Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
authorpaulson
Mon Oct 22 11:54:22 2001 +0200 (2001-10-22)
changeset 1186856db9f3a6b3e
parent 11867 76401b2ee871
child 11869 66d4f20eb3fc
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
to their abstract counterparts, while other binary numerals work correctly.
src/HOL/Integ/Bin.ML
src/HOL/Integ/Bin.thy
src/HOL/Integ/Int.ML
src/HOL/Integ/Int.thy
src/HOL/Integ/IntArith.ML
src/HOL/Integ/IntArith.thy
src/HOL/Integ/IntDef.ML
src/HOL/Integ/IntDef.thy
src/HOL/Integ/IntDiv.ML
src/HOL/Integ/IntDiv.thy
src/HOL/Integ/IntPower.ML
src/HOL/Integ/IntPower.thy
src/HOL/Integ/NatSimprocs.ML
src/HOL/Integ/int_arith1.ML
src/HOL/Integ/int_arith2.ML
src/HOL/Integ/int_factor_simprocs.ML
src/HOL/Integ/nat_bin.ML
src/HOL/Integ/nat_simprocs.ML
src/HOL/Library/Multiset.thy
src/HOL/List.ML
src/HOL/Nat.ML
src/HOL/NatDef.ML
src/HOL/NumberTheory/Chinese.thy
src/HOL/NumberTheory/EulerFermat.thy
src/HOL/NumberTheory/Fib.thy
src/HOL/NumberTheory/IntFact.thy
src/HOL/NumberTheory/IntPrimes.thy
src/HOL/NumberTheory/WilsonBij.thy
src/HOL/NumberTheory/WilsonRuss.thy
src/HOL/Numeral.thy
src/HOL/ROOT.ML
src/HOL/Real/ex/Sqrt_Irrational.thy
src/HOL/SVC_Oracle.ML
src/HOL/UNITY/Comp/Counter.ML
src/HOL/UNITY/Comp/Counter.thy
src/HOL/UNITY/Comp/Counterc.ML
src/HOL/UNITY/Comp/Counterc.thy
src/HOL/UNITY/Simple/Lift.ML
src/HOL/UNITY/Simple/Lift.thy
src/HOL/UNITY/Simple/Mutex.ML
src/HOL/UNITY/Simple/Mutex.thy
src/HOL/UNITY/SubstAx.ML
src/HOL/UNITY/Union.ML
src/HOL/ex/BinEx.thy
src/HOL/ex/IntRing.thy
src/HOL/ex/svc_test.ML
     1.1 --- a/src/HOL/Integ/Bin.ML	Mon Oct 22 11:01:30 2001 +0200
     1.2 +++ b/src/HOL/Integ/Bin.ML	Mon Oct 22 11:54:22 2001 +0200
     1.3 @@ -98,7 +98,7 @@
     1.4  (**** The carry/borrow functions, bin_succ and bin_pred ****)
     1.5  
     1.6  
     1.7 -(**** number_of ****)
     1.8 +(** number_of **)
     1.9  
    1.10  Goal "number_of(NCons w b) = (number_of(w BIT b)::int)";
    1.11  by (induct_tac "w" 1);
    1.12 @@ -107,12 +107,12 @@
    1.13  
    1.14  Addsimps [number_of_NCons];
    1.15  
    1.16 -Goal "number_of(bin_succ w) = int 1 + number_of w";
    1.17 +Goal "number_of(bin_succ w) = (1 + number_of w :: int)";
    1.18  by (induct_tac "w" 1);
    1.19  by (ALLGOALS (asm_simp_tac (simpset() addsimps zadd_ac)));
    1.20  qed "number_of_succ";
    1.21  
    1.22 -Goal "number_of(bin_pred w) = - (int 1) + number_of w";
    1.23 +Goal "number_of(bin_pred w) = (- 1 + number_of w :: int)";
    1.24  by (induct_tac "w" 1);
    1.25  by (ALLGOALS (asm_simp_tac (simpset() addsimps zadd_ac)));
    1.26  qed "number_of_pred";
    1.27 @@ -127,17 +127,15 @@
    1.28  			    zadd_assoc]) 1);
    1.29  qed "number_of_minus";
    1.30  
    1.31 - 
    1.32 -bind_thms ("bin_add_simps", [bin_add_BIT_BIT, number_of_succ, number_of_pred]);
    1.33 -
    1.34  (*This proof is complicated by the mutual recursion*)
    1.35  Goal "! w. number_of(bin_add v w) = (number_of v + number_of w::int)";
    1.36 -by (induct_tac "v" 1);
    1.37 -by (simp_tac (simpset() addsimps bin_add_simps) 1);
    1.38 -by (simp_tac (simpset() addsimps bin_add_simps) 1);
    1.39 +by (induct_tac "v" 1); 
    1.40 +by (Simp_tac 1);
    1.41 +by (simp_tac (simpset() addsimps [number_of_pred]) 1);
    1.42  by (rtac allI 1);
    1.43  by (induct_tac "w" 1);
    1.44 -by (ALLGOALS (asm_simp_tac (simpset() addsimps bin_add_simps @ zadd_ac)));
    1.45 +by (ALLGOALS (asm_simp_tac (simpset() addsimps 
    1.46 +               [bin_add_BIT_BIT, number_of_succ, number_of_pred] @ zadd_ac)));
    1.47  qed_spec_mp "number_of_add";
    1.48  
    1.49  
    1.50 @@ -147,7 +145,8 @@
    1.51  by (simp_tac (simpset() addsimps [number_of_add, number_of_minus]) 1);
    1.52  qed "diff_number_of_eq";
    1.53  
    1.54 -bind_thms ("bin_mult_simps", [zmult_zminus, number_of_minus, number_of_add]);
    1.55 +bind_thms ("bin_mult_simps", 
    1.56 +           [int_Suc0_eq_1, zmult_zminus, number_of_minus, number_of_add]);
    1.57  
    1.58  Goal "number_of(bin_mult v w) = (number_of v * number_of w::int)";
    1.59  by (induct_tac "v" 1);
    1.60 @@ -167,54 +166,15 @@
    1.61  qed "double_number_of_BIT";
    1.62  
    1.63  
    1.64 -(** Simplification rules with integer constants **)
    1.65 -
    1.66 -Goal "Numeral0 + z = (z::int)";
    1.67 -by (Simp_tac 1);
    1.68 -qed "zadd_0";
    1.69 -
    1.70 -Goal "z + Numeral0 = (z::int)";
    1.71 -by (Simp_tac 1);
    1.72 -qed "zadd_0_right";
    1.73 -
    1.74 -Addsimps [zadd_0, zadd_0_right];
    1.75 -
    1.76 -
    1.77 -(** Converting simple cases of (int n) to numerals **)
    1.78 -
    1.79 -Goal "int 0 = Numeral0";
    1.80 -by (rtac sym 1);
    1.81 -by (rtac number_of_Pls 1);
    1.82 -qed "int_0";
    1.83 -
    1.84 -Goal "int 1 = Numeral1";
    1.85 -by (Simp_tac 1);
    1.86 -qed "int_1";
    1.87 +(** Converting numerals 0 and 1 to their abstract versions **)
    1.88  
    1.89 -Goal "int (Suc n) = Numeral1 + int n";
    1.90 -by (simp_tac (simpset() addsimps [zadd_int]) 1);
    1.91 -qed "int_Suc";
    1.92 -
    1.93 -Goal "- (Numeral0) = (Numeral0::int)";
    1.94 +Goal "Numeral0 = (0::int)";
    1.95  by (Simp_tac 1);
    1.96 -qed "zminus_0";
    1.97 -
    1.98 -Addsimps [zminus_0];
    1.99 -
   1.100 +qed "int_numeral_0_eq_0";
   1.101  
   1.102 -Goal "(Numeral0::int) - x = -x";
   1.103 -by (simp_tac (simpset() addsimps [zdiff_def]) 1);
   1.104 -qed "zdiff0";
   1.105 -
   1.106 -Goal "x - (Numeral0::int) = x";
   1.107 -by (simp_tac (simpset() addsimps [zdiff_def]) 1);
   1.108 -qed "zdiff0_right";
   1.109 -
   1.110 -Goal "x - x = (Numeral0::int)";
   1.111 -by (simp_tac (simpset() addsimps [zdiff_def]) 1);
   1.112 -qed "zdiff_self";
   1.113 -
   1.114 -Addsimps [zdiff0, zdiff0_right, zdiff_self];
   1.115 +Goal "Numeral1 = (1::int)";
   1.116 +by (simp_tac (simpset() addsimps [int_1, int_Suc0_eq_1]) 1);
   1.117 +qed "int_numeral_1_eq_1";
   1.118  
   1.119  
   1.120  (** Special simplification, for constants only **)
   1.121 @@ -240,99 +200,30 @@
   1.122  
   1.123  (** Special-case simplification for small constants **)
   1.124  
   1.125 -Goal "Numeral0 * z = (Numeral0::int)";
   1.126 -by (Simp_tac 1);
   1.127 -qed "zmult_0";
   1.128 -
   1.129 -Goal "z * Numeral0 = (Numeral0::int)";
   1.130 -by (Simp_tac 1);
   1.131 -qed "zmult_0_right";
   1.132 -
   1.133 -Goal "Numeral1 * z = (z::int)";
   1.134 -by (Simp_tac 1);
   1.135 -qed "zmult_1";
   1.136 -
   1.137 -Goal "z * Numeral1 = (z::int)";
   1.138 -by (Simp_tac 1);
   1.139 -qed "zmult_1_right";
   1.140 -
   1.141  Goal "-1 * z = -(z::int)";
   1.142 -by (simp_tac (simpset() addsimps zcompare_rls@[zmult_zminus]) 1);
   1.143 +by (simp_tac (simpset() addsimps zcompare_rls@[int_Suc0_eq_1, zmult_zminus]) 1);
   1.144  qed "zmult_minus1";
   1.145  
   1.146  Goal "z * -1 = -(z::int)";
   1.147 -by (simp_tac (simpset() addsimps zcompare_rls@[zmult_zminus_right]) 1);
   1.148 +by (stac zmult_commute 1 THEN rtac zmult_minus1 1);
   1.149  qed "zmult_minus1_right";
   1.150  
   1.151 -Addsimps [zmult_0, zmult_0_right, 
   1.152 -	  zmult_1, zmult_1_right,
   1.153 -	  zmult_minus1, zmult_minus1_right];
   1.154 +Addsimps [zmult_minus1, zmult_minus1_right];
   1.155  
   1.156  (*Negation of a coefficient*)
   1.157  Goal "- (number_of w) * z = number_of(bin_minus w) * (z::int)";
   1.158  by (simp_tac (simpset() addsimps [number_of_minus, zmult_zminus]) 1);
   1.159  qed "zminus_number_of_zmult";
   1.160 -
   1.161  Addsimps [zminus_number_of_zmult];
   1.162  
   1.163 -
   1.164 -(** Inequality reasoning **)
   1.165 -
   1.166 -Goal "(m*n = (Numeral0::int)) = (m = Numeral0 | n = Numeral0)";
   1.167 -by (stac (int_0 RS sym) 1 THEN rtac zmult_eq_int0_iff 1);
   1.168 -qed "zmult_eq_0_iff";
   1.169 -AddIffs [zmult_eq_0_iff];
   1.170 -
   1.171 -Goal "(w < z + (Numeral1::int)) = (w<z | w=z)";
   1.172 -by (simp_tac (simpset() addsimps [zless_add_int_Suc_eq]) 1);
   1.173 -qed "zless_add1_eq";
   1.174 -
   1.175 -Goal "(w + (Numeral1::int) <= z) = (w<z)";
   1.176 -by (simp_tac (simpset() addsimps [add_int_Suc_zle_eq]) 1);
   1.177 -qed "add1_zle_eq";
   1.178 -
   1.179 -Goal "((Numeral1::int) + w <= z) = (w<z)";
   1.180 -by (stac zadd_commute 1);
   1.181 -by (rtac add1_zle_eq 1);
   1.182 -qed "add1_left_zle_eq";
   1.183 -
   1.184 -Goal "neg x = (x < Numeral0)";
   1.185 -by (simp_tac (simpset() addsimps [neg_eq_less_int0]) 1); 
   1.186 -qed "neg_eq_less_0"; 
   1.187 -
   1.188 -Goal "(~neg x) = (Numeral0 <= x)";
   1.189 -by (simp_tac (simpset() addsimps [not_neg_eq_ge_int0]) 1); 
   1.190 -qed "not_neg_eq_ge_0"; 
   1.191 +(*Integer unary minus for the abstract constant 1. Cannot be inserted
   1.192 +  as a simprule until later: it is number_of_Min re-oriented!*)
   1.193 +Goal "- 1 = (-1::int)";
   1.194 +by (Simp_tac 1);
   1.195 +qed "zminus_1_eq_m1";
   1.196  
   1.197 -Goal "Numeral0 <= int m";
   1.198 -by (Simp_tac 1);
   1.199 -qed "zero_zle_int";
   1.200 -AddIffs [zero_zle_int];
   1.201 -
   1.202 -
   1.203 -(** Needed because (int 0) rewrites to Numeral0.   (* FIXME !? *)
   1.204 -    Can these be generalized without evaluating large numbers?**)
   1.205 -
   1.206 -Goal "~ (int k < Numeral0)";
   1.207 -by (Simp_tac 1);
   1.208 -qed "int_less_0_conv";
   1.209 -
   1.210 -Goal "(int k <= Numeral0) = (k=0)";
   1.211 -by (Simp_tac 1);
   1.212 -qed "int_le_0_conv";
   1.213 -
   1.214 -Goal "(int k = Numeral0) = (k=0)";
   1.215 -by (Simp_tac 1);
   1.216 -qed "int_eq_0_conv";
   1.217 -
   1.218 -Goal "(Numeral0 < int k) = (0<k)";
   1.219 -by (Simp_tac 1);
   1.220 -qed "zero_less_int_conv";
   1.221 -
   1.222 -Addsimps [int_less_0_conv, int_le_0_conv, int_eq_0_conv, zero_less_int_conv];
   1.223 -
   1.224 -Goal "(0 < nat z) = (Numeral0 < z)";
   1.225 -by (cut_inst_tac [("w","Numeral0")] zless_nat_conj 1);
   1.226 +Goal "(0 < nat z) = (0 < z)";
   1.227 +by (cut_inst_tac [("w","0")] zless_nat_conj 1);
   1.228  by Auto_tac;  
   1.229  qed "zero_less_nat_eq";
   1.230  Addsimps [zero_less_nat_eq];
   1.231 @@ -345,7 +236,7 @@
   1.232  Goalw [iszero_def]
   1.233    "((number_of x::int) = number_of y) = \
   1.234  \  iszero (number_of (bin_add x (bin_minus y)))"; 
   1.235 -by (simp_tac (simpset() delsimps [thm "number_of_reorient"] 
   1.236 +by (simp_tac (simpset()
   1.237                   addsimps zcompare_rls @ [number_of_add, number_of_minus]) 1); 
   1.238  qed "eq_number_of_eq"; 
   1.239  
   1.240 @@ -360,9 +251,11 @@
   1.241  Goalw [iszero_def]
   1.242       "iszero (number_of (w BIT x)) = (~x & iszero (number_of w::int))"; 
   1.243  by (int_case_tac "number_of w" 1); 
   1.244 -by (ALLGOALS (asm_simp_tac 
   1.245 -	      (simpset() addsimps zcompare_rls @ 
   1.246 -				  [zminus_zadd_distrib RS sym, zadd_int]))); 
   1.247 +by (ALLGOALS 
   1.248 +    (asm_simp_tac 
   1.249 +     (simpset() addsimps zcompare_rls @ 
   1.250 +  	                 [zero_reorient, zminus_zadd_distrib RS sym, 
   1.251 +                          int_Suc0_eq_1 RS sym, zadd_int]))); 
   1.252  qed "iszero_number_of_BIT"; 
   1.253  
   1.254  Goal "iszero (number_of (w BIT False)) = iszero (number_of w::int)"; 
   1.255 @@ -383,20 +276,22 @@
   1.256  by (simp_tac (simpset() addsimps bin_mult_simps) 1);
   1.257  qed "less_number_of_eq_neg"; 
   1.258  
   1.259 -Goal "~ neg (number_of Pls)"; 
   1.260 -by (Simp_tac 1); 
   1.261 +(*But if Numeral0 is rewritten to 0 then this rule can't be applied:
   1.262 +  Numeral0 IS (number_of Pls) *)
   1.263 +Goal "~ neg (number_of Pls)";
   1.264 +by (simp_tac (simpset() addsimps [neg_eq_less_0]) 1);  
   1.265  qed "not_neg_number_of_Pls"; 
   1.266  
   1.267  Goal "neg (number_of Min)"; 
   1.268 -by (Simp_tac 1);
   1.269 +by (simp_tac (simpset() addsimps [neg_eq_less_0, int_0_less_1]) 1);
   1.270  qed "neg_number_of_Min"; 
   1.271  
   1.272  Goal "neg (number_of (w BIT x)) = neg (number_of w)"; 
   1.273  by (Asm_simp_tac 1); 
   1.274  by (int_case_tac "number_of w" 1); 
   1.275  by (ALLGOALS (asm_simp_tac 
   1.276 -	      (simpset() addsimps [zadd_int, neg_eq_less_int0, 
   1.277 -				   symmetric zdiff_def] @ zcompare_rls))); 
   1.278 +	      (simpset() addsimps [int_Suc0_eq_1 RS sym, zadd_int, 
   1.279 +                         neg_eq_less_0, symmetric zdiff_def] @ zcompare_rls)));
   1.280  qed "neg_number_of_BIT"; 
   1.281  
   1.282  
   1.283 @@ -413,7 +308,114 @@
   1.284   "abs(number_of x::int) = \
   1.285  \ (if number_of x < (0::int) then -number_of x else number_of x)";
   1.286  by(rtac refl 1);
   1.287 -qed "abs_number_of";
   1.288 +qed "zabs_number_of";
   1.289 +
   1.290 +(*0 and 1 require special rewrites because they aren't numerals*)
   1.291 +Goal "abs (0::int) = 0";
   1.292 +by (simp_tac (simpset() addsimps [zabs_def]) 1); 
   1.293 +qed "zabs_0";
   1.294 +
   1.295 +Goal "abs (1::int) = 1";
   1.296 +by (simp_tac (simpset() delsimps [int_0, int_1] 
   1.297 +                       addsimps [int_0 RS sym, int_1 RS sym, zabs_def]) 1); 
   1.298 +qed "zabs_1";
   1.299 +
   1.300 +(*Re-orientation of the equation nnn=x*)
   1.301 +Goal "(number_of w = x) = (x = number_of w)";
   1.302 +by Auto_tac;  
   1.303 +qed "number_of_reorient";
   1.304 +
   1.305 +
   1.306 +structure Bin_Simprocs =
   1.307 +  struct
   1.308 +  fun prove_conv name tacs sg (hyps: thm list) (t,u) =
   1.309 +    if t aconv u then None
   1.310 +    else
   1.311 +    let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))
   1.312 +    in Some
   1.313 +       (prove_goalw_cterm [] ct (K tacs)
   1.314 +	handle ERROR => error 
   1.315 +	    ("The error(s) above occurred while trying to prove " ^
   1.316 +	     string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))
   1.317 +    end
   1.318 +
   1.319 +  (*version without the hyps argument*)
   1.320 +  fun prove_conv_nohyps name tacs sg = prove_conv name tacs sg []
   1.321 +
   1.322 +  fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc
   1.323 +  fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context())) (s, HOLogic.termT)
   1.324 +  val prep_pats = map prep_pat
   1.325 +
   1.326 +  fun is_numeral (Const("Numeral.number_of", _) $ w) = true
   1.327 +    | is_numeral _ = false
   1.328 +
   1.329 +  fun simplify_meta_eq f_number_of_eq f_eq =
   1.330 +      mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
   1.331 +
   1.332 +  structure IntAbstractNumeralsData =
   1.333 +    struct
   1.334 +    val dest_eq		= HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
   1.335 +    val is_numeral	= is_numeral
   1.336 +    val numeral_0_eq_0    = int_numeral_0_eq_0
   1.337 +    val numeral_1_eq_1    = int_numeral_1_eq_1
   1.338 +    val prove_conv	= prove_conv_nohyps "int_abstract_numerals"
   1.339 +    fun norm_tac simps	= ALLGOALS (simp_tac (HOL_ss addsimps simps))
   1.340 +    val simplify_meta_eq  = simplify_meta_eq 
   1.341 +    end
   1.342 +
   1.343 +  structure IntAbstractNumerals = AbstractNumeralsFun (IntAbstractNumeralsData)
   1.344 +
   1.345 +
   1.346 +  (*For addition, we already have rules for the operand 0.
   1.347 +    Multiplication is omitted because there are already special rules for 
   1.348 +    both 0 and 1 as operands.  Unary minus is trivial, just have - 1 = -1.
   1.349 +    For the others, having three patterns is a compromise between just having
   1.350 +    one (many spurious calls) and having nine (just too many!) *)
   1.351 +  val eval_numerals = 
   1.352 +    map prep_simproc
   1.353 +     [("int_add_eval_numerals",
   1.354 +       prep_pats ["(m::int) + 1", "(m::int) + number_of v"], 
   1.355 +       IntAbstractNumerals.proc (number_of_add RS sym)),
   1.356 +      ("int_diff_eval_numerals",
   1.357 +       prep_pats ["(m::int) - 1", "(m::int) - number_of v"], 
   1.358 +       IntAbstractNumerals.proc diff_number_of_eq),
   1.359 +      ("int_eq_eval_numerals",
   1.360 +       prep_pats ["(m::int) = 0", "(m::int) = 1", "(m::int) = number_of v"], 
   1.361 +       IntAbstractNumerals.proc eq_number_of_eq),
   1.362 +      ("int_less_eval_numerals",
   1.363 +       prep_pats ["(m::int) < 0", "(m::int) < 1", "(m::int) < number_of v"], 
   1.364 +       IntAbstractNumerals.proc less_number_of_eq_neg),
   1.365 +      ("int_le_eval_numerals",
   1.366 +       prep_pats ["(m::int) <= 0", "(m::int) <= 1", "(m::int) <= number_of v"],
   1.367 +       IntAbstractNumerals.proc le_number_of_eq_not_less)]
   1.368 +
   1.369 +  (*reorientation simprules using ==, for the following simproc*)
   1.370 +  val meta_zero_reorient = zero_reorient RS eq_reflection
   1.371 +  val meta_one_reorient = one_reorient RS eq_reflection
   1.372 +  val meta_number_of_reorient = number_of_reorient RS eq_reflection
   1.373 +
   1.374 +  (*reorientation simplification procedure: reorients (polymorphic) 
   1.375 +    0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
   1.376 +  fun reorient_proc sg _ (_ $ t $ u) =
   1.377 +    case u of
   1.378 +	Const("0", _) => None
   1.379 +      | Const("1", _) => None
   1.380 +      | Const("Numeral.number_of", _) $ _ => None
   1.381 +      | _ => Some (case t of
   1.382 +		  Const("0", _) => meta_zero_reorient
   1.383 +		| Const("1", _) => meta_one_reorient
   1.384 +		| Const("Numeral.number_of", _) $ _ => meta_number_of_reorient)
   1.385 +
   1.386 +  val reorient_simproc = 
   1.387 +      prep_simproc ("reorient_simproc",
   1.388 +	            prep_pats ["0=x", "1=x", "number_of w = x"], 
   1.389 +                    reorient_proc)
   1.390 +
   1.391 +  end;
   1.392 +
   1.393 +
   1.394 +Addsimprocs Bin_Simprocs.eval_numerals;
   1.395 +Addsimprocs [Bin_Simprocs.reorient_simproc];
   1.396  
   1.397  
   1.398  (*Delete the original rewrites, with their clumsy conditional expressions*)
   1.399 @@ -423,21 +425,24 @@
   1.400  (*Hide the binary representation of integer constants*)
   1.401  Delsimps [number_of_Pls, number_of_Min, number_of_BIT];
   1.402  
   1.403 +
   1.404 +
   1.405  (*Simplification of arithmetic operations on integer constants.
   1.406    Note that bin_pred_Pls, etc. were added to the simpset by primrec.*)
   1.407  
   1.408 -bind_thms ("NCons_simps", [NCons_Pls_0, NCons_Pls_1, NCons_Min_0, NCons_Min_1, NCons_BIT]);
   1.409 +bind_thms ("NCons_simps", 
   1.410 +           [NCons_Pls_0, NCons_Pls_1, NCons_Min_0, NCons_Min_1, NCons_BIT]);
   1.411  
   1.412  bind_thms ("bin_arith_extra_simps",
   1.413      [number_of_add RS sym,
   1.414 -     number_of_minus RS sym,
   1.415 +     number_of_minus RS sym, zminus_1_eq_m1,
   1.416       number_of_mult RS sym,
   1.417       bin_succ_1, bin_succ_0, 
   1.418       bin_pred_1, bin_pred_0, 
   1.419       bin_minus_1, bin_minus_0,  
   1.420       bin_add_Pls_right, bin_add_Min_right,
   1.421       bin_add_BIT_0, bin_add_BIT_10, bin_add_BIT_11,
   1.422 -     diff_number_of_eq, abs_number_of,
   1.423 +     diff_number_of_eq, zabs_number_of, zabs_0, zabs_1,
   1.424       bin_mult_1, bin_mult_0] @ NCons_simps);
   1.425  
   1.426  (*For making a minimal simpset, one must include these default simprules
   1.427 @@ -454,7 +459,8 @@
   1.428      [eq_number_of_eq, iszero_number_of_Pls, nonzero_number_of_Min,
   1.429       iszero_number_of_0, iszero_number_of_1,
   1.430       less_number_of_eq_neg,
   1.431 -     not_neg_number_of_Pls, neg_number_of_Min, neg_number_of_BIT,
   1.432 +     not_neg_number_of_Pls, not_neg_0, not_neg_1, not_iszero_1, 
   1.433 +     neg_number_of_Min, neg_number_of_BIT,
   1.434       le_number_of_eq_not_less]);
   1.435  
   1.436  Addsimps bin_arith_extra_simps;
   1.437 @@ -484,3 +490,14 @@
   1.438  
   1.439  Addsimps [add_number_of_left, mult_number_of_left,
   1.440  	  add_number_of_diff1, add_number_of_diff2]; 
   1.441 +
   1.442 +
   1.443 +(** Inserting these natural simprules earlier would break many proofs! **) 
   1.444 +
   1.445 +(* int (Suc n) = 1 + int n *)
   1.446 +Addsimps [int_Suc];
   1.447 +
   1.448 +(* Numeral0 -> 0 and Numeral1 -> 1 *)
   1.449 +Addsimps [int_numeral_0_eq_0, int_numeral_1_eq_1];
   1.450 +
   1.451 +
     2.1 --- a/src/HOL/Integ/Bin.thy	Mon Oct 22 11:01:30 2001 +0200
     2.2 +++ b/src/HOL/Integ/Bin.thy	Mon Oct 22 11:54:22 2001 +0200
     2.3 @@ -15,12 +15,7 @@
     2.4  for the numerical interpretation.
     2.5  
     2.6  The representation expects that (m mod 2) is 0 or 1, even if m is negative;
     2.7 -For instance, ~5 div 2 = ~3 and ~5 mod 2 = 1; thus ~5 = (~3)*2 + 1
     2.8 -
     2.9 -Division is not defined yet!  To do it efficiently requires computing the
    2.10 -quotient and remainder using ML and checking the answer using multiplication
    2.11 -by proof.  Then uniqueness of the quotient and remainder yields theorems
    2.12 -quoting the previously computed values.  (Or code an oracle...)
    2.13 +For instance, -5 div 2 = -3 and -5 mod 2 = 1; thus -5 = (-3)*2 + 1
    2.14  *)
    2.15  
    2.16  Bin = Int + Numeral +
    2.17 @@ -41,11 +36,11 @@
    2.18  instance
    2.19    int :: number
    2.20  
    2.21 -primrec
    2.22 -  number_of_Pls  "number_of Pls = int 0"
    2.23 -  number_of_Min  "number_of Min = - (int 1)"
    2.24 -  number_of_BIT  "number_of(w BIT x) = (if x then int 1 else int 0) +
    2.25 -	                             (number_of w) + (number_of w)" 
    2.26 +primrec (*the type constraint is essential!*)
    2.27 +  number_of_Pls  "number_of Pls = 0"
    2.28 +  number_of_Min  "number_of Min = - (1::int)"
    2.29 +  number_of_BIT  "number_of(w BIT x) = (if x then 1 else 0) +
    2.30 +	                               (number_of w) + (number_of w)" 
    2.31  
    2.32  primrec
    2.33    bin_succ_Pls  "bin_succ Pls = Pls BIT True" 
    2.34 @@ -86,5 +81,4 @@
    2.35  	            (if x then (bin_add (NCons (bin_mult v w) False) w)
    2.36  	                  else (NCons (bin_mult v w) False))"
    2.37  
    2.38 -
    2.39  end
     3.1 --- a/src/HOL/Integ/Int.ML	Mon Oct 22 11:01:30 2001 +0200
     3.2 +++ b/src/HOL/Integ/Int.ML	Mon Oct 22 11:54:22 2001 +0200
     3.3 @@ -9,6 +9,56 @@
     3.4  *)
     3.5  
     3.6  
     3.7 +Goal "int 0 = (0::int)";
     3.8 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
     3.9 +qed "int_0";
    3.10 +
    3.11 +Goal "int 1 = 1";
    3.12 +by (simp_tac (simpset() addsimps [One_int_def]) 1); 
    3.13 +qed "int_1";
    3.14 +
    3.15 +Goal "int (Suc 0) = 1";
    3.16 +by (simp_tac (simpset() addsimps [One_int_def, One_nat_def]) 1); 
    3.17 +qed "int_Suc0_eq_1";
    3.18 +
    3.19 +Goalw [zdiff_def,zless_def] "neg x = (x < 0)";
    3.20 +by Auto_tac; 
    3.21 +qed "neg_eq_less_0"; 
    3.22 +
    3.23 +Goalw [zle_def] "(~neg x) = (0 <= x)";
    3.24 +by (simp_tac (simpset() addsimps [neg_eq_less_0]) 1); 
    3.25 +qed "not_neg_eq_ge_0"; 
    3.26 +
    3.27 +(** Needed to simplify inequalities when Numeral1 can get simplified to 1 **)
    3.28 +
    3.29 +Goal "~ neg 0";
    3.30 +by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);  
    3.31 +qed "not_neg_0"; 
    3.32 +
    3.33 +Goal "~ neg 1";
    3.34 +by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);  
    3.35 +qed "not_neg_1"; 
    3.36 +
    3.37 +Goal "iszero 0";
    3.38 +by (simp_tac (simpset() addsimps [iszero_def]) 1);
    3.39 +qed "iszero_0"; 
    3.40 +
    3.41 +Goal "~ iszero 1";
    3.42 +by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def, 
    3.43 +                                  iszero_def]) 1);
    3.44 +qed "not_iszero_1"; 
    3.45 +
    3.46 +Goal "0 < (1::int)";
    3.47 +by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1); 
    3.48 +qed "int_0_less_1";
    3.49 +
    3.50 +Goal "0 \\<noteq> (1::int)";
    3.51 +by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1); 
    3.52 +qed "int_0_neq_1";
    3.53 +
    3.54 +Addsimps [int_0, int_1, int_0_neq_1];
    3.55 +
    3.56 +
    3.57  (*** Abel_Cancel simproc on the integers ***)
    3.58  
    3.59  (* Lemmas needed for the simprocs *)
    3.60 @@ -23,7 +73,7 @@
    3.61    everything gets cancelled on the right.*)
    3.62  Goal "((x::int) + (y + z) = y) = (x = -z)";
    3.63  by (stac zadd_left_commute 1);
    3.64 -by (res_inst_tac [("t", "y")] (zadd_int0_right RS subst) 1
    3.65 +by (res_inst_tac [("t", "y")] (zadd_0_right RS subst) 1
    3.66      THEN stac zadd_left_cancel 1);
    3.67  by (simp_tac (simpset() addsimps [eq_zdiff_eq RS sym]) 1);
    3.68  qed "zadd_cancel_end";
    3.69 @@ -36,7 +86,7 @@
    3.70  
    3.71    val sg_ref 		= Sign.self_ref (Theory.sign_of (the_context ()))
    3.72    val T		= HOLogic.intT
    3.73 -  val zero	= Const ("IntDef.int", HOLogic.natT --> T) $ HOLogic.zero
    3.74 +  val zero		= Const ("0", HOLogic.intT)
    3.75    val restrict_to_left  = restrict_to_left
    3.76    val add_cancel_21	= zadd_cancel_21
    3.77    val add_cancel_end	= zadd_cancel_end
    3.78 @@ -44,8 +94,8 @@
    3.79    val add_assoc		= zadd_assoc
    3.80    val add_commute	= zadd_commute
    3.81    val add_left_commute	= zadd_left_commute
    3.82 -  val add_0		= zadd_int0
    3.83 -  val add_0_right	= zadd_int0_right
    3.84 +  val add_0		= zadd_0
    3.85 +  val add_0_right	= zadd_0_right
    3.86  
    3.87    val eq_diff_eq	= eq_zdiff_eq
    3.88    val eqI_rules		= [zless_eqI, zeq_eqI, zle_eqI]
    3.89 @@ -56,8 +106,8 @@
    3.90    val diff_def		= zdiff_def
    3.91    val minus_add_distrib	= zminus_zadd_distrib
    3.92    val minus_minus	= zminus_zminus
    3.93 -  val minus_0		= zminus_int0
    3.94 -  val add_inverses	= [zadd_zminus_inverse, zadd_zminus_inverse2];
    3.95 +  val minus_0		= zminus_0
    3.96 +  val add_inverses	= [zadd_zminus_inverse, zadd_zminus_inverse2]
    3.97    val cancel_simps	= [zadd_zminus_cancel, zminus_zadd_cancel]
    3.98  end;
    3.99  
   3.100 @@ -86,29 +136,26 @@
   3.101  by (simp_tac (simpset() addsimps [zle_def, zless_def]) 1);
   3.102  qed "zle_eq_not_neg";
   3.103  
   3.104 +(** Inequality reasoning **)
   3.105  
   3.106 -(*** Inequality lemmas involving int (Suc m) ***)
   3.107 -
   3.108 -Goal "(w < z + int (Suc m)) = (w < z + int m | w = z + int m)";
   3.109 +Goal "(w < z + (1::int)) = (w<z | w=z)";
   3.110  by (auto_tac (claset(),
   3.111 -	      simpset() addsimps [zless_iff_Suc_zadd, zadd_assoc, zadd_int]));
   3.112 -by (cut_inst_tac [("m","m")] int_Suc_int_1 1);
   3.113 -by (cut_inst_tac [("m","n")] int_Suc_int_1 1);
   3.114 -by (Asm_full_simp_tac 1);
   3.115 -by (case_tac "n" 1);
   3.116 -by Auto_tac;
   3.117 -by (cut_inst_tac [("m","m")] int_Suc_int_1 1);
   3.118 -by (full_simp_tac (simpset() addsimps zadd_ac) 1);
   3.119 -by (asm_full_simp_tac (simpset() addsimps [zadd_assoc RS sym]) 1);
   3.120 -by (auto_tac (claset(),
   3.121 -	      simpset() addsimps [zless_iff_Suc_zadd, zadd_assoc, zadd_int]));
   3.122 -qed "zless_add_int_Suc_eq";
   3.123 +	      simpset() addsimps [zless_iff_Suc_zadd, int_Suc,
   3.124 +                                  gr0_conv_Suc, zero_reorient]));
   3.125 +by (res_inst_tac [("x","Suc n")] exI 1); 
   3.126 +by (simp_tac (simpset() addsimps [int_Suc]) 1); 
   3.127 +qed "zless_add1_eq";
   3.128  
   3.129 -Goal "(w + int (Suc m) <= z) = (w + int m < z)";
   3.130 -by (simp_tac (simpset() addsimps [zle_def, zless_add_int_Suc_eq]) 1);
   3.131 -by (auto_tac (claset() addIs [zle_anti_sym] addEs [zless_asym],
   3.132 +Goal "(w + (1::int) <= z) = (w<z)";
   3.133 +by (asm_full_simp_tac (simpset() addsimps [zle_def, zless_add1_eq]) 1); 
   3.134 +by (auto_tac (claset() addIs [zle_anti_sym],
   3.135  	      simpset() addsimps [order_less_imp_le, symmetric zle_def]));
   3.136 -qed "add_int_Suc_zle_eq";
   3.137 +qed "add1_zle_eq";
   3.138 +
   3.139 +Goal "((1::int) + w <= z) = (w<z)";
   3.140 +by (stac zadd_commute 1);
   3.141 +by (rtac add1_zle_eq 1);
   3.142 +qed "add1_left_zle_eq";
   3.143  
   3.144  
   3.145  (*** Monotonicity results ***)
   3.146 @@ -194,7 +241,21 @@
   3.147  by Auto_tac;
   3.148  qed "zminus_equation";
   3.149  
   3.150 -Goal "- (int (Suc n)) < int 0";
   3.151 +
   3.152 +(** Instances of the equations above, for zero **)
   3.153 +
   3.154 +(*instantiate a variable to zero and simplify*)
   3.155 +fun zero_instance v th = simplify (simpset()) (inst v "0" th);
   3.156 +
   3.157 +Addsimps [zero_instance "x" zless_zminus,
   3.158 +          zero_instance "y" zminus_zless,
   3.159 +          zero_instance "x" zle_zminus,
   3.160 +          zero_instance "y" zminus_zle,
   3.161 +          zero_instance "x" equation_zminus,
   3.162 +          zero_instance "y" zminus_equation];
   3.163 +
   3.164 +
   3.165 +Goal "- (int (Suc n)) < 0";
   3.166  by (simp_tac (simpset() addsimps [zless_def]) 1);
   3.167  qed "negative_zless_0"; 
   3.168  
   3.169 @@ -204,7 +265,7 @@
   3.170  qed "negative_zless"; 
   3.171  AddIffs [negative_zless]; 
   3.172  
   3.173 -Goal "- int n <= int 0";
   3.174 +Goal "- int n <= 0";
   3.175  by (simp_tac (simpset() addsimps [zminus_zle]) 1);
   3.176  qed "negative_zle_0"; 
   3.177  
   3.178 @@ -213,7 +274,7 @@
   3.179  qed "negative_zle"; 
   3.180  AddIffs [negative_zle]; 
   3.181  
   3.182 -Goal "~(int 0 <= - (int (Suc n)))";
   3.183 +Goal "~(0 <= - (int (Suc n)))";
   3.184  by (stac zle_zminus 1);
   3.185  by (Simp_tac 1);
   3.186  qed "not_zle_0_negative"; 
   3.187 @@ -242,19 +303,11 @@
   3.188  
   3.189  
   3.190  Goal "(w <= z) = (EX n. z = w + int n)";
   3.191 -by (auto_tac (claset() addSIs [not_sym RS not0_implies_Suc],
   3.192 +by (auto_tac (claset() addIs [inst "x" "0::nat" exI]
   3.193 +		       addSIs [not_sym RS not0_implies_Suc],
   3.194  	      simpset() addsimps [zless_iff_Suc_zadd, int_le_less]));
   3.195  qed "zle_iff_zadd";
   3.196  
   3.197 -
   3.198 -Goalw [zdiff_def,zless_def] "neg x = (x < int 0)";
   3.199 -by Auto_tac; 
   3.200 -qed "neg_eq_less_int0"; 
   3.201 -
   3.202 -Goalw [zle_def] "(~neg x) = (int 0 <= x)";
   3.203 -by (simp_tac (simpset() addsimps [neg_eq_less_int0]) 1); 
   3.204 -qed "not_neg_eq_ge_int0"; 
   3.205 -
   3.206  Goal "abs (int m) = int m";
   3.207  by (simp_tac (simpset() addsimps [zabs_def]) 1); 
   3.208  qed "abs_int_eq";
   3.209 @@ -266,23 +319,28 @@
   3.210  Goalw [nat_def] "nat(int n) = n";
   3.211  by Auto_tac;
   3.212  qed "nat_int";
   3.213 +Addsimps [nat_int];
   3.214  
   3.215 -Goalw [nat_def] "nat(- int n) = 0";
   3.216 +Goalw [nat_def] "nat(- (int n)) = 0";
   3.217  by (auto_tac (claset(),
   3.218 -	      simpset() addsimps [neg_eq_less_int0, zminus_zless])); 
   3.219 +     simpset() addsimps [neg_eq_less_0, zero_reorient, zminus_zless])); 
   3.220  qed "nat_zminus_int";
   3.221 +Addsimps [nat_zminus_int];
   3.222  
   3.223 -Addsimps [nat_int, nat_zminus_int];
   3.224 +Goalw [Zero_int_def] "nat 0 = 0";
   3.225 +by (rtac nat_int 1);
   3.226 +qed "nat_zero";
   3.227 +Addsimps [nat_zero];
   3.228  
   3.229  Goal "~ neg z ==> int (nat z) = z"; 
   3.230 -by (dtac (not_neg_eq_ge_int0 RS iffD1) 1); 
   3.231 +by (dtac (not_neg_eq_ge_0 RS iffD1) 1); 
   3.232  by (dtac zle_imp_zless_or_eq 1); 
   3.233  by (auto_tac (claset(), simpset() addsimps [zless_iff_Suc_zadd])); 
   3.234  qed "not_neg_nat"; 
   3.235  
   3.236  Goal "neg x ==> EX n. x = - (int (Suc n))"; 
   3.237  by (auto_tac (claset(), 
   3.238 -	      simpset() addsimps [neg_eq_less_int0, zless_iff_Suc_zadd,
   3.239 +	      simpset() addsimps [neg_eq_less_0, zless_iff_Suc_zadd,
   3.240  				  zdiff_eq_eq RS sym, zdiff_def])); 
   3.241  qed "negD"; 
   3.242  
   3.243 @@ -295,28 +353,28 @@
   3.244  by (etac (not_neg_nat RS subst) 2);
   3.245  by (auto_tac (claset(), simpset() addsimps [neg_nat])); 
   3.246  by (auto_tac (claset() addDs [order_less_trans], 
   3.247 -	      simpset() addsimps [neg_eq_less_int0])); 
   3.248 +	      simpset() addsimps [neg_eq_less_0])); 
   3.249  qed "zless_nat_eq_int_zless";
   3.250  
   3.251 -Goal "z <= int 0 ==> nat z = 0"; 
   3.252 +Goal "z <= 0 ==> nat z = 0"; 
   3.253  by (auto_tac (claset(), 
   3.254 -	      simpset() addsimps [order_le_less, neg_eq_less_int0, 
   3.255 +	      simpset() addsimps [order_le_less, neg_eq_less_0, 
   3.256  				  zle_def, neg_nat])); 
   3.257  qed "nat_le_int0"; 
   3.258  
   3.259 -(*An alternative condition is  int 0 <= w  *)
   3.260 -Goal "int 0 < z ==> (nat w < nat z) = (w < z)";
   3.261 +(*An alternative condition is  0 <= w  *)
   3.262 +Goal "0 < z ==> (nat w < nat z) = (w < z)";
   3.263  by (stac (zless_int RS sym) 1);
   3.264 -by (asm_simp_tac (simpset() addsimps [not_neg_nat, not_neg_eq_ge_int0, 
   3.265 +by (asm_simp_tac (simpset() addsimps [not_neg_nat, not_neg_eq_ge_0, 
   3.266  				      order_le_less]) 1);
   3.267  by (case_tac "neg w" 1);
   3.268  by (asm_simp_tac (simpset() addsimps [not_neg_nat]) 2);
   3.269 -by (asm_full_simp_tac (simpset() addsimps [neg_eq_less_int0, neg_nat]) 1);
   3.270 +by (asm_full_simp_tac (simpset() addsimps [neg_eq_less_0, neg_nat]) 1);
   3.271  by (blast_tac (claset() addIs [order_less_trans]) 1);
   3.272  val lemma = result();
   3.273  
   3.274 -Goal "(nat w < nat z) = (int 0 < z & w < z)";
   3.275 -by (case_tac "int 0 < z" 1);
   3.276 +Goal "(nat w < nat z) = (0 < z & w < z)";
   3.277 +by (case_tac "0 < z" 1);
   3.278  by (auto_tac (claset(), 
   3.279  	      simpset() addsimps [lemma, nat_le_int0, linorder_not_less])); 
   3.280  qed "zless_nat_conj";
   3.281 @@ -339,35 +397,36 @@
   3.282  
   3.283  Goal "i <= (j::int) ==> i * int k <= j * int k";
   3.284  by (induct_tac "k" 1);
   3.285 -by (stac int_Suc_int_1 2);
   3.286 +by (stac int_Suc 2);
   3.287  by (ALLGOALS 
   3.288 -    (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono])));
   3.289 +    (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono, 
   3.290 +                                       int_Suc0_eq_1])));
   3.291  val lemma = result();
   3.292  
   3.293 -Goal "[| i <= j;  int 0 <= k |] ==> i*k <= j*k";
   3.294 +Goal "[| i <= j;  (0::int) <= k |] ==> i*k <= j*k";
   3.295  by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
   3.296  by (etac lemma 2);
   3.297 -by (full_simp_tac (simpset() addsimps [not_neg_eq_ge_int0]) 1);
   3.298 +by (full_simp_tac (simpset() addsimps [not_neg_eq_ge_0]) 1);
   3.299  qed "zmult_zle_mono1";
   3.300  
   3.301 -Goal "[| i <= j;  k <= int 0 |] ==> j*k <= i*k";
   3.302 +Goal "[| i <= j;  k <= (0::int) |] ==> j*k <= i*k";
   3.303  by (rtac (zminus_zle_zminus RS iffD1) 1);
   3.304  by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
   3.305  				      zmult_zle_mono1, zle_zminus]) 1);
   3.306  qed "zmult_zle_mono1_neg";
   3.307  
   3.308 -Goal "[| i <= j;  int 0 <= k |] ==> k*i <= k*j";
   3.309 +Goal "[| i <= j;  (0::int) <= k |] ==> k*i <= k*j";
   3.310  by (dtac zmult_zle_mono1 1);
   3.311  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
   3.312  qed "zmult_zle_mono2";
   3.313  
   3.314 -Goal "[| i <= j;  k <= int 0 |] ==> k*j <= k*i";
   3.315 +Goal "[| i <= j;  k <= (0::int) |] ==> k*j <= k*i";
   3.316  by (dtac zmult_zle_mono1_neg 1);
   3.317  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
   3.318  qed "zmult_zle_mono2_neg";
   3.319  
   3.320  (* <= monotonicity, BOTH arguments*)
   3.321 -Goal "[| i <= j;  k <= l;  int 0 <= j;  int 0 <= k |] ==> i*k <= j*l";
   3.322 +Goal "[| i <= j;  k <= l;  (0::int) <= j;  (0::int) <= k |] ==> i*k <= j*l";
