;  m:nat;  n:nat |] ==> n #+ (m#-n) = m";
 by (rtac (notless RS rev_mp) 1);
 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
 by (resolve_tac prems 1);
 by (resolve_tac prems 1);
-by (ALLGOALS (ASM_SIMP_TAC
-	      (arith_ss addrews (prems@[succ_mem_succ_iff, Ord_0_mem_succ, 
+by (ALLGOALS (asm_simp_tac
+	      (arith_ss addsimps (prems@[succ_mem_succ_iff, Ord_0_mem_succ, 
 				  naturals_are_ordinals]))));
 val add_diff_inverse = result();
 
 
 (*Subtraction is the inverse of addition. *)
 val [mnat,nnat] = goal Arith.thy
     "[| m:nat;  n:nat |] ==> (n#+m) #-n = m";
 by (rtac (nnat RS nat_induct) 1);
-by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mnat])));
+by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mnat])));
 val diff_add_inverse = result();
 
 val [mnat,nnat] = goal Arith.thy
     "[| m:nat;  n:nat |] ==> n #- (n#+m) = 0";
 by (rtac (nnat RS nat_induct) 1);
-by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mnat])));
+by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mnat])));
 val diff_add_0 = result();
 
 
 (*** Remainder ***)
 
 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
-val prems = goal Arith.thy
-    "[| 0:n; ~ m:n;  m:nat;  n:nat |] ==> m #- n : m";
-by (cut_facts_tac prems 1);
+goal Arith.thy "!!m n. [| 0:n; ~ m:n;  m:nat;  n:nat |] ==> m #- n : m";
 by (etac rev_mp 1);
 by (etac rev_mp 1);
 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (resolve_tac prems 1);
-by (resolve_tac prems 1);
-by (ALLGOALS (ASM_SIMP_TAC
-	      (nat_ss addrews (prems@[diff_leq,diff_succ_succ]))));
+by (ALLGOALS (asm_simp_tac (nat_ss addsimps [diff_leq,diff_succ_succ])));
 val div_termination = result();
 
 val div_rls =
     [Ord_transrec_type, apply_type, div_termination, if_type] @ 
     nat_typechecks;

 val prems = goalw Arith.thy [mod_def]
     "[| 0:n;  m:nat;  n:nat |] ==> m mod n : nat";
 by (REPEAT (ares_tac (prems @ div_rls) 1 ORELSE etac Ord_trans 1));
 val mod_type = result();
 
-val div_ss = ZF_ss addrews [naturals_are_ordinals,div_termination];
+val div_ss = ZF_ss addsimps [naturals_are_ordinals,div_termination];
 
 val prems = goal Arith.thy "[| 0:n;  m:n;  m:nat;  n:nat |] ==> m mod n = m";
 by (rtac (mod_def RS def_transrec RS trans) 1);
-by (SIMP_TAC (div_ss addrews prems) 1);
+by (simp_tac (div_ss addsimps prems) 1);
 val mod_less = result();
 
 val prems = goal Arith.thy
     "[| 0:n;  ~m:n;  m:nat;  n:nat |] ==> m mod n = (m#-n) mod n";
 by (rtac (mod_def RS def_transrec RS trans) 1);
-by (SIMP_TAC (div_ss addrews prems) 1);
+by (simp_tac (div_ss addsimps prems) 1);
 val mod_geq = result();
 
 (*** Quotient ***)
 
 (*Type checking depends upon termination!*)

 val div_type = result();
 
 val prems = goal Arith.thy
     "[| 0:n;  m:n;  m:nat;  n:nat |] ==> m div n = 0";
 by (rtac (div_def RS def_transrec RS trans) 1);
-by (SIMP_TAC (div_ss addrews prems) 1);
+by (simp_tac (div_ss addsimps prems) 1);
 val div_less = result();
 
 val prems = goal Arith.thy
     "[| 0:n;  ~m:n;  m:nat;  n:nat |] ==> m div n = succ((m#-n) div n)";
 by (rtac (div_def RS def_transrec RS trans) 1);
-by (SIMP_TAC (div_ss addrews prems) 1);
+by (simp_tac (div_ss addsimps prems) 1);
 val div_geq = result();
 
 (*Main Result.*)
 val prems = goal Arith.thy
     "[| 0:n;  m:nat;  n:nat |] ==> (m div n)#*n #+ m mod n = m";
 by (res_inst_tac [("i","m")] complete_induct 1);
 by (resolve_tac prems 1);
 by (res_inst_tac [("Q","x:n")] (excluded_middle RS disjE) 1);
 by (ALLGOALS 
-    (ASM_SIMP_TAC
-     (arith_ss addrews ([mod_type,div_type] @ prems @
+    (asm_simp_tac
+     (arith_ss addsimps ([mod_type,div_type] @ prems @
         [mod_less,mod_geq, div_less, div_geq,
 	 add_assoc, add_diff_inverse, div_termination]))));
 val mod_div_equality = result();
 
 
 (**** Additional theorems about "less than" ****)
 
 val [mnat,nnat] = goal Arith.thy
     "[| m:nat;  n:nat |] ==> ~ (m #+ n) : n";
 by (rtac (mnat RS nat_induct) 1);
-by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mem_not_refl])));
+by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mem_not_refl])));
 by (rtac notI 1);
 by (etac notE 1);
 by (etac (succI1 RS Ord_trans) 1);
 by (rtac (nnat RS naturals_are_ordinals) 1);
 val add_not_less_self = result();