src/HOL/Integ/IntArith.ML
author paulson
Fri Jun 16 13:21:17 2000 +0200 (2000-06-16)
changeset 9079 8e9b7095bf59
parent 9063 0d7628966069
child 9214 9454f30eacc7
permissions -rw-r--r--
some missing simprules for integer linear arithmetic
     1 (*  Title:      HOL/Integ/IntArith.thy
     2     ID:         $Id$
     3     Authors:    Larry Paulson and Tobias Nipkow
     4 
     5 Simprocs and decision procedure for linear arithmetic.
     6 *)
     7 
     8 (*** Simprocs for numeric literals ***)
     9 
    10 (** Combining of literal coefficients in sums of products **)
    11 
    12 Goal "(x < y) = (x-y < (#0::int))";
    13 by (simp_tac (simpset() addsimps zcompare_rls) 1);
    14 qed "zless_iff_zdiff_zless_0";
    15 
    16 Goal "(x = y) = (x-y = (#0::int))";
    17 by (simp_tac (simpset() addsimps zcompare_rls) 1);
    18 qed "eq_iff_zdiff_eq_0";
    19 
    20 Goal "(x <= y) = (x-y <= (#0::int))";
    21 by (simp_tac (simpset() addsimps zcompare_rls) 1);
    22 qed "zle_iff_zdiff_zle_0";
    23 
    24 
    25 (** For combine_numerals **)
    26 
    27 Goal "i*u + (j*u + k) = (i+j)*u + (k::int)";
    28 by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib]) 1);
    29 qed "left_zadd_zmult_distrib";
    30 
    31 
    32 (** For cancel_numerals **)
    33 
    34 val rel_iff_rel_0_rls = map (inst "y" "?u+?v")
    35                           [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, 
    36 			   zle_iff_zdiff_zle_0] @
    37 		        map (inst "y" "n")
    38                           [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, 
    39 			   zle_iff_zdiff_zle_0];
    40 
    41 Goal "!!i::int. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
    42 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
    43 		                     zadd_ac@rel_iff_rel_0_rls) 1);
    44 qed "eq_add_iff1";
    45 
    46 Goal "!!i::int. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
    47 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
    48                                      zadd_ac@rel_iff_rel_0_rls) 1);
    49 qed "eq_add_iff2";
    50 
    51 Goal "!!i::int. (i*u + m < j*u + n) = ((i-j)*u + m < n)";
    52 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
    53                                      zadd_ac@rel_iff_rel_0_rls) 1);
    54 qed "less_add_iff1";
    55 
    56 Goal "!!i::int. (i*u + m < j*u + n) = (m < (j-i)*u + n)";
    57 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
    58                                      zadd_ac@rel_iff_rel_0_rls) 1);
    59 qed "less_add_iff2";
    60 
    61 Goal "!!i::int. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)";
    62 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
    63                                      zadd_ac@rel_iff_rel_0_rls) 1);
    64 qed "le_add_iff1";
    65 
    66 Goal "!!i::int. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)";
    67 by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]
    68                                      @zadd_ac@rel_iff_rel_0_rls) 1);
    69 qed "le_add_iff2";
    70 
    71 (*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
    72 Goal "u = u' ==> (t==u) == (t==u')";
    73 by Auto_tac;
    74 qed "eq_cong2";
    75 
    76 
    77 structure Int_Numeral_Simprocs =
    78 struct
    79 
    80 (*Utilities*)
    81 
    82 fun mk_numeral n = HOLogic.number_of_const HOLogic.intT $ 
    83                    NumeralSyntax.mk_bin n;
    84 
    85 (*Decodes a binary INTEGER*)
    86 fun dest_numeral (Const("Numeral.number_of", _) $ w) = 
    87      (NumeralSyntax.dest_bin w
    88       handle Match => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
    89   | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
    90 
    91 fun find_first_numeral past (t::terms) =
    92 	((dest_numeral t, rev past @ terms)