   3.323  by (etac (zmult_zle_mono1 RS order_trans) 1);
   3.324  by (assume_tac 1);
   3.325  by (etac zmult_zle_mono2 1);
   3.326 @@ -379,64 +438,65 @@
   3.327  
   3.328  Goal "i<j ==> 0<k --> int k * i < int k * j";
   3.329  by (induct_tac "k" 1);
   3.330 -by (stac int_Suc_int_1 2);
   3.331 +by (stac int_Suc 2);
   3.332  by (case_tac "n=0" 2);
   3.333  by (ALLGOALS (asm_full_simp_tac
   3.334  	      (simpset() addsimps [zadd_zmult_distrib, zadd_zless_mono, 
   3.335 -				   order_le_less])));
   3.336 -val lemma = result() RS mp;
   3.337 +				   int_Suc0_eq_1, order_le_less])));
   3.338 +val lemma = result();
   3.339  
   3.340 -Goal "[| i<j;  int 0 < k |] ==> k*i < k*j";
   3.341 +Goal "[| i<j;  (0::int) < k |] ==> k*i < k*j";
   3.342  by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
   3.343 -by (etac lemma 2);
   3.344 -by (asm_simp_tac (simpset() addsimps [not_neg_eq_ge_int0, 
   3.345 +by (etac (lemma RS mp) 2);
   3.346 +by (asm_simp_tac (simpset() addsimps [not_neg_eq_ge_0, 
   3.347  				      order_le_less]) 1);
   3.348  by (forward_tac [conjI RS (zless_nat_conj RS iffD2)] 1);
   3.349  by Auto_tac;
   3.350  qed "zmult_zless_mono2";
   3.351  
   3.352 -Goal "[| i<j;  int 0 < k |] ==> i*k < j*k";
   3.353 +Goal "[| i<j;  (0::int) < k |] ==> i*k < j*k";
   3.354  by (dtac zmult_zless_mono2 1);
   3.355  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
   3.356  qed "zmult_zless_mono1";
   3.357  
   3.358  (* < monotonicity, BOTH arguments*)
   3.359 -Goal "[| i < j;  k < l;  int 0 < j;  int 0 < k |] ==> i*k < j*l";
   3.360 +Goal "[| i < j;  k < l;  (0::int) < j;  (0::int) < k |] ==> i*k < j*l";
   3.361  by (etac (zmult_zless_mono1 RS order_less_trans) 1);
   3.362  by (assume_tac 1);
   3.363  by (etac zmult_zless_mono2 1);
   3.364  by (assume_tac 1);
   3.365  qed "zmult_zless_mono";
   3.366  
   3.367 -Goal "[| i<j;  k < int 0 |] ==> j*k < i*k";
   3.368 +Goal "[| i<j;  k < (0::int) |] ==> j*k < i*k";
   3.369  by (rtac (zminus_zless_zminus RS iffD1) 1);
   3.370  by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
   3.371  				      zmult_zless_mono1, zless_zminus]) 1);
   3.372  qed "zmult_zless_mono1_neg";
   3.373  
   3.374 -Goal "[| i<j;  k < int 0 |] ==> k*j < k*i";
   3.375 +Goal "[| i<j;  k < (0::int) |] ==> k*j < k*i";
   3.376  by (rtac (zminus_zless_zminus RS iffD1) 1);
   3.377  by (asm_simp_tac (simpset() addsimps [zmult_zminus RS sym,
   3.378  				      zmult_zless_mono2, zless_zminus]) 1);
   3.379  qed "zmult_zless_mono2_neg";
   3.380  
   3.381  
   3.382 -Goal "(m*n = int 0) = (m = int 0 | n = int 0)";
   3.383 -by (case_tac "m < int 0" 1);
   3.384 +Goal "(m*n = (0::int)) = (m = 0 | n = 0)";
   3.385 +by (case_tac "m < (0::int)" 1);
   3.386  by (auto_tac (claset(), 
   3.387  	      simpset() addsimps [linorder_not_less, order_le_less, 
   3.388  				  linorder_neq_iff])); 
   3.389  by (REPEAT 
   3.390      (force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1], 
   3.391  		simpset()) 1));
   3.392 -qed "zmult_eq_int0_iff";
   3.393 +qed "zmult_eq_0_iff";
   3.394 +AddIffs [zmult_eq_0_iff];
   3.395  
   3.396  
   3.397  (** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
   3.398      but not (yet?) for k*m < n*k. **)
   3.399  
   3.400 -Goal "(m*k < n*k) = ((int 0 < k & m<n) | (k < int 0 & n<m))";
   3.401 -by (case_tac "k = int 0" 1);
   3.402 +Goal "(m*k < n*k) = (((0::int) < k & m<n) | (k < 0 & n<m))";
   3.403 +by (case_tac "k = (0::int)" 1);
   3.404  by (auto_tac (claset(), simpset() addsimps [linorder_neq_iff, 
   3.405                                zmult_zless_mono1, zmult_zless_mono1_neg]));  
   3.406  by (auto_tac (claset(), 
   3.407 @@ -449,22 +509,22 @@
   3.408  qed "zmult_zless_cancel2";
   3.409  
   3.410  
   3.411 -Goal "(k*m < k*n) = ((int 0 < k & m<n) | (k < int 0 & n<m))";
   3.412 +Goal "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))";
   3.413  by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute, 
   3.414                                    zmult_zless_cancel2]) 1);
   3.415  qed "zmult_zless_cancel1";
   3.416  
   3.417 -Goal "(m*k <= n*k) = ((int 0 < k --> m<=n) & (k < int 0 --> n<=m))";
   3.418 +Goal "(m*k <= n*k) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
   3.419  by (simp_tac (simpset() addsimps [linorder_not_less RS sym, 
   3.420                                    zmult_zless_cancel2]) 1);
   3.421  qed "zmult_zle_cancel2";
   3.422  
   3.423 -Goal "(k*m <= k*n) = ((int 0 < k --> m<=n) & (k < int 0 --> n<=m))";
   3.424 +Goal "(k*m <= k*n) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
   3.425  by (simp_tac (simpset() addsimps [linorder_not_less RS sym, 
   3.426                                    zmult_zless_cancel1]) 1);
   3.427  qed "zmult_zle_cancel1";
   3.428  
   3.429 -Goal "(m*k = n*k) = (k = int 0 | m=n)";
   3.430 +Goal "(m*k = n*k) = (k = (0::int) | m=n)";
   3.431  by (cut_facts_tac [linorder_less_linear] 1);
   3.432  by Safe_tac;
   3.433  by Auto_tac;  
   3.434 @@ -475,8 +535,15 @@
   3.435  
   3.436  qed "zmult_cancel2";
   3.437  
   3.438 -Goal "(k*m = k*n) = (k = int 0 | m=n)";
   3.439 +Goal "(k*m = k*n) = (k = (0::int) | m=n)";
   3.440  by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute, 
   3.441                                    zmult_cancel2]) 1);
   3.442  qed "zmult_cancel1";
   3.443  Addsimps [zmult_cancel1, zmult_cancel2];
   3.444 +
   3.445 +
   3.446 +(*Analogous to zadd_int*)
   3.447 +Goal "n<=m --> int m - int n = int (m-n)";
   3.448 +by (induct_thm_tac diff_induct "m n" 1);
   3.449 +by (auto_tac (claset(), simpset() addsimps [int_Suc, symmetric zdiff_def])); 
   3.450 +qed_spec_mp "zdiff_int";
     4.1 --- a/src/HOL/Integ/Int.thy	Mon Oct 22 11:01:30 2001 +0200
     4.2 +++ b/src/HOL/Integ/Int.thy	Mon Oct 22 11:54:22 2001 +0200
     4.3 @@ -9,7 +9,7 @@
     4.4  Int = IntDef + 
     4.5  
     4.6  instance int :: order (zle_refl,zle_trans,zle_anti_sym,int_less_le)
     4.7 -instance int :: plus_ac0 (zadd_commute,zadd_assoc,zadd_zero)
     4.8 +instance int :: plus_ac0 (zadd_commute,zadd_assoc,zadd_0)
     4.9  instance int :: linorder (zle_linear)
    4.10  
    4.11  constdefs
     5.1 --- a/src/HOL/Integ/IntArith.ML	Mon Oct 22 11:01:30 2001 +0200
     5.2 +++ b/src/HOL/Integ/IntArith.ML	Mon Oct 22 11:54:22 2001 +0200
     5.3 @@ -3,6 +3,7 @@
     5.4      Authors:    Larry Paulson and Tobias Nipkow
     5.5  *)
     5.6  
     5.7 +
     5.8  Goal "abs(abs(x::int)) = abs(x)";
     5.9  by(arith_tac 1);
    5.10  qed "abs_abs";
    5.11 @@ -20,7 +21,7 @@
    5.12  
    5.13  (*** Intermediate value theorems ***)
    5.14  
    5.15 -Goal "(ALL i<n::nat. abs(f(i+1) - f i) <= Numeral1) --> \
    5.16 +Goal "(ALL i<n::nat. abs(f(i+1) - f i) <= 1) --> \
    5.17  \     f 0 <= k --> k <= f n --> (EX i <= n. f i = (k::int))";
    5.18  by(induct_tac "n" 1);
    5.19   by(Asm_simp_tac 1);
    5.20 @@ -40,7 +41,7 @@
    5.21  
    5.22  bind_thm("nat0_intermed_int_val", ObjectLogic.rulify_no_asm lemma);
    5.23  
    5.24 -Goal "[| !i. m <= i & i < n --> abs(f(i + 1::nat) - f i) <= Numeral1; m < n; \
    5.25 +Goal "[| !i. m <= i & i < n --> abs(f(i + 1::nat) - f i) <= 1; m < n; \
    5.26  \        f m <= k; k <= f n |] ==> ? i. m <= i & i <= n & f i = (k::int)";
    5.27  by(cut_inst_tac [("n","n-m"),("f", "%i. f(i+m)"),("k","k")]lemma 1);
    5.28  by(Asm_full_simp_tac 1);
    5.29 @@ -56,22 +57,22 @@
    5.30  
    5.31  (*** Some convenient biconditionals for products of signs ***)
    5.32  
    5.33 -Goal "[| (Numeral0::int) < i; Numeral0 < j |] ==> Numeral0 < i*j";
    5.34 +Goal "[| (0::int) < i; 0 < j |] ==> 0 < i*j";
    5.35  by (dtac zmult_zless_mono1 1);
    5.36  by Auto_tac; 
    5.37  qed "zmult_pos";
    5.38  
    5.39 -Goal "[| i < (Numeral0::int); j < Numeral0 |] ==> Numeral0 < i*j";
    5.40 +Goal "[| i < (0::int); j < 0 |] ==> 0 < i*j";
    5.41  by (dtac zmult_zless_mono1_neg 1);
    5.42  by Auto_tac; 
    5.43  qed "zmult_neg";
    5.44  
    5.45 -Goal "[| (Numeral0::int) < i; j < Numeral0 |] ==> i*j < Numeral0";
    5.46 +Goal "[| (0::int) < i; j < 0 |] ==> i*j < 0";
    5.47  by (dtac zmult_zless_mono1_neg 1);
    5.48  by Auto_tac; 
    5.49  qed "zmult_pos_neg";
    5.50  
    5.51 -Goal "((Numeral0::int) < x*y) = (Numeral0 < x & Numeral0 < y | x < Numeral0 & y < Numeral0)";
    5.52 +Goal "((0::int) < x*y) = (0 < x & 0 < y | x < 0 & y < 0)";
    5.53  by (auto_tac (claset(), 
    5.54                simpset() addsimps [order_le_less, linorder_not_less,
    5.55  	                          zmult_pos, zmult_neg]));
    5.56 @@ -84,13 +85,13 @@
    5.57                simpset() addsimps [zmult_commute]));  
    5.58  qed "int_0_less_mult_iff";
    5.59  
    5.60 -Goal "((Numeral0::int) <= x*y) = (Numeral0 <= x & Numeral0 <= y | x <= Numeral0 & y <= Numeral0)";
    5.61 +Goal "((0::int) <= x*y) = (0 <= x & 0 <= y | x <= 0 & y <= 0)";
    5.62  by (auto_tac (claset(), 
    5.63                simpset() addsimps [order_le_less, linorder_not_less,  
    5.64                                    int_0_less_mult_iff]));
    5.65  qed "int_0_le_mult_iff";
    5.66  
    5.67 -Goal "(x*y < (Numeral0::int)) = (Numeral0 < x & y < Numeral0 | x < Numeral0 & Numeral0 < y)";
    5.68 +Goal "(x*y < (0::int)) = (0 < x & y < 0 | x < 0 & 0 < y)";
    5.69  by (auto_tac (claset(), 
    5.70                simpset() addsimps [int_0_le_mult_iff, 
    5.71                                    linorder_not_le RS sym]));
    5.72 @@ -98,30 +99,31 @@
    5.73                simpset() addsimps [linorder_not_le]));
    5.74  qed "zmult_less_0_iff";
    5.75  
    5.76 -Goal "(x*y <= (Numeral0::int)) = (Numeral0 <= x & y <= Numeral0 | x <= Numeral0 & Numeral0 <= y)";
    5.77 +Goal "(x*y <= (0::int)) = (0 <= x & y <= 0 | x <= 0 & 0 <= y)";
    5.78  by (auto_tac (claset() addDs [order_less_not_sym], 
    5.79                simpset() addsimps [int_0_less_mult_iff, 
    5.80                                    linorder_not_less RS sym]));
    5.81  qed "zmult_le_0_iff";
    5.82  
    5.83  Goal "abs (x * y) = abs x * abs (y::int)";
    5.84 -by (simp_tac (simpset () addsplits [zabs_split] 
    5.85 +by (simp_tac (simpset () delsimps [thm "number_of_reorient"] addsplits [zabs_split] 
    5.86 +                         addsplits [zabs_split] 
    5.87                           addsimps [zmult_less_0_iff, zle_def]) 1);
    5.88  qed "abs_mult";
    5.89  
    5.90 -Goal "(abs x = Numeral0) = (x = (Numeral0::int))";
    5.91 +Goal "(abs x = 0) = (x = (0::int))";
    5.92  by (simp_tac (simpset () addsplits [zabs_split]) 1);
    5.93  qed "abs_eq_0";
    5.94  AddIffs [abs_eq_0];
    5.95  
    5.96 -Goal "(Numeral0 < abs x) = (x ~= (Numeral0::int))";
    5.97 +Goal "(0 < abs x) = (x ~= (0::int))";
    5.98  by (simp_tac (simpset () addsplits [zabs_split]) 1);
    5.99  by (arith_tac 1);
   5.100  qed "zero_less_abs_iff";
   5.101  AddIffs [zero_less_abs_iff];
   5.102  
   5.103 -Goal "Numeral0 <= x * (x::int)";
   5.104 -by (subgoal_tac "(- x) * x <= Numeral0" 1);
   5.105 +Goal "0 <= x * (x::int)";
   5.106 +by (subgoal_tac "(- x) * x <= 0" 1);
   5.107   by (Asm_full_simp_tac 1);
   5.108  by (simp_tac (HOL_basic_ss addsimps [zmult_le_0_iff]) 1);
   5.109  by Auto_tac;
   5.110 @@ -132,48 +134,48 @@
   5.111  
   5.112  (*** Products and 1, by T. M. Rasmussen ***)
   5.113  
   5.114 -Goal "(m = m*(n::int)) = (n = Numeral1 | m = Numeral0)";
   5.115 +Goal "(m = m*(n::int)) = (n = 1 | m = 0)";
   5.116  by Auto_tac;
   5.117 -by (subgoal_tac "m*Numeral1 = m*n" 1);
   5.118 +by (subgoal_tac "m*1 = m*n" 1);
   5.119  by (dtac (zmult_cancel1 RS iffD1) 1); 
   5.120  by Auto_tac;
   5.121  qed "zmult_eq_self_iff";
   5.122  
   5.123 -Goal "[| Numeral1 < m; Numeral1 < n |] ==> Numeral1 < m*(n::int)";
   5.124 -by (res_inst_tac [("y","Numeral1*n")] order_less_trans 1);
   5.125 +Goal "[| 1 < m; 1 < n |] ==> 1 < m*(n::int)";
   5.126 +by (res_inst_tac [("y","1*n")] order_less_trans 1);
   5.127  by (rtac zmult_zless_mono1 2);
   5.128  by (ALLGOALS Asm_simp_tac);
   5.129  qed "zless_1_zmult";
   5.130  
   5.131 -Goal "[| Numeral0 < n; n ~= Numeral1 |] ==> Numeral1 < (n::int)";
   5.132 +Goal "[| 0 < n; n ~= 1 |] ==> 1 < (n::int)";
   5.133  by (arith_tac 1);
   5.134  val lemma = result();
   5.135  
   5.136 -Goal "Numeral0 < (m::int) ==> (m * n = Numeral1) = (m = Numeral1 & n = Numeral1)";
   5.137 +Goal "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)";
   5.138  by Auto_tac;
   5.139 -by (case_tac "m=Numeral1" 1);
   5.140 -by (case_tac "n=Numeral1" 2);
   5.141 -by (case_tac "m=Numeral1" 4);
   5.142 -by (case_tac "n=Numeral1" 5);
   5.143 +by (case_tac "m=1" 1);
   5.144 +by (case_tac "n=1" 2);
   5.145 +by (case_tac "m=1" 4);
   5.146 +by (case_tac "n=1" 5);
   5.147  by Auto_tac;
   5.148  by distinct_subgoals_tac;
   5.149 -by (subgoal_tac "Numeral1<m*n" 1);
   5.150 +by (subgoal_tac "1<m*n" 1);
   5.151  by (Asm_full_simp_tac 1);
   5.152  by (rtac zless_1_zmult 1);
   5.153  by (ALLGOALS (rtac lemma));
   5.154  by Auto_tac;  
   5.155 -by (subgoal_tac "Numeral0<m*n" 1);
   5.156 +by (subgoal_tac "0<m*n" 1);
   5.157  by (Asm_simp_tac 2);
   5.158  by (dtac (int_0_less_mult_iff RS iffD1) 1);
   5.159  by Auto_tac;  
   5.160  qed "pos_zmult_eq_1_iff";
   5.161  
   5.162 -Goal "(m*n = (Numeral1::int)) = ((m = Numeral1 & n = Numeral1) | (m = -1 & n = -1))";
   5.163 -by (case_tac "Numeral0<m" 1);
   5.164 +Goal "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))";
   5.165 +by (case_tac "0<m" 1);
   5.166  by (asm_simp_tac (simpset() addsimps [pos_zmult_eq_1_iff]) 1);
   5.167 -by (case_tac "m=Numeral0" 1);
   5.168 -by (Asm_simp_tac 1);
   5.169 -by (subgoal_tac "Numeral0 < -m" 1);
   5.170 +by (case_tac "m=0" 1);
   5.171 +by (asm_simp_tac (simpset () delsimps [thm "number_of_reorient"]) 1);
   5.172 +by (subgoal_tac "0 < -m" 1);
   5.173  by (arith_tac 2);
   5.174  by (dres_inst_tac [("n","-n")] pos_zmult_eq_1_iff 1); 
   5.175  by Auto_tac;  
     6.1 --- a/src/HOL/Integ/IntArith.thy	Mon Oct 22 11:01:30 2001 +0200
     6.2 +++ b/src/HOL/Integ/IntArith.thy	Mon Oct 22 11:54:22 2001 +0200
     6.3 @@ -3,5 +3,4 @@
     6.4  
     6.5  use "int_arith1.ML"	setup int_arith_setup
     6.6  use "int_arith2.ML"	lemmas [arith_split] = zabs_split
     6.7 -
     6.8  end
     7.1 --- a/src/HOL/Integ/IntDef.ML	Mon Oct 22 11:01:30 2001 +0200
     7.2 +++ b/src/HOL/Integ/IntDef.ML	Mon Oct 22 11:54:22 2001 +0200
     7.3 @@ -7,9 +7,6 @@
     7.4  *)
     7.5  
     7.6  
     7.7 -(*Rewrite the overloaded 0::int and 1::int*)    (* FIXME *)
     7.8 -Addsimps [Zero_int_def, One_int_def];
     7.9 -
    7.10  Goalw  [intrel_def] "(((x1,y1),(x2,y2)): intrel) = (x1+y2 = x2+y1)";
    7.11  by (Blast_tac 1);
    7.12  qed "intrel_iff";
    7.13 @@ -90,11 +87,10 @@
    7.14  by (Asm_full_simp_tac 1);
    7.15  qed "inj_zminus";
    7.16  
    7.17 -Goalw [int_def] "- (int 0) = int 0";
    7.18 +Goalw [int_def, Zero_int_def] "- 0 = (0::int)";
    7.19  by (simp_tac (simpset() addsimps [zminus]) 1);
    7.20 -qed "zminus_int0";
    7.21 -
    7.22 -Addsimps [zminus_int0];
    7.23 +qed "zminus_0";
    7.24 +Addsimps [zminus_0];
    7.25  
    7.26  
    7.27  (**** neg: the test for negative integers ****)
    7.28 @@ -159,61 +155,57 @@
    7.29  by (simp_tac (simpset() addsimps [zadd_int, zadd_assoc RS sym]) 1);
    7.30  qed "zadd_int_left";
    7.31  
    7.32 -Goal "int (Suc m) = int 1 + (int m)";
    7.33 -by (simp_tac (simpset() addsimps [zadd_int]) 1);
    7.34 -qed "int_Suc_int_1";
    7.35 +Goal "int (Suc m) = 1 + (int m)";
    7.36 +by (simp_tac (simpset() addsimps [One_int_def, zadd_int]) 1);
    7.37 +qed "int_Suc";
    7.38  
    7.39 -Goalw [int_def] "int 0 + z = z";
    7.40 +(*also for the instance declaration int :: plus_ac0*)
    7.41 +Goalw [Zero_int_def, int_def] "(0::int) + z = z";
    7.42  by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
    7.43  by (asm_simp_tac (simpset() addsimps [zadd]) 1);
    7.44 -qed "zadd_int0";
    7.45 +qed "zadd_0";
    7.46 +Addsimps [zadd_0];
    7.47  
    7.48 -Goal "z + int 0 = z";
    7.49 -by (rtac (zadd_commute RS trans) 1);
    7.50 -by (rtac zadd_int0 1);
    7.51 -qed "zadd_int0_right";
    7.52 +Goal "z + (0::int) = z";
    7.53 +by (rtac ([zadd_commute, zadd_0] MRS trans) 1);
    7.54 +qed "zadd_0_right";
    7.55 +Addsimps [zadd_0_right];
    7.56  
    7.57 -Goalw [int_def] "z + (- z) = int 0";
    7.58 +Goalw [int_def, Zero_int_def] "z + (- z) = (0::int)";
    7.59  by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
    7.60  by (asm_simp_tac (simpset() addsimps [zminus, zadd, add_commute]) 1);
    7.61  qed "zadd_zminus_inverse";
    7.62  
    7.63 -Goal "(- z) + z = int 0";
    7.64 +Goal "(- z) + z = (0::int)";
    7.65  by (rtac (zadd_commute RS trans) 1);
    7.66  by (rtac zadd_zminus_inverse 1);
    7.67  qed "zadd_zminus_inverse2";
    7.68  
    7.69 -Addsimps [zadd_int0, zadd_int0_right,
    7.70 -	  zadd_zminus_inverse, zadd_zminus_inverse2];
    7.71 -
    7.72 -(*for the instance declaration int :: plus_ac0*)
    7.73 -Goal "0 + z = (z::int)";
    7.74 -by (Simp_tac 1);
    7.75 -qed "zadd_zero";
    7.76 +Addsimps [zadd_zminus_inverse, zadd_zminus_inverse2];
    7.77  
    7.78  Goal "z + (- z + w) = (w::int)";
    7.79 -by (simp_tac (simpset() addsimps [zadd_assoc RS sym]) 1);
    7.80 +by (simp_tac (simpset() addsimps [zadd_assoc RS sym, zadd_0]) 1);
    7.81  qed "zadd_zminus_cancel";
    7.82  
    7.83  Goal "(-z) + (z + w) = (w::int)";
    7.84 -by (simp_tac (simpset() addsimps [zadd_assoc RS sym]) 1);
    7.85 +by (simp_tac (simpset() addsimps [zadd_assoc RS sym, zadd_0]) 1);
    7.86  qed "zminus_zadd_cancel";
    7.87  
    7.88  Addsimps [zadd_zminus_cancel, zminus_zadd_cancel];
    7.89  
    7.90 -Goal "int 0 - x = -x";
    7.91 +Goal "(0::int) - x = -x";
    7.92  by (simp_tac (simpset() addsimps [zdiff_def]) 1);
    7.93 -qed "zdiff_int0";
    7.94 +qed "zdiff0";
    7.95  
    7.96 -Goal "x - int 0 = x";
    7.97 +Goal "x - (0::int) = x";
    7.98  by (simp_tac (simpset() addsimps [zdiff_def]) 1);
    7.99 -qed "zdiff_int0_right";
   7.100 +qed "zdiff0_right";
   7.101  
   7.102 -Goal "x - x = int 0";
   7.103 -by (simp_tac (simpset() addsimps [zdiff_def]) 1);
   7.104 +Goal "x - x = (0::int)";
   7.105 +by (simp_tac (simpset() addsimps [zdiff_def, Zero_int_def]) 1);
   7.106  qed "zdiff_self";
   7.107  
   7.108 -Addsimps [zdiff_int0, zdiff_int0_right, zdiff_self];
   7.109 +Addsimps [zdiff0, zdiff0_right, zdiff_self];
   7.110  
   7.111  
   7.112  (** Lemmas **)
   7.113 @@ -321,25 +313,27 @@
   7.114  by (simp_tac (simpset() addsimps [zmult]) 1);
   7.115  qed "zmult_int";
   7.116  
   7.117 -Goalw [int_def] "int 0 * z = int 0";
   7.118 +Goalw [Zero_int_def, int_def] "0 * z = (0::int)";
   7.119  by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
   7.120  by (asm_simp_tac (simpset() addsimps [zmult]) 1);
   7.121 -qed "zmult_int0";
   7.122 +qed "zmult_0";
   7.123 +Addsimps [zmult_0];
   7.124  
   7.125 -Goalw [int_def] "int (Suc 0) * z = z";
   7.126 +Goalw [One_int_def, int_def] "(1::int) * z = z";
   7.127  by (res_inst_tac [("z","z")] eq_Abs_Integ 1);
   7.128  by (asm_simp_tac (simpset() addsimps [zmult]) 1);
   7.129 -qed "zmult_int1";
   7.130 -
   7.131 -Goal "z * int 0 = int 0";
   7.132 -by (rtac ([zmult_commute, zmult_int0] MRS trans) 1);
   7.133 -qed "zmult_int0_right";
   7.134 +qed "zmult_1";
   7.135 +Addsimps [zmult_1];
   7.136  
   7.137 -Goal "z * int (Suc 0) = z";
   7.138 -by (rtac ([zmult_commute, zmult_int1] MRS trans) 1);
   7.139 -qed "zmult_int1_right";
   7.140 +Goal "z * 0 = (0::int)";
   7.141 +by (rtac ([zmult_commute, zmult_0] MRS trans) 1);
   7.142 +qed "zmult_0_right";
   7.143 +Addsimps [zmult_0_right];
   7.144  
   7.145 -Addsimps [zmult_int0, zmult_int0_right, zmult_int1, zmult_int1_right];
   7.146 +Goal "z * (1::int) = z";
   7.147 +by (rtac ([zmult_commute, zmult_1] MRS trans) 1);
   7.148 +qed "zmult_1_right";
   7.149 +Addsimps [zmult_1_right];
   7.150  
   7.151  
   7.152  (* Theorems about less and less_equal *)
   7.153 @@ -413,12 +407,27 @@
   7.154  qed "int_int_eq"; 
   7.155  AddIffs [int_int_eq]; 
   7.156  
   7.157 +Goal "(int n = 0) = (n = 0)";
   7.158 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
   7.159 +qed "int_eq_0_conv";
   7.160 +Addsimps [int_eq_0_conv];
   7.161 +
   7.162  Goal "(int m < int n) = (m<n)";
   7.163  by (simp_tac (simpset() addsimps [less_iff_Suc_add, zless_iff_Suc_zadd, 
   7.164  				  zadd_int]) 1);
   7.165  qed "zless_int";
   7.166  Addsimps [zless_int];
   7.167  
   7.168 +Goal "~ (int k < 0)";
   7.169 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
   7.170 +qed "int_less_0_conv";
   7.171 +Addsimps [int_less_0_conv];
   7.172 +
   7.173 +Goal "(0 < int n) = (0 < n)";
   7.174 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
   7.175 +qed "zero_less_int_conv";
   7.176 +Addsimps [zero_less_int_conv];
   7.177 +
   7.178  
   7.179  (*** Properties of <= ***)
   7.180  
   7.181 @@ -427,6 +436,16 @@
   7.182  qed "zle_int";
   7.183  Addsimps [zle_int];
   7.184  
   7.185 +Goal "(0 <= int n)";
   7.186 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
   7.187 +qed "zero_zle_int";
   7.188 +Addsimps [zero_zle_int];
   7.189 +
   7.190 +Goal "(int n <= 0) = (n = 0)";
   7.191 +by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
   7.192 +qed "int_le_0_conv";
   7.193 +Addsimps [int_le_0_conv];
   7.194 +
   7.195  Goalw [zle_def] "z <= w ==> z < w | z=(w::int)";
   7.196  by (cut_facts_tac [zless_linear] 1);
   7.197  by (blast_tac (claset() addEs [zless_asym]) 1);
   7.198 @@ -506,11 +525,13 @@
   7.199  qed "zle_zdiff_eq";
   7.200  
   7.201  Goalw [zdiff_def] "(x-y = z) = (x = z + (y::int))";
   7.202 -by (auto_tac (claset(), simpset() addsimps [zadd_assoc]));
   7.203 +by (auto_tac (claset(), 
   7.204 +              simpset() addsimps [zadd_assoc, symmetric Zero_int_def]));
   7.205  qed "zdiff_eq_eq";
   7.206  
   7.207  Goalw [zdiff_def] "(x = z-y) = (x + (y::int) = z)";
   7.208 -by (auto_tac (claset(), simpset() addsimps [zadd_assoc]));
   7.209 +by (auto_tac (claset(), 
   7.210 +              simpset() addsimps [zadd_assoc, symmetric Zero_int_def]));
   7.211  qed "eq_zdiff_eq";
   7.212  
   7.213  (*This list of rewrites simplifies (in)equalities by bringing subtractions
   7.214 @@ -528,7 +549,8 @@
   7.215  Goal "!!w::int. (z + w' = z + w) = (w' = w)";
   7.216  by Safe_tac;
   7.217  by (dres_inst_tac [("f", "%x. x + (-z)")] arg_cong 1);
   7.218 -by (asm_full_simp_tac (simpset() addsimps zadd_ac) 1);
   7.219 +by (asm_full_simp_tac (simpset() addsimps symmetric Zero_int_def :: 
   7.220 +                                          zadd_ac) 1);
   7.221  qed "zadd_left_cancel";
   7.222  
   7.223  Addsimps [zadd_left_cancel];
     8.1 --- a/src/HOL/Integ/IntDef.thy	Mon Oct 22 11:01:30 2001 +0200
     8.2 +++ b/src/HOL/Integ/IntDef.thy	Mon Oct 22 11:54:22 2001 +0200
     8.3 @@ -31,7 +31,7 @@
     8.4  
     8.5    (*For simplifying equalities*)
     8.6    iszero :: int => bool
     8.7 -  "iszero z == z = int 0"
     8.8 +  "iszero z == z = (0::int)"
     8.9    
    8.10  defs (*of overloaded constants*)
    8.11    
     9.1 --- a/src/HOL/Integ/IntDiv.ML	Mon Oct 22 11:01:30 2001 +0200
     9.2 +++ b/src/HOL/Integ/IntDiv.ML	Mon Oct 22 11:54:22 2001 +0200
     9.3 @@ -34,21 +34,21 @@
     9.4  
     9.5  (*** Uniqueness and monotonicity of quotients and remainders ***)
     9.6  
     9.7 -Goal "[| b*q' + r'  <= b*q + r;  Numeral0 <= r';  Numeral0 < b;  r < b |] \
     9.8 +Goal "[| b*q' + r'  <= b*q + r;  0 <= r';  0 < b;  r < b |] \
     9.9  \     ==> q' <= (q::int)";
    9.10  by (subgoal_tac "r' + b * (q'-q) <= r" 1);
    9.11  by (simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 2);
    9.12 -by (subgoal_tac "Numeral0 < b * (Numeral1 + q - q')" 1);
    9.13 +by (subgoal_tac "0 < b * (1 + q - q')" 1);
    9.14  by (etac order_le_less_trans 2);
    9.15  by (full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2,
    9.16  				       zadd_zmult_distrib2]) 2);
    9.17 -by (subgoal_tac "b * q' < b * (Numeral1 + q)" 1);
    9.18 +by (subgoal_tac "b * q' < b * (1 + q)" 1);
    9.19  by (full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2,
    9.20  				       zadd_zmult_distrib2]) 2);
    9.21  by (asm_full_simp_tac (simpset() addsimps [zmult_zless_cancel1]) 1); 
    9.22  qed "unique_quotient_lemma";
    9.23  
    9.24 -Goal "[| b*q' + r' <= b*q + r;  r <= Numeral0;  b < Numeral0;  b < r' |] \
    9.25 +Goal "[| b*q' + r' <= b*q + r;  r <= 0;  b < 0;  b < r' |] \
    9.26  \     ==> q <= (q'::int)";
    9.27  by (res_inst_tac [("b", "-b"), ("r", "-r'"), ("r'", "-r")] 
    9.28      unique_quotient_lemma 1);
    9.29 @@ -57,7 +57,7 @@
    9.30  qed "unique_quotient_lemma_neg";
    9.31  
    9.32  
    9.33 -Goal "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= Numeral0 |] \
    9.34 +Goal "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= 0 |] \
    9.35  \     ==> q = q'";
    9.36  by (asm_full_simp_tac 
    9.37      (simpset() addsimps split_ifs@
    9.38 @@ -72,7 +72,7 @@
    9.39  qed "unique_quotient";
    9.40  
    9.41  
    9.42 -Goal "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= Numeral0 |] \
    9.43 +Goal "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= 0 |] \
    9.44  \     ==> r = r'";
    9.45  by (subgoal_tac "q = q'" 1);
    9.46  by (blast_tac (claset() addIs [unique_quotient]) 2);
    9.47 @@ -83,9 +83,9 @@
    9.48  (*** Correctness of posDivAlg, the division algorithm for a>=0 and b>0 ***)
    9.49  
    9.50  
    9.51 -Goal "adjust a b (q,r) = (let diff = r-b in \
    9.52 -\                         if Numeral0 <= diff then (2*q + Numeral1, diff)  \
    9.53 -\                                       else (2*q, r))";
    9.54 +Goal "adjust b (q,r) = (let diff = r-b in \
    9.55 +\                         if 0 <= diff then (2*q + 1, diff)  \
    9.56 +\                                      else (2*q, r))";
    9.57  by (simp_tac (simpset() addsimps [Let_def,adjust_def]) 1);
    9.58  qed "adjust_eq";
    9.59  Addsimps [adjust_eq];
    9.60 @@ -101,9 +101,9 @@
    9.61  bind_thm ("posDivAlg_raw_eqn", lemma RS hd posDivAlg.simps);
    9.62  
    9.63  (**use with simproc to avoid re-proving the premise*)
    9.64 -Goal "Numeral0 < b ==> \
    9.65 +Goal "0 < b ==> \
    9.66  \     posDivAlg (a,b) =      \
    9.67 -\      (if a<b then (Numeral0,a) else adjust a b (posDivAlg(a, 2*b)))";
    9.68 +\      (if a<b then (0,a) else adjust b (posDivAlg(a, 2*b)))";
    9.69  by (rtac (posDivAlg_raw_eqn RS trans) 1);
    9.70  by (Asm_simp_tac 1);
    9.71  qed "posDivAlg_eqn";
    9.72 @@ -112,7 +112,7 @@
    9.73  
    9.74  
    9.75  (*Correctness of posDivAlg: it computes quotients correctly*)
    9.76 -Goal "Numeral0 <= a --> Numeral0 < b --> quorem ((a, b), posDivAlg (a, b))";
    9.77 +Goal "0 <= a --> 0 < b --> quorem ((a, b), posDivAlg (a, b))";
    9.78  by (induct_thm_tac posDivAlg_induct "a b" 1);
    9.79  by Auto_tac;
    9.80  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
    9.81 @@ -139,9 +139,9 @@
    9.82  bind_thm ("negDivAlg_raw_eqn", lemma RS hd negDivAlg.simps);
    9.83  
    9.84  (**use with simproc to avoid re-proving the premise*)
    9.85 -Goal "Numeral0 < b ==> \
    9.86 +Goal "0 < b ==> \
    9.87  \     negDivAlg (a,b) =      \
    9.88 -\      (if Numeral0<=a+b then (-1,a+b) else adjust a b (negDivAlg(a, 2*b)))";
    9.89 +\      (if 0<=a+b then (-1,a+b) else adjust b (negDivAlg(a, 2*b)))";
    9.90  by (rtac (negDivAlg_raw_eqn RS trans) 1);
    9.91  by (Asm_simp_tac 1);
    9.92  qed "negDivAlg_eqn";
    9.93 @@ -151,7 +151,7 @@
    9.94  
    9.95  (*Correctness of negDivAlg: it computes quotients correctly
    9.96    It doesn't work if a=0 because the 0/b=0 rather than -1*)
    9.97 -Goal "a < Numeral0 --> Numeral0 < b --> quorem ((a, b), negDivAlg (a, b))";
    9.98 +Goal "a < 0 --> 0 < b --> quorem ((a, b), negDivAlg (a, b))";
    9.99  by (induct_thm_tac negDivAlg_induct "a b" 1);
   9.100  by Auto_tac;
   9.101  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
   9.102 @@ -168,18 +168,18 @@
   9.103  (*** Existence shown by proving the division algorithm to be correct ***)
   9.104  
   9.105  (*the case a=0*)
   9.106 -Goal "b ~= Numeral0 ==> quorem ((Numeral0,b), (Numeral0,Numeral0))";
   9.107 +Goal "b ~= 0 ==> quorem ((0,b), (0,0))";
   9.108  by (auto_tac (claset(), 
   9.109  	      simpset() addsimps [quorem_def, linorder_neq_iff]));
   9.110  qed "quorem_0";
   9.111  
   9.112 -Goal "posDivAlg (Numeral0, b) = (Numeral0, Numeral0)";
   9.113 +Goal "posDivAlg (0, b) = (0, 0)";
   9.114  by (stac posDivAlg_raw_eqn 1);
   9.115  by Auto_tac;
   9.116  qed "posDivAlg_0";
   9.117  Addsimps [posDivAlg_0];
   9.118  
   9.119 -Goal "negDivAlg (-1, b) = (-1, b-Numeral1)";
   9.120 +Goal "negDivAlg (-1, b) = (-1, b - 1)";
   9.121  by (stac negDivAlg_raw_eqn 1);
   9.122  by Auto_tac;
   9.123  qed "negDivAlg_minus1";
   9.124 @@ -194,7 +194,7 @@
   9.125  by (auto_tac (claset(), simpset() addsimps split_ifs@[quorem_def]));
   9.126  qed "quorem_neg";
   9.127  
   9.128 -Goal "b ~= Numeral0 ==> quorem ((a,b), divAlg(a,b))";
   9.129 +Goal "b ~= 0 ==> quorem ((a,b), divAlg(a,b))";
   9.130  by (auto_tac (claset(), 
   9.131  	      simpset() addsimps [quorem_0, divAlg_def]));
   9.132  by (REPEAT_FIRST (resolve_tac [quorem_neg, posDivAlg_correct,
   9.133 @@ -206,11 +206,11 @@
   9.134  (** Arbitrary definitions for division by zero.  Useful to simplify 
   9.135      certain equations **)
   9.136  
   9.137 -Goal "a div (Numeral0::int) = Numeral0";
   9.138 +Goal "a div (0::int) = 0";
   9.139  by (simp_tac (simpset() addsimps [div_def, divAlg_def, posDivAlg_raw_eqn]) 1);
   9.140  qed "DIVISION_BY_ZERO_ZDIV";  (*NOT for adding to default simpset*)
   9.141  
   9.142 -Goal "a mod (Numeral0::int) = a";
   9.143 +Goal "a mod (0::int) = a";
   9.144  by (simp_tac (simpset() addsimps [mod_def, divAlg_def, posDivAlg_raw_eqn]) 1);
   9.145  qed "DIVISION_BY_ZERO_ZMOD";  (*NOT for adding to default simpset*)
   9.146  
   9.147 @@ -222,20 +222,20 @@
   9.148  (** Basic laws about division and remainder **)
   9.149  
   9.150  Goal "(a::int) = b * (a div b) + (a mod b)";
   9.151 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.152 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.153  by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
   9.154  by (auto_tac (claset(), 
   9.155  	      simpset() addsimps [quorem_def, div_def, mod_def]));
   9.156  qed "zmod_zdiv_equality";  
   9.157  
   9.158 -Goal "(Numeral0::int) < b ==> Numeral0 <= a mod b & a mod b < b";
   9.159 +Goal "(0::int) < b ==> 0 <= a mod b & a mod b < b";
   9.160  by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
   9.161  by (auto_tac (claset(), 
   9.162  	      simpset() addsimps [quorem_def, mod_def]));
   9.163  bind_thm ("pos_mod_sign", result() RS conjunct1);
   9.164  bind_thm ("pos_mod_bound", result() RS conjunct2);
   9.165  
   9.166 -Goal "b < (Numeral0::int) ==> a mod b <= Numeral0 & b < a mod b";
   9.167 +Goal "b < (0::int) ==> a mod b <= 0 & b < a mod b";
   9.168  by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
   9.169  by (auto_tac (claset(), 
   9.170  	      simpset() addsimps [quorem_def, div_def, mod_def]));
   9.171 @@ -245,7 +245,7 @@
   9.172  
   9.173  (** proving general properties of div and mod **)
   9.174  
   9.175 -Goal "b ~= Numeral0 ==> quorem ((a, b), (a div b, a mod b))";
   9.176 +Goal "b ~= 0 ==> quorem ((a, b), (a div b, a mod b))";
   9.177  by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
   9.178  by (auto_tac
   9.179      (claset(),
   9.180 @@ -254,42 +254,42 @@
   9.181  			 neg_mod_sign, neg_mod_bound]));
   9.182  qed "quorem_div_mod";
   9.183  
   9.184 -Goal "[| quorem((a,b),(q,r));  b ~= Numeral0 |] ==> a div b = q";
   9.185 +Goal "[| quorem((a,b),(q,r));  b ~= 0 |] ==> a div b = q";
   9.186  by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_quotient]) 1);
   9.187  qed "quorem_div";
   9.188  
   9.189 -Goal "[| quorem((a,b),(q,r));  b ~= Numeral0 |] ==> a mod b = r";
   9.190 +Goal "[| quorem((a,b),(q,r));  b ~= 0 |] ==> a mod b = r";
   9.191  by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_remainder]) 1);
   9.192  qed "quorem_mod";
   9.193  
   9.194 -Goal "[| (Numeral0::int) <= a;  a < b |] ==> a div b = Numeral0";
   9.195 +Goal "[| (0::int) <= a;  a < b |] ==> a div b = 0";
   9.196  by (rtac quorem_div 1);
   9.197  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.198  qed "div_pos_pos_trivial";
   9.199  
   9.200 -Goal "[| a <= (Numeral0::int);  b < a |] ==> a div b = Numeral0";
   9.201 +Goal "[| a <= (0::int);  b < a |] ==> a div b = 0";
   9.202  by (rtac quorem_div 1);
   9.203  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.204  qed "div_neg_neg_trivial";
   9.205  
   9.206 -Goal "[| (Numeral0::int) < a;  a+b <= Numeral0 |] ==> a div b = -1";
   9.207 +Goal "[| (0::int) < a;  a+b <= 0 |] ==> a div b = -1";
   9.208  by (rtac quorem_div 1);
   9.209  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.210  qed "div_pos_neg_trivial";
   9.211  
   9.212 -(*There is no div_neg_pos_trivial because  Numeral0 div b = Numeral0 would supersede it*)
   9.213 +(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
   9.214  
   9.215 -Goal "[| (Numeral0::int) <= a;  a < b |] ==> a mod b = a";
   9.216 -by (res_inst_tac [("q","Numeral0")] quorem_mod 1);
   9.217 +Goal "[| (0::int) <= a;  a < b |] ==> a mod b = a";
   9.218 +by (res_inst_tac [("q","0")] quorem_mod 1);
   9.219  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.220  qed "mod_pos_pos_trivial";
   9.221  
   9.222 -Goal "[| a <= (Numeral0::int);  b < a |] ==> a mod b = a";
   9.223 -by (res_inst_tac [("q","Numeral0")] quorem_mod 1);
   9.224 +Goal "[| a <= (0::int);  b < a |] ==> a mod b = a";
   9.225 +by (res_inst_tac [("q","0")] quorem_mod 1);
   9.226  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.227  qed "mod_neg_neg_trivial";
   9.228  
   9.229 -Goal "[| (Numeral0::int) < a;  a+b <= Numeral0 |] ==> a mod b = a+b";
   9.230 +Goal "[| (0::int) < a;  a+b <= 0 |] ==> a mod b = a+b";
   9.231  by (res_inst_tac [("q","-1")] quorem_mod 1);
   9.232  by (auto_tac (claset(), simpset() addsimps [quorem_def]));
   9.233  qed "mod_pos_neg_trivial";
   9.234 @@ -299,7 +299,7 @@
   9.235  
   9.236  (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
   9.237  Goal "(-a) div (-b) = a div (b::int)";
   9.238 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.239 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.240  by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg)) 
   9.241  	  RS quorem_div) 1);
   9.242  by Auto_tac;
   9.243 @@ -308,7 +308,7 @@
   9.244  
   9.245  (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
   9.246  Goal "(-a) mod (-b) = - (a mod (b::int))";
   9.247 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.248 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.249  by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg)) 
   9.250  	  RS quorem_mod) 1);
   9.251  by Auto_tac;
   9.252 @@ -319,8 +319,8 @@
   9.253  (*** div, mod and unary minus ***)
