src/ZF/ex/Bin.ML

 by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
 by (rewrite_goals_tac bin.con_defs);
 by (simp_tac rank_ss 1);
 qed "bin_rec_Minus";
 
-goal Bin.thy "bin_rec(w$$x,a,b,h) = h(w, x, bin_rec(w,a,b,h))";
+goal Bin.thy "bin_rec(Bcons(w,x),a,b,h) = h(w, x, bin_rec(w,a,b,h))";
 by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
 by (rewrite_goals_tac bin.con_defs);
 by (simp_tac rank_ss 1);
 qed "bin_rec_Bcons";
 
 (*Type checking*)
 val prems = goal Bin.thy
     "[| w: bin;    \
 \       a: C(Plus);   b: C(Minus);       \
-\       !!w x r. [| w: bin;  x: bool;  r: C(w) |] ==> h(w,x,r): C(w$$x)  \
+\       !!w x r. [| w: bin; x: bool; r: C(w) |] ==> h(w,x,r): C(Bcons(w,x))  \
 \    |] ==> bin_rec(w,a,b,h) : C(w)";
 by (bin_ind_tac "w" prems 1);
 by (ALLGOALS 
-    (asm_simp_tac (ZF_ss addsimps (prems@[bin_rec_Plus,bin_rec_Minus,
-					 bin_rec_Bcons]))));
+    (asm_simp_tac (ZF_ss addsimps (prems@[bin_rec_Plus, bin_rec_Minus,
+					  bin_rec_Bcons]))));
 qed "bin_rec_type";
 
 (** Versions for use with definitions **)
 
 val [rew] = goal Bin.thy

 by (rewtac rew);
 by (rtac bin_rec_Minus 1);
 qed "def_bin_rec_Minus";
 
 val [rew] = goal Bin.thy
-    "[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(w$$x) = h(w,x,j(w))";
+    "[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Bcons(w,x)) = h(w,x,j(w))";
 by (rewtac rew);
 by (rtac bin_rec_Bcons 1);
 qed "def_bin_rec_Bcons";
 
 fun bin_recs def = map standard
 	([def] RL [def_bin_rec_Plus, def_bin_rec_Minus, def_bin_rec_Bcons]);
+
+goalw Bin.thy [norm_Bcons_def] "norm_Bcons(Plus,0) = Plus";
+by (asm_simp_tac (ZF_ss addsimps (bin.case_eqns @ bool_simps)) 1);
+qed "norm_Bcons_Plus_0";
+
+goalw Bin.thy [norm_Bcons_def] "norm_Bcons(Plus,1) = Bcons(Plus,1)";
+by (asm_simp_tac (ZF_ss addsimps (bin.case_eqns @ bool_simps)) 1);
+qed "norm_Bcons_Plus_1";
+
+goalw Bin.thy [norm_Bcons_def] "norm_Bcons(Minus,0) = Bcons(Minus,0)";
+by (asm_simp_tac (ZF_ss addsimps (bin.case_eqns @ bool_simps)) 1);
+qed "norm_Bcons_Minus_0";
+
+goalw Bin.thy [norm_Bcons_def] "norm_Bcons(Minus,1) = Minus";
+by (asm_simp_tac (ZF_ss addsimps (bin.case_eqns @ bool_simps)) 1);
+qed "norm_Bcons_Minus_1";
+
+goalw Bin.thy [norm_Bcons_def]
+    "norm_Bcons(Bcons(w,x),b) = Bcons(Bcons(w,x),b)";
+by (asm_simp_tac (ZF_ss addsimps bin.case_eqns) 1);
+qed "norm_Bcons_Bcons";
+
+val norm_Bcons_simps = [norm_Bcons_Plus_0, norm_Bcons_Plus_1, 
+			norm_Bcons_Minus_0, norm_Bcons_Minus_1,
+			norm_Bcons_Bcons];
 
 (** Type checking **)
 
 val bin_typechecks0 = bin_rec_type :: bin.intrs;
 

     "!!w. w: bin ==> integ_of_bin(w) : integ";
 by (typechk_tac (bin_typechecks0@integ_typechecks@
 		 nat_typechecks@[bool_into_nat]));
 qed "integ_of_bin_type";
 