    93 	 handle TERM _ => find_first_numeral (t::past) terms)
    94   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
    95 
    96 val zero = mk_numeral 0;
    97 val mk_plus = HOLogic.mk_binop "op +";
    98 
    99 val uminus_const = Const ("uminus", HOLogic.intT --> HOLogic.intT);
   100 
   101 (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
   102 fun mk_sum []        = zero
   103   | mk_sum [t,u]     = mk_plus (t, u)
   104   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   105 
   106 (*this version ALWAYS includes a trailing zero*)
   107 fun long_mk_sum []        = zero
   108   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   109 
   110 val dest_plus = HOLogic.dest_bin "op +" HOLogic.intT;
   111 
   112 (*decompose additions AND subtractions as a sum*)
   113 fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
   114         dest_summing (pos, t, dest_summing (pos, u, ts))
   115   | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
   116         dest_summing (pos, t, dest_summing (not pos, u, ts))
   117   | dest_summing (pos, t, ts) =
   118 	if pos then t::ts else uminus_const$t :: ts;
   119 
   120 fun dest_sum t = dest_summing (true, t, []);
   121 
   122 val mk_diff = HOLogic.mk_binop "op -";
   123 val dest_diff = HOLogic.dest_bin "op -" HOLogic.intT;
   124 
   125 val one = mk_numeral 1;
   126 val mk_times = HOLogic.mk_binop "op *";
   127 
   128 fun mk_prod [] = one
   129   | mk_prod [t] = t
   130   | mk_prod (t :: ts) = if t = one then mk_prod ts
   131                         else mk_times (t, mk_prod ts);
   132 
   133 val dest_times = HOLogic.dest_bin "op *" HOLogic.intT;
   134 
   135 fun dest_prod t =
   136       let val (t,u) = dest_times t 
   137       in  dest_prod t @ dest_prod u  end
   138       handle TERM _ => [t];
   139 
   140 (*DON'T do the obvious simplifications; that would create special cases*) 
   141 fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
   142 
   143 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   144 fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
   145   | dest_coeff sign t =
   146     let val ts = sort Term.term_ord (dest_prod t)
   147 	val (n, ts') = find_first_numeral [] ts
   148                           handle TERM _ => (1, ts)
   149     in (sign*n, mk_prod ts') end;
   150 
   151 (*Find first coefficient-term THAT MATCHES u*)
   152 fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 
   153   | find_first_coeff past u (t::terms) =
   154 	let val (n,u') = dest_coeff 1 t
   155 	in  if u aconv u' then (n, rev past @ terms)
   156 			  else find_first_coeff (t::past) u terms
   157 	end
   158 	handle TERM _ => find_first_coeff (t::past) u terms;
   159 
   160 
   161 (*Simplify #1*n and n*#1 to n*)
   162 val add_0s = [zadd_0, zadd_0_right];
   163 val mult_1s = [zmult_1, zmult_1_right, zmult_minus1, zmult_minus1_right];
   164 
   165 (*To perform binary arithmetic*)
   166 val bin_simps = [add_number_of_left] @ bin_arith_simps @ bin_rel_simps;
   167 
   168 (*To evaluate binary negations of coefficients*)
   169 val zminus_simps = NCons_simps @
   170                    [number_of_minus RS sym, 
   171 		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
   172 		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
   173 
   174 (*To let us treat subtraction as addition*)
   175 val diff_simps = [zdiff_def, zminus_zadd_distrib, zminus_zminus];
   176 
   177 (*Apply the given rewrite (if present) just once*)
   178 fun trans_tac None      = all_tac
   179   | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
   180 
   181 fun prove_conv name tacs sg (t, u) =
   182   if t aconv u then None
   183   else
   184   let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))
   185   in Some
   186      (prove_goalw_cterm [] ct (K tacs)
   187       handle ERROR => error 