   9.254  
   9.255  Goal "quorem((a,b),(q,r)) \
   9.256 -\     ==> quorem ((-a,b), (if r=Numeral0 then -q else -q-Numeral1), \
   9.257 -\                         (if r=Numeral0 then Numeral0 else b-r))";
   9.258 +\     ==> quorem ((-a,b), (if r=0 then -q else -q - 1), \
   9.259 +\                         (if r=0 then 0 else b-r))";
   9.260  by (auto_tac
   9.261      (claset(),
   9.262       simpset() addsimps split_ifs@
   9.263 @@ -328,14 +328,14 @@
   9.264  			 zdiff_zmult_distrib2]));
   9.265  val lemma = result();
   9.266  
   9.267 -Goal "b ~= (Numeral0::int) \
   9.268 +Goal "b ~= (0::int) \
   9.269  \     ==> (-a) div b = \
   9.270 -\         (if a mod b = Numeral0 then - (a div b) else  - (a div b) - Numeral1)";
   9.271 +\         (if a mod b = 0 then - (a div b) else  - (a div b) - 1)";
   9.272  by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_div]) 1);
   9.273  qed "zdiv_zminus1_eq_if";
   9.274  
   9.275 -Goal "(-a::int) mod b = (if a mod b = Numeral0 then Numeral0 else  b - (a mod b))";
   9.276 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.277 +Goal "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))";
   9.278 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.279  by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_mod]) 1);
   9.280  qed "zmod_zminus1_eq_if";
   9.281  
   9.282 @@ -349,32 +349,32 @@
   9.283  by Auto_tac;  
   9.284  qed "zmod_zminus2";
   9.285  
   9.286 -Goal "b ~= (Numeral0::int) \
   9.287 +Goal "b ~= (0::int) \
   9.288  \     ==> a div (-b) = \
   9.289 -\         (if a mod b = Numeral0 then - (a div b) else  - (a div b) - Numeral1)";
   9.290 +\         (if a mod b = 0 then - (a div b) else  - (a div b) - 1)";
   9.291  by (asm_simp_tac (simpset() addsimps [zdiv_zminus1_eq_if, zdiv_zminus2]) 1); 
   9.292  qed "zdiv_zminus2_eq_if";
   9.293  
   9.294 -Goal "a mod (-b::int) = (if a mod b = Numeral0 then Numeral0 else  (a mod b) - b)";
   9.295 +Goal "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)";
   9.296  by (asm_simp_tac (simpset() addsimps [zmod_zminus1_eq_if, zmod_zminus2]) 1); 
   9.297  qed "zmod_zminus2_eq_if";
   9.298  
   9.299  
   9.300  (*** division of a number by itself ***)
   9.301  
   9.302 -Goal "[| (Numeral0::int) < a; a = r + a*q; r < a |] ==> Numeral1 <= q";
   9.303 -by (subgoal_tac "Numeral0 < a*q" 1);
   9.304 +Goal "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 <= q";
   9.305 +by (subgoal_tac "0 < a*q" 1);
   9.306  by (arith_tac 2);
   9.307  by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
   9.308  val lemma1 = result();
   9.309  
   9.310 -Goal "[| (Numeral0::int) < a; a = r + a*q; Numeral0 <= r |] ==> q <= Numeral1";
   9.311 -by (subgoal_tac "Numeral0 <= a*(Numeral1-q)" 1);
   9.312 +Goal "[| (0::int) < a; a = r + a*q; 0 <= r |] ==> q <= 1";
   9.313 +by (subgoal_tac "0 <= a*(1-q)" 1);
   9.314  by (asm_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 2);
   9.315  by (asm_full_simp_tac (simpset() addsimps [int_0_le_mult_iff]) 1);
   9.316  val lemma2 = result();
   9.317  
   9.318 -Goal "[| quorem((a,a),(q,r));  a ~= (Numeral0::int) |] ==> q = Numeral1";
   9.319 +Goal "[| quorem((a,a),(q,r));  a ~= (0::int) |] ==> q = 1";
   9.320  by (asm_full_simp_tac 
   9.321      (simpset() addsimps split_ifs@[quorem_def, linorder_neq_iff]) 1);
   9.322  by (rtac order_antisym 1);
   9.323 @@ -386,20 +386,20 @@
   9.324  	      simpset() addsimps [zadd_commute, zmult_zminus]) 1));
   9.325  qed "self_quotient";
   9.326  
   9.327 -Goal "[| quorem((a,a),(q,r));  a ~= (Numeral0::int) |] ==> r = Numeral0";
   9.328 +Goal "[| quorem((a,a),(q,r));  a ~= (0::int) |] ==> r = 0";
   9.329  by (ftac self_quotient 1);
   9.330  by (assume_tac 1);
   9.331  by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
   9.332  qed "self_remainder";
   9.333  
   9.334 -Goal "a ~= Numeral0 ==> a div a = (Numeral1::int)";
   9.335 +Goal "a ~= 0 ==> a div a = (1::int)";
   9.336  by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_quotient]) 1);
   9.337  qed "zdiv_self";
   9.338  Addsimps [zdiv_self];
   9.339  
   9.340  (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
   9.341 -Goal "a mod a = (Numeral0::int)";
   9.342 -by (zdiv_undefined_case_tac "a = Numeral0" 1);
   9.343 +Goal "a mod a = (0::int)";
   9.344 +by (zdiv_undefined_case_tac "a = 0" 1);
   9.345  by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_remainder]) 1);
   9.346  qed "zmod_self";
   9.347  Addsimps [zmod_self];
   9.348 @@ -407,65 +407,65 @@
   9.349  
   9.350  (*** Computation of division and remainder ***)
   9.351  
   9.352 -Goal "(Numeral0::int) div b = Numeral0";
   9.353 +Goal "(0::int) div b = 0";
   9.354  by (simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.355  qed "zdiv_zero";
   9.356  
   9.357 -Goal "(Numeral0::int) < b ==> -1 div b = -1";
   9.358 +Goal "(0::int) < b ==> -1 div b = -1";
   9.359  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.360  qed "div_eq_minus1";
   9.361  
   9.362 -Goal "(Numeral0::int) mod b = Numeral0";
   9.363 +Goal "(0::int) mod b = 0";
   9.364  by (simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.365  qed "zmod_zero";
   9.366  
   9.367  Addsimps [zdiv_zero, zmod_zero];
   9.368  
   9.369 -Goal "(Numeral0::int) < b ==> -1 div b = -1";
   9.370 +Goal "(0::int) < b ==> -1 div b = -1";
   9.371  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.372  qed "zdiv_minus1";
   9.373  
   9.374 -Goal "(Numeral0::int) < b ==> -1 mod b = b-Numeral1";
   9.375 +Goal "(0::int) < b ==> -1 mod b = b - 1";
   9.376  by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.377  qed "zmod_minus1";
   9.378  
   9.379  (** a positive, b positive **)
   9.380  
   9.381 -Goal "[| Numeral0 < a;  Numeral0 <= b |] ==> a div b = fst (posDivAlg(a,b))";
   9.382 +Goal "[| 0 < a;  0 <= b |] ==> a div b = fst (posDivAlg(a,b))";
   9.383  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.384  qed "div_pos_pos";
   9.385  
   9.386 -Goal "[| Numeral0 < a;  Numeral0 <= b |] ==> a mod b = snd (posDivAlg(a,b))";
   9.387 +Goal "[| 0 < a;  0 <= b |] ==> a mod b = snd (posDivAlg(a,b))";
   9.388  by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.389  qed "mod_pos_pos";
   9.390  
   9.391  (** a negative, b positive **)
   9.392  
   9.393 -Goal "[| a < Numeral0;  Numeral0 < b |] ==> a div b = fst (negDivAlg(a,b))";
   9.394 +Goal "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg(a,b))";
   9.395  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.396  qed "div_neg_pos";
   9.397  
   9.398 -Goal "[| a < Numeral0;  Numeral0 < b |] ==> a mod b = snd (negDivAlg(a,b))";
   9.399 +Goal "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg(a,b))";
   9.400  by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.401  qed "mod_neg_pos";
   9.402  
   9.403  (** a positive, b negative **)
   9.404  
   9.405 -Goal "[| Numeral0 < a;  b < Numeral0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))";
   9.406 +Goal "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))";
   9.407  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.408  qed "div_pos_neg";
   9.409  
   9.410 -Goal "[| Numeral0 < a;  b < Numeral0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))";
   9.411 +Goal "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))";
   9.412  by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.413  qed "mod_pos_neg";
   9.414  
   9.415  (** a negative, b negative **)
   9.416  
   9.417 -Goal "[| a < Numeral0;  b <= Numeral0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))";
   9.418 +Goal "[| a < 0;  b <= 0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))";
   9.419  by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
   9.420  qed "div_neg_neg";
   9.421  
   9.422 -Goal "[| a < Numeral0;  b <= Numeral0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))";
   9.423 +Goal "[| a < 0;  b <= 0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))";
   9.424  by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
   9.425  qed "mod_neg_neg";
   9.426  
   9.427 @@ -478,20 +478,20 @@
   9.428  
   9.429  (** Special-case simplification **)
   9.430  
   9.431 -Goal "a mod (Numeral1::int) = Numeral0";
   9.432 -by (cut_inst_tac [("a","a"),("b","Numeral1")] pos_mod_sign 1);
   9.433 -by (cut_inst_tac [("a","a"),("b","Numeral1")] pos_mod_bound 2);
   9.434 +Goal "a mod (1::int) = 0";
   9.435 +by (cut_inst_tac [("a","a"),("b","1")] pos_mod_sign 1);
   9.436 +by (cut_inst_tac [("a","a"),("b","1")] pos_mod_bound 2);
   9.437  by Auto_tac;
   9.438  qed "zmod_1";
   9.439  Addsimps [zmod_1];
   9.440  
   9.441 -Goal "a div (Numeral1::int) = a";
   9.442 -by (cut_inst_tac [("a","a"),("b","Numeral1")] zmod_zdiv_equality 1);
   9.443 +Goal "a div (1::int) = a";
   9.444 +by (cut_inst_tac [("a","a"),("b","1")] zmod_zdiv_equality 1);
   9.445  by Auto_tac;
   9.446  qed "zdiv_1";
   9.447  Addsimps [zdiv_1];
   9.448  
   9.449 -Goal "a mod (-1::int) = Numeral0";
   9.450 +Goal "a mod (-1::int) = 0";
   9.451  by (cut_inst_tac [("a","a"),("b","-1")] neg_mod_sign 1);
   9.452  by (cut_inst_tac [("a","a"),("b","-1")] neg_mod_bound 2);
   9.453  by Auto_tac;
   9.454 @@ -504,10 +504,19 @@
   9.455  qed "zdiv_minus1_right";
   9.456  Addsimps [zdiv_minus1_right];
   9.457  
   9.458 +(** The last remaining cases: 1 div z and 1 mod z **)
   9.459 +
   9.460 +Addsimps (map (fn th => int_0_less_1 RS inst "b" "number_of ?w" th) 
   9.461 +              [div_pos_pos, div_pos_neg, mod_pos_pos, mod_pos_neg]);
   9.462 +
   9.463 +Addsimps (map (read_instantiate_sg (sign_of (the_context ()))
   9.464 +	       [("a", "1"), ("b", "number_of ?w")])
   9.465 +	  [posDivAlg_eqn, negDivAlg_eqn]);
   9.466 +
   9.467  
   9.468  (*** Monotonicity in the first argument (divisor) ***)
   9.469  
   9.470 -Goal "[| a <= a';  Numeral0 < (b::int) |] ==> a div b <= a' div b";
   9.471 +Goal "[| a <= a';  0 < (b::int) |] ==> a div b <= a' div b";
   9.472  by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
   9.473  by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
   9.474  by (rtac unique_quotient_lemma 1);
   9.475 @@ -516,7 +525,7 @@
   9.476  by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
   9.477  qed "zdiv_mono1";
   9.478  
   9.479 -Goal "[| a <= a';  (b::int) < Numeral0 |] ==> a' div b <= a div b";
   9.480 +Goal "[| a <= a';  (b::int) < 0 |] ==> a' div b <= a div b";
   9.481  by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
   9.482  by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
   9.483  by (rtac unique_quotient_lemma_neg 1);
   9.484 @@ -528,14 +537,14 @@
   9.485  
   9.486  (*** Monotonicity in the second argument (dividend) ***)
   9.487  
   9.488 -Goal "[| b*q + r = b'*q' + r';  Numeral0 <= b'*q' + r';  \
   9.489 -\        r' < b';  Numeral0 <= r;  Numeral0 < b';  b' <= b |]  \
   9.490 +Goal "[| b*q + r = b'*q' + r';  0 <= b'*q' + r';  \
   9.491 +\        r' < b';  0 <= r;  0 < b';  b' <= b |]  \
   9.492  \     ==> q <= (q'::int)";
   9.493 -by (subgoal_tac "Numeral0 <= q'" 1);
   9.494 - by (subgoal_tac "Numeral0 < b'*(q' + Numeral1)" 2);
   9.495 +by (subgoal_tac "0 <= q'" 1);
   9.496 + by (subgoal_tac "0 < b'*(q' + 1)" 2);
   9.497    by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 3);
   9.498   by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 2);
   9.499 -by (subgoal_tac "b*q < b*(q' + Numeral1)" 1);
   9.500 +by (subgoal_tac "b*q < b*(q' + 1)" 1);
   9.501   by (asm_full_simp_tac (simpset() addsimps [zmult_zless_cancel1]) 1); 
   9.502  by (subgoal_tac "b*q = r' - r + b'*q'" 1);
   9.503   by (Simp_tac 2);
   9.504 @@ -545,9 +554,9 @@
   9.505  by Auto_tac;
   9.506  qed "zdiv_mono2_lemma";
   9.507  
   9.508 -Goal "[| (Numeral0::int) <= a;  Numeral0 < b';  b' <= b |]  \
   9.509 +Goal "[| (0::int) <= a;  0 < b';  b' <= b |]  \
   9.510  \     ==> a div b <= a div b'";
   9.511 -by (subgoal_tac "b ~= Numeral0" 1);
   9.512 +by (subgoal_tac "b ~= 0" 1);
   9.513  by (arith_tac 2);
   9.514  by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
   9.515  by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
   9.516 @@ -557,14 +566,14 @@
   9.517  by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
   9.518  qed "zdiv_mono2";
   9.519  
   9.520 -Goal "[| b*q + r = b'*q' + r';  b'*q' + r' < Numeral0;  \
   9.521 -\        r < b;  Numeral0 <= r';  Numeral0 < b';  b' <= b |]  \
   9.522 +Goal "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;  \
   9.523 +\        r < b;  0 <= r';  0 < b';  b' <= b |]  \
   9.524  \     ==> q' <= (q::int)";
   9.525 -by (subgoal_tac "q' < Numeral0" 1);
   9.526 - by (subgoal_tac "b'*q' < Numeral0" 2);
   9.527 +by (subgoal_tac "q' < 0" 1);
   9.528 + by (subgoal_tac "b'*q' < 0" 2);
   9.529    by (arith_tac 3);
   9.530   by (asm_full_simp_tac (simpset() addsimps [zmult_less_0_iff]) 2);
   9.531 -by (subgoal_tac "b*q' < b*(q + Numeral1)" 1);
   9.532 +by (subgoal_tac "b*q' < b*(q + 1)" 1);
   9.533   by (asm_full_simp_tac (simpset() addsimps [zmult_zless_cancel1]) 1); 
   9.534  by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
   9.535  by (subgoal_tac "b*q' <= b'*q'" 1);
   9.536 @@ -574,7 +583,7 @@
   9.537  by (arith_tac 1);
   9.538  qed "zdiv_mono2_neg_lemma";
   9.539  
   9.540 -Goal "[| a < (Numeral0::int);  Numeral0 < b';  b' <= b |]  \
   9.541 +Goal "[| a < (0::int);  0 < b';  b' <= b |]  \
   9.542  \     ==> a div b' <= a div b";
   9.543  by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
   9.544  by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
   9.545 @@ -589,7 +598,7 @@
   9.546  
   9.547  (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
   9.548  
   9.549 -Goal "[| quorem((b,c),(q,r));  c ~= Numeral0 |] \
   9.550 +Goal "[| quorem((b,c),(q,r));  c ~= 0 |] \
   9.551  \     ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))";
   9.552  by (auto_tac
   9.553      (claset(),
   9.554 @@ -602,12 +611,12 @@
   9.555  val lemma = result();
   9.556  
   9.557  Goal "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)";
   9.558 -by (zdiv_undefined_case_tac "c = Numeral0" 1);
   9.559 +by (zdiv_undefined_case_tac "c = 0" 1);
   9.560  by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_div]) 1);
   9.561  qed "zdiv_zmult1_eq";
   9.562  
   9.563  Goal "(a*b) mod c = a*(b mod c) mod (c::int)";
   9.564 -by (zdiv_undefined_case_tac "c = Numeral0" 1);
   9.565 +by (zdiv_undefined_case_tac "c = 0" 1);
   9.566  by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_mod]) 1);
   9.567  qed "zmod_zmult1_eq";
   9.568  
   9.569 @@ -623,27 +632,27 @@
   9.570  by (rtac zmod_zmult1_eq 1);
   9.571  qed "zmod_zmult_distrib";
   9.572  
   9.573 -Goal "b ~= (Numeral0::int) ==> (a*b) div b = a";
   9.574 +Goal "b ~= (0::int) ==> (a*b) div b = a";
   9.575  by (asm_simp_tac (simpset() addsimps [zdiv_zmult1_eq]) 1);
   9.576  qed "zdiv_zmult_self1";
   9.577  
   9.578 -Goal "b ~= (Numeral0::int) ==> (b*a) div b = a";
   9.579 +Goal "b ~= (0::int) ==> (b*a) div b = a";
   9.580  by (stac zmult_commute 1 THEN etac zdiv_zmult_self1 1);
   9.581  qed "zdiv_zmult_self2";
   9.582  
   9.583  Addsimps [zdiv_zmult_self1, zdiv_zmult_self2];
   9.584  
   9.585 -Goal "(a*b) mod b = (Numeral0::int)";
   9.586 +Goal "(a*b) mod b = (0::int)";
   9.587  by (simp_tac (simpset() addsimps [zmod_zmult1_eq]) 1);
   9.588  qed "zmod_zmult_self1";
   9.589  
   9.590 -Goal "(b*a) mod b = (Numeral0::int)";
   9.591 +Goal "(b*a) mod b = (0::int)";
   9.592  by (simp_tac (simpset() addsimps [zmult_commute, zmod_zmult1_eq]) 1);
   9.593  qed "zmod_zmult_self2";
   9.594  
   9.595  Addsimps [zmod_zmult_self1, zmod_zmult_self2];
   9.596  
   9.597 -Goal "(m mod d = Numeral0) = (EX q::int. m = d*q)";
   9.598 +Goal "(m mod d = 0) = (EX q::int. m = d*q)";
   9.599  by (cut_inst_tac [("a","m"),("b","d")] zmod_zdiv_equality 1);
   9.600  by Auto_tac;  
   9.601  qed "zmod_eq_0_iff";
   9.602 @@ -652,7 +661,7 @@
   9.603  
   9.604  (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
   9.605  
   9.606 -Goal "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c ~= Numeral0 |] \
   9.607 +Goal "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c ~= 0 |] \
   9.608  \     ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))";
   9.609  by (auto_tac
   9.610      (claset(),
   9.611 @@ -666,19 +675,19 @@
   9.612  
   9.613  (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   9.614  Goal "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)";
   9.615 -by (zdiv_undefined_case_tac "c = Numeral0" 1);
   9.616 +by (zdiv_undefined_case_tac "c = 0" 1);
   9.617  by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
   9.618  			       MRS lemma RS quorem_div]) 1);
   9.619  qed "zdiv_zadd1_eq";
   9.620  
   9.621  Goal "(a+b) mod (c::int) = (a mod c + b mod c) mod c";
   9.622 -by (zdiv_undefined_case_tac "c = Numeral0" 1);
   9.623 +by (zdiv_undefined_case_tac "c = 0" 1);
   9.624  by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
   9.625  			       MRS lemma RS quorem_mod]) 1);
   9.626  qed "zmod_zadd1_eq";
   9.627  
   9.628 -Goal "(a mod b) div b = (Numeral0::int)";
   9.629 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.630 +Goal "(a mod b) div b = (0::int)";
   9.631 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.632  by (auto_tac (claset(), 
   9.633         simpset() addsimps [linorder_neq_iff, 
   9.634  			   pos_mod_sign, pos_mod_bound, div_pos_pos_trivial, 
   9.635 @@ -687,7 +696,7 @@
   9.636  Addsimps [mod_div_trivial];
   9.637  
   9.638  Goal "(a mod b) mod b = a mod (b::int)";
   9.639 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.640 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.641  by (auto_tac (claset(), 
   9.642         simpset() addsimps [linorder_neq_iff, 
   9.643  			   pos_mod_sign, pos_mod_bound, mod_pos_pos_trivial, 
   9.644 @@ -710,22 +719,22 @@
   9.645  qed "zmod_zadd_right_eq";
   9.646  
   9.647  
   9.648 -Goal "a ~= (Numeral0::int) ==> (a+b) div a = b div a + Numeral1";
   9.649 +Goal "a ~= (0::int) ==> (a+b) div a = b div a + 1";
   9.650  by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
   9.651  qed "zdiv_zadd_self1";
   9.652  
   9.653 -Goal "a ~= (Numeral0::int) ==> (b+a) div a = b div a + Numeral1";
   9.654 +Goal "a ~= (0::int) ==> (b+a) div a = b div a + 1";
   9.655  by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
   9.656  qed "zdiv_zadd_self2";
   9.657  Addsimps [zdiv_zadd_self1, zdiv_zadd_self2];
   9.658  
   9.659  Goal "(a+b) mod a = b mod (a::int)";
   9.660 -by (zdiv_undefined_case_tac "a = Numeral0" 1);
   9.661 +by (zdiv_undefined_case_tac "a = 0" 1);
   9.662  by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
   9.663  qed "zmod_zadd_self1";
   9.664  
   9.665  Goal "(b+a) mod a = b mod (a::int)";
   9.666 -by (zdiv_undefined_case_tac "a = Numeral0" 1);
   9.667 +by (zdiv_undefined_case_tac "a = 0" 1);
   9.668  by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
   9.669  qed "zmod_zadd_self2";
   9.670  Addsimps [zmod_zadd_self1, zmod_zadd_self2];
   9.671 @@ -739,8 +748,8 @@
   9.672  
   9.673  (** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
   9.674  
   9.675 -Goal "[| (Numeral0::int) < c;  b < r;  r <= Numeral0 |] ==> b*c < b*(q mod c) + r";
   9.676 -by (subgoal_tac "b * (c - q mod c) < r * Numeral1" 1);
   9.677 +Goal "[| (0::int) < c;  b < r;  r <= 0 |] ==> b*c < b*(q mod c) + r";
   9.678 +by (subgoal_tac "b * (c - q mod c) < r * 1" 1);
   9.679  by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
   9.680  by (rtac order_le_less_trans 1);
   9.681  by (etac zmult_zless_mono1 2);
   9.682 @@ -748,23 +757,24 @@
   9.683  by (auto_tac
   9.684      (claset(),
   9.685       simpset() addsimps zcompare_rls@
   9.686 -                        [zadd_commute, add1_zle_eq, pos_mod_bound]));
   9.687 +                        [inst "z" "1" zadd_commute, add1_zle_eq, 
   9.688 +                         pos_mod_bound]));
   9.689  val lemma1 = result();
   9.690  
   9.691 -Goal "[| (Numeral0::int) < c;   b < r;  r <= Numeral0 |] ==> b * (q mod c) + r <= Numeral0";
   9.692 -by (subgoal_tac "b * (q mod c) <= Numeral0" 1);
   9.693 +Goal "[| (0::int) < c;   b < r;  r <= 0 |] ==> b * (q mod c) + r <= 0";
   9.694 +by (subgoal_tac "b * (q mod c) <= 0" 1);
   9.695  by (arith_tac 1);
   9.696  by (asm_simp_tac (simpset() addsimps [zmult_le_0_iff, pos_mod_sign]) 1);
   9.697  val lemma2 = result();
   9.698  
   9.699 -Goal "[| (Numeral0::int) < c;  Numeral0 <= r;  r < b |] ==> Numeral0 <= b * (q mod c) + r";
   9.700 -by (subgoal_tac "Numeral0 <= b * (q mod c)" 1);
   9.701 +Goal "[| (0::int) < c;  0 <= r;  r < b |] ==> 0 <= b * (q mod c) + r";
   9.702 +by (subgoal_tac "0 <= b * (q mod c)" 1);
   9.703  by (arith_tac 1);
   9.704  by (asm_simp_tac (simpset() addsimps [int_0_le_mult_iff, pos_mod_sign]) 1);
   9.705  val lemma3 = result();
   9.706  
   9.707 -Goal "[| (Numeral0::int) < c; Numeral0 <= r; r < b |] ==> b * (q mod c) + r < b * c";
   9.708 -by (subgoal_tac "r * Numeral1 < b * (c - q mod c)" 1);
   9.709 +Goal "[| (0::int) < c; 0 <= r; r < b |] ==> b * (q mod c) + r < b * c";
   9.710 +by (subgoal_tac "r * 1 < b * (c - q mod c)" 1);
   9.711  by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
   9.712  by (rtac order_less_le_trans 1);
   9.713  by (etac zmult_zless_mono1 1);
   9.714 @@ -772,10 +782,11 @@
   9.715  by (auto_tac
   9.716      (claset(),
   9.717       simpset() addsimps zcompare_rls@
   9.718 -                        [zadd_commute, add1_zle_eq, pos_mod_bound]));
   9.719 +                        [inst "z" "1" zadd_commute, add1_zle_eq,
   9.720 +                         pos_mod_bound]));
   9.721  val lemma4 = result();
   9.722  
   9.723 -Goal "[| quorem ((a,b), (q,r));  b ~= Numeral0;  Numeral0 < c |] \
   9.724 +Goal "[| quorem ((a,b), (q,r));  b ~= 0;  0 < c |] \
   9.725  \     ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))";
   9.726  by (auto_tac  
   9.727      (claset(),
   9.728 @@ -786,15 +797,15 @@
   9.729  			 lemma1, lemma2, lemma3, lemma4]));
   9.730  val lemma = result();
   9.731  
   9.732 -Goal "(Numeral0::int) < c ==> a div (b*c) = (a div b) div c";
   9.733 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.734 +Goal "(0::int) < c ==> a div (b*c) = (a div b) div c";
   9.735 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.736  by (force_tac (claset(),
   9.737  	       simpset() addsimps [quorem_div_mod RS lemma RS quorem_div, 
   9.738  				   zmult_eq_0_iff]) 1);
   9.739  qed "zdiv_zmult2_eq";
   9.740  
   9.741 -Goal "(Numeral0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b";
   9.742 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.743 +Goal "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b";
   9.744 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.745  by (force_tac (claset(),
   9.746  	       simpset() addsimps [quorem_div_mod RS lemma RS quorem_mod, 
   9.747  				   zmult_eq_0_iff]) 1);
   9.748 @@ -803,26 +814,26 @@
   9.749  
   9.750  (*** Cancellation of common factors in "div" ***)
   9.751  
   9.752 -Goal "[| (Numeral0::int) < b;  c ~= Numeral0 |] ==> (c*a) div (c*b) = a div b";
   9.753 +Goal "[| (0::int) < b;  c ~= 0 |] ==> (c*a) div (c*b) = a div b";
   9.754  by (stac zdiv_zmult2_eq 1);
   9.755  by Auto_tac;
   9.756  val lemma1 = result();
   9.757  
   9.758 -Goal "[| b < (Numeral0::int);  c ~= Numeral0 |] ==> (c*a) div (c*b) = a div b";
   9.759 +Goal "[| b < (0::int);  c ~= 0 |] ==> (c*a) div (c*b) = a div b";
   9.760  by (subgoal_tac "(c * (-a)) div (c * (-b)) = (-a) div (-b)" 1);
   9.761  by (rtac lemma1 2);
   9.762  by Auto_tac;
   9.763  val lemma2 = result();
   9.764  
   9.765 -Goal "c ~= (Numeral0::int) ==> (c*a) div (c*b) = a div b";
   9.766 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.767 +Goal "c ~= (0::int) ==> (c*a) div (c*b) = a div b";
   9.768 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.769  by (auto_tac
   9.770      (claset(), 
   9.771       simpset() addsimps [read_instantiate [("x", "b")] linorder_neq_iff, 
   9.772  			 lemma1, lemma2]));
   9.773  qed "zdiv_zmult_zmult1";
   9.774  
   9.775 -Goal "c ~= (Numeral0::int) ==> (a*c) div (b*c) = a div b";
   9.776 +Goal "c ~= (0::int) ==> (a*c) div (b*c) = a div b";
   9.777  by (dtac zdiv_zmult_zmult1 1);
   9.778  by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
   9.779  qed "zdiv_zmult_zmult2";
   9.780 @@ -831,20 +842,20 @@
   9.781  
   9.782  (*** Distribution of factors over "mod" ***)
   9.783  
   9.784 -Goal "[| (Numeral0::int) < b;  c ~= Numeral0 |] ==> (c*a) mod (c*b) = c * (a mod b)";
   9.785 +Goal "[| (0::int) < b;  c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)";
   9.786  by (stac zmod_zmult2_eq 1);
   9.787  by Auto_tac;
   9.788  val lemma1 = result();
   9.789  
   9.790 -Goal "[| b < (Numeral0::int);  c ~= Numeral0 |] ==> (c*a) mod (c*b) = c * (a mod b)";
   9.791 +Goal "[| b < (0::int);  c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)";
   9.792  by (subgoal_tac "(c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))" 1);
   9.793  by (rtac lemma1 2);
   9.794  by Auto_tac;
   9.795  val lemma2 = result();
   9.796  
   9.797  Goal "(c*a) mod (c*b) = (c::int) * (a mod b)";
   9.798 -by (zdiv_undefined_case_tac "b = Numeral0" 1);
   9.799 -by (zdiv_undefined_case_tac "c = Numeral0" 1);
   9.800 +by (zdiv_undefined_case_tac "b = 0" 1);
   9.801 +by (zdiv_undefined_case_tac "c = 0" 1);
   9.802  by (auto_tac
   9.803      (claset(), 
   9.804       simpset() addsimps [read_instantiate [("x", "b")] linorder_neq_iff, 
   9.805 @@ -861,16 +872,16 @@
   9.806  
   9.807  (** computing "div" by shifting **)
   9.808  
   9.809 -Goal "(Numeral0::int) <= a ==> (Numeral1 + 2*b) div (2*a) = b div a";
   9.810 -by (zdiv_undefined_case_tac "a = Numeral0" 1);
   9.811 -by (subgoal_tac "Numeral1 <= a" 1);
   9.812 +Goal "(0::int) <= a ==> (1 + 2*b) div (2*a) = b div a";
   9.813 +by (zdiv_undefined_case_tac "a = 0" 1);
   9.814 +by (subgoal_tac "1 <= a" 1);
   9.815   by (arith_tac 2);
   9.816 -by (subgoal_tac "Numeral1 < a * 2" 1);
   9.817 +by (subgoal_tac "1 < a * 2" 1);
   9.818   by (arith_tac 2);
   9.819 -by (subgoal_tac "2*(Numeral1 + b mod a) <= 2*a" 1);
   9.820 +by (subgoal_tac "2*(1 + b mod a) <= 2*a" 1);
   9.821   by (rtac zmult_zle_mono2 2);
   9.822  by (auto_tac (claset(),
   9.823 -	      simpset() addsimps [zadd_commute, zmult_commute, 
   9.824 +	      simpset() addsimps [inst "z" "1" zadd_commute, zmult_commute, 
   9.825  				  add1_zle_eq, pos_mod_bound]));
   9.826  by (stac zdiv_zadd1_eq 1);
   9.827  by (asm_simp_tac (simpset() addsimps [zdiv_zmult_zmult2, zmod_zmult_zmult2, 
   9.828 @@ -881,18 +892,18 @@
   9.829                      pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
   9.830  by (auto_tac (claset(),
   9.831  	      simpset() addsimps [mod_pos_pos_trivial]));
   9.832 -by (subgoal_tac "Numeral0 <= b mod a" 1);
   9.833 +by (subgoal_tac "0 <= b mod a" 1);
   9.834   by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
   9.835  by (arith_tac 1);
   9.836  qed "pos_zdiv_mult_2";
   9.837  
   9.838  
   9.839 -Goal "a <= (Numeral0::int) ==> (Numeral1 + 2*b) div (2*a) = (b+Numeral1) div a";
   9.840 -by (subgoal_tac "(Numeral1 + 2*(-b-Numeral1)) div (2 * (-a)) = (-b-Numeral1) div (-a)" 1);
   9.841 +Goal "a <= (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a";
   9.842 +by (subgoal_tac "(1 + 2*(-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)" 1);
   9.843  by (rtac pos_zdiv_mult_2 2);
   9.844  by (auto_tac (claset(),
   9.845  	      simpset() addsimps [zmult_zminus_right]));
   9.846 -by (subgoal_tac "(-1 - (2 * b)) = - (Numeral1 + (2 * b))" 1);
   9.847 +by (subgoal_tac "(-1 - (2 * b)) = - (1 + (2 * b))" 1);
   9.848  by (Simp_tac 2);
   9.849  by (asm_full_simp_tac (HOL_ss
   9.850  		       addsimps [zdiv_zminus_zminus, zdiff_def,
   9.851 @@ -902,17 +913,19 @@
   9.852  
   9.853  (*Not clear why this must be proved separately; probably number_of causes
   9.854    simplification problems*)
   9.855 -Goal "~ Numeral0 <= x ==> x <= (Numeral0::int)";
   9.856 +Goal "~ 0 <= x ==> x <= (0::int)";
   9.857  by Auto_tac;
   9.858  val lemma = result();
   9.859  
   9.860  Goal "number_of (v BIT b) div number_of (w BIT False) = \
   9.861 -\         (if ~b | (Numeral0::int) <= number_of w                   \
   9.862 +\         (if ~b | (0::int) <= number_of w                   \
   9.863  \          then number_of v div (number_of w)    \
   9.864 -\          else (number_of v + (Numeral1::int)) div (number_of w))";
   9.865 +\          else (number_of v + (1::int)) div (number_of w))";
   9.866  by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
   9.867 +by (subgoal_tac "2 ~= (0::int)" 1);
   9.868 + by (Simp_tac 2);  (*we do this because the next step can't simplify numerals*)
   9.869  by (asm_simp_tac (simpset()
   9.870 -           delsimps [thm "number_of_reorient"]@bin_arith_extra_simps@bin_rel_simps
   9.871 +           delsimps bin_arith_extra_simps
   9.872   	   addsimps [zdiv_zmult_zmult1, pos_zdiv_mult_2, lemma, 
   9.873                       neg_zdiv_mult_2]) 1);
   9.874  qed "zdiv_number_of_BIT";
   9.875 @@ -921,16 +934,16 @@
   9.876  
   9.877  (** computing "mod" by shifting (proofs resemble those for "div") **)
   9.878  
   9.879 -Goal "(Numeral0::int) <= a ==> (Numeral1 + 2*b) mod (2*a) = Numeral1 + 2 * (b mod a)";
   9.880 -by (zdiv_undefined_case_tac "a = Numeral0" 1);
   9.881 -by (subgoal_tac "Numeral1 <= a" 1);
   9.882 +Goal "(0::int) <= a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)";
   9.883 +by (zdiv_undefined_case_tac "a = 0" 1);
   9.884 +by (subgoal_tac "1 <= a" 1);
   9.885   by (arith_tac 2);
   9.886 -by (subgoal_tac "Numeral1 < a * 2" 1);
   9.887 +by (subgoal_tac "1 < a * 2" 1);
   9.888   by (arith_tac 2);
   9.889 -by (subgoal_tac "2*(Numeral1 + b mod a) <= 2*a" 1);
   9.890 +by (subgoal_tac "2*(1 + b mod a) <= 2*a" 1);
   9.891   by (rtac zmult_zle_mono2 2);
   9.892  by (auto_tac (claset(),
   9.893 -	      simpset() addsimps [zadd_commute, zmult_commute, 
   9.894 +	      simpset() addsimps [inst "z" "1" zadd_commute, zmult_commute, 
   9.895  				  add1_zle_eq, pos_mod_bound]));
   9.896  by (stac zmod_zadd1_eq 1);
   9.897  by (asm_simp_tac (simpset() addsimps [zmod_zmult_zmult2, 
   9.898 @@ -941,19 +954,18 @@
   9.899                      pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
   9.900  by (auto_tac (claset(),
   9.901  	      simpset() addsimps [mod_pos_pos_trivial]));
   9.902 -by (subgoal_tac "Numeral0 <= b mod a" 1);
   9.903 +by (subgoal_tac "0 <= b mod a" 1);
   9.904   by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
   9.905  by (arith_tac 1);
   9.906  qed "pos_zmod_mult_2";
   9.907  
   9.908  
   9.909 -Goal "a <= (Numeral0::int) ==> (Numeral1 + 2*b) mod (2*a) = 2 * ((b+Numeral1) mod a) - Numeral1";
   9.910 +Goal "a <= (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1";
   9.911  by (subgoal_tac 
   9.912 -    "(Numeral1 + 2*(-b-Numeral1)) mod (2*(-a)) = Numeral1 + 2*((-b-Numeral1) mod (-a))" 1);
   9.913 +    "(1 + 2*(-b - 1)) mod (2*(-a)) = 1 + 2*((-b - 1) mod (-a))" 1);
   9.914  by (rtac pos_zmod_mult_2 2);
   9.915 -by (auto_tac (claset(),
   9.916 -	      simpset() addsimps [zmult_zminus_right]));
   9.917 -by (subgoal_tac "(-1 - (2 * b)) = - (Numeral1 + (2 * b))" 1);
   9.918 +by (auto_tac (claset(), simpset() addsimps [zmult_zminus_right]));
   9.919 +by (subgoal_tac "(-1 - (2 * b)) = - (1 + (2 * b))" 1);
   9.920  by (Simp_tac 2);
   9.921  by (asm_full_simp_tac (HOL_ss
   9.922  		       addsimps [zmod_zminus_zminus, zdiff_def,
   9.923 @@ -964,15 +976,16 @@
   9.924  
   9.925  Goal "number_of (v BIT b) mod number_of (w BIT False) = \
   9.926  \         (if b then \
   9.927 -\               if (Numeral0::int) <= number_of w \
   9.928 -\               then 2 * (number_of v mod number_of w) + Numeral1    \
   9.929 -\               else 2 * ((number_of v + (Numeral1::int)) mod number_of w) - Numeral1  \
   9.930 +\               if (0::int) <= number_of w \
   9.931 +\               then 2 * (number_of v mod number_of w) + 1    \
   9.932 +\               else 2 * ((number_of v + (1::int)) mod number_of w) - 1  \
   9.933  \          else 2 * (number_of v mod number_of w))";
   9.934  by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
   9.935  by (asm_simp_tac (simpset()
   9.936  		  delsimps bin_arith_extra_simps@bin_rel_simps
   9.937  		  addsimps [zmod_zmult_zmult1,
   9.938  			    pos_zmod_mult_2, lemma, neg_zmod_mult_2]) 1);
   9.939 +by (Simp_tac 1); 
   9.940  qed "zmod_number_of_BIT";
   9.941  
   9.942  Addsimps [zmod_number_of_BIT];
   9.943 @@ -980,7 +993,7 @@
   9.944  
   9.945  (** Quotients of signs **)
   9.946  
   9.947 -Goal "[| a < (Numeral0::int);  Numeral0 < b |] ==> a div b < Numeral0";
   9.948 +Goal "[| a < (0::int);  0 < b |] ==> a div b < 0";
   9.949  by (subgoal_tac "a div b <= -1" 1);
   9.950  by (Force_tac 1);
   9.951  by (rtac order_trans 1);
   9.952 @@ -988,12 +1001,12 @@
   9.953  by (auto_tac (claset(), simpset() addsimps [zdiv_minus1]));
   9.954  qed "div_neg_pos_less0";
   9.955  
   9.956 -Goal "[| (Numeral0::int) <= a;  b < Numeral0 |] ==> a div b <= Numeral0";
   9.957 +Goal "[| (0::int) <= a;  b < 0 |] ==> a div b <= 0";
   9.958  by (dtac zdiv_mono1_neg 1);
   9.959  by Auto_tac;
   9.960  qed "div_nonneg_neg_le0";
   9.961  
   9.962 -Goal "(Numeral0::int) < b ==> (Numeral0 <= a div b) = (Numeral0 <= a)";
   9.963 +Goal "(0::int) < b ==> (0 <= a div b) = (0 <= a)";
   9.964  by Auto_tac;
   9.965  by (dtac zdiv_mono1 2);
   9.966  by (auto_tac (claset(), simpset() addsimps [linorder_neq_iff]));
   9.967 @@ -1001,20 +1014,20 @@
   9.968  by (blast_tac (claset() addIs [div_neg_pos_less0]) 1);
   9.969  qed "pos_imp_zdiv_nonneg_iff";
   9.970  
   9.971 -Goal "b < (Numeral0::int) ==> (Numeral0 <= a div b) = (a <= (Numeral0::int))";
   9.972 +Goal "b < (0::int) ==> (0 <= a div b) = (a <= (0::int))";
   9.973  by (stac (zdiv_zminus_zminus RS sym) 1);
   9.974  by (stac pos_imp_zdiv_nonneg_iff 1);
   9.975  by Auto_tac;
   9.976  qed "neg_imp_zdiv_nonneg_iff";
   9.977  
   9.978  (*But not (a div b <= 0 iff a<=0); consider a=1, b=2 when a div b = 0.*)
   9.979 -Goal "(Numeral0::int) < b ==> (a div b < Numeral0) = (a < Numeral0)";
   9.980 +Goal "(0::int) < b ==> (a div b < 0) = (a < 0)";
   9.981  by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
   9.982  				      pos_imp_zdiv_nonneg_iff]) 1);
   9.983  qed "pos_imp_zdiv_neg_iff";
   9.984  
   9.985  (*Again the law fails for <=: consider a = -1, b = -2 when a div b = 0*)
   9.986 -Goal "b < (Numeral0::int) ==> (a div b < Numeral0) = (Numeral0 < a)";
   9.987 +Goal "b < (0::int) ==> (a div b < 0) = (0 < a)";
   9.988  by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
   9.989  				      neg_imp_zdiv_nonneg_iff]) 1);
   9.990  qed "neg_imp_zdiv_neg_iff";
    10.1 --- a/src/HOL/Integ/IntDiv.thy	Mon Oct 22 11:01:30 2001 +0200
    10.2 +++ b/src/HOL/Integ/IntDiv.thy	Mon Oct 22 11:54:22 2001 +0200
    10.3 @@ -12,27 +12,27 @@
    10.4    quorem :: "(int*int) * (int*int) => bool"
    10.5      "quorem == %((a,b), (q,r)).