+goalw Bin.thy [norm_Bcons_def]
+    "!!w. [| w: bin; b: bool |] ==> norm_Bcons(w,b) : bin";
+by (etac bin.elim 1);
+by (ALLGOALS (asm_simp_tac (ZF_ss addsimps bin.case_eqns)));
+by (typechk_tac (bin_typechecks0@bool_typechecks));
+qed "norm_Bcons_type";
+
 goalw Bin.thy [bin_succ_def]
     "!!w. w: bin ==> bin_succ(w) : bin";
-by (typechk_tac (bin_typechecks0@bool_typechecks));
+by (typechk_tac ([norm_Bcons_type]@bin_typechecks0@bool_typechecks));
 qed "bin_succ_type";
 
 goalw Bin.thy [bin_pred_def]
     "!!w. w: bin ==> bin_pred(w) : bin";
-by (typechk_tac (bin_typechecks0@bool_typechecks));
+by (typechk_tac ([norm_Bcons_type]@bin_typechecks0@bool_typechecks));
 qed "bin_pred_type";
 
 goalw Bin.thy [bin_minus_def]
     "!!w. w: bin ==> bin_minus(w) : bin";
 by (typechk_tac ([bin_pred_type]@bin_typechecks0@bool_typechecks));
 qed "bin_minus_type";
 
 goalw Bin.thy [bin_add_def]
     "!!v w. [| v: bin; w: bin |] ==> bin_add(v,w) : bin";
-by (typechk_tac ([bin_succ_type,bin_pred_type]@bin_typechecks0@
-		 bool_typechecks@ZF_typechecks));
+by (typechk_tac ([norm_Bcons_type, bin_succ_type, bin_pred_type]@
+		 bin_typechecks0@ bool_typechecks@ZF_typechecks));
 qed "bin_add_type";
 
 goalw Bin.thy [bin_mult_def]
     "!!v w. [| v: bin; w: bin |] ==> bin_mult(v,w) : bin";
-by (typechk_tac ([bin_minus_type,bin_add_type]@bin_typechecks0@
-		 bool_typechecks));
+by (typechk_tac ([norm_Bcons_type, bin_minus_type, bin_add_type]@
+		 bin_typechecks0@ bool_typechecks));
 qed "bin_mult_type";
 
 val bin_typechecks = bin_typechecks0 @
-    [integ_of_bin_type, bin_succ_type, bin_pred_type, 
+    [integ_of_bin_type, norm_Bcons_type, bin_succ_type, bin_pred_type, 
      bin_minus_type, bin_add_type, bin_mult_type];
 
 val bin_ss = integ_ss 
     addsimps([bool_1I, bool_0I,
 	     bin_rec_Plus, bin_rec_Minus, bin_rec_Bcons] @ 

     "!!z v w. [| z: integ;  v: integ;  w: integ |]   \
 \    ==> z $+ (v $+ w) = v $+ (z $+ w)";
 by (REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1));
 qed "zadd_assoc_swap";
 
-val zadd_cong = 
-    read_instantiate_sg (sign_of Integ.thy) [("t","op $+")] subst_context2;
-
-val zadd_kill = (refl RS zadd_cong);
-val zadd_assoc_swap_kill = zadd_kill RSN (4, zadd_assoc_swap RS trans);
-
 (*Pushes 'constants' of the form $#m to the right -- LOOPS if two!*)
 bind_thm ("zadd_assoc_znat", (znat_type RS zadd_assoc_swap));
 
-goal Integ.thy 
-    "!!z w. [| z: integ;  w: integ |]   \
-\    ==> w $+ (z $+ (w $+ z)) = w $+ (w $+ (z $+ z))";
-by (REPEAT (ares_tac [zadd_kill, zadd_assoc_swap] 1));
-qed "zadd_swap_pairs";
-
 
 val carry_ss = bin_ss addsimps 
                (bin_recs bin_succ_def @ bin_recs bin_pred_def);
 
