generic of_nat and of_int functions, and generalization of iszero
authorpaulson
Tue Feb 10 12:02:11 2004 +0100 (2004-02-10)
changeset 1437869c4d5997669
parent 14377 f454b3004f8f
child 14379 ea10a8c3e9cf
generic of_nat and of_int functions, and generalization of iszero
and neg
src/HOL/Complex/CStar.ML
src/HOL/Complex/ComplexBin.ML
src/HOL/Complex/NSComplexBin.ML
src/HOL/Complex/NSInduct.ML
src/HOL/Complex/ex/NSPrimes.ML
src/HOL/Hyperreal/HSeries.ML
src/HOL/Hyperreal/HTranscendental.ML
src/HOL/Hyperreal/HyperArith.thy
src/HOL/Hyperreal/HyperDef.thy
src/HOL/Hyperreal/HyperNat.thy
src/HOL/Hyperreal/HyperPow.thy
src/HOL/Hyperreal/NSA.thy
src/HOL/Hyperreal/NatStar.ML
src/HOL/Hyperreal/SEQ.ML
src/HOL/Hyperreal/Star.thy
src/HOL/Integ/Bin.thy
src/HOL/Integ/Int.thy
src/HOL/Integ/IntArith.thy
src/HOL/Integ/IntDef.thy
src/HOL/Integ/IntDiv.thy
src/HOL/Integ/NatBin.thy
src/HOL/Integ/NatSimprocs.thy
src/HOL/Integ/Presburger.thy
src/HOL/Integ/int_arith1.ML
src/HOL/Integ/int_factor_simprocs.ML
src/HOL/IsaMakefile
src/HOL/Isar_examples/document/root.bib
src/HOL/NumberTheory/IntPrimes.thy
src/HOL/Presburger.thy
src/HOL/Real/PReal.thy
src/HOL/Real/RatArith.thy
src/HOL/Real/Rational.thy
src/HOL/Real/RealArith.thy
src/HOL/Real/RealDef.thy
src/HOL/Real/rat_arith.ML
src/HOL/UNITY/Simple/Lift.thy
src/HOL/ex/BinEx.thy
     1.1 --- a/src/HOL/Complex/CStar.ML	Thu Feb 05 10:45:28 2004 +0100
     1.2 +++ b/src/HOL/Complex/CStar.ML	Tue Feb 10 12:02:11 2004 +0100
     1.3 @@ -404,10 +404,9 @@
     1.4  Goal "( *fcNat* (%n. z ^ n)) N = (hcomplex_of_complex z) hcpow N";
     1.5  *)
     1.6  
     1.7 -Goalw [hypnat_of_nat_def] 
     1.8 -   "( *fc* (%z. z ^ n)) Z = Z hcpow hypnat_of_nat n";
     1.9 +Goal "( *fc* (%z. z ^ n)) Z = Z hcpow hypnat_of_nat n";
    1.10  by (res_inst_tac [("z","Z")] eq_Abs_hcomplex 1);
    1.11 -by (auto_tac (claset(), simpset() addsimps [hcpow,starfunC]));
    1.12 +by (auto_tac (claset(), simpset() addsimps [hcpow,starfunC,hypnat_of_nat_eq]));
    1.13  qed "starfunC_hcpow";
    1.14  
    1.15  Goal "( *fc* (%h. f (x + h))) xa  = ( *fc* f) (hcomplex_of_complex  x + xa)";
     2.1 --- a/src/HOL/Complex/ComplexBin.ML	Thu Feb 05 10:45:28 2004 +0100
     2.2 +++ b/src/HOL/Complex/ComplexBin.ML	Tue Feb 10 12:02:11 2004 +0100
     2.3 @@ -70,7 +70,7 @@
     2.4  (** Equals (=) **)
     2.5  
     2.6  Goal "((number_of v :: complex) = number_of v') = \
     2.7 -\     iszero (number_of (bin_add v (bin_minus v')))";
     2.8 +\     iszero (number_of (bin_add v (bin_minus v')) :: int)";
     2.9  by (simp_tac
    2.10      (HOL_ss addsimps [complex_number_of_def, 
    2.11  	              complex_of_real_eq_iff, eq_real_number_of]) 1);
     3.1 --- a/src/HOL/Complex/NSComplexBin.ML	Thu Feb 05 10:45:28 2004 +0100
     3.2 +++ b/src/HOL/Complex/NSComplexBin.ML	Tue Feb 10 12:02:11 2004 +0100
     3.3 @@ -100,7 +100,7 @@
     3.4  (** Equals (=) **)
     3.5  
     3.6  Goal "((number_of v :: hcomplex) = number_of v') = \
     3.7 -\     iszero (number_of (bin_add v (bin_minus v')))";
     3.8 +\     iszero (number_of (bin_add v (bin_minus v')) :: int)";
     3.9  by (simp_tac
    3.10      (HOL_ss addsimps [hcomplex_number_of_def, 
    3.11  	              hcomplex_of_complex_eq_iff, eq_complex_number_of]) 1);
     4.1 --- a/src/HOL/Complex/NSInduct.ML	Thu Feb 05 10:45:28 2004 +0100
     4.2 +++ b/src/HOL/Complex/NSInduct.ML	Tue Feb 10 12:02:11 2004 +0100
     4.3 @@ -11,8 +11,8 @@
     4.4  by (Ultra_tac 1);
     4.5  qed "starPNat";
     4.6  
     4.7 -Goalw [hypnat_of_nat_def] "( *pNat* P) (hypnat_of_nat n) = P n";
     4.8 -by (auto_tac (claset(),simpset() addsimps [starPNat]));
     4.9 +Goal "( *pNat* P) (hypnat_of_nat n) = P n";
    4.10 +by (auto_tac (claset(),simpset() addsimps [starPNat, hypnat_of_nat_eq]));
    4.11  qed "starPNat_hypnat_of_nat";
    4.12  Addsimps [starPNat_hypnat_of_nat];
    4.13  
    4.14 @@ -27,7 +27,7 @@
    4.15  by (dres_inst_tac [("x","hypnat_of_nat n")] spec 1);
    4.16  by (rtac ccontr 1);
    4.17  by (auto_tac (claset(),simpset() addsimps [starPNat,
    4.18 -    hypnat_of_nat_def,hypnat_add]));
    4.19 +    hypnat_of_nat_eq,hypnat_add]));
    4.20  qed "hypnat_induct_obj";
    4.21  
    4.22  Goal
     5.1 --- a/src/HOL/Complex/ex/NSPrimes.ML	Thu Feb 05 10:45:28 2004 +0100
     5.2 +++ b/src/HOL/Complex/ex/NSPrimes.ML	Tue Feb 10 12:02:11 2004 +0100
     5.3 @@ -175,7 +175,7 @@
     5.4  (* behaves as expected! *)
     5.5  Goal "( *sNat* insert x A) = insert (hypnat_of_nat x) ( *sNat* A)";
     5.6  by (auto_tac (claset(),simpset() addsimps [starsetNat_def,
     5.7 -    hypnat_of_nat_def]));
     5.8 +    hypnat_of_nat_eq]));
     5.9  by (ALLGOALS(res_inst_tac [("z","xa")] eq_Abs_hypnat));
    5.10  by Auto_tac;
    5.11  by (TRYALL(dtac bspec));
    5.12 @@ -251,9 +251,8 @@
    5.13  by Auto_tac;
    5.14  qed "lemma_infinite_set_singleton";
    5.15  
    5.16 -Goalw [SHNat_def,hypnat_of_nat_def] 
    5.17 -   "inj f ==> Abs_hypnat(hypnatrel `` {f}) ~: Nats";
    5.18 -by Auto_tac;
    5.19 +Goal "inj f ==> Abs_hypnat(hypnatrel `` {f}) ~: Nats";
    5.20 +by (auto_tac (claset(), simpset() addsimps [SHNat_eq,hypnat_of_nat_eq])); 
    5.21  by (subgoal_tac "ALL m n. m ~= n --> f n ~= f m" 1);
    5.22  by Auto_tac;
    5.23  by (dtac injD 2);
    5.24 @@ -357,7 +356,7 @@
    5.25  by Auto_tac;
    5.26  by (dtac inj_fun_not_hypnat_in_SHNat 1);
    5.27  by (dtac range_subset_mem_starsetNat 1);
    5.28 -by (auto_tac (claset(),simpset() addsimps [SHNat_def]));
    5.29 +by (auto_tac (claset(),simpset() addsimps [SHNat_eq]));
    5.30  qed "hypnat_infinite_has_nonstandard";
    5.31  
    5.32  Goal "*sNat* A =  hypnat_of_nat ` A ==> finite A";
    5.33 @@ -385,7 +384,7 @@
    5.34  Goal "~ (ALL n:- {0}. hypnat_of_nat n hdvd 1)";
    5.35  by Auto_tac;
    5.36  by (res_inst_tac [("x","2")] bexI 1);
    5.37 -by (auto_tac (claset(),simpset() addsimps [hypnat_of_nat_def,
    5.38 +by (auto_tac (claset(),simpset() addsimps [hypnat_of_nat_eq,
    5.39      hypnat_one_def,hdvd,dvd_def]));
    5.40  val lemma_not_dvd_hypnat_one = result();
    5.41  Addsimps [lemma_not_dvd_hypnat_one];
    5.42 @@ -417,9 +416,9 @@
    5.43  qed "zero_not_prime";
    5.44  Addsimps [zero_not_prime];
    5.45  
    5.46 -Goalw [starprime_def,starsetNat_def,hypnat_of_nat_def]
    5.47 -   "hypnat_of_nat 0 ~: starprime";
    5.48 -by (auto_tac (claset() addSIs [bexI],simpset()));
    5.49 +Goal "hypnat_of_nat 0 ~: starprime";
    5.50 +by (auto_tac (claset() addSIs [bexI],
    5.51 +      simpset() addsimps [starprime_def,starsetNat_def,hypnat_of_nat_eq]));
    5.52  qed "hypnat_of_nat_zero_not_prime";
    5.53  Addsimps [hypnat_of_nat_zero_not_prime];
    5.54  
    5.55 @@ -443,9 +442,9 @@
    5.56  qed "one_not_prime2";
    5.57  Addsimps [one_not_prime2];
    5.58  
    5.59 -Goalw [starprime_def,starsetNat_def,hypnat_of_nat_def]
    5.60 -   "hypnat_of_nat 1 ~: starprime";
    5.61 -by (auto_tac (claset() addSIs [bexI],simpset()));
    5.62 +Goal "hypnat_of_nat 1 ~: starprime";
    5.63 +by (auto_tac (claset() addSIs [bexI],
    5.64 +     simpset() addsimps [starprime_def,starsetNat_def,hypnat_of_nat_eq]));
    5.65  qed "hypnat_of_nat_one_not_prime";
    5.66  Addsimps [hypnat_of_nat_one_not_prime];
    5.67  
     6.1 --- a/src/HOL/Hyperreal/HSeries.ML	Thu Feb 05 10:45:28 2004 +0100
     6.2 +++ b/src/HOL/Hyperreal/HSeries.ML	Tue Feb 10 12:02:11 2004 +0100
     6.3 @@ -5,6 +5,8 @@
     6.4                    for hyperreals
     6.5  *) 
     6.6  
     6.7 +val hypreal_of_nat_eq = thm"hypreal_of_nat_eq";
     6.8 +
     6.9  Goalw [sumhr_def]
    6.10       "sumhr(M,N,f) =  \
    6.11  \       Abs_hypreal(UN X:Rep_hypnat(M). UN Y: Rep_hypnat(N). \
    6.12 @@ -125,12 +127,11 @@
    6.13  qed "sumhr_hrabs";
    6.14  
    6.15  (* other general version also needed *)
    6.16 -Goalw [hypnat_of_nat_def]
    6.17 -     "(ALL r. m <= r & r < n --> f r = g r) --> \
    6.18 +Goal "(ALL r. m <= r & r < n --> f r = g r) --> \
    6.19  \     sumhr(hypnat_of_nat m, hypnat_of_nat n, f) = \
    6.20  \     sumhr(hypnat_of_nat m, hypnat_of_nat n, g)";
    6.21  by (Step_tac 1 THEN dtac sumr_fun_eq 1);
    6.22 -by (auto_tac (claset(), simpset() addsimps [sumhr]));
    6.23 +by (auto_tac (claset(), simpset() addsimps [sumhr, hypnat_of_nat_eq]));
    6.24  qed "sumhr_fun_hypnat_eq";
    6.25  
    6.26  Goalw [hypnat_zero_def,hypreal_of_real_def]
    6.27 @@ -163,12 +164,11 @@
    6.28  by (auto_tac (claset(), simpset() addsimps [sumhr, hypreal_minus,sumr_minus]));
    6.29  qed "sumhr_minus";
    6.30  
    6.31 -Goalw [hypnat_of_nat_def]
    6.32 -     "sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))";
    6.33 +Goal "sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))";
    6.34  by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
    6.35  by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
    6.36  by (auto_tac (claset(),
    6.37 -              simpset() addsimps [sumhr, hypnat_add,sumr_shift_bounds]));
    6.38 +              simpset() addsimps [sumhr, hypnat_add,sumr_shift_bounds, hypnat_of_nat_eq]));
    6.39  qed "sumhr_shift_bounds";
    6.40  
    6.41  (*------------------------------------------------------------------
    6.42 @@ -196,12 +196,11 @@
    6.43  qed "sumhr_minus_one_realpow_zero";
    6.44  Addsimps [sumhr_minus_one_realpow_zero];
    6.45  
    6.46 -Goalw [hypnat_of_nat_def,hypreal_of_real_def]
    6.47 -     "(ALL n. m <= Suc n --> f n = r) & m <= na \
    6.48 +Goal "(ALL n. m <= Suc n --> f n = r) & m <= na \
    6.49  \          ==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) = \
    6.50  \          (hypreal_of_nat (na - m) * hypreal_of_real r)";
    6.51  by (auto_tac (claset() addSDs [sumr_interval_const],
    6.52 -              simpset() addsimps [sumhr,hypreal_of_nat_def,
    6.53 +              simpset() addsimps [hypreal_of_real_def, sumhr,hypreal_of_nat_eq, hypnat_of_nat_eq,
    6.54                                    hypreal_of_real_def, hypreal_mult]));
    6.55  qed "sumhr_interval_const";
    6.56  
     7.1 --- a/src/HOL/Hyperreal/HTranscendental.ML	Thu Feb 05 10:45:28 2004 +0100
     7.2 +++ b/src/HOL/Hyperreal/HTranscendental.ML	Tue Feb 10 12:02:11 2004 +0100
     7.3 @@ -122,7 +122,7 @@
     7.4  Goal "( *f* sqrt)(x pow (hypnat_of_nat 2)) = abs(x)";
     7.5  by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
     7.6  by (auto_tac (claset(),simpset() addsimps [starfun,
     7.7 -    hypreal_hrabs,hypnat_of_nat_def,hyperpow]));
     7.8 +    hypreal_hrabs,hypnat_of_nat_eq,hyperpow]));
     7.9  qed "hypreal_sqrt_hyperpow_hrabs";
    7.10  Addsimps [hypreal_sqrt_hyperpow_hrabs];
    7.11  
     8.1 --- a/src/HOL/Hyperreal/HyperArith.thy	Thu Feb 05 10:45:28 2004 +0100
     8.2 +++ b/src/HOL/Hyperreal/HyperArith.thy	Tue Feb 10 12:02:11 2004 +0100
     8.3 @@ -77,7 +77,7 @@
     8.4  
     8.5  lemma eq_hypreal_number_of [simp]:
     8.6       "((number_of v :: hypreal) = number_of v') =
     8.7 -      iszero (number_of (bin_add v (bin_minus v')))"
     8.8 +      iszero (number_of (bin_add v (bin_minus v')) :: int)"
     8.9  apply (simp only: hypreal_number_of_def hypreal_of_real_eq_iff eq_real_number_of)
    8.10  done
    8.11  
    8.12 @@ -87,7 +87,7 @@
    8.13  (*"neg" is used in rewrite rules for binary comparisons*)
    8.14  lemma less_hypreal_number_of [simp]:
    8.15       "((number_of v :: hypreal) < number_of v') =
    8.16 -      neg (number_of (bin_add v (bin_minus v')))"
    8.17 +      neg (number_of (bin_add v (bin_minus v')) :: int)"
    8.18  by (simp only: hypreal_number_of_def hypreal_of_real_less_iff less_real_number_of)
    8.19  
    8.20  
    8.21 @@ -139,10 +139,6 @@
    8.22  
    8.23  setup hypreal_arith_setup
    8.24  
    8.25 -text{*Used once in NSA*}
    8.26 -lemma hypreal_add_minus_eq_minus: "x + y = (0::hypreal) ==> x = -y"
    8.27 -by arith
    8.28 -
    8.29  lemma hypreal_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::hypreal) \<le> x + y"
    8.30  by arith
    8.31  
    8.32 @@ -192,7 +188,7 @@
    8.33  
    8.34  lemma hrabs_number_of [simp]:
    8.35       "abs (number_of v :: hypreal) =
    8.36 -        (if neg (number_of v) then number_of (bin_minus v)
    8.37 +        (if neg (number_of v :: int) then number_of (bin_minus v)
    8.38           else number_of v)"
    8.39  by (simp add: hrabs_def)
    8.40  
    8.41 @@ -229,8 +225,11 @@
    8.42  
    8.43  constdefs
    8.44  
    8.45 -  hypreal_of_nat :: "nat => hypreal"
    8.46 -  "hypreal_of_nat (n::nat) == hypreal_of_real (real n)"
    8.47 +  hypreal_of_nat   :: "nat => hypreal"
    8.48 +   "hypreal_of_nat m  == of_nat m"
    8.49 +
    8.50 +lemma SNat_eq: "Nats = {n. \<exists>N. n = hypreal_of_nat N}"
    8.51 +by (force simp add: hypreal_of_nat_def Nats_def) 
    8.52  
    8.53  
    8.54  lemma hypreal_of_nat_add [simp]:
    8.55 @@ -252,46 +251,42 @@
    8.56  (* is a hyperreal c.f. NS extension                           *)
    8.57  (*------------------------------------------------------------*)
    8.58  
    8.59 -lemma hypreal_of_nat_iff:
    8.60 -     "hypreal_of_nat  m = Abs_hypreal(hyprel``{%n. real m})"
    8.61 -by (simp add: hypreal_of_nat_def hypreal_of_real_def real_of_nat_def)
    8.62 +lemma hypreal_of_nat_eq:
    8.63 +     "hypreal_of_nat (n::nat) = hypreal_of_real (real n)"
    8.64 +apply (induct n) 
    8.65 +apply (simp_all add: hypreal_of_nat_def real_of_nat_def)
    8.66 +done
    8.67  
    8.68 -lemma inj_hypreal_of_nat: "inj hypreal_of_nat"
    8.69 -by (simp add: inj_on_def hypreal_of_nat_def)
    8.70 +lemma hypreal_of_nat:
    8.71 +     "hypreal_of_nat m = Abs_hypreal(hyprel``{%n. real m})"
    8.72 +apply (induct m) 
    8.73 +apply (simp_all add: hypreal_of_nat_def real_of_nat_def hypreal_zero_def 
    8.74 +                     hypreal_one_def hypreal_add)
    8.75 +done
    8.76  
    8.77  lemma hypreal_of_nat_Suc:
    8.78       "hypreal_of_nat (Suc n) = hypreal_of_nat n + (1::hypreal)"
    8.79 -by (simp add: hypreal_of_nat_def real_of_nat_Suc)
    8.80 +by (simp add: hypreal_of_nat_def)
    8.81  
    8.82  (*"neg" is used in rewrite rules for binary comparisons*)
    8.83  lemma hypreal_of_nat_number_of [simp]:
    8.84       "hypreal_of_nat (number_of v :: nat) =
    8.85 -         (if neg (number_of v) then 0
    8.86 +         (if neg (number_of v :: int) then 0
    8.87            else (number_of v :: hypreal))"
    8.88 -by (simp add: hypreal_of_nat_def)
    8.89 +by (simp add: hypreal_of_nat_eq)
    8.90  
    8.91  lemma hypreal_of_nat_zero [simp]: "hypreal_of_nat 0 = 0"
    8.92 -by (simp del: numeral_0_eq_0 add: numeral_0_eq_0 [symmetric])
    8.93 +by (simp add: hypreal_of_nat_def) 
    8.94  
    8.95  lemma hypreal_of_nat_one [simp]: "hypreal_of_nat 1 = 1"
    8.96 -by (simp add: hypreal_of_nat_Suc)
    8.97 -
    8.98 -lemma hypreal_of_nat: "hypreal_of_nat m = Abs_hypreal(hyprel `` {%n. real m})"
    8.99 -apply (induct_tac "m")
   8.100 -apply (auto simp add: hypreal_zero_def hypreal_of_nat_Suc hypreal_zero_num hypreal_one_num hypreal_add real_of_nat_Suc)
   8.101 -done
   8.102 +by (simp add: hypreal_of_nat_def) 
   8.103  
   8.104 -lemma hypreal_of_nat_le_iff:
   8.105 -      "(hypreal_of_nat n \<le> hypreal_of_nat m) = (n \<le> m)"
   8.106 -apply (auto simp add: linorder_not_less [symmetric])
   8.107 -done
   8.108 -declare hypreal_of_nat_le_iff [simp]
   8.109 +lemma hypreal_of_nat_le_iff [simp]:
   8.110 +     "(hypreal_of_nat n \<le> hypreal_of_nat m) = (n \<le> m)"
   8.111 +by (simp add: hypreal_of_nat_def) 
   8.112  
   8.113 -lemma hypreal_of_nat_ge_zero: "0 \<le> hypreal_of_nat n"
   8.114 -apply (simp (no_asm) add: hypreal_of_nat_zero [symmetric] 
   8.115 -         del: hypreal_of_nat_zero)
   8.116 -done
   8.117 -declare hypreal_of_nat_ge_zero [simp]
   8.118 +lemma hypreal_of_nat_ge_zero [simp]: "0 \<le> hypreal_of_nat n"
   8.119 +by (simp add: hypreal_of_nat_def) 
   8.120  
   8.121  
   8.122  (*
   8.123 @@ -302,7 +297,6 @@
   8.124  
   8.125  ML
   8.126  {*
   8.127 -val hypreal_add_minus_eq_minus = thm "hypreal_add_minus_eq_minus";
   8.128  val hypreal_le_add_order = thm"hypreal_le_add_order";
   8.129  
   8.130  val hypreal_of_nat_def = thm"hypreal_of_nat_def";
   8.131 @@ -316,8 +310,6 @@
   8.132  val hypreal_of_nat_add = thm "hypreal_of_nat_add";
   8.133  val hypreal_of_nat_mult = thm "hypreal_of_nat_mult";
   8.134  val hypreal_of_nat_less_iff = thm "hypreal_of_nat_less_iff";
   8.135 -val hypreal_of_nat_iff = thm "hypreal_of_nat_iff";
   8.136 -val inj_hypreal_of_nat = thm "inj_hypreal_of_nat";
   8.137  val hypreal_of_nat_Suc = thm "hypreal_of_nat_Suc";
   8.138  val hypreal_of_nat_number_of = thm "hypreal_of_nat_number_of";
   8.139  val hypreal_of_nat_zero = thm "hypreal_of_nat_zero";
     9.1 --- a/src/HOL/Hyperreal/HyperDef.thy	Thu Feb 05 10:45:28 2004 +0100
     9.2 +++ b/src/HOL/Hyperreal/HyperDef.thy	Tue Feb 10 12:02:11 2004 +0100
     9.3 @@ -176,6 +176,11 @@
     9.4       "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
     9.5  by (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric])
     9.6  
     9.7 +lemma cofinite_mem_FreeUltrafilterNat: "finite (-X) ==> X \<in> FreeUltrafilterNat"
     9.8 +apply (drule FreeUltrafilterNat_finite)  
     9.9 +apply (simp add: FreeUltrafilterNat_Compl_iff2 [symmetric])
    9.10 +done
    9.11 +
    9.12  lemma FreeUltrafilterNat_UNIV [simp]: "(UNIV::nat set) \<in> FreeUltrafilterNat"
    9.13  by (rule FreeUltrafilterNat_mem [THEN FreeUltrafilter_Ultrafilter, THEN Ultrafilter_Filter, THEN mem_FiltersetD4])
    9.14  
    9.15 @@ -768,7 +773,7 @@
    9.16  
    9.17  lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} |  
    9.18        (\<exists>y. {n::nat. x = real n} = {y})"
    9.19 -by (force dest: inj_real_of_nat [THEN injD])
    9.20 +by (force)
    9.21  
    9.22  lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
    9.23  by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto)
    9.24 @@ -789,7 +794,7 @@
    9.25  lemma lemma_epsilon_empty_singleton_disj:
    9.26       "{n::nat. x = inverse(real(Suc n))} = {} |  
    9.27        (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})"
    9.28 -by (auto simp add: inj_real_of_nat [THEN inj_eq])
    9.29 +by (auto)
    9.30  
    9.31  lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}"
    9.32  by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto)
    10.1 --- a/src/HOL/Hyperreal/HyperNat.thy	Thu Feb 05 10:45:28 2004 +0100
    10.2 +++ b/src/HOL/Hyperreal/HyperNat.thy	Tue Feb 10 12:02:11 2004 +0100
    10.3 @@ -24,36 +24,9 @@
    10.4  
    10.5  consts whn :: hypnat
    10.6  
    10.7 -constdefs
    10.8 -
    10.9 -  (* embedding the naturals in the hypernaturals *)
   10.10 -  hypnat_of_nat   :: "nat => hypnat"
   10.11 -  "hypnat_of_nat m  == Abs_hypnat(hypnatrel``{%n::nat. m})"
   10.12 -
   10.13 -  (* hypernaturals as members of the hyperreals; the set is defined as  *)
   10.14 -  (* the nonstandard extension of set of naturals embedded in the reals *)
   10.15 -  HNat         :: "hypreal set"
   10.16 -  "HNat == *s* {n. \<exists>no::nat. n = real no}"
   10.17 -
   10.18 -  (* the set of infinite hypernatural numbers *)
   10.19 -  HNatInfinite :: "hypnat set"
   10.20 -  "HNatInfinite == {n. n \<notin> Nats}"
   10.21 -
   10.22 -  (* explicit embedding of the hypernaturals in the hyperreals *)
   10.23 -  hypreal_of_hypnat :: "hypnat => hypreal"
   10.24 -  "hypreal_of_hypnat N  == Abs_hypreal(\<Union>X \<in> Rep_hypnat(N).