    10.6                        a = b*q + r &
    10.7 -                      (if Numeral0 < b then Numeral0<=r & r<b else b<r & r <= Numeral0)"
    10.8 +                      (if 0 < b then 0<=r & r<b else b<r & r <= 0)"
    10.9  
   10.10 -  adjust :: "[int, int, int*int] => int*int"
   10.11 -    "adjust a b == %(q,r). if Numeral0 <= r-b then (2*q + Numeral1, r-b)
   10.12 -                           else (2*q, r)"
   10.13 +  adjust :: "[int, int*int] => int*int"
   10.14 +    "adjust b == %(q,r). if 0 <= r-b then (2*q + 1, r-b)
   10.15 +                         else (2*q, r)"
   10.16  
   10.17  (** the division algorithm **)
   10.18  
   10.19  (*for the case a>=0, b>0*)
   10.20  consts posDivAlg :: "int*int => int*int"
   10.21 -recdef posDivAlg "inv_image less_than (%(a,b). nat(a - b + Numeral1))"
   10.22 +recdef posDivAlg "inv_image less_than (%(a,b). nat(a - b + 1))"
   10.23      "posDivAlg (a,b) =
   10.24 -       (if (a<b | b<=Numeral0) then (Numeral0,a)
   10.25 -        else adjust a b (posDivAlg(a, 2*b)))"
   10.26 +       (if (a<b | b<=0) then (0,a)
   10.27 +        else adjust b (posDivAlg(a, 2*b)))"
   10.28  
   10.29  (*for the case a<0, b>0*)
   10.30  consts negDivAlg :: "int*int => int*int"
   10.31  recdef negDivAlg "inv_image less_than (%(a,b). nat(- a - b))"
   10.32      "negDivAlg (a,b) =
   10.33 -       (if (Numeral0<=a+b | b<=Numeral0) then (-1,a+b)
   10.34 -        else adjust a b (negDivAlg(a, 2*b)))"
   10.35 +       (if (0<=a+b | b<=0) then (-1,a+b)
   10.36 +        else adjust b (negDivAlg(a, 2*b)))"
   10.37  
   10.38  (*for the general case b~=0*)
   10.39  
   10.40 @@ -44,12 +44,12 @@
   10.41      including the special case a=0, b<0, because negDivAlg requires a<0*)
   10.42    divAlg :: "int*int => int*int"
   10.43      "divAlg ==
   10.44 -       %(a,b). if Numeral0<=a then
   10.45 -                  if Numeral0<=b then posDivAlg (a,b)
   10.46 -                  else if a=Numeral0 then (Numeral0,Numeral0)
   10.47 +       %(a,b). if 0<=a then
   10.48 +                  if 0<=b then posDivAlg (a,b)
   10.49 +                  else if a=0 then (0,0)
   10.50                         else negateSnd (negDivAlg (-a,-b))
   10.51                 else 
   10.52 -                  if Numeral0<b then negDivAlg (a,b)
   10.53 +                  if 0<b then negDivAlg (a,b)
   10.54                    else         negateSnd (posDivAlg (-a,-b))"
   10.55  
   10.56  instance
    11.1 --- a/src/HOL/Integ/IntPower.ML	Mon Oct 22 11:01:30 2001 +0200
    11.2 +++ b/src/HOL/Integ/IntPower.ML	Mon Oct 22 11:54:22 2001 +0200
    11.3 @@ -15,7 +15,7 @@
    11.4  by (rtac (zmod_zmult_distrib RS sym) 1);
    11.5  qed "zpower_zmod";
    11.6  
    11.7 -Goal "Numeral1^y = (Numeral1::int)";
    11.8 +Goal "1^y = (1::int)";
    11.9  by (induct_tac "y" 1);
   11.10  by Auto_tac;
   11.11  qed "zpower_1";
    12.1 --- a/src/HOL/Integ/IntPower.thy	Mon Oct 22 11:01:30 2001 +0200
    12.2 +++ b/src/HOL/Integ/IntPower.thy	Mon Oct 22 11:54:22 2001 +0200
    12.3 @@ -12,7 +12,7 @@
    12.4    int :: {power}
    12.5  
    12.6  primrec
    12.7 -  power_0   "p ^ 0 = Numeral1"
    12.8 +  power_0   "p ^ 0 = 1"
    12.9    power_Suc "p ^ (Suc n) = (p::int) * (p ^ n)"
   12.10  
   12.11  end
    13.1 --- a/src/HOL/Integ/NatSimprocs.ML	Mon Oct 22 11:01:30 2001 +0200
    13.2 +++ b/src/HOL/Integ/NatSimprocs.ML	Mon Oct 22 11:54:22 2001 +0200
    13.3 @@ -6,10 +6,11 @@
    13.4  Simprocs for nat numerals (see also nat_simprocs.ML).
    13.5  *)
    13.6  
    13.7 -(** For simplifying  Suc m - # n **)
    13.8 +(** For simplifying  Suc m - #n **)
    13.9  
   13.10  Goal "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)";
   13.11 -by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1);
   13.12 +by (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]
   13.13 +			    addsplits [nat_diff_split]) 1);
   13.14  qed "Suc_diff_eq_diff_pred";
   13.15  
   13.16  (*Now just instantiating n to (number_of v) does the right simplification,
   13.17 @@ -48,8 +49,8 @@
   13.18  by (stac add_eq_if 1);
   13.19  by (asm_simp_tac
   13.20      (simpset() addsplits [nat.split]
   13.21 -               addsimps [Let_def, neg_imp_number_of_eq_0,
   13.22 -                         neg_number_of_bin_pred_iff_0]) 1);
   13.23 +               addsimps [numeral_1_eq_Suc_0 RS sym, Let_def, 
   13.24 +                   neg_imp_number_of_eq_0, neg_number_of_bin_pred_iff_0]) 1);
   13.25  qed "nat_case_add_eq_if";
   13.26  
   13.27  Addsimps [nat_case_number_of, nat_case_add_eq_if];
   13.28 @@ -71,8 +72,8 @@
   13.29  by (stac add_eq_if 1);
   13.30  by (asm_simp_tac
   13.31      (simpset() addsplits [nat.split]
   13.32 -               addsimps [Let_def, neg_imp_number_of_eq_0,
   13.33 -                         neg_number_of_bin_pred_iff_0]) 1);
   13.34 +               addsimps [numeral_1_eq_Suc_0 RS sym, Let_def, 
   13.35 +                  neg_imp_number_of_eq_0, neg_number_of_bin_pred_iff_0]) 1);
   13.36  qed "nat_rec_add_eq_if";
   13.37  
   13.38  Addsimps [nat_rec_number_of, nat_rec_add_eq_if];
   13.39 @@ -80,7 +81,7 @@
   13.40  
   13.41  (** For simplifying  # m - Suc n **)
   13.42  
   13.43 -Goal "m - Suc n = (m - Numeral1) - n";
   13.44 +Goal "m - Suc n = (m - 1) - n";
   13.45  by (simp_tac (numeral_ss addsplits [nat_diff_split]) 1);
   13.46  qed "diff_Suc_eq_diff_pred";
   13.47  
   13.48 @@ -93,7 +94,7 @@
   13.49  (** Evens and Odds, for Mutilated Chess Board **)
   13.50  
   13.51  (*Case analysis on b<2*)
   13.52 -Goal "(n::nat) < 2 ==> n = Numeral0 | n = Numeral1";
   13.53 +Goal "(n::nat) < 2 ==> n = 0 | n = Suc 0";
   13.54  by (arith_tac 1);
   13.55  qed "less_2_cases";
   13.56  
   13.57 @@ -101,16 +102,16 @@
   13.58  by (subgoal_tac "m mod 2 < 2" 1);
   13.59  by (Asm_simp_tac 2);
   13.60  be (less_2_cases RS disjE) 1;
   13.61 -by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc])));
   13.62 +by (ALLGOALS (asm_simp_tac (simpset() addsimps [Let_def, mod_Suc, nat_1])));
   13.63  qed "mod2_Suc_Suc";
   13.64  Addsimps [mod2_Suc_Suc];
   13.65  
   13.66 -Goal "!!m::nat. (0 < m mod 2) = (m mod 2 = Numeral1)";
   13.67 +Goal "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)";
   13.68  by (subgoal_tac "m mod 2 < 2" 1);
   13.69  by (Asm_simp_tac 2);
   13.70  by (auto_tac (claset(), simpset() delsimps [mod_less_divisor]));
   13.71  qed "mod2_gr_0";
   13.72 -Addsimps [mod2_gr_0, rename_numerals mod2_gr_0];
   13.73 +Addsimps [mod2_gr_0];
   13.74  
   13.75  (** Removal of small numerals: Numeral0, Numeral1 and (in additive positions) 2 **)
   13.76  
    14.1 --- a/src/HOL/Integ/int_arith1.ML	Mon Oct 22 11:01:30 2001 +0200
    14.2 +++ b/src/HOL/Integ/int_arith1.ML	Mon Oct 22 11:54:22 2001 +0200
    14.3 @@ -5,19 +5,21 @@
    14.4  Simprocs and decision procedure for linear arithmetic.
    14.5  *)
    14.6  
    14.7 +Addsimps [int_numeral_0_eq_0, int_numeral_1_eq_1];
    14.8 +
    14.9  (*** Simprocs for numeric literals ***)
   14.10  
   14.11  (** Combining of literal coefficients in sums of products **)
   14.12  
   14.13 -Goal "(x < y) = (x-y < (Numeral0::int))";
   14.14 +Goal "(x < y) = (x-y < (0::int))";
   14.15  by (simp_tac (simpset() addsimps zcompare_rls) 1);
   14.16  qed "zless_iff_zdiff_zless_0";
   14.17  
   14.18 -Goal "(x = y) = (x-y = (Numeral0::int))";
   14.19 +Goal "(x = y) = (x-y = (0::int))";
   14.20  by (simp_tac (simpset() addsimps zcompare_rls) 1);
   14.21  qed "eq_iff_zdiff_eq_0";
   14.22  
   14.23 -Goal "(x <= y) = (x-y <= (Numeral0::int))";
   14.24 +Goal "(x <= y) = (x-y <= (0::int))";
   14.25  by (simp_tac (simpset() addsimps zcompare_rls) 1);
   14.26  qed "zle_iff_zdiff_zle_0";
   14.27  
   14.28 @@ -77,12 +79,22 @@
   14.29  structure Int_Numeral_Simprocs =
   14.30  struct
   14.31  
   14.32 +(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
   14.33 +  isn't complicated by the abstract 0 and 1.*)
   14.34 +val numeral_syms = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym];
   14.35 +val numeral_sym_ss = HOL_ss addsimps numeral_syms;
   14.36 +
   14.37 +fun rename_numerals th = 
   14.38 +    simplify numeral_sym_ss (Thm.transfer (the_context ()) th);
   14.39 +
   14.40  (*Utilities*)
   14.41  
   14.42  fun mk_numeral n = HOLogic.number_of_const HOLogic.intT $ HOLogic.mk_bin n;
   14.43  
   14.44  (*Decodes a binary INTEGER*)
   14.45 -fun dest_numeral (Const("Numeral.number_of", _) $ w) = 
   14.46 +fun dest_numeral (Const("0", _)) = 0
   14.47 +  | dest_numeral (Const("1", _)) = 1
   14.48 +  | dest_numeral (Const("Numeral.number_of", _) $ w) = 
   14.49       (HOLogic.dest_binum w
   14.50        handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
   14.51    | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
   14.52 @@ -97,7 +109,7 @@
   14.53  
   14.54  val uminus_const = Const ("uminus", HOLogic.intT --> HOLogic.intT);
   14.55  
   14.56 -(*Thus mk_sum[t] yields t+Numeral0; longer sums don't have a trailing zero*)
   14.57 +(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   14.58  fun mk_sum []        = zero
   14.59    | mk_sum [t,u]     = mk_plus (t, u)
   14.60    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   14.61 @@ -157,16 +169,20 @@
   14.62  	handle TERM _ => find_first_coeff (t::past) u terms;
   14.63  
   14.64  
   14.65 -(*Simplify Numeral1*n and n*Numeral1 to n*)
   14.66 -val add_0s = [zadd_0, zadd_0_right];
   14.67 -val mult_1s = [zmult_1, zmult_1_right, zmult_minus1, zmult_minus1_right];
   14.68 +(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
   14.69 +val add_0s =  map rename_numerals [zadd_0, zadd_0_right];
   14.70 +val mult_1s = map rename_numerals [zmult_1, zmult_1_right] @
   14.71 +              [zmult_minus1, zmult_minus1_right];
   14.72  
   14.73 -(*To perform binary arithmetic*)
   14.74 -val bin_simps = [add_number_of_left] @ bin_arith_simps @ bin_rel_simps;
   14.75 +(*To perform binary arithmetic.  The "left" rewriting handles patterns
   14.76 +  created by the simprocs, such as 3 * (5 * x). *)
   14.77 +val bin_simps = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym,
   14.78 +                 add_number_of_left, mult_number_of_left] @ 
   14.79 +                bin_arith_simps @ bin_rel_simps;
   14.80  
   14.81  (*To evaluate binary negations of coefficients*)
   14.82  val zminus_simps = NCons_simps @
   14.83 -                   [number_of_minus RS sym, 
   14.84 +                   [zminus_1_eq_m1, number_of_minus RS sym, 
   14.85  		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
   14.86  		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
   14.87  
   14.88 @@ -189,28 +205,10 @@
   14.89  fun trans_tac None      = all_tac
   14.90    | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
   14.91  
   14.92 -fun prove_conv name tacs sg (hyps: thm list) (t,u) =
   14.93 -  if t aconv u then None
   14.94 -  else
   14.95 -  let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))
   14.96 -  in Some
   14.97 -     (prove_goalw_cterm [] ct (K tacs)
   14.98 -      handle ERROR => error 
   14.99 -	  ("The error(s) above occurred while trying to prove " ^
  14.100 -	   string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))
  14.101 -  end;
  14.102 -
  14.103 -(*version without the hyps argument*)
  14.104 -fun prove_conv_nohyps name tacs sg = prove_conv name tacs sg [];
  14.105 -
  14.106  fun simplify_meta_eq rules =
  14.107      mk_meta_eq o
  14.108      simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
  14.109  
  14.110 -fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
  14.111 -fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context())) (s, HOLogic.termT);
  14.112 -val prep_pats = map prep_pat;
  14.113 -
  14.114  structure CancelNumeralsCommon =
  14.115    struct
  14.116    val mk_sum    	= mk_sum
  14.117 @@ -220,8 +218,8 @@
  14.118    val find_first_coeff	= find_first_coeff []
  14.119    val trans_tac         = trans_tac
  14.120    val norm_tac = 
  14.121 -     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
  14.122 -                                         zminus_simps@zadd_ac))
  14.123 +     ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
  14.124 +                                         diff_simps@zminus_simps@zadd_ac))
  14.125       THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps))
  14.126       THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@
  14.127                                                zadd_ac@zmult_ac))
  14.128 @@ -232,7 +230,7 @@
  14.129  
  14.130  structure EqCancelNumerals = CancelNumeralsFun
  14.131   (open CancelNumeralsCommon
  14.132 -  val prove_conv = prove_conv "inteq_cancel_numerals"
  14.133 +  val prove_conv = Bin_Simprocs.prove_conv "inteq_cancel_numerals"
  14.134    val mk_bal   = HOLogic.mk_eq
  14.135    val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
  14.136    val bal_add1 = eq_add_iff1 RS trans
  14.137 @@ -241,7 +239,7 @@
  14.138  
  14.139  structure LessCancelNumerals = CancelNumeralsFun
  14.140   (open CancelNumeralsCommon
  14.141 -  val prove_conv = prove_conv "intless_cancel_numerals"
  14.142 +  val prove_conv = Bin_Simprocs.prove_conv "intless_cancel_numerals"
  14.143    val mk_bal   = HOLogic.mk_binrel "op <"
  14.144    val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT
  14.145    val bal_add1 = less_add_iff1 RS trans
  14.146 @@ -250,7 +248,7 @@
  14.147  
  14.148  structure LeCancelNumerals = CancelNumeralsFun
  14.149   (open CancelNumeralsCommon
  14.150 -  val prove_conv = prove_conv "intle_cancel_numerals"
  14.151 +  val prove_conv = Bin_Simprocs.prove_conv "intle_cancel_numerals"
  14.152    val mk_bal   = HOLogic.mk_binrel "op <="
  14.153    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT
  14.154    val bal_add1 = le_add_iff1 RS trans
  14.155 @@ -258,19 +256,22 @@
  14.156  );
  14.157  
  14.158  val cancel_numerals = 
  14.159 -  map prep_simproc
  14.160 +  map Bin_Simprocs.prep_simproc
  14.161     [("inteq_cancel_numerals",
  14.162 -     prep_pats ["(l::int) + m = n", "(l::int) = m + n", 
  14.163 +     Bin_Simprocs.prep_pats
  14.164 +               ["(l::int) + m = n", "(l::int) = m + n", 
  14.165  		"(l::int) - m = n", "(l::int) = m - n", 
  14.166  		"(l::int) * m = n", "(l::int) = m * n"], 
  14.167       EqCancelNumerals.proc),
  14.168      ("intless_cancel_numerals", 
  14.169 -     prep_pats ["(l::int) + m < n", "(l::int) < m + n", 
  14.170 +     Bin_Simprocs.prep_pats
  14.171 +               ["(l::int) + m < n", "(l::int) < m + n", 
  14.172  		"(l::int) - m < n", "(l::int) < m - n", 
  14.173  		"(l::int) * m < n", "(l::int) < m * n"], 
  14.174       LessCancelNumerals.proc),
  14.175      ("intle_cancel_numerals", 
  14.176 -     prep_pats ["(l::int) + m <= n", "(l::int) <= m + n", 
  14.177 +     Bin_Simprocs.prep_pats
  14.178 +               ["(l::int) + m <= n", "(l::int) <= m + n", 
  14.179  		"(l::int) - m <= n", "(l::int) <= m - n", 
  14.180  		"(l::int) * m <= n", "(l::int) <= m * n"], 
  14.181       LeCancelNumerals.proc)];
  14.182 @@ -284,11 +285,11 @@
  14.183    val mk_coeff		= mk_coeff
  14.184    val dest_coeff	= dest_coeff 1
  14.185    val left_distrib	= left_zadd_zmult_distrib RS trans
  14.186 -  val prove_conv        = prove_conv_nohyps "int_combine_numerals"
  14.187 +  val prove_conv        = Bin_Simprocs.prove_conv_nohyps "int_combine_numerals"
  14.188    val trans_tac          = trans_tac
  14.189    val norm_tac = 
  14.190 -     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
  14.191 -                                         zminus_simps@zadd_ac))
  14.192 +     ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
  14.193 +                                         diff_simps@zminus_simps@zadd_ac))
  14.194       THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps))
  14.195       THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@
  14.196                                                zadd_ac@zmult_ac))
  14.197 @@ -299,9 +300,9 @@
  14.198  
  14.199  structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
  14.200    
  14.201 -val combine_numerals = 
  14.202 -    prep_simproc ("int_combine_numerals",
  14.203 -		  prep_pats ["(i::int) + j", "(i::int) - j"],
  14.204 +val combine_numerals = Bin_Simprocs.prep_simproc
  14.205 +                 ("int_combine_numerals",
  14.206 +		  Bin_Simprocs.prep_pats ["(i::int) + j", "(i::int) - j"],
  14.207  		  CombineNumerals.proc);
  14.208  
  14.209  end;
  14.210 @@ -316,7 +317,7 @@
  14.211  print_depth 22;
  14.212  set timing;
  14.213  set trace_simp;
  14.214 -fun test s = (Goal s; by (Simp_tac 1)); 
  14.215 +fun test s = (Goal s, by (Simp_tac 1)); 
  14.216  
  14.217  test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
  14.218  
  14.219 @@ -403,17 +404,20 @@
  14.220  local
  14.221  
  14.222  (* reduce contradictory <= to False *)
  14.223 -val add_rules = simp_thms @ bin_arith_simps @ bin_rel_simps @
  14.224 -                [zadd_0, zadd_0_right, zdiff_def,
  14.225 -		 zadd_zminus_inverse, zadd_zminus_inverse2, 
  14.226 -		 zmult_0, zmult_0_right, 
  14.227 -		 zmult_1, zmult_1_right, 
  14.228 -		 zmult_minus1, zmult_minus1_right,
  14.229 -		 zminus_zadd_distrib, zminus_zminus, zmult_assoc,
  14.230 -                 Zero_int_def, One_int_def, int_0, int_1, zadd_int RS sym, int_Suc];
  14.231 +val add_rules = 
  14.232 +    simp_thms @ bin_arith_simps @ bin_rel_simps @ 
  14.233 +    [int_numeral_0_eq_0, int_numeral_1_eq_1,
  14.234 +     zadd_0, zadd_0_right, zdiff_def,
  14.235 +     zadd_zminus_inverse, zadd_zminus_inverse2, 
  14.236 +     zmult_0, zmult_0_right, 
  14.237 +     zmult_1, zmult_1_right, 
  14.238 +     zmult_minus1, zmult_minus1_right,
  14.239 +     zminus_zadd_distrib, zminus_zminus, zmult_assoc,
  14.240 +     int_0, int_1, zadd_int RS sym, int_Suc];
  14.241  
  14.242  val simprocs = [Int_Times_Assoc.conv, Int_Numeral_Simprocs.combine_numerals]@
  14.243 -               Int_Numeral_Simprocs.cancel_numerals;
  14.244 +               Int_Numeral_Simprocs.cancel_numerals @ 
  14.245 +               Bin_Simprocs.eval_numerals;
  14.246  
  14.247  val add_mono_thms_int =
  14.248    map (fn s => prove_goal (the_context ()) s
  14.249 @@ -457,7 +461,7 @@
  14.250  by (fast_arith_tac 1);
  14.251  Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";
  14.252  by (fast_arith_tac 1);
  14.253 -Goal "!!a::int. [| a < b; c < d |] ==> a+c+ Numeral1 < b+d";
  14.254 +Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";
  14.255  by (fast_arith_tac 1);
  14.256  Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
  14.257  by (fast_arith_tac 1);
    15.1 --- a/src/HOL/Integ/int_arith2.ML	Mon Oct 22 11:01:30 2001 +0200
    15.2 +++ b/src/HOL/Integ/int_arith2.ML	Mon Oct 22 11:54:22 2001 +0200
    15.3 @@ -3,19 +3,17 @@
    15.4      Authors:    Larry Paulson and Tobias Nipkow
    15.5  *)
    15.6  
    15.7 -(** Simplification of inequalities involving numerical constants **)
    15.8 -
    15.9 -Goal "(w <= z - (Numeral1::int)) = (w<(z::int))";
   15.10 +Goal "(w <= z - (1::int)) = (w<(z::int))";
   15.11  by (arith_tac 1);
   15.12  qed "zle_diff1_eq";
   15.13  Addsimps [zle_diff1_eq];
   15.14  
   15.15 -Goal "(w < z + Numeral1) = (w<=(z::int))";
   15.16 +Goal "(w < z + 1) = (w<=(z::int))";
   15.17  by (arith_tac 1);
   15.18  qed "zle_add1_eq_le";
   15.19  Addsimps [zle_add1_eq_le];
   15.20  
   15.21 -Goal "(z = z + w) = (w = (Numeral0::int))";
   15.22 +Goal "(z = z + w) = (w = (0::int))";
   15.23  by (arith_tac 1);
   15.24  qed "zadd_left_cancel0";
   15.25  Addsimps [zadd_left_cancel0];
   15.26 @@ -23,13 +21,13 @@
   15.27  
   15.28  (* nat *)
   15.29  
   15.30 -Goal "Numeral0 <= z ==> int (nat z) = z"; 
   15.31 +Goal "0 <= z ==> int (nat z) = z"; 
   15.32  by (asm_full_simp_tac
   15.33      (simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat]) 1); 
   15.34  qed "nat_0_le"; 
   15.35  
   15.36 -Goal "z <= Numeral0 ==> nat z = 0"; 
   15.37 -by (case_tac "z = Numeral0" 1);
   15.38 +Goal "z <= 0 ==> nat z = 0"; 
   15.39 +by (case_tac "z = 0" 1);
   15.40  by (asm_simp_tac (simpset() addsimps [nat_le_int0]) 1); 
   15.41  by (asm_full_simp_tac 
   15.42      (simpset() addsimps [neg_eq_less_0, neg_nat, linorder_neq_iff]) 1);
   15.43 @@ -37,19 +35,19 @@
   15.44  
   15.45  Addsimps [nat_0_le, nat_le_0];
   15.46  
   15.47 -val [major,minor] = Goal "[| Numeral0 <= z;  !!m. z = int m ==> P |] ==> P"; 
   15.48 +val [major,minor] = Goal "[| 0 <= z;  !!m. z = int m ==> P |] ==> P"; 
   15.49  by (rtac (major RS nat_0_le RS sym RS minor) 1);
   15.50  qed "nonneg_eq_int"; 
   15.51  
   15.52 -Goal "(nat w = m) = (if Numeral0 <= w then w = int m else m=0)";
   15.53 +Goal "(nat w = m) = (if 0 <= w then w = int m else m=0)";
   15.54  by Auto_tac;
   15.55  qed "nat_eq_iff";
   15.56  
   15.57 -Goal "(m = nat w) = (if Numeral0 <= w then w = int m else m=0)";
   15.58 +Goal "(m = nat w) = (if 0 <= w then w = int m else m=0)";
   15.59  by Auto_tac;
   15.60  qed "nat_eq_iff2";
   15.61  
   15.62 -Goal "Numeral0 <= w ==> (nat w < m) = (w < int m)";
   15.63 +Goal "0 <= w ==> (nat w < m) = (w < int m)";
   15.64  by (rtac iffI 1);
   15.65  by (asm_full_simp_tac 
   15.66      (simpset() delsimps [zless_int] addsimps [zless_int RS sym]) 2);
   15.67 @@ -57,48 +55,41 @@
   15.68  by (Simp_tac 1);
   15.69  qed "nat_less_iff";
   15.70  
   15.71 -Goal "(int m = z) = (m = nat z & Numeral0 <= z)";
   15.72 +Goal "(int m = z) = (m = nat z & 0 <= z)";
   15.73  by (auto_tac (claset(), simpset() addsimps [nat_eq_iff2]));  
   15.74  qed "int_eq_iff";
   15.75  
   15.76 -Addsimps [inst "z" "number_of ?v" int_eq_iff];
   15.77 -
   15.78  
   15.79  (*Users don't want to see (int 0), int(Suc 0) or w + - z*)
   15.80 -Addsimps [int_0, int_Suc, symmetric zdiff_def];
   15.81 +Addsimps (map symmetric [Zero_int_def, One_int_def, zdiff_def]);
   15.82  
   15.83 -Goal "nat Numeral0 = 0";
   15.84 +Goal "nat 0 = 0";
   15.85  by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   15.86  qed "nat_0";
   15.87  
   15.88 -Goal "nat Numeral1 = Suc 0";
   15.89 -by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   15.90 +Goal "nat 1 = Suc 0";
   15.91 +by (stac nat_eq_iff 1);
   15.92 +by (Simp_tac 1);
   15.93  qed "nat_1";
   15.94  
   15.95  Goal "nat 2 = Suc (Suc 0)";
   15.96 -by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   15.97 +by (stac nat_eq_iff 1);
   15.98 +by (Simp_tac 1);
   15.99  qed "nat_2";
  15.100  
  15.101 -Goal "Numeral0 <= w ==> (nat w < nat z) = (w<z)";
  15.102 +Goal "0 <= w ==> (nat w < nat z) = (w<z)";
  15.103  by (case_tac "neg z" 1);
  15.104  by (auto_tac (claset(), simpset() addsimps [nat_less_iff]));
  15.105  by (auto_tac (claset() addIs [zless_trans], 
  15.106  	      simpset() addsimps [neg_eq_less_0, zle_def]));
  15.107  qed "nat_less_eq_zless";
  15.108  
  15.109 -Goal "Numeral0 < w | Numeral0 <= z ==> (nat w <= nat z) = (w<=z)";
  15.110 +Goal "0 < w | 0 <= z ==> (nat w <= nat z) = (w<=z)";
  15.111  by (auto_tac (claset(), 
  15.112  	      simpset() addsimps [linorder_not_less RS sym, 
  15.113  				  zless_nat_conj]));
  15.114  qed "nat_le_eq_zle";
  15.115  
  15.116 -(*Analogous to zadd_int, but more easily provable using the arithmetic in Bin*)
  15.117 -Goal "n<=m --> int m - int n = int (m-n)";
  15.118 -by (induct_thm_tac diff_induct "m n" 1);
  15.119 -by Auto_tac;
  15.120 -qed_spec_mp "zdiff_int";
  15.121 -
  15.122 -
  15.123  (*** abs: absolute value, as an integer ****)
  15.124  
  15.125  (* Simpler: use zabs_def as rewrite rule;
  15.126 @@ -106,13 +97,17 @@
  15.127  *)
  15.128  
  15.129  Goalw [zabs_def]
  15.130 - "P(abs(i::int)) = ((Numeral0 <= i --> P i) & (i < Numeral0 --> P(-i)))";
  15.131 + "P(abs(i::int)) = ((0 <= i --> P i) & (i < 0 --> P(-i)))";
  15.132  by(Simp_tac 1);
  15.133  qed "zabs_split";
  15.134  
  15.135 -Goal "Numeral0 <= abs (z::int)";
  15.136 +Goal "0 <= abs (z::int)";
  15.137  by (simp_tac (simpset() addsimps [zabs_def]) 1); 
  15.138  qed "zero_le_zabs";
  15.139  AddIffs [zero_le_zabs];
  15.140  
  15.141 +
  15.142 +(*Not sure why this simprule is required!*)
  15.143 +Addsimps [inst "z" "number_of ?v" int_eq_iff];
  15.144 +
  15.145  (*continued in IntArith.ML ...*)
    16.1 --- a/src/HOL/Integ/int_factor_simprocs.ML	Mon Oct 22 11:01:30 2001 +0200
    16.2 +++ b/src/HOL/Integ/int_factor_simprocs.ML	Mon Oct 22 11:54:22 2001 +0200
    16.3 @@ -10,27 +10,27 @@
    16.4  
    16.5  (** Factor cancellation theorems for "int" **)
    16.6  
    16.7 -Goal "!!k::int. (k*m <= k*n) = ((Numeral0 < k --> m<=n) & (k < Numeral0 --> n<=m))";
    16.8 +Goal "!!k::int. (k*m <= k*n) = ((0 < k --> m<=n) & (k < 0 --> n<=m))";
    16.9  by (stac zmult_zle_cancel1 1);
   16.10  by Auto_tac;  
   16.11  qed "int_mult_le_cancel1";
   16.12  
   16.13 -Goal "!!k::int. (k*m < k*n) = ((Numeral0 < k & m<n) | (k < Numeral0 & n<m))";
   16.14 +Goal "!!k::int. (k*m < k*n) = ((0 < k & m<n) | (k < 0 & n<m))";
   16.15  by (stac zmult_zless_cancel1 1);
   16.16  by Auto_tac;  
   16.17  qed "int_mult_less_cancel1";
   16.18  
   16.19 -Goal "!!k::int. (k*m = k*n) = (k = Numeral0 | m=n)";
   16.20 +Goal "!!k::int. (k*m = k*n) = (k = 0 | m=n)";
   16.21  by Auto_tac;  
   16.22  qed "int_mult_eq_cancel1";
   16.23  
   16.24 -Goal "!!k::int. k~=Numeral0 ==> (k*m) div (k*n) = (m div n)";
   16.25 +Goal "!!k::int. k~=0 ==> (k*m) div (k*n) = (m div n)";
   16.26  by (stac zdiv_zmult_zmult1 1); 
   16.27  by Auto_tac;  
   16.28  qed "int_mult_div_cancel1";
   16.29  
   16.30  (*For ExtractCommonTermFun, cancelling common factors*)
   16.31 -Goal "(k*m) div (k*n) = (if k = (Numeral0::int) then Numeral0 else m div n)";
   16.32 +Goal "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)";
   16.33  by (simp_tac (simpset() addsimps [int_mult_div_cancel1]) 1); 
   16.34  qed "int_mult_div_cancel_disj";
   16.35  
   16.36 @@ -54,7 +54,7 @@
   16.37  
   16.38  structure DivCancelNumeralFactor = CancelNumeralFactorFun
   16.39   (open CancelNumeralFactorCommon
   16.40 -  val prove_conv = prove_conv "intdiv_cancel_numeral_factor"
   16.41 +  val prove_conv = Bin_Simprocs.prove_conv "intdiv_cancel_numeral_factor"
   16.42    val mk_bal   = HOLogic.mk_binop "Divides.op div"
   16.43    val dest_bal = HOLogic.dest_bin "Divides.op div" HOLogic.intT
   16.44    val cancel = int_mult_div_cancel1 RS trans
   16.45 @@ -63,7 +63,7 @@
   16.46  
   16.47  structure EqCancelNumeralFactor = CancelNumeralFactorFun
   16.48   (open CancelNumeralFactorCommon
   16.49 -  val prove_conv = prove_conv "inteq_cancel_numeral_factor"
   16.50 +  val prove_conv = Bin_Simprocs.prove_conv "inteq_cancel_numeral_factor"
   16.51    val mk_bal   = HOLogic.mk_eq
   16.52    val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
   16.53    val cancel = int_mult_eq_cancel1 RS trans
   16.54 @@ -72,7 +72,7 @@
   16.55  
   16.56  structure LessCancelNumeralFactor = CancelNumeralFactorFun
   16.57   (open CancelNumeralFactorCommon
   16.58 -  val prove_conv = prove_conv "intless_cancel_numeral_factor"
   16.59 +  val prove_conv = Bin_Simprocs.prove_conv "intless_cancel_numeral_factor"
   16.60    val mk_bal   = HOLogic.mk_binrel "op <"
   16.61    val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT
   16.62    val cancel = int_mult_less_cancel1 RS trans
   16.63 @@ -81,7 +81,7 @@
   16.64  
   16.65  structure LeCancelNumeralFactor = CancelNumeralFactorFun
   16.66   (open CancelNumeralFactorCommon
   16.67 -  val prove_conv = prove_conv "intle_cancel_numeral_factor"
   16.68 +  val prove_conv = Bin_Simprocs.prove_conv "intle_cancel_numeral_factor"
   16.69    val mk_bal   = HOLogic.mk_binrel "op <="
   16.70    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT
   16.71    val cancel = int_mult_le_cancel1 RS trans
   16.72 @@ -89,18 +89,18 @@
   16.73  )
   16.74  
   16.75  val int_cancel_numeral_factors = 
   16.76 -  map prep_simproc
   16.77 +  map Bin_Simprocs.prep_simproc
   16.78     [("inteq_cancel_numeral_factors",
   16.79 -     prep_pats ["(l::int) * m = n", "(l::int) = m * n"], 
   16.80 +     Bin_Simprocs.prep_pats ["(l::int) * m = n", "(l::int) = m * n"], 
   16.81       EqCancelNumeralFactor.proc),
   16.82      ("intless_cancel_numeral_factors", 
   16.83 -     prep_pats ["(l::int) * m < n", "(l::int) < m * n"], 
   16.84 +     Bin_Simprocs.prep_pats ["(l::int) * m < n", "(l::int) < m * n"], 
   16.85       LessCancelNumeralFactor.proc),
   16.86      ("intle_cancel_numeral_factors", 
   16.87 -     prep_pats ["(l::int) * m <= n", "(l::int) <= m * n"], 
   16.88 +     Bin_Simprocs.prep_pats ["(l::int) * m <= n", "(l::int) <= m * n"], 
   16.89       LeCancelNumeralFactor.proc),
   16.90      ("intdiv_cancel_numeral_factors", 
   16.91 -     prep_pats ["((l::int) * m) div n", "(l::int) div (m * n)"], 
   16.92 +     Bin_Simprocs.prep_pats ["((l::int) * m) div n", "(l::int) div (m * n)"], 
   16.93       DivCancelNumeralFactor.proc)];
   16.94  
   16.95  end;
   16.96 @@ -181,7 +181,7 @@
   16.97  
   16.98  structure EqCancelFactor = ExtractCommonTermFun
   16.99   (open CancelFactorCommon
  16.100 -  val prove_conv = prove_conv "int_eq_cancel_factor"
  16.101 +  val prove_conv = Bin_Simprocs.prove_conv "int_eq_cancel_factor"
  16.102    val mk_bal   = HOLogic.mk_eq
  16.103    val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
  16.104    val simplify_meta_eq  = cancel_simplify_meta_eq int_mult_eq_cancel1
  16.105 @@ -189,19 +189,19 @@
  16.106  
  16.107  structure DivideCancelFactor = ExtractCommonTermFun
  16.108   (open CancelFactorCommon
  16.109 -  val prove_conv = prove_conv "int_divide_cancel_factor"
  16.110 +  val prove_conv = Bin_Simprocs.prove_conv "int_divide_cancel_factor"
  16.111    val mk_bal   = HOLogic.mk_binop "Divides.op div"
  16.112    val dest_bal = HOLogic.dest_bin "Divides.op div" HOLogic.intT
  16.113    val simplify_meta_eq  = cancel_simplify_meta_eq int_mult_div_cancel_disj
  16.114  );
  16.115  
  16.116  val int_cancel_factor = 
  16.117 -  map prep_simproc
  16.118 +  map Bin_Simprocs.prep_simproc
  16.119     [("int_eq_cancel_factor",
  16.120 -     prep_pats ["(l::int) * m = n", "(l::int) = m * n"], 
  16.121 +     Bin_Simprocs.prep_pats ["(l::int) * m = n", "(l::int) = m * n"], 
  16.122       EqCancelFactor.proc),
  16.123      ("int_divide_cancel_factor", 
  16.124 -     prep_pats ["((l::int) * m) div n", "(l::int) div (m * n)"], 
  16.125 +     Bin_Simprocs.prep_pats ["((l::int) * m) div n", "(l::int) div (m * n)"], 
  16.126       DivideCancelFactor.proc)];
  16.127  
  16.128  end;
    17.1 --- a/src/HOL/Integ/nat_bin.ML	Mon Oct 22 11:01:30 2001 +0200
    17.2 +++ b/src/HOL/Integ/nat_bin.ML	Mon Oct 22 11:54:22 2001 +0200
    17.3 @@ -13,29 +13,29 @@
    17.4  Goal "nat (number_of w) = number_of w";
    17.5  by (simp_tac (simpset() addsimps [nat_number_of_def]) 1);
    17.6  qed "nat_number_of";
    17.7 -Addsimps [nat_number_of];
    17.8 -
    17.9 -(*These rewrites should one day be re-oriented...*)
   17.10 +Addsimps [nat_number_of, nat_0, nat_1];
   17.11  
   17.12  Goal "Numeral0 = (0::nat)";
   17.13 -by (simp_tac (HOL_basic_ss addsimps [nat_0, nat_number_of_def]) 1);
   17.14 +by (simp_tac (simpset() addsimps [nat_number_of_def]) 1); 
   17.15  qed "numeral_0_eq_0";
   17.16  
   17.17  Goal "Numeral1 = (1::nat)";
   17.18 -by (simp_tac (HOL_basic_ss addsimps [nat_1, nat_number_of_def, One_nat_def]) 1);
   17.19 +by (simp_tac (simpset() addsimps [nat_1, nat_number_of_def]) 1); 
   17.20  qed "numeral_1_eq_1";
   17.21  
   17.22 -Goal "2 = Suc 1";
   17.23 -by (simp_tac (HOL_basic_ss addsimps [nat_2, nat_number_of_def, One_nat_def]) 1);
   17.24 +Goal "Numeral1 = Suc 0";
   17.25 +by (simp_tac (simpset() addsimps [numeral_1_eq_1]) 1); 
   17.26 +qed "numeral_1_eq_Suc_0";
   17.27 +
   17.28 +Goalw [nat_number_of_def, One_nat_def] "2 = Suc 1";
   17.29 +by (rtac nat_2 1); 
   17.30  qed "numeral_2_eq_2";
   17.31  
   17.32 -bind_thm ("zero_eq_numeral_0", numeral_0_eq_0 RS sym);
   17.33 -
   17.34  (** int (coercion from nat to int) **)
   17.35  
   17.36  (*"neg" is used in rewrite rules for binary comparisons*)
   17.37  Goal "int (number_of v :: nat) = \
   17.38 -\        (if neg (number_of v) then Numeral0 \
   17.39 +\        (if neg (number_of v) then 0 \
   17.40  \         else (number_of v :: int))";
   17.41  by (simp_tac
   17.42      (simpset_of Int.thy addsimps [neg_nat, nat_number_of_def, 
   17.43 @@ -45,45 +45,39 @@
   17.44  
   17.45  
   17.46  val nat_bin_arith_setup =
   17.47 - [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
   17.48 -   {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
   17.49 -    inj_thms = inj_thms,
   17.50 -    lessD = lessD,
   17.51 -    simpset = simpset addsimps [int_nat_number_of,
   17.52 - not_neg_number_of_Pls,neg_number_of_Min,neg_number_of_BIT]})];
   17.53 + [Fast_Arith.map_data 
   17.54 +   (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
   17.55 +     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
   17.56 +      inj_thms = inj_thms,
   17.57 +      lessD = lessD,
   17.58 +      simpset = simpset addsimps [int_nat_number_of, not_neg_number_of_Pls,
   17.59 +                                  neg_number_of_Min,neg_number_of_BIT]})];
   17.60  
   17.61  (** Successor **)
   17.62  
   17.63 -Goal "(Numeral0::int) <= z ==> Suc (nat z) = nat (Numeral1 + z)";
   17.64 +Goal "(0::int) <= z ==> Suc (nat z) = nat (1 + z)";
   17.65  by (rtac sym 1);
   17.66 -by (asm_simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   17.67 +by (asm_simp_tac (simpset() addsimps [nat_eq_iff, int_Suc]) 1);
   17.68  qed "Suc_nat_eq_nat_zadd1";
   17.69  
   17.70 -Goal "Suc (number_of v) = \
   17.71 -\       (if neg (number_of v) then Numeral1 else number_of (bin_succ v))";