+
+(*norm_Bcons preserves the integer value of its argument*)
+goal Bin.thy
+    "!!w. [| w: bin; b: bool |] ==>	\
+\         integ_of_bin(norm_Bcons(w,b)) = integ_of_bin(Bcons(w,b))";
+by (etac bin.elim 1);
+by (asm_simp_tac (ZF_ss addsimps norm_Bcons_simps) 3);
+by (ALLGOALS (etac boolE));
+by (ALLGOALS (asm_simp_tac (bin_ss addsimps (norm_Bcons_simps))));
+qed "integ_of_bin_norm_Bcons";
+
 goal Bin.thy
     "!!w. w: bin ==> integ_of_bin(bin_succ(w)) = $#1 $+ integ_of_bin(w)";
 by (etac bin.induct 1);
-by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
-by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
-by (etac boolE 1);
-by (ALLGOALS (asm_simp_tac (carry_ss addsimps [zadd_assoc])));
-by (REPEAT (ares_tac (zadd_swap_pairs::typechecks) 1));
+by (simp_tac carry_ss 1);
+by (simp_tac carry_ss 1);
+by (etac boolE 1);
+by (ALLGOALS
+    (asm_simp_tac (carry_ss addsimps integ_of_bin_norm_Bcons::zadd_ac)));
 qed "integ_of_bin_succ";
 
 goal Bin.thy
     "!!w. w: bin ==> integ_of_bin(bin_pred(w)) = $~ ($#1) $+ integ_of_bin(w)";
 by (etac bin.induct 1);
-by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
-by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
-by (etac boolE 1);
-by (ALLGOALS 
-    (asm_simp_tac 
-     (carry_ss addsimps [zadd_assoc RS sym,
-			zadd_zminus_inverse, zadd_zminus_inverse2])));
-by (REPEAT (ares_tac ([zadd_commute, zadd_cong, refl]@typechecks) 1));
+by (simp_tac carry_ss 1);
+by (simp_tac carry_ss 1);
+by (etac boolE 1);
+by (ALLGOALS
+    (asm_simp_tac (carry_ss addsimps integ_of_bin_norm_Bcons::zadd_ac)));
 qed "integ_of_bin_pred";
 
 (*These two results replace the definitions of bin_succ and bin_pred*)
 
 

 		    [integ_of_bin_succ, integ_of_bin_pred]);
 
 goal Bin.thy
     "!!w. w: bin ==> integ_of_bin(bin_minus(w)) = $~ integ_of_bin(w)";
 by (etac bin.induct 1);
-by (simp_tac (bin_minus_ss addsimps [zminus_0]) 1);
-by (simp_tac (bin_minus_ss addsimps [zadd_0_right]) 1);
+by (simp_tac bin_minus_ss 1);
+by (simp_tac bin_minus_ss 1);
 by (etac boolE 1);
 by (ALLGOALS 
     (asm_simp_tac (bin_minus_ss addsimps [zminus_zadd_distrib, zadd_assoc])));
 qed "integ_of_bin_minus";
 

 
 goalw Bin.thy [bin_add_def] "!!w. w: bin ==> bin_add(Minus,w) = bin_pred(w)";
 by (asm_simp_tac bin_ss 1);
 qed "bin_add_Minus";
 
-goalw Bin.thy [bin_add_def] "bin_add(v$$x,Plus) = v$$x";
+goalw Bin.thy [bin_add_def] "bin_add(Bcons(v,x),Plus) = Bcons(v,x)";
 by (simp_tac bin_ss 1);
 qed "bin_add_Bcons_Plus";
 
-goalw Bin.thy [bin_add_def] "bin_add(v$$x,Minus) = bin_pred(v$$x)";
+goalw Bin.thy [bin_add_def] "bin_add(Bcons(v,x),Minus) = bin_pred(Bcons(v,x))";
 by (simp_tac bin_ss 1);
 qed "bin_add_Bcons_Minus";
 
 goalw Bin.thy [bin_add_def]
     "!!w y. [| w: bin;  y: bool |] ==> \
-\           bin_add(v$$x, w$$y) = \
-\           bin_add(v, cond(x and y, bin_succ(w), w)) $$ (x xor y)";
+\           bin_add(Bcons(v,x), Bcons(w,y)) = \
+\           norm_Bcons(bin_add(v, cond(x and y, bin_succ(w), w)), x xor y)";
 by (asm_simp_tac bin_ss 1);
 qed "bin_add_Bcons_Bcons";
 