   188 	  ("The error(s) above occurred while trying to prove " ^
   189 	   string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))
   190   end;
   191 
   192 fun simplify_meta_eq rules =
   193     mk_meta_eq o
   194     simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
   195 
   196 fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
   197 fun prep_pat s = Thm.read_cterm (Theory.sign_of Int.thy) (s, HOLogic.termT);
   198 val prep_pats = map prep_pat;
   199 
   200 structure CancelNumeralsCommon =
   201   struct
   202   val mk_sum    	= mk_sum
   203   val dest_sum		= dest_sum
   204   val mk_coeff		= mk_coeff
   205   val dest_coeff	= dest_coeff 1
   206   val find_first_coeff	= find_first_coeff []
   207   val trans_tac         = trans_tac
   208   val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
   209                                                      zminus_simps@zadd_ac))
   210                  THEN ALLGOALS
   211                     (simp_tac (HOL_ss addsimps [zmult_zminus_right RS sym]@
   212                                                bin_simps@zadd_ac@zmult_ac))
   213   val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   214   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   215   end;
   216 
   217 
   218 structure EqCancelNumerals = CancelNumeralsFun
   219  (open CancelNumeralsCommon
   220   val prove_conv = prove_conv "inteq_cancel_numerals"
   221   val mk_bal   = HOLogic.mk_eq
   222   val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
   223   val bal_add1 = eq_add_iff1 RS trans
   224   val bal_add2 = eq_add_iff2 RS trans
   225 );
   226 
   227 structure LessCancelNumerals = CancelNumeralsFun
   228  (open CancelNumeralsCommon
   229   val prove_conv = prove_conv "intless_cancel_numerals"
   230   val mk_bal   = HOLogic.mk_binrel "op <"
   231   val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT
   232   val bal_add1 = less_add_iff1 RS trans
   233   val bal_add2 = less_add_iff2 RS trans
   234 );
   235 
   236 structure LeCancelNumerals = CancelNumeralsFun
   237  (open CancelNumeralsCommon
   238   val prove_conv = prove_conv "intle_cancel_numerals"
   239   val mk_bal   = HOLogic.mk_binrel "op <="
   240   val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT
   241   val bal_add1 = le_add_iff1 RS trans
   242   val bal_add2 = le_add_iff2 RS trans
   243 );
   244 
   245 val cancel_numerals = 
   246   map prep_simproc
   247    [("inteq_cancel_numerals",
   248      prep_pats ["(l::int) + m = n", "(l::int) = m + n", 
   249 		"(l::int) - m = n", "(l::int) = m - n", 
   250 		"(l::int) * m = n", "(l::int) = m * n"], 
   251      EqCancelNumerals.proc),
   252     ("intless_cancel_numerals", 
   253      prep_pats ["(l::int) + m < n", "(l::int) < m + n", 
   254 		"(l::int) - m < n", "(l::int) < m - n", 
   255 		"(l::int) * m < n", "(l::int) < m * n"], 
   256      LessCancelNumerals.proc),
   257     ("intle_cancel_numerals", 
   258      prep_pats ["(l::int) + m <= n", "(l::int) <= m + n", 
   259 		"(l::int) - m <= n", "(l::int) <= m - n", 
   260 		"(l::int) * m <= n", "(l::int) <= m * n"], 
   261      LeCancelNumerals.proc)];
   262 
   263 
   264 structure CombineNumeralsData =
   265   struct
   266   val mk_sum    	= long_mk_sum    (*to work for e.g. #2*x + #3*x *)
   267   val dest_sum		= dest_sum
   268   val mk_coeff		= mk_coeff
   269   val dest_coeff	= dest_coeff 1
   270   val left_distrib	= left_zadd_zmult_distrib RS trans
   271   val prove_conv	= prove_conv "int_combine_numerals"
   272   val trans_tac          = trans_tac
   273   val norm_tac = ALLGOALS
   274                    (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
   275                                               zminus_simps@zadd_ac))
   276                  THEN ALLGOALS
   277                     (simp_tac (HOL_ss addsimps [zmult_zminus_right RS sym]@