   17.72 +Goal "Suc (number_of v + n) = \
   17.73 +\       (if neg (number_of v) then 1+n else number_of (bin_succ v) + n)";
   17.74  by (simp_tac
   17.75      (simpset_of Int.thy addsimps [neg_nat, nat_1, not_neg_eq_ge_0, 
   17.76  				  nat_number_of_def, int_Suc, 
   17.77  				  Suc_nat_eq_nat_zadd1, number_of_succ]) 1);
   17.78 +qed "Suc_nat_number_of_add";
   17.79 +
   17.80 +Goal "Suc (number_of v) = \
   17.81 +\       (if neg (number_of v) then 1 else number_of (bin_succ v))";
   17.82 +by (cut_inst_tac [("n","0")] Suc_nat_number_of_add 1);
   17.83 +by (asm_full_simp_tac (simpset() delcongs [if_weak_cong]) 1); 
   17.84  qed "Suc_nat_number_of";
   17.85  Addsimps [Suc_nat_number_of];
   17.86  
   17.87 -Goal "Suc (number_of v + n) = \
   17.88 -\       (if neg (number_of v) then Numeral1+n else number_of (bin_succ v) + n)";
   17.89 -by (Simp_tac 1);
   17.90 -qed "Suc_nat_number_of_add";
   17.91 -
   17.92 -Goal "Suc Numeral0 = Numeral1";
   17.93 -by (Simp_tac 1);
   17.94 -qed "Suc_numeral_0_eq_1";
   17.95 -
   17.96 -Goal "Suc Numeral1 = 2";
   17.97 -by (Simp_tac 1);
   17.98 -qed "Suc_numeral_1_eq_2";
   17.99 -
  17.100  (** Addition **)
  17.101  
  17.102 -Goal "[| (Numeral0::int) <= z;  Numeral0 <= z' |] ==> nat (z+z') = nat z + nat z'";
  17.103 +Goal "[| (0::int) <= z;  0 <= z' |] ==> nat (z+z') = nat z + nat z'";
  17.104  by (rtac (inj_int RS injD) 1);
  17.105  by (asm_simp_tac (simpset() addsimps [zadd_int RS sym]) 1);
  17.106  qed "nat_add_distrib";
  17.107 @@ -103,7 +97,7 @@
  17.108  
  17.109  (** Subtraction **)
  17.110  
  17.111 -Goal "[| (Numeral0::int) <= z';  z' <= z |] ==> nat (z-z') = nat z - nat z'";
  17.112 +Goal "[| (0::int) <= z';  z' <= z |] ==> nat (z-z') = nat z - nat z'";
  17.113  by (rtac (inj_int RS injD) 1);
  17.114  by (asm_simp_tac (simpset() addsimps [zdiff_int RS sym, nat_le_eq_zle]) 1);
  17.115  qed "nat_diff_distrib";
  17.116 @@ -122,7 +116,7 @@
  17.117       "(number_of v :: nat) - number_of v' = \
  17.118  \       (if neg (number_of v') then number_of v \
  17.119  \        else let d = number_of (bin_add v (bin_minus v')) in    \
  17.120 -\             if neg d then Numeral0 else nat d)";
  17.121 +\             if neg d then 0 else nat d)";
  17.122  by (simp_tac
  17.123      (simpset_of Int.thy delcongs [if_weak_cong]
  17.124  			addsimps [not_neg_eq_ge_0, nat_0,
  17.125 @@ -134,22 +128,22 @@
  17.126  
  17.127  (** Multiplication **)
  17.128  
  17.129 -Goal "(Numeral0::int) <= z ==> nat (z*z') = nat z * nat z'";
  17.130 -by (case_tac "Numeral0 <= z'" 1);
  17.131 +Goal "(0::int) <= z ==> nat (z*z') = nat z * nat z'";
  17.132 +by (case_tac "0 <= z'" 1);
  17.133  by (asm_full_simp_tac (simpset() addsimps [zmult_le_0_iff]) 2);
  17.134  by (rtac (inj_int RS injD) 1);
  17.135  by (asm_simp_tac (simpset() addsimps [zmult_int RS sym,
  17.136  				      int_0_le_mult_iff]) 1);
  17.137  qed "nat_mult_distrib";
  17.138  
  17.139 -Goal "z <= (Numeral0::int) ==> nat(z*z') = nat(-z) * nat(-z')"; 
  17.140 +Goal "z <= (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"; 
  17.141  by (rtac trans 1); 
  17.142  by (rtac nat_mult_distrib 2); 
  17.143  by Auto_tac;  
  17.144  qed "nat_mult_distrib_neg";
  17.145  
  17.146  Goal "(number_of v :: nat) * number_of v' = \
  17.147 -\      (if neg (number_of v) then Numeral0 else number_of (bin_mult v v'))";
  17.148 +\      (if neg (number_of v) then 0 else number_of (bin_mult v v'))";
  17.149  by (simp_tac
  17.150      (simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, 
  17.151  				  nat_mult_distrib RS sym, number_of_mult, 
  17.152 @@ -161,15 +155,15 @@
  17.153  
  17.154  (** Quotient **)
  17.155  
  17.156 -Goal "(Numeral0::int) <= z ==> nat (z div z') = nat z div nat z'";
  17.157 -by (case_tac "Numeral0 <= z'" 1);
  17.158 +Goal "(0::int) <= z ==> nat (z div z') = nat z div nat z'";
  17.159 +by (case_tac "0 <= z'" 1);
  17.160  by (auto_tac (claset(), 
  17.161  	      simpset() addsimps [div_nonneg_neg_le0, DIVISION_BY_ZERO_DIV]));
  17.162 -by (zdiv_undefined_case_tac "z' = Numeral0" 1);
  17.163 +by (zdiv_undefined_case_tac "z' = 0" 1);
  17.164   by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_DIV]) 1);
  17.165  by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
  17.166  by (rename_tac "m m'" 1);
  17.167 -by (subgoal_tac "Numeral0 <= int m div int m'" 1);
  17.168 +by (subgoal_tac "0 <= int m div int m'" 1);
  17.169   by (asm_full_simp_tac 
  17.170       (simpset() addsimps [numeral_0_eq_0, pos_imp_zdiv_nonneg_iff]) 2);
  17.171  by (rtac (inj_int RS injD) 1);
  17.172 @@ -184,7 +178,7 @@
  17.173  qed "nat_div_distrib";
  17.174  
  17.175  Goal "(number_of v :: nat)  div  number_of v' = \
  17.176 -\         (if neg (number_of v) then Numeral0 \
  17.177 +\         (if neg (number_of v) then 0 \
  17.178  \          else nat (number_of v div number_of v'))";
  17.179  by (simp_tac
  17.180      (simpset_of Int.thy addsimps [not_neg_eq_ge_0, nat_number_of_def, neg_nat, 
  17.181 @@ -197,12 +191,12 @@
  17.182  (** Remainder **)
  17.183  
  17.184  (*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
  17.185 -Goal "[| (Numeral0::int) <= z;  Numeral0 <= z' |] ==> nat (z mod z') = nat z mod nat z'";
  17.186 -by (zdiv_undefined_case_tac "z' = Numeral0" 1);
  17.187 +Goal "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'";
  17.188 +by (zdiv_undefined_case_tac "z' = 0" 1);
  17.189   by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_MOD]) 1);
  17.190  by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
  17.191  by (rename_tac "m m'" 1);
  17.192 -by (subgoal_tac "Numeral0 <= int m mod int m'" 1);
  17.193 +by (subgoal_tac "0 <= int m mod int m'" 1);
  17.194   by (asm_full_simp_tac 
  17.195       (simpset() addsimps [nat_less_iff, numeral_0_eq_0, pos_mod_sign]) 2);
  17.196  by (rtac (inj_int RS injD) 1);
  17.197 @@ -217,7 +211,7 @@
  17.198  qed "nat_mod_distrib";
  17.199  
  17.200  Goal "(number_of v :: nat)  mod  number_of v' = \
  17.201 -\       (if neg (number_of v) then Numeral0 \
  17.202 +\       (if neg (number_of v) then 0 \
  17.203  \        else if neg (number_of v') then number_of v \
  17.204  \        else nat (number_of v mod number_of v'))";
  17.205  by (simp_tac
  17.206 @@ -228,12 +222,36 @@
  17.207  
  17.208  Addsimps [mod_nat_number_of];
  17.209  
  17.210 +structure NatAbstractNumeralsData =
  17.211 +  struct
  17.212 +  val dest_eq		= HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
  17.213 +  val is_numeral	= Bin_Simprocs.is_numeral
  17.214 +  val numeral_0_eq_0    = numeral_0_eq_0
  17.215 +  val numeral_1_eq_1    = numeral_1_eq_Suc_0
  17.216 +  val prove_conv   = Bin_Simprocs.prove_conv_nohyps "nat_abstract_numerals"
  17.217 +  fun norm_tac simps	= ALLGOALS (simp_tac (HOL_ss addsimps simps))
  17.218 +  val simplify_meta_eq  = Bin_Simprocs.simplify_meta_eq 
  17.219 +  end
  17.220 +
  17.221 +structure NatAbstractNumerals = AbstractNumeralsFun (NatAbstractNumeralsData)
  17.222 +
  17.223 +val nat_eval_numerals = 
  17.224 +  map Bin_Simprocs.prep_simproc
  17.225 +   [("nat_div_eval_numerals",
  17.226 +     Bin_Simprocs.prep_pats ["(Suc 0) div m"],
  17.227 +     NatAbstractNumerals.proc div_nat_number_of),
  17.228 +    ("nat_mod_eval_numerals",
  17.229 +     Bin_Simprocs.prep_pats ["(Suc 0) mod m"],
  17.230 +     NatAbstractNumerals.proc mod_nat_number_of)];
  17.231 +
  17.232 +Addsimprocs nat_eval_numerals;
  17.233 +
  17.234  
  17.235  (*** Comparisons ***)
  17.236  
  17.237  (** Equals (=) **)
  17.238  
  17.239 -Goal "[| (Numeral0::int) <= z;  Numeral0 <= z' |] ==> (nat z = nat z') = (z=z')";
  17.240 +Goal "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')";
  17.241  by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
  17.242  qed "eq_nat_nat_iff";
  17.243  
  17.244 @@ -262,7 +280,7 @@
  17.245      (simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, 
  17.246  				  nat_less_eq_zless, less_number_of_eq_neg,
  17.247  				  nat_0]) 1);
  17.248 -by (simp_tac (simpset_of Int.thy addsimps [neg_eq_less_int0, zminus_zless, 
  17.249 +by (simp_tac (simpset_of Int.thy addsimps [neg_eq_less_0, zminus_zless, 
  17.250  				number_of_minus, zless_nat_eq_int_zless]) 1);
  17.251  qed "less_nat_number_of";
  17.252  
  17.253 @@ -278,24 +296,14 @@
  17.254  
  17.255  Addsimps [le_nat_number_of_eq_not_less];
  17.256  
  17.257 -(*** New versions of existing theorems involving 0, 1, 2 ***)
  17.258  
  17.259 -(*Maps n to # n for n = 0, 1, 2*)
  17.260 -val numeral_sym_ss =
  17.261 -    HOL_ss addsimps [numeral_0_eq_0 RS sym,
  17.262 -		     numeral_1_eq_1 RS sym,
  17.263 -		     numeral_2_eq_2 RS sym,
  17.264 -		     Suc_numeral_1_eq_2, Suc_numeral_0_eq_1];
  17.265 -
  17.266 -fun rename_numerals th = simplify numeral_sym_ss (Thm.transfer (the_context ()) th);
  17.267 -
  17.268 -(*Maps # n to n for n = 0, 1, 2*)
  17.269 +(*Maps #n to n for n = 0, 1, 2*)
  17.270  bind_thms ("numerals", [numeral_0_eq_0, numeral_1_eq_1, numeral_2_eq_2]);
  17.271  val numeral_ss = simpset() addsimps numerals;
  17.272  
  17.273  (** Nat **)
  17.274  
  17.275 -Goal "Numeral0 < n ==> n = Suc(n - Numeral1)";
  17.276 +Goal "0 < n ==> n = Suc(n - 1)";
  17.277  by (asm_full_simp_tac numeral_ss 1);
  17.278  qed "Suc_pred'";
  17.279  
  17.280 @@ -303,111 +311,48 @@
  17.281    NOT suitable for rewriting because n recurs in the condition.*)
  17.282  bind_thm ("expand_Suc", inst "n" "number_of ?v" Suc_pred');
  17.283  
  17.284 -(** NatDef & Nat **)
  17.285 -
  17.286 -Addsimps (map rename_numerals [min_0L, min_0R, max_0L, max_0R]);
  17.287 -
  17.288 -AddIffs (map rename_numerals
  17.289 -	 [Suc_not_Zero, Zero_not_Suc, zero_less_Suc, not_less0, less_one, 
  17.290 -	  le0, le_0_eq, neq0_conv, not_gr0]);
  17.291 -
  17.292  (** Arith **)
  17.293  
  17.294 -(*Identity laws for + - * *)	 
  17.295 -val basic_renamed_arith_simps =
  17.296 -    map rename_numerals
  17.297 -        [diff_0, diff_0_eq_0, add_0, add_0_right, 
  17.298 -	 mult_0, mult_0_right, mult_1, mult_1_right];
  17.299 -	 
  17.300 -(*Non-trivial simplifications*)	 
  17.301 -val other_renamed_arith_simps =
  17.302 -    map rename_numerals
  17.303 -	[diff_is_0_eq, zero_less_diff,
  17.304 -	 mult_is_0, zero_less_mult_iff, mult_eq_1_iff];
  17.305 -
  17.306 -Addsimps (basic_renamed_arith_simps @ other_renamed_arith_simps);
  17.307 -
  17.308 -AddIffs (map rename_numerals [add_is_0, add_gr_0]);
  17.309 -
  17.310 -Goal "Suc n = n + Numeral1";
  17.311 +Goal "Suc n = n + 1";
  17.312  by (asm_simp_tac numeral_ss 1);
  17.313  qed "Suc_eq_add_numeral_1";
  17.314  
  17.315  (* These two can be useful when m = number_of... *)
  17.316  
  17.317 -Goal "(m::nat) + n = (if m=Numeral0 then n else Suc ((m - Numeral1) + n))";
  17.318 +Goal "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))";
  17.319  by (case_tac "m" 1);
  17.320  by (ALLGOALS (asm_simp_tac numeral_ss));
  17.321  qed "add_eq_if";
  17.322  
  17.323 -Goal "(m::nat) * n = (if m=Numeral0 then Numeral0 else n + ((m - Numeral1) * n))";
  17.324 +Goal "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))";
  17.325  by (case_tac "m" 1);
  17.326  by (ALLGOALS (asm_simp_tac numeral_ss));
  17.327  qed "mult_eq_if";
  17.328  
  17.329 -Goal "(p ^ m :: nat) = (if m=Numeral0 then Numeral1 else p * (p ^ (m - Numeral1)))";
  17.330 +Goal "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))";
  17.331  by (case_tac "m" 1);
  17.332  by (ALLGOALS (asm_simp_tac numeral_ss));
  17.333  qed "power_eq_if";
  17.334  
  17.335 -Goal "[| Numeral0<n; Numeral0<m |] ==> m - n < (m::nat)";
  17.336 +Goal "[| 0<n; 0<m |] ==> m - n < (m::nat)";
  17.337  by (asm_full_simp_tac (numeral_ss addsimps [diff_less]) 1);
  17.338  qed "diff_less'";
  17.339  
  17.340  Addsimps [inst "n" "number_of ?v" diff_less'];
  17.341  
  17.342 -(*various theorems that aren't in the default simpset*)
  17.343 -bind_thm ("add_is_one'", rename_numerals add_is_1);
  17.344 -bind_thm ("zero_induct'", rename_numerals zero_induct);
  17.345 -bind_thm ("diff_self_eq_0'", rename_numerals diff_self_eq_0);
  17.346 -bind_thm ("mult_eq_self_implies_10'", rename_numerals mult_eq_self_implies_10);
  17.347 -
  17.348 -(** Divides **)
  17.349 -
  17.350 -Addsimps (map rename_numerals [mod_1, mod_0, div_1, div_0]);
  17.351 -AddIffs (map rename_numerals [dvd_1_left, dvd_0_right]);
  17.352 -
  17.353 -(*useful?*)
  17.354 -bind_thm ("mod_self'", rename_numerals mod_self);
  17.355 -bind_thm ("div_self'", rename_numerals div_self);
  17.356 -bind_thm ("div_less'", rename_numerals div_less);
  17.357 -bind_thm ("mod_mult_self_is_zero'", rename_numerals mod_mult_self_is_0);
  17.358 -
  17.359  (** Power **)
  17.360  
  17.361 -Goal "(p::nat) ^ Numeral0 = Numeral1";
  17.362 -by (simp_tac numeral_ss 1);
  17.363 -qed "power_zero";
  17.364 -
  17.365 -Goal "(p::nat) ^ Numeral1 = p";
  17.366 -by (simp_tac numeral_ss 1);
  17.367 -qed "power_one";
  17.368 -Addsimps [power_zero, power_one];
  17.369 -
  17.370  Goal "(p::nat) ^ 2 = p*p";
  17.371  by (simp_tac numeral_ss 1);
  17.372  qed "power_two";
  17.373  
  17.374 -Goal "Numeral0 < (i::nat) ==> Numeral0 < i^n";
  17.375 -by (asm_simp_tac numeral_ss 1);
  17.376 -qed "zero_less_power'";
  17.377 -Addsimps [zero_less_power'];
  17.378 -
  17.379 -bind_thm ("binomial_zero", rename_numerals binomial_0);
  17.380 -bind_thm ("binomial_Suc'", rename_numerals binomial_Suc);
  17.381 -bind_thm ("binomial_n_n'", rename_numerals binomial_n_n);
  17.382 -
  17.383 -(*binomial_0_Suc doesn't work well on numerals*)
  17.384 -Addsimps (map rename_numerals [binomial_n_0, binomial_zero, binomial_1]);
  17.385 -
  17.386 -Addsimps [rename_numerals card_Pow];
  17.387  
  17.388  (*** Comparisons involving (0::nat) ***)
  17.389  
  17.390  Goal "(number_of v = (0::nat)) = \
  17.391  \     (if neg (number_of v) then True else iszero (number_of v))";
  17.392 -by (simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym]) 1);
  17.393 -qed "eq_number_of_0";
  17.394 +by (simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym, iszero_0]) 1);
  17.395 +qed "eq_number_of_0"; 
  17.396  
  17.397  Goal "((0::nat) = number_of v) = \
  17.398  \     (if neg (number_of v) then True else iszero (number_of v))";
  17.399 @@ -422,7 +367,7 @@
  17.400  Addsimps [eq_number_of_0, eq_0_number_of, less_0_number_of];
  17.401  
  17.402  Goal "neg (number_of v) ==> number_of v = (0::nat)";
  17.403 -by (asm_simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym]) 1);
  17.404 +by (asm_simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym, iszero_0]) 1);
  17.405  qed "neg_imp_number_of_eq_0";
  17.406  
  17.407  
  17.408 @@ -489,13 +434,13 @@
  17.409  	  le_number_of_Suc, le_Suc_number_of];
  17.410  
  17.411  (* Push int(.) inwards: *)
  17.412 -Addsimps [int_Suc,zadd_int RS sym];
  17.413 +Addsimps [zadd_int RS sym];
  17.414  
  17.415  Goal "(m+m = n+n) = (m = (n::int))";
  17.416  by Auto_tac;
  17.417  val lemma1 = result();
  17.418  
  17.419 -Goal "m+m ~= int (Suc 0) + n + n";
  17.420 +Goal "m+m ~= (1::int) + n + n";
  17.421  by Auto_tac;
  17.422  by (dres_inst_tac [("f", "%x. x mod 2")] arg_cong 1);
  17.423  by (full_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1); 
  17.424 @@ -536,7 +481,7 @@
  17.425  (*** Further lemmas about "nat" ***)
  17.426  
  17.427  Goal "nat (abs (w * z)) = nat (abs w) * nat (abs z)";
  17.428 -by (case_tac "z=Numeral0 | w=Numeral0" 1);
  17.429 +by (case_tac "z=0 | w=0" 1);
  17.430  by Auto_tac;  
  17.431  by (simp_tac (simpset() addsimps [zabs_def, nat_mult_distrib RS sym, 
  17.432                            nat_mult_distrib_neg RS sym, zmult_less_0_iff]) 1);
    18.1 --- a/src/HOL/Integ/nat_simprocs.ML	Mon Oct 22 11:01:30 2001 +0200
    18.2 +++ b/src/HOL/Integ/nat_simprocs.ML	Mon Oct 22 11:54:22 2001 +0200
    18.3 @@ -66,19 +66,19 @@
    18.4  
    18.5  (** For cancel_numeral_factors **)
    18.6  
    18.7 -Goal "(Numeral0::nat) < k ==> (k*m <= k*n) = (m<=n)";
    18.8 +Goal "(0::nat) < k ==> (k*m <= k*n) = (m<=n)";
    18.9  by Auto_tac;  
   18.10  qed "nat_mult_le_cancel1";
   18.11  
   18.12 -Goal "(Numeral0::nat) < k ==> (k*m < k*n) = (m<n)";
   18.13 +Goal "(0::nat) < k ==> (k*m < k*n) = (m<n)";
   18.14  by Auto_tac;  
   18.15  qed "nat_mult_less_cancel1";
   18.16  
   18.17 -Goal "(Numeral0::nat) < k ==> (k*m = k*n) = (m=n)";
   18.18 +Goal "(0::nat) < k ==> (k*m = k*n) = (m=n)";
   18.19  by Auto_tac;  
   18.20  qed "nat_mult_eq_cancel1";
   18.21  
   18.22 -Goal "(Numeral0::nat) < k ==> (k*m) div (k*n) = (m div n)";
   18.23 +Goal "(0::nat) < k ==> (k*m) div (k*n) = (m div n)";
   18.24  by Auto_tac;  
   18.25  qed "nat_mult_div_cancel1";
   18.26  
   18.27 @@ -105,6 +105,14 @@
   18.28  structure Nat_Numeral_Simprocs =
   18.29  struct
   18.30  
   18.31 +(*Maps n to #n for n = 0, 1, 2*)
   18.32 +val numeral_syms =
   18.33 +       [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym, numeral_2_eq_2 RS sym];
   18.34 +val numeral_sym_ss = HOL_ss addsimps numeral_syms;
   18.35 +
   18.36 +fun rename_numerals th = 
   18.37 +    simplify numeral_sym_ss (Thm.transfer (the_context ()) th);
   18.38 +
   18.39  (*Utilities*)
   18.40  
   18.41  fun mk_numeral n = HOLogic.number_of_const HOLogic.natT $ HOLogic.mk_bin n;
   18.42 @@ -125,13 +133,13 @@
   18.43  val zero = mk_numeral 0;
   18.44  val mk_plus = HOLogic.mk_binop "op +";
   18.45  
   18.46 -(*Thus mk_sum[t] yields t+Numeral0; longer sums don't have a trailing zero*)
   18.47 +(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   18.48  fun mk_sum []        = zero
   18.49    | mk_sum [t,u]     = mk_plus (t, u)
   18.50    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   18.51  
   18.52  (*this version ALWAYS includes a trailing zero*)
   18.53 -fun long_mk_sum []        = zero
   18.54 +fun long_mk_sum []        = HOLogic.zero
   18.55    | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   18.56  
   18.57  val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT;
   18.58 @@ -153,9 +161,8 @@
   18.59  
   18.60  val trans_tac = Int_Numeral_Simprocs.trans_tac;
   18.61  
   18.62 -val prove_conv = Int_Numeral_Simprocs.prove_conv;
   18.63 -
   18.64 -val bin_simps = [add_nat_number_of, nat_number_of_add_left,
   18.65 +val bin_simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
   18.66 +                 add_nat_number_of, nat_number_of_add_left,
   18.67                   diff_nat_number_of, le_nat_number_of_eq_not_less,
   18.68                   less_nat_number_of, mult_nat_number_of, 
   18.69                   thm "Let_number_of", nat_number_of] @
   18.70 @@ -204,14 +211,14 @@
   18.71          handle TERM _ => find_first_coeff (t::past) u terms;
   18.72  
   18.73  
   18.74 -(*Simplify Numeral1*n and n*Numeral1 to n*)
   18.75 -val add_0s = map rename_numerals [add_0, add_0_right];
   18.76 +(*Simplify 1*n and n*1 to n*)
   18.77 +val add_0s  = map rename_numerals [add_0, add_0_right];
   18.78  val mult_1s = map rename_numerals [mult_1, mult_1_right];
   18.79  
   18.80  (*Final simplification: cancel + and *; replace Numeral0 by 0 and Numeral1 by 1*)
   18.81  val simplify_meta_eq =
   18.82      Int_Numeral_Simprocs.simplify_meta_eq
   18.83 -         [numeral_0_eq_0, numeral_1_eq_1, add_0, add_0_right,
   18.84 +         [numeral_0_eq_0, numeral_1_eq_Suc_0, add_0, add_0_right,
   18.85           mult_0, mult_0_right, mult_1, mult_1_right];
   18.86  
   18.87  
   18.88 @@ -243,19 +250,19 @@
   18.89    val find_first_coeff  = find_first_coeff []
   18.90    val trans_tac          = trans_tac
   18.91    val norm_tac = ALLGOALS
   18.92 -                   (simp_tac (HOL_ss addsimps add_0s@mult_1s@
   18.93 -                                       [add_0, Suc_eq_add_numeral_1]@add_ac))
   18.94 +                   (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
   18.95 +                                [Suc_eq_add_numeral_1] @ add_ac))
   18.96                   THEN ALLGOALS (simp_tac
   18.97                                  (HOL_ss addsimps bin_simps@add_ac@mult_ac))
   18.98    val numeral_simp_tac  = ALLGOALS
   18.99 -                (simp_tac (HOL_ss addsimps [numeral_0_eq_0 RS sym]@add_0s@bin_simps))
  18.100 +                (simp_tac (HOL_ss addsimps add_0s@bin_simps))
  18.101    val simplify_meta_eq  = simplify_meta_eq
  18.102    end;
  18.103  
  18.104  
  18.105  structure EqCancelNumerals = CancelNumeralsFun
  18.106   (open CancelNumeralsCommon
  18.107 -  val prove_conv = prove_conv "nateq_cancel_numerals"
  18.108 +  val prove_conv = Bin_Simprocs.prove_conv "nateq_cancel_numerals"
  18.109    val mk_bal   = HOLogic.mk_eq
  18.110    val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  18.111    val bal_add1 = nat_eq_add_iff1 RS trans
  18.112 @@ -264,7 +271,7 @@
  18.113  
  18.114  structure LessCancelNumerals = CancelNumeralsFun
  18.115   (open CancelNumeralsCommon
  18.116 -  val prove_conv = prove_conv "natless_cancel_numerals"
  18.117 +  val prove_conv = Bin_Simprocs.prove_conv "natless_cancel_numerals"
  18.118    val mk_bal   = HOLogic.mk_binrel "op <"
  18.119    val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT
  18.120    val bal_add1 = nat_less_add_iff1 RS trans
  18.121 @@ -273,7 +280,7 @@
  18.122  
  18.123  structure LeCancelNumerals = CancelNumeralsFun
  18.124   (open CancelNumeralsCommon
  18.125 -  val prove_conv = prove_conv "natle_cancel_numerals"
  18.126 +  val prove_conv = Bin_Simprocs.prove_conv "natle_cancel_numerals"
  18.127    val mk_bal   = HOLogic.mk_binrel "op <="
  18.128    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT
  18.129    val bal_add1 = nat_le_add_iff1 RS trans
  18.130 @@ -282,7 +289,7 @@
  18.131  
  18.132  structure DiffCancelNumerals = CancelNumeralsFun
  18.133   (open CancelNumeralsCommon
  18.134 -  val prove_conv = prove_conv "natdiff_cancel_numerals"
  18.135 +  val prove_conv = Bin_Simprocs.prove_conv "natdiff_cancel_numerals"
  18.136    val mk_bal   = HOLogic.mk_binop "op -"
  18.137    val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT
  18.138    val bal_add1 = nat_diff_add_eq1 RS trans
  18.139 @@ -324,16 +331,15 @@
  18.140    val mk_coeff          = mk_coeff
  18.141    val dest_coeff        = dest_coeff
  18.142    val left_distrib      = left_add_mult_distrib RS trans
  18.143 -  val prove_conv = 
  18.144 -       Int_Numeral_Simprocs.prove_conv_nohyps "nat_combine_numerals"
  18.145 +  val prove_conv	= Bin_Simprocs.prove_conv_nohyps "nat_combine_numerals"
  18.146    val trans_tac          = trans_tac
  18.147    val norm_tac = ALLGOALS
  18.148 -                   (simp_tac (HOL_ss addsimps add_0s@mult_1s@
  18.149 -                                       [add_0, Suc_eq_add_numeral_1]@add_ac))
  18.150 +                   (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
  18.151 +                              [Suc_eq_add_numeral_1] @ add_ac))
  18.152                   THEN ALLGOALS (simp_tac
  18.153                                  (HOL_ss addsimps bin_simps@add_ac@mult_ac))
  18.154    val numeral_simp_tac  = ALLGOALS
  18.155 -                (simp_tac (HOL_ss addsimps [numeral_0_eq_0 RS sym]@add_0s@bin_simps))
  18.156 +                (simp_tac (HOL_ss addsimps add_0s@bin_simps))
  18.157    val simplify_meta_eq  = simplify_meta_eq
  18.158    end;
  18.159  
  18.160 @@ -352,8 +358,8 @@
  18.161    val mk_coeff		= mk_coeff
  18.162    val dest_coeff	= dest_coeff
  18.163    val trans_tac         = trans_tac
  18.164 -  val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps 
  18.165 -                                             [Suc_eq_add_numeral_1]@mult_1s))
  18.166 +  val norm_tac = ALLGOALS 
  18.167 +                    (simp_tac (HOL_ss addsimps [Suc_eq_add_numeral_1]@mult_1s))
  18.168                   THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@mult_ac))
  18.169    val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps bin_simps))
  18.170    val simplify_meta_eq  = simplify_meta_eq
  18.171 @@ -361,7 +367,7 @@
  18.172  
  18.173  structure DivCancelNumeralFactor = CancelNumeralFactorFun
  18.174   (open CancelNumeralFactorCommon
  18.175 -  val prove_conv = prove_conv "natdiv_cancel_numeral_factor"
  18.176 +  val prove_conv = Bin_Simprocs.prove_conv "natdiv_cancel_numeral_factor"
  18.177    val mk_bal   = HOLogic.mk_binop "Divides.op div"
  18.178    val dest_bal = HOLogic.dest_bin "Divides.op div" HOLogic.natT
  18.179    val cancel = nat_mult_div_cancel1 RS trans
  18.180 @@ -370,7 +376,7 @@
  18.181  
  18.182  structure EqCancelNumeralFactor = CancelNumeralFactorFun
  18.183   (open CancelNumeralFactorCommon
  18.184 -  val prove_conv = prove_conv "nateq_cancel_numeral_factor"
  18.185 +  val prove_conv = Bin_Simprocs.prove_conv "nateq_cancel_numeral_factor"
  18.186    val mk_bal   = HOLogic.mk_eq
  18.187    val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  18.188    val cancel = nat_mult_eq_cancel1 RS trans
  18.189 @@ -379,7 +385,7 @@
  18.190  
  18.191  structure LessCancelNumeralFactor = CancelNumeralFactorFun
  18.192   (open CancelNumeralFactorCommon
  18.193 -  val prove_conv = prove_conv "natless_cancel_numeral_factor"
  18.194 +  val prove_conv = Bin_Simprocs.prove_conv "natless_cancel_numeral_factor"
  18.195    val mk_bal   = HOLogic.mk_binrel "op <"
  18.196    val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT
  18.197    val cancel = nat_mult_less_cancel1 RS trans
  18.198 @@ -388,7 +394,7 @@
  18.199  
  18.200  structure LeCancelNumeralFactor = CancelNumeralFactorFun
  18.201   (open CancelNumeralFactorCommon
  18.202 -  val prove_conv = prove_conv "natle_cancel_numeral_factor"
  18.203 +  val prove_conv = Bin_Simprocs.prove_conv "natle_cancel_numeral_factor"
  18.204    val mk_bal   = HOLogic.mk_binrel "op <="
  18.205    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT
  18.206    val cancel = nat_mult_le_cancel1 RS trans
  18.207 @@ -443,7 +449,7 @@
  18.208  
  18.209  structure EqCancelFactor = ExtractCommonTermFun
  18.210   (open CancelFactorCommon
  18.211 -  val prove_conv = prove_conv "nat_eq_cancel_factor"
  18.212 +  val prove_conv = Bin_Simprocs.prove_conv "nat_eq_cancel_factor"
  18.213    val mk_bal   = HOLogic.mk_eq
  18.214    val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  18.215    val simplify_meta_eq  = cancel_simplify_meta_eq nat_mult_eq_cancel_disj
  18.216 @@ -451,7 +457,7 @@
  18.217  
  18.218  structure LessCancelFactor = ExtractCommonTermFun
  18.219   (open CancelFactorCommon
  18.220 -  val prove_conv = prove_conv "nat_less_cancel_factor"
  18.221 +  val prove_conv = Bin_Simprocs.prove_conv "nat_less_cancel_factor"
  18.222    val mk_bal   = HOLogic.mk_binrel "op <"
  18.223    val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT
  18.224    val simplify_meta_eq  = cancel_simplify_meta_eq nat_mult_less_cancel_disj
  18.225 @@ -459,7 +465,7 @@
  18.226  
  18.227  structure LeCancelFactor = ExtractCommonTermFun
  18.228   (open CancelFactorCommon
  18.229 -  val prove_conv = prove_conv "nat_leq_cancel_factor"
  18.230 +  val prove_conv = Bin_Simprocs.prove_conv "nat_leq_cancel_factor"
  18.231    val mk_bal   = HOLogic.mk_binrel "op <="
  18.232    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT
  18.233    val simplify_meta_eq  = cancel_simplify_meta_eq nat_mult_le_cancel_disj
  18.234 @@ -467,7 +473,7 @@
  18.235  
  18.236  structure DivideCancelFactor = ExtractCommonTermFun
  18.237   (open CancelFactorCommon
  18.238 -  val prove_conv = prove_conv "nat_divide_cancel_factor"
  18.239 +  val prove_conv = Bin_Simprocs.prove_conv "nat_divide_cancel_factor"
  18.240    val mk_bal   = HOLogic.mk_binop "Divides.op div"
  18.241    val dest_bal = HOLogic.dest_bin "Divides.op div" HOLogic.natT
  18.242    val simplify_meta_eq  = cancel_simplify_meta_eq nat_mult_div_cancel_disj
  18.243 @@ -514,7 +520,7 @@
  18.244  test "Suc u - 2 = y";
  18.245  test "Suc (Suc (Suc u)) - 2 = y";
  18.246  test "(i + j + 2 + (k::nat)) - 1 = y";
  18.247 -test "(i + j + Numeral1 + (k::nat)) - 2 = y";
  18.248 +test "(i + j + 1 + (k::nat)) - 2 = y";
  18.249  
  18.250  test "(2*x + (u*v) + y) - v*3*u = (w::nat)";
  18.251  test "(2*x*u*v + 5 + (u*v)*4 + y) - v*u*4 = (w::nat)";
  18.252 @@ -590,7 +596,8 @@
  18.253  
  18.254  (* reduce contradictory <= to False *)
  18.255  val add_rules =
  18.256 -  [add_nat_number_of, diff_nat_number_of, mult_nat_number_of,
  18.257 +  [thm "Let_number_of", thm "Let_0", thm "Let_1", nat_0, nat_1, 
  18.258 +   add_nat_number_of, diff_nat_number_of, mult_nat_number_of,
  18.259     eq_nat_number_of, less_nat_number_of, le_nat_number_of_eq_not_less,
  18.260     le_Suc_number_of,le_number_of_Suc,
  18.261     less_Suc_number_of,less_number_of_Suc,
  18.262 @@ -610,7 +617,6 @@
  18.263     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
  18.264      inj_thms = inj_thms, lessD = lessD,
  18.265      simpset = simpset addsimps add_rules
  18.266 -                      addsimps basic_renamed_arith_simps
  18.267                        addsimprocs simprocs})];
  18.268  
  18.269  end;
    19.1 --- a/src/HOL/Library/Multiset.thy	Mon Oct 22 11:01:30 2001 +0200
    19.2 +++ b/src/HOL/Library/Multiset.thy	Mon Oct 22 11:54:22 2001 +0200
    19.3 @@ -274,18 +274,8 @@
    19.4    apply (simp (no_asm) add: expand_fun_eq)
    19.5    apply (rule conjI)
    19.6     apply force
    19.7 -  apply clarify
    19.8 -  apply (rule conjI)
    19.9 -   apply blast
   19.10 -  apply clarify
   19.11 -  apply (rule iffI)
   19.12 -   apply (rule conjI)
   19.13 -    apply clarify
   19.14 -    apply (rule conjI)
   19.15 -     apply (simp add: eq_sym_conv)   (* FIXME blast fails !? *)
   19.16 -    apply fast
   19.17 -   apply simp
   19.18 -  apply force
   19.19 +  apply safe
   19.20 +  apply (simp_all add: eq_sym_conv)
   19.21    done
   19.22  
   19.23  (*
    20.1 --- a/src/HOL/List.ML	Mon Oct 22 11:01:30 2001 +0200
    20.2 +++ b/src/HOL/List.ML	Mon Oct 22 11:54:22 2001 +0200
    20.3 @@ -232,23 +232,23 @@
    20.4  local
    20.5  
    20.6  val list_eq_pattern =
    20.7 -  Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT);
    20.8 +  Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)
    20.9  
   20.10  fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
   20.11        (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
   20.12    | last (Const("List.op @",_) $ _ $ ys) = last ys
   20.13 -  | last t = t;
   20.14 +  | last t = t
   20.15  
   20.16  fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
   20.17 -  | list1 _ = false;
   20.18 +  | list1 _ = false
   20.19  
   20.20  fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
   20.21        (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
   20.22    | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
   20.23 -  | butlast xs = Const("List.list.Nil",fastype_of xs);
   20.24 +  | butlast xs = Const("List.list.Nil",fastype_of xs)
   20.25  
   20.26  val rearr_tac =
   20.27 -  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]);
   20.28 +  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons])
   20.29  
   20.30  fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
   20.31    let
   20.32 @@ -272,9 +272,9 @@
   20.33       if lastl aconv lastr
   20.34       then rearr append_same_eq
   20.35       else None
   20.36 -  end;
   20.37 +  end
   20.38  in
   20.39 -val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq;
   20.40 +val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq
   20.41  end;
   20.42  
   20.43  Addsimprocs [list_eq_simproc];
   20.44 @@ -1501,24 +1501,19 @@
   20.45  Addsimps [sublist_upt_eq_take];
   20.46  
   20.47  
   20.48 -(*** Versions of some theorems above using binary numerals ***)
   20.49 -
   20.50 -AddIffs (map rename_numerals
   20.51 -	  [length_0_conv, length_greater_0_conv, sum_eq_0_conv]);
   20.52 -
   20.53 -Goal "take n (x#xs) = (if n = Numeral0 then [] else x # take (n - Numeral1) xs)";
   20.54 +Goal "take n (x#xs) = (if n=0 then [] else x # take (n - 1) xs)";
   20.55  by (case_tac "n" 1);
   20.56  by (ALLGOALS 
   20.57      (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
   20.58  qed "take_Cons'";
   20.59  
   20.60 -Goal "drop n (x#xs) = (if n = Numeral0 then x#xs else drop (n - Numeral1) xs)";
   20.61 +Goal "drop n (x#xs) = (if n=0 then x#xs else drop (n - 1) xs)";
   20.62  by (case_tac "n" 1);
   20.63  by (ALLGOALS
   20.64      (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
   20.65  qed "drop_Cons'";
   20.66  
   20.67 -Goal "(x#xs)!n = (if n = Numeral0 then x else xs!(n - Numeral1))";
   20.68 +Goal "(x#xs)!n = (if n=0 then x else xs!(n - 1))";
   20.69  by (case_tac "n" 1);
   20.70  by (ALLGOALS
   20.71      (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
    21.1 --- a/src/HOL/Nat.ML	Mon Oct 22 11:01:30 2001 +0200
    21.2 +++ b/src/HOL/Nat.ML	Mon Oct 22 11:54:22 2001 +0200
    21.3 @@ -243,9 +243,8 @@
    21.4  by (Auto_tac);
    21.5  qed "add_is_1";
    21.6  
    21.7 -Goal "(Suc 0 = m+n) = (m=Suc 0 & n=0 | m=0 & n= Suc 0)";
    21.8 -by (case_tac "m" 1);
    21.9 -by (Auto_tac);
   21.10 +Goal "(Suc 0 = m+n) = (m = Suc 0 & n=0 | m=0 & n = Suc 0)";
   21.11 +by (rtac ([eq_commute, add_is_1] MRS trans) 1);
   21.12  qed "one_is_add";
   21.13  
   21.14  Goal "!!m::nat. (0<m+n) = (0<m | 0<n)";
    22.1 --- a/src/HOL/NatDef.ML	Mon Oct 22 11:01:30 2001 +0200
    22.2 +++ b/src/HOL/NatDef.ML	Mon Oct 22 11:54:22 2001 +0200
    22.3 @@ -450,25 +450,15 @@
    22.4  bind_thms ("not_less_simps", [not_less_less_Suc_eq, leD RS not_less_less_Suc_eq]);
    22.5  
    22.6  
    22.7 -(** Re-orientation of the equations 0=x and Suc n = x. *
    22.8 -
    22.9 -  The condition "True" is a hack to prevent looping for e.g. Suc m = Suc n.