-val bin_add_rews = [bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
-		    bin_add_Bcons_Minus, bin_add_Bcons_Bcons,
-		    integ_of_bin_succ, integ_of_bin_pred];
-
-val bin_add_ss = bin_ss addsimps ([bool_subset_nat RS subsetD] @ bin_add_rews);
+val bin_add_simps = [bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
+		     bin_add_Bcons_Minus, bin_add_Bcons_Bcons,
+		     integ_of_bin_succ, integ_of_bin_pred,
+		     integ_of_bin_norm_Bcons];
+
+val bin_add_ss = 
+    bin_ss addsimps ([bool_subset_nat RS subsetD] @ bin_add_simps);
 
 goal Bin.thy
     "!!v. v: bin ==> \
 \         ALL w: bin. integ_of_bin(bin_add(v,w)) = \
 \                     integ_of_bin(v) $+ integ_of_bin(w)";
 by (etac bin.induct 1);
 by (simp_tac bin_add_ss 1);
 by (simp_tac bin_add_ss 1);
 by (rtac ballI 1);
 by (bin_ind_tac "wa" [] 1);
-by (asm_simp_tac (bin_add_ss addsimps [zadd_0_right]) 1);
 by (asm_simp_tac bin_add_ss 1);
-by (REPEAT (ares_tac (zadd_commute::typechecks) 1));
-by (etac boolE 1);
-by (asm_simp_tac (bin_add_ss addsimps [zadd_assoc, zadd_swap_pairs]) 2);
-by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill]@typechecks) 2));
-by (etac boolE 1);
-by (ALLGOALS (asm_simp_tac (bin_add_ss addsimps [zadd_assoc,zadd_swap_pairs])));
-by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill RS sym]@
-		      typechecks) 1));
-qed "integ_of_bin_add_lemma";
-
-val integ_of_bin_add = integ_of_bin_add_lemma RS bspec;
+by (asm_simp_tac (bin_add_ss addsimps zadd_ac) 1);
+by (etac boolE 1);
+by (asm_simp_tac (bin_add_ss addsimps zadd_ac) 2);
+by (etac boolE 1);
+by (ALLGOALS (asm_simp_tac (bin_add_ss addsimps zadd_ac)));
+val integ_of_bin_add_lemma = result();
+
+bind_thm("integ_of_bin_add", integ_of_bin_add_lemma RS bspec);
 
 
 (*** bin_add: binary multiplication ***)
 
 val bin_mult_ss =
     bin_ss addsimps (bin_recs bin_mult_def @ 
-		       [integ_of_bin_minus, integ_of_bin_add]);
-
+		       [integ_of_bin_minus, integ_of_bin_add,
+			integ_of_bin_norm_Bcons]);
 
 val major::prems = goal Bin.thy
     "[| v: bin; w: bin |] ==>	\
 \    integ_of_bin(bin_mult(v,w)) = \
 \    integ_of_bin(v) $* integ_of_bin(w)";
 by (cut_facts_tac prems 1);
 by (bin_ind_tac "v" [major] 1);
-by (asm_simp_tac (bin_mult_ss addsimps [zmult_0]) 1);
-by (asm_simp_tac (bin_mult_ss addsimps [zmult_1,zmult_zminus]) 1);
+by (asm_simp_tac bin_mult_ss 1);
+by (asm_simp_tac bin_mult_ss 1);
 by (etac boolE 1);
 by (asm_simp_tac (bin_mult_ss addsimps [zadd_zmult_distrib]) 2);
 by (asm_simp_tac 
-    (bin_mult_ss addsimps [zadd_zmult_distrib, zmult_1, zadd_assoc]) 1);
-by (REPEAT (ares_tac ([zadd_commute, zadd_assoc_swap_kill RS sym]@
-		      typechecks) 1));
+    (bin_mult_ss addsimps ([zadd_zmult_distrib, zmult_1] @ zadd_ac)) 1);
 qed "integ_of_bin_mult";
 
 (**** Computations ****)
 