   10.25 -                                       hyprel``{%n::nat. real (X n)})"
   10.26  
   10.27  defs (overloaded)
   10.28  
   10.29 -  (** the overloaded constant "Nats" **)
   10.30 -
   10.31 -  (* set of naturals embedded in the hyperreals*)
   10.32 -  SNat_def:  "Nats == {n. \<exists>N. n = hypreal_of_nat N}"
   10.33 -
   10.34 -  (* set of naturals embedded in the hypernaturals*)
   10.35 -  SHNat_def: "Nats == {n. \<exists>N. n = hypnat_of_nat N}"
   10.36 -
   10.37    (** hypernatural arithmetic **)
   10.38  
   10.39    hypnat_zero_def:  "0 == Abs_hypnat(hypnatrel``{%n::nat. 0})"
   10.40 @@ -151,18 +124,6 @@
   10.41  
   10.42  declare Rep_hypnat_nonempty [simp]
   10.43  
   10.44 -subsection{*@{term hypnat_of_nat}:
   10.45 -            the Injection from @{typ nat} to @{typ hypnat}*}
   10.46 -
   10.47 -lemma inj_hypnat_of_nat: "inj(hypnat_of_nat)"
   10.48 -apply (rule inj_onI)
   10.49 -apply (unfold hypnat_of_nat_def)
   10.50 -apply (drule inj_on_Abs_hypnat [THEN inj_onD])
   10.51 -apply (rule hypnatrel_in_hypnat)+
   10.52 -apply (drule eq_equiv_class)
   10.53 -apply (rule equiv_hypnatrel)
   10.54 -apply (simp_all split: split_if_asm)
   10.55 -done
   10.56  
   10.57  lemma eq_Abs_hypnat:
   10.58      "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P"
   10.59 @@ -439,6 +400,13 @@
   10.60  apply (auto simp add: hypnat_zero_def hypnat_mult, ultra+)
   10.61  done
   10.62  
   10.63 +lemma hypnat_diff_is_0_eq [simp]: "((m::hypnat) - n = 0) = (m \<le> n)"
   10.64 +apply (rule eq_Abs_hypnat [of m])
   10.65 +apply (rule eq_Abs_hypnat [of n])
   10.66 +apply (simp add: hypnat_le hypnat_minus hypnat_zero_def)
   10.67 +done
   10.68 +
   10.69 +
   10.70  
   10.71  subsection{*Theorems for Ordering*}
   10.72  
   10.73 @@ -496,42 +464,73 @@
   10.74  apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto) 
   10.75  done
   10.76  
   10.77 +lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"
   10.78 +by (simp add: linorder_not_le [symmetric] add_commute [of x]) 
   10.79 +
   10.80 +lemma hypnat_diff_split:
   10.81 +    "P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
   10.82 +    -- {* elimination of @{text -} on @{text hypnat} *}
   10.83 +proof (cases "a<b" rule: case_split)
   10.84 +  case True
   10.85 +    thus ?thesis
   10.86 +      by (auto simp add: hypnat_add_self_not_less order_less_imp_le 
   10.87 +                         hypnat_diff_is_0_eq [THEN iffD2])
   10.88 +next
   10.89 +  case False
   10.90 +    thus ?thesis
   10.91 +      by (auto simp add: linorder_not_less dest: order_le_less_trans); 
   10.92 +qed
   10.93 +
   10.94 +
   10.95  subsection{*The Embedding @{term hypnat_of_nat} Preserves Ring and 
   10.96        Order Properties*}
   10.97  
   10.98 +constdefs
   10.99 +
  10.100 +  hypnat_of_nat   :: "nat => hypnat"
  10.101 +  "hypnat_of_nat m  == of_nat m"
  10.102 +
  10.103 +  (* the set of infinite hypernatural numbers *)
  10.104 +  HNatInfinite :: "hypnat set"
  10.105 +  "HNatInfinite == {n. n \<notin> Nats}"
  10.106 +
  10.107 +
  10.108  lemma hypnat_of_nat_add:
  10.109        "hypnat_of_nat ((z::nat) + w) = hypnat_of_nat z + hypnat_of_nat w"
  10.110 -by (simp add: hypnat_of_nat_def hypnat_add)
  10.111 -
  10.112 -lemma hypnat_of_nat_minus:
  10.113 -      "hypnat_of_nat ((z::nat) - w) = hypnat_of_nat z - hypnat_of_nat w"
  10.114 -by (simp add: hypnat_of_nat_def hypnat_minus)
  10.115 +by (simp add: hypnat_of_nat_def)
  10.116  
  10.117  lemma hypnat_of_nat_mult:
  10.118        "hypnat_of_nat (z * w) = hypnat_of_nat z * hypnat_of_nat w"
  10.119 -by (simp add: hypnat_of_nat_def hypnat_mult)
  10.120 +by (simp add: hypnat_of_nat_def)
  10.121  
  10.122  lemma hypnat_of_nat_less_iff [simp]:
  10.123        "(hypnat_of_nat z < hypnat_of_nat w) = (z < w)"
  10.124 -by (simp add: hypnat_less hypnat_of_nat_def)
  10.125 +by (simp add: hypnat_of_nat_def)
  10.126  
  10.127  lemma hypnat_of_nat_le_iff [simp]:
  10.128        "(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)"
  10.129 -by (simp add: linorder_not_less [symmetric])
  10.130 +by (simp add: hypnat_of_nat_def)
  10.131  
  10.132 -lemma hypnat_of_nat_one: "hypnat_of_nat (Suc 0) = (1::hypnat)"
  10.133 -by (simp add: hypnat_of_nat_def hypnat_one_def)
  10.134 +lemma hypnat_of_nat_eq_iff [simp]:
  10.135 +      "(hypnat_of_nat z = hypnat_of_nat w) = (z = w)"
  10.136 +by (simp add: hypnat_of_nat_def)
  10.137  
  10.138 -lemma hypnat_of_nat_zero: "hypnat_of_nat 0 = 0"
  10.139 -by (simp add: hypnat_of_nat_def hypnat_zero_def)
  10.140 +lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"
  10.141 +by (simp add: hypnat_of_nat_def)
  10.142  
  10.143 -lemma hypnat_of_nat_zero_iff: "(hypnat_of_nat n = 0) = (n = 0)"
  10.144 -by (auto intro: FreeUltrafilterNat_P 
  10.145 -         simp add: hypnat_of_nat_def hypnat_zero_def)
  10.146 +lemma hypnat_of_nat_zero [simp]: "hypnat_of_nat 0 = 0"
  10.147 +by (simp add: hypnat_of_nat_def)
  10.148 +
  10.149 +lemma hypnat_of_nat_zero_iff [simp]: "(hypnat_of_nat n = 0) = (n = 0)"
  10.150 +by (simp add: hypnat_of_nat_def)
  10.151  
  10.152 -lemma hypnat_of_nat_Suc:
  10.153 +lemma hypnat_of_nat_Suc [simp]:
  10.154       "hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)"
  10.155 -by (auto simp add: hypnat_add hypnat_of_nat_def hypnat_one_def)
  10.156 +by (simp add: hypnat_of_nat_def)
  10.157 +
  10.158 +lemma hypnat_of_nat_minus:
  10.159 +      "hypnat_of_nat ((j::nat) - k) = hypnat_of_nat j - hypnat_of_nat k"
  10.160 +by (simp add: hypnat_of_nat_def split: nat_diff_split hypnat_diff_split)
  10.161  
  10.162  
  10.163  subsection{*Existence of an Infinite Hypernatural Number*}
  10.164 @@ -546,111 +545,64 @@
  10.165  follows because member @{term FreeUltrafilterNat} is not finite.
  10.166  See @{text HyperDef.thy} for similar argument.*}
  10.167  
  10.168 -lemma not_ex_hypnat_of_nat_eq_omega:
  10.169 -      "~ (\<exists>x. hypnat_of_nat x = whn)"
  10.170 -apply (simp add: hypnat_omega_def hypnat_of_nat_def)
  10.171 -apply (auto dest: FreeUltrafilterNat_not_finite)
  10.172 -done
  10.173 -
  10.174 -lemma hypnat_of_nat_not_eq_omega: "hypnat_of_nat x \<noteq> whn"
  10.175 -by (cut_tac not_ex_hypnat_of_nat_eq_omega, auto)
  10.176 -declare hypnat_of_nat_not_eq_omega [THEN not_sym, simp]
  10.177 -
  10.178  
  10.179  subsection{*Properties of the set @{term Nats} of Embedded Natural Numbers*}
  10.180  
  10.181 -(* Infinite hypernatural not in embedded Nats *)
  10.182 -lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats"
  10.183 -by (simp add: SHNat_def)
  10.184 -
  10.185 -(*-----------------------------------------------------------------------
  10.186 -     Closure laws for members of (embedded) set standard naturals Nats
  10.187 - -----------------------------------------------------------------------*)
  10.188 -lemma SHNat_add:
  10.189 -     "!!x::hypnat. [| x \<in> Nats; y \<in> Nats |] ==> x + y \<in> Nats"
  10.190 -apply (simp add: SHNat_def, safe)
  10.191 -apply (rule_tac x = "N + Na" in exI)
  10.192 -apply (simp add: hypnat_of_nat_add)
  10.193 -done
  10.194 -
  10.195 -lemma SHNat_minus:
  10.196 -     "!!x::hypnat. [| x \<in> Nats; y \<in> Nats |] ==> x - y \<in> Nats"
  10.197 -apply (simp add: SHNat_def, safe)
  10.198 -apply (rule_tac x = "N - Na" in exI)
  10.199 -apply (simp add: hypnat_of_nat_minus)
  10.200 -done
  10.201 -
  10.202 -lemma SHNat_mult:
  10.203 -     "!!x::hypnat. [| x \<in> Nats; y \<in> Nats |] ==> x * y \<in> Nats"
  10.204 -apply (simp add: SHNat_def, safe)
  10.205 -apply (rule_tac x = "N * Na" in exI)
  10.206 -apply (simp (no_asm) add: hypnat_of_nat_mult)
  10.207 +lemma of_nat_eq_add [rule_format]:
  10.208 +     "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
  10.209 +apply (induct n) 
  10.210 +apply (auto simp add: add_assoc) 
  10.211 +apply (case_tac x) 
  10.212 +apply (auto simp add: add_commute [of 1]) 
  10.213  done
  10.214  
  10.215 -lemma SHNat_add_cancel: "!!x::hypnat. [| x + y \<in> Nats; y \<in> Nats |] ==> x \<in> Nats"
  10.216 -by (drule_tac x = "x+y" in SHNat_minus, auto)
  10.217 -
  10.218 -lemma SHNat_hypnat_of_nat [simp]: "hypnat_of_nat x \<in> Nats"
  10.219 -by (simp add: SHNat_def, blast)
  10.220 -
  10.221 -lemma SHNat_hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) \<in> Nats"
  10.222 -by simp
  10.223 -
  10.224 -lemma SHNat_hypnat_of_nat_zero [simp]: "hypnat_of_nat 0 \<in> Nats"
  10.225 -by simp
  10.226 -
  10.227 -lemma SHNat_one [simp]: "(1::hypnat) \<in> Nats"
  10.228 -by (simp add: hypnat_of_nat_one [symmetric])
  10.229 -
  10.230 -lemma SHNat_zero [simp]: "(0::hypnat) \<in> Nats"
  10.231 -by (simp add: hypnat_of_nat_zero [symmetric])
  10.232 -
  10.233 -lemma SHNat_iff: "(x \<in> Nats) = (\<exists>y. x = hypnat_of_nat  y)"
  10.234 -by (simp add: SHNat_def)
  10.235 -
  10.236 -lemma SHNat_hypnat_of_nat_iff:
  10.237 -      "Nats = hypnat_of_nat ` (UNIV::nat set)"
  10.238 -by (auto simp add: SHNat_def)
  10.239 -
  10.240 -lemma leSuc_Un_eq: "{n. n \<le> Suc m} = {n. n \<le> m} Un {n. n = Suc m}"
  10.241 -by (auto simp add: le_Suc_eq)
  10.242 -
  10.243 -lemma finite_nat_le_segment: "finite {n::nat. n \<le> m}"
  10.244 -apply (induct_tac "m")
  10.245 -apply (auto simp add: leSuc_Un_eq)
  10.246 +lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"
  10.247 +apply (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split)
  10.248  done
  10.249  
  10.250  lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat"
  10.251 -by (insert finite_nat_le_segment
  10.252 -                [THEN FreeUltrafilterNat_finite, 
  10.253 -                 THEN FreeUltrafilterNat_Compl_mem, of m], ultra)
  10.254 -
  10.255 -(*????hyperdef*)
  10.256 -lemma cofinite_mem_FreeUltrafilterNat: "finite (-X) ==> X \<in> FreeUltrafilterNat"
  10.257 -apply (drule FreeUltrafilterNat_finite)  
  10.258 -apply (simp add: FreeUltrafilterNat_Compl_iff2 [symmetric])
  10.259 +apply (insert finite_atMost [of m]) 
  10.260 +apply (simp add: atMost_def) 
  10.261 +apply (drule FreeUltrafilterNat_finite) 
  10.262 +apply (drule FreeUltrafilterNat_Compl_mem) 
  10.263 +apply ultra
  10.264  done
  10.265  
  10.266  lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}"
  10.267  by (simp add: Collect_neg_eq [symmetric] linorder_not_le) 
  10.268  
  10.269 +
  10.270 +lemma hypnat_of_nat_eq:
  10.271 +     "hypnat_of_nat m  = Abs_hypnat(hypnatrel``{%n::nat. m})"
  10.272 +apply (induct m) 
  10.273 +apply (simp_all add: hypnat_zero_def hypnat_one_def hypnat_add); 
  10.274 +done
  10.275 +
  10.276 +lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}"
  10.277 +by (force simp add: hypnat_of_nat_def Nats_def) 
  10.278 +
  10.279  lemma hypnat_omega_gt_SHNat:
  10.280       "n \<in> Nats ==> n < whn"
  10.281 -apply (auto simp add: SHNat_def hypnat_of_nat_def hypnat_less_def 
  10.282 -                      hypnat_le_def hypnat_omega_def)
  10.283 +apply (auto simp add: hypnat_of_nat_eq hypnat_less_def hypnat_le_def
  10.284 +                      hypnat_omega_def SHNat_eq)
  10.285   prefer 2 apply (force dest: FreeUltrafilterNat_not_finite)
  10.286  apply (auto intro!: exI)
  10.287  apply (rule cofinite_mem_FreeUltrafilterNat)
  10.288  apply (simp add: Compl_Collect_le finite_nat_segment) 
  10.289  done
  10.290  
  10.291 -lemma hypnat_of_nat_less_whn: "hypnat_of_nat n < whn"
  10.292 -by (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"], auto)
  10.293 -declare hypnat_of_nat_less_whn [simp]
  10.294 +(* Infinite hypernatural not in embedded Nats *)
  10.295 +lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats"
  10.296 +apply (blast dest: hypnat_omega_gt_SHNat) 
  10.297 +done
  10.298  
  10.299 -lemma hypnat_of_nat_le_whn: "hypnat_of_nat n \<le> whn"
  10.300 +lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
  10.301 +apply (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"])
  10.302 +apply (simp add: hypnat_of_nat_def) 
  10.303 +done
  10.304 +
  10.305 +lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn"
  10.306  by (rule hypnat_of_nat_less_whn [THEN order_less_imp_le])
  10.307 -declare hypnat_of_nat_le_whn [simp]
  10.308  
  10.309  lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"
  10.310  by (simp add: hypnat_omega_gt_SHNat)
  10.311 @@ -661,37 +613,22 @@
  10.312  
  10.313  subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*}
  10.314  
  10.315 -lemma HNatInfinite_whn: "whn \<in> HNatInfinite"
  10.316 -by (simp add: HNatInfinite_def SHNat_def)
  10.317 -declare HNatInfinite_whn [simp]
  10.318 -
  10.319 -lemma SHNat_not_HNatInfinite: "x \<in> Nats ==> x \<notin> HNatInfinite"
  10.320 -by (simp add: HNatInfinite_def)
  10.321 -
  10.322 -lemma not_HNatInfinite_SHNat: "x \<notin> HNatInfinite ==> x \<in> Nats"
  10.323 +lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite"
  10.324  by (simp add: HNatInfinite_def)
  10.325  
  10.326 -lemma not_SHNat_HNatInfinite: "x \<notin> Nats ==> x \<in> HNatInfinite"
  10.327 -by (simp add: HNatInfinite_def)
  10.328 -
  10.329 -lemma HNatInfinite_not_SHNat: "x \<in> HNatInfinite ==> x \<notin> Nats"
  10.330 +lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"
  10.331  by (simp add: HNatInfinite_def)
  10.332  
  10.333 -lemma SHNat_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"
  10.334 -by (blast intro!: SHNat_not_HNatInfinite not_HNatInfinite_SHNat)
  10.335 -
  10.336 -lemma not_SHNat_HNatInfinite_iff: "(x \<notin> Nats) = (x \<in> HNatInfinite)"
  10.337 -by (blast intro!: not_SHNat_HNatInfinite HNatInfinite_not_SHNat)
  10.338 -
  10.339 -lemma SHNat_HNatInfinite_disj: "x \<in> Nats | x \<in> HNatInfinite"
  10.340 -by (simp add: SHNat_not_HNatInfinite_iff)
  10.341 +lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)"
  10.342 +by (simp add: HNatInfinite_def)
  10.343  
  10.344  
  10.345  subsection{*Alternative Characterization of the Set of Infinite Hypernaturals:
  10.346  @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*}
  10.347  
  10.348  (*??delete? similar reasoning in hypnat_omega_gt_SHNat above*)
  10.349 -lemma HNatInfinite_FreeUltrafilterNat_lemma: "\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat
  10.350 +lemma HNatInfinite_FreeUltrafilterNat_lemma:
  10.351 +     "\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat
  10.352        ==> {n. N < f n} \<in> FreeUltrafilterNat"
  10.353  apply (induct_tac "N")
  10.354  apply (drule_tac x = 0 in spec)
  10.355 @@ -700,15 +637,14 @@
  10.356  done
  10.357  
  10.358  lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"
  10.359 -apply (unfold HNatInfinite_def SHNat_def hypnat_of_nat_def, safe)
  10.360 -apply (drule_tac [2] x = "Abs_hypnat (hypnatrel `` {%n. N}) " in bspec)
  10.361 +apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq)
  10.362  apply (rule_tac z = x in eq_Abs_hypnat)
  10.363 -apply (rule_tac z = n in eq_Abs_hypnat)
  10.364 -apply (auto simp add: hypnat_less)
  10.365 -apply (auto  elim: HNatInfinite_FreeUltrafilterNat_lemma 
  10.366 -           simp add: FreeUltrafilterNat_Compl_iff1 Collect_neg_eq [symmetric])
  10.367 +apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma 
  10.368 +            simp add: hypnat_less FreeUltrafilterNat_Compl_iff1 
  10.369 +                      Collect_neg_eq [symmetric])
  10.370  done
  10.371  
  10.372 +
  10.373  subsection{*Alternative Characterization of @{term HNatInfinite} using 
  10.374  Free Ultrafilter*}
  10.375  
  10.376 @@ -716,9 +652,8 @@
  10.377       "x \<in> HNatInfinite 
  10.378        ==> \<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat"
  10.379  apply (rule eq_Abs_hypnat [of x])
  10.380 -apply (auto simp add: HNatInfinite_iff SHNat_iff hypnat_of_nat_def)
  10.381 +apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
  10.382  apply (rule bexI [OF _ lemma_hypnatrel_refl], clarify) 
  10.383 -apply (drule_tac x = "hypnat_of_nat u" in bspec, simp) 
  10.384  apply (auto simp add: hypnat_of_nat_def hypnat_less)
  10.385  done
  10.386  
  10.387 @@ -726,26 +661,24 @@
  10.388       "\<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat
  10.389        ==> x \<in> HNatInfinite"
  10.390  apply (rule eq_Abs_hypnat [of x])
  10.391 -apply (auto simp add: hypnat_less HNatInfinite_iff SHNat_iff hypnat_of_nat_def)
  10.392 +apply (auto simp add: hypnat_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
  10.393  apply (drule spec, ultra, auto) 
  10.394  done
  10.395  
  10.396  lemma HNatInfinite_FreeUltrafilterNat_iff:
  10.397       "(x \<in> HNatInfinite) = 
  10.398        (\<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat)"
  10.399 -apply (blast intro: HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite)
  10.400 -done
  10.401 +by (blast intro: HNatInfinite_FreeUltrafilterNat 
  10.402 +                 FreeUltrafilterNat_HNatInfinite)
  10.403  
  10.404 -lemma HNatInfinite_gt_one: "x \<in> HNatInfinite ==> (1::hypnat) < x"
  10.405 +lemma HNatInfinite_gt_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) < x"
  10.406  by (auto simp add: HNatInfinite_iff)
  10.407 -declare HNatInfinite_gt_one [simp]
  10.408  
  10.409 -lemma zero_not_mem_HNatInfinite: "0 \<notin> HNatInfinite"
  10.410 +lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite"
  10.411  apply (auto simp add: HNatInfinite_iff)
  10.412  apply (drule_tac a = " (1::hypnat) " in equals0D)
  10.413  apply simp
  10.414  done
  10.415 -declare zero_not_mem_HNatInfinite [simp]
  10.416  
  10.417  lemma HNatInfinite_not_eq_zero: "x \<in> HNatInfinite ==> 0 < x"
  10.418  apply (drule HNatInfinite_gt_one) 
  10.419 @@ -758,78 +691,60 @@
  10.420  
  10.421  subsection{*Closure Rules*}
  10.422  
  10.423 -lemma HNatInfinite_add: "[| x \<in> HNatInfinite; y \<in> HNatInfinite |]
  10.424 -            ==> x + y \<in> HNatInfinite"
  10.425 +lemma HNatInfinite_add:
  10.426 +     "[| x \<in> HNatInfinite; y \<in> HNatInfinite |] ==> x + y \<in> HNatInfinite"
  10.427  apply (auto simp add: HNatInfinite_iff)
  10.428  apply (drule bspec, assumption)
  10.429 -apply (drule bspec [OF _ SHNat_zero])
  10.430 +apply (drule bspec [OF _ Nats_0])
  10.431  apply (drule add_strict_mono, assumption, simp)
  10.432  done
  10.433  
  10.434 -lemma HNatInfinite_SHNat_add: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite"
  10.435 -apply (rule ccontr, drule not_HNatInfinite_SHNat)
  10.436 -apply (drule_tac x = "x + y" in SHNat_minus)
  10.437 -apply (auto simp add: SHNat_not_HNatInfinite_iff)
  10.438 +lemma HNatInfinite_SHNat_add:
  10.439 +     "[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite"
  10.440 +apply (auto simp add: HNatInfinite_not_Nats_iff) 
  10.441 +apply (drule_tac a = "x + y" in Nats_diff)
  10.442 +apply (auto ); 
  10.443  done
  10.444  
  10.445 -lemma HNatInfinite_SHNat_diff: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> x - y \<in> HNatInfinite"
  10.446 -apply (rule ccontr, drule not_HNatInfinite_SHNat)
  10.447 -apply (drule_tac x = "x - y" in SHNat_add)
  10.448 -apply (subgoal_tac [2] "y \<le> x")
  10.449 -apply (auto dest!: hypnat_le_add_diff_inverse2 simp add: not_SHNat_HNatInfinite_iff [symmetric])
  10.450 -apply (auto intro!: order_less_imp_le simp add: not_SHNat_HNatInfinite_iff HNatInfinite_iff)
  10.451 -done
  10.452 +lemma HNatInfinite_Nats_imp_less: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> y < x"
  10.453 +by (simp add: HNatInfinite_iff) 
  10.454 +
  10.455 +lemma HNatInfinite_SHNat_diff:
  10.456 +  assumes x: "x \<in> HNatInfinite" and y: "y \<in> Nats" 
  10.457 +  shows "x - y \<in> HNatInfinite"
  10.458 +proof -
  10.459 +  have "y < x" by (simp add: HNatInfinite_Nats_imp_less prems)
  10.460 +  hence "x - y + y = x" by (simp add: order_less_imp_le)
  10.461 +  with x show ?thesis
  10.462 +    by (force simp add: HNatInfinite_not_Nats_iff 
  10.463 +              dest: Nats_add [of "x-y", OF _ y]) 
  10.464 +qed
  10.465  
  10.466  lemma HNatInfinite_add_one: "x \<in> HNatInfinite ==> x + (1::hypnat) \<in> HNatInfinite"
  10.467  by (auto intro: HNatInfinite_SHNat_add)
  10.468  
  10.469 -lemma HNatInfinite_minus_one: "x \<in> HNatInfinite ==> x - (1::hypnat) \<in> HNatInfinite"
  10.470 -apply (rule ccontr, drule not_HNatInfinite_SHNat)
  10.471 -apply (drule_tac x = "x - (1::hypnat) " and y = " (1::hypnat) " in SHNat_add)
  10.472 -apply (auto simp add: not_SHNat_HNatInfinite_iff [symmetric])
  10.473 -done
  10.474 -
  10.475  lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)"
  10.476  apply (rule_tac x = "x - (1::hypnat) " in exI)
  10.477  apply auto
  10.478  done
  10.479  
  10.480  
  10.481 -subsection{*@{term HNat}: the Hypernaturals Embedded in the Hyperreals*}
  10.482 -
  10.483 +subsection{*Embedding of the Hypernaturals into the Hyperreals*}
  10.484  text{*Obtained using the nonstandard extension of the naturals*}
  10.485  
  10.486 -lemma HNat_hypreal_of_nat: "hypreal_of_nat N \<in> HNat"
  10.487 -apply (simp add: HNat_def starset_def hypreal_of_nat_def hypreal_of_real_def, auto, ultra)
  10.488 -apply (rule_tac x = N in exI, auto)
  10.489 -done
  10.490 -declare HNat_hypreal_of_nat [simp]
  10.491 -
  10.492 -lemma HNat_add: "[| x \<in> HNat; y \<in> HNat |] ==> x + y \<in> HNat"
  10.493 -apply (simp add: HNat_def starset_def)
  10.494 -apply (rule_tac z = x in eq_Abs_hypreal)
  10.495 -apply (rule_tac z = y in eq_Abs_hypreal)
  10.496 -apply (auto dest!: bspec intro: lemma_hyprel_refl simp add: hypreal_add, ultra)
  10.497 -apply (rule_tac x = "no+noa" in exI, auto)
  10.498 -done
  10.499 -
  10.500 -lemma HNat_mult:
  10.501 -     "[| x \<in> HNat; y \<in> HNat |] ==> x * y \<in> HNat"
  10.502 -apply (simp add: HNat_def starset_def)
  10.503 -apply (rule_tac z = x in eq_Abs_hypreal)
  10.504 -apply (rule_tac z = y in eq_Abs_hypreal)
  10.505 -apply (auto dest!: bspec intro: lemma_hyprel_refl simp add: hypreal_mult, ultra)
  10.506 -apply (rule_tac x = "no*noa" in exI, auto)
  10.507 -done
  10.508 +constdefs
  10.509 +  hypreal_of_hypnat :: "hypnat => hypreal"
  10.510 +   "hypreal_of_hypnat N  == 
  10.511 +      Abs_hypreal(\<Union>X \<in> Rep_hypnat(N). hyprel``{%n::nat. real (X n)})"
  10.512  
  10.513  
  10.514 -subsection{*Embedding of the Hypernaturals into the Hyperreals*}
  10.515 +lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"
  10.516 +by (simp add: hypreal_of_nat_def) 
  10.517  
  10.518  (*WARNING: FRAGILE!*)
  10.519 -lemma lemma_hyprel_FUFN: "(Ya \<in> hyprel ``{%n. f(n)}) =
  10.520 -      ({n. f n = Ya n} \<in> FreeUltrafilterNat)"
  10.521 -apply auto
  10.522 -done
  10.523 +lemma lemma_hyprel_FUFN:
  10.524 +     "(Ya \<in> hyprel ``{%n. f(n)}) = ({n. f n = Ya n} \<in> FreeUltrafilterNat)"
  10.525 +by force
  10.526  
  10.527  lemma hypreal_of_hypnat:
  10.528        "hypreal_of_hypnat (Abs_hypnat(hypnatrel``{%n. X n})) =
  10.529 @@ -840,16 +755,13 @@
  10.530         simp add: lemma_hyprel_FUFN)
  10.531  done
  10.532  
  10.533 -lemma inj_hypreal_of_hypnat: "inj(hypreal_of_hypnat)"
  10.534 -apply (rule inj_onI)
  10.535 -apply (rule_tac z = x in eq_Abs_hypnat)
  10.536 -apply (rule_tac z = y in eq_Abs_hypnat)
  10.537 +lemma hypreal_of_hypnat_inject [simp]:
  10.538 +     "(hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"
  10.539 +apply (rule eq_Abs_hypnat [of m])
  10.540 +apply (rule eq_Abs_hypnat [of n])
  10.541  apply (auto simp add: hypreal_of_hypnat)
  10.542  done
  10.543  
  10.544 -declare inj_hypreal_of_hypnat [THEN inj_eq, simp]
  10.545 -declare inj_hypnat_of_nat [THEN inj_eq, simp]
  10.546 -
  10.547  lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0"
  10.548  by (simp add: hypnat_zero_def hypreal_zero_def hypreal_of_hypnat)
  10.549  
  10.550 @@ -886,43 +798,21 @@
  10.551  apply (simp add: hypreal_of_hypnat hypreal_zero_num hypreal_le)
  10.552  done
  10.553  
  10.554 -(*????DELETE??*)
  10.555 -lemma hypnat_eq_zero: "\<forall>n. N \<le> n ==> N = (0::hypnat)"
  10.556 -apply (drule_tac x = 0 in spec)
  10.557 -apply (rule_tac z = N in eq_Abs_hypnat)
  10.558 -apply (auto simp add: hypnat_le hypnat_zero_def)
  10.559 -done
  10.560 -
  10.561 -(*????DELETE??*)
  10.562 -lemma hypnat_not_all_eq_zero: "~ (\<forall>n. n = (0::hypnat))"
  10.563 -by auto
  10.564 -