   22.10 -  Conditional rewrite rules are tried after unconditional ones.
   22.11 -**)
   22.12 +(** Re-orientation of the equations 0=x and 1=x. 
   22.13 +    No longer added as simprules (they loop) 
   22.14 +    but via reorient_simproc in Bin **)
   22.15  
   22.16  (*Polymorphic, not just for "nat"*)
   22.17 -Goal "True ==> (0 = x) = (x = 0)";
   22.18 +Goal "(0 = x) = (x = 0)";
   22.19  by Auto_tac;  
   22.20  qed "zero_reorient";
   22.21 -Addsimps [zero_reorient];
   22.22  
   22.23 -(*Polymorphic, not just for "nat"*)
   22.24 -Goal "True ==> (1 = x) = (x = 1)";
   22.25 +Goal "(1 = x) = (x = 1)";
   22.26  by Auto_tac;  
   22.27  qed "one_reorient";
   22.28 -Addsimps [one_reorient];
   22.29 -
   22.30 -Goal "True ==> (Suc (Suc 0) = x) = (x = Suc (Suc 0))";  (* FIXME !? *)
   22.31 -by Auto_tac;
   22.32 -qed "two_reorient";
   22.33 -Addsimps [two_reorient];
    23.1 --- a/src/HOL/NumberTheory/Chinese.thy	Mon Oct 22 11:01:30 2001 +0200
    23.2 +++ b/src/HOL/NumberTheory/Chinese.thy	Mon Oct 22 11:54:22 2001 +0200
    23.3 @@ -45,11 +45,11 @@
    23.4  defs
    23.5    m_cond_def:
    23.6      "m_cond n mf ==
    23.7 -      (\<forall>i. i \<le> n --> Numeral0 < mf i) \<and>
    23.8 -      (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i, mf j) = Numeral1)"
    23.9 +      (\<forall>i. i \<le> n --> 0 < mf i) \<and>
   23.10 +      (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i, mf j) = 1)"
   23.11  
   23.12    km_cond_def:
   23.13 -    "km_cond n kf mf == \<forall>i. i \<le> n --> zgcd (kf i, mf i) = Numeral1"
   23.14 +    "km_cond n kf mf == \<forall>i. i \<le> n --> zgcd (kf i, mf i) = 1"
   23.15  
   23.16    lincong_sol_def:
   23.17      "lincong_sol n kf bf mf x == \<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i)"
   23.18 @@ -63,8 +63,8 @@
   23.19    xilin_sol_def:
   23.20      "xilin_sol i n kf bf mf ==
   23.21        if 0 < n \<and> i \<le> n \<and> m_cond n mf \<and> km_cond n kf mf then
   23.22 -        (SOME x. Numeral0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i))
   23.23 -      else Numeral0"
   23.24 +        (SOME x. 0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i))
   23.25 +      else 0"
   23.26  
   23.27    x_sol_def:
   23.28      "x_sol n kf bf mf == funsum (\<lambda>i. xilin_sol i n kf bf mf * mhf mf n i) 0 n"
   23.29 @@ -72,15 +72,15 @@
   23.30  
   23.31  text {* \medskip @{term funprod} and @{term funsum} *}
   23.32  
   23.33 -lemma funprod_pos: "(\<forall>i. i \<le> n --> Numeral0 < mf i) ==> Numeral0 < funprod mf 0 n"
   23.34 +lemma funprod_pos: "(\<forall>i. i \<le> n --> 0 < mf i) ==> 0 < funprod mf 0 n"
   23.35    apply (induct n)
   23.36     apply auto
   23.37    apply (simp add: int_0_less_mult_iff)
   23.38    done
   23.39  
   23.40  lemma funprod_zgcd [rule_format (no_asm)]:
   23.41 -  "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i, mf m) = Numeral1) -->
   23.42 -    zgcd (funprod mf k l, mf m) = Numeral1"
   23.43 +  "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i, mf m) = 1) -->
   23.44 +    zgcd (funprod mf k l, mf m) = 1"
   23.45    apply (induct l)
   23.46     apply simp_all
   23.47    apply (rule impI)+
   23.48 @@ -110,14 +110,14 @@
   23.49    done
   23.50  
   23.51  lemma funsum_zero [rule_format (no_asm)]:
   23.52 -    "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = Numeral0) --> (funsum f k l) = Numeral0"
   23.53 +    "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = 0) --> (funsum f k l) = 0"
   23.54    apply (induct l)
   23.55     apply auto
   23.56    done
   23.57  
   23.58  lemma funsum_oneelem [rule_format (no_asm)]:
   23.59    "k \<le> j --> j \<le> k + l -->
   23.60 -    (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = Numeral0) -->
   23.61 +    (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = 0) -->
   23.62      funsum f k l = f j"
   23.63    apply (induct l)
   23.64     prefer 2
   23.65 @@ -127,9 +127,9 @@
   23.66     apply (subgoal_tac "k = j")
   23.67      apply (simp_all (no_asm_simp))
   23.68    apply (case_tac "Suc (k + n) = j")
   23.69 -   apply (subgoal_tac "funsum f k n = Numeral0")
   23.70 +   apply (subgoal_tac "funsum f k n = 0")
   23.71      apply (rule_tac [2] funsum_zero)
   23.72 -    apply (subgoal_tac [3] "f (Suc (k + n)) = Numeral0")
   23.73 +    apply (subgoal_tac [3] "f (Suc (k + n)) = 0")
   23.74       apply (subgoal_tac [3] "j \<le> k + n")
   23.75        prefer 4
   23.76        apply arith
   23.77 @@ -175,7 +175,7 @@
   23.78  
   23.79  lemma unique_xi_sol:
   23.80    "0 < n ==> i \<le> n ==> m_cond n mf ==> km_cond n kf mf
   23.81 -    ==> \<exists>!x. Numeral0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)"
   23.82 +    ==> \<exists>!x. 0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)"
   23.83    apply (rule zcong_lineq_unique)
   23.84     apply (tactic {* stac (thm "zgcd_zmult_cancel") 2 *})
   23.85      apply (unfold m_cond_def km_cond_def mhf_def)
   23.86 @@ -227,7 +227,7 @@
   23.87  
   23.88  lemma chinese_remainder:
   23.89    "0 < n ==> m_cond n mf ==> km_cond n kf mf
   23.90 -    ==> \<exists>!x. Numeral0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x"
   23.91 +    ==> \<exists>!x. 0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x"
   23.92    apply safe
   23.93     apply (rule_tac [2] m = "funprod mf 0 n" in zcong_zless_imp_eq)
   23.94         apply (rule_tac [6] zcong_funprod)
   23.95 @@ -242,7 +242,7 @@
   23.96          apply (tactic {* stac (thm "zmod_zmult_distrib" RS sym) 7 *})
   23.97          apply (tactic {* stac (thm "zcong_zmod" RS sym) 7 *})
   23.98          apply (subgoal_tac [7]
   23.99 -          "Numeral0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
  23.100 +          "0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
  23.101            \<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)")
  23.102           prefer 7
  23.103           apply (simp add: zmult_ac)
    24.1 --- a/src/HOL/NumberTheory/EulerFermat.thy	Mon Oct 22 11:01:30 2001 +0200
    24.2 +++ b/src/HOL/NumberTheory/EulerFermat.thy	Mon Oct 22 11:54:22 2001 +0200
    24.3 @@ -29,33 +29,33 @@
    24.4  inductive "RsetR m"
    24.5    intros
    24.6      empty [simp]: "{} \<in> RsetR m"
    24.7 -    insert: "A \<in> RsetR m ==> zgcd (a, m) = Numeral1 ==>
    24.8 +    insert: "A \<in> RsetR m ==> zgcd (a, m) = 1 ==>
    24.9        \<forall>a'. a' \<in> A --> \<not> zcong a a' m ==> insert a A \<in> RsetR m"
   24.10  
   24.11  recdef BnorRset
   24.12    "measure ((\<lambda>(a, m). nat a) :: int * int => nat)"
   24.13    "BnorRset (a, m) =
   24.14 -   (if Numeral0 < a then
   24.15 -    let na = BnorRset (a - Numeral1, m)
   24.16 -    in (if zgcd (a, m) = Numeral1 then insert a na else na)
   24.17 +   (if 0 < a then
   24.18 +    let na = BnorRset (a - 1, m)
   24.19 +    in (if zgcd (a, m) = 1 then insert a na else na)
   24.20      else {})"
   24.21  
   24.22  defs
   24.23 -  norRRset_def: "norRRset m == BnorRset (m - Numeral1, m)"
   24.24 +  norRRset_def: "norRRset m == BnorRset (m - 1, m)"
   24.25    noXRRset_def: "noXRRset m x == (\<lambda>a. a * x) ` norRRset m"
   24.26    phi_def: "phi m == card (norRRset m)"
   24.27    is_RRset_def: "is_RRset A m == A \<in> RsetR m \<and> card A = phi m"
   24.28    RRset2norRR_def:
   24.29      "RRset2norRR A m a ==
   24.30 -     (if Numeral1 < m \<and> is_RRset A m \<and> a \<in> A then
   24.31 +     (if 1 < m \<and> is_RRset A m \<and> a \<in> A then
   24.32          SOME b. zcong a b m \<and> b \<in> norRRset m
   24.33 -      else Numeral0)"
   24.34 +      else 0)"
   24.35  
   24.36  constdefs
   24.37    zcongm :: "int => int => int => bool"
   24.38    "zcongm m == \<lambda>a b. zcong a b m"
   24.39  
   24.40 -lemma abs_eq_1_iff [iff]: "(abs z = (Numeral1::int)) = (z = Numeral1 \<or> z = -1)"
   24.41 +lemma abs_eq_1_iff [iff]: "(abs z = (1::int)) = (z = 1 \<or> z = -1)"
   24.42    -- {* LCP: not sure why this lemma is needed now *}
   24.43    apply (auto simp add: zabs_def)
   24.44    done
   24.45 @@ -67,7 +67,7 @@
   24.46  
   24.47  lemma BnorRset_induct:
   24.48    "(!!a m. P {} a m) ==>
   24.49 -    (!!a m. Numeral0 < (a::int) ==> P (BnorRset (a - Numeral1, m::int)) (a - Numeral1) m
   24.50 +    (!!a m. 0 < (a::int) ==> P (BnorRset (a - 1, m::int)) (a - 1) m
   24.51        ==> P (BnorRset(a,m)) a m)
   24.52      ==> P (BnorRset(u,v)) u v"
   24.53  proof -
   24.54 @@ -75,7 +75,7 @@
   24.55    show ?thesis
   24.56      apply (rule BnorRset.induct)
   24.57      apply safe
   24.58 -     apply (case_tac [2] "Numeral0 < a")
   24.59 +     apply (case_tac [2] "0 < a")
   24.60        apply (rule_tac [2] rule_context)
   24.61         apply simp_all
   24.62       apply (simp_all add: BnorRset.simps rule_context)
   24.63 @@ -94,7 +94,7 @@
   24.64    apply (auto dest: Bnor_mem_zle)
   24.65    done
   24.66  
   24.67 -lemma Bnor_mem_zg [rule_format]: "b \<in> BnorRset (a, m) --> Numeral0 < b"
   24.68 +lemma Bnor_mem_zg [rule_format]: "b \<in> BnorRset (a, m) --> 0 < b"
   24.69    apply (induct a m rule: BnorRset_induct)
   24.70     prefer 2
   24.71     apply (subst BnorRset.simps)
   24.72 @@ -103,7 +103,7 @@
   24.73    done
   24.74  
   24.75  lemma Bnor_mem_if [rule_format]:
   24.76 -    "zgcd (b, m) = Numeral1 --> Numeral0 < b --> b \<le> a --> b \<in> BnorRset (a, m)"
   24.77 +    "zgcd (b, m) = 1 --> 0 < b --> b \<le> a --> b \<in> BnorRset (a, m)"
   24.78    apply (induct a m rule: BnorRset.induct)
   24.79    apply auto
   24.80     apply (case_tac "a = b")
   24.81 @@ -128,7 +128,7 @@
   24.82      apply (rule_tac [3] allI)
   24.83      apply (rule_tac [3] impI)
   24.84      apply (rule_tac [3] zcong_not)
   24.85 -       apply (subgoal_tac [6] "a' \<le> a - Numeral1")
   24.86 +       apply (subgoal_tac [6] "a' \<le> a - 1")
   24.87          apply (rule_tac [7] Bnor_mem_zle)
   24.88          apply (rule_tac [5] Bnor_mem_zg)
   24.89          apply auto
   24.90 @@ -142,13 +142,13 @@
   24.91     apply auto
   24.92    done
   24.93  
   24.94 -lemma aux: "a \<le> b - Numeral1 ==> a < (b::int)"
   24.95 +lemma aux: "a \<le> b - 1 ==> a < (b::int)"
   24.96    apply auto
   24.97    done
   24.98  
   24.99  lemma norR_mem_unique:
  24.100 -  "Numeral1 < m ==>
  24.101 -    zgcd (a, m) = Numeral1 ==> \<exists>!b. [a = b] (mod m) \<and> b \<in> norRRset m"
  24.102 +  "1 < m ==>
  24.103 +    zgcd (a, m) = 1 ==> \<exists>!b. [a = b] (mod m) \<and> b \<in> norRRset m"
  24.104    apply (unfold norRRset_def)
  24.105    apply (cut_tac a = a and m = m in zcong_zless_unique)
  24.106     apply auto
  24.107 @@ -158,7 +158,7 @@
  24.108    apply (rule_tac "x" = "b" in exI)
  24.109    apply safe
  24.110    apply (rule Bnor_mem_if)
  24.111 -    apply (case_tac [2] "b = Numeral0")
  24.112 +    apply (case_tac [2] "b = 0")
  24.113       apply (auto intro: order_less_le [THEN iffD2])
  24.114     prefer 2
  24.115     apply (simp only: zcong_def)
  24.116 @@ -173,7 +173,7 @@
  24.117  text {* \medskip @{term noXRRset} *}
  24.118  
  24.119  lemma RRset_gcd [rule_format]:
  24.120 -    "is_RRset A m ==> a \<in> A --> zgcd (a, m) = Numeral1"
  24.121 +    "is_RRset A m ==> a \<in> A --> zgcd (a, m) = 1"
  24.122    apply (unfold is_RRset_def)
  24.123    apply (rule RsetR.induct)
  24.124      apply auto
  24.125 @@ -181,7 +181,7 @@
  24.126  
  24.127  lemma RsetR_zmult_mono:
  24.128    "A \<in> RsetR m ==>
  24.129 -    Numeral0 < m ==> zgcd (x, m) = Numeral1 ==> (\<lambda>a. a * x) ` A \<in> RsetR m"
  24.130 +    0 < m ==> zgcd (x, m) = 1 ==> (\<lambda>a. a * x) ` A \<in> RsetR m"
  24.131    apply (erule RsetR.induct)
  24.132     apply simp_all
  24.133    apply (rule RsetR.insert)
  24.134 @@ -191,8 +191,8 @@
  24.135    done
  24.136  
  24.137  lemma card_nor_eq_noX:
  24.138 -  "Numeral0 < m ==>
  24.139 -    zgcd (x, m) = Numeral1 ==> card (noXRRset m x) = card (norRRset m)"
  24.140 +  "0 < m ==>
  24.141 +    zgcd (x, m) = 1 ==> card (noXRRset m x) = card (norRRset m)"
  24.142    apply (unfold norRRset_def noXRRset_def)
  24.143    apply (rule card_image)
  24.144     apply (auto simp add: inj_on_def Bnor_fin)
  24.145 @@ -200,7 +200,7 @@
  24.146    done
  24.147  
  24.148  lemma noX_is_RRset:
  24.149 -    "Numeral0 < m ==> zgcd (x, m) = Numeral1 ==> is_RRset (noXRRset m x) m"
  24.150 +    "0 < m ==> zgcd (x, m) = 1 ==> is_RRset (noXRRset m x) m"
  24.151    apply (unfold is_RRset_def phi_def)
  24.152    apply (auto simp add: card_nor_eq_noX)
  24.153    apply (unfold noXRRset_def norRRset_def)
  24.154 @@ -210,7 +210,7 @@
  24.155    done
  24.156  
  24.157  lemma aux_some:
  24.158 -  "Numeral1 < m ==> is_RRset A m ==> a \<in> A
  24.159 +  "1 < m ==> is_RRset A m ==> a \<in> A
  24.160      ==> zcong a (SOME b. [a = b] (mod m) \<and> b \<in> norRRset m) m \<and>
  24.161        (SOME b. [a = b] (mod m) \<and> b \<in> norRRset m) \<in> norRRset m"
  24.162    apply (rule norR_mem_unique [THEN ex1_implies_ex, THEN someI_ex])
  24.163 @@ -219,7 +219,7 @@
  24.164    done
  24.165  
  24.166  lemma RRset2norRR_correct:
  24.167 -  "Numeral1 < m ==> is_RRset A m ==> a \<in> A ==>
  24.168 +  "1 < m ==> is_RRset A m ==> a \<in> A ==>
  24.169      [a = RRset2norRR A m a] (mod m) \<and> RRset2norRR A m a \<in> norRRset m"
  24.170    apply (unfold RRset2norRR_def)
  24.171    apply simp
  24.172 @@ -238,7 +238,7 @@
  24.173    done
  24.174  
  24.175  lemma RRset_zcong_eq [rule_format]:
  24.176 -  "Numeral1 < m ==>
  24.177 +  "1 < m ==>
  24.178      is_RRset A m ==> [a = b] (mod m) ==> a \<in> A --> b \<in> A --> a = b"
  24.179    apply (unfold is_RRset_def)
  24.180    apply (rule RsetR.induct)
  24.181 @@ -252,7 +252,7 @@
  24.182    done
  24.183  
  24.184  lemma RRset2norRR_inj:
  24.185 -    "Numeral1 < m ==> is_RRset A m ==> inj_on (RRset2norRR A m) A"
  24.186 +    "1 < m ==> is_RRset A m ==> inj_on (RRset2norRR A m) A"
  24.187    apply (unfold RRset2norRR_def inj_on_def)
  24.188    apply auto
  24.189    apply (subgoal_tac "\<exists>b. ([x = b] (mod m) \<and> b \<in> norRRset m) \<and>
  24.190 @@ -267,7 +267,7 @@
  24.191    done
  24.192  
  24.193  lemma RRset2norRR_eq_norR:
  24.194 -    "Numeral1 < m ==> is_RRset A m ==> RRset2norRR A m ` A = norRRset m"
  24.195 +    "1 < m ==> is_RRset A m ==> RRset2norRR A m ` A = norRRset m"
  24.196    apply (rule card_seteq)
  24.197      prefer 3
  24.198      apply (subst card_image)
  24.199 @@ -286,7 +286,7 @@
  24.200    done
  24.201  
  24.202  lemma Bnor_prod_power [rule_format]:
  24.203 -  "x \<noteq> Numeral0 ==> a < m --> setprod ((\<lambda>a. a * x) ` BnorRset (a, m)) =
  24.204 +  "x \<noteq> 0 ==> a < m --> setprod ((\<lambda>a. a * x) ` BnorRset (a, m)) =
  24.205        setprod (BnorRset(a, m)) * x^card (BnorRset (a, m))"
  24.206    apply (induct a m rule: BnorRset_induct)
  24.207     prefer 2
  24.208 @@ -313,7 +313,7 @@
  24.209    done
  24.210  
  24.211  lemma Bnor_prod_zgcd [rule_format]:
  24.212 -    "a < m --> zgcd (setprod (BnorRset (a, m)), m) = Numeral1"
  24.213 +    "a < m --> zgcd (setprod (BnorRset (a, m)), m) = 1"
  24.214    apply (induct a m rule: BnorRset_induct)
  24.215     prefer 2
  24.216     apply (subst BnorRset.simps)
  24.217 @@ -324,12 +324,12 @@
  24.218    done
  24.219  
  24.220  theorem Euler_Fermat:
  24.221 -    "Numeral0 < m ==> zgcd (x, m) = Numeral1 ==> [x^(phi m) = Numeral1] (mod m)"
  24.222 +    "0 < m ==> zgcd (x, m) = 1 ==> [x^(phi m) = 1] (mod m)"
  24.223    apply (unfold norRRset_def phi_def)
  24.224 -  apply (case_tac "x = Numeral0")
  24.225 -   apply (case_tac [2] "m = Numeral1")
  24.226 +  apply (case_tac "x = 0")
  24.227 +   apply (case_tac [2] "m = 1")
  24.228      apply (rule_tac [3] iffD1)
  24.229 -     apply (rule_tac [3] k = "setprod (BnorRset (m - Numeral1, m))"
  24.230 +     apply (rule_tac [3] k = "setprod (BnorRset (m - 1, m))"
  24.231         in zcong_cancel2)
  24.232        prefer 5
  24.233        apply (subst Bnor_prod_power [symmetric])
  24.234 @@ -352,7 +352,7 @@
  24.235  
  24.236  lemma Bnor_prime [rule_format (no_asm)]:
  24.237    "p \<in> zprime ==>
  24.238 -    a < p --> (\<forall>b. Numeral0 < b \<and> b \<le> a --> zgcd (b, p) = Numeral1)
  24.239 +    a < p --> (\<forall>b. 0 < b \<and> b \<le> a --> zgcd (b, p) = 1)
  24.240      --> card (BnorRset (a, p)) = nat a"
  24.241    apply (unfold zprime_def)
  24.242    apply (induct a p rule: BnorRset.induct)
  24.243 @@ -361,7 +361,7 @@
  24.244    apply auto
  24.245    done
  24.246  
  24.247 -lemma phi_prime: "p \<in> zprime ==> phi p = nat (p - Numeral1)"
  24.248 +lemma phi_prime: "p \<in> zprime ==> phi p = nat (p - 1)"
  24.249    apply (unfold phi_def norRRset_def)
  24.250    apply (rule Bnor_prime)
  24.251      apply auto
  24.252 @@ -370,7 +370,7 @@
  24.253    done
  24.254  
  24.255  theorem Little_Fermat:
  24.256 -    "p \<in> zprime ==> \<not> p dvd x ==> [x^(nat (p - Numeral1)) = Numeral1] (mod p)"
  24.257 +    "p \<in> zprime ==> \<not> p dvd x ==> [x^(nat (p - 1)) = 1] (mod p)"
  24.258    apply (subst phi_prime [symmetric])
  24.259     apply (rule_tac [2] Euler_Fermat)
  24.260      apply (erule_tac [3] zprime_imp_zrelprime)
    25.1 --- a/src/HOL/NumberTheory/Fib.thy	Mon Oct 22 11:01:30 2001 +0200
    25.2 +++ b/src/HOL/NumberTheory/Fib.thy	Mon Oct 22 11:54:22 2001 +0200
    25.3 @@ -67,8 +67,8 @@
    25.4  *}
    25.5  
    25.6  lemma fib_Cassini: "int (fib (Suc (Suc n)) * fib n) =
    25.7 -  (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - Numeral1
    25.8 -   else int (fib (Suc n) * fib (Suc n)) + Numeral1)"
    25.9 +  (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
   25.10 +   else int (fib (Suc n) * fib (Suc n)) + 1)"
   25.11    apply (induct n rule: fib.induct)
   25.12      apply (simp add: fib.Suc_Suc)
   25.13     apply (simp add: fib.Suc_Suc mod_Suc)
    26.1 --- a/src/HOL/NumberTheory/IntFact.thy	Mon Oct 22 11:01:30 2001 +0200
    26.2 +++ b/src/HOL/NumberTheory/IntFact.thy	Mon Oct 22 11:54:22 2001 +0200
    26.3 @@ -22,18 +22,18 @@
    26.4    d22set :: "int => int set"
    26.5  
    26.6  recdef zfact  "measure ((\<lambda>n. nat n) :: int => nat)"
    26.7 -  "zfact n = (if n \<le> Numeral0 then Numeral1 else n * zfact (n - Numeral1))"
    26.8 +  "zfact n = (if n \<le> 0 then 1 else n * zfact (n - 1))"
    26.9  
   26.10  defs
   26.11 -  setprod_def: "setprod A == (if finite A then fold (op *) Numeral1 A else Numeral1)"
   26.12 +  setprod_def: "setprod A == (if finite A then fold (op *) 1 A else 1)"
   26.13  
   26.14  recdef d22set  "measure ((\<lambda>a. nat a) :: int => nat)"
   26.15 -  "d22set a = (if Numeral1 < a then insert a (d22set (a - Numeral1)) else {})"
   26.16 +  "d22set a = (if 1 < a then insert a (d22set (a - 1)) else {})"
   26.17  
   26.18  
   26.19  text {* \medskip @{term setprod} --- product of finite set *}
   26.20  
   26.21 -lemma setprod_empty [simp]: "setprod {} = Numeral1"
   26.22 +lemma setprod_empty [simp]: "setprod {} = 1"
   26.23    apply (simp add: setprod_def)
   26.24    done
   26.25  
   26.26 @@ -54,7 +54,7 @@
   26.27  
   26.28  lemma d22set_induct:
   26.29    "(!!a. P {} a) ==>
   26.30 -    (!!a. Numeral1 < (a::int) ==> P (d22set (a - Numeral1)) (a - Numeral1)
   26.31 +    (!!a. 1 < (a::int) ==> P (d22set (a - 1)) (a - 1)
   26.32        ==> P (d22set a) a)
   26.33      ==> P (d22set u) u"
   26.34  proof -
   26.35 @@ -62,14 +62,14 @@
   26.36    show ?thesis
   26.37      apply (rule d22set.induct)
   26.38      apply safe
   26.39 -     apply (case_tac [2] "Numeral1 < a")
   26.40 +     apply (case_tac [2] "1 < a")
   26.41        apply (rule_tac [2] rule_context)
   26.42         apply (simp_all (no_asm_simp))
   26.43       apply (simp_all (no_asm_simp) add: d22set.simps rule_context)
   26.44      done
   26.45  qed
   26.46  
   26.47 -lemma d22set_g_1 [rule_format]: "b \<in> d22set a --> Numeral1 < b"
   26.48 +lemma d22set_g_1 [rule_format]: "b \<in> d22set a --> 1 < b"
   26.49    apply (induct a rule: d22set_induct)
   26.50     prefer 2
   26.51     apply (subst d22set.simps)
   26.52 @@ -87,7 +87,7 @@
   26.53    apply (auto dest: d22set_le)
   26.54    done
   26.55  
   26.56 -lemma d22set_mem [rule_format]: "Numeral1 < b --> b \<le> a --> b \<in> d22set a"
   26.57 +lemma d22set_mem [rule_format]: "1 < b --> b \<le> a --> b \<in> d22set a"
   26.58    apply (induct a rule: d22set.induct)
   26.59    apply auto
   26.60     apply (simp_all add: d22set.simps)
   26.61 @@ -109,7 +109,7 @@
   26.62     apply (simp add: d22set.simps zfact.simps)
   26.63    apply (subst d22set.simps)
   26.64    apply (subst zfact.simps)
   26.65 -  apply (case_tac "Numeral1 < a")
   26.66 +  apply (case_tac "1 < a")
   26.67     prefer 2
   26.68     apply (simp add: d22set.simps zfact.simps)
   26.69    apply (simp add: d22set_fin d22set_le_swap)
    27.1 --- a/src/HOL/NumberTheory/IntPrimes.thy	Mon Oct 22 11:01:30 2001 +0200
    27.2 +++ b/src/HOL/NumberTheory/IntPrimes.thy	Mon Oct 22 11:54:22 2001 +0200
    27.3 @@ -29,7 +29,7 @@
    27.4    "measure ((\<lambda>(m, n, r', r, s', s, t', t). nat r)
    27.5      :: int * int * int * int *int * int * int * int => nat)"
    27.6    "xzgcda (m, n, r', r, s', s, t', t) =
    27.7 -    (if r \<le> Numeral0 then (r', s', t')
    27.8 +    (if r \<le> 0 then (r', s', t')
    27.9       else xzgcda (m, n, r, r' mod r, s, s' - (r' div r) * s, t, t' - (r' div r) * t))"
   27.10    (hints simp: pos_mod_bound)
   27.11  
   27.12 @@ -38,13 +38,13 @@
   27.13    "zgcd == \<lambda>(x,y). int (gcd (nat (abs x), nat (abs y)))"
   27.14  
   27.15  defs
   27.16 -  xzgcd_def: "xzgcd m n == xzgcda (m, n, m, n, Numeral1, Numeral0, Numeral0, Numeral1)"
   27.17 -  zprime_def: "zprime == {p. Numeral1 < p \<and> (\<forall>m. m dvd p --> m = Numeral1 \<or> m = p)}"
   27.18 +  xzgcd_def: "xzgcd m n == xzgcda (m, n, m, n, 1, 0, 0, 1)"
   27.19 +  zprime_def: "zprime == {p. 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)}"
   27.20    zcong_def: "[a = b] (mod m) == m dvd (a - b)"
   27.21  
   27.22  
   27.23  lemma zabs_eq_iff:
   27.24 -    "(abs (z::int) = w) = (z = w \<and> Numeral0 <= z \<or> z = -w \<and> z < Numeral0)"
   27.25 +    "(abs (z::int) = w) = (z = w \<and> 0 <= z \<or> z = -w \<and> z < 0)"
   27.26    apply (auto simp add: zabs_def)
   27.27    done
   27.28  
   27.29 @@ -64,17 +64,17 @@
   27.30  
   27.31  subsection {* Divides relation *}
   27.32  
   27.33 -lemma zdvd_0_right [iff]: "(m::int) dvd Numeral0"
   27.34 +lemma zdvd_0_right [iff]: "(m::int) dvd 0"
   27.35    apply (unfold dvd_def)
   27.36    apply (blast intro: zmult_0_right [symmetric])
   27.37    done
   27.38  
   27.39 -lemma zdvd_0_left [iff]: "(Numeral0 dvd (m::int)) = (m = Numeral0)"
   27.40 +lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
   27.41    apply (unfold dvd_def)
   27.42    apply auto
   27.43    done
   27.44  
   27.45 -lemma zdvd_1_left [iff]: "Numeral1 dvd (m::int)"
   27.46 +lemma zdvd_1_left [iff]: "1 dvd (m::int)"
   27.47    apply (unfold dvd_def)
   27.48    apply simp
   27.49    done
   27.50 @@ -104,7 +104,7 @@
   27.51    done
   27.52  
   27.53  lemma zdvd_anti_sym:
   27.54 -    "Numeral0 < m ==> Numeral0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
   27.55 +    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
   27.56    apply (unfold dvd_def)
   27.57    apply auto
   27.58    apply (simp add: zmult_assoc zmult_eq_self_iff int_0_less_mult_iff zmult_eq_1_iff)
   27.59 @@ -186,19 +186,19 @@
   27.60    apply (simp add: zdvd_zadd zdvd_zmult2)
   27.61    done
   27.62  
   27.63 -lemma zdvd_iff_zmod_eq_0: "(k dvd n) = (n mod (k::int) = Numeral0)"
   27.64 +lemma zdvd_iff_zmod_eq_0: "(k dvd n) = (n mod (k::int) = 0)"
   27.65    apply (unfold dvd_def)
   27.66    apply auto
   27.67    done
   27.68  
   27.69 -lemma zdvd_not_zless: "Numeral0 < m ==> m < n ==> \<not> n dvd (m::int)"
   27.70 +lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
   27.71    apply (unfold dvd_def)
   27.72    apply auto
   27.73 -  apply (subgoal_tac "Numeral0 < n")
   27.74 +  apply (subgoal_tac "0 < n")
   27.75     prefer 2
   27.76     apply (blast intro: zless_trans)
   27.77    apply (simp add: int_0_less_mult_iff)
   27.78 -  apply (subgoal_tac "n * k < n * Numeral1")
   27.79 +  apply (subgoal_tac "n * k < n * 1")
   27.80     apply (drule zmult_zless_cancel1 [THEN iffD1])
   27.81     apply auto
   27.82    done
   27.83 @@ -221,7 +221,7 @@
   27.84        nat_mult_distrib [symmetric] nat_eq_iff2)
   27.85    done
   27.86  
   27.87 -lemma nat_dvd_iff: "(nat z dvd m) = (if Numeral0 \<le> z then (z dvd int m) else m = 0)"
   27.88 +lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
   27.89    apply (auto simp add: dvd_def zmult_int [symmetric])
   27.90    apply (rule_tac x = "nat k" in exI)
   27.91    apply (cut_tac k = m in int_less_0_conv)
   27.92 @@ -245,11 +245,11 @@
   27.93  
   27.94  subsection {* Euclid's Algorithm and GCD *}
   27.95  
   27.96 -lemma zgcd_0 [simp]: "zgcd (m, Numeral0) = abs m"
   27.97 +lemma zgcd_0 [simp]: "zgcd (m, 0) = abs m"
   27.98    apply (simp add: zgcd_def zabs_def)
   27.99    done
  27.100  
  27.101 -lemma zgcd_0_left [simp]: "zgcd (Numeral0, m) = abs m"
  27.102 +lemma zgcd_0_left [simp]: "zgcd (0, m) = abs m"
  27.103    apply (simp add: zgcd_def zabs_def)
  27.104    done
  27.105  
  27.106 @@ -261,7 +261,7 @@
  27.107    apply (simp add: zgcd_def)
  27.108    done
  27.109  
  27.110 -lemma zgcd_non_0: "Numeral0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"
  27.111 +lemma zgcd_non_0: "0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"
  27.112    apply (frule_tac b = n and a = m in pos_mod_sign)
  27.113    apply (simp add: zgcd_def zabs_def nat_mod_distrib)
  27.114    apply (cut_tac a = "-m" and b = n in zmod_zminus1_eq_if)
  27.115 @@ -273,17 +273,17 @@
  27.116    done
  27.117  
  27.118  lemma zgcd_eq: "zgcd (m, n) = zgcd (n, m mod n)"
  27.119 -  apply (tactic {* zdiv_undefined_case_tac "n = Numeral0" 1 *})
  27.120 +  apply (tactic {* zdiv_undefined_case_tac "n = 0" 1 *})
  27.121    apply (auto simp add: linorder_neq_iff zgcd_non_0)
  27.122    apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0)
  27.123     apply auto
  27.124    done
  27.125  
  27.126 -lemma zgcd_1 [simp]: "zgcd (m, Numeral1) = Numeral1"
  27.127 +lemma zgcd_1 [simp]: "zgcd (m, 1) = 1"
  27.128    apply (simp add: zgcd_def zabs_def)
  27.129    done
  27.130  
  27.131 -lemma zgcd_0_1_iff [simp]: "(zgcd (Numeral0, m) = Numeral1) = (abs m = Numeral1)"
  27.132 +lemma zgcd_0_1_iff [simp]: "(zgcd (0, m) = 1) = (abs m = 1)"
  27.133    apply (simp add: zgcd_def zabs_def)
  27.134    done
  27.135  
  27.136 @@ -303,7 +303,7 @@
  27.137    apply (simp add: zgcd_def gcd_commute)
  27.138    done
  27.139  
  27.140 -lemma zgcd_1_left [simp]: "zgcd (Numeral1, m) = Numeral1"
  27.141 +lemma zgcd_1_left [simp]: "zgcd (1, m) = 1"
  27.142    apply (simp add: zgcd_def gcd_1_left)
  27.143    done
  27.144  
  27.145 @@ -320,7 +320,7 @@
  27.146  lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
  27.147    -- {* addition is an AC-operator *}
  27.148  
  27.149 -lemma zgcd_zmult_distrib2: "Numeral0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"
  27.150 +lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"
  27.151    apply (simp del: zmult_zminus_right
  27.152      add: zmult_zminus_right [symmetric] nat_mult_distrib zgcd_def zabs_def
  27.153      zmult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
  27.154 @@ -330,29 +330,29 @@
  27.155    apply (simp add: zabs_def zgcd_zmult_distrib2)
  27.156    done
  27.157  
  27.158 -lemma zgcd_self [simp]: "Numeral0 \<le> m ==> zgcd (m, m) = m"
  27.159 -  apply (cut_tac k = m and m = "Numeral1" and n = "Numeral1" in zgcd_zmult_distrib2)
  27.160 +lemma zgcd_self [simp]: "0 \<le> m ==> zgcd (m, m) = m"
  27.161 +  apply (cut_tac k = m and m = "1" and n = "1" in zgcd_zmult_distrib2)
  27.162     apply simp_all
  27.163    done
  27.164  
  27.165 -lemma zgcd_zmult_eq_self [simp]: "Numeral0 \<le> k ==> zgcd (k, k * n) = k"
  27.166 -  apply (cut_tac k = k and m = "Numeral1" and n = n in zgcd_zmult_distrib2)
  27.167 +lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd (k, k * n) = k"
  27.168 +  apply (cut_tac k = k and m = "1" and n = n in zgcd_zmult_distrib2)
  27.169     apply simp_all
  27.170    done
  27.171  
  27.172 -lemma zgcd_zmult_eq_self2 [simp]: "Numeral0 \<le> k ==> zgcd (k * n, k) = k"
  27.173 -  apply (cut_tac k = k and m = n and n = "Numeral1" in zgcd_zmult_distrib2)
  27.174 +lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n, k) = k"
  27.175 +  apply (cut_tac k = k and m = n and n = "1" in zgcd_zmult_distrib2)
  27.176     apply simp_all
  27.177    done
  27.178  
  27.179 -lemma aux: "zgcd (n, k) = Numeral1 ==> k dvd m * n ==> Numeral0 \<le> m ==> k dvd m"
  27.180 +lemma aux: "zgcd (n, k) = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"
  27.181    apply (subgoal_tac "m = zgcd (m * n, m * k)")
  27.182     apply (erule ssubst, rule zgcd_greatest_iff [THEN iffD2])
  27.183     apply (simp_all add: zgcd_zmult_distrib2 [symmetric] int_0_le_mult_iff)
  27.184    done
  27.185  
  27.186 -lemma zrelprime_zdvd_zmult: "zgcd (n, k) = Numeral1 ==> k dvd m * n ==> k dvd m"
  27.187 -  apply (case_tac "Numeral0 \<le> m")
  27.188 +lemma zrelprime_zdvd_zmult: "zgcd (n, k) = 1 ==> k dvd m * n ==> k dvd m"
  27.189 +  apply (case_tac "0 \<le> m")
  27.190     apply (blast intro: aux)
  27.191    apply (subgoal_tac "k dvd -m")
  27.192     apply (rule_tac [2] aux)
  27.193 @@ -360,20 +360,20 @@
  27.194    done
  27.195  
  27.196  lemma zprime_imp_zrelprime:
  27.197 -    "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = Numeral1"
  27.198 +    "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = 1"
  27.199    apply (unfold zprime_def)
  27.200    apply auto
  27.201    done
  27.202  
  27.203  lemma zless_zprime_imp_zrelprime:
  27.204 -    "p \<in> zprime ==> Numeral0 < n ==> n < p ==> zgcd (n, p) = Numeral1"
  27.205 +    "p \<in> zprime ==> 0 < n ==> n < p ==> zgcd (n, p) = 1"
  27.206    apply (erule zprime_imp_zrelprime)
  27.207    apply (erule zdvd_not_zless)
  27.208    apply assumption
  27.209    done
  27.210  
  27.211  lemma zprime_zdvd_zmult:
  27.212 -    "Numeral0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
  27.213 +    "0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
  27.214    apply safe
  27.215    apply (rule zrelprime_zdvd_zmult)
  27.216     apply (rule zprime_imp_zrelprime)
  27.217 @@ -392,7 +392,7 @@
  27.218    done
  27.219  
  27.220  lemma zgcd_zmult_zdvd_zgcd:
  27.221 -    "zgcd (k, n) = Numeral1 ==> zgcd (k * m, n) dvd zgcd (m, n)"
  27.222 +    "zgcd (k, n) = 1 ==> zgcd (k * m, n) dvd zgcd (m, n)"
  27.223    apply (simp add: zgcd_greatest_iff)
  27.224    apply (rule_tac n = k in zrelprime_zdvd_zmult)
  27.225     prefer 2
  27.226 @@ -402,16 +402,16 @@
  27.227    apply (simp (no_asm) add: zgcd_ac)
  27.228    done
  27.229  
  27.230 -lemma zgcd_zmult_cancel: "zgcd (k, n) = Numeral1 ==> zgcd (k * m, n) = zgcd (m, n)"
  27.231 +lemma zgcd_zmult_cancel: "zgcd (k, n) = 1 ==> zgcd (k * m, n) = zgcd (m, n)"
  27.232    apply (simp add: zgcd_def nat_abs_mult_distrib gcd_mult_cancel)
  27.233    done
  27.234  
  27.235  lemma zgcd_zgcd_zmult:
  27.236 -    "zgcd (k, m) = Numeral1 ==> zgcd (n, m) = Numeral1 ==> zgcd (k * n, m) = Numeral1"
  27.237 +    "zgcd (k, m) = 1 ==> zgcd (n, m) = 1 ==> zgcd (k * n, m) = 1"
  27.238    apply (simp (no_asm_simp) add: zgcd_zmult_cancel)
  27.239    done
  27.240  
  27.241 -lemma zdvd_iff_zgcd: "Numeral0 < m ==> (m dvd n) = (zgcd (n, m) = m)"
  27.242 +lemma zdvd_iff_zgcd: "0 < m ==> (m dvd n) = (zgcd (n, m) = m)"
  27.243    apply safe
  27.244     apply (rule_tac [2] n = "zgcd (n, m)" in zdvd_trans)
  27.245      apply (rule_tac [3] zgcd_zdvd1)
  27.246 @@ -423,7 +423,7 @@
  27.247  
  27.248  subsection {* Congruences *}
  27.249  
  27.250 -lemma zcong_1 [simp]: "[a = b] (mod Numeral1)"
  27.251 +lemma zcong_1 [simp]: "[a = b] (mod 1)"
  27.252    apply (unfold zcong_def)
  27.253    apply auto
  27.254    done
  27.255 @@ -494,19 +494,19 @@
  27.256    done
  27.257  
  27.258  lemma zcong_square:
  27.259 -  "p \<in> zprime ==> Numeral0 < a ==> [a * a = Numeral1] (mod p)
  27.260 -    ==> [a = Numeral1] (mod p) \<or> [a = p - Numeral1] (mod p)"
  27.261 +  "p \<in> zprime ==> 0 < a ==> [a * a = 1] (mod p)
  27.262 +    ==> [a = 1] (mod p) \<or> [a = p - 1] (mod p)"
  27.263    apply (unfold zcong_def)
  27.264    apply (rule zprime_zdvd_zmult)
  27.265 -    apply (rule_tac [3] s = "a * a - Numeral1 + p * (Numeral1 - a)" in subst)
  27.266 +    apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
  27.267       prefer 4
  27.268       apply (simp add: zdvd_reduce)
  27.269      apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)
  27.270    done
  27.271  
  27.272  lemma zcong_cancel:
  27.273 -  "Numeral0 \<le> m ==>
  27.274 -    zgcd (k, m) = Numeral1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