 (** extra rules for bin_succ, bin_pred **)
 
 val [bin_succ_Plus, bin_succ_Minus, _] = bin_recs bin_succ_def;
 val [bin_pred_Plus, bin_pred_Minus, _] = bin_recs bin_pred_def;
 
-goal Bin.thy "bin_succ(w$$1) = bin_succ(w) $$ 0";
+goal Bin.thy "bin_succ(Bcons(w,1)) = Bcons(bin_succ(w), 0)";
 by (simp_tac carry_ss 1);
 qed "bin_succ_Bcons1";
 
-goal Bin.thy "bin_succ(w$$0) = w$$1";
+goal Bin.thy "bin_succ(Bcons(w,0)) = norm_Bcons(w,1)";
 by (simp_tac carry_ss 1);
 qed "bin_succ_Bcons0";
 
-goal Bin.thy "bin_pred(w$$1) = w$$0";
+goal Bin.thy "bin_pred(Bcons(w,1)) = norm_Bcons(w,0)";
 by (simp_tac carry_ss 1);
 qed "bin_pred_Bcons1";
 
-goal Bin.thy "bin_pred(w$$0) = bin_pred(w) $$ 1";
+goal Bin.thy "bin_pred(Bcons(w,0)) = Bcons(bin_pred(w), 1)";
 by (simp_tac carry_ss 1);
 qed "bin_pred_Bcons0";
 
 (** extra rules for bin_minus **)
 
 val [bin_minus_Plus, bin_minus_Minus, _] = bin_recs bin_minus_def;
 
-goal Bin.thy "bin_minus(w$$1) = bin_pred(bin_minus(w) $$ 0)";
+goal Bin.thy "bin_minus(Bcons(w,1)) = bin_pred(Bcons(bin_minus(w), 0))";
 by (simp_tac bin_minus_ss 1);
 qed "bin_minus_Bcons1";
 
-goal Bin.thy "bin_minus(w$$0) = bin_minus(w) $$ 0";
+goal Bin.thy "bin_minus(Bcons(w,0)) = Bcons(bin_minus(w), 0)";
 by (simp_tac bin_minus_ss 1);
 qed "bin_minus_Bcons0";
 
 (** extra rules for bin_add **)
 
 goal Bin.thy 
-    "!!w. w: bin ==> bin_add(v$$1, w$$1) = bin_add(v, bin_succ(w)) $$ 0";
+    "!!w. w: bin ==> bin_add(Bcons(v,1), Bcons(w,1)) = \
+\                    norm_Bcons(bin_add(v, bin_succ(w)), 0)";
 by (asm_simp_tac bin_add_ss 1);
 qed "bin_add_Bcons_Bcons11";
 
 goal Bin.thy 
-    "!!w. w: bin ==> bin_add(v$$1, w$$0) = bin_add(v,w) $$ 1";
+    "!!w. w: bin ==> bin_add(Bcons(v,1), Bcons(w,0)) =  \
+\                    norm_Bcons(bin_add(v,w), 1)";
 by (asm_simp_tac bin_add_ss 1);
 qed "bin_add_Bcons_Bcons10";
 
 goal Bin.thy 
-    "!!w y.[| w: bin;  y: bool |] ==> bin_add(v$$0, w$$y) = bin_add(v,w) $$ y";
+    "!!w y. [| w: bin;  y: bool |] ==> \
+\           bin_add(Bcons(v,0), Bcons(w,y)) = norm_Bcons(bin_add(v,w), y)";
 by (asm_simp_tac bin_add_ss 1);
 qed "bin_add_Bcons_Bcons0";
 
 (** extra rules for bin_mult **)
 
 val [bin_mult_Plus, bin_mult_Minus, _] = bin_recs bin_mult_def;
 
-goal Bin.thy "bin_mult(v$$1, w) = bin_add(bin_mult(v,w)$$0, w)";
+goal Bin.thy
+    "bin_mult(Bcons(v,1), w) = bin_add(norm_Bcons(bin_mult(v,w),0), w)";
 by (simp_tac bin_mult_ss 1);
 qed "bin_mult_Bcons1";
 