  10.565 -(*????DELETE??*)
  10.566 -lemma hypnat_le_one_eq_one: "n \<noteq> 0 ==> (n \<le> (1::hypnat)) = (n = (1::hypnat))"
  10.567 -by (auto simp add: order_le_less)
  10.568 -
  10.569 -(*WHERE DO THESE BELONG???*)
  10.570 -lemma HNatInfinite_inverse_Infinitesimal: "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
  10.571 +lemma HNatInfinite_inverse_Infinitesimal [simp]:
  10.572 +     "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
  10.573  apply (rule eq_Abs_hypnat [of n])
  10.574 -apply (auto simp add: hypreal_of_hypnat hypreal_inverse HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)
  10.575 +apply (auto simp add: hypreal_of_hypnat hypreal_inverse 
  10.576 +      HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)
  10.577  apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)
  10.578  apply (drule_tac x = "m + 1" in spec, ultra)
  10.579  done
  10.580 -declare HNatInfinite_inverse_Infinitesimal [simp]
  10.581 -
  10.582 -lemma HNatInfinite_inverse_not_zero: "n \<in> HNatInfinite ==> hypreal_of_hypnat n \<noteq> 0"
  10.583 -by (simp add: HNatInfinite_not_eq_zero)
  10.584 -
  10.585  
  10.586  
  10.587  ML
  10.588  {*
  10.589  val hypnat_of_nat_def = thm"hypnat_of_nat_def";
  10.590 -val HNat_def = thm"HNat_def";
  10.591  val HNatInfinite_def = thm"HNatInfinite_def";
  10.592  val hypreal_of_hypnat_def = thm"hypreal_of_hypnat_def";
  10.593 -val SNat_def = thm"SNat_def";
  10.594 -val SHNat_def = thm"SHNat_def";
  10.595  val hypnat_zero_def = thm"hypnat_zero_def";
  10.596  val hypnat_one_def = thm"hypnat_one_def";
  10.597  val hypnat_omega_def = thm"hypnat_omega_def";
  10.598 @@ -939,7 +829,6 @@
  10.599  val lemma_hypnatrel_refl = thm "lemma_hypnatrel_refl";
  10.600  val hypnat_empty_not_mem = thm "hypnat_empty_not_mem";
  10.601  val Rep_hypnat_nonempty = thm "Rep_hypnat_nonempty";
  10.602 -val inj_hypnat_of_nat = thm "inj_hypnat_of_nat";
  10.603  val eq_Abs_hypnat = thm "eq_Abs_hypnat";
  10.604  val hypnat_add_congruent2 = thm "hypnat_add_congruent2";
  10.605  val hypnat_add = thm "hypnat_add";
  10.606 @@ -1000,39 +889,14 @@
  10.607  val hypnat_of_nat_Suc = thm "hypnat_of_nat_Suc";
  10.608  val hypnat_omega = thm "hypnat_omega";
  10.609  val Rep_hypnat_omega = thm "Rep_hypnat_omega";
  10.610 -val not_ex_hypnat_of_nat_eq_omega = thm "not_ex_hypnat_of_nat_eq_omega";
  10.611 -val hypnat_of_nat_not_eq_omega = thm "hypnat_of_nat_not_eq_omega";
  10.612  val SHNAT_omega_not_mem = thm "SHNAT_omega_not_mem";
  10.613 -val SHNat_add = thm "SHNat_add";
  10.614 -val SHNat_minus = thm "SHNat_minus";
  10.615 -val SHNat_mult = thm "SHNat_mult";
  10.616 -val SHNat_add_cancel = thm "SHNat_add_cancel";
  10.617 -val SHNat_hypnat_of_nat = thm "SHNat_hypnat_of_nat";
  10.618 -val SHNat_hypnat_of_nat_one = thm "SHNat_hypnat_of_nat_one";
  10.619 -val SHNat_hypnat_of_nat_zero = thm "SHNat_hypnat_of_nat_zero";
  10.620 -val SHNat_one = thm "SHNat_one";
  10.621 -val SHNat_zero = thm "SHNat_zero";
  10.622 -val SHNat_iff = thm "SHNat_iff";
  10.623 -val SHNat_hypnat_of_nat_iff = thm "SHNat_hypnat_of_nat_iff";
  10.624 -val leSuc_Un_eq = thm "leSuc_Un_eq";
  10.625 -val finite_nat_le_segment = thm "finite_nat_le_segment";
  10.626 -val lemma_unbounded_set = thm "lemma_unbounded_set";
  10.627  val cofinite_mem_FreeUltrafilterNat = thm "cofinite_mem_FreeUltrafilterNat";
  10.628 -val Compl_Collect_le = thm "Compl_Collect_le";
  10.629  val hypnat_omega_gt_SHNat = thm "hypnat_omega_gt_SHNat";
  10.630  val hypnat_of_nat_less_whn = thm "hypnat_of_nat_less_whn";
  10.631  val hypnat_of_nat_le_whn = thm "hypnat_of_nat_le_whn";
  10.632  val hypnat_zero_less_hypnat_omega = thm "hypnat_zero_less_hypnat_omega";
  10.633  val hypnat_one_less_hypnat_omega = thm "hypnat_one_less_hypnat_omega";
  10.634  val HNatInfinite_whn = thm "HNatInfinite_whn";
  10.635 -val SHNat_not_HNatInfinite = thm "SHNat_not_HNatInfinite";
  10.636 -val not_HNatInfinite_SHNat = thm "not_HNatInfinite_SHNat";
  10.637 -val not_SHNat_HNatInfinite = thm "not_SHNat_HNatInfinite";
  10.638 -val HNatInfinite_not_SHNat = thm "HNatInfinite_not_SHNat";
  10.639 -val SHNat_not_HNatInfinite_iff = thm "SHNat_not_HNatInfinite_iff";
  10.640 -val not_SHNat_HNatInfinite_iff = thm "not_SHNat_HNatInfinite_iff";
  10.641 -val SHNat_HNatInfinite_disj = thm "SHNat_HNatInfinite_disj";
  10.642 -val HNatInfinite_FreeUltrafilterNat_lemma = thm "HNatInfinite_FreeUltrafilterNat_lemma";
  10.643  val HNatInfinite_iff = thm "HNatInfinite_iff";
  10.644  val HNatInfinite_FreeUltrafilterNat = thm "HNatInfinite_FreeUltrafilterNat";
  10.645  val FreeUltrafilterNat_HNatInfinite = thm "FreeUltrafilterNat_HNatInfinite";
  10.646 @@ -1045,26 +909,16 @@
  10.647  val HNatInfinite_SHNat_add = thm "HNatInfinite_SHNat_add";
  10.648  val HNatInfinite_SHNat_diff = thm "HNatInfinite_SHNat_diff";
  10.649  val HNatInfinite_add_one = thm "HNatInfinite_add_one";
  10.650 -val HNatInfinite_minus_one = thm "HNatInfinite_minus_one";
  10.651  val HNatInfinite_is_Suc = thm "HNatInfinite_is_Suc";
  10.652  val HNat_hypreal_of_nat = thm "HNat_hypreal_of_nat";
  10.653 -val HNat_add = thm "HNat_add";
  10.654 -val HNat_mult = thm "HNat_mult";
  10.655 -val lemma_hyprel_FUFN = thm "lemma_hyprel_FUFN";
  10.656  val hypreal_of_hypnat = thm "hypreal_of_hypnat";
  10.657 -val inj_hypreal_of_hypnat = thm "inj_hypreal_of_hypnat";
  10.658  val hypreal_of_hypnat_zero = thm "hypreal_of_hypnat_zero";
  10.659  val hypreal_of_hypnat_one = thm "hypreal_of_hypnat_one";
  10.660  val hypreal_of_hypnat_add = thm "hypreal_of_hypnat_add";
  10.661  val hypreal_of_hypnat_mult = thm "hypreal_of_hypnat_mult";
  10.662  val hypreal_of_hypnat_less_iff = thm "hypreal_of_hypnat_less_iff";
  10.663 -val hypreal_of_hypnat_eq_zero_iff = thm "hypreal_of_hypnat_eq_zero_iff";
  10.664  val hypreal_of_hypnat_ge_zero = thm "hypreal_of_hypnat_ge_zero";
  10.665 -val hypnat_eq_zero = thm "hypnat_eq_zero";
  10.666 -val hypnat_not_all_eq_zero = thm "hypnat_not_all_eq_zero";
  10.667 -val hypnat_le_one_eq_one = thm "hypnat_le_one_eq_one";
  10.668  val HNatInfinite_inverse_Infinitesimal = thm "HNatInfinite_inverse_Infinitesimal";
  10.669 -val HNatInfinite_inverse_not_zero = thm "HNatInfinite_inverse_not_zero";
  10.670  *}
  10.671  
  10.672  end
    11.1 --- a/src/HOL/Hyperreal/HyperPow.thy	Thu Feb 05 10:45:28 2004 +0100
    11.2 +++ b/src/HOL/Hyperreal/HyperPow.thy	Tue Feb 10 12:02:11 2004 +0100
    11.3 @@ -163,8 +163,8 @@
    11.4  lemma hyperpow_not_zero [rule_format (no_asm)]:
    11.5       "r \<noteq> (0::hypreal) --> r pow n \<noteq> 0"
    11.6  apply (simp (no_asm) add: hypreal_zero_def)
    11.7 -apply (rule_tac z = n in eq_Abs_hypnat)
    11.8 -apply (rule_tac z = r in eq_Abs_hypreal)
    11.9 +apply (rule eq_Abs_hypnat [of n])
   11.10 +apply (rule eq_Abs_hypreal [of r])
   11.11  apply (auto simp add: hyperpow)
   11.12  apply (drule FreeUltrafilterNat_Compl_mem, ultra)
   11.13  done
   11.14 @@ -172,29 +172,29 @@
   11.15  lemma hyperpow_inverse:
   11.16       "r \<noteq> (0::hypreal) --> inverse(r pow n) = (inverse r) pow n"
   11.17  apply (simp (no_asm) add: hypreal_zero_def)
   11.18 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.19 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.20 +apply (rule eq_Abs_hypnat [of n])
   11.21 +apply (rule eq_Abs_hypreal [of r])
   11.22  apply (auto dest!: FreeUltrafilterNat_Compl_mem simp add: hypreal_inverse hyperpow)
   11.23  apply (rule FreeUltrafilterNat_subset)
   11.24  apply (auto dest: realpow_not_zero intro: power_inverse)
   11.25  done
   11.26  
   11.27  lemma hyperpow_hrabs: "abs r pow n = abs (r pow n)"
   11.28 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.29 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.30 +apply (rule eq_Abs_hypnat [of n])
   11.31 +apply (rule eq_Abs_hypreal [of r])
   11.32  apply (auto simp add: hypreal_hrabs hyperpow power_abs [symmetric])
   11.33  done
   11.34  
   11.35  lemma hyperpow_add: "r pow (n + m) = (r pow n) * (r pow m)"
   11.36 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.37 -apply (rule_tac z = m in eq_Abs_hypnat)
   11.38 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.39 +apply (rule eq_Abs_hypnat [of n])
   11.40 +apply (rule eq_Abs_hypnat [of m])
   11.41 +apply (rule eq_Abs_hypreal [of r])
   11.42  apply (auto simp add: hyperpow hypnat_add hypreal_mult power_add)
   11.43  done
   11.44  
   11.45  lemma hyperpow_one: "r pow (1::hypnat) = r"
   11.46  apply (unfold hypnat_one_def)
   11.47 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.48 +apply (rule eq_Abs_hypreal [of r])
   11.49  apply (auto simp add: hyperpow)
   11.50  done
   11.51  declare hyperpow_one [simp]
   11.52 @@ -202,38 +202,38 @@
   11.53  lemma hyperpow_two:
   11.54       "r pow ((1::hypnat) + (1::hypnat)) = r * r"
   11.55  apply (unfold hypnat_one_def)
   11.56 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.57 +apply (rule eq_Abs_hypreal [of r])
   11.58  apply (auto simp add: hyperpow hypnat_add hypreal_mult)
   11.59  done
   11.60  
   11.61  lemma hyperpow_gt_zero: "(0::hypreal) < r ==> 0 < r pow n"
   11.62  apply (simp add: hypreal_zero_def)
   11.63 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.64 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.65 +apply (rule eq_Abs_hypnat [of n])
   11.66 +apply (rule eq_Abs_hypreal [of r])
   11.67  apply (auto elim!: FreeUltrafilterNat_subset zero_less_power
   11.68                     simp add: hyperpow hypreal_less hypreal_le)
   11.69  done
   11.70  
   11.71  lemma hyperpow_ge_zero: "(0::hypreal) \<le> r ==> 0 \<le> r pow n"
   11.72  apply (simp add: hypreal_zero_def)
   11.73 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.74 -apply (rule_tac z = r in eq_Abs_hypreal)
   11.75 +apply (rule eq_Abs_hypnat [of n])
   11.76 +apply (rule eq_Abs_hypreal [of r])
   11.77  apply (auto elim!: FreeUltrafilterNat_subset zero_le_power 
   11.78              simp add: hyperpow hypreal_le)
   11.79  done
   11.80  
   11.81  lemma hyperpow_le: "[|(0::hypreal) < x; x \<le> y|] ==> x pow n \<le> y pow n"
   11.82  apply (simp add: hypreal_zero_def)
   11.83 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.84 -apply (rule_tac z = x in eq_Abs_hypreal)
   11.85 -apply (rule_tac z = y in eq_Abs_hypreal)
   11.86 +apply (rule eq_Abs_hypnat [of n])
   11.87 +apply (rule eq_Abs_hypreal [of x])
   11.88 +apply (rule eq_Abs_hypreal [of y])
   11.89  apply (auto simp add: hyperpow hypreal_le hypreal_less)
   11.90  apply (erule FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset], assumption)
   11.91  apply (auto intro: power_mono)
   11.92  done
   11.93  
   11.94  lemma hyperpow_eq_one: "1 pow n = (1::hypreal)"
   11.95 -apply (rule_tac z = n in eq_Abs_hypnat)
   11.96 +apply (rule eq_Abs_hypnat [of n])
   11.97  apply (auto simp add: hypreal_one_def hyperpow)
   11.98  done
   11.99  declare hyperpow_eq_one [simp]
  11.100 @@ -241,15 +241,15 @@
  11.101  lemma hrabs_hyperpow_minus_one: "abs(-1 pow n) = (1::hypreal)"
  11.102  apply (subgoal_tac "abs ((- (1::hypreal)) pow n) = (1::hypreal) ")
  11.103  apply simp
  11.104 -apply (rule_tac z = n in eq_Abs_hypnat)
  11.105 +apply (rule eq_Abs_hypnat [of n])
  11.106  apply (auto simp add: hypreal_one_def hyperpow hypreal_minus hypreal_hrabs)
  11.107  done
  11.108  declare hrabs_hyperpow_minus_one [simp]
  11.109  
  11.110  lemma hyperpow_mult: "(r * s) pow n = (r pow n) * (s pow n)"
  11.111 -apply (rule_tac z = n in eq_Abs_hypnat)
  11.112 -apply (rule_tac z = r in eq_Abs_hypreal)
  11.113 -apply (rule_tac z = s in eq_Abs_hypreal)
  11.114 +apply (rule eq_Abs_hypnat [of n])
  11.115 +apply (rule eq_Abs_hypreal [of r])
  11.116 +apply (rule eq_Abs_hypreal [of s])
  11.117  apply (auto simp add: hyperpow hypreal_mult power_mult_distrib)
  11.118  done
  11.119  
  11.120 @@ -297,9 +297,9 @@
  11.121  
  11.122  lemma hyperpow_less_le:
  11.123       "[|(0::hypreal) \<le> r; r \<le> 1; n < N|] ==> r pow N \<le> r pow n"
  11.124 -apply (rule_tac z = n in eq_Abs_hypnat)
  11.125 -apply (rule_tac z = N in eq_Abs_hypnat)
  11.126 -apply (rule_tac z = r in eq_Abs_hypreal)
  11.127 +apply (rule eq_Abs_hypnat [of n])
  11.128 +apply (rule eq_Abs_hypnat [of N])
  11.129 +apply (rule eq_Abs_hypreal [of r])
  11.130  apply (auto simp add: hyperpow hypreal_le hypreal_less hypnat_less 
  11.131              hypreal_zero_def hypreal_one_def)
  11.132  apply (erule FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset])
  11.133 @@ -314,8 +314,7 @@
  11.134  
  11.135  lemma hyperpow_realpow:
  11.136        "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)"
  11.137 -apply (unfold hypreal_of_real_def hypnat_of_nat_def)
  11.138 -apply (auto simp add: hyperpow)
  11.139 +apply (simp add: hypreal_of_real_def hypnat_of_nat_eq hyperpow)
  11.140  done
  11.141  
  11.142  lemma hyperpow_SReal: "(hypreal_of_real r) pow (hypnat_of_nat n) \<in> Reals"
  11.143 @@ -355,9 +354,8 @@
  11.144  
  11.145  lemma hrealpow_hyperpow_Infinitesimal_iff:
  11.146       "(x ^ n \<in> Infinitesimal) = (x pow (hypnat_of_nat n) \<in> Infinitesimal)"
  11.147 -apply (unfold hypnat_of_nat_def)
  11.148 -apply (rule_tac z = x in eq_Abs_hypreal)
  11.149 -apply (auto simp add: hrealpow hyperpow)
  11.150 +apply (rule eq_Abs_hypreal [of x])
  11.151 +apply (simp add: hrealpow hyperpow hypnat_of_nat_eq)
  11.152  done
  11.153  
  11.154  lemma Infinitesimal_hrealpow:
    12.1 --- a/src/HOL/Hyperreal/NSA.thy	Thu Feb 05 10:45:28 2004 +0100
    12.2 +++ b/src/HOL/Hyperreal/NSA.thy	Tue Feb 10 12:02:11 2004 +0100
    12.3 @@ -33,7 +33,7 @@
    12.4     "x @= y       == (x + -y) \<in> Infinitesimal"
    12.5  
    12.6  
    12.7 -defs
    12.8 +defs (overloaded)
    12.9  
   12.10     (*standard real numbers as a subset of the hyperreals*)
   12.11     SReal_def:      "Reals == {x. \<exists>r. x = hypreal_of_real r}"
   12.12 @@ -47,7 +47,8 @@
   12.13       Closure laws for members of (embedded) set standard real Reals
   12.14   --------------------------------------------------------------------*)
   12.15  
   12.16 -lemma SReal_add: "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x + y \<in> Reals"
   12.17 +lemma SReal_add [simp]:
   12.18 +     "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x + y \<in> Reals"
   12.19  apply (auto simp add: SReal_def)
   12.20  apply (rule_tac x = "r + ra" in exI, simp)
   12.21  done
   12.22 @@ -1911,6 +1912,11 @@
   12.23  apply (blast dest!: reals_Archimedean intro: order_less_trans)
   12.24  done
   12.25  
   12.26 +lemma of_nat_in_Reals [simp]: "(of_nat n::hypreal) \<in> \<real>"
   12.27 +apply (induct n)
   12.28 +apply (simp_all add: SReal_add);
   12.29 +done 
   12.30 + 
   12.31  lemma lemma_Infinitesimal2: "(\<forall>r \<in> Reals. 0 < r --> x < r) =
   12.32        (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
   12.33  apply safe
   12.34 @@ -1918,13 +1924,14 @@
   12.35  apply (simp (no_asm_use) add: SReal_inverse)
   12.36  apply (rule real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive, THEN hypreal_of_real_less_iff [THEN iffD2], THEN [2] impE])
   12.37  prefer 2 apply assumption
   12.38 -apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_def)
   12.39 +apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
   12.40  apply (auto dest!: reals_Archimedean simp add: SReal_iff)
   12.41  apply (drule hypreal_of_real_less_iff [THEN iffD2])
   12.42 -apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_def)
   12.43 +apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
   12.44  apply (blast intro: order_less_trans)
   12.45  done
   12.46  
   12.47 +
   12.48  lemma Infinitesimal_hypreal_of_nat_iff:
   12.49       "Infinitesimal = {x. \<forall>n. abs x < inverse (hypreal_of_nat (Suc n))}"
   12.50  apply (simp add: Infinitesimal_def)
    13.1 --- a/src/HOL/Hyperreal/NatStar.ML	Thu Feb 05 10:45:28 2004 +0100
    13.2 +++ b/src/HOL/Hyperreal/NatStar.ML	Tue Feb 10 12:02:11 2004 +0100
    13.3 @@ -6,6 +6,9 @@
    13.4                    nat=>nat functions
    13.5  *) 
    13.6  
    13.7 +val hypnat_of_nat_eq = thm"hypnat_of_nat_eq";
    13.8 +val SHNat_eq = thm"SHNat_eq";
    13.9 +
   13.10  Goalw [starsetNat_def] 
   13.11        "*sNat*(UNIV::nat set) = (UNIV::hypnat set)";
   13.12  by (auto_tac (claset(), simpset() addsimps [FreeUltrafilterNat_Nat_set]));
   13.13 @@ -111,28 +114,26 @@
   13.14  by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
   13.15  qed "NatStar_subset";
   13.16  
   13.17 -Goalw [starsetNat_def,hypnat_of_nat_def] 
   13.18 -          "a : A ==> hypnat_of_nat a : *sNat* A";
   13.19 +Goal "a : A ==> hypnat_of_nat a : *sNat* A";
   13.20  by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],
   13.21 -         simpset()));
   13.22 +         simpset() addsimps [starsetNat_def,hypnat_of_nat_eq]));
   13.23  qed "NatStar_mem";
   13.24  
   13.25  Goalw [starsetNat_def] "hypnat_of_nat ` A <= *sNat* A";
   13.26 -by (auto_tac (claset(), simpset() addsimps [hypnat_of_nat_def]));
   13.27 +by (auto_tac (claset(), simpset() addsimps [hypnat_of_nat_eq]));
   13.28  by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
   13.29  qed "NatStar_hypreal_of_real_image_subset";
   13.30  
   13.31  Goal "Nats <= *sNat* (UNIV:: nat set)";
   13.32 -by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_iff,
   13.33 -                          NatStar_hypreal_of_real_image_subset]) 1);
   13.34 +by (auto_tac (claset(), simpset() addsimps [starsetNat_def,SHNat_eq,hypnat_of_nat_eq]));
   13.35  qed "NatStar_SHNat_subset";
   13.36  
   13.37  Goalw [starsetNat_def] 
   13.38       "*sNat* X Int Nats = hypnat_of_nat ` X";
   13.39  by (auto_tac (claset(),
   13.40           simpset() addsimps 
   13.41 -           [hypnat_of_nat_def,SHNat_def]));
   13.42 -by (fold_tac [hypnat_of_nat_def]);
   13.43 +           [hypnat_of_nat_eq,SHNat_eq]));
   13.44 +by (simp_tac (simpset() addsimps [hypnat_of_nat_eq RS sym]) 1);
   13.45  by (rtac imageI 1 THEN rtac ccontr 1);
   13.46  by (dtac bspec 1);
   13.47  by (rtac lemma_hypnatrel_refl 1);
   13.48 @@ -289,7 +290,7 @@
   13.49  
   13.50  Goal "( *fNat2* (%x. k)) z = hypnat_of_nat  k";
   13.51  by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
   13.52 -by (auto_tac (claset(), simpset() addsimps [starfunNat2, hypnat_of_nat_def]));
   13.53 +by (auto_tac (claset(), simpset() addsimps [starfunNat2, hypnat_of_nat_eq]));
   13.54  qed "starfunNat2_const_fun";
   13.55  
   13.56  Addsimps [starfunNat2_const_fun];
   13.57 @@ -312,13 +313,13 @@
   13.58  
   13.59  Goal "( *fNat* f) (hypnat_of_nat a) = hypreal_of_real (f a)";
   13.60  by (auto_tac (claset(),
   13.61 -      simpset() addsimps [starfunNat,hypnat_of_nat_def,hypreal_of_real_def]));
   13.62 +      simpset() addsimps [starfunNat,hypnat_of_nat_eq,hypreal_of_real_def]));
   13.63  qed "starfunNat_eq";
   13.64  
   13.65  Addsimps [starfunNat_eq];
   13.66  
   13.67  Goal "( *fNat2* f) (hypnat_of_nat a) = hypnat_of_nat (f a)";
   13.68 -by (auto_tac (claset(), simpset() addsimps [starfunNat2,hypnat_of_nat_def]));
   13.69 +by (auto_tac (claset(), simpset() addsimps [starfunNat2,hypnat_of_nat_eq]));
   13.70  qed "starfunNat2_eq";
   13.71  
   13.72  Addsimps [starfunNat2_eq];
   13.73 @@ -337,7 +338,7 @@
   13.74  by (Step_tac 1);
   13.75  by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
   13.76  by (auto_tac (claset(),
   13.77 -         simpset() addsimps [starfunNat, hypnat_of_nat_def,hypreal_le,
   13.78 +         simpset() addsimps [starfunNat, hypnat_of_nat_eq,hypreal_le,
   13.79                               hypreal_less, hypnat_le,hypnat_less]));
   13.80  by (Ultra_tac 1);
   13.81  by Auto_tac;
   13.82 @@ -349,7 +350,7 @@
   13.83  by (Step_tac 1);
   13.84  by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
   13.85  by (auto_tac (claset(),
   13.86 -         simpset() addsimps [starfunNat, hypnat_of_nat_def,hypreal_le,
   13.87 +         simpset() addsimps [starfunNat, hypnat_of_nat_eq,hypreal_le,
   13.88                               hypreal_less, hypnat_le,hypnat_less]));
   13.89  by (Ultra_tac 1);
   13.90  by Auto_tac;
   13.91 @@ -384,12 +385,12 @@
   13.92  Goal "( *fNat* (%n. (X n) ^ m)) N = ( *fNat* X) N pow hypnat_of_nat m";
   13.93  by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);
   13.94  by (auto_tac (claset(),
   13.95 -         simpset() addsimps [hyperpow, hypnat_of_nat_def,starfunNat]));
   13.96 +         simpset() addsimps [hyperpow, hypnat_of_nat_eq,starfunNat]));
   13.97  qed "starfunNat_pow2";
   13.98  
   13.99 -Goalw [hypnat_of_nat_def] "( *f* (%r. r ^ n)) R = (R) pow hypnat_of_nat n";
  13.100 +Goal "( *f* (%r. r ^ n)) R = (R) pow hypnat_of_nat n";
  13.101  by (res_inst_tac [("z","R")] eq_Abs_hypreal 1);
  13.102 -by (auto_tac (claset(), simpset() addsimps [hyperpow,starfun]));
  13.103 +by (auto_tac (claset(), simpset() addsimps [hyperpow,starfun,hypnat_of_nat_eq]));
  13.104  qed "starfun_pow";
  13.105  
  13.106  (*----------------------------------------------------- 
  13.107 @@ -469,7 +470,7 @@
  13.108  qed "starfunNat_n_minus";
  13.109  
  13.110  Goal "( *fNatn* f) (hypnat_of_nat n) = Abs_hypreal(hyprel `` {%i. f i n})";
  13.111 -by (auto_tac (claset(), simpset() addsimps [starfunNat_n,hypnat_of_nat_def]));
  13.112 +by (auto_tac (claset(), simpset() addsimps [starfunNat_n,hypnat_of_nat_eq]));
  13.113  qed "starfunNat_n_eq";
  13.114  Addsimps [starfunNat_n_eq];
  13.115  
  13.116 @@ -477,7 +478,7 @@
  13.117  by Auto_tac;
  13.118  by (rtac ext 1 THEN rtac ccontr 1);
  13.119  by (dres_inst_tac [("x","hypnat_of_nat(x)")] fun_cong 1);
  13.120 -by (auto_tac (claset(), simpset() addsimps [starfunNat,hypnat_of_nat_def]));
  13.121 +by (auto_tac (claset(), simpset() addsimps [starfunNat,hypnat_of_nat_eq]));
  13.122  qed "starfun_eq_iff";
  13.123  
  13.124  
    14.1 --- a/src/HOL/Hyperreal/SEQ.ML	Thu Feb 05 10:45:28 2004 +0100
    14.2 +++ b/src/HOL/Hyperreal/SEQ.ML	Tue Feb 10 12:02:11 2004 +0100
    14.3 @@ -4,6 +4,8 @@
    14.4      Description : Theory of sequence and series of real numbers
    14.5  *) 
    14.6  
    14.7 +val Nats_1 = thm"Nats_1";
    14.8 +
    14.9  fun CLAIM_SIMP str thms =
   14.10                 prove_goal (the_context()) str
   14.11                 (fn prems => [cut_facts_tac prems 1,
   14.12 @@ -92,7 +94,7 @@
   14.13  by (induct_tac "u" 1);
   14.14  by (auto_tac (claset(),simpset() addsimps [lemma_NSLIMSEQ2]));
   14.15  by (auto_tac (claset() addIs [(lemma_NSLIMSEQ3 RS finite_subset),
   14.16 -    finite_nat_le_segment], simpset()));
   14.17 +    rewrite_rule [atMost_def] finite_atMost], simpset()));
   14.18  by (dtac lemma_NSLIMSEQ1 1 THEN Step_tac 1);
   14.19  by (ALLGOALS(Asm_simp_tac));
   14.20  qed "NSLIMSEQ_finite_set";
   14.21 @@ -1050,7 +1052,7 @@
   14.22       simpset() addsimps [NSCauchy_def, NSLIMSEQ_def,starfunNat_shift_one]));
   14.23  by (dtac bspec 1 THEN assume_tac 1);
   14.24  by (dtac bspec 1 THEN assume_tac 1);
   14.25 -by (dtac (SHNat_one RSN (2,HNatInfinite_SHNat_add)) 1);
   14.26 +by (dtac (Nats_1 RSN (2,HNatInfinite_SHNat_add)) 1);
   14.27  by (blast_tac (claset() addIs [approx_trans3]) 1);
   14.28  qed "NSLIMSEQ_Suc";
   14.29  
   14.30 @@ -1067,7 +1069,7 @@
   14.31        simpset() addsimps [NSCauchy_def, NSLIMSEQ_def,starfunNat_shift_one]));
   14.32  by (dtac bspec 1 THEN assume_tac 1);
   14.33  by (dtac bspec 1 THEN assume_tac 1);
   14.34 -by (ftac (SHNat_one RSN (2,HNatInfinite_SHNat_diff)) 1);
   14.35 +by (ftac (Nats_1 RSN (2,HNatInfinite_SHNat_diff)) 1);
   14.36  by (rotate_tac 2 1);
   14.37  by (auto_tac (claset() addSDs [bspec] addIs [approx_trans3],
   14.38      simpset()));
    15.1 --- a/src/HOL/Hyperreal/Star.thy	Thu Feb 05 10:45:28 2004 +0100
    15.2 +++ b/src/HOL/Hyperreal/Star.thy	Tue Feb 10 12:02:11 2004 +0100
    15.3 @@ -456,14 +456,15 @@
    15.4     In this theory since hypreal_hrabs proved here. (To Check:) Maybe
    15.5     move both if possible?
    15.6   -------------------------------------------------------------------*)
    15.7 -lemma Infinitesimal_FreeUltrafilterNat_iff2: "(x:Infinitesimal) =
    15.8 +lemma Infinitesimal_FreeUltrafilterNat_iff2:
    15.9 +     "(x:Infinitesimal) =
   15.10        (EX X:Rep_hypreal(x).