   278                                                bin_simps@zadd_ac@zmult_ac))
   279   val numeral_simp_tac	= ALLGOALS 
   280                     (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   281   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   282   end;
   283 
   284 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   285   
   286 val combine_numerals = 
   287     prep_simproc ("int_combine_numerals",
   288 		  prep_pats ["(i::int) + j", "(i::int) - j"],
   289 		  CombineNumerals.proc);
   290 
   291 end;
   292 
   293 
   294 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
   295 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
   296 
   297 (*The Abel_Cancel simprocs are now obsolete*)
   298 Delsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv];
   299 
   300 (*examples:
   301 print_depth 22;
   302 set timing;
   303 set trace_simp;
   304 fun test s = (Goal s; by (Simp_tac 1)); 
   305 
   306 test "l + #2 + #2 + #2 + (l + #2) + (oo + #2) = (uu::int)";
   307 
   308 test "#2*u = (u::int)";
   309 test "(i + j + #12 + (k::int)) - #15 = y";
   310 test "(i + j + #12 + (k::int)) - #5 = y";
   311 
   312 test "y - b < (b::int)";
   313 test "y - (#3*b + c) < (b::int) - #2*c";
   314 
   315 test "(#2*x - (u*v) + y) - v*#3*u = (w::int)";
   316 test "(#2*x*u*v + (u*v)*#4 + y) - v*u*#4 = (w::int)";
   317 test "(#2*x*u*v + (u*v)*#4 + y) - v*u = (w::int)";
   318 test "u*v - (x*u*v + (u*v)*#4 + y) = (w::int)";
   319 
   320 test "(i + j + #12 + (k::int)) = u + #15 + y";
   321 test "(i + j*#2 + #12 + (k::int)) = j + #5 + y";
   322 
   323 test "#2*y + #3*z + #6*w + #2*y + #3*z + #2*u = #2*y' + #3*z' + #6*w' + #2*y' + #3*z' + u + (vv::int)";
   324 
   325 test "a + -(b+c) + b = (d::int)";
   326 test "a + -(b+c) - b = (d::int)";
   327 
   328 (*negative numerals*)
   329 test "(i + j + #-2 + (k::int)) - (u + #5 + y) = zz";
   330 test "(i + j + #-3 + (k::int)) < u + #5 + y";
   331 test "(i + j + #3 + (k::int)) < u + #-6 + y";
   332 test "(i + j + #-12 + (k::int)) - #15 = y";
   333 test "(i + j + #12 + (k::int)) - #-15 = y";
   334 test "(i + j + #-12 + (k::int)) - #-15 = y";
   335 *)
   336 
   337 
   338 (** Constant folding for integer plus and times **)
   339 
   340 (*We do not need
   341     structure Nat_Plus_Assoc = Assoc_Fold (Nat_Plus_Assoc_Data);
   342     structure Int_Plus_Assoc = Assoc_Fold (Int_Plus_Assoc_Data);
   343   because combine_numerals does the same thing*)
   344 
   345 structure Int_Times_Assoc_Data : ASSOC_FOLD_DATA =
   346 struct
   347   val ss		= HOL_ss
   348   val eq_reflection	= eq_reflection
   349   val thy    = Bin.thy
   350   val T	     = HOLogic.intT
   351   val plus   = Const ("op *", [HOLogic.intT,HOLogic.intT] ---> HOLogic.intT);
   352   val add_ac = zmult_ac
   353 end;
   354 
   355 structure Int_Times_Assoc = Assoc_Fold (Int_Times_Assoc_Data);
   356 
   357 Addsimprocs [Int_Times_Assoc.conv];
   358 
   359 
   360 (** The same for the naturals **)
   361 
   362 structure Nat_Times_Assoc_Data : ASSOC_FOLD_DATA =
   363 struct
   364   val ss		= HOL_ss
   365   val eq_reflection	= eq_reflection
   366   val thy    = Bin.thy
   367   val T	     = HOLogic.natT
   368   val plus   = Const ("op *", [HOLogic.natT,HOLogic.natT] ---> HOLogic.natT);
   369   val add_ac = mult_ac
   370 end;
   371 
   372 structure Nat_Times_Assoc = Assoc_Fold (Nat_Times_Assoc_Data);
   373 
   374 Addsimprocs [Nat_Times_Assoc.conv];
   375 
   376 
   377 
   378 (*** decision procedure for linear arithmetic ***)
   379 
   380 (*---------------------------------------------------------------------------*)
   381 (* Linear arithmetic                                                         *)
   382 (*---------------------------------------------------------------------------*)