  27.275 +  "0 \<le> m ==>
  27.276 +    zgcd (k, m) = 1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
  27.277    apply safe
  27.278     prefer 2
  27.279     apply (blast intro: zcong_scalar)
  27.280 @@ -523,19 +523,19 @@
  27.281    done
  27.282  
  27.283  lemma zcong_cancel2:
  27.284 -  "Numeral0 \<le> m ==>
  27.285 -    zgcd (k, m) = Numeral1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
  27.286 +  "0 \<le> m ==>
  27.287 +    zgcd (k, m) = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
  27.288    apply (simp add: zmult_commute zcong_cancel)
  27.289    done
  27.290  
  27.291  lemma zcong_zgcd_zmult_zmod:
  27.292 -  "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = Numeral1
  27.293 +  "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = 1
  27.294      ==> [a = b] (mod m * n)"
  27.295    apply (unfold zcong_def dvd_def)
  27.296    apply auto
  27.297    apply (subgoal_tac "m dvd n * ka")
  27.298     apply (subgoal_tac "m dvd ka")
  27.299 -    apply (case_tac [2] "Numeral0 \<le> ka")
  27.300 +    apply (case_tac [2] "0 \<le> ka")
  27.301       prefer 3
  27.302       apply (subst zdvd_zminus_iff [symmetric])
  27.303       apply (rule_tac n = n in zrelprime_zdvd_zmult)
  27.304 @@ -550,8 +550,8 @@
  27.305    done
  27.306  
  27.307  lemma zcong_zless_imp_eq:
  27.308 -  "Numeral0 \<le> a ==>
  27.309 -    a < m ==> Numeral0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
  27.310 +  "0 \<le> a ==>
  27.311 +    a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
  27.312    apply (unfold zcong_def dvd_def)
  27.313    apply auto
  27.314    apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
  27.315 @@ -566,38 +566,38 @@
  27.316    done
  27.317  
  27.318  lemma zcong_square_zless:
  27.319 -  "p \<in> zprime ==> Numeral0 < a ==> a < p ==>
  27.320 -    [a * a = Numeral1] (mod p) ==> a = Numeral1 \<or> a = p - Numeral1"
  27.321 +  "p \<in> zprime ==> 0 < a ==> a < p ==>
  27.322 +    [a * a = 1] (mod p) ==> a = 1 \<or> a = p - 1"
  27.323    apply (cut_tac p = p and a = a in zcong_square)
  27.324       apply (simp add: zprime_def)
  27.325      apply (auto intro: zcong_zless_imp_eq)
  27.326    done
  27.327  
  27.328  lemma zcong_not:
  27.329 -    "Numeral0 < a ==> a < m ==> Numeral0 < b ==> b < a ==> \<not> [a = b] (mod m)"
  27.330 +    "0 < a ==> a < m ==> 0 < b ==> b < a ==> \<not> [a = b] (mod m)"
  27.331    apply (unfold zcong_def)
  27.332    apply (rule zdvd_not_zless)
  27.333     apply auto
  27.334    done
  27.335  
  27.336  lemma zcong_zless_0:
  27.337 -    "Numeral0 \<le> a ==> a < m ==> [a = Numeral0] (mod m) ==> a = Numeral0"
  27.338 +    "0 \<le> a ==> a < m ==> [a = 0] (mod m) ==> a = 0"
  27.339    apply (unfold zcong_def dvd_def)
  27.340    apply auto
  27.341 -  apply (subgoal_tac "Numeral0 < m")
  27.342 +  apply (subgoal_tac "0 < m")
  27.343     apply (rotate_tac -1)
  27.344     apply (simp add: int_0_le_mult_iff)
  27.345 -   apply (subgoal_tac "m * k < m * Numeral1")
  27.346 +   apply (subgoal_tac "m * k < m * 1")
  27.347      apply (drule zmult_zless_cancel1 [THEN iffD1])
  27.348      apply (auto simp add: linorder_neq_iff)
  27.349    done
  27.350  
  27.351  lemma zcong_zless_unique:
  27.352 -    "Numeral0 < m ==> (\<exists>!b. Numeral0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
  27.353 +    "0 < m ==> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
  27.354    apply auto
  27.355     apply (subgoal_tac [2] "[b = y] (mod m)")
  27.356 -    apply (case_tac [2] "b = Numeral0")
  27.357 -     apply (case_tac [3] "y = Numeral0")
  27.358 +    apply (case_tac [2] "b = 0")
  27.359 +     apply (case_tac [3] "y = 0")
  27.360        apply (auto intro: zcong_trans zcong_zless_0 zcong_zless_imp_eq order_less_le
  27.361          simp add: zcong_sym)
  27.362    apply (unfold zcong_def dvd_def)
  27.363 @@ -616,8 +616,8 @@
  27.364    done
  27.365  
  27.366  lemma zgcd_zcong_zgcd:
  27.367 -  "Numeral0 < m ==>
  27.368 -    zgcd (a, m) = Numeral1 ==> [a = b] (mod m) ==> zgcd (b, m) = Numeral1"
  27.369 +  "0 < m ==>
  27.370 +    zgcd (a, m) = 1 ==> [a = b] (mod m) ==> zgcd (b, m) = 1"
  27.371    apply (auto simp add: zcong_iff_lin)
  27.372    done
  27.373  
  27.374 @@ -643,7 +643,7 @@
  27.375    apply (simp add: zadd_commute)
  27.376    done
  27.377  
  27.378 -lemma zcong_zmod_eq: "Numeral0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
  27.379 +lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
  27.380    apply auto
  27.381     apply (rule_tac m = m in zcong_zless_imp_eq)
  27.382         prefer 5
  27.383 @@ -659,13 +659,13 @@
  27.384    apply (auto simp add: zcong_def)
  27.385    done
  27.386  
  27.387 -lemma zcong_zero [iff]: "[a = b] (mod Numeral0) = (a = b)"
  27.388 +lemma zcong_zero [iff]: "[a = b] (mod 0) = (a = b)"
  27.389    apply (auto simp add: zcong_def)
  27.390    done
  27.391  
  27.392  lemma "[a = b] (mod m) = (a mod m = b mod m)"
  27.393 -  apply (tactic {* zdiv_undefined_case_tac "m = Numeral0" 1 *})
  27.394 -  apply (case_tac "Numeral0 < m")
  27.395 +  apply (tactic {* zdiv_undefined_case_tac "m = 0" 1 *})
  27.396 +  apply (case_tac "0 < m")
  27.397     apply (simp add: zcong_zmod_eq)
  27.398    apply (rule_tac t = m in zminus_zminus [THEN subst])
  27.399    apply (subst zcong_zminus)
  27.400 @@ -677,7 +677,7 @@
  27.401  subsection {* Modulo *}
  27.402  
  27.403  lemma zmod_zdvd_zmod:
  27.404 -    "Numeral0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
  27.405 +    "0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
  27.406    apply (unfold dvd_def)
  27.407    apply auto
  27.408    apply (subst zcong_zmod_eq [symmetric])
  27.409 @@ -696,14 +696,14 @@
  27.410  declare xzgcda.simps [simp del]
  27.411  
  27.412  lemma aux1:
  27.413 -  "zgcd (r', r) = k --> Numeral0 < r -->
  27.414 +  "zgcd (r', r) = k --> 0 < r -->
  27.415      (\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn))"
  27.416    apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
  27.417      z = s and aa = t' and ab = t in xzgcda.induct)
  27.418    apply (subst zgcd_eq)
  27.419    apply (subst xzgcda.simps)
  27.420    apply auto
  27.421 -  apply (case_tac "r' mod r = Numeral0")
  27.422 +  apply (case_tac "r' mod r = 0")
  27.423     prefer 2
  27.424     apply (frule_tac a = "r'" in pos_mod_sign)
  27.425     apply auto
  27.426 @@ -716,14 +716,14 @@
  27.427    done
  27.428  
  27.429  lemma aux2:
  27.430 -  "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> Numeral0 < r -->
  27.431 +  "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> 0 < r -->
  27.432      zgcd (r', r) = k"
  27.433    apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
  27.434      z = s and aa = t' and ab = t in xzgcda.induct)
  27.435    apply (subst zgcd_eq)
  27.436    apply (subst xzgcda.simps)
  27.437    apply (auto simp add: linorder_not_le)
  27.438 -  apply (case_tac "r' mod r = Numeral0")
  27.439 +  apply (case_tac "r' mod r = 0")
  27.440     prefer 2
  27.441     apply (frule_tac a = "r'" in pos_mod_sign)
  27.442     apply auto
  27.443 @@ -735,7 +735,7 @@
  27.444    done
  27.445  
  27.446  lemma xzgcd_correct:
  27.447 -    "Numeral0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
  27.448 +    "0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
  27.449    apply (unfold xzgcd_def)
  27.450    apply (rule iffI)
  27.451     apply (rule_tac [2] aux2 [THEN mp, THEN mp])
  27.452 @@ -768,17 +768,17 @@
  27.453    by (rule iffD2 [OF order_less_le conjI])
  27.454  
  27.455  lemma xzgcda_linear [rule_format]:
  27.456 -  "Numeral0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
  27.457 +  "0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
  27.458      r' = s' * m + t' * n -->  r = s * m + t * n --> rn = sn * m + tn * n"
  27.459    apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
  27.460      z = s and aa = t' and ab = t in xzgcda.induct)
  27.461    apply (subst xzgcda.simps)
  27.462    apply (simp (no_asm))
  27.463    apply (rule impI)+
  27.464 -  apply (case_tac "r' mod r = Numeral0")
  27.465 +  apply (case_tac "r' mod r = 0")
  27.466     apply (simp add: xzgcda.simps)
  27.467     apply clarify
  27.468 -  apply (subgoal_tac "Numeral0 < r' mod r")
  27.469 +  apply (subgoal_tac "0 < r' mod r")
  27.470     apply (rule_tac [2] order_le_neq_implies_less)
  27.471     apply (rule_tac [2] pos_mod_sign)
  27.472      apply (cut_tac m = m and n = n and r' = r' and r = r and s' = s' and
  27.473 @@ -787,7 +787,7 @@
  27.474    done
  27.475  
  27.476  lemma xzgcd_linear:
  27.477 -    "Numeral0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
  27.478 +    "0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
  27.479    apply (unfold xzgcd_def)
  27.480    apply (erule xzgcda_linear)
  27.481      apply assumption
  27.482 @@ -795,7 +795,7 @@
  27.483    done
  27.484  
  27.485  lemma zgcd_ex_linear:
  27.486 -    "Numeral0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"
  27.487 +    "0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"
  27.488    apply (simp add: xzgcd_correct)
  27.489    apply safe
  27.490    apply (rule exI)+
  27.491 @@ -804,8 +804,8 @@
  27.492    done
  27.493  
  27.494  lemma zcong_lineq_ex:
  27.495 -    "Numeral0 < n ==> zgcd (a, n) = Numeral1 ==> \<exists>x. [a * x = Numeral1] (mod n)"
  27.496 -  apply (cut_tac m = a and n = n and k = "Numeral1" in zgcd_ex_linear)
  27.497 +    "0 < n ==> zgcd (a, n) = 1 ==> \<exists>x. [a * x = 1] (mod n)"
  27.498 +  apply (cut_tac m = a and n = n and k = "1" in zgcd_ex_linear)
  27.499      apply safe
  27.500    apply (rule_tac x = s in exI)
  27.501    apply (rule_tac b = "s * a + t * n" in zcong_trans)
  27.502 @@ -816,8 +816,8 @@
  27.503    done
  27.504  
  27.505  lemma zcong_lineq_unique:
  27.506 -  "Numeral0 < n ==>
  27.507 -    zgcd (a, n) = Numeral1 ==> \<exists>!x. Numeral0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
  27.508 +  "0 < n ==>
  27.509 +    zgcd (a, n) = 1 ==> \<exists>!x. 0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
  27.510    apply auto
  27.511     apply (rule_tac [2] zcong_zless_imp_eq)
  27.512         apply (tactic {* stac (thm "zcong_cancel2" RS sym) 6 *})
  27.513 @@ -833,7 +833,7 @@
  27.514    apply (subst zcong_zmod)
  27.515    apply (subst zmod_zmult1_eq [symmetric])
  27.516    apply (subst zcong_zmod [symmetric])
  27.517 -  apply (subgoal_tac "[a * x * b = Numeral1 * b] (mod n)")
  27.518 +  apply (subgoal_tac "[a * x * b = 1 * b] (mod n)")
  27.519     apply (rule_tac [2] zcong_zmult)
  27.520      apply (simp_all add: zmult_assoc)
  27.521    done
    28.1 --- a/src/HOL/NumberTheory/WilsonBij.thy	Mon Oct 22 11:01:30 2001 +0200
    28.2 +++ b/src/HOL/NumberTheory/WilsonBij.thy	Mon Oct 22 11:54:22 2001 +0200
    28.3 @@ -20,19 +20,19 @@
    28.4  constdefs
    28.5    reciR :: "int => int => int => bool"
    28.6    "reciR p ==
    28.7 -    \<lambda>a b. zcong (a * b) Numeral1 p \<and> Numeral1 < a \<and> a < p - Numeral1 \<and> Numeral1 < b \<and> b < p - Numeral1"
    28.8 +    \<lambda>a b. zcong (a * b) 1 p \<and> 1 < a \<and> a < p - 1 \<and> 1 < b \<and> b < p - 1"
    28.9    inv :: "int => int => int"
   28.10    "inv p a ==
   28.11 -    if p \<in> zprime \<and> Numeral0 < a \<and> a < p then
   28.12 -      (SOME x. Numeral0 \<le> x \<and> x < p \<and> zcong (a * x) Numeral1 p)
   28.13 -    else Numeral0"
   28.14 +    if p \<in> zprime \<and> 0 < a \<and> a < p then
   28.15 +      (SOME x. 0 \<le> x \<and> x < p \<and> zcong (a * x) 1 p)
   28.16 +    else 0"
   28.17  
   28.18  
   28.19  text {* \medskip Inverse *}
   28.20  
   28.21  lemma inv_correct:
   28.22 -  "p \<in> zprime ==> Numeral0 < a ==> a < p
   28.23 -    ==> Numeral0 \<le> inv p a \<and> inv p a < p \<and> [a * inv p a = Numeral1] (mod p)"
   28.24 +  "p \<in> zprime ==> 0 < a ==> a < p
   28.25 +    ==> 0 \<le> inv p a \<and> inv p a < p \<and> [a * inv p a = 1] (mod p)"
   28.26    apply (unfold inv_def)
   28.27    apply (simp (no_asm_simp))
   28.28    apply (rule zcong_lineq_unique [THEN ex1_implies_ex, THEN someI_ex])
   28.29 @@ -46,53 +46,53 @@
   28.30  lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2, standard]
   28.31  
   28.32  lemma inv_not_0:
   28.33 -  "p \<in> zprime ==> Numeral1 < a ==> a < p - Numeral1 ==> inv p a \<noteq> Numeral0"
   28.34 +  "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 0"
   28.35    -- {* same as @{text WilsonRuss} *}
   28.36    apply safe
   28.37    apply (cut_tac a = a and p = p in inv_is_inv)
   28.38       apply (unfold zcong_def)
   28.39       apply auto
   28.40 -  apply (subgoal_tac "\<not> p dvd Numeral1")
   28.41 +  apply (subgoal_tac "\<not> p dvd 1")
   28.42     apply (rule_tac [2] zdvd_not_zless)
   28.43 -    apply (subgoal_tac "p dvd Numeral1")
   28.44 +    apply (subgoal_tac "p dvd 1")
   28.45       prefer 2
   28.46       apply (subst zdvd_zminus_iff [symmetric])
   28.47       apply auto
   28.48    done
   28.49  
   28.50  lemma inv_not_1:
   28.51 -  "p \<in> zprime ==> Numeral1 < a ==> a < p - Numeral1 ==> inv p a \<noteq> Numeral1"
   28.52 +  "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 1"
   28.53    -- {* same as @{text WilsonRuss} *}
   28.54    apply safe
   28.55    apply (cut_tac a = a and p = p in inv_is_inv)
   28.56       prefer 4
   28.57       apply simp
   28.58 -     apply (subgoal_tac "a = Numeral1")
   28.59 +     apply (subgoal_tac "a = 1")
   28.60        apply (rule_tac [2] zcong_zless_imp_eq)
   28.61            apply auto
   28.62    done
   28.63  
   28.64 -lemma aux: "[a * (p - Numeral1) = Numeral1] (mod p) = [a = p - Numeral1] (mod p)"
   28.65 +lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
   28.66    -- {* same as @{text WilsonRuss} *}
   28.67    apply (unfold zcong_def)
   28.68    apply (simp add: zdiff_zdiff_eq zdiff_zdiff_eq2 zdiff_zmult_distrib2)
   28.69 -  apply (rule_tac s = "p dvd -((a + Numeral1) + (p * -a))" in trans)
   28.70 +  apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
   28.71     apply (simp add: zmult_commute zminus_zdiff_eq)
   28.72    apply (subst zdvd_zminus_iff)
   28.73    apply (subst zdvd_reduce)
   28.74 -  apply (rule_tac s = "p dvd (a + Numeral1) + (p * -Numeral1)" in trans)
   28.75 +  apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
   28.76     apply (subst zdvd_reduce)
   28.77     apply auto
   28.78    done
   28.79  
   28.80  lemma inv_not_p_minus_1:
   28.81 -  "p \<in> zprime ==> Numeral1 < a ==> a < p - Numeral1 ==> inv p a \<noteq> p - Numeral1"
   28.82 +  "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> p - 1"
   28.83    -- {* same as @{text WilsonRuss} *}
   28.84    apply safe
   28.85    apply (cut_tac a = a and p = p in inv_is_inv)
   28.86       apply auto
   28.87    apply (simp add: aux)
   28.88 -  apply (subgoal_tac "a = p - Numeral1")
   28.89 +  apply (subgoal_tac "a = p - 1")
   28.90     apply (rule_tac [2] zcong_zless_imp_eq)
   28.91         apply auto
   28.92    done
   28.93 @@ -102,9 +102,9 @@
   28.94    but use ``@{text correct}'' theorems.
   28.95  *}
   28.96  
   28.97 -lemma inv_g_1: "p \<in> zprime ==> Numeral1 < a ==> a < p - Numeral1 ==> Numeral1 < inv p a"
   28.98 -  apply (subgoal_tac "inv p a \<noteq> Numeral1")
   28.99 -   apply (subgoal_tac "inv p a \<noteq> Numeral0")
  28.100 +lemma inv_g_1: "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> 1 < inv p a"
  28.101 +  apply (subgoal_tac "inv p a \<noteq> 1")
  28.102 +   apply (subgoal_tac "inv p a \<noteq> 0")
  28.103      apply (subst order_less_le)
  28.104      apply (subst zle_add1_eq_le [symmetric])
  28.105      apply (subst order_less_le)
  28.106 @@ -116,7 +116,7 @@
  28.107    done
  28.108  
  28.109  lemma inv_less_p_minus_1:
  28.110 -  "p \<in> zprime ==> Numeral1 < a ==> a < p - Numeral1 ==> inv p a < p - Numeral1"
  28.111 +  "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
  28.112    -- {* ditto *}
  28.113    apply (subst order_less_le)
  28.114    apply (simp add: inv_not_p_minus_1 inv_less)
  28.115 @@ -125,11 +125,11 @@
  28.116  
  28.117  text {* \medskip Bijection *}
  28.118  
  28.119 -lemma aux1: "Numeral1 < x ==> Numeral0 \<le> (x::int)"
  28.120 +lemma aux1: "1 < x ==> 0 \<le> (x::int)"
  28.121    apply auto
  28.122    done
  28.123  
  28.124 -lemma aux2: "Numeral1 < x ==> Numeral0 < (x::int)"
  28.125 +lemma aux2: "1 < x ==> 0 < (x::int)"
  28.126    apply auto
  28.127    done
  28.128  
  28.129 @@ -137,7 +137,7 @@
  28.130    apply auto
  28.131    done
  28.132  
  28.133 -lemma aux4: "x \<le> p - 2 ==> x < (p::int)-Numeral1"
  28.134 +lemma aux4: "x \<le> p - 2 ==> x < (p::int) - 1"
  28.135    apply auto
  28.136    done
  28.137  
  28.138 @@ -167,7 +167,7 @@
  28.139    apply auto
  28.140    apply (rule d22set_mem)
  28.141     apply (erule inv_g_1)
  28.142 -    apply (subgoal_tac [3] "inv p xa < p - Numeral1")
  28.143 +    apply (subgoal_tac [3] "inv p xa < p - 1")
  28.144       apply (erule_tac [4] inv_less_p_minus_1)
  28.145        apply (auto intro: d22set_g_1 d22set_le aux4)
  28.146    done
  28.147 @@ -229,28 +229,28 @@
  28.148  subsection {* Wilson *}
  28.149  
  28.150  lemma bijER_zcong_prod_1:
  28.151 -    "p \<in> zprime ==> A \<in> bijER (reciR p) ==> [setprod A = Numeral1] (mod p)"
  28.152 +    "p \<in> zprime ==> A \<in> bijER (reciR p) ==> [setprod A = 1] (mod p)"
  28.153    apply (unfold reciR_def)
  28.154    apply (erule bijER.induct)
  28.155 -    apply (subgoal_tac [2] "a = Numeral1 \<or> a = p - Numeral1")
  28.156 +    apply (subgoal_tac [2] "a = 1 \<or> a = p - 1")
  28.157       apply (rule_tac [3] zcong_square_zless)
  28.158          apply auto
  28.159    apply (subst setprod_insert)
  28.160      prefer 3
  28.161      apply (subst setprod_insert)
  28.162        apply (auto simp add: fin_bijER)
  28.163 -  apply (subgoal_tac "zcong ((a * b) * setprod A) (Numeral1 * Numeral1) p")
  28.164 +  apply (subgoal_tac "zcong ((a * b) * setprod A) (1 * 1) p")
  28.165     apply (simp add: zmult_assoc)
  28.166    apply (rule zcong_zmult)
  28.167     apply auto
  28.168    done
  28.169  
  28.170 -theorem Wilson_Bij: "p \<in> zprime ==> [zfact (p - Numeral1) = -1] (mod p)"
  28.171 -  apply (subgoal_tac "zcong ((p - Numeral1) * zfact (p - 2)) (-1 * Numeral1) p")
  28.172 +theorem Wilson_Bij: "p \<in> zprime ==> [zfact (p - 1) = -1] (mod p)"
  28.173 +  apply (subgoal_tac "zcong ((p - 1) * zfact (p - 2)) (-1 * 1) p")
  28.174     apply (rule_tac [2] zcong_zmult)
  28.175      apply (simp add: zprime_def)
  28.176      apply (subst zfact.simps)
  28.177 -    apply (rule_tac t = "p - Numeral1 - Numeral1" and s = "p - 2" in subst)
  28.178 +    apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
  28.179       apply auto
  28.180     apply (simp add: zcong_def)
  28.181    apply (subst d22set_prod_zfact [symmetric])
    29.1 --- a/src/HOL/NumberTheory/WilsonRuss.thy	Mon Oct 22 11:01:30 2001 +0200
    29.2 +++ b/src/HOL/NumberTheory/WilsonRuss.thy	Mon Oct 22 11:54:22 2001 +0200
    29.3 @@ -25,20 +25,20 @@
    29.4  recdef wset
    29.5    "measure ((\<lambda>(a, p). nat a) :: int * int => nat)"
    29.6    "wset (a, p) =
    29.7 -    (if Numeral1 < a then
    29.8 -      let ws = wset (a - Numeral1, p)
    29.9 +    (if 1 < a then
   29.10 +      let ws = wset (a - 1, p)
   29.11        in (if a \<in> ws then ws else insert a (insert (inv p a) ws)) else {})"
   29.12  
   29.13  
   29.14  text {* \medskip @{term [source] inv} *}
   29.15  
   29.16 -lemma aux: "Numeral1 < m ==> Suc (nat (m - 2)) = nat (m - Numeral1)"
   29.17 +lemma aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
   29.18    apply (subst int_int_eq [symmetric])
   29.19    apply auto
   29.20    done
   29.21  
   29.22  lemma inv_is_inv:
   29.23 -    "p \<in> zprime \<Longrightarrow> Numeral0 < a \<Longrightarrow> a < p ==> [a * inv p a = Numeral1] (mod p)"
   29.24 +    "p \<in> zprime \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"
   29.25    apply (unfold inv_def)
   29.26    apply (subst zcong_zmod)
   29.27    apply (subst zmod_zmult1_eq [symmetric])
   29.28 @@ -52,71 +52,71 @@
   29.29    done
   29.30  
   29.31  lemma inv_distinct:
   29.32 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> a \<noteq> inv p a"
   29.33 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> a \<noteq> inv p a"
   29.34    apply safe
   29.35    apply (cut_tac a = a and p = p in zcong_square)
   29.36       apply (cut_tac [3] a = a and p = p in inv_is_inv)
   29.37          apply auto
   29.38 -   apply (subgoal_tac "a = Numeral1")
   29.39 +   apply (subgoal_tac "a = 1")
   29.40      apply (rule_tac [2] m = p in zcong_zless_imp_eq)
   29.41 -        apply (subgoal_tac [7] "a = p - Numeral1")
   29.42 +        apply (subgoal_tac [7] "a = p - 1")
   29.43           apply (rule_tac [8] m = p in zcong_zless_imp_eq)
   29.44               apply auto
   29.45    done
   29.46  
   29.47  lemma inv_not_0:
   29.48 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> inv p a \<noteq> Numeral0"
   29.49 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 0"
   29.50    apply safe
   29.51    apply (cut_tac a = a and p = p in inv_is_inv)
   29.52       apply (unfold zcong_def)
   29.53       apply auto
   29.54 -  apply (subgoal_tac "\<not> p dvd Numeral1")
   29.55 +  apply (subgoal_tac "\<not> p dvd 1")
   29.56     apply (rule_tac [2] zdvd_not_zless)
   29.57 -    apply (subgoal_tac "p dvd Numeral1")
   29.58 +    apply (subgoal_tac "p dvd 1")
   29.59       prefer 2
   29.60       apply (subst zdvd_zminus_iff [symmetric])
   29.61       apply auto
   29.62    done
   29.63  
   29.64  lemma inv_not_1:
   29.65 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> inv p a \<noteq> Numeral1"
   29.66 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 1"
   29.67    apply safe
   29.68    apply (cut_tac a = a and p = p in inv_is_inv)
   29.69       prefer 4
   29.70       apply simp
   29.71 -     apply (subgoal_tac "a = Numeral1")
   29.72 +     apply (subgoal_tac "a = 1")
   29.73        apply (rule_tac [2] zcong_zless_imp_eq)
   29.74            apply auto
   29.75    done
   29.76  
   29.77 -lemma aux: "[a * (p - Numeral1) = Numeral1] (mod p) = [a = p - Numeral1] (mod p)"
   29.78 +lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
   29.79    apply (unfold zcong_def)
   29.80    apply (simp add: zdiff_zdiff_eq zdiff_zdiff_eq2 zdiff_zmult_distrib2)
   29.81 -  apply (rule_tac s = "p dvd -((a + Numeral1) + (p * -a))" in trans)
   29.82 +  apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
   29.83     apply (simp add: zmult_commute zminus_zdiff_eq)
   29.84    apply (subst zdvd_zminus_iff)
   29.85    apply (subst zdvd_reduce)
   29.86 -  apply (rule_tac s = "p dvd (a + Numeral1) + (p * -Numeral1)" in trans)
   29.87 +  apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
   29.88     apply (subst zdvd_reduce)
   29.89     apply auto
   29.90    done
   29.91  
   29.92  lemma inv_not_p_minus_1:
   29.93 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> inv p a \<noteq> p - Numeral1"
   29.94 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
   29.95    apply safe
   29.96    apply (cut_tac a = a and p = p in inv_is_inv)
   29.97       apply auto
   29.98    apply (simp add: aux)
   29.99 -  apply (subgoal_tac "a = p - Numeral1")
  29.100 +  apply (subgoal_tac "a = p - 1")
  29.101     apply (rule_tac [2] zcong_zless_imp_eq)
  29.102         apply auto
  29.103    done
  29.104  
  29.105  lemma inv_g_1:
  29.106 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> Numeral1 < inv p a"
  29.107 -  apply (case_tac "Numeral0\<le> inv p a")
  29.108 -   apply (subgoal_tac "inv p a \<noteq> Numeral1")
  29.109 -    apply (subgoal_tac "inv p a \<noteq> Numeral0")
  29.110 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> 1 < inv p a"
  29.111 +  apply (case_tac "0\<le> inv p a")
  29.112 +   apply (subgoal_tac "inv p a \<noteq> 1")
  29.113 +    apply (subgoal_tac "inv p a \<noteq> 0")
  29.114       apply (subst order_less_le)
  29.115       apply (subst zle_add1_eq_le [symmetric])
  29.116       apply (subst order_less_le)
  29.117 @@ -128,7 +128,7 @@
  29.118    done
  29.119  
  29.120  lemma inv_less_p_minus_1:
  29.121 -    "p \<in> zprime \<Longrightarrow> Numeral1 < a \<Longrightarrow> a < p - Numeral1 ==> inv p a < p - Numeral1"
  29.122 +    "p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
  29.123    apply (case_tac "inv p a < p")
  29.124     apply (subst order_less_le)
  29.125     apply (simp add: inv_not_p_minus_1)
  29.126 @@ -138,23 +138,23 @@
  29.127    done
  29.128  
  29.129  lemma aux: "5 \<le> p ==>
  29.130 -    nat (p - 2) * nat (p - 2) = Suc (nat (p - Numeral1) * nat (p - 3))"
  29.131 +    nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
  29.132    apply (subst int_int_eq [symmetric])
  29.133    apply (simp add: zmult_int [symmetric])
  29.134    apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
  29.135    done
  29.136  
  29.137  lemma zcong_zpower_zmult:
  29.138 -    "[x^y = Numeral1] (mod p) \<Longrightarrow> [x^(y * z) = Numeral1] (mod p)"
  29.139 +    "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
  29.140    apply (induct z)
  29.141     apply (auto simp add: zpower_zadd_distrib)
  29.142 -  apply (subgoal_tac "zcong (x^y * x^(y * n)) (Numeral1 * Numeral1) p")
  29.143 +  apply (subgoal_tac "zcong (x^y * x^(y * n)) (1 * 1) p")
  29.144     apply (rule_tac [2] zcong_zmult)
  29.145      apply simp_all
  29.146    done
  29.147  
  29.148  lemma inv_inv: "p \<in> zprime \<Longrightarrow>
  29.149 -    5 \<le> p \<Longrightarrow> Numeral0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
  29.150 +    5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
  29.151    apply (unfold inv_def)
  29.152    apply (subst zpower_zmod)
  29.153    apply (subst zpower_zpower)
  29.154 @@ -165,7 +165,7 @@
  29.155        apply (subst zcong_zmod [symmetric])
  29.156        apply (subst aux)
  29.157         apply (subgoal_tac [2]
  29.158 -	 "zcong (a * a^(nat (p - Numeral1) * nat (p - 3))) (a * Numeral1) p")
  29.159 +	 "zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
  29.160          apply (rule_tac [3] zcong_zmult)
  29.161           apply (rule_tac [4] zcong_zpower_zmult)
  29.162           apply (erule_tac [4] Little_Fermat)
  29.163 @@ -180,7 +180,7 @@
  29.164  
  29.165  lemma wset_induct:
  29.166    "(!!a p. P {} a p) \<Longrightarrow>
  29.167 -    (!!a p. Numeral1 < (a::int) \<Longrightarrow> P (wset (a - Numeral1, p)) (a - Numeral1) p
  29.168 +    (!!a p. 1 < (a::int) \<Longrightarrow> P (wset (a - 1, p)) (a - 1) p
  29.169        ==> P (wset (a, p)) a p)
  29.170      ==> P (wset (u, v)) u v"
  29.171  proof -
  29.172 @@ -188,7 +188,7 @@
  29.173    show ?thesis
  29.174      apply (rule wset.induct)
  29.175      apply safe
  29.176 -     apply (case_tac [2] "Numeral1 < a")
  29.177 +     apply (case_tac [2] "1 < a")
  29.178        apply (rule_tac [2] rule_context)
  29.179          apply simp_all
  29.180        apply (simp_all add: wset.simps rule_context)
  29.181 @@ -196,27 +196,27 @@
  29.182  qed
  29.183  
  29.184  lemma wset_mem_imp_or [rule_format]:
  29.185 -  "Numeral1 < a \<Longrightarrow> b \<notin> wset (a - Numeral1, p)
  29.186 +  "1 < a \<Longrightarrow> b \<notin> wset (a - 1, p)
  29.187      ==> b \<in> wset (a, p) --> b = a \<or> b = inv p a"
  29.188    apply (subst wset.simps)
  29.189    apply (unfold Let_def)
  29.190    apply simp
  29.191    done
  29.192  
  29.193 -lemma wset_mem_mem [simp]: "Numeral1 < a ==> a \<in> wset (a, p)"
  29.194 +lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset (a, p)"
  29.195    apply (subst wset.simps)
  29.196    apply (unfold Let_def)
  29.197    apply simp
  29.198    done
  29.199  
  29.200 -lemma wset_subset: "Numeral1 < a \<Longrightarrow> b \<in> wset (a - Numeral1, p) ==> b \<in> wset (a, p)"
  29.201 +lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1, p) ==> b \<in> wset (a, p)"
  29.202    apply (subst wset.simps)
  29.203    apply (unfold Let_def)
  29.204    apply auto
  29.205    done
  29.206  
  29.207  lemma wset_g_1 [rule_format]:
  29.208 -    "p \<in> zprime --> a < p - Numeral1 --> b \<in> wset (a, p) --> Numeral1 < b"
  29.209 +    "p \<in> zprime --> a < p - 1 --> b \<in> wset (a, p) --> 1 < b"
  29.210    apply (induct a p rule: wset_induct)
  29.211     apply auto
  29.212    apply (case_tac "b = a")
  29.213 @@ -230,7 +230,7 @@
  29.214    done
  29.215  
  29.216  lemma wset_less [rule_format]:
  29.217 -    "p \<in> zprime --> a < p - Numeral1 --> b \<in> wset (a, p) --> b < p - Numeral1"
  29.218 +    "p \<in> zprime --> a < p - 1 --> b \<in> wset (a, p) --> b < p - 1"
  29.219    apply (induct a p rule: wset_induct)
  29.220     apply auto
  29.221    apply (case_tac "b = a")
  29.222 @@ -245,7 +245,7 @@
  29.223  
  29.224  lemma wset_mem [rule_format]:
  29.225    "p \<in> zprime -->
  29.226 -    a < p - Numeral1 --> Numeral1 < b --> b \<le> a --> b \<in> wset (a, p)"
  29.227 +    a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset (a, p)"
  29.228    apply (induct a p rule: wset.induct)
  29.229    apply auto
  29.230     apply (subgoal_tac "b = a")
  29.231 @@ -256,7 +256,7 @@
  29.232    done
  29.233  
  29.234  lemma wset_mem_inv_mem [rule_format]:
  29.235 -  "p \<in> zprime --> 5 \<le> p --> a < p - Numeral1 --> b \<in> wset (a, p)
  29.236 +  "p \<in> zprime --> 5 \<le> p --> a < p - 1 --> b \<in> wset (a, p)
  29.237      --> inv p b \<in> wset (a, p)"
  29.238    apply (induct a p rule: wset_induct)
  29.239     apply auto
  29.240 @@ -274,7 +274,7 @@
  29.241    done
  29.242  
  29.243  lemma wset_inv_mem_mem:
  29.244 -  "p \<in> zprime \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - Numeral1 \<Longrightarrow> Numeral1 < b \<Longrightarrow> b < p - Numeral1
  29.245 +  "p \<in> zprime \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - 1 \<Longrightarrow> 1 < b \<Longrightarrow> b < p - 1
  29.246      \<Longrightarrow> inv p b \<in> wset (a, p) \<Longrightarrow> b \<in> wset (a, p)"
  29.247    apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
  29.248     apply (rule_tac [2] wset_mem_inv_mem)
  29.249 @@ -292,7 +292,7 @@
  29.250  
  29.251  lemma wset_zcong_prod_1 [rule_format]:
  29.252    "p \<in> zprime -->
  29.253 -    5 \<le> p --> a < p - Numeral1 --> [setprod (wset (a, p)) = Numeral1] (mod p)"
  29.254 +    5 \<le> p --> a < p - 1 --> [setprod (wset (a, p)) = 1] (mod p)"
  29.255    apply (induct a p rule: wset_induct)
  29.256     prefer 2
  29.257     apply (subst wset.simps)
  29.258 @@ -301,13 +301,13 @@
  29.259    apply (subst setprod_insert)
  29.260      apply (tactic {* stac (thm "setprod_insert") 3 *})
  29.261        apply (subgoal_tac [5]
  29.262 -	"zcong (a * inv p a * setprod (wset (a - Numeral1, p))) (Numeral1 * Numeral1) p")
  29.263 +	"zcong (a * inv p a * setprod (wset (a - 1, p))) (1 * 1) p")
  29.264         prefer 5
  29.265         apply (simp add: zmult_assoc)
  29.266        apply (rule_tac [5] zcong_zmult)
  29.267         apply (rule_tac [5] inv_is_inv)
  29.268           apply (tactic "Clarify_tac 4")
  29.269 -         apply (subgoal_tac [4] "a \<in> wset (a - Numeral1, p)")
  29.270 +         apply (subgoal_tac [4] "a \<in> wset (a - 1, p)")
  29.271            apply (rule_tac [5] wset_inv_mem_mem)
  29.272                 apply (simp_all add: wset_fin)
  29.273    apply (rule inv_distinct)
  29.274 @@ -323,7 +323,7 @@
  29.275        apply (erule_tac [4] wset_g_1)
  29.276         prefer 6
  29.277         apply (subst zle_add1_eq_le [symmetric])
  29.278 -       apply (subgoal_tac "p - 2 + Numeral1 = p - Numeral1")
  29.279 +       apply (subgoal_tac "p - 2 + 1 = p - 1")
  29.280          apply (simp (no_asm_simp))
  29.281          apply (erule wset_less)
  29.282           apply auto
  29.283 @@ -348,12 +348,12 @@
  29.284    done
  29.285  
  29.286  theorem Wilson_Russ:
  29.287 -    "p \<in> zprime ==> [zfact (p - Numeral1) = -1] (mod p)"
  29.288 -  apply (subgoal_tac "[(p - Numeral1) * zfact (p - 2) = -1 * Numeral1] (mod p)")
  29.289 +    "p \<in> zprime ==> [zfact (p - 1) = -1] (mod p)"
  29.290 +  apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
  29.291     apply (rule_tac [2] zcong_zmult)
  29.292      apply (simp only: zprime_def)
  29.293      apply (subst zfact.simps)
  29.294 -    apply (rule_tac t = "p - Numeral1 - Numeral1" and s = "p - 2" in subst)
  29.295 +    apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
  29.296       apply auto
  29.297     apply (simp only: zcong_def)
  29.298     apply (simp (no_asm_simp))
    30.1 --- a/src/HOL/Numeral.thy	Mon Oct 22 11:01:30 2001 +0200
    30.2 +++ b/src/HOL/Numeral.thy	Mon Oct 22 11:54:22 2001 +0200
    30.3 @@ -40,10 +40,10 @@
    30.4  lemma Let_number_of [simp]: "Let (number_of v) f == f (number_of v)"
    30.5    by (simp add: Let_def)
    30.6  
    30.7 -(*The condition "True" is a hack to prevent looping.