   15.11          ALL m. {n. abs(X n) < inverse(real(Suc m))}
   15.12                 : FreeUltrafilterNat)"
   15.13 -apply (rule_tac z = x in eq_Abs_hypreal)
   15.14 +apply (rule eq_Abs_hypreal [of x])
   15.15  apply (auto intro!: bexI lemma_hyprel_refl 
   15.16 -            simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def 
   15.17 -     hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_def)
   15.18 +            simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def
   15.19 +     hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_eq)
   15.20  apply (drule_tac x = n in spec, ultra)
   15.21  done
   15.22  
    16.1 --- a/src/HOL/Integ/Bin.thy	Thu Feb 05 10:45:28 2004 +0100
    16.2 +++ b/src/HOL/Integ/Bin.thy	Tue Feb 10 12:02:11 2004 +0100
    16.3 @@ -8,7 +8,7 @@
    16.4  
    16.5  header{*Arithmetic on Binary Integers*}
    16.6  
    16.7 -theory Bin = Int + Numeral:
    16.8 +theory Bin = IntDef + Numeral:
    16.9  
   16.10  text{*The sign @{term Pls} stands for an infinite string of leading Falses.*}
   16.11  text{*The sign @{term Min} stands for an infinite string of leading Trues.*}
   16.12 @@ -258,7 +258,7 @@
   16.13  
   16.14  lemma eq_number_of_eq:
   16.15    "((number_of x::int) = number_of y) =
   16.16 -   iszero (number_of (bin_add x (bin_minus y)))"
   16.17 +   iszero (number_of (bin_add x (bin_minus y)) :: int)"
   16.18  apply (unfold iszero_def)
   16.19  apply (simp add: compare_rls number_of_add number_of_minus)
   16.20  done
   16.21 @@ -272,14 +272,15 @@
   16.22  done
   16.23  
   16.24  lemma iszero_number_of_BIT:
   16.25 -     "iszero (number_of (w BIT x)) = (~x & iszero (number_of w::int))"
   16.26 +     "iszero (number_of (w BIT x)::int) = (~x & iszero (number_of w::int))"
   16.27  apply (unfold iszero_def)
   16.28  apply (cases "(number_of w)::int" rule: int_cases) 
   16.29  apply (simp_all (no_asm_simp) add: compare_rls zero_reorient
   16.30           zminus_zadd_distrib [symmetric] int_Suc0_eq_1 [symmetric] zadd_int)
   16.31  done
   16.32  
   16.33 -lemma iszero_number_of_0: "iszero (number_of (w BIT False)) = iszero (number_of w::int)"
   16.34 +lemma iszero_number_of_0:
   16.35 +     "iszero (number_of (w BIT False)::int) = iszero (number_of w::int)"
   16.36  by (simp only: iszero_number_of_BIT simp_thms)
   16.37  
   16.38  lemma iszero_number_of_1: "~ iszero (number_of (w BIT True)::int)"
   16.39 @@ -291,24 +292,23 @@
   16.40  
   16.41  lemma less_number_of_eq_neg:
   16.42      "((number_of x::int) < number_of y)
   16.43 -     = neg (number_of (bin_add x (bin_minus y)))"
   16.44 +     = neg (number_of (bin_add x (bin_minus y)) ::int )"
   16.45 +by (simp add: neg_def number_of_add number_of_minus compare_rls) 
   16.46  
   16.47 -apply (unfold zless_def zdiff_def)
   16.48 -apply (simp add: bin_mult_simps)
   16.49 -done
   16.50  
   16.51  (*But if Numeral0 is rewritten to 0 then this rule can't be applied:
   16.52    Numeral0 IS (number_of Pls) *)
   16.53 -lemma not_neg_number_of_Pls: "~ neg (number_of bin.Pls)"
   16.54 -by (simp add: neg_eq_less_0)
   16.55 +lemma not_neg_number_of_Pls: "~ neg (number_of bin.Pls ::int)"
   16.56 +by (simp add: neg_def)
   16.57  
   16.58 -lemma neg_number_of_Min: "neg (number_of bin.Min)"
   16.59 -by (simp add: neg_eq_less_0 int_0_less_1)
   16.60 +lemma neg_number_of_Min: "neg (number_of bin.Min ::int)"
   16.61 +by (simp add: neg_def int_0_less_1)
   16.62  
   16.63 -lemma neg_number_of_BIT: "neg (number_of (w BIT x)) = neg (number_of w)"
   16.64 +lemma neg_number_of_BIT:
   16.65 +     "neg (number_of (w BIT x)::int) = neg (number_of w ::int)"
   16.66  apply simp
   16.67  apply (cases "(number_of w)::int" rule: int_cases) 
   16.68 -apply (simp_all (no_asm_simp) add: int_Suc0_eq_1 [symmetric] zadd_int neg_eq_less_0 zdiff_def [symmetric] compare_rls)
   16.69 +apply (simp_all (no_asm_simp) add: int_Suc0_eq_1 [symmetric] zadd_int neg_def zdiff_def [symmetric] compare_rls)
   16.70  done
   16.71  
   16.72  
   16.73 @@ -326,9 +326,7 @@
   16.74  lemma zabs_number_of:
   16.75   "abs(number_of x::int) =
   16.76    (if number_of x < (0::int) then -number_of x else number_of x)"
   16.77 -apply (unfold zabs_def)
   16.78 -apply (rule refl)
   16.79 -done
   16.80 +by (simp add: zabs_def)
   16.81  
   16.82  (*0 and 1 require special rewrites because they aren't numerals*)
   16.83  lemma zabs_0: "abs (0::int) = 0"
    17.1 --- a/src/HOL/Integ/Int.thy	Thu Feb 05 10:45:28 2004 +0100
    17.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    17.3 @@ -1,409 +0,0 @@
    17.4 -(*  Title:      Integ/Int.thy
    17.5 -    ID:         $Id$
    17.6 -    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    17.7 -    Copyright   1998  University of Cambridge
    17.8 -*)
    17.9 -
   17.10 -header {*Type "int" is an Ordered Ring and Other Lemmas*}
   17.11 -
   17.12 -theory Int = IntDef:
   17.13 -
   17.14 -constdefs
   17.15 -   nat  :: "int => nat"
   17.16 -    "nat(Z) == if neg Z then 0 else (THE m. Z = int m)"
   17.17 -
   17.18 -defs (overloaded)
   17.19 -    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
   17.20 -
   17.21 -
   17.22 -lemma int_0 [simp]: "int 0 = (0::int)"
   17.23 -by (simp add: Zero_int_def)
   17.24 -
   17.25 -lemma int_1 [simp]: "int 1 = 1"
   17.26 -by (simp add: One_int_def)
   17.27 -
   17.28 -lemma int_Suc0_eq_1: "int (Suc 0) = 1"
   17.29 -by (simp add: One_int_def One_nat_def)
   17.30 -
   17.31 -lemma neg_eq_less_0: "neg x = (x < 0)"
   17.32 -by (unfold zdiff_def zless_def, auto)
   17.33 -
   17.34 -lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
   17.35 -apply (unfold zle_def)
   17.36 -apply (simp add: neg_eq_less_0)
   17.37 -done
   17.38 -
   17.39 -subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
   17.40 -
   17.41 -lemma not_neg_0: "~ neg 0"
   17.42 -by (simp add: One_int_def neg_eq_less_0)
   17.43 -
   17.44 -lemma not_neg_1: "~ neg 1"
   17.45 -by (simp add: One_int_def neg_eq_less_0)
   17.46 -
   17.47 -lemma iszero_0: "iszero 0"
   17.48 -by (simp add: iszero_def)
   17.49 -
   17.50 -lemma not_iszero_1: "~ iszero 1"
   17.51 -by (simp only: Zero_int_def One_int_def One_nat_def iszero_def int_int_eq)
   17.52 -
   17.53 -
   17.54 -subsection{*nat: magnitide of an integer, as a natural number*}
   17.55 -
   17.56 -lemma nat_int [simp]: "nat(int n) = n"
   17.57 -by (unfold nat_def, auto)
   17.58 -
   17.59 -lemma nat_zero [simp]: "nat 0 = 0"
   17.60 -apply (unfold Zero_int_def)
   17.61 -apply (rule nat_int)
   17.62 -done
   17.63 -
   17.64 -lemma neg_nat: "neg z ==> nat z = 0"
   17.65 -by (unfold nat_def, auto)
   17.66 -
   17.67 -lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
   17.68 -apply (drule not_neg_eq_ge_0 [THEN iffD1])
   17.69 -apply (drule zle_imp_zless_or_eq)
   17.70 -apply (auto simp add: zless_iff_Suc_zadd)
   17.71 -done
   17.72 -
   17.73 -lemma nat_0_le [simp]: "0 \<le> z ==> int (nat z) = z"
   17.74 -by (simp add: neg_eq_less_0 zle_def not_neg_nat)
   17.75 -
   17.76 -lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
   17.77 -by (auto simp add: order_le_less neg_eq_less_0 zle_def neg_nat)
   17.78 -
   17.79 -text{*An alternative condition is @{term "0 \<le> w"} *}
   17.80 -lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
   17.81 -apply (subst zless_int [symmetric])
   17.82 -apply (simp (no_asm_simp) add: not_neg_nat not_neg_eq_ge_0 order_le_less)
   17.83 -apply (case_tac "neg w")
   17.84 - apply (simp add: neg_eq_less_0 neg_nat)
   17.85 - apply (blast intro: order_less_trans)
   17.86 -apply (simp add: not_neg_nat)
   17.87 -done
   17.88 -
   17.89 -lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
   17.90 -apply (case_tac "0 < z")
   17.91 -apply (auto simp add: nat_mono_iff linorder_not_less)
   17.92 -done
   17.93 -
   17.94 -subsection{*Monotonicity results*}
   17.95 -
   17.96 -text{*Most are available in theory @{text Ring_and_Field}, but they are
   17.97 -      not yet available: we must prove @{text zadd_zless_mono} before we
   17.98 -      can prove that the integers are a ring.*}
   17.99 -
  17.100 -lemma zadd_right_cancel_zless [simp]: "(v+z < w+z) = (v < (w::int))"
  17.101 -by (simp add: zless_def zdiff_def zadd_ac) 
  17.102 -
  17.103 -lemma zadd_left_cancel_zless [simp]: "(z+v < z+w) = (v < (w::int))"
  17.104 -by (simp add: zless_def zdiff_def zadd_ac) 
  17.105 -
  17.106 -lemma zadd_right_cancel_zle [simp] : "(v+z \<le> w+z) = (v \<le> (w::int))"
  17.107 -by (simp add: linorder_not_less [symmetric]) 
  17.108 -
  17.109 -lemma zadd_left_cancel_zle [simp] : "(z+v \<le> z+w) = (v \<le> (w::int))"
  17.110 -by (simp add: linorder_not_less [symmetric]) 
  17.111 -
  17.112 -(*"v\<le>w ==> v+z \<le> w+z"*)
  17.113 -lemmas zadd_zless_mono1 = zadd_right_cancel_zless [THEN iffD2, standard]
  17.114 -
  17.115 -(*"v\<le>w ==> z+v \<le> z+w"*)
  17.116 -lemmas zadd_zless_mono2 = zadd_left_cancel_zless [THEN iffD2, standard]
  17.117 -
  17.118 -(*"v\<le>w ==> v+z \<le> w+z"*)
  17.119 -lemmas zadd_zle_mono1 = zadd_right_cancel_zle [THEN iffD2, standard]
  17.120 -
  17.121 -(*"v\<le>w ==> z+v \<le> z+w"*)
  17.122 -lemmas zadd_zle_mono2 = zadd_left_cancel_zle [THEN iffD2, standard]
  17.123 -
  17.124 -lemma zadd_zle_mono: "[| w'\<le>w; z'\<le>z |] ==> w' + z' \<le> w + (z::int)"
  17.125 -by (erule zadd_zle_mono1 [THEN zle_trans], simp)
  17.126 -
  17.127 -lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
  17.128 -by (erule zadd_zless_mono1 [THEN order_less_le_trans], simp)
  17.129 -
  17.130 -
  17.131 -subsection{*Strict Monotonicity of Multiplication*}
  17.132 -
  17.133 -text{*strict, in 1st argument; proof is by induction on k>0*}
  17.134 -lemma zmult_zless_mono2_lemma: "i<j ==> 0<k --> int k * i < int k * j"
  17.135 -apply (induct_tac "k", simp) 
  17.136 -apply (simp add: int_Suc)
  17.137 -apply (case_tac "n=0")
  17.138 -apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
  17.139 -apply (simp_all add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
  17.140 -done
  17.141 -
  17.142 -lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
  17.143 -apply (rule_tac t = k in not_neg_nat [THEN subst])
  17.144 -apply (erule_tac [2] zmult_zless_mono2_lemma [THEN mp])
  17.145 -apply (simp add: not_neg_eq_ge_0 order_le_less)
  17.146 -apply (frule conjI [THEN zless_nat_conj [THEN iffD2]], auto)
  17.147 -done
  17.148 -
  17.149 -
  17.150 -text{*The Integers Form an Ordered Ring*}
  17.151 -instance int :: ordered_ring
  17.152 -proof
  17.153 -  fix i j k :: int
  17.154 -  show "0 < (1::int)" by (rule int_0_less_1)
  17.155 -  show "i \<le> j ==> k + i \<le> k + j" by simp
  17.156 -  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: zmult_zless_mono2)
  17.157 -  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
  17.158 -qed
  17.159 -
  17.160 -
  17.161 -subsection{*Comparison laws*}
  17.162 -
  17.163 -text{*Legacy bindings: all are in theory @{text Ring_and_Field}.*}
  17.164 -
  17.165 -lemma zminus_zless_zminus: "(- x < - y) = (y < (x::int))"
  17.166 -  by (rule Ring_and_Field.neg_less_iff_less)
  17.167 -
  17.168 -lemma zminus_zle_zminus: "(- x \<le> - y) = (y \<le> (x::int))"
  17.169 -  by (rule Ring_and_Field.neg_le_iff_le)
  17.170 -
  17.171 -
  17.172 -text{*The next several equations can make the simplifier loop!*}
  17.173 -
  17.174 -lemma zless_zminus: "(x < - y) = (y < - (x::int))"
  17.175 -  by (rule Ring_and_Field.less_minus_iff)
  17.176 -
  17.177 -lemma zminus_zless: "(- x < y) = (- y < (x::int))"
  17.178 -  by (rule Ring_and_Field.minus_less_iff)
  17.179 -
  17.180 -lemma zle_zminus: "(x \<le> - y) = (y \<le> - (x::int))"
  17.181 -  by (rule Ring_and_Field.le_minus_iff)
  17.182 -
  17.183 -lemma zminus_zle: "(- x \<le> y) = (- y \<le> (x::int))"
  17.184 -  by (rule Ring_and_Field.minus_le_iff)
  17.185 -
  17.186 -lemma equation_zminus: "(x = - y) = (y = - (x::int))"
  17.187 -  by (rule Ring_and_Field.equation_minus_iff)
  17.188 -
  17.189 -lemma zminus_equation: "(- x = y) = (- (y::int) = x)"
  17.190 -  by (rule Ring_and_Field.minus_equation_iff)
  17.191 -
  17.192 -
  17.193 -subsection{*Lemmas about the Function @{term int} and Orderings*}
  17.194 -
  17.195 -lemma negative_zless_0: "- (int (Suc n)) < 0"
  17.196 -by (simp add: zless_def)
  17.197 -
  17.198 -lemma negative_zless [iff]: "- (int (Suc n)) < int m"
  17.199 -by (rule negative_zless_0 [THEN order_less_le_trans], simp)
  17.200 -
  17.201 -lemma negative_zle_0: "- int n \<le> 0"
  17.202 -by (simp add: zminus_zle)
  17.203 -
  17.204 -lemma negative_zle [iff]: "- int n \<le> int m"
  17.205 -by (simp add: zless_def zle_def zdiff_def zadd_int)
  17.206 -
  17.207 -lemma not_zle_0_negative [simp]: "~(0 \<le> - (int (Suc n)))"
  17.208 -by (subst zle_zminus, simp)
  17.209 -
  17.210 -lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
  17.211 -apply safe 
  17.212 -apply (drule_tac [2] zle_zminus [THEN iffD1])
  17.213 -apply (auto dest: zle_trans [OF _ negative_zle_0]) 
  17.214 -done
  17.215 -
  17.216 -lemma not_int_zless_negative [simp]: "~(int n < - int m)"
  17.217 -by (simp add: zle_def [symmetric])
  17.218 -
  17.219 -lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
  17.220 -apply (rule iffI)
  17.221 -apply (rule int_zle_neg [THEN iffD1])
  17.222 -apply (drule sym)
  17.223 -apply (simp_all (no_asm_simp))
  17.224 -done
  17.225 -
  17.226 -lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
  17.227 -by (force intro: exI [of _ "0::nat"] 
  17.228 -            intro!: not_sym [THEN not0_implies_Suc]
  17.229 -            simp add: zless_iff_Suc_zadd int_le_less)
  17.230 -
  17.231 -lemma abs_int_eq [simp]: "abs (int m) = int m"
  17.232 -by (simp add: zabs_def)
  17.233 -
  17.234 -text{*This version is proved for all ordered rings, not just integers!
  17.235 -      It is proved here because attribute @{text arith_split} is not available
  17.236 -      in theory @{text Ring_and_Field}.
  17.237 -      But is it really better than just rewriting with @{text abs_if}?*}
  17.238 -lemma abs_split [arith_split]:
  17.239 -     "P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
  17.240 -by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
  17.241 -
  17.242 -
  17.243 -subsection{*Misc Results*}
  17.244 -
  17.245 -lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
  17.246 -apply (unfold nat_def)
  17.247 -apply (auto simp add: neg_eq_less_0 zero_reorient zminus_zless)
  17.248 -done
  17.249 -
  17.250 -lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
  17.251 -apply (case_tac "neg z")
  17.252 -apply (erule_tac [2] not_neg_nat [THEN subst])
  17.253 -apply (auto simp add: neg_nat)
  17.254 -apply (auto dest: order_less_trans simp add: neg_eq_less_0)
  17.255 -done
  17.256 -
  17.257 -lemma zless_eq_neg: "(w<z) = neg(w-z)"
  17.258 -by (simp add: zless_def)
  17.259 -
  17.260 -lemma eq_eq_iszero: "(w=z) = iszero(w-z)"
  17.261 -by (simp add: iszero_def diff_eq_eq)
  17.262 -
  17.263 -lemma zle_eq_not_neg: "(w\<le>z) = (~ neg(z-w))"
  17.264 -by (simp add: zle_def zless_def)
  17.265 -
  17.266 -
  17.267 -subsection{*Monotonicity of Multiplication*}
  17.268 -
  17.269 -lemma zmult_zle_mono1: "[| i \<le> j;  (0::int) \<le> k |] ==> i*k \<le> j*k"
  17.270 -  by (rule Ring_and_Field.mult_right_mono)
  17.271 -
  17.272 -lemma zmult_zle_mono1_neg: "[| i \<le> j;  k \<le> (0::int) |] ==> j*k \<le> i*k"
  17.273 -  by (rule Ring_and_Field.mult_right_mono_neg)
  17.274 -
  17.275 -lemma zmult_zle_mono2: "[| i \<le> j;  (0::int) \<le> k |] ==> k*i \<le> k*j"
  17.276 -  by (rule Ring_and_Field.mult_left_mono)
  17.277 -
  17.278 -lemma zmult_zle_mono2_neg: "[| i \<le> j;  k \<le> (0::int) |] ==> k*j \<le> k*i"
  17.279 -  by (rule Ring_and_Field.mult_left_mono_neg)
  17.280 -
  17.281 -(* \<le> monotonicity, BOTH arguments*)
  17.282 -lemma zmult_zle_mono:
  17.283 -     "[| i \<le> j;  k \<le> l;  (0::int) \<le> j;  (0::int) \<le> k |] ==> i*k \<le> j*l"
  17.284 -  by (rule Ring_and_Field.mult_mono)
  17.285 -
  17.286 -lemma zmult_zless_mono1: "[| i<j;  (0::int) < k |] ==> i*k < j*k"
  17.287 -  by (rule Ring_and_Field.mult_strict_right_mono)
  17.288 -
  17.289 -lemma zmult_zless_mono1_neg: "[| i<j;  k < (0::int) |] ==> j*k < i*k"
  17.290 -  by (rule Ring_and_Field.mult_strict_right_mono_neg)
  17.291 -
  17.292 -lemma zmult_zless_mono2_neg: "[| i<j;  k < (0::int) |] ==> k*j < k*i"
  17.293 -  by (rule Ring_and_Field.mult_strict_left_mono_neg)
  17.294 -
  17.295 -lemma zmult_eq_0_iff [iff]: "(m*n = (0::int)) = (m = 0 | n = 0)"
  17.296 -  by simp
  17.297 -
  17.298 -lemma zmult_zless_cancel2: "(m*k < n*k) = (((0::int) < k & m<n) | (k<0 & n<m))"
  17.299 -  by (rule Ring_and_Field.mult_less_cancel_right)
  17.300 -
  17.301 -lemma zmult_zless_cancel1:
  17.302 -     "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))"
  17.303 -  by (rule Ring_and_Field.mult_less_cancel_left)
  17.304 -
  17.305 -lemma zmult_zle_cancel2:
  17.306 -     "(m*k \<le> n*k) = (((0::int) < k --> m\<le>n) & (k < 0 --> n\<le>m))"
  17.307 -  by (rule Ring_and_Field.mult_le_cancel_right)
  17.308 -
  17.309 -lemma zmult_zle_cancel1:
  17.310 -     "(k*m \<le> k*n) = (((0::int) < k --> m\<le>n) & (k < 0 --> n\<le>m))"
  17.311 -  by (rule Ring_and_Field.mult_le_cancel_left)
  17.312 -
  17.313 -lemma zmult_cancel2: "(m*k = n*k) = (k = (0::int) | m=n)"
  17.314 - by (rule Ring_and_Field.mult_cancel_right)
  17.315 -
  17.316 -lemma zmult_cancel1 [simp]: "(k*m = k*n) = (k = (0::int) | m=n)"
  17.317 - by (rule Ring_and_Field.mult_cancel_left)
  17.318 -
  17.319 -
  17.320 -text{*A case theorem distinguishing non-negative and negative int*}
  17.321 -
  17.322 -lemma negD: "neg x ==> \<exists>n. x = - (int (Suc n))"
  17.323 -by (auto simp add: neg_eq_less_0 zless_iff_Suc_zadd 
  17.324 -                   diff_eq_eq [symmetric] zdiff_def)
  17.325 -
  17.326 -lemma int_cases [cases type: int, case_names nonneg neg]: 
  17.327 -     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
  17.328 -apply (case_tac "neg z")
  17.329 -apply (fast dest!: negD)
  17.330 -apply (drule not_neg_nat [symmetric], auto) 
  17.331 -done
  17.332 -
  17.333 -lemma int_induct [induct type: int, case_names nonneg neg]: 
  17.334 -     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
  17.335 -  by (cases z) auto
  17.336 -
  17.337 -
  17.338 -(*Legacy ML bindings, but no longer the structure Int.*)
  17.339 -ML
  17.340 -{*
  17.341 -val zabs_def = thm "zabs_def"
  17.342 -val nat_def  = thm "nat_def"
  17.343 -
  17.344 -val int_0 = thm "int_0";
  17.345 -val int_1 = thm "int_1";
  17.346 -val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
  17.347 -val neg_eq_less_0 = thm "neg_eq_less_0";
  17.348 -val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
  17.349 -val not_neg_0 = thm "not_neg_0";
  17.350 -val not_neg_1 = thm "not_neg_1";
  17.351 -val iszero_0 = thm "iszero_0";
  17.352 -val not_iszero_1 = thm "not_iszero_1";
  17.353 -val int_0_less_1 = thm "int_0_less_1";
  17.354 -val int_0_neq_1 = thm "int_0_neq_1";
  17.355 -val zless_eq_neg = thm "zless_eq_neg";
  17.356 -val eq_eq_iszero = thm "eq_eq_iszero";
  17.357 -val zle_eq_not_neg = thm "zle_eq_not_neg";
  17.358 -val zadd_right_cancel_zless = thm "zadd_right_cancel_zless";
  17.359 -val zadd_left_cancel_zless = thm "zadd_left_cancel_zless";
  17.360 -val zadd_right_cancel_zle = thm "zadd_right_cancel_zle";
  17.361 -val zadd_left_cancel_zle = thm "zadd_left_cancel_zle";
  17.362 -val zadd_zless_mono1 = thm "zadd_zless_mono1";
  17.363 -val zadd_zless_mono2 = thm "zadd_zless_mono2";
  17.364 -val zadd_zle_mono1 = thm "zadd_zle_mono1";
  17.365 -val zadd_zle_mono2 = thm "zadd_zle_mono2";
  17.366 -val zadd_zle_mono = thm "zadd_zle_mono";
  17.367 -val zadd_zless_mono = thm "zadd_zless_mono";
  17.368 -val zminus_zless_zminus = thm "zminus_zless_zminus";
  17.369 -val zminus_zle_zminus = thm "zminus_zle_zminus";
  17.370 -val zless_zminus = thm "zless_zminus";
  17.371 -val zminus_zless = thm "zminus_zless";
  17.372 -val zle_zminus = thm "zle_zminus";
  17.373 -val zminus_zle = thm "zminus_zle";
  17.374 -val equation_zminus = thm "equation_zminus";
  17.375 -val zminus_equation = thm "zminus_equation";
  17.376 -val negative_zless = thm "negative_zless";
  17.377 -val negative_zle = thm "negative_zle";
  17.378 -val not_zle_0_negative = thm "not_zle_0_negative";
  17.379 -val not_int_zless_negative = thm "not_int_zless_negative";
  17.380 -val negative_eq_positive = thm "negative_eq_positive";
  17.381 -val zle_iff_zadd = thm "zle_iff_zadd";
  17.382 -val abs_int_eq = thm "abs_int_eq";
  17.383 -val abs_split = thm"abs_split";
  17.384 -val nat_int = thm "nat_int";
  17.385 -val nat_zminus_int = thm "nat_zminus_int";
  17.386 -val nat_zero = thm "nat_zero";
  17.387 -val not_neg_nat = thm "not_neg_nat";
  17.388 -val neg_nat = thm "neg_nat";
  17.389 -val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
  17.390 -val nat_0_le = thm "nat_0_le";
  17.391 -val nat_le_0 = thm "nat_le_0";
  17.392 -val zless_nat_conj = thm "zless_nat_conj";
  17.393 -val int_cases = thm "int_cases";
  17.394 -val zmult_zle_mono1 = thm "zmult_zle_mono1";
  17.395 -val zmult_zle_mono1_neg = thm "zmult_zle_mono1_neg";
  17.396 -val zmult_zle_mono2 = thm "zmult_zle_mono2";
  17.397 -val zmult_zle_mono2_neg = thm "zmult_zle_mono2_neg";
  17.398 -val zmult_zle_mono = thm "zmult_zle_mono";
  17.399 -val zmult_zless_mono2 = thm "zmult_zless_mono2";
  17.400 -val zmult_zless_mono1 = thm "zmult_zless_mono1";
  17.401 -val zmult_zless_mono1_neg = thm "zmult_zless_mono1_neg";
  17.402 -val zmult_zless_mono2_neg = thm "zmult_zless_mono2_neg";
  17.403 -val zmult_eq_0_iff = thm "zmult_eq_0_iff";
  17.404 -val zmult_zless_cancel2 = thm "zmult_zless_cancel2";
  17.405 -val zmult_zless_cancel1 = thm "zmult_zless_cancel1";
  17.406 -val zmult_zle_cancel2 = thm "zmult_zle_cancel2";
  17.407 -val zmult_zle_cancel1 = thm "zmult_zle_cancel1";
  17.408 -val zmult_cancel2 = thm "zmult_cancel2";
  17.409 -val zmult_cancel1 = thm "zmult_cancel1";
  17.410 -*}
  17.411 -
  17.412 -end
    18.1 --- a/src/HOL/Integ/IntArith.thy	Thu Feb 05 10:45:28 2004 +0100
    18.2 +++ b/src/HOL/Integ/IntArith.thy	Tue Feb 10 12:02:11 2004 +0100
    18.3 @@ -12,15 +12,11 @@
    18.4  subsection{*Inequality Reasoning for the Arithmetic Simproc*}
    18.5  
    18.6  lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
    18.7 -  proof (auto simp add: zle_def zless_iff_Suc_zadd) 
    18.8 -  fix m n
    18.9 -  assume "w + 1 = w + (1 + int m) + (1 + int n)"
   18.10 -  hence "(w + 1) + (1 + int (m + n)) = (w + 1) + 0" 
   18.11 -    by (simp add: add_ac zadd_int [symmetric])
   18.12 -  hence "int (Suc(m+n)) = 0" 
   18.13 -    by (simp only: Ring_and_Field.add_left_cancel int_Suc)
   18.14 -  thus False by (simp only: int_eq_0_conv)
   18.15 -  qed
   18.16 +apply (rule eq_Abs_Integ [of z])
   18.17 +apply (rule eq_Abs_Integ [of w])
   18.18 +apply (simp add: linorder_not_le [symmetric] zle int_def zadd One_int_def)
   18.19 +done
   18.20 +
   18.21  
   18.22  use "int_arith1.ML"
   18.23  setup int_arith_setup
   18.24 @@ -86,11 +82,11 @@
   18.25  by (subst nat_eq_iff, simp)
   18.26  
   18.27  lemma nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
   18.28 -apply (case_tac "neg z")
   18.29 +apply (case_tac "z < 0")
   18.30  apply (auto simp add: nat_less_iff)
   18.31 -apply (auto intro: zless_trans simp add: neg_eq_less_0 zle_def)
   18.32  done
   18.33  
   18.34 +
   18.35  lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
   18.36  by (auto simp add: linorder_not_less [symmetric] zless_nat_conj)
   18.37  
   18.38 @@ -229,12 +225,12 @@
   18.39  lemma zmult_eq_self_iff: "(m = m*(n::int)) = (n = 1 | m = 0)"
   18.40  apply auto
   18.41  apply (subgoal_tac "m*1 = m*n")
   18.42 -apply (drule zmult_cancel1 [THEN iffD1], auto)
   18.43 +apply (drule mult_cancel_left [THEN iffD1], auto)
   18.44  done
   18.45  
   18.46  lemma zless_1_zmult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::int)"
   18.47  apply (rule_tac y = "1*n" in order_less_trans)
   18.48 -apply (rule_tac [2] zmult_zless_mono1)
   18.49 +apply (rule_tac [2] mult_strict_right_mono)
   18.50  apply (simp_all (no_asm_simp))
   18.51  done
   18.52  
    19.1 --- a/src/HOL/Integ/IntDef.thy	Thu Feb 05 10:45:28 2004 +0100
    19.2 +++ b/src/HOL/Integ/IntDef.thy	Tue Feb 10 12:02:11 2004 +0100
    19.3 @@ -27,13 +27,6 @@
    19.4  
    19.5    int :: "nat => int"
    19.6    "int m == Abs_Integ(intrel `` {(m,0)})"
    19.7 -
    19.8 -  neg   :: "int => bool"
    19.9 -  "neg(Z) == \<exists>x y. x<y & (x,y::nat):Rep_Integ(Z)"
   19.10 -
   19.11 -  (*For simplifying equalities*)
   19.12 -  iszero :: "int => bool"
   19.13 -  "iszero z == z = (0::int)"
   19.14    
   19.15  defs (overloaded)
   19.16    
   19.17 @@ -48,16 +41,17 @@
   19.18  		 intrel``{(x1+x2, y1+y2)})"
   19.19  
   19.20    zdiff_def:  "z - (w::int) == z + (-w)"
   19.21 -
   19.22 -  zless_def:  "z<w == neg(z - w)"
   19.23 -
   19.24 -  zle_def:    "z <= (w::int) == ~(w < z)"
   19.25 -
   19.26    zmult_def:
   19.27     "z * w == 
   19.28         Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).   