-goal Bin.thy "bin_mult(v$$0, w) = bin_mult(v,w)$$0";
+goal Bin.thy "bin_mult(Bcons(v,0), w) = norm_Bcons(bin_mult(v,w),0)";
 by (simp_tac bin_mult_ss 1);
 qed "bin_mult_Bcons0";
 
 
 (*** The computation simpset ***)
 
 val bin_comp_ss = integ_ss 
-    addsimps [bin_succ_Plus, bin_succ_Minus,
-	     bin_succ_Bcons1, bin_succ_Bcons0,
-	     bin_pred_Plus, bin_pred_Minus,
-	     bin_pred_Bcons1, bin_pred_Bcons0,
-	     bin_minus_Plus, bin_minus_Minus,
-	     bin_minus_Bcons1, bin_minus_Bcons0,
-	     bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus, 
-	     bin_add_Bcons_Minus, bin_add_Bcons_Bcons0, 
-	     bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11,
-	     bin_mult_Plus, bin_mult_Minus,
-	     bin_mult_Bcons1, bin_mult_Bcons0]
+    addsimps [integ_of_bin_add RS sym,   (*invoke bin_add*)
+	      integ_of_bin_minus RS sym, (*invoke bin_minus*)
+	      integ_of_bin_mult RS sym,	 (*invoke bin_mult*)
+	      bin_succ_Plus, bin_succ_Minus,
+	      bin_succ_Bcons1, bin_succ_Bcons0,
+	      bin_pred_Plus, bin_pred_Minus,
+	      bin_pred_Bcons1, bin_pred_Bcons0,
+	      bin_minus_Plus, bin_minus_Minus,
+	      bin_minus_Bcons1, bin_minus_Bcons0,
+	      bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus, 
+	      bin_add_Bcons_Minus, bin_add_Bcons_Bcons0, 
+	      bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11,
+	      bin_mult_Plus, bin_mult_Minus,
+	      bin_mult_Bcons1, bin_mult_Bcons0] @
+             norm_Bcons_simps
     setsolver (type_auto_tac ([bool_1I, bool_0I] @ bin_typechecks0));
 
 (*** Examples of performing binary arithmetic by simplification ***)
 
 proof_timing := true;
 (*All runtimes below are on a SPARCserver 10*)
 
-(* 13+19 = 32 *)
-goal Bin.thy
-    "bin_add(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = Plus$$1$$0$$0$$0$$0$$0";
-by (simp_tac bin_comp_ss 1);	(*0.6 secs*)
+goal Bin.thy "#13  $+  #19 = #32";
+by (simp_tac bin_comp_ss 1);	(*0.4 secs*)
 result();
 
 bin_add(binary_of_int 13, binary_of_int 19);
 
-(* 1234+5678 = 6912 *)
-goal Bin.thy
-    "bin_add(Plus$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0, \
-\	     Plus$$1$$0$$1$$1$$0$$0$$0$$1$$0$$1$$1$$1$$0) = \
-\    Plus$$1$$1$$0$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0";
-by (simp_tac bin_comp_ss 1);	(*2.6 secs*)
+goal Bin.thy "#1234  $+  #5678 = #6912";
+by (simp_tac bin_comp_ss 1);	(*1.3 secs*)
 result();
 
 bin_add(binary_of_int 1234, binary_of_int 5678);
 
-(* 1359-2468 = ~1109 *)
-goal Bin.thy
-    "bin_add(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1,		\
-\	     Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = 	\
-\    Minus$$1$$0$$1$$1$$1$$0$$1$$0$$1$$0$$1$$1";
-by (simp_tac bin_comp_ss 1);	(*2.3 secs*)
+goal Bin.thy "#1359  $+  #~2468 = #~1109";
+by (simp_tac bin_comp_ss 1);	(*1.2 secs*)
 result();
 
 bin_add(binary_of_int 1359, binary_of_int ~2468);
 