   383 
   384 (*
   385 Instantiation of the generic linear arithmetic package for int.
   386 *)
   387 
   388 (* Update parameters of arithmetic prover *)
   389 let
   390 
   391 (* reduce contradictory <= to False *)
   392 val add_rules = simp_thms @ bin_arith_simps @ bin_rel_simps @
   393                 [int_0, zadd_0, zadd_0_right, zdiff_def,
   394 		 zadd_zminus_inverse, zadd_zminus_inverse2, 
   395 		 zmult_0, zmult_0_right, 
   396 		 zmult_1, zmult_1_right, 
   397 		 zmult_minus1, zmult_minus1_right,
   398 		 zminus_zadd_distrib, zminus_zminus];
   399 
   400 val simprocs = [Int_Times_Assoc.conv, Int_Numeral_Simprocs.combine_numerals]@
   401                Int_Numeral_Simprocs.cancel_numerals;
   402 
   403 val add_mono_thms =
   404   map (fn s => prove_goal Int.thy s
   405                  (fn prems => [cut_facts_tac prems 1,
   406                       asm_simp_tac (simpset() addsimps [zadd_zle_mono]) 1]))
   407     ["(i <= j) & (k <= l) ==> i + k <= j + (l::int)",
   408      "(i  = j) & (k <= l) ==> i + k <= j + (l::int)",
   409      "(i <= j) & (k  = l) ==> i + k <= j + (l::int)",
   410      "(i  = j) & (k  = l) ==> i + k  = j + (l::int)"
   411     ];
   412 
   413 in
   414 LA_Data_Ref.add_mono_thms := !LA_Data_Ref.add_mono_thms @ add_mono_thms;
   415 LA_Data_Ref.lessD := !LA_Data_Ref.lessD @ [add1_zle_eq RS iffD2];
   416 LA_Data_Ref.ss_ref := !LA_Data_Ref.ss_ref addsimps add_rules
   417                       addsimprocs simprocs
   418                       addcongs [if_weak_cong];
   419 LA_Data_Ref.discrete := !LA_Data_Ref.discrete @ [("IntDef.int",true)]
   420 end;
   421 
   422 let
   423 val int_arith_simproc_pats =
   424   map (fn s => Thm.read_cterm (Theory.sign_of Int.thy) (s, HOLogic.boolT))
   425       ["(m::int) < n","(m::int) <= n", "(m::int) = n"];
   426 
   427 val fast_int_arith_simproc = mk_simproc
   428   "fast_int_arith" int_arith_simproc_pats Fast_Arith.lin_arith_prover;
   429 in
   430 Addsimprocs [fast_int_arith_simproc]
   431 end;
   432 
   433 (* Some test data
   434 Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
   435 by (fast_arith_tac 1);
   436 Goal "!!a::int. [| a < b; c < d |] ==> a-d+ #2 <= b+(-c)";
   437 by (fast_arith_tac 1);
   438 Goal "!!a::int. [| a < b; c < d |] ==> a+c+ #1 < b+d";
   439 by (fast_arith_tac 1);
   440 Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
   441 by (fast_arith_tac 1);
   442 Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \
   443 \     ==> a+a <= j+j";
   444 by (fast_arith_tac 1);
   445 Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \
   446 \     ==> a+a - - #-1 < j+j - #3";
   447 by (fast_arith_tac 1);
   448 Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
   449 by (arith_tac 1);
   450 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   451 \     ==> a <= l";
   452 by (fast_arith_tac 1);
   453 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   454 \     ==> a+a+a+a <= l+l+l+l";