   19.29  		 intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
   19.30  
   19.31 +  zless_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
   19.32 +
   19.33 +  zle_def:
   19.34 +  "z \<le> (w::int) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
   19.35 +                            (x1,y1) \<in> Rep_Integ z & (x2,y2) \<in> Rep_Integ w"
   19.36 +
   19.37  lemma intrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in>  intrel) = (x1+y2 = x2+y1)"
   19.38  by (unfold intrel_def, blast)
   19.39  
   19.40 @@ -121,8 +115,8 @@
   19.41  done
   19.42  
   19.43  lemma zminus_zminus [simp]: "- (- z) = (z::int)"
   19.44 -apply (rule_tac z = z in eq_Abs_Integ)
   19.45 -apply (simp (no_asm_simp) add: zminus)
   19.46 +apply (rule eq_Abs_Integ [of z])
   19.47 +apply (simp add: zminus)
   19.48  done
   19.49  
   19.50  lemma inj_zminus: "inj(%z::int. -z)"
   19.51 @@ -134,16 +128,6 @@
   19.52  by (simp add: int_def Zero_int_def zminus)
   19.53  
   19.54  
   19.55 -subsection{*neg: the test for negative integers*}
   19.56 -
   19.57 -
   19.58 -lemma not_neg_int [simp]: "~ neg(int n)"
   19.59 -by (simp add: neg_def int_def)
   19.60 -
   19.61 -lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
   19.62 -by (simp add: neg_def int_def zminus)
   19.63 -
   19.64 -
   19.65  subsection{*zadd: addition on Integ*}
   19.66  
   19.67  lemma zadd: 
   19.68 @@ -155,22 +139,22 @@
   19.69  done
   19.70  
   19.71  lemma zminus_zadd_distrib [simp]: "- (z + w) = (- z) + (- w::int)"
   19.72 -apply (rule_tac z = z in eq_Abs_Integ)
   19.73 -apply (rule_tac z = w in eq_Abs_Integ)
   19.74 -apply (simp (no_asm_simp) add: zminus zadd)
   19.75 +apply (rule eq_Abs_Integ [of z])
   19.76 +apply (rule eq_Abs_Integ [of w])
   19.77 +apply (simp add: zminus zadd)
   19.78  done
   19.79  
   19.80  lemma zadd_commute: "(z::int) + w = w + z"
   19.81 -apply (rule_tac z = z in eq_Abs_Integ)
   19.82 -apply (rule_tac z = w in eq_Abs_Integ)
   19.83 -apply (simp (no_asm_simp) add: add_ac zadd)
   19.84 +apply (rule eq_Abs_Integ [of z])
   19.85 +apply (rule eq_Abs_Integ [of w])
   19.86 +apply (simp add: add_ac zadd)
   19.87  done
   19.88  
   19.89  lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
   19.90 -apply (rule_tac z = z1 in eq_Abs_Integ)
   19.91 -apply (rule_tac z = z2 in eq_Abs_Integ)
   19.92 -apply (rule_tac z = z3 in eq_Abs_Integ)
   19.93 -apply (simp (no_asm_simp) add: zadd add_assoc)
   19.94 +apply (rule eq_Abs_Integ [of z1])
   19.95 +apply (rule eq_Abs_Integ [of z2])
   19.96 +apply (rule eq_Abs_Integ [of z3])
   19.97 +apply (simp add: zadd add_assoc)
   19.98  done
   19.99  
  19.100  (*For AC rewriting*)
  19.101 @@ -197,8 +181,8 @@
  19.102  (*also for the instance declaration int :: plus_ac0*)
  19.103  lemma zadd_0 [simp]: "(0::int) + z = z"
  19.104  apply (unfold Zero_int_def int_def)
  19.105 -apply (rule_tac z = z in eq_Abs_Integ)
  19.106 -apply (simp (no_asm_simp) add: zadd)
  19.107 +apply (rule eq_Abs_Integ [of z])
  19.108 +apply (simp add: zadd)
  19.109  done
  19.110  
  19.111  lemma zadd_0_right [simp]: "z + (0::int) = z"
  19.112 @@ -206,8 +190,8 @@
  19.113  
  19.114  lemma zadd_zminus_inverse [simp]: "z + (- z) = (0::int)"
  19.115  apply (unfold int_def Zero_int_def)
  19.116 -apply (rule_tac z = z in eq_Abs_Integ)
  19.117 -apply (simp (no_asm_simp) add: zminus zadd add_commute)
  19.118 +apply (rule eq_Abs_Integ [of z])
  19.119 +apply (simp add: zminus zadd add_commute)
  19.120  done
  19.121  
  19.122  lemma zadd_zminus_inverse2 [simp]: "(- z) + z = (0::int)"
  19.123 @@ -236,57 +220,52 @@
  19.124  lemma zadd_assoc_cong: "(z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
  19.125  by (simp add: zadd_assoc [symmetric])
  19.126  
  19.127 -lemma zadd_assoc_swap: "(z::int) + (v + w) = v + (z + w)"
  19.128 -by (rule zadd_commute [THEN zadd_assoc_cong])
  19.129 -
  19.130  
  19.131  subsection{*zmult: multiplication on Integ*}
  19.132  
  19.133 -(*Congruence property for multiplication*)
  19.134 +text{*Congruence property for multiplication*}
  19.135  lemma zmult_congruent2: "congruent2 intrel  
  19.136          (%p1 p2. (%(x1,y1). (%(x2,y2).    
  19.137                      intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
  19.138  apply (rule equiv_intrel [THEN congruent2_commuteI])
  19.139 -apply (rule_tac [2] p=w in PairE)  
  19.140 -apply (force simp add: add_ac mult_ac, clarify) 
  19.141 -apply (simp (no_asm_simp) del: equiv_intrel_iff add: add_ac mult_ac)
  19.142 + apply (force simp add: add_ac mult_ac) 
  19.143 +apply (clarify, simp del: equiv_intrel_iff add: add_ac mult_ac)
  19.144  apply (rename_tac x1 x2 y1 y2 z1 z2)
  19.145  apply (rule equiv_class_eq [OF equiv_intrel intrel_iff [THEN iffD2]])
  19.146 -apply (simp add: intrel_def)
  19.147 -apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2", arith)
  19.148 +apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2")
  19.149 +apply (simp add: mult_ac, arith) 
  19.150  apply (simp add: add_mult_distrib [symmetric])
  19.151  done
  19.152  
  19.153  lemma zmult: 
  19.154     "Abs_Integ((intrel``{(x1,y1)})) * Abs_Integ((intrel``{(x2,y2)})) =    
  19.155      Abs_Integ(intrel `` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
  19.156 -apply (unfold zmult_def)
  19.157 -apply (simp (no_asm_simp) add: UN_UN_split_split_eq zmult_congruent2 equiv_intrel [THEN UN_equiv_class2])
  19.158 -done
  19.159 +by (simp add: zmult_def UN_UN_split_split_eq zmult_congruent2 
  19.160 +              equiv_intrel [THEN UN_equiv_class2])
  19.161  
  19.162  lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
  19.163 -apply (rule_tac z = z in eq_Abs_Integ)
  19.164 -apply (rule_tac z = w in eq_Abs_Integ)
  19.165 -apply (simp (no_asm_simp) add: zminus zmult add_ac)
  19.166 +apply (rule eq_Abs_Integ [of z])
  19.167 +apply (rule eq_Abs_Integ [of w])
  19.168 +apply (simp add: zminus zmult add_ac)
  19.169  done
  19.170  
  19.171  lemma zmult_commute: "(z::int) * w = w * z"
  19.172 -apply (rule_tac z = z in eq_Abs_Integ)
  19.173 -apply (rule_tac z = w in eq_Abs_Integ)
  19.174 -apply (simp (no_asm_simp) add: zmult add_ac mult_ac)
  19.175 +apply (rule eq_Abs_Integ [of z])
  19.176 +apply (rule eq_Abs_Integ [of w])
  19.177 +apply (simp add: zmult add_ac mult_ac)
  19.178  done
  19.179  
  19.180  lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
  19.181 -apply (rule_tac z = z1 in eq_Abs_Integ)
  19.182 -apply (rule_tac z = z2 in eq_Abs_Integ)
  19.183 -apply (rule_tac z = z3 in eq_Abs_Integ)
  19.184 -apply (simp (no_asm_simp) add: add_mult_distrib2 zmult add_ac mult_ac)
  19.185 +apply (rule eq_Abs_Integ [of z1])
  19.186 +apply (rule eq_Abs_Integ [of z2])
  19.187 +apply (rule eq_Abs_Integ [of z3])
  19.188 +apply (simp add: add_mult_distrib2 zmult add_ac mult_ac)
  19.189  done
  19.190  
  19.191  lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
  19.192 -apply (rule_tac z = z1 in eq_Abs_Integ)
  19.193 -apply (rule_tac z = z2 in eq_Abs_Integ)
  19.194 -apply (rule_tac z = w in eq_Abs_Integ)
  19.195 +apply (rule eq_Abs_Integ [of z1])
  19.196 +apply (rule eq_Abs_Integ [of z2])
  19.197 +apply (rule eq_Abs_Integ [of w])
  19.198  apply (simp add: add_mult_distrib2 zadd zmult add_ac mult_ac)
  19.199  done
  19.200  
  19.201 @@ -314,14 +293,14 @@
  19.202  
  19.203  lemma zmult_0 [simp]: "0 * z = (0::int)"
  19.204  apply (unfold Zero_int_def int_def)
  19.205 -apply (rule_tac z = z in eq_Abs_Integ)
  19.206 -apply (simp (no_asm_simp) add: zmult)
  19.207 +apply (rule eq_Abs_Integ [of z])
  19.208 +apply (simp add: zmult)
  19.209  done
  19.210  
  19.211  lemma zmult_1 [simp]: "(1::int) * z = z"
  19.212  apply (unfold One_int_def int_def)
  19.213 -apply (rule_tac z = z in eq_Abs_Integ)
  19.214 -apply (simp (no_asm_simp) add: zmult)
  19.215 +apply (rule eq_Abs_Integ [of z])
  19.216 +apply (simp add: zmult)
  19.217  done
  19.218  
  19.219  lemma zmult_0_right [simp]: "z * 0 = (0::int)"
  19.220 @@ -352,64 +331,73 @@
  19.221  qed
  19.222  
  19.223  
  19.224 -subsection{*Theorems about the Ordering*}
  19.225 +subsection{*The @{text "\<le>"} Ordering*}
  19.226 +
  19.227 +lemma zle: 
  19.228 +  "(Abs_Integ(intrel``{(x1,y1)}) \<le> Abs_Integ(intrel``{(x2,y2)})) =  
  19.229 +   (x1 + y2 \<le> x2 + y1)"
  19.230 +by (force simp add: zle_def)
  19.231 +
  19.232 +lemma zle_refl: "w \<le> (w::int)"
  19.233 +apply (rule eq_Abs_Integ [of w])
  19.234 +apply (force simp add: zle)
  19.235 +done
  19.236 +
  19.237 +lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
  19.238 +apply (rule eq_Abs_Integ [of i]) 
  19.239 +apply (rule eq_Abs_Integ [of j]) 
  19.240 +apply (rule eq_Abs_Integ [of k]) 
  19.241 +apply (simp add: zle) 
  19.242 +done
  19.243 +
  19.244 +lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
  19.245 +apply (rule eq_Abs_Integ [of w]) 
  19.246 +apply (rule eq_Abs_Integ [of z]) 
  19.247 +apply (simp add: zle) 
  19.248 +done
  19.249 +
  19.250 +(* Axiom 'order_less_le' of class 'order': *)
  19.251 +lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
  19.252 +by (simp add: zless_def)
  19.253 +
  19.254 +instance int :: order
  19.255 +proof qed
  19.256 + (assumption |
  19.257 +  rule zle_refl zle_trans zle_anti_sym zless_le)+
  19.258 +
  19.259 +(* Axiom 'linorder_linear' of class 'linorder': *)
  19.260 +lemma zle_linear: "(z::int) \<le> w | w \<le> z"
  19.261 +apply (rule eq_Abs_Integ [of z])
  19.262 +apply (rule eq_Abs_Integ [of w])
  19.263 +apply (simp add: zle linorder_linear) 
  19.264 +done
  19.265 +
  19.266 +instance int :: plus_ac0
  19.267 +proof qed (rule zadd_commute zadd_assoc zadd_0)+
  19.268 +
  19.269 +instance int :: linorder
  19.270 +proof qed (rule zle_linear)
  19.271 +
  19.272 +
  19.273 +lemmas zless_linear = linorder_less_linear [where 'a = int]
  19.274 +
  19.275 +
  19.276 +lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
  19.277 +by (simp add: Zero_int_def)
  19.278  
  19.279  (*This lemma allows direct proofs of other <-properties*)
  19.280  lemma zless_iff_Suc_zadd: 
  19.281      "(w < z) = (\<exists>n. z = w + int(Suc n))"
  19.282 -apply (unfold zless_def neg_def zdiff_def int_def)
  19.283 -apply (rule_tac z = z in eq_Abs_Integ)
  19.284 -apply (rule_tac z = w in eq_Abs_Integ, clarify)
  19.285 -apply (simp add: zadd zminus)
  19.286 +apply (rule eq_Abs_Integ [of z])
  19.287 +apply (rule eq_Abs_Integ [of w])
  19.288 +apply (simp add: linorder_not_le [where 'a = int, symmetric] 
  19.289 +                 linorder_not_le [where 'a = nat] 
  19.290 +                 zle int_def zdiff_def zadd zminus) 
  19.291  apply (safe dest!: less_imp_Suc_add)
  19.292  apply (rule_tac x = k in exI)
  19.293  apply (simp_all add: add_ac)
  19.294  done
  19.295  
  19.296 -lemma zless_zadd_Suc: "z < z + int (Suc n)"
  19.297 -by (auto simp add: zless_iff_Suc_zadd zadd_assoc zadd_int)
  19.298 -
  19.299 -lemma zless_trans: "[| z1<z2; z2<z3 |] ==> z1 < (z3::int)"
  19.300 -by (auto simp add: zless_iff_Suc_zadd zadd_assoc zadd_int)
  19.301 -
  19.302 -lemma zless_not_sym: "!!w::int. z<w ==> ~w<z"
  19.303 -apply (safe dest!: zless_iff_Suc_zadd [THEN iffD1])
  19.304 -apply (rule_tac z = z in eq_Abs_Integ, safe)
  19.305 -apply (simp add: int_def zadd)
  19.306 -done
  19.307 -
  19.308 -(* [| n<m;  ~P ==> m<n |] ==> P *)
  19.309 -lemmas zless_asym = zless_not_sym [THEN swap, standard]
  19.310 -
  19.311 -lemma zless_not_refl: "!!z::int. ~ z<z"
  19.312 -apply (rule zless_asym [THEN notI])
  19.313 -apply (assumption+)
  19.314 -done
  19.315 -
  19.316 -(* z<z ==> R *)
  19.317 -lemmas zless_irrefl = zless_not_refl [THEN notE, standard, elim!]
  19.318 -
  19.319 -
  19.320 -(*"Less than" is a linear ordering*)
  19.321 -lemma zless_linear: 
  19.322 -    "z<w | z=w | w<(z::int)"
  19.323 -apply (unfold zless_def neg_def zdiff_def)
  19.324 -apply (rule_tac z = z in eq_Abs_Integ)
  19.325 -apply (rule_tac z = w in eq_Abs_Integ, safe)
  19.326 -apply (simp add: zadd zminus Image_iff Bex_def)
  19.327 -apply (rule_tac m1 = "x+ya" and n1 = "xa+y" in less_linear [THEN disjE])
  19.328 -apply (force simp add: add_ac)+
  19.329 -done
  19.330 -
  19.331 -lemma int_neq_iff: "!!w::int. (w ~= z) = (w<z | z<w)"
  19.332 -by (cut_tac zless_linear, blast)
  19.333 -
  19.334 -(*** eliminates ~= in premises ***)
  19.335 -lemmas int_neqE = int_neq_iff [THEN iffD1, THEN disjE, standard]
  19.336 -
  19.337 -lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
  19.338 -by (simp add: Zero_int_def)
  19.339 -
  19.340  lemma zless_int [simp]: "(int m < int n) = (m<n)"
  19.341  by (simp add: less_iff_Suc_add zless_iff_Suc_zadd zadd_int)
  19.342  
  19.343 @@ -425,84 +413,553 @@
  19.344  lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
  19.345  by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
  19.346  
  19.347 -
  19.348 -subsection{*Properties of the @{text "\<le>"} Relation*}
  19.349 +lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
  19.350 +by (simp add: linorder_not_less [symmetric])
  19.351  
  19.352 -lemma zle_int [simp]: "(int m <= int n) = (m<=n)"
  19.353 -by (simp add: zle_def le_def)
  19.354 +lemma zero_zle_int [simp]: "(0 \<le> int n)"
  19.355 +by (simp add: Zero_int_def)
  19.356  
  19.357 -lemma zero_zle_int [simp]: "(0 <= int n)"
  19.358 +lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
  19.359 +by (simp add: Zero_int_def)
  19.360 +
  19.361 +lemma int_0 [simp]: "int 0 = (0::int)"
  19.362  by (simp add: Zero_int_def)
  19.363  
  19.364 -lemma int_le_0_conv [simp]: "(int n <= 0) = (n = 0)"
  19.365 -by (simp add: Zero_int_def)
  19.366 +lemma int_1 [simp]: "int 1 = 1"
  19.367 +by (simp add: One_int_def)
  19.368 +
  19.369 +lemma int_Suc0_eq_1: "int (Suc 0) = 1"
  19.370 +by (simp add: One_int_def One_nat_def)
  19.371 +
  19.372 +subsection{*Monotonicity results*}
  19.373 +
  19.374 +lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)" 
  19.375 +apply (rule eq_Abs_Integ [of i]) 
  19.376 +apply (rule eq_Abs_Integ [of j]) 
  19.377 +apply (rule eq_Abs_Integ [of k]) 
  19.378 +apply (simp add: zle zadd) 
  19.379 +done
  19.380 +
  19.381 +lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)" 
  19.382 +apply (rule eq_Abs_Integ [of i]) 
  19.383 +apply (rule eq_Abs_Integ [of j]) 
  19.384 +apply (rule eq_Abs_Integ [of k]) 
  19.385 +apply (simp add: linorder_not_le [where 'a = int, symmetric] 
  19.386 +                 linorder_not_le [where 'a = nat]  zle zadd)
  19.387 +done
  19.388 +
  19.389 +lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
  19.390 +by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono]) 
  19.391 +
  19.392 +
  19.393 +subsection{*Strict Monotonicity of Multiplication*}
  19.394 +
  19.395 +text{*strict, in 1st argument; proof is by induction on k>0*}
  19.396 +lemma zmult_zless_mono2_lemma [rule_format]:
  19.397 +     "i<j ==> 0<k --> int k * i < int k * j"
  19.398 +apply (induct_tac "k", simp) 
  19.399 +apply (simp add: int_Suc)
  19.400 +apply (case_tac "n=0")
  19.401 +apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
  19.402 +apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
  19.403 +done
  19.404  
  19.405 -lemma zle_imp_zless_or_eq: "z <= w ==> z < w | z=(w::int)"
  19.406 -apply (unfold zle_def)
  19.407 -apply (cut_tac zless_linear)
  19.408 -apply (blast elim: zless_asym)
  19.409 +lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
  19.410 +apply (rule eq_Abs_Integ [of k]) 
  19.411 +apply (auto simp add: zle zadd int_def Zero_int_def)
  19.412 +apply (rule_tac x="x-y" in exI, simp) 
  19.413 +done
  19.414 +
  19.415 +lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
  19.416 +apply (frule order_less_imp_le [THEN zero_le_imp_eq_int]) 
  19.417 +apply (auto simp add: zmult_zless_mono2_lemma) 
  19.418 +done
  19.419 +
  19.420 +
  19.421 +defs (overloaded)
  19.422 +    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
  19.423 +
  19.424 +
  19.425 +text{*The Integers Form an Ordered Ring*}
  19.426 +instance int :: ordered_ring
  19.427 +proof
  19.428 +  fix i j k :: int
  19.429 +  show "0 < (1::int)" by (rule int_0_less_1)
  19.430 +  show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
  19.431 +  show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
  19.432 +  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
  19.433 +qed
  19.434 +
  19.435 +
  19.436 +subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
  19.437 +
  19.438 +constdefs
  19.439 +   nat  :: "int => nat"
  19.440 +    "nat(Z) == if Z<0 then 0 else (THE m. Z = int m)"
  19.441 +
  19.442 +lemma nat_int [simp]: "nat(int n) = n"
  19.443 +by (unfold nat_def, auto)
  19.444 +
  19.445 +lemma nat_zero [simp]: "nat 0 = 0"
  19.446 +apply (unfold Zero_int_def)
  19.447 +apply (rule nat_int)
  19.448 +done
  19.449 +
  19.450 +lemma nat_0_le [simp]: "0 \<le> z ==> int (nat z) = z"
  19.451 +apply (rule eq_Abs_Integ [of z]) 
  19.452 +apply (simp add: nat_def linorder_not_le [symmetric] zle int_def Zero_int_def)
  19.453 +apply (subgoal_tac "(THE m. x = m + y) = x-y")
  19.454 +apply (auto simp add: the_equality) 
  19.455  done
  19.456  
  19.457 -lemma zless_or_eq_imp_zle: "z<w | z=w ==> z <= (w::int)"
  19.458 -apply (unfold zle_def)
  19.459 -apply (cut_tac zless_linear)
  19.460 -apply (blast elim: zless_asym)
  19.461 +lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
  19.462 +by (simp add: nat_def  order_less_le eq_commute [of 0])
  19.463 +
  19.464 +text{*An alternative condition is @{term "0 \<le> w"} *}
  19.465 +lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
  19.466 +apply (subst zless_int [symmetric])
  19.467 +apply (simp add: order_le_less)
  19.468 +apply (case_tac "w < 0")
  19.469 + apply (simp add: order_less_imp_le)
  19.470 + apply (blast intro: order_less_trans)
  19.471 +apply (simp add: linorder_not_less)
  19.472 +done
  19.473 +
  19.474 +lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
  19.475 +apply (case_tac "0 < z")
  19.476 +apply (auto simp add: nat_mono_iff linorder_not_less)
  19.477 +done
  19.478 +
  19.479 +
  19.480 +subsection{*Lemmas about the Function @{term int} and Orderings*}
  19.481 +
  19.482 +lemma negative_zless_0: "- (int (Suc n)) < 0"
  19.483 +by (simp add: zless_def)
  19.484 +
  19.485 +lemma negative_zless [iff]: "- (int (Suc n)) < int m"
  19.486 +by (rule negative_zless_0 [THEN order_less_le_trans], simp)
  19.487 +
  19.488 +lemma negative_zle_0: "- int n \<le> 0"
  19.489 +by (simp add: minus_le_iff)
  19.490 +
  19.491 +lemma negative_zle [iff]: "- int n \<le> int m"
  19.492 +by (rule order_trans [OF negative_zle_0 zero_zle_int])
  19.493 +
  19.494 +lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
  19.495 +by (subst le_minus_iff, simp)
  19.496 +
  19.497 +lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
  19.498 +apply safe 
  19.499 +apply (drule_tac [2] le_minus_iff [THEN iffD1])
  19.500 +apply (auto dest: zle_trans [OF _ negative_zle_0]) 
  19.501  done
  19.502  
  19.503 -lemma int_le_less: "(x <= (y::int)) = (x < y | x=y)"
  19.504 -apply (rule iffI) 
  19.505 -apply (erule zle_imp_zless_or_eq) 
  19.506 -apply (erule zless_or_eq_imp_zle) 
  19.507 +lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
  19.508 +by (simp add: linorder_not_less)
  19.509 +
  19.510 +lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
  19.511 +by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
  19.512 +
  19.513 +lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
  19.514 +by (force intro: exI [of _ "0::nat"] 
  19.515 +            intro!: not_sym [THEN not0_implies_Suc]
  19.516 +            simp add: zless_iff_Suc_zadd order_le_less)
  19.517 +
  19.518 +
  19.519 +text{*This version is proved for all ordered rings, not just integers!
  19.520 +      It is proved here because attribute @{text arith_split} is not available
  19.521 +      in theory @{text Ring_and_Field}.