    30.8 -  Conditional rewrite rules are tried after unconditional ones, so a rule
    30.9 -  like eq_nat_number_of will be tried first to eliminate #mm=#nn. *)
   30.10 -lemma number_of_reorient [simp]: "True ==> (number_of w = x) = (x = number_of w)"
   30.11 -  by auto
   30.12 +lemma Let_0 [simp]: "Let 0 f == f 0"
   30.13 +  by (simp add: Let_def)
   30.14 +
   30.15 +lemma Let_1 [simp]: "Let 1 f == f 1"
   30.16 +  by (simp add: Let_def)
   30.17  
   30.18  end
    31.1 --- a/src/HOL/ROOT.ML	Mon Oct 22 11:01:30 2001 +0200
    31.2 +++ b/src/HOL/ROOT.ML	Mon Oct 22 11:54:22 2001 +0200
    31.3 @@ -29,6 +29,7 @@
    31.4  use "~~/src/Provers/Arith/abel_cancel.ML";
    31.5  use "~~/src/Provers/Arith/assoc_fold.ML";
    31.6  use "~~/src/Provers/quantifier1.ML";
    31.7 +use "~~/src/Provers/Arith/abstract_numerals.ML";
    31.8  use "~~/src/Provers/Arith/cancel_numerals.ML";
    31.9  use "~~/src/Provers/Arith/combine_numerals.ML";
   31.10  use "~~/src/Provers/Arith/cancel_numeral_factor.ML";
    32.1 --- a/src/HOL/Real/ex/Sqrt_Irrational.thy	Mon Oct 22 11:01:30 2001 +0200
    32.2 +++ b/src/HOL/Real/ex/Sqrt_Irrational.thy	Mon Oct 22 11:54:22 2001 +0200
    32.3 @@ -119,7 +119,7 @@
    32.4    {
    32.5      fix m :: nat assume dvd: "m dvd 2"
    32.6      hence "m \<le> 2" by (simp add: dvd_imp_le)
    32.7 -    moreover from dvd have "m \<noteq> 0" by (auto dest: dvd_0_left iff del: neq0_conv)
    32.8 +    moreover from dvd have "m \<noteq> 0" by (auto iff del: neq0_conv)
    32.9      ultimately have "m = 1 \<or> m = 2" by arith
   32.10    }
   32.11    thus "2 \<in> prime" by (simp add: prime_def)
    33.1 --- a/src/HOL/SVC_Oracle.ML	Mon Oct 22 11:01:30 2001 +0200
    33.2 +++ b/src/HOL/SVC_Oracle.ML	Mon Oct 22 11:54:22 2001 +0200
    33.3 @@ -41,30 +41,29 @@
    33.4  	  | _ => (case gen_assoc Pattern.aeconv (!pairs, t) of
    33.5  		      Some v => v
    33.6  		    | None   => insert t)
    33.7 +    (*abstraction of a numeric literal*)
    33.8 +    fun lit (t as Const("0", _)) = t
    33.9 +      | lit (t as Const("1", _)) = t
   33.10 +      | lit (t as Const("Numeral.number_of", _) $ w) = t
   33.11 +      | lit t = replace t
   33.12      (*abstraction of a real/rational expression*)
   33.13      fun rat ((c as Const("op +", _)) $ x $ y) = c $ (rat x) $ (rat y)
   33.14        | rat ((c as Const("op -", _)) $ x $ y) = c $ (rat x) $ (rat y)
   33.15        | rat ((c as Const("op /", _)) $ x $ y) = c $ (rat x) $ (rat y)
   33.16        | rat ((c as Const("op *", _)) $ x $ y) = c $ (rat x) $ (rat y)
   33.17        | rat ((c as Const("uminus", _)) $ x) = c $ (rat x)
   33.18 -      | rat ((c as Const("0", _))) = c
   33.19 -      | rat ((c as Const("1", _))) = c
   33.20 -      | rat (t as Const("Numeral.number_of", _) $ w) = t
   33.21 -      | rat t = replace t
   33.22 +      | rat t = lit t
   33.23      (*abstraction of an integer expression: no div, mod*)
   33.24      fun int ((c as Const("op +", _)) $ x $ y) = c $ (int x) $ (int y)
   33.25        | int ((c as Const("op -", _)) $ x $ y) = c $ (int x) $ (int y)
   33.26        | int ((c as Const("op *", _)) $ x $ y) = c $ (int x) $ (int y)
   33.27        | int ((c as Const("uminus", _)) $ x) = c $ (int x)
   33.28 -      | int (t as Const("Numeral.number_of", _) $ w) = t
   33.29 -      | int t = replace t
   33.30 +      | int t = lit t
   33.31      (*abstraction of a natural number expression: no minus*)
   33.32      fun nat ((c as Const("op +", _)) $ x $ y) = c $ (nat x) $ (nat y)
   33.33        | nat ((c as Const("op *", _)) $ x $ y) = c $ (nat x) $ (nat y)
   33.34        | nat ((c as Const("Suc", _)) $ x) = c $ (nat x)
   33.35 -      | nat (t as Const("0", _)) = t
   33.36 -      | nat (t as Const("Numeral.number_of", _) $ w) = t
   33.37 -      | nat t = replace t
   33.38 +      | nat t = lit t
   33.39      (*abstraction of a relation: =, <, <=*)
   33.40      fun rel (T, c $ x $ y) =
   33.41  	    if T = HOLogic.realT then c $ (rat x) $ (rat y)
    34.1 --- a/src/HOL/UNITY/Comp/Counter.ML	Mon Oct 22 11:01:30 2001 +0200
    34.2 +++ b/src/HOL/UNITY/Comp/Counter.ML	Mon Oct 22 11:54:22 2001 +0200
    34.3 @@ -100,7 +100,7 @@
    34.4  
    34.5  (* Compositional Proof *)
    34.6  
    34.7 -Goal "(ALL i. i < I --> s (c i) = Numeral0) --> sum I s = Numeral0";
    34.8 +Goal "(ALL i. i < I --> s (c i) = 0) --> sum I s = 0";
    34.9  by (induct_tac "I" 1);
   34.10  by Auto_tac;
   34.11  qed "sum_0'";
    35.1 --- a/src/HOL/UNITY/Comp/Counter.thy	Mon Oct 22 11:01:30 2001 +0200
    35.2 +++ b/src/HOL/UNITY/Comp/Counter.thy	Mon Oct 22 11:54:22 2001 +0200
    35.3 @@ -21,21 +21,21 @@
    35.4    sumj :: "[nat, nat, state]=>int"
    35.5  
    35.6  primrec (* sum I s = sigma_{i<I}. s (c i) *)
    35.7 -  "sum 0 s = Numeral0"
    35.8 +  "sum 0 s = 0"
    35.9    "sum (Suc i) s = s (c i) + sum i s"
   35.10  
   35.11  primrec
   35.12 -  "sumj 0 i s = Numeral0"
   35.13 +  "sumj 0 i s = 0"
   35.14    "sumj (Suc n) i s = (if n=i then sum n s else s (c n) + sumj n i s)"
   35.15    
   35.16  types command = "(state*state)set"
   35.17  
   35.18  constdefs
   35.19    a :: "nat=>command"
   35.20 - "a i == {(s, s'). s'=s(c i:= s (c i) + Numeral1, C:= s C + Numeral1)}"
   35.21 + "a i == {(s, s'). s'=s(c i:= s (c i) + 1, C:= s C + 1)}"
   35.22  
   35.23    Component :: "nat => state program"
   35.24    "Component i ==
   35.25 -    mk_program({s. s C = Numeral0 & s (c i) = Numeral0}, {a i},
   35.26 +    mk_program({s. s C = 0 & s (c i) = 0}, {a i},
   35.27  	       UN G: preserves (%s. s (c i)). Acts G)"
   35.28  end  
    36.1 --- a/src/HOL/UNITY/Comp/Counterc.ML	Mon Oct 22 11:01:30 2001 +0200
    36.2 +++ b/src/HOL/UNITY/Comp/Counterc.ML	Mon Oct 22 11:54:22 2001 +0200
    36.3 @@ -38,7 +38,7 @@
    36.4  qed_spec_mp "sumj_ext";
    36.5  
    36.6  
    36.7 -Goal "(ALL i. i<I --> c s i = Numeral0) -->  sum I s = Numeral0";
    36.8 +Goal "(ALL i. i<I --> c s i = 0) -->  sum I s = 0";
    36.9  by (induct_tac "I" 1);
   36.10  by Auto_tac;
   36.11  qed "sum0";
    37.1 --- a/src/HOL/UNITY/Comp/Counterc.thy	Mon Oct 22 11:01:30 2001 +0200
    37.2 +++ b/src/HOL/UNITY/Comp/Counterc.thy	Mon Oct 22 11:54:22 2001 +0200
    37.3 @@ -24,20 +24,20 @@
    37.4    sumj :: "[nat, nat, state]=>int"
    37.5  
    37.6  primrec (* sum I s = sigma_{i<I}. c s i *)
    37.7 -  "sum 0 s = Numeral0"
    37.8 +  "sum 0 s = 0"
    37.9    "sum (Suc i) s = (c s) i + sum i s"
   37.10  
   37.11  primrec
   37.12 -  "sumj 0 i s = Numeral0"
   37.13 +  "sumj 0 i s = 0"
   37.14    "sumj (Suc n) i s = (if n=i then sum n s else (c s) n + sumj n i s)"
   37.15    
   37.16  types command = "(state*state)set"
   37.17  
   37.18  constdefs
   37.19    a :: "nat=>command"
   37.20 - "a i == {(s, s'). (c s') i = (c s) i + Numeral1 & (C s') = (C s) + Numeral1}"
   37.21 + "a i == {(s, s'). (c s') i = (c s) i + 1 & (C s') = (C s) + 1}"
   37.22   
   37.23    Component :: "nat => state program"
   37.24 -  "Component i == mk_program({s. C s = Numeral0 & (c s) i = Numeral0}, {a i},
   37.25 +  "Component i == mk_program({s. C s = 0 & (c s) i = 0}, {a i},
   37.26  	       UN G: preserves (%s. (c s) i). Acts G)"
   37.27  end  
    38.1 --- a/src/HOL/UNITY/Simple/Lift.ML	Mon Oct 22 11:01:30 2001 +0200
    38.2 +++ b/src/HOL/UNITY/Simple/Lift.ML	Mon Oct 22 11:54:22 2001 +0200
    38.3 @@ -142,7 +142,7 @@
    38.4  
    38.5  
    38.6  (*lem_lift_4_1 *)
    38.7 -Goal "Numeral0 < N ==> \
    38.8 +Goal "0 < N ==> \
    38.9  \     Lift : (moving Int Req n Int {s. metric n s = N} Int \
   38.10  \             {s. floor s ~: req s} Int {s. up s})   \
   38.11  \            LeadsTo \
   38.12 @@ -157,7 +157,7 @@
   38.13  
   38.14  
   38.15  (*lem_lift_4_3 *)
   38.16 -Goal "Numeral0 < N ==> \
   38.17 +Goal "0 < N ==> \
   38.18  \     Lift : (moving Int Req n Int {s. metric n s = N} Int \
   38.19  \             {s. floor s ~: req s} - {s. up s})   \
   38.20  \            LeadsTo (moving Int Req n Int {s. metric n s < N})";
   38.21 @@ -170,7 +170,7 @@
   38.22  qed "E_thm12b";
   38.23  
   38.24  (*lift_4*)
   38.25 -Goal "Numeral0<N ==> Lift : (moving Int Req n Int {s. metric n s = N} Int \
   38.26 +Goal "0<N ==> Lift : (moving Int Req n Int {s. metric n s = N} Int \
   38.27  \                           {s. floor s ~: req s}) LeadsTo     \
   38.28  \                          (moving Int Req n Int {s. metric n s < N})";
   38.29  by (rtac ([subset_imp_LeadsTo, [E_thm12a, E_thm12b] MRS LeadsTo_Un] 
   38.30 @@ -182,7 +182,7 @@
   38.31  (** towards lift_5 **)
   38.32  
   38.33  (*lem_lift_5_3*)
   38.34 -Goal "Numeral0<N   \
   38.35 +Goal "0<N   \
   38.36  \ ==> Lift : (closed Int Req n Int {s. metric n s = N} Int goingup) LeadsTo \
   38.37  \            (moving Int Req n Int {s. metric n s < N})";
   38.38  by (cut_facts_tac [bounded] 1);
   38.39 @@ -192,7 +192,7 @@
   38.40  
   38.41  
   38.42  (*lem_lift_5_1 has ~goingup instead of goingdown*)
   38.43 -Goal "Numeral0<N ==>   \
   38.44 +Goal "0<N ==>   \
   38.45  \     Lift : (closed Int Req n Int {s. metric n s = N} Int goingdown) LeadsTo \
   38.46  \                  (moving Int Req n Int {s. metric n s < N})";
   38.47  by (cut_facts_tac [bounded] 1);
   38.48 @@ -203,14 +203,14 @@
   38.49  
   38.50  (*lem_lift_5_0 proves an intersection involving ~goingup and goingup,
   38.51    i.e. the trivial disjunction, leading to an asymmetrical proof.*)
   38.52 -Goal "Numeral0<N ==> Req n Int {s. metric n s = N} <= goingup Un goingdown";
   38.53 +Goal "0<N ==> Req n Int {s. metric n s = N} <= goingup Un goingdown";
   38.54  by (Clarify_tac 1);
   38.55  by (auto_tac (claset(), metric_ss));
   38.56  qed "E_thm16c";
   38.57  
   38.58  
   38.59  (*lift_5*)
   38.60 -Goal "Numeral0<N ==> Lift : (closed Int Req n Int {s. metric n s = N}) LeadsTo   \
   38.61 +Goal "0<N ==> Lift : (closed Int Req n Int {s. metric n s = N}) LeadsTo   \
   38.62  \                          (moving Int Req n Int {s. metric n s < N})";
   38.63  by (rtac ([subset_imp_LeadsTo, [E_thm16a, E_thm16b] MRS LeadsTo_Un] 
   38.64  	  MRS LeadsTo_Trans) 1);
   38.65 @@ -222,7 +222,7 @@
   38.66  (** towards lift_3 **)
   38.67  
   38.68  (*lemma used to prove lem_lift_3_1*)
   38.69 -Goal "[| metric n s = Numeral0;  Min <= floor s;  floor s <= Max |] ==> floor s = n";
   38.70 +Goal "[| metric n s = 0;  Min <= floor s;  floor s <= Max |] ==> floor s = n";
   38.71  by (auto_tac (claset(), metric_ss));
   38.72  qed "metric_eq_0D";
   38.73  
   38.74 @@ -230,7 +230,7 @@
   38.75  
   38.76  
   38.77  (*lem_lift_3_1*)
   38.78 -Goal "Lift : (moving Int Req n Int {s. metric n s = Numeral0}) LeadsTo   \
   38.79 +Goal "Lift : (moving Int Req n Int {s. metric n s = 0}) LeadsTo   \
   38.80  \                  (stopped Int atFloor n)";
   38.81  by (cut_facts_tac [bounded] 1);
   38.82  by (ensures_tac "request_act" 1);
   38.83 @@ -246,7 +246,7 @@
   38.84  qed "E_thm13";
   38.85  
   38.86  (*lem_lift_3_6*)
   38.87 -Goal "Numeral0 < N ==> \
   38.88 +Goal "0 < N ==> \
   38.89  \     Lift : \
   38.90  \       (stopped Int Req n Int {s. metric n s = N} Int {s. floor s : req s}) \
   38.91  \       LeadsTo (opened Int Req n Int {s. metric n s = N})";
   38.92 @@ -264,7 +264,7 @@
   38.93  
   38.94  (** the final steps **)
   38.95  
   38.96 -Goal "Numeral0 < N ==> \
   38.97 +Goal "0 < N ==> \
   38.98  \     Lift : \
   38.99  \       (moving Int Req n Int {s. metric n s = N} Int {s. floor s : req s})   \
  38.100  \       LeadsTo (moving Int Req n Int {s. metric n s < N})";
  38.101 @@ -274,7 +274,7 @@
  38.102  
  38.103  
  38.104  (*Now we observe that our integer metric is really a natural number*)
  38.105 -Goal "Lift : Always {s. Numeral0 <= metric n s}";
  38.106 +Goal "Lift : Always {s. 0 <= metric n s}";
  38.107  by (rtac (bounded RS Always_weaken) 1);
  38.108  by (auto_tac (claset(), metric_ss));
  38.109  qed "Always_nonneg";
  38.110 @@ -283,8 +283,8 @@
  38.111  
  38.112  Goal "Lift : (moving Int Req n) LeadsTo (stopped Int atFloor n)";
  38.113  by (rtac (Always_nonneg RS integ_0_le_induct) 1);
  38.114 -by (case_tac "Numeral0 < z" 1);
  38.115 -(*If z <= Numeral0 then actually z = Numeral0*)
  38.116 +by (case_tac "0 < z" 1);
  38.117 +(*If z <= 0 then actually z = 0*)
  38.118  by (force_tac (claset() addIs [R_thm11, order_antisym], 
  38.119  	       simpset() addsimps [linorder_not_less]) 2);
  38.120  by (rtac ([asm_rl, Un_upper1] MRS LeadsTo_weaken_R) 1);
    39.1 --- a/src/HOL/UNITY/Simple/Lift.thy	Mon Oct 22 11:01:30 2001 +0200
    39.2 +++ b/src/HOL/UNITY/Simple/Lift.thy	Mon Oct 22 11:54:22 2001 +0200
    39.3 @@ -87,25 +87,25 @@
    39.4    req_up :: "(state*state) set"
    39.5      "req_up ==
    39.6           {(s,s'). s' = s (|stop  :=False,
    39.7 -			   floor := floor s + Numeral1,
    39.8 +			   floor := floor s + 1,
    39.9  			   up    := True|)
   39.10  		       & s : (ready Int goingup)}"
   39.11  
   39.12    req_down :: "(state*state) set"
   39.13      "req_down ==
   39.14           {(s,s'). s' = s (|stop  :=False,
   39.15 -			   floor := floor s - Numeral1,
   39.16 +			   floor := floor s - 1,
   39.17  			   up    := False|)
   39.18  		       & s : (ready Int goingdown)}"
   39.19  
   39.20    move_up :: "(state*state) set"
   39.21      "move_up ==
   39.22 -         {(s,s'). s' = s (|floor := floor s + Numeral1|)
   39.23 +         {(s,s'). s' = s (|floor := floor s + 1|)
   39.24  		       & ~ stop s & up s & floor s ~: req s}"
   39.25  
   39.26    move_down :: "(state*state) set"
   39.27      "move_down ==
   39.28 -         {(s,s'). s' = s (|floor := floor s - Numeral1|)
   39.29 +         {(s,s'). s' = s (|floor := floor s - 1|)
   39.30  		       & ~ stop s & ~ up s & floor s ~: req s}"
   39.31  
   39.32    (*This action is omitted from prior treatments, which therefore are
   39.33 @@ -156,7 +156,7 @@
   39.34               else
   39.35               if n < floor s then (if up s then (Max - floor s) + (Max-n)
   39.36  		                  else floor s - n)
   39.37 -             else Numeral0"
   39.38 +             else 0"
   39.39  
   39.40  locale floor =
   39.41    fixes 
    40.1 --- a/src/HOL/UNITY/Simple/Mutex.ML	Mon Oct 22 11:01:30 2001 +0200
    40.2 +++ b/src/HOL/UNITY/Simple/Mutex.ML	Mon Oct 22 11:54:22 2001 +0200
    40.3 @@ -42,7 +42,7 @@
    40.4  getgoal 1;  
    40.5  
    40.6  
    40.7 -Goal "((Numeral1::int) <= i & i <= 3) = (i = Numeral1 | i = 2 | i = 3)";
    40.8 +Goal "((1::int) <= i & i <= 3) = (i = 1 | i = 2 | i = 3)";
    40.9  by (arith_tac 1);
   40.10  qed "eq_123";
   40.11  
   40.12 @@ -53,7 +53,7 @@
   40.13  by (constrains_tac 1);
   40.14  qed "U_F0";
   40.15  
   40.16 -Goal "Mutex : {s. m s=Numeral1} LeadsTo {s. p s = v s & m s = 2}";
   40.17 +Goal "Mutex : {s. m s=1} LeadsTo {s. p s = v s & m s = 2}";
   40.18  by (ensures_tac "U1" 1);
   40.19  qed "U_F1";
   40.20  
   40.21 @@ -75,12 +75,12 @@
   40.22  by (auto_tac (claset() addSEs [less_SucE], simpset()));
   40.23  val U_lemma2 = result();
   40.24  
   40.25 -Goal "Mutex : {s. m s = Numeral1} LeadsTo {s. p s}";
   40.26 +Goal "Mutex : {s. m s = 1} LeadsTo {s. p s}";
   40.27  by (rtac ([U_F1 RS LeadsTo_weaken_R, U_lemma2] MRS LeadsTo_Trans) 1);
   40.28  by (Blast_tac 1);
   40.29  val U_lemma1 = result();
   40.30  
   40.31 -Goal "Mutex : {s. Numeral1 <= m s & m s <= 3} LeadsTo {s. p s}";
   40.32 +Goal "Mutex : {s. 1 <= m s & m s <= 3} LeadsTo {s. p s}";
   40.33  by (simp_tac (simpset() addsimps [eq_123, Collect_disj_eq, LeadsTo_Un_distrib,
   40.34  				  U_lemma1, U_lemma2, U_F3] ) 1);
   40.35  val U_lemma123 = result();
   40.36 @@ -99,7 +99,7 @@
   40.37  by (constrains_tac 1);
   40.38  qed "V_F0";
   40.39  
   40.40 -Goal "Mutex : {s. n s=Numeral1} LeadsTo {s. p s = (~ u s) & n s = 2}";
   40.41 +Goal "Mutex : {s. n s=1} LeadsTo {s. p s = (~ u s) & n s = 2}";
   40.42  by (ensures_tac "V1" 1);
   40.43  qed "V_F1";
   40.44  
   40.45 @@ -121,12 +121,12 @@
   40.46  by (auto_tac (claset() addSEs [less_SucE], simpset()));
   40.47  val V_lemma2 = result();
   40.48  
   40.49 -Goal "Mutex : {s. n s = Numeral1} LeadsTo {s. ~ p s}";
   40.50 +Goal "Mutex : {s. n s = 1} LeadsTo {s. ~ p s}";
   40.51  by (rtac ([V_F1 RS LeadsTo_weaken_R, V_lemma2] MRS LeadsTo_Trans) 1);
   40.52  by (Blast_tac 1);
   40.53  val V_lemma1 = result();
   40.54  
   40.55 -Goal "Mutex : {s. Numeral1 <= n s & n s <= 3} LeadsTo {s. ~ p s}";
   40.56 +Goal "Mutex : {s. 1 <= n s & n s <= 3} LeadsTo {s. ~ p s}";
   40.57  by (simp_tac (simpset() addsimps [eq_123, Collect_disj_eq, LeadsTo_Un_distrib,
   40.58  				  V_lemma1, V_lemma2, V_F3] ) 1);
   40.59  val V_lemma123 = result();
   40.60 @@ -142,7 +142,7 @@
   40.61  (** Absence of starvation **)
   40.62  
   40.63  (*Misra's F6*)
   40.64 -Goal "Mutex : {s. m s = Numeral1} LeadsTo {s. m s = 3}";
   40.65 +Goal "Mutex : {s. m s = 1} LeadsTo {s. m s = 3}";
   40.66  by (rtac (LeadsTo_cancel2 RS LeadsTo_Un_duplicate) 1);
   40.67  by (rtac U_F2 2);
   40.68  by (simp_tac (simpset() addsimps [Collect_conj_eq] ) 1);
   40.69 @@ -154,7 +154,7 @@
   40.70  qed "m1_Leadsto_3";
   40.71  
   40.72  (*The same for V*)
   40.73 -Goal "Mutex : {s. n s = Numeral1} LeadsTo {s. n s = 3}";
   40.74 +Goal "Mutex : {s. n s = 1} LeadsTo {s. n s = 3}";
   40.75  by (rtac (LeadsTo_cancel2 RS LeadsTo_Un_duplicate) 1);
   40.76  by (rtac V_F2 2);
   40.77  by (simp_tac (simpset() addsimps [Collect_conj_eq] ) 1);
    41.1 --- a/src/HOL/UNITY/Simple/Mutex.thy	Mon Oct 22 11:01:30 2001 +0200
    41.2 +++ b/src/HOL/UNITY/Simple/Mutex.thy	Mon Oct 22 11:54:22 2001 +0200
    41.3 @@ -22,10 +22,10 @@
    41.4    (** The program for process U **)
    41.5    
    41.6    U0 :: command
    41.7 -    "U0 == {(s,s'). s' = s (|u:=True, m:=Numeral1|) & m s = Numeral0}"
    41.8 +    "U0 == {(s,s'). s' = s (|u:=True, m:=1|) & m s = 0}"
    41.9  
   41.10    U1 :: command
   41.11 -    "U1 == {(s,s'). s' = s (|p:= v s, m:=2|) & m s = Numeral1}"
   41.12 +    "U1 == {(s,s'). s' = s (|p:= v s, m:=2|) & m s = 1}"
   41.13  
   41.14    U2 :: command
   41.15      "U2 == {(s,s'). s' = s (|m:=3|) & ~ p s & m s = 2}"
   41.16 @@ -34,15 +34,15 @@
   41.17      "U3 == {(s,s'). s' = s (|u:=False, m:=4|) & m s = 3}"
   41.18  
   41.19    U4 :: command
   41.20 -    "U4 == {(s,s'). s' = s (|p:=True, m:=Numeral0|) & m s = 4}"
   41.21 +    "U4 == {(s,s'). s' = s (|p:=True, m:=0|) & m s = 4}"
   41.22  
   41.23    (** The program for process V **)
   41.24    
   41.25    V0 :: command
   41.26 -    "V0 == {(s,s'). s' = s (|v:=True, n:=Numeral1|) & n s = Numeral0}"
   41.27 +    "V0 == {(s,s'). s' = s (|v:=True, n:=1|) & n s = 0}"
   41.28  
   41.29    V1 :: command
   41.30 -    "V1 == {(s,s'). s' = s (|p:= ~ u s, n:=2|) & n s = Numeral1}"
   41.31 +    "V1 == {(s,s'). s' = s (|p:= ~ u s, n:=2|) & n s = 1}"
   41.32  
   41.33    V2 :: command
   41.34      "V2 == {(s,s'). s' = s (|n:=3|) & p s & n s = 2}"
   41.35 @@ -51,10 +51,10 @@
   41.36      "V3 == {(s,s'). s' = s (|v:=False, n:=4|) & n s = 3}"
   41.37  
   41.38    V4 :: command
   41.39 -    "V4 == {(s,s'). s' = s (|p:=False, n:=Numeral0|) & n s = 4}"
   41.40 +    "V4 == {(s,s'). s' = s (|p:=False, n:=0|) & n s = 4}"
   41.41  
   41.42    Mutex :: state program
   41.43 -    "Mutex == mk_program ({s. ~ u s & ~ v s & m s = Numeral0 & n s = Numeral0},
   41.44 +    "Mutex == mk_program ({s. ~ u s & ~ v s & m s = 0 & n s = 0},
   41.45  		 	  {U0, U1, U2, U3, U4, V0, V1, V2, V3, V4},
   41.46  			  UNIV)"
   41.47  
   41.48 @@ -62,15 +62,15 @@
   41.49    (** The correct invariants **)
   41.50  
   41.51    IU :: state set
   41.52 -    "IU == {s. (u s = (Numeral1 <= m s & m s <= 3)) & (m s = 3 --> ~ p s)}"
   41.53 +    "IU == {s. (u s = (1 <= m s & m s <= 3)) & (m s = 3 --> ~ p s)}"
   41.54  
   41.55    IV :: state set
   41.56 -    "IV == {s. (v s = (Numeral1 <= n s & n s <= 3)) & (n s = 3 --> p s)}"
   41.57 +    "IV == {s. (v s = (1 <= n s & n s <= 3)) & (n s = 3 --> p s)}"
   41.58  
   41.59    (** The faulty invariant (for U alone) **)
   41.60  
   41.61    bad_IU :: state set
   41.62 -    "bad_IU == {s. (u s = (Numeral1 <= m s & m s <= 3)) &
   41.63 +    "bad_IU == {s. (u s = (1 <= m s & m s <= 3)) &
   41.64  	           (3 <= m s & m s <= 4 --> ~ p s)}"
   41.65  
   41.66  end
    42.1 --- a/src/HOL/UNITY/SubstAx.ML	Mon Oct 22 11:01:30 2001 +0200
    42.2 +++ b/src/HOL/UNITY/SubstAx.ML	Mon Oct 22 11:54:22 2001 +0200
    42.3 @@ -341,9 +341,9 @@
    42.4  by (auto_tac (claset() addIs prems, simpset()));
    42.5  qed "LessThan_induct";
    42.6  
    42.7 -(*Integer version.  Could generalize from Numeral0 to any lower bound*)
    42.8 +(*Integer version.  Could generalize from 0 to any lower bound*)
    42.9  val [reach, prem] =
   42.10 -Goal "[| F : Always {s. (Numeral0::int) <= f s};  \
   42.11 +Goal "[| F : Always {s. (0::int) <= f s};  \
   42.12  \        !! z. F : (A Int {s. f s = z}) LeadsTo                     \
   42.13  \                           ((A Int {s. f s < z}) Un B) |] \
   42.14  \     ==> F : A LeadsTo B";
    43.1 --- a/src/HOL/UNITY/Union.ML	Mon Oct 22 11:01:30 2001 +0200
    43.2 +++ b/src/HOL/UNITY/Union.ML	Mon Oct 22 11:54:22 2001 +0200
    43.3 @@ -352,7 +352,7 @@
    43.4  
    43.5  bind_thm ("ok_sym", ok_commute RS iffD1);
    43.6  
    43.7 -Goal "OK {(Numeral0::int,F),(Numeral1,G),(2,H)} snd = (F ok G & (F Join G) ok H)";
    43.8 +Goal "OK {(0::int,F),(1,G),(2,H)} snd = (F ok G & (F Join G) ok H)";
    43.9  by (asm_full_simp_tac
   43.10      (simpset() addsimps [Ball_def, conj_disj_distribR, ok_def, Join_def, 
   43.11                     OK_def, insert_absorb, all_conj_distrib, eq_commute]) 1); 
    44.1 --- a/src/HOL/ex/BinEx.thy	Mon Oct 22 11:01:30 2001 +0200
    44.2 +++ b/src/HOL/ex/BinEx.thy	Mon Oct 22 11:54:22 2001 +0200
    44.3 @@ -60,7 +60,7 @@
    44.4  lemma "(13557456::int) < 18678654"
    44.5    by simp
    44.6  
    44.7 -lemma "(999999::int) \<le> (1000001 + Numeral1) - 2"
    44.8 +lemma "(999999::int) \<le> (1000001 + 1) - 2"
    44.9    by simp
   44.10  
   44.11  lemma "(1234567::int) \<le> 1234567"
   44.12 @@ -72,7 +72,7 @@
   44.13  lemma "(10::int) div 3 = 3"
   44.14    by simp
   44.15  
   44.16 -lemma "(10::int) mod 3 = Numeral1"
   44.17 +lemma "(10::int) mod 3 = 1"
   44.18    by simp
   44.19  
   44.20  text {* A negative divisor *}
   44.21 @@ -104,7 +104,7 @@
   44.22  
   44.23  text {* A few bigger examples *}
   44.24  
   44.25 -lemma "(8452::int) mod 3 = Numeral1"
   44.26 +lemma "(8452::int) mod 3 = 1"
   44.27    by simp
   44.28  
   44.29  lemma "(59485::int) div 434 = 137"
   44.30 @@ -119,7 +119,7 @@
   44.31  lemma "10000000 div 2 = (5000000::int)"
   44.32    by simp
   44.33  
   44.34 -lemma "10000001 mod 2 = (Numeral1::int)"
   44.35 +lemma "10000001 mod 2 = (1::int)"
   44.36    by simp
   44.37  
   44.38  lemma "10000055 div 32 = (312501::int)"
   44.39 @@ -161,10 +161,10 @@
   44.40  lemma "(32::nat) - 14 = 18"
   44.41    by simp
   44.42  
   44.43 -lemma "(14::nat) - 15 = Numeral0"
   44.44 +lemma "(14::nat) - 15 = 0"
   44.45    by simp
   44.46  
   44.47 -lemma "(14::nat) - 1576644 = Numeral0"
   44.48 +lemma "(14::nat) - 1576644 = 0"
   44.49    by simp
   44.50  
   44.51  lemma "(48273776::nat) - 3873737 = 44400039"
   44.52 @@ -185,49 +185,49 @@
   44.53  lemma "(10::nat) div 3 = 3"
   44.54    by simp
   44.55  
   44.56 -lemma "(10::nat) mod 3 = Numeral1"
   44.57 +lemma "(10::nat) mod 3 = 1"
   44.58    by simp
   44.59  
   44.60  lemma "(10000::nat) div 9 = 1111"
   44.61    by simp
   44.62  
   44.63 -lemma "(10000::nat) mod 9 = Numeral1"
   44.64 +lemma "(10000::nat) mod 9 = 1"
   44.65    by simp
   44.66  
   44.67  lemma "(10000::nat) div 16 = 625"
   44.68    by simp
   44.69  
   44.70 -lemma "(10000::nat) mod 16 = Numeral0"
   44.71 +lemma "(10000::nat) mod 16 = 0"
   44.72    by simp
   44.73  
   44.74  
   44.75  text {* \medskip Testing the cancellation of complementary terms *}
   44.76  
   44.77 -lemma "y + (x + -x) = (Numeral0::int) + y"
   44.78 +lemma "y + (x + -x) = (0::int) + y"
   44.79    by simp
   44.80  
   44.81 -lemma "y + (-x + (- y + x)) = (Numeral0::int)"
   44.82 +lemma "y + (-x + (- y + x)) = (0::int)"
   44.83    by simp
   44.84  
   44.85 -lemma "-x + (y + (- y + x)) = (Numeral0::int)"
   44.86 +lemma "-x + (y + (- y + x)) = (0::int)"
   44.87    by simp
   44.88  
   44.89 -lemma "x + (x + (- x + (- x + (- y + - z)))) = (Numeral0::int) - y - z"
   44.90 +lemma "x + (x + (- x + (- x + (- y + - z)))) = (0::int) - y - z"
   44.91    by simp
   44.92  
   44.93 -lemma "x + x - x - x - y - z = (Numeral0::int) - y - z"
   44.94 +lemma "x + x - x - x - y - z = (0::int) - y - z"
   44.95    by simp
   44.96  
   44.97 -lemma "x + y + z - (x + z) = y - (Numeral0::int)"
   44.98 +lemma "x + y + z - (x + z) = y - (0::int)"
   44.99    by simp
  44.100  
  44.101 -lemma "x + (y + (y + (y + (-x + -x)))) = (Numeral0::int) + y - x + y + y"
  44.102 +lemma "x + (y + (y + (y + (-x + -x)))) = (0::int) + y - x + y + y"
  44.103    by simp
  44.104  
  44.105 -lemma "x + (y + (y + (y + (-y + -x)))) = y + (Numeral0::int) + y"
  44.106 +lemma "x + (y + (y + (y + (-y + -x)))) = y + (0::int) + y"
  44.107    by simp
  44.108  
  44.109 -lemma "x + y - x + z - x - y - z + x < (Numeral1::int)"
  44.110 +lemma "x + y - x + z - x - y - z + x < (1::int)"
  44.111    by simp
  44.112  
  44.113  
  44.114 @@ -302,7 +302,7 @@
  44.115      apply simp_all
  44.116    done
  44.117  
  44.118 -lemma normal_Pls_eq_0: "w \<in> normal ==> (w = Pls) = (number_of w = (Numeral0::int))"
  44.119 +lemma normal_Pls_eq_0: "w \<in> normal ==> (w = Pls) = (number_of w = (0::int))"
  44.120    apply (erule normal.induct)
  44.121       apply auto
  44.122    done
  44.123 @@ -313,7 +313,7 @@
  44.124    apply (rule normal.intros)
  44.125    apply assumption
  44.126    apply (simp add: normal_Pls_eq_0)
  44.127 -  apply (simp only: number_of_minus iszero_def zminus_equation [of _ "int 0"])
  44.128 +  apply (simp only: number_of_minus iszero_def zminus_equation [of _ "0"])
  44.129    apply (rule not_sym)
  44.130    apply simp
  44.131    done
    45.1 --- a/src/HOL/ex/IntRing.thy	Mon Oct 22 11:01:30 2001 +0200
    45.2 +++ b/src/HOL/ex/IntRing.thy	Mon Oct 22 11:54:22 2001 +0200
    45.3 @@ -10,7 +10,7 @@
    45.4  IntRing = Ring + Lagrange +
    45.5  
    45.6  instance int :: add_semigroup (zadd_assoc)
    45.7 -instance int :: add_monoid (Zero_int_def,zadd_int0,zadd_int0_right)
    45.8 +instance int :: add_monoid (zadd_0,zadd_0_right)
    45.9  instance int :: add_group {|Auto_tac|}
   45.10  instance int :: add_agroup (zadd_commute)
   45.11  instance int :: ring (zmult_assoc,zadd_zmult_distrib2,zadd_zmult_distrib)
    46.1 --- a/src/HOL/ex/svc_test.ML	Mon Oct 22 11:01:30 2001 +0200
    46.2 +++ b/src/HOL/ex/svc_test.ML	Mon Oct 22 11:54:22 2001 +0200
    46.3 @@ -233,12 +233,12 @@
    46.4  
    46.5  Goal "x ~= 14 & x ~= 13 & x ~= 12 & x ~= 11 & x ~= 10 & x ~= 9 & \
    46.6  \     x ~= 8 & x ~= 7 & x ~= 6 & x ~= 5 & x ~= 4 & x ~= 3 & \
    46.7 -\     x ~= 2 & x ~= Numeral1 & Numeral0 < x & x < 16 --> 15 = (x::int)";
    46.8 +\     x ~= 2 & x ~= 1 & 0 < x & x < 16 --> 15 = (x::int)";
    46.9  by (svc_tac 1);
   46.10  qed "";
   46.11  
   46.12  (*merely to test polarity handling in the presence of biconditionals*)
   46.13 -Goal "(x < (y::int)) = (x+Numeral1 <= y)";
   46.14 +Goal "(x < (y::int)) = (x+1 <= y)";
   46.15  by (svc_tac 1);
   46.16  qed "";
   46.17  
   46.18 @@ -249,6 +249,6 @@
   46.19  by (svc_tac 1);
   46.20  qed "";
   46.21  
   46.22 -Goal "(n::nat) < 2 ==> (n = Numeral0) | (n = Numeral1)";
   46.23 +Goal "(n::nat) < 2 ==> (n = 0) | (n = 1)";
   46.24  by (svc_tac 1);
   46.25  qed "";