   455 by (fast_arith_tac 1);
   456 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   457 \     ==> a+a+a+a+a <= l+l+l+l+i";
   458 by (fast_arith_tac 1);
   459 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   460 \     ==> a+a+a+a+a+a <= l+l+l+l+i+l";
   461 by (fast_arith_tac 1);
   462 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   463 \     ==> #6*a <= #5*l+i";
   464 by (fast_arith_tac 1);
   465 *)
   466 
   467 (*---------------------------------------------------------------------------*)
   468 (* End of linear arithmetic                                                  *)
   469 (*---------------------------------------------------------------------------*)
   470 
   471 (** Simplification of inequalities involving numerical constants **)
   472 
   473 Goal "(w <= z - (#1::int)) = (w<(z::int))";
   474 by (arith_tac 1);
   475 qed "zle_diff1_eq";
   476 Addsimps [zle_diff1_eq];
   477 
   478 Goal "(w < z + #1) = (w<=(z::int))";
   479 by (arith_tac 1);
   480 qed "zle_add1_eq_le";
   481 Addsimps [zle_add1_eq_le];
   482 
   483 Goal "(z = z + w) = (w = (#0::int))";
   484 by (arith_tac 1);
   485 qed "zadd_left_cancel0";
   486 Addsimps [zadd_left_cancel0];
   487 
   488 
   489 (* nat *)
   490 
   491 Goal "#0 <= z ==> int (nat z) = z"; 
   492 by (asm_full_simp_tac
   493     (simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat]) 1); 
   494 qed "nat_0_le"; 
   495 
   496 Goal "z <= #0 ==> nat z = 0"; 
   497 by (case_tac "z = #0" 1);
   498 by (asm_simp_tac (simpset() addsimps [nat_le_int0]) 1); 
   499 by (asm_full_simp_tac 
   500     (simpset() addsimps [neg_eq_less_0, neg_nat, linorder_neq_iff]) 1);
   501 qed "nat_le_0"; 
   502 
   503 Addsimps [nat_0_le, nat_le_0];
   504 
   505 val [major,minor] = Goal "[| #0 <= z;  !!m. z = int m ==> P |] ==> P"; 
   506 by (rtac (major RS nat_0_le RS sym RS minor) 1);
   507 qed "nonneg_eq_int"; 
   508 
   509 Goal "#0 <= w ==> (nat w = m) = (w = int m)";
   510 by Auto_tac;
   511 qed "nat_eq_iff";
   512 
   513 Goal "#0 <= w ==> (m = nat w) = (w = int m)";
   514 by Auto_tac;
   515 qed "nat_eq_iff2";
   516 
   517 Goal "#0 <= w ==> (nat w < m) = (w < int m)";
   518 by (rtac iffI 1);
   519 by (asm_full_simp_tac 
   520     (simpset() delsimps [zless_int] addsimps [zless_int RS sym]) 2);
   521 by (etac (nat_0_le RS subst) 1);
   522 by (Simp_tac 1);
   523 qed "nat_less_iff";
   524 
   525 
   526 (*Users don't want to see (int 0), int(Suc 0) or w + - z*)
   527 Addsimps [int_0, int_Suc, symmetric zdiff_def];
   528 
   529 Goal "nat #0 = 0";
   530 by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   531 qed "nat_0";
   532 
   533 Goal "nat #1 = 1";
   534 by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   535 qed "nat_1";
   536 
   537 Goal "nat #2 = 2";
   538 by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);
   539 qed "nat_2";