  19.522 +      But is it really better than just rewriting with @{text abs_if}?*}
  19.523 +lemma abs_split [arith_split]:
  19.524 +     "P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
  19.525 +by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
  19.526 +
  19.527 +lemma abs_int_eq [simp]: "abs (int m) = int m"
  19.528 +by (simp add: zabs_def)
  19.529 +
  19.530 +
  19.531 +subsection{*Misc Results*}
  19.532 +
  19.533 +lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
  19.534 +by (auto simp add: nat_def zero_reorient minus_less_iff)
  19.535 +
  19.536 +lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
  19.537 +apply (case_tac "0 \<le> z")
  19.538 +apply (erule nat_0_le [THEN subst], simp) 
  19.539 +apply (simp add: linorder_not_le)
  19.540 +apply (auto dest: order_less_trans simp add: order_less_imp_le)
  19.541  done
  19.542  
  19.543 -lemma zle_refl: "w <= (w::int)"
  19.544 -by (simp add: int_le_less)
  19.545 +
  19.546 +
  19.547 +subsection{*Monotonicity of Multiplication*}
  19.548 +
  19.549 +lemma zmult_zle_mono2: "[| i \<le> j;  (0::int) \<le> k |] ==> k*i \<le> k*j"
  19.550 +  by (rule Ring_and_Field.mult_left_mono)
  19.551 +
  19.552 +lemma zmult_zless_cancel2: "(m*k < n*k) = (((0::int) < k & m<n) | (k<0 & n<m))"
  19.553 +  by (rule Ring_and_Field.mult_less_cancel_right)
  19.554 +
  19.555 +lemma zmult_zless_cancel1:
  19.556 +     "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))"
  19.557 +  by (rule Ring_and_Field.mult_less_cancel_left)
  19.558  
  19.559 -(* Axiom 'linorder_linear' of class 'linorder': *)
  19.560 -lemma zle_linear: "(z::int) <= w | w <= z"
  19.561 -apply (simp add: int_le_less)
  19.562 -apply (cut_tac zless_linear, blast)
  19.563 +lemma zmult_zle_cancel1:
  19.564 +     "(k*m \<le> k*n) = (((0::int) < k --> m\<le>n) & (k < 0 --> n\<le>m))"
  19.565 +  by (rule Ring_and_Field.mult_le_cancel_left)
  19.566 +
  19.567 +
  19.568 +
  19.569 +text{*A case theorem distinguishing non-negative and negative int*}
  19.570 +
  19.571 +lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
  19.572 +by (auto simp add: zless_iff_Suc_zadd 
  19.573 +                   diff_eq_eq [symmetric] zdiff_def)
  19.574 +
  19.575 +lemma int_cases [cases type: int, case_names nonneg neg]: 
  19.576 +     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
  19.577 +apply (case_tac "z < 0", blast dest!: negD)
  19.578 +apply (simp add: linorder_not_less)
  19.579 +apply (blast dest: nat_0_le [THEN sym])
  19.580  done
  19.581  
  19.582 -(* Axiom 'order_trans of class 'order': *)
  19.583 -lemma zle_trans: "[| i <= j; j <= k |] ==> i <= (k::int)"
  19.584 -apply (drule zle_imp_zless_or_eq) 
  19.585 -apply (drule zle_imp_zless_or_eq) 
  19.586 -apply (rule zless_or_eq_imp_zle) 
  19.587 -apply (blast intro: zless_trans) 
  19.588 +lemma int_induct [induct type: int, case_names nonneg neg]: 
  19.589 +     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
  19.590 +  by (cases z) auto
  19.591 +
  19.592 +
  19.593 +subsection{*The Constants @{term neg} and @{term iszero}*}
  19.594 +
  19.595 +constdefs
  19.596 +
  19.597 +  neg   :: "'a::ordered_ring => bool"
  19.598 +  "neg(Z) == Z < 0"
  19.599 +
  19.600 +  (*For simplifying equalities*)
  19.601 +  iszero :: "'a::semiring => bool"
  19.602 +  "iszero z == z = (0)"
  19.603 +  
  19.604 +
  19.605 +lemma not_neg_int [simp]: "~ neg(int n)"
  19.606 +by (simp add: neg_def)
  19.607 +
  19.608 +lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
  19.609 +by (simp add: neg_def neg_less_0_iff_less)
  19.610 +
  19.611 +lemmas neg_eq_less_0 = neg_def
  19.612 +
  19.613 +lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
  19.614 +by (simp add: neg_def linorder_not_less)
  19.615 +
  19.616 +subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
  19.617 +
  19.618 +lemma not_neg_0: "~ neg 0"
  19.619 +by (simp add: One_int_def neg_def)
  19.620 +
  19.621 +lemma not_neg_1: "~ neg 1"
  19.622 +by (simp add: neg_def linorder_not_less zero_le_one) 
  19.623 +
  19.624 +lemma iszero_0: "iszero 0"
  19.625 +by (simp add: iszero_def)
  19.626 +
  19.627 +lemma not_iszero_1: "~ iszero 1"
  19.628 +by (simp add: iszero_def eq_commute) 
  19.629 +
  19.630 +lemma neg_nat: "neg z ==> nat z = 0"
  19.631 +by (simp add: nat_def neg_def) 
  19.632 +
  19.633 +lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
  19.634 +by (simp add: linorder_not_less neg_def)
  19.635 +
  19.636 +
  19.637 +subsection{*Embedding of the Naturals into any Semiring: @{term of_nat}*}
  19.638 +
  19.639 +consts of_nat :: "nat => 'a::semiring"
  19.640 +
  19.641 +primrec
  19.642 +  of_nat_0:   "of_nat 0 = 0"
  19.643 +  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
  19.644 +
  19.645 +lemma of_nat_1 [simp]: "of_nat 1 = 1"
  19.646 +by simp
  19.647 +
  19.648 +lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
  19.649 +apply (induct m)
  19.650 +apply (simp_all add: add_ac) 
  19.651 +done
  19.652 +
  19.653 +lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
  19.654 +apply (induct m) 
  19.655 +apply (simp_all add: mult_ac add_ac right_distrib) 
  19.656 +done
  19.657 +
  19.658 +lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semiring)"
  19.659 +apply (induct m, simp_all) 
  19.660 +apply (erule order_trans) 
  19.661 +apply (rule less_add_one [THEN order_less_imp_le]) 
  19.662  done
  19.663  
  19.664 -lemma zle_anti_sym: "[| z <= w; w <= z |] ==> z = (w::int)"
  19.665 -apply (drule zle_imp_zless_or_eq) 
  19.666 -apply (drule zle_imp_zless_or_eq) 
  19.667 -apply (blast elim: zless_asym) 
  19.668 +lemma less_imp_of_nat_less:
  19.669 +     "m < n ==> of_nat m < (of_nat n::'a::ordered_semiring)"
  19.670 +apply (induct m n rule: diff_induct, simp_all) 
  19.671 +apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force) 
  19.672 +done
  19.673 +
  19.674 +lemma of_nat_less_imp_less:
  19.675 +     "of_nat m < (of_nat n::'a::ordered_semiring) ==> m < n"
  19.676 +apply (induct m n rule: diff_induct, simp_all) 
  19.677 +apply (insert zero_le_imp_of_nat) 
  19.678 +apply (force simp add: linorder_not_less [symmetric]) 
  19.679  done
  19.680  
  19.681 -(* Axiom 'order_less_le' of class 'order': *)
  19.682 -lemma int_less_le: "((w::int) < z) = (w <= z & w ~= z)"
  19.683 -apply (simp add: zle_def int_neq_iff)
  19.684 -apply (blast elim!: zless_asym)
  19.685 +lemma of_nat_less_iff [simp]:
  19.686 +     "(of_nat m < (of_nat n::'a::ordered_semiring)) = (m<n)"
  19.687 +by (blast intro: of_nat_less_imp_less less_imp_of_nat_less ) 
  19.688 +
  19.689 +text{*Special cases where either operand is zero*}
  19.690 +declare of_nat_less_iff [of 0, simplified, simp]
  19.691 +declare of_nat_less_iff [of _ 0, simplified, simp]
  19.692 +
  19.693 +lemma of_nat_le_iff [simp]:
  19.694 +     "(of_nat m \<le> (of_nat n::'a::ordered_semiring)) = (m \<le> n)"
  19.695 +by (simp add: linorder_not_less [symmetric]) 
  19.696 +
  19.697 +text{*Special cases where either operand is zero*}
  19.698 +declare of_nat_le_iff [of 0, simplified, simp]
  19.699 +declare of_nat_le_iff [of _ 0, simplified, simp]
  19.700 +
  19.701 +lemma of_nat_eq_iff [simp]:
  19.702 +     "(of_nat m = (of_nat n::'a::ordered_semiring)) = (m = n)"
  19.703 +by (simp add: order_eq_iff) 
  19.704 +
  19.705 +text{*Special cases where either operand is zero*}
  19.706 +declare of_nat_eq_iff [of 0, simplified, simp]
  19.707 +declare of_nat_eq_iff [of _ 0, simplified, simp]
  19.708 +
  19.709 +lemma of_nat_diff [simp]:
  19.710 +     "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring)"
  19.711 +by (simp del: of_nat_add
  19.712 +	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split) 
  19.713 +
  19.714 +
  19.715 +subsection{*The Set of Natural Numbers*}
  19.716 +
  19.717 +constdefs
  19.718 +   Nats  :: "'a::semiring set"
  19.719 +    "Nats == range of_nat"
  19.720 +
  19.721 +syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
  19.722 +
  19.723 +lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
  19.724 +by (simp add: Nats_def) 
  19.725 +
  19.726 +lemma Nats_0 [simp]: "0 \<in> Nats"
  19.727 +apply (simp add: Nats_def) 
  19.728 +apply (rule range_eqI) 
  19.729 +apply (rule of_nat_0 [symmetric])
  19.730 +done
  19.731 +
  19.732 +lemma Nats_1 [simp]: "1 \<in> Nats"
  19.733 +apply (simp add: Nats_def) 
  19.734 +apply (rule range_eqI) 
  19.735 +apply (rule of_nat_1 [symmetric])
  19.736 +done
  19.737 +
  19.738 +lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
  19.739 +apply (auto simp add: Nats_def) 
  19.740 +apply (rule range_eqI) 
  19.741 +apply (rule of_nat_add [symmetric])
  19.742 +done
  19.743 +
  19.744 +lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
  19.745 +apply (auto simp add: Nats_def) 
  19.746 +apply (rule range_eqI) 
  19.747 +apply (rule of_nat_mult [symmetric])
  19.748  done
  19.749  
  19.750 -instance int :: order
  19.751 -proof qed (assumption | rule zle_refl zle_trans zle_anti_sym int_less_le)+
  19.752 +text{*Agreement with the specific embedding for the integers*}
  19.753 +lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
  19.754 +proof
  19.755 +  fix n
  19.756 +  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac) 
  19.757 +qed
  19.758 +
  19.759 +
  19.760 +subsection{*Embedding of the Integers into any Ring: @{term of_int}*}
  19.761 +
  19.762 +constdefs
  19.763 +   of_int :: "int => 'a::ring"
  19.764 +   "of_int z ==
  19.765 +      (THE a. \<exists>i j. (i,j) \<in> Rep_Integ z & a = (of_nat i) - (of_nat j))"
  19.766 +
  19.767 +
  19.768 +lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
  19.769 +apply (simp add: of_int_def)
  19.770 +apply (rule the_equality, auto) 
  19.771 +apply (simp add: compare_rls add_ac of_nat_add [symmetric]
  19.772 +            del: of_nat_add) 
  19.773 +done
  19.774 +
  19.775 +lemma of_int_0 [simp]: "of_int 0 = 0"
  19.776 +by (simp add: of_int Zero_int_def int_def)
  19.777 +
  19.778 +lemma of_int_1 [simp]: "of_int 1 = 1"
  19.779 +by (simp add: of_int One_int_def int_def)
  19.780 +
  19.781 +lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
  19.782 +apply (rule eq_Abs_Integ [of w])
  19.783 +apply (rule eq_Abs_Integ [of z])
  19.784 +apply (simp add: compare_rls of_int zadd) 
  19.785 +done
  19.786 +
  19.787 +lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
  19.788 +apply (rule eq_Abs_Integ [of z])
  19.789 +apply (simp add: compare_rls of_int zminus) 
  19.790 +done
  19.791  
  19.792 -instance int :: plus_ac0
  19.793 -proof qed (rule zadd_commute zadd_assoc zadd_0)+
  19.794 +lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
  19.795 +by (simp add: diff_minus)
  19.796 +
  19.797 +lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
  19.798 +apply (rule eq_Abs_Integ [of w])
  19.799 +apply (rule eq_Abs_Integ [of z])
  19.800 +apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib 
  19.801 +                 zmult add_ac) 
  19.802 +done
  19.803 +
  19.804 +lemma of_int_le_iff [simp]:
  19.805 +     "(of_int w \<le> (of_int z::'a::ordered_ring)) = (w \<le> z)"
  19.806 +apply (rule eq_Abs_Integ [of w])
  19.807 +apply (rule eq_Abs_Integ [of z])
  19.808 +apply (simp add: compare_rls of_int zle zdiff_def zadd zminus 
  19.809 +                 of_nat_add [symmetric]   del: of_nat_add) 
  19.810 +done
  19.811 +
  19.812 +text{*Special cases where either operand is zero*}
  19.813 +declare of_int_le_iff [of 0, simplified, simp]
  19.814 +declare of_int_le_iff [of _ 0, simplified, simp]
  19.815  
  19.816 -instance int :: linorder
  19.817 -proof qed (rule zle_linear)
  19.818 +lemma of_int_less_iff [simp]:
  19.819 +     "(of_int w < (of_int z::'a::ordered_ring)) = (w < z)"
  19.820 +by (simp add: linorder_not_le [symmetric])
  19.821 +
  19.822 +text{*Special cases where either operand is zero*}
  19.823 +declare of_int_less_iff [of 0, simplified, simp]
  19.824 +declare of_int_less_iff [of _ 0, simplified, simp]
  19.825 +
  19.826 +lemma of_int_eq_iff [simp]:
  19.827 +     "(of_int w = (of_int z::'a::ordered_ring)) = (w = z)"
  19.828 +by (simp add: order_eq_iff) 
  19.829 +
  19.830 +text{*Special cases where either operand is zero*}
  19.831 +declare of_int_eq_iff [of 0, simplified, simp]
  19.832 +declare of_int_eq_iff [of _ 0, simplified, simp]
  19.833 +
  19.834 +
  19.835 +subsection{*The Set of Integers*}
  19.836 +
  19.837 +constdefs
  19.838 +   Ints  :: "'a::ring set"
  19.839 +    "Ints == range of_int"
  19.840  
  19.841  
  19.842 -lemma zadd_left_cancel [simp]: "!!w::int. (z + w' = z + w) = (w' = w)"
  19.843 -  by (rule add_left_cancel) 
  19.844 +syntax (xsymbols)
  19.845 +  Ints      :: "'a set"                   ("\<int>")
  19.846 +
  19.847 +lemma Ints_0 [simp]: "0 \<in> Ints"
  19.848 +apply (simp add: Ints_def) 
  19.849 +apply (rule range_eqI) 
  19.850 +apply (rule of_int_0 [symmetric])
  19.851 +done
  19.852 +
  19.853 +lemma Ints_1 [simp]: "1 \<in> Ints"
  19.854 +apply (simp add: Ints_def) 
  19.855 +apply (rule range_eqI) 
  19.856 +apply (rule of_int_1 [symmetric])
  19.857 +done
  19.858 +
  19.859 +lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
  19.860 +apply (auto simp add: Ints_def) 
  19.861 +apply (rule range_eqI) 
  19.862 +apply (rule of_int_add [symmetric])
  19.863 +done
  19.864 +
  19.865 +lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
  19.866 +apply (auto simp add: Ints_def) 
  19.867 +apply (rule range_eqI) 
  19.868 +apply (rule of_int_minus [symmetric])
  19.869 +done
  19.870 +
  19.871 +lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
  19.872 +apply (auto simp add: Ints_def) 
  19.873 +apply (rule range_eqI) 
  19.874 +apply (rule of_int_diff [symmetric])
  19.875 +done
  19.876 +
  19.877 +lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
  19.878 +apply (auto simp add: Ints_def) 
  19.879 +apply (rule range_eqI) 
  19.880 +apply (rule of_int_mult [symmetric])
  19.881 +done
  19.882 +
  19.883 +text{*Collapse nested embeddings*}
  19.884 +lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
  19.885 +by (induct n, auto) 
  19.886 +
  19.887 +lemma of_int_int_eq [simp]: "of_int (int n) = int n"
  19.888 +by (simp add: int_eq_of_nat) 
  19.889  
  19.890  
  19.891 +lemma Ints_cases [case_names of_int, cases set: Ints]:
  19.892 +  "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
  19.893 +proof (unfold Ints_def)
  19.894 +  assume "!!z. q = of_int z ==> C"
  19.895 +  assume "q \<in> range of_int" thus C ..
  19.896 +qed
  19.897 +
  19.898 +lemma Ints_induct [case_names of_int, induct set: Ints]:
  19.899 +  "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
  19.900 +  by (rule Ints_cases) auto
  19.901 +
  19.902 +
  19.903 +
  19.904 +(*Legacy ML bindings, but no longer the structure Int.*)
  19.905  ML
  19.906  {*
  19.907 +val zabs_def = thm "zabs_def"
  19.908 +val nat_def  = thm "nat_def"
  19.909 +
  19.910 +val int_0 = thm "int_0";
  19.911 +val int_1 = thm "int_1";
  19.912 +val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
  19.913 +val neg_eq_less_0 = thm "neg_eq_less_0";
  19.914 +val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
  19.915 +val not_neg_0 = thm "not_neg_0";
  19.916 +val not_neg_1 = thm "not_neg_1";
  19.917 +val iszero_0 = thm "iszero_0";
  19.918 +val not_iszero_1 = thm "not_iszero_1";
  19.919 +val int_0_less_1 = thm "int_0_less_1";
  19.920 +val int_0_neq_1 = thm "int_0_neq_1";
  19.921 +val negative_zless = thm "negative_zless";
  19.922 +val negative_zle = thm "negative_zle";
  19.923 +val not_zle_0_negative = thm "not_zle_0_negative";
  19.924 +val not_int_zless_negative = thm "not_int_zless_negative";
  19.925 +val negative_eq_positive = thm "negative_eq_positive";
  19.926 +val zle_iff_zadd = thm "zle_iff_zadd";
  19.927 +val abs_int_eq = thm "abs_int_eq";
  19.928 +val abs_split = thm"abs_split";
  19.929 +val nat_int = thm "nat_int";
  19.930 +val nat_zminus_int = thm "nat_zminus_int";
  19.931 +val nat_zero = thm "nat_zero";
  19.932 +val not_neg_nat = thm "not_neg_nat";
  19.933 +val neg_nat = thm "neg_nat";
  19.934 +val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
  19.935 +val nat_0_le = thm "nat_0_le";
  19.936 +val nat_le_0 = thm "nat_le_0";
  19.937 +val zless_nat_conj = thm "zless_nat_conj";
  19.938 +val int_cases = thm "int_cases";
  19.939 +
  19.940  val int_def = thm "int_def";
  19.941 -val neg_def = thm "neg_def";
  19.942 -val iszero_def = thm "iszero_def";
  19.943  val Zero_int_def = thm "Zero_int_def";
  19.944  val One_int_def = thm "One_int_def";
  19.945  val zadd_def = thm "zadd_def";
  19.946 @@ -524,8 +981,6 @@
  19.947  val zminus_zminus = thm "zminus_zminus";
  19.948  val inj_zminus = thm "inj_zminus";
  19.949  val zminus_0 = thm "zminus_0";
  19.950 -val not_neg_int = thm "not_neg_int";
  19.951 -val neg_zminus_int = thm "neg_zminus_int";
  19.952  val zadd = thm "zadd";
  19.953  val zminus_zadd_distrib = thm "zminus_zadd_distrib";
  19.954  val zadd_commute = thm "zadd_commute";
  19.955 @@ -545,8 +1000,6 @@
  19.956  val zdiff0 = thm "zdiff0";
  19.957  val zdiff0_right = thm "zdiff0_right";
  19.958  val zdiff_self = thm "zdiff_self";
  19.959 -val zadd_assoc_cong = thm "zadd_assoc_cong";
  19.960 -val zadd_assoc_swap = thm "zadd_assoc_swap";
  19.961  val zmult_congruent2 = thm "zmult_congruent2";
  19.962  val zmult = thm "zmult";
  19.963  val zmult_zminus = thm "zmult_zminus";
  19.964 @@ -564,15 +1017,6 @@
  19.965  val zmult_0_right = thm "zmult_0_right";
  19.966  val zmult_1_right = thm "zmult_1_right";
  19.967  val zless_iff_Suc_zadd = thm "zless_iff_Suc_zadd";
  19.968 -val zless_zadd_Suc = thm "zless_zadd_Suc";
  19.969 -val zless_trans = thm "zless_trans";
  19.970 -val zless_not_sym = thm "zless_not_sym";
  19.971 -val zless_asym = thm "zless_asym";
  19.972 -val zless_not_refl = thm "zless_not_refl";
  19.973 -val zless_irrefl = thm "zless_irrefl";
  19.974 -val zless_linear = thm "zless_linear";
  19.975 -val int_neq_iff = thm "int_neq_iff";
  19.976 -val int_neqE = thm "int_neqE";
  19.977  val int_int_eq = thm "int_int_eq";
  19.978  val int_eq_0_conv = thm "int_eq_0_conv";
  19.979  val zless_int = thm "zless_int";
  19.980 @@ -581,15 +1025,48 @@
  19.981  val zle_int = thm "zle_int";
  19.982  val zero_zle_int = thm "zero_zle_int";
  19.983  val int_le_0_conv = thm "int_le_0_conv";
  19.984 -val zle_imp_zless_or_eq = thm "zle_imp_zless_or_eq";
  19.985 -val zless_or_eq_imp_zle = thm "zless_or_eq_imp_zle";
  19.986 -val int_le_less = thm "int_le_less";
  19.987  val zle_refl = thm "zle_refl";
  19.988  val zle_linear = thm "zle_linear";
  19.989  val zle_trans = thm "zle_trans";
  19.990  val zle_anti_sym = thm "zle_anti_sym";
  19.991 -val int_less_le = thm "int_less_le";
  19.992 -val zadd_left_cancel = thm "zadd_left_cancel";
  19.993 +
  19.994 +val Ints_def = thm "Ints_def";
  19.995 +val Nats_def = thm "Nats_def";
  19.996 +
  19.997 +val of_nat_0 = thm "of_nat_0";
  19.998 +val of_nat_Suc = thm "of_nat_Suc";
  19.999 +val of_nat_1 = thm "of_nat_1";
 19.1000 +val of_nat_add = thm "of_nat_add";
 19.1001 +val of_nat_mult = thm "of_nat_mult";
 19.1002 +val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
 19.1003 +val less_imp_of_nat_less = thm "less_imp_of_nat_less";
 19.1004 +val of_nat_less_imp_less = thm "of_nat_less_imp_less";
 19.1005 +val of_nat_less_iff = thm "of_nat_less_iff";
 19.1006 +val of_nat_le_iff = thm "of_nat_le_iff";
 19.1007 +val of_nat_eq_iff = thm "of_nat_eq_iff";
 19.1008 +val Nats_0 = thm "Nats_0";
 19.1009 +val Nats_1 = thm "Nats_1";
 19.1010 +val Nats_add = thm "Nats_add";
 19.1011 +val Nats_mult = thm "Nats_mult";
 19.1012 +val of_int = thm "of_int";
 19.1013 +val of_int_0 = thm "of_int_0";
 19.1014 +val of_int_1 = thm "of_int_1";
 19.1015 +val of_int_add = thm "of_int_add";
 19.1016 +val of_int_minus = thm "of_int_minus";
 19.1017 +val of_int_diff = thm "of_int_diff";
 19.1018 +val of_int_mult = thm "of_int_mult";
 19.1019 +val of_int_le_iff = thm "of_int_le_iff";
 19.1020 +val of_int_less_iff = thm "of_int_less_iff";
 19.1021 +val of_int_eq_iff = thm "of_int_eq_iff";
 19.1022 +val Ints_0 = thm "Ints_0";
 19.1023 +val Ints_1 = thm "Ints_1";
 19.1024 +val Ints_add = thm "Ints_add";
 19.1025 +val Ints_minus = thm "Ints_minus";
 19.1026 +val Ints_diff = thm "Ints_diff";
 19.1027 +val Ints_mult = thm "Ints_mult";
 19.1028 +val of_int_of_nat_eq = thm"of_int_of_nat_eq";
 19.1029 +val Ints_cases = thm "Ints_cases";
 19.1030 +val Ints_induct = thm "Ints_induct";
 19.1031  *}
 19.1032  
 19.1033  end
    20.1 --- a/src/HOL/Integ/IntDiv.thy	Thu Feb 05 10:45:28 2004 +0100
    20.2 +++ b/src/HOL/Integ/IntDiv.thy	Tue Feb 10 12:02:11 2004 +0100
    20.3 @@ -531,7 +531,7 @@
    20.4   prefer 2 apply simp
    20.5  apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
    20.6  apply (subst zadd_commute, rule zadd_zless_mono, arith)
    20.7 -apply (rule zmult_zle_mono1, auto)
    20.8 +apply (rule mult_right_mono, auto)
    20.9  done
   20.10  
   20.11  lemma zdiv_mono2:
   20.12 @@ -561,7 +561,7 @@
   20.13   apply (simp add: zmult_zless_cancel1)
   20.14  apply (simp add: zadd_zmult_distrib2)
   20.15  apply (subgoal_tac "b*q' \<le> b'*q'")
   20.16 - prefer 2 apply (simp add: zmult_zle_mono1_neg)
   20.17 + prefer 2 apply (simp add: mult_right_mono_neg)
   20.18  apply (subgoal_tac "b'*q' < b + b*q")
   20.19   apply arith
   20.20  apply simp 
   20.21 @@ -702,8 +702,8 @@
   20.22  apply (subgoal_tac "b * (c - q mod c) < r * 1")
   20.23  apply (simp add: zdiff_zmult_distrib2)
   20.24  apply (rule order_le_less_trans)
   20.25 -apply (erule_tac [2] zmult_zless_mono1)
   20.26 -apply (rule zmult_zle_mono2_neg)
   20.27 +apply (erule_tac [2] mult_strict_right_mono)
   20.28 +apply (rule mult_left_mono_neg)
   20.29  apply (auto simp add: compare_rls zadd_commute [of 1]
   20.30                        add1_zle_eq pos_mod_bound)
   20.31  done
   20.32 @@ -724,7 +724,7 @@
   20.33  apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
   20.34  apply (simp add: zdiff_zmult_distrib2)
   20.35  apply (rule order_less_le_trans)
   20.36 -apply (erule zmult_zless_mono1)
   20.37 +apply (erule mult_strict_right_mono)
   20.38  apply (rule_tac [2] zmult_zle_mono2)
   20.39  apply (auto simp add: compare_rls zadd_commute [of 1]
   20.40                        add1_zle_eq pos_mod_bound)
   20.41 @@ -1111,7 +1111,7 @@
   20.42    apply (unfold dvd_def, auto)
   20.43    apply (subgoal_tac "0 < n")
   20.44     prefer 2
   20.45 -   apply (blast intro: zless_trans)
   20.46 +   apply (blast intro: order_less_trans)
   20.47    apply (simp add: zero_less_mult_iff)
   20.48    apply (subgoal_tac "n * k < n * 1")
   20.49     apply (drule zmult_zless_cancel1 [THEN iffD1], auto)
   20.50 @@ -1150,7 +1150,7 @@
   20.51  
   20.52  lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
   20.53    apply (auto simp add: dvd_def)
   20.54 -   apply (drule zminus_equation [THEN iffD1])
   20.55 +   apply (drule minus_equation_iff [THEN iffD1])
   20.56     apply (rule_tac [!] x = "-k" in exI, auto)
   20.57    done
   20.58  
    21.1 --- a/src/HOL/Integ/NatBin.thy	Thu Feb 05 10:45:28 2004 +0100
    21.2 +++ b/src/HOL/Integ/NatBin.thy	Tue Feb 10 12:02:11 2004 +0100
    21.3 @@ -80,7 +80,7 @@
    21.4  (*"neg" is used in rewrite rules for binary comparisons*)
    21.5  lemma int_nat_number_of:
    21.6       "int (number_of v :: nat) =  
    21.7 -         (if neg (number_of v) then 0  
    21.8 +         (if neg (number_of v :: int) then 0  
    21.9            else (number_of v :: int))"
   21.10  by (simp del: nat_number_of
   21.11  	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)
   21.12 @@ -96,14 +96,14 @@
   21.13  
   21.14  lemma Suc_nat_number_of_add:
   21.15       "Suc (number_of v + n) =  
   21.16 -        (if neg (number_of v) then 1+n else number_of (bin_succ v) + n)" 
   21.17 +        (if neg (number_of v :: int) then 1+n else number_of (bin_succ v) + n)" 
   21.18  by (simp del: nat_number_of 
   21.19           add: nat_number_of_def neg_nat
   21.20                Suc_nat_eq_nat_zadd1 number_of_succ) 
   21.21  
   21.22  lemma Suc_nat_number_of:
   21.23       "Suc (number_of v) =  
   21.24 -        (if neg (number_of v) then 1 else number_of (bin_succ v))"
   21.25 +        (if neg (number_of v :: int) then 1 else number_of (bin_succ v))"
   21.26  apply (cut_tac n = 0 in Suc_nat_number_of_add)
   21.27  apply (simp cong del: if_weak_cong)
   21.28  done
   21.29 @@ -115,8 +115,8 @@
   21.30  (*"neg" is used in rewrite rules for binary comparisons*)
   21.31  lemma add_nat_number_of:
   21.32       "(number_of v :: nat) + number_of v' =  
   21.33 -         (if neg (number_of v) then number_of v'  
   21.34 -          else if neg (number_of v') then number_of v  
   21.35 +         (if neg (number_of v :: int) then number_of v'  
   21.36 +          else if neg (number_of v' :: int) then number_of v  
   21.37            else number_of (bin_add v v'))"
   21.38  by (force dest!: neg_nat
   21.39            simp del: nat_number_of
   21.40 @@ -138,7 +138,7 @@
   21.41  