-(* 93746-46375 = 47371 *)
-goal Bin.thy
-    "bin_add(Plus$$1$$0$$1$$1$$0$$1$$1$$1$$0$$0$$0$$1$$1$$0$$0$$1$$0, \
-\	     Minus$$0$$1$$0$$0$$1$$0$$1$$0$$1$$1$$0$$1$$1$$0$$0$$1) = \
-\    Plus$$0$$1$$0$$1$$1$$1$$0$$0$$1$$0$$0$$0$$0$$1$$0$$1$$1";
-by (simp_tac bin_comp_ss 1);	(*3.9 secs*)
+goal Bin.thy "#93746  $+  #~46375 = #47371";
+by (simp_tac bin_comp_ss 1);	(*1.9 secs*)
 result();
 
 bin_add(binary_of_int 93746, binary_of_int ~46375);
 
-(* negation of 65745 *)
-goal Bin.thy
-    "bin_minus(Plus$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1$$1$$0$$1$$0$$0$$0$$1) = \
-\    Minus$$0$$1$$1$$1$$1$$1$$1$$1$$1$$0$$0$$1$$0$$1$$1$$1$$1";
-by (simp_tac bin_comp_ss 1);	(*0.6 secs*)
+goal Bin.thy "$~ #65745 = #~65745";
+by (simp_tac bin_comp_ss 1);	(*0.4 secs*)
 result();
 
 bin_minus(binary_of_int 65745);
 
 (* negation of ~54321 *)
-goal Bin.thy
-    "bin_minus(Minus$$0$$0$$1$$0$$1$$0$$1$$1$$1$$1$$0$$0$$1$$1$$1$$1) = \
-\    Plus$$0$$1$$1$$0$$1$$0$$1$$0$$0$$0$$0$$1$$1$$0$$0$$0$$1";
+goal Bin.thy "$~ #~54321 = #54321";
+by (simp_tac bin_comp_ss 1);	(*0.5 secs*)
+result();
+
+bin_minus(binary_of_int ~54321);
+
+goal Bin.thy "#13  $*  #19 = #247";
 by (simp_tac bin_comp_ss 1);	(*0.7 secs*)
 result();
 
-bin_minus(binary_of_int ~54321);
-
-(* 13*19 = 247 *)
-goal Bin.thy "bin_mult(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = \
-\               Plus$$1$$1$$1$$1$$0$$1$$1$$1";
-by (simp_tac bin_comp_ss 1);	(*1.5 secs*)
-result();
-
 bin_mult(binary_of_int 13, binary_of_int 19);
 
-(* ~84 * 51 = ~4284 *)
-goal Bin.thy
-    "bin_mult(Minus$$0$$1$$0$$1$$1$$0$$0, Plus$$1$$1$$0$$0$$1$$1) = \
-\    Minus$$0$$1$$1$$1$$1$$0$$1$$0$$0$$0$$1$$0$$0";
-by (simp_tac bin_comp_ss 1);	(*2.6 secs*)
+goal Bin.thy "#~84  $*  #51 = #~4284";
+by (simp_tac bin_comp_ss 1);	(*1.3 secs*)
 result();
 
 bin_mult(binary_of_int ~84, binary_of_int 51);
 
-(* 255*255 = 65025;  the worst case for 8-bit operands *)
-goal Bin.thy
-    "bin_mult(Plus$$1$$1$$1$$1$$1$$1$$1$$1, \
-\             Plus$$1$$1$$1$$1$$1$$1$$1$$1) = \
-\        Plus$$1$$1$$1$$1$$1$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1";
-by (simp_tac bin_comp_ss 1);	(*9.8 secs*)
+(*The worst case for 8-bit operands *)
+goal Bin.thy "#255  $*  #255 = #65025";
+by (simp_tac bin_comp_ss 1);	(*4.3 secs*)
 result();
 
 bin_mult(binary_of_int 255, binary_of_int 255);
 
-(* 1359 * ~2468 = ~3354012 *)
-goal Bin.thy
-    "bin_mult(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1, 		\
-\	      Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = 	\
-\    Minus$$0$$0$$1$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0$$0$$1$$1$$0$$0$$1$$0$$0";
-by (simp_tac bin_comp_ss 1);	(*13.7 secs*)
+goal Bin.thy "#1359  $*  #~2468 = #~3354012";
+by (simp_tac bin_comp_ss 1);	(*6.1 secs*)
 result();
 
 bin_mult(binary_of_int 1359, binary_of_int ~2468);