   540 
   541 Goal "#0 <= w ==> (nat w < nat z) = (w<z)";
   542 by (case_tac "neg z" 1);
   543 by (auto_tac (claset(), simpset() addsimps [nat_less_iff]));
   544 by (auto_tac (claset() addIs [zless_trans], 
   545 	      simpset() addsimps [neg_eq_less_0, zle_def]));
   546 qed "nat_less_eq_zless";
   547 
   548 Goal "#0 < w | #0 <= z ==> (nat w <= nat z) = (w<=z)";
   549 by (auto_tac (claset(), 
   550 	      simpset() addsimps [linorder_not_less RS sym, 
   551 				  zless_nat_conj]));
   552 qed "nat_le_eq_zle";
   553 
   554 (*Analogous to zadd_int, but more easily provable using the arithmetic in Bin*)
   555 Goal "n<=m --> int m - int n = int (m-n)";
   556 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   557 by Auto_tac;
   558 qed_spec_mp "zdiff_int";
   559 
   560 
   561 (*** Some convenient biconditionals for products of signs ***)
   562 
   563 Goal "[| (#0::int) < i; #0 < j |] ==> #0 < i*j";
   564 by (dtac zmult_zless_mono1 1);
   565 by Auto_tac; 
   566 qed "zmult_pos";
   567 
   568 Goal "[| i < (#0::int); j < #0 |] ==> #0 < i*j";
   569 by (dtac zmult_zless_mono1_neg 1);
   570 by Auto_tac; 
   571 qed "zmult_neg";
   572 
   573 Goal "[| (#0::int) < i; j < #0 |] ==> i*j < #0";
   574 by (dtac zmult_zless_mono1_neg 1);
   575 by Auto_tac; 
   576 qed "zmult_pos_neg";
   577 
   578 Goal "((#0::int) < x*y) = (#0 < x & #0 < y | x < #0 & y < #0)";
   579 by (auto_tac (claset(), 
   580               simpset() addsimps [order_le_less, linorder_not_less,
   581 	                          zmult_pos, zmult_neg]));
   582 by (ALLGOALS (rtac ccontr)); 
   583 by (auto_tac (claset(), 
   584 	      simpset() addsimps [order_le_less, linorder_not_less]));
   585 by (ALLGOALS (etac rev_mp)); 
   586 by (ALLGOALS (dtac zmult_pos_neg THEN' assume_tac));
   587 by (auto_tac (claset() addDs [order_less_not_sym], 
   588               simpset() addsimps [zmult_commute]));  
   589 qed "int_0_less_mult_iff";
   590 
   591 Goal "((#0::int) <= x*y) = (#0 <= x & #0 <= y | x <= #0 & y <= #0)";
   592 by (auto_tac (claset(), 
   593               simpset() addsimps [order_le_less, linorder_not_less,  
   594                                   int_0_less_mult_iff]));
   595 qed "int_0_le_mult_iff";
   596 
   597 Goal "(x*y < (#0::int)) = (#0 < x & y < #0 | x < #0 & #0 < y)";
   598 by (auto_tac (claset(), 
   599               simpset() addsimps [int_0_le_mult_iff, 
   600                                   linorder_not_le RS sym]));
   601 by (auto_tac (claset() addDs [order_less_not_sym],  
   602               simpset() addsimps [linorder_not_le]));
   603 qed "zmult_less_0_iff";
   604 
   605 Goal "(x*y <= (#0::int)) = (#0 <= x & y <= #0 | x <= #0 & #0 <= y)";
   606 by (auto_tac (claset() addDs [order_less_not_sym], 
   607               simpset() addsimps [int_0_less_mult_iff, 
   608                                   linorder_not_less RS sym]));
   609 qed "zmult_le_0_iff";
   610