   21.42  lemma diff_nat_number_of: 
   21.43       "(number_of v :: nat) - number_of v' =  
   21.44 -        (if neg (number_of v') then number_of v  
   21.45 +        (if neg (number_of v' :: int) then number_of v  
   21.46           else let d = number_of (bin_add v (bin_minus v')) in     
   21.47                if neg d then 0 else nat d)"
   21.48  by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def) 
   21.49 @@ -150,7 +150,7 @@
   21.50  
   21.51  lemma mult_nat_number_of:
   21.52       "(number_of v :: nat) * number_of v' =  
   21.53 -       (if neg (number_of v) then 0 else number_of (bin_mult v v'))"
   21.54 +       (if neg (number_of v :: int) then 0 else number_of (bin_mult v v'))"
   21.55  by (force dest!: neg_nat
   21.56            simp del: nat_number_of
   21.57            simp add: nat_number_of_def nat_mult_distrib [symmetric]) 
   21.58 @@ -162,7 +162,7 @@
   21.59  
   21.60  lemma div_nat_number_of:
   21.61       "(number_of v :: nat)  div  number_of v' =  
   21.62 -          (if neg (number_of v) then 0  
   21.63 +          (if neg (number_of v :: int) then 0  
   21.64             else nat (number_of v div number_of v'))"
   21.65  by (force dest!: neg_nat
   21.66            simp del: nat_number_of
   21.67 @@ -175,8 +175,8 @@
   21.68  
   21.69  lemma mod_nat_number_of:
   21.70       "(number_of v :: nat)  mod  number_of v' =  
   21.71 -        (if neg (number_of v) then 0  
   21.72 -         else if neg (number_of v') then number_of v  
   21.73 +        (if neg (number_of v :: int) then 0  
   21.74 +         else if neg (number_of v' :: int) then number_of v  
   21.75           else nat (number_of v mod number_of v'))"
   21.76  by (force dest!: neg_nat
   21.77            simp del: nat_number_of
   21.78 @@ -242,9 +242,9 @@
   21.79  (*"neg" is used in rewrite rules for binary comparisons*)
   21.80  lemma eq_nat_number_of:
   21.81       "((number_of v :: nat) = number_of v') =  
   21.82 -      (if neg (number_of v) then (iszero (number_of v') | neg (number_of v'))  
   21.83 -       else if neg (number_of v') then iszero (number_of v)  
   21.84 -       else iszero (number_of (bin_add v (bin_minus v'))))"
   21.85 +      (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))  
   21.86 +       else if neg (number_of v' :: int) then iszero (number_of v :: int)  
   21.87 +       else iszero (number_of (bin_add v (bin_minus v')) :: int))"
   21.88  apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
   21.89                    eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
   21.90              split add: split_if cong add: imp_cong)
   21.91 @@ -259,8 +259,8 @@
   21.92  (*"neg" is used in rewrite rules for binary comparisons*)
   21.93  lemma less_nat_number_of:
   21.94       "((number_of v :: nat) < number_of v') =  
   21.95 -         (if neg (number_of v) then neg (number_of (bin_minus v'))  
   21.96 -          else neg (number_of (bin_add v (bin_minus v'))))"
   21.97 +         (if neg (number_of v :: int) then neg (number_of (bin_minus v') :: int)  
   21.98 +          else neg (number_of (bin_add v (bin_minus v')) :: int))"
   21.99  apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
  21.100                  nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
  21.101              cong add: imp_cong, simp) 
  21.102 @@ -397,24 +397,24 @@
  21.103  
  21.104  lemma eq_number_of_0:
  21.105       "(number_of v = (0::nat)) =  
  21.106 -      (if neg (number_of v) then True else iszero (number_of v))"
  21.107 +      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
  21.108  apply (simp add: numeral_0_eq_0 [symmetric] iszero_0)
  21.109  done
  21.110  
  21.111  lemma eq_0_number_of:
  21.112       "((0::nat) = number_of v) =  
  21.113 -      (if neg (number_of v) then True else iszero (number_of v))"
  21.114 +      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
  21.115  apply (rule trans [OF eq_sym_conv eq_number_of_0])
  21.116  done
  21.117  
  21.118  lemma less_0_number_of:
  21.119 -     "((0::nat) < number_of v) = neg (number_of (bin_minus v))"
  21.120 +     "((0::nat) < number_of v) = neg (number_of (bin_minus v) :: int)"
  21.121  by (simp add: numeral_0_eq_0 [symmetric])
  21.122  
  21.123  (*Simplification already handles n<0, n<=0 and 0<=n.*)
  21.124  declare eq_number_of_0 [simp] eq_0_number_of [simp] less_0_number_of [simp]
  21.125  
  21.126 -lemma neg_imp_number_of_eq_0: "neg (number_of v) ==> number_of v = (0::nat)"
  21.127 +lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
  21.128  by (simp add: numeral_0_eq_0 [symmetric] iszero_0)
  21.129  
  21.130  
  21.131 @@ -530,7 +530,7 @@
  21.132  
  21.133  lemma power_nat_number_of:
  21.134       "(number_of v :: nat) ^ n =  
  21.135 -       (if neg (number_of v) then 0^n else nat ((number_of v :: int) ^ n))"
  21.136 +       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
  21.137  by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
  21.138           split add: split_if cong: imp_cong)
  21.139  
  21.140 @@ -605,7 +605,7 @@
  21.141  
  21.142  lemma nat_number_of_BIT_True:
  21.143    "number_of (w BIT True) =
  21.144 -    (if neg (number_of w) then 0
  21.145 +    (if neg (number_of w :: int) then 0
  21.146       else let n = number_of w in Suc (n + n))"
  21.147    apply (simp only: nat_number_of_def Let_def split: split_if)
  21.148    apply (intro conjI impI)
  21.149 @@ -618,7 +618,7 @@
  21.150  lemma nat_number_of_BIT_False:
  21.151      "number_of (w BIT False) = (let n::nat = number_of w in n + n)"
  21.152    apply (simp only: nat_number_of_def Let_def)
  21.153 -  apply (cases "neg (number_of w)")
  21.154 +  apply (cases "neg (number_of w :: int)")
  21.155     apply (simp add: neg_nat neg_number_of_BIT)
  21.156    apply (rule int_int_eq [THEN iffD1])
  21.157    apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
  21.158 @@ -637,8 +637,8 @@
  21.159  
  21.160  lemma nat_number_of_add_left:
  21.161       "number_of v + (number_of v' + (k::nat)) =  
  21.162 -         (if neg (number_of v) then number_of v' + k  
  21.163 -          else if neg (number_of v') then number_of v + k  
  21.164 +         (if neg (number_of v :: int) then number_of v' + k  
  21.165 +          else if neg (number_of v' :: int) then number_of v + k  
  21.166            else number_of (bin_add v v') + k)"
  21.167  apply simp
  21.168  done
  21.169 @@ -831,6 +831,8 @@
  21.170    "uminus" :: "int => int"      ("`~")
  21.171    "op +" :: "int => int => int" ("(_ `+/ _)")
  21.172    "op *" :: "int => int => int" ("(_ `*/ _)")
  21.173 +  "op <" :: "int => int => bool" ("(_ </ _)")
  21.174 +  "op <=" :: "int => int => bool" ("(_ <=/ _)")
  21.175    "neg"                         ("(_ < 0)")
  21.176  
  21.177  end
    22.1 --- a/src/HOL/Integ/NatSimprocs.thy	Thu Feb 05 10:45:28 2004 +0100
    22.2 +++ b/src/HOL/Integ/NatSimprocs.thy	Tue Feb 10 12:02:11 2004 +0100
    22.3 @@ -20,7 +20,7 @@
    22.4  (*Now just instantiating n to (number_of v) does the right simplification,
    22.5    but with some redundant inequality tests.*)
    22.6  lemma neg_number_of_bin_pred_iff_0:
    22.7 -     "neg (number_of (bin_pred v)) = (number_of v = (0::nat))"
    22.8 +     "neg (number_of (bin_pred v)::int) = (number_of v = (0::nat))"
    22.9  apply (subgoal_tac "neg (number_of (bin_pred v)) = (number_of v < Suc 0) ")
   22.10  apply (simp only: less_Suc_eq_le le_0_eq)
   22.11  apply (subst less_number_of_Suc, simp)
   22.12 @@ -29,7 +29,7 @@
   22.13  text{*No longer required as a simprule because of the @{text inverse_fold}
   22.14     simproc*}
   22.15  lemma Suc_diff_number_of:
   22.16 -     "neg (number_of (bin_minus v)) ==>  
   22.17 +     "neg (number_of (bin_minus v)::int) ==>  
   22.18        Suc m - (number_of v) = m - (number_of (bin_pred v))"
   22.19  apply (subst Suc_diff_eq_diff_pred, simp, simp)
   22.20  apply (force simp only: diff_nat_number_of less_0_number_of [symmetric] 
    23.1 --- a/src/HOL/Integ/Presburger.thy	Thu Feb 05 10:45:28 2004 +0100
    23.2 +++ b/src/HOL/Integ/Presburger.thy	Tue Feb 10 12:02:11 2004 +0100
    23.3 @@ -961,7 +961,7 @@
    23.4    apply (case_tac "0 \<le> k")
    23.5    apply rules
    23.6    apply (simp add: linorder_not_le)
    23.7 -  apply (drule zmult_zless_mono2_neg [OF iffD2 [OF zero_less_int_conv]])
    23.8 +  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
    23.9    apply assumption
   23.10    apply (simp add: mult_ac)
   23.11    done
    24.1 --- a/src/HOL/Integ/int_arith1.ML	Thu Feb 05 10:45:28 2004 +0100
    24.2 +++ b/src/HOL/Integ/int_arith1.ML	Tue Feb 10 12:02:11 2004 +0100
    24.3 @@ -28,6 +28,11 @@
    24.4  val bin_mult_Min = thm"bin_mult_Min";
    24.5  val bin_mult_BIT = thm"bin_mult_BIT";
    24.6  
    24.7 +val neg_def = thm "neg_def";
    24.8 +val iszero_def = thm "iszero_def";
    24.9 +val not_neg_int = thm "not_neg_int";
   24.10 +val neg_zminus_int = thm "neg_zminus_int";
   24.11 +
   24.12  val zadd_ac = thms "Ring_and_Field.add_ac"
   24.13  val zmult_ac = thms "Ring_and_Field.mult_ac"
   24.14  val NCons_Pls_0 = thm"NCons_Pls_0";
    25.1 --- a/src/HOL/Integ/int_factor_simprocs.ML	Thu Feb 05 10:45:28 2004 +0100
    25.2 +++ b/src/HOL/Integ/int_factor_simprocs.ML	Tue Feb 10 12:02:11 2004 +0100
    25.3 @@ -5,25 +5,11 @@
    25.4  
    25.5  Factor cancellation simprocs for the integers.
    25.6  
    25.7 -This file can't be combined with int_arith1,2 because it requires IntDiv.thy.
    25.8 +This file can't be combined with int_arith1 because it requires IntDiv.thy.
    25.9  *)
   25.10  
   25.11  (** Factor cancellation theorems for "int" **)
   25.12  
   25.13 -Goal "!!k::int. (k*m <= k*n) = ((0 < k --> m<=n) & (k < 0 --> n<=m))";
   25.14 -by (stac zmult_zle_cancel1 1);
   25.15 -by Auto_tac;
   25.16 -qed "int_mult_le_cancel1";
   25.17 -
   25.18 -Goal "!!k::int. (k*m < k*n) = ((0 < k & m<n) | (k < 0 & n<m))";
   25.19 -by (stac zmult_zless_cancel1 1);
   25.20 -by Auto_tac;
   25.21 -qed "int_mult_less_cancel1";
   25.22 -
   25.23 -Goal "!!k::int. (k*m = k*n) = (k = 0 | m=n)";
   25.24 -by Auto_tac;
   25.25 -qed "int_mult_eq_cancel1";
   25.26 -
   25.27  Goal "!!k::int. k~=0 ==> (k*m) div (k*n) = (m div n)";
   25.28  by (stac zdiv_zmult_zmult1 1);
   25.29  by Auto_tac;
   25.30 @@ -34,6 +20,7 @@
   25.31  by (simp_tac (simpset() addsimps [int_mult_div_cancel1]) 1);
   25.32  qed "int_mult_div_cancel_disj";
   25.33  
   25.34 +
   25.35  local
   25.36    open Int_Numeral_Simprocs
   25.37  in
   25.38 @@ -65,7 +52,7 @@
   25.39    val prove_conv = Bin_Simprocs.prove_conv
   25.40    val mk_bal   = HOLogic.mk_eq
   25.41    val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
   25.42 -  val cancel = int_mult_eq_cancel1 RS trans
   25.43 +  val cancel = mult_cancel_left RS trans
   25.44    val neg_exchanges = false
   25.45  )
   25.46  
   25.47 @@ -74,7 +61,7 @@
   25.48    val prove_conv = Bin_Simprocs.prove_conv
   25.49    val mk_bal   = HOLogic.mk_binrel "op <"
   25.50    val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT
   25.51 -  val cancel = int_mult_less_cancel1 RS trans
   25.52 +  val cancel = mult_less_cancel_left RS trans
   25.53    val neg_exchanges = true
   25.54  )
   25.55  
   25.56 @@ -83,7 +70,7 @@
   25.57    val prove_conv = Bin_Simprocs.prove_conv
   25.58    val mk_bal   = HOLogic.mk_binrel "op <="
   25.59    val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT
   25.60 -  val cancel = int_mult_le_cancel1 RS trans
   25.61 +  val cancel = mult_le_cancel_left RS trans
   25.62    val neg_exchanges = true
   25.63  )
   25.64  
   25.65 @@ -179,7 +166,7 @@
   25.66    val prove_conv = Bin_Simprocs.prove_conv
   25.67    val mk_bal   = HOLogic.mk_eq
   25.68    val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
   25.69 -  val simplify_meta_eq  = cancel_simplify_meta_eq int_mult_eq_cancel1
   25.70 +  val simplify_meta_eq  = cancel_simplify_meta_eq mult_cancel_left
   25.71  );
   25.72  
   25.73  structure DivideCancelFactor = ExtractCommonTermFun
    26.1 --- a/src/HOL/IsaMakefile	Thu Feb 05 10:45:28 2004 +0100
    26.2 +++ b/src/HOL/IsaMakefile	Tue Feb 10 12:02:11 2004 +0100
    26.3 @@ -86,7 +86,7 @@
    26.4    Hilbert_Choice.thy Hilbert_Choice_lemmas.ML HOL.ML \
    26.5    HOL.thy HOL_lemmas.ML Inductive.thy Integ/Bin.thy \
    26.6    Integ/cooper_dec.ML Integ/cooper_proof.ML \
    26.7 -  Integ/Equiv.thy Integ/Int.thy Integ/IntArith.thy Integ/IntDef.thy \
    26.8 +  Integ/Equiv.thy Integ/IntArith.thy Integ/IntDef.thy \
    26.9    Integ/IntDiv.thy Integ/NatBin.thy Integ/NatSimprocs.thy Integ/int_arith1.ML \
   26.10    Integ/int_factor_simprocs.ML Integ/nat_simprocs.ML \
   26.11    Integ/Presburger.thy Integ/presburger.ML Integ/qelim.ML \
    27.1 --- a/src/HOL/Isar_examples/document/root.bib	Thu Feb 05 10:45:28 2004 +0100
    27.2 +++ b/src/HOL/Isar_examples/document/root.bib	Tue Feb 10 12:02:11 2004 +0100
    27.3 @@ -71,7 +71,7 @@
    27.4    institution =  CUCL,
    27.5    year = 	 1996,
    27.6    number =	 394,
    27.7 -  note = {\url{http://www.ftp.cl.cam.ac.uk/ftp/papers/reports/}}
    27.8 +  note = {\url{http://www.cl.cam.ac.uk/users/lcp/papers/Reports/mutil.pdf}}
    27.9  }
   27.10  
   27.11  @Proceedings{tphols98,
    28.1 --- a/src/HOL/NumberTheory/IntPrimes.thy	Thu Feb 05 10:45:28 2004 +0100
    28.2 +++ b/src/HOL/NumberTheory/IntPrimes.thy	Tue Feb 10 12:02:11 2004 +0100
    28.3 @@ -338,7 +338,7 @@
    28.4      a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
    28.5    apply (unfold zcong_def dvd_def, auto)
    28.6    apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
    28.7 -  apply (cut_tac z = a and w = b in zless_linear, auto)
    28.8 +  apply (cut_tac x = a and y = b in linorder_less_linear, auto)
    28.9     apply (subgoal_tac [2] "(a - b) mod m = a - b")
   28.10      apply (rule_tac [3] mod_pos_pos_trivial, auto)
   28.11    apply (subgoal_tac "(m + (a - b)) mod m = m + (a - b)")
    29.1 --- a/src/HOL/Presburger.thy	Thu Feb 05 10:45:28 2004 +0100
    29.2 +++ b/src/HOL/Presburger.thy	Tue Feb 10 12:02:11 2004 +0100
    29.3 @@ -961,7 +961,7 @@
    29.4    apply (case_tac "0 \<le> k")
    29.5    apply rules
    29.6    apply (simp add: linorder_not_le)
    29.7 -  apply (drule zmult_zless_mono2_neg [OF iffD2 [OF zero_less_int_conv]])
    29.8 +  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
    29.9    apply assumption
   29.10    apply (simp add: mult_ac)
   29.11    done
    30.1 --- a/src/HOL/Real/PReal.thy	Thu Feb 05 10:45:28 2004 +0100
    30.2 +++ b/src/HOL/Real/PReal.thy	Tue Feb 10 12:02:11 2004 +0100
    30.3 @@ -700,7 +700,7 @@
    30.4    assumes A: "A \<in> preal"
    30.5        and "\<forall>x\<in>A. x + u \<in> A"
    30.6        and "0 \<le> z"
    30.7 -     shows "\<exists>b\<in>A. b + (rat z) * u \<in> A"
    30.8 +     shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
    30.9  proof (cases z rule: int_cases)
   30.10    case (nonneg n)
   30.11    show ?thesis
   30.12 @@ -709,8 +709,8 @@
   30.13        from preal_nonempty [OF A]
   30.14        show ?case  by force 
   30.15      case (Suc k)
   30.16 -      from this obtain b where "b \<in> A" "b + rat (int k) * u \<in> A" ..
   30.17 -      hence "b + rat (int k)*u + u \<in> A" by (simp add: prems)
   30.18 +      from this obtain b where "b \<in> A" "b + of_int (int k) * u \<in> A" ..
   30.19 +      hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
   30.20        thus ?case by (force simp add: left_distrib add_ac prems) 
   30.21    qed
   30.22  next
   30.23 @@ -718,7 +718,6 @@
   30.24    with prems show ?thesis by simp
   30.25  qed
   30.26  
   30.27 -
   30.28  lemma Gleason9_34_contra:
   30.29    assumes A: "A \<in> preal"
   30.30      shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
   30.31 @@ -736,10 +735,10 @@
   30.32    have apos [simp]: "0 < a" 
   30.33      by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) 
   30.34    let ?k = "a*d"
   30.35 -  have frle: "Fract a b \<le> rat ?k * (Fract c d)" 
   30.36 +  have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)" 
   30.37    proof -
   30.38      have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
   30.39 -      by (simp add: rat_def mult_rat le_rat order_less_imp_not_eq2 mult_ac) 
   30.40 +      by (simp add: mult_rat le_rat order_less_imp_not_eq2 mult_ac) 
   30.41      moreover
   30.42      have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
   30.43        by (rule mult_mono, 
   30.44 @@ -751,11 +750,11 @@
   30.45    have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)  
   30.46    from Gleason9_34_exists [OF A closed k]
   30.47    obtain z where z: "z \<in> A" 
   30.48 -             and mem: "z + rat ?k * Fract c d \<in> A" ..
   30.49 -  have less: "z + rat ?k * Fract c d < Fract a b"
   30.50 +             and mem: "z + of_int ?k * Fract c d \<in> A" ..
   30.51 +  have less: "z + of_int ?k * Fract c d < Fract a b"
   30.52      by (rule not_in_preal_ub [OF A notin mem ypos])
   30.53    have "0<z" by (rule preal_imp_pos [OF A z])
   30.54 -  with frle and less show False by arith 
   30.55 +  with frle and less show False by (simp add: Fract_of_int_eq) 
   30.56  qed
   30.57  
   30.58  
    31.1 --- a/src/HOL/Real/RatArith.thy	Thu Feb 05 10:45:28 2004 +0100
    31.2 +++ b/src/HOL/Real/RatArith.thy	Tue Feb 10 12:02:11 2004 +0100
    31.3 @@ -13,15 +13,11 @@
    31.4  
    31.5  instance rat :: number ..
    31.6  
    31.7 -defs
    31.8 +defs (overloaded)
    31.9    rat_number_of_def:
   31.10 -    "number_of v == Fract (number_of v) 1"
   31.11 +    "(number_of v :: rat) == of_int (number_of v)"
   31.12       (*::bin=>rat         ::bin=>int*)
   31.13  
   31.14 -theorem number_of_rat: "number_of b = rat (number_of b)"
   31.15 -  by (simp add: rat_number_of_def rat_def)
   31.16 -
   31.17 -declare number_of_rat [symmetric, simp]
   31.18  
   31.19  lemma rat_numeral_0_eq_0: "Numeral0 = (0::rat)"
   31.20  by (simp add: rat_number_of_def zero_rat [symmetric])
   31.21 @@ -33,25 +29,26 @@
   31.22  subsection{*Arithmetic Operations On Numerals*}
   31.23  
   31.24  lemma add_rat_number_of [simp]:
   31.25 -     "(number_of v :: rat) + number_of v' = number_of (bin_add v v')"
   31.26 -by (simp add: rat_number_of_def add_rat)
   31.27 +     "(number_of v :: rat) + number_of v' = number_of (bin_add v v')" 
   31.28 +by (simp only: rat_number_of_def of_int_add number_of_add)
   31.29  
   31.30  lemma minus_rat_number_of [simp]:
   31.31       "- (number_of w :: rat) = number_of (bin_minus w)"
   31.32 -by (simp add: rat_number_of_def minus_rat)
   31.33 +by (simp only: rat_number_of_def of_int_minus number_of_minus)
   31.34  
   31.35  lemma diff_rat_number_of [simp]: 
   31.36     "(number_of v :: rat) - number_of w = number_of (bin_add v (bin_minus w))"
   31.37 -by (simp add: rat_number_of_def diff_rat)
   31.38 +by (simp only: add_rat_number_of minus_rat_number_of diff_minus)
   31.39  
   31.40  lemma mult_rat_number_of [simp]:
   31.41       "(number_of v :: rat) * number_of v' = number_of (bin_mult v v')"
   31.42 -by (simp add: rat_number_of_def mult_rat)
   31.43 +by (simp only: rat_number_of_def of_int_mult number_of_mult)
   31.44  
   31.45  text{*Lemmas for specialist use, NOT as default simprules*}
   31.46  lemma rat_mult_2: "2 * z = (z+z::rat)"
   31.47  proof -
   31.48 -  have eq: "(2::rat) = 1 + 1" by (simp add: rat_numeral_1_eq_1 [symmetric])
   31.49 +  have eq: "(2::rat) = 1 + 1"
   31.50 +    by (simp del: rat_number_of_def add: rat_numeral_1_eq_1 [symmetric])
   31.51    thus ?thesis by (simp add: eq left_distrib)
   31.52  qed
   31.53  
   31.54 @@ -63,20 +60,20 @@
   31.55  
   31.56  lemma eq_rat_number_of [simp]:
   31.57       "((number_of v :: rat) = number_of v') =  
   31.58 -      iszero (number_of (bin_add v (bin_minus v')))"
   31.59 -by (simp add: rat_number_of_def eq_rat)
   31.60 +      iszero (number_of (bin_add v (bin_minus v')) :: int)"
   31.61 +by (simp add: rat_number_of_def)
   31.62  
   31.63  text{*@{term neg} is used in rewrite rules for binary comparisons*}
   31.64  lemma less_rat_number_of [simp]:
   31.65       "((number_of v :: rat) < number_of v') =  
   31.66 -      neg (number_of (bin_add v (bin_minus v')))"
   31.67 -by (simp add: rat_number_of_def less_rat)
   31.68 +      neg (number_of (bin_add v (bin_minus v')) :: int)"
   31.69 +by (simp add: rat_number_of_def)
   31.70  
   31.71  
   31.72  text{*New versions of existing theorems involving 0, 1*}
   31.73  
   31.74  lemma rat_minus_1_eq_m1 [simp]: "- 1 = (-1::rat)"
   31.75 -by (simp add: rat_numeral_1_eq_1 [symmetric])
   31.76 +by (simp del: rat_number_of_def add: rat_numeral_1_eq_1 [symmetric])
   31.77  
   31.78  lemma rat_mult_minus1 [simp]: "-1 * z = -(z::rat)"
   31.79  proof -
   31.80 @@ -143,13 +140,15 @@
   31.81  
   31.82  lemma abs_nat_number_of [simp]: 
   31.83       "abs (number_of v :: rat) =  
   31.84 -        (if neg (number_of v) then number_of (bin_minus v)  
   31.85 +        (if neg (number_of v :: int)  then number_of (bin_minus v)  
   31.86           else number_of v)"
   31.87 -by (simp add: abs_if)
   31.88 +by (simp add: abs_if) 
   31.89  
   31.90  lemma abs_minus_one [simp]: "abs (-1) = (1::rat)"
   31.91  by (simp add: abs_if)
   31.92  
   31.93 +declare rat_number_of_def [simp]
   31.94 +
   31.95  
   31.96  ML
   31.97  {*
    32.1 --- a/src/HOL/Real/Rational.thy	Thu Feb 05 10:45:28 2004 +0100
    32.2 +++ b/src/HOL/Real/Rational.thy	Tue Feb 10 12:02:11 2004 +0100
    32.3 @@ -465,11 +465,11 @@
    32.4  
    32.5  theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
    32.6      (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
    32.7 -  by (simp add: less_rat_def le_rat eq_rat int_less_le)
    32.8 +  by (simp add: less_rat_def le_rat eq_rat order_less_le)
    32.9  
   32.10  theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
   32.11    by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
   32.12 -     (auto simp add: mult_less_0_iff zero_less_mult_iff int_le_less 
   32.13 +     (auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less 
   32.14                  split: abs_split)
   32.15  
   32.16  
   32.17 @@ -619,7 +619,7 @@
   32.18      proof -
   32.19        let ?E = "e * f" and ?F = "f * f"
   32.20        from neq gt have "0 < ?E"
   32.21 -        by (auto simp add: zero_rat less_rat le_rat int_less_le eq_rat)
   32.22 +        by (auto simp add: zero_rat less_rat le_rat order_less_le eq_rat)
   32.23        moreover from neq have "0 < ?F"
   32.24          by (auto simp add: zero_less_mult_iff)
   32.25        moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
   32.26 @@ -642,72 +642,6 @@
   32.27  qed
   32.28  
   32.29  
   32.30 -subsection {* Embedding integers: The Injection @{term rat} *}
   32.31 -
   32.32 -constdefs
   32.33 -  rat :: "int => rat"    (* FIXME generalize int to any numeric subtype (?) *)
   32.34 -  "rat z == Fract z 1"
   32.35 -  int_set :: "rat set"    ("\<int>")    (* FIXME generalize rat to any numeric supertype (?) *)
   32.36 -  "\<int> == range rat"
   32.37 -
   32.38 -lemma int_set_cases [case_names rat, cases set: int_set]:
   32.39 -  "q \<in> \<int> ==> (!!z. q = rat z ==> C) ==> C"
   32.40 -proof (unfold int_set_def)
   32.41 -  assume "!!z. q = rat z ==> C"
   32.42 -  assume "q \<in> range rat" thus C ..
   32.43 -qed
   32.44 -
   32.45 -lemma int_set_induct [case_names rat, induct set: int_set]:
   32.46 -  "q \<in> \<int> ==> (!!z. P (rat z)) ==> P q"
   32.47 -  by (rule int_set_cases) auto
   32.48 -
   32.49 -lemma rat_of_int_zero [simp]: "rat (0::int) = (0::rat)"
   32.50 -by (simp add: rat_def zero_rat [symmetric])
   32.51 -
   32.52 -lemma rat_of_int_one [simp]: "rat (1::int) = (1::rat)"
   32.53 -by (simp add: rat_def one_rat [symmetric])
   32.54 -
   32.55 -lemma rat_of_int_add_distrib [simp]: "rat (x + y) = rat (x::int) + rat y"
   32.56 -by (simp add: rat_def add_rat)
   32.57 -
   32.58 -lemma rat_of_int_minus_distrib [simp]: "rat (-x) = -rat (x::int)"
   32.59 -by (simp add: rat_def minus_rat)
   32.60 -
   32.61 -lemma rat_of_int_diff_distrib [simp]: "rat (x - y) = rat (x::int) - rat y"
   32.62 -by (simp add: rat_def diff_rat)
   32.63 -
   32.64 -lemma rat_of_int_mult_distrib [simp]: "rat (x * y) = rat (x::int) * rat y"
   32.65 -by (simp add: rat_def mult_rat)
   32.66 -
   32.67 -lemma rat_inject [simp]: "(rat z = rat w) = (z = w)"
   32.68 -proof
   32.69 -  assume "rat z = rat w"
   32.70 -  hence "Fract z 1 = Fract w 1" by (unfold rat_def)
   32.71 -  hence "\<lfloor>fract z 1\<rfloor> = \<lfloor>fract w 1\<rfloor>" ..
   32.72 -  thus "z = w" by auto
   32.73 -next
   32.74 -  assume "z = w"
   32.75 -  thus "rat z = rat w" by simp
   32.76 -qed
   32.77 -
   32.78 -
   32.79 -lemma rat_of_int_zero_cancel [simp]: "(rat x = 0) = (x = 0)"
   32.80 -proof -
   32.81 -  have "(rat x = 0) = (rat x = rat 0)" by simp
   32.82 -  also have "... = (x = 0)" by (rule rat_inject)
   32.83 -  finally show ?thesis .
   32.84 -qed
   32.85 -
   32.86 -lemma rat_of_int_less_iff [simp]: "rat (x::int) < rat y = (x < y)"
   32.87 -by (simp add: rat_def less_rat) 
   32.88 -
   32.89 -lemma rat_of_int_le_iff [simp]: "(rat (x::int) \<le> rat y) = (x \<le> y)"
   32.90 -by (simp add: linorder_not_less [symmetric])
   32.91 -
   32.92 -lemma zero_less_rat_of_int_iff [simp]: "(0 < rat y) = (0 < y)"
   32.93 -by (insert rat_of_int_less_iff [of 0 y], simp)
   32.94 -
   32.95 -
   32.96  subsection {* Various Other Results *}
   32.97  
   32.98  lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
   32.99 @@ -733,4 +667,30 @@
  32.100       "0 < b ==> (0 < Fract a b) = (0 < a)"
  32.101  by (simp add: zero_rat less_rat order_less_imp_not_eq2 zero_less_mult_iff) 
  32.102  
  32.103 +lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
  32.104 +apply (insert add_rat [of concl: m n 1 1])
  32.105 +apply (simp add: one_rat  [symmetric]) 
  32.106 +done
  32.107 +
  32.108 +lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
  32.109 +apply (induct k) 
  32.110 +apply (simp add: zero_rat) 
  32.111 +apply (simp add: Fract_add_one) 
  32.112 +done
  32.113 +
  32.114 +lemma Fract_of_int_eq: "Fract k 1 = of_int k"
  32.115 +proof (cases k rule: int_cases) 
  32.116 +  case (nonneg n)
  32.117 +    thus ?thesis by (simp add: int_eq_of_nat Fract_of_nat_eq)
  32.118 +next 
  32.119 +  case (neg n)
  32.120 +    hence "Fract k 1 = - (Fract (of_nat (Suc n)) 1)"
  32.121 +      by (simp only: minus_rat int_eq_of_nat) 
  32.122 +    also have "... =  - (of_nat (Suc n))"
  32.123 +      by (simp only: Fract_of_nat_eq)
  32.124 +    finally show ?thesis 
  32.125 +      by (simp add: only: prems int_eq_of_nat of_int_minus of_int_of_nat_eq) 
  32.126 +qed 
  32.127 +
  32.128 +
  32.129  end
    33.1 --- a/src/HOL/Real/RealArith.thy	Thu Feb 05 10:45:28 2004 +0100
    33.2 +++ b/src/HOL/Real/RealArith.thy	Tue Feb 10 12:02:11 2004 +0100
    33.3 @@ -62,13 +62,13 @@
    33.4  
    33.5  lemma eq_real_number_of [simp]:
    33.6       "((number_of v :: real) = number_of v') =  
    33.7 -      iszero (number_of (bin_add v (bin_minus v')))"
    33.8 +      iszero (number_of (bin_add v (bin_minus v')) :: int)"
    33.9  by (simp only: real_number_of_def real_of_int_inject eq_number_of_eq)
   33.10  
   33.11  text{*@{term neg} is used in rewrite rules for binary comparisons*}
   33.12  lemma less_real_number_of [simp]:
   33.13       "((number_of v :: real) < number_of v') =  
   33.14 -      neg (number_of (bin_add v (bin_minus v')))"
   33.15 +      neg (number_of (bin_add v (bin_minus v')) :: int)"
   33.16  by (simp only: real_number_of_def real_of_int_less_iff less_number_of_eq_neg)
   33.17  
   33.18  
   33.19 @@ -134,12 +134,12 @@
   33.20  have. *}
   33.21  lemma real_of_nat_number_of [simp]:
   33.22       "real (number_of v :: nat) =  
   33.23 -        (if neg (number_of v) then 0  
   33.24 +        (if neg (number_of v :: int) then 0  
   33.25           else (number_of v :: real))"
   33.26  proof cases
   33.27 -  assume "neg (number_of v)" thus ?thesis by simp
   33.28 +  assume "neg (number_of v :: int)" thus ?thesis by simp
   33.29  next
   33.30 -  assume neg: "~ neg (number_of v)"
   33.31 +  assume neg: "~ neg (number_of v :: int)"
   33.32    thus ?thesis
   33.33      by (simp only: nat_number_of_def real_of_nat_real_of_int [OF neg], simp) 
   33.34  qed
   33.35 @@ -222,7 +222,7 @@
   33.36  
   33.37  lemma abs_nat_number_of [simp]: 
   33.38       "abs (number_of v :: real) =  
   33.39 -        (if neg (number_of v) then number_of (bin_minus v)  
   33.40 +        (if neg (number_of v :: int) then number_of (bin_minus v)  
   33.41           else number_of v)"
   33.42  by (simp add: real_abs_def bin_arith_simps minus_real_number_of
   33.43         less_real_number_of real_of_int_le_iff)
    34.1 --- a/src/HOL/Real/RealDef.thy	Thu Feb 05 10:45:28 2004 +0100
    34.2 +++ b/src/HOL/Real/RealDef.thy	Tue Feb 10 12:02:11 2004 +0100
    34.3 @@ -23,9 +23,8 @@
    34.4  instance real :: inverse ..
    34.5  
    34.6  consts
    34.7 -   (*Overloaded constants denoting the Nat and Real subsets of enclosing
    34.8 +   (*Overloaded constant denoting the Real subset of enclosing
    34.9       types such as hypreal and complex*)
   34.10 -   Nats  :: "'a set"
   34.11     Reals :: "'a set"
   34.12  
   34.13     (*overloaded constant for injecting other types into "real"*)
   34.14 @@ -85,16 +84,6 @@
   34.15  
   34.16  syntax (xsymbols)
   34.17    Reals     :: "'a set"                   ("\<real>")
   34.18 -  Nats      :: "'a set"                   ("\<nat>")
   34.19 -
   34.20 -
   34.21 -defs (overloaded)
   34.22 -  real_of_int_def:
   34.23 -   "real z == Abs_REAL(\<Union>(i,j) \<in> Rep_Integ z. realrel ``
   34.24 -		       {(preal_of_rat(rat(int(Suc i))),
   34.25 -			 preal_of_rat(rat(int(Suc j))))})"
   34.26 -
   34.27 -  real_of_nat_def:   "real n == real (int n)"
   34.28  
   34.29  
   34.30  subsection{*Proving that realrel is an equivalence relation*}
   34.31 @@ -172,30 +161,6 @@
   34.32  apply (simp add: Rep_REAL_inverse)
   34.33  done
   34.34  
   34.35 -lemma real_eq_iff:
   34.36 -     "[|(x1,y1) \<in> Rep_REAL w; (x2,y2) \<in> Rep_REAL z|] 
   34.37 -      ==> (z = w) = (x1+y2 = x2+y1)"
   34.38 -apply (insert quotient_eq_iff
   34.39 -                [OF equiv_realrel, 
   34.40 -                 of "Rep_REAL w" "Rep_REAL z" "(x1,y1)" "(x2,y2)"])
   34.41 -apply (simp add: Rep_REAL [unfolded REAL_def] Rep_REAL_inject eq_commute) 
   34.42 -done 
   34.43 -
   34.44 -lemma mem_REAL_imp_eq:
   34.45 -     "[|R \<in> REAL; (x1,y1) \<in> R; (x2,y2) \<in> R|] ==> x1+y2 = x2+y1" 
   34.46 -apply (auto simp add: REAL_def realrel_def quotient_def)
   34.47 -apply (blast dest: preal_trans_lemma) 
   34.48 -done
   34.49 -
   34.50 -lemma Rep_REAL_cancel_right:
   34.51 -     "((x + z, y + z) \<in> Rep_REAL R) = ((x, y) \<in> Rep_REAL R)"
   34.52 -apply (rule_tac z = R in eq_Abs_REAL, simp) 
   34.53 -apply (rename_tac u v) 
   34.54 -apply (subgoal_tac "(u + (y + z) = x + z + v) = ((u + y) + z = (x + v) + z)")
   34.55 - prefer 2 apply (simp add: preal_add_ac) 
   34.56 -apply (simp add: preal_add_right_cancel_iff) 
   34.57 -done
   34.58 -
   34.59  
   34.60  subsection{*Congruence property for addition*}
   34.61  
   34.62 @@ -218,21 +183,21 @@
   34.63  done
   34.64  
   34.65  lemma real_add_commute: "(z::real) + w = w + z"
   34.66 -apply (rule_tac z = z in eq_Abs_REAL)
   34.67 -apply (rule_tac z = w in eq_Abs_REAL)
   34.68 +apply (rule eq_Abs_REAL [of z])
   34.69 +apply (rule eq_Abs_REAL [of w])
   34.70  apply (simp add: preal_add_ac real_add)
   34.71  done
   34.72  
   34.73  lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
   34.74 -apply (rule_tac z = z1 in eq_Abs_REAL)
   34.75 -apply (rule_tac z = z2 in eq_Abs_REAL)
   34.76 -apply (rule_tac z = z3 in eq_Abs_REAL)
   34.77 +apply (rule eq_Abs_REAL [of z1])
   34.78 +apply (rule eq_Abs_REAL [of z2])
   34.79 +apply (rule eq_Abs_REAL [of z3])
   34.80  apply (simp add: real_add preal_add_assoc)
   34.81  done
   34.82  
   34.83  lemma real_add_zero_left: "(0::real) + z = z"
   34.84  apply (unfold real_of_preal_def real_zero_def)
   34.85 -apply (rule_tac z = z in eq_Abs_REAL)
   34.86 +apply (rule eq_Abs_REAL [of z])
   34.87  apply (simp add: real_add preal_add_ac)
   34.88  done
   34.89  
   34.90 @@ -263,7 +228,7 @@
   34.91  
   34.92  lemma real_add_minus_left: "(-z) + z = (0::real)"
   34.93  apply (unfold real_zero_def)
   34.94 -apply (rule_tac z = z in eq_Abs_REAL)
   34.95 +apply (rule eq_Abs_REAL [of z])
   34.96  apply (simp add: real_minus real_add preal_add_commute)
   34.97  done
   34.98  
   34.99 @@ -283,7 +248,7 @@
  34.100  
  34.101  lemma real_mult_congruent2:
  34.102      "congruent2 realrel (%p1 p2.
  34.103 -          (%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"
  34.104 +        (%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"
  34.105  apply (rule equiv_realrel [THEN congruent2_commuteI], clarify)
  34.106  apply (unfold split_def)
  34.107  apply (simp add: preal_mult_commute preal_add_commute)
  34.108 @@ -298,29 +263,29 @@
  34.109  done
  34.110  
  34.111  lemma real_mult_commute: "(z::real) * w = w * z"
  34.112 -apply (rule_tac z = z in eq_Abs_REAL)
  34.113 -apply (rule_tac z = w in eq_Abs_REAL)
  34.114 +apply (rule eq_Abs_REAL [of z])
  34.115 +apply (rule eq_Abs_REAL [of w])
  34.116  apply (simp add: real_mult preal_add_ac preal_mult_ac)
  34.117  done
  34.118  
  34.119  lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
  34.120 -apply (rule_tac z = z1 in eq_Abs_REAL)
  34.121 -apply (rule_tac z = z2 in eq_Abs_REAL)
  34.122 -apply (rule_tac z = z3 in eq_Abs_REAL)
  34.123 +apply (rule eq_Abs_REAL [of z1])
  34.124 +apply (rule eq_Abs_REAL [of z2])
  34.125 +apply (rule eq_Abs_REAL [of z3])
  34.126  apply (simp add: preal_add_mult_distrib2 real_mult preal_add_ac preal_mult_ac)
  34.127  done
  34.128  
  34.129  lemma real_mult_1: "(1::real) * z = z"
  34.130  apply (unfold real_one_def)
  34.131 -apply (rule_tac z = z in eq_Abs_REAL)
  34.132 +apply (rule eq_Abs_REAL [of z])
  34.133  apply (simp add: real_mult preal_add_mult_distrib2 preal_mult_1_right
  34.134                   preal_mult_ac preal_add_ac)
  34.135  done
  34.136  
  34.137  lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
  34.138 -apply (rule_tac z = z1 in eq_Abs_REAL)
  34.139 -apply (rule_tac z = z2 in eq_Abs_REAL)
  34.140 -apply (rule_tac z = w in eq_Abs_REAL)
  34.141 +apply (rule eq_Abs_REAL [of z1])
  34.142 +apply (rule eq_Abs_REAL [of z2])
  34.143 +apply (rule eq_Abs_REAL [of w])
  34.144  apply (simp add: preal_add_mult_distrib2 real_add real_mult preal_add_ac preal_mult_ac)
  34.145  done
  34.146  
  34.147 @@ -344,7 +309,7 @@
  34.148  
  34.149  lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
  34.150  apply (unfold real_zero_def real_one_def)
  34.151 -apply (rule_tac z = x in eq_Abs_REAL)
  34.152 +apply (rule eq_Abs_REAL [of x])
  34.153  apply (cut_tac x = xa and y = y in linorder_less_linear)
  34.154  apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
  34.155  apply (rule_tac
  34.156 @@ -420,63 +385,69 @@
  34.157  subsection{*The @{text "\<le>"} Ordering*}
  34.158  
  34.159  lemma real_le_refl: "w \<le> (w::real)"
  34.160 -apply (rule_tac z = w in eq_Abs_REAL)
  34.161 +apply (rule eq_Abs_REAL [of w])
  34.162  apply (force simp add: real_le_def)
  34.163  done
  34.164  
  34.165 -lemma real_le_trans_lemma:
  34.166 -  assumes le1: "x1 + y2 \<le> x2 + y1"
  34.167 -      and le2: "u1 + v2 \<le> u2 + v1"
  34.168 -      and eq: "x2 + v1 = u1 + y2"
  34.169 -  shows "x1 + v2 + u1 + y2 \<le> u2 + u1 + y2 + (y1::preal)"
  34.170 +text{*The arithmetic decision procedure is not set up for type preal.
  34.171 +  This lemma is currently unused, but it could simplify the proofs of the
  34.172 +  following two lemmas.*}
  34.173 +lemma preal_eq_le_imp_le:
  34.174 +  assumes eq: "a+b = c+d" and le: "c \<le> a"
  34.175 +  shows "b \<le> (d::preal)"
  34.176 +proof -
  34.177 +  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
  34.178 +  hence "a+b \<le> a+d" by (simp add: prems)
  34.179 +  thus "b \<le> d" by (simp add: preal_cancels)
  34.180 +qed
  34.181 +
  34.182 +lemma real_le_lemma:
  34.183 +  assumes l: "u1 + v2 \<le> u2 + v1"
  34.184 +      and "x1 + v1 = u1 + y1"
  34.185 +      and "x2 + v2 = u2 + y2"
  34.186 +  shows "x1 + y2 \<le> x2 + (y1::preal)"
  34.187  proof -
  34.188 -  have "x1 + v2 + u1 + y2 = (x1 + y2) + (u1 + v2)" by (simp add: preal_add_ac)
  34.189 -  also have "... \<le> (x2 + y1) + (u1 + v2)"
  34.190 -         by (simp add: prems preal_add_le_cancel_right)
  34.191 -  also have "... \<le> (x2 + y1) + (u2 + v1)"
  34.192 +  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
  34.193 +  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
  34.194 +  also have "... \<le> (x2+y1) + (u2+v1)"
  34.195           by (simp add: prems preal_add_le_cancel_left)
  34.196 -  also have "... = (x2 + v1) + (u2 + y1)" by (simp add: preal_add_ac)
  34.197 -  also have "... = (u1 + y2) + (u2 + y1)" by (simp add: prems)
  34.198 -  also have "... = u2 + u1 + y2 + y1" by (simp add: preal_add_ac)
  34.199 -  finally show ?thesis .
  34.200 +  finally show ?thesis by (simp add: preal_add_le_cancel_right)
  34.201 +qed						 
  34.202 +
  34.203 +lemma real_le: 
  34.204 +  "(Abs_REAL(realrel``{(x1,y1)}) \<le> Abs_REAL(realrel``{(x2,y2)})) =  
  34.205 +   (x1 + y2 \<le> x2 + y1)"
  34.206 +apply (simp add: real_le_def) 
  34.207 +apply (auto intro: real_le_lemma);
  34.208 +done
  34.209 +
  34.210 +lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
  34.211 +apply (rule eq_Abs_REAL [of z])
  34.212 +apply (rule eq_Abs_REAL [of w])
  34.213 +apply (simp add: real_le order_antisym) 
  34.214 +done
  34.215 +
  34.216 +lemma real_trans_lemma:
  34.217 +  assumes "x + v \<le> u + y"
  34.218 +      and "u + v' \<le> u' + v"
  34.219 +      and "x2 + v2 = u2 + y2"
  34.220 +  shows "x + v' \<le> u' + (y::preal)"
  34.221 +proof -
  34.222 +  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
  34.223 +  also have "... \<le> (u+y) + (u+v')" 
  34.224 +    by (simp add: preal_add_le_cancel_right prems) 
  34.225 +  also have "... \<le> (u+y) + (u'+v)" 
  34.226 +    by (simp add: preal_add_le_cancel_left prems) 
  34.227 +  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
  34.228 +  finally show ?thesis by (simp add: preal_add_le_cancel_right)
  34.229  qed						 
  34.230  
  34.231  lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
  34.232 -apply (simp add: real_le_def, clarify)
  34.233 -apply (rename_tac x1 u1 y1 v1 x2 u2 y2 v2) 
  34.234 -apply (drule mem_REAL_imp_eq [OF Rep_REAL], assumption)  
  34.235 -apply (rule_tac x=x1 in exI) 
  34.236 -apply (rule_tac x=y1 in exI) 
  34.237 -apply (rule_tac x="u2 + (x2 + v1)" in exI) 
  34.238 -apply (rule_tac x="v2 + (u1 + y2)" in exI) 
  34.239 -apply (simp add: Rep_REAL_cancel_right preal_add_le_cancel_right 
  34.240 -                 preal_add_assoc [symmetric] real_le_trans_lemma)
  34.241 -done
  34.242 -
  34.243 -lemma real_le_anti_sym_lemma: 
  34.244 -  assumes le1: "x1 + y2 \<le> x2 + y1"
  34.245 -      and le2: "u1 + v2 \<le> u2 + v1"
  34.246 -      and eq1: "x1 + v2 = u2 + y1"
  34.247 -      and eq2: "x2 + v1 = u1 + y2"
  34.248 -  shows "x2 + y1 = x1 + (y2::preal)"
  34.249 -proof (rule order_antisym)
  34.250 -  show "x1 + y2 \<le> x2 + y1" .
  34.251 -  have "(x2 + y1) + (u1+u2) = x2 + u1 + (u2 + y1)" by (simp add: preal_add_ac)
  34.252 -  also have "... = x2 + u1 + (x1 + v2)" by (simp add: prems)
  34.253 -  also have "... = (x2 + x1) + (u1 + v2)" by (simp add: preal_add_ac)
  34.254 -  also have "... \<le> (x2 + x1) + (u2 + v1)" 
  34.255 -                                  by (simp add: preal_add_le_cancel_left)
  34.256 -  also have "... = (x1 + u2) + (x2 + v1)" by (simp add: preal_add_ac)
  34.257 -  also have "... = (x1 + u2) + (u1 + y2)" by (simp add: prems)
  34.258 -  also have "... = (x1 + y2) + (u1 + u2)" by (simp add: preal_add_ac)
  34.259 -  finally show "x2 + y1 \<le> x1 + y2" by (simp add: preal_add_le_cancel_right)
  34.260 -qed  
  34.261 -
  34.262 -lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
  34.263 -apply (simp add: real_le_def, clarify) 
  34.264 -apply (rule real_eq_iff [THEN iffD2], assumption+)
  34.265 -apply (drule mem_REAL_imp_eq [OF Rep_REAL], assumption)+
  34.266 -apply (blast intro: real_le_anti_sym_lemma) 
  34.267 +apply (rule eq_Abs_REAL [of i])
  34.268 +apply (rule eq_Abs_REAL [of j])
  34.269 +apply (rule eq_Abs_REAL [of k])
  34.270 +apply (simp add: real_le) 
  34.271 +apply (blast intro: real_trans_lemma) 
  34.272  done
  34.273  
  34.274  (* Axiom 'order_less_le' of class 'order': *)
  34.275 @@ -488,124 +459,50 @@
  34.276   (assumption |
  34.277    rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
  34.278  
  34.279 -text{*Simplifies a strange formula that occurs quantified.*}
  34.280 -lemma preal_strange_le_eq: "(x1 + x2 \<le> x2 + y1) = (x1 \<le> (y1::preal))"
  34.281 -by (simp add: preal_add_commute [of x1] preal_add_le_cancel_left) 
  34.282 -
  34.283 -text{*This is the nicest way to prove linearity*}
  34.284 -lemma real_le_linear_0: "(z::real) \<le> 0 | 0 \<le> z"
  34.285 -apply (rule_tac z = z in eq_Abs_REAL)
  34.286 -apply (auto simp add: real_le_def real_zero_def preal_add_ac 
  34.287 -                      preal_cancels preal_strange_le_eq)
  34.288 -apply (cut_tac x=x and y=y in linorder_linear, auto) 
  34.289 -done
  34.290 -
  34.291 -lemma real_minus_zero_le_iff: "(0 \<le> -R) = (R \<le> (0::real))"
  34.292 -apply (rule_tac z = R in eq_Abs_REAL)
  34.293 -apply (force simp add: real_le_def real_zero_def real_minus preal_add_ac 
  34.294 -                       preal_cancels preal_strange_le_eq)
  34.295 +(* Axiom 'linorder_linear' of class 'linorder': *)
  34.296 +lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
  34.297 +apply (rule eq_Abs_REAL [of z])
  34.298 +apply (rule eq_Abs_REAL [of w]) 
  34.299 +apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
  34.300 +apply (cut_tac x="x+ya" and y="xa+y" in linorder_linear) 
  34.301 +apply (auto ); 
  34.302  done
  34.303  
  34.304 -lemma real_le_imp_diff_le_0: "x \<le> y ==> x-y \<le> (0::real)"
  34.305 -apply (rule_tac z = x in eq_Abs_REAL)
  34.306 -apply (rule_tac z = y in eq_Abs_REAL)
  34.307 -apply (auto simp add: real_le_def real_zero_def real_diff_def real_minus 
  34.308 -    real_add preal_add_ac preal_cancels preal_strange_le_eq)
  34.309 -apply (rule exI)+
  34.310 -apply (rule conjI, assumption)
  34.311 -apply (subgoal_tac " x + (x2 + y1 + ya) = (x + y1) + (x2 + ya)")
  34.312 - prefer 2 apply (simp (no_asm) only: preal_add_ac) 
  34.313 -apply (subgoal_tac "x1 + y2 + (xa + y) = (x1 + y) + (xa + y2)")
  34.314 - prefer 2 apply (simp (no_asm) only: preal_add_ac) 
  34.315 -apply simp 
  34.316 -done
  34.317 -
  34.318 -lemma real_diff_le_0_imp_le: "x-y \<le> (0::real) ==> x \<le> y"
  34.319 -apply (rule_tac z = x in eq_Abs_REAL)
  34.320 -apply (rule_tac z = y in eq_Abs_REAL)
  34.321 -apply (auto simp add: real_le_def real_zero_def real_diff_def real_minus 
  34.322 -    real_add preal_add_ac preal_cancels preal_strange_le_eq)
  34.323 -apply (rule exI)+
  34.324 -apply (rule conjI, rule_tac [2] conjI)
  34.325 - apply (rule_tac [2] refl)+
  34.326 -apply (subgoal_tac "(x + ya) + (x1 + y1) \<le> (xa + y) + (x1 + y1)") 
  34.327 -apply (simp add: preal_cancels)
  34.328 -apply (subgoal_tac "x1 + (x + (y1 + ya)) \<le> y1 + (x1 + (xa + y))")
  34.329 - apply (simp add: preal_add_ac) 
  34.330 -apply (simp add: preal_cancels)
  34.331 -done
  34.332 -
  34.333 -lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
  34.334 -by (blast intro!: real_diff_le_0_imp_le real_le_imp_diff_le_0)
  34.335 -
  34.336 -
  34.337 -(* Axiom 'linorder_linear' of class 'linorder': *)
  34.338 -lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
  34.339 -apply (insert real_le_linear_0 [of "z-w"])
  34.340 -apply (auto simp add: real_le_eq_diff [of w] real_le_eq_diff [of z] 
  34.341 -                      real_minus_zero_le_iff [symmetric])
  34.342 -done
  34.343  
  34.344  instance real :: linorder
  34.345    by (intro_classes, rule real_le_linear)
  34.346  
  34.347  
  34.348 +lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
  34.349 +apply (rule eq_Abs_REAL [of x])
  34.350 +apply (rule eq_Abs_REAL [of y]) 
  34.351 +apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
  34.352 +                      preal_add_ac)
  34.353 +apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
  34.354 +done 
  34.355 +
  34.356  lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)"
  34.357  apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"])
  34.358  apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)")
  34.359   prefer 2 apply (simp add: diff_minus add_ac, simp) 
  34.360  done
  34.361  
  34.362 -
  34.363 -lemma real_minus_zero_le_iff2: "(-R \<le> 0) = (0 \<le> (R::real))"
  34.364 -apply (rule_tac z = R in eq_Abs_REAL)
  34.365 -apply (force simp add: real_le_def real_zero_def real_minus preal_add_ac 
  34.366 -                       preal_cancels preal_strange_le_eq)
  34.367 -done
  34.368 -
  34.369 -lemma real_minus_zero_less_iff: "(0 < -R) = (R < (0::real))"
  34.370 -by (simp add: linorder_not_le [symmetric] real_minus_zero_le_iff2) 
  34.371 -
  34.372 -lemma real_minus_zero_less_iff2: "(-R < 0) = ((0::real) < R)"
  34.373 -by (simp add: linorder_not_le [symmetric] real_minus_zero_le_iff) 
  34.374 -
  34.375  lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
  34.376  by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
  34.377  
  34.378 -text{*Used a few times in Lim and Transcendental*}
  34.379  lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
  34.380  by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
  34.381  
  34.382 -text{*Handles other strange cases that arise in these proofs.*}
  34.383 -lemma forall_imp_less: "\<forall>u v. u \<le> v \<longrightarrow> x + v \<noteq> u + (y::preal) ==> y < x";
  34.384 -apply (drule_tac x=x in spec) 
  34.385 -apply (drule_tac x=y in spec) 
  34.386 -apply (simp add: preal_add_commute linorder_not_le) 
  34.387 -done
  34.388 -
  34.389 -text{*The arithmetic decision procedure is not set up for type preal.*}
  34.390 -lemma preal_eq_le_imp_le:
  34.391 -  assumes eq: "a+b = c+d" and le: "c \<le> a"
  34.392 -  shows "b \<le> (d::preal)"
  34.393 -proof -
  34.394 -  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
  34.395 -  hence "a+b \<le> a+d" by (simp add: prems)
  34.396 -  thus "b \<le> d" by (simp add: preal_cancels)
  34.397 -qed
  34.398 -
  34.399  lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
  34.400 -apply (simp add: linorder_not_le [symmetric])
  34.401 +apply (rule eq_Abs_REAL [of x])
  34.402 +apply (rule eq_Abs_REAL [of y])
  34.403 +apply (simp add: linorder_not_le [where 'a = real, symmetric] 
  34.404 +                 linorder_not_le [where 'a = preal] 
  34.405 +                  real_zero_def real_le real_mult)
  34.406    --{*Reduce to the (simpler) @{text "\<le>"} relation *}
  34.407 -apply (rule_tac z = x in eq_Abs_REAL)
  34.408 -apply (rule_tac z = y in eq_Abs_REAL)
  34.409 -apply (auto simp add: real_zero_def real_le_def real_mult preal_add_ac 
  34.410 -                      preal_cancels preal_strange_le_eq)
  34.411 -apply (drule preal_eq_le_imp_le, assumption)
  34.412 -apply (auto  dest!: forall_imp_less less_add_left_Ex 
  34.413 +apply (auto  dest!: less_add_left_Ex 
  34.414       simp add: preal_add_ac preal_mult_ac 
  34.415 -         preal_add_mult_distrib preal_add_mult_distrib2)
  34.416 -apply (insert preal_self_less_add_right)
  34.417 -apply (simp add: linorder_not_le [symmetric])
  34.418 +          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
  34.419  done
  34.420  
  34.421  lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
  34.422 @@ -617,13 +514,11 @@
  34.423  
  34.424  text{*lemma for proving @{term "0<(1::real)"}*}
  34.425  lemma real_zero_le_one: "0 \<le> (1::real)"
  34.426 -apply (auto simp add: real_zero_def real_one_def real_le_def preal_add_ac 
  34.427 -                      preal_cancels)
  34.428 -apply (rule_tac x="preal_of_rat 1 + preal_of_rat 1" in exI) 
  34.429 -apply (rule_tac x="preal_of_rat 1" in exI) 
  34.430 -apply (auto simp add: preal_add_ac preal_self_less_add_left order_less_imp_le)
  34.431 +apply (simp add: real_zero_def real_one_def real_le 
  34.432 +                 preal_self_less_add_left order_less_imp_le)
  34.433  done
  34.434  
  34.435 +
  34.436  subsection{*The Reals Form an Ordered Field*}
  34.437  
  34.438  instance real :: ordered_field
  34.439 @@ -658,7 +553,7 @@
  34.440  lemma real_of_preal_trichotomy:
  34.441        "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
  34.442  apply (unfold real_of_preal_def real_zero_def)
  34.443 -apply (rule_tac z = x in eq_Abs_REAL)
  34.444 +apply (rule eq_Abs_REAL [of x])
  34.445  apply (auto simp add: real_minus preal_add_ac)
  34.446  apply (cut_tac x = x and y = y in linorder_less_linear)
  34.447  apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
  34.448 @@ -824,126 +719,43 @@
  34.449  
  34.450  subsection{*Embedding the Integers into the Reals*}
  34.451  
  34.452 -lemma real_of_int_congruent: 
  34.453 -  "congruent intrel (%p. (%(i,j). realrel ``  
  34.454 -   {(preal_of_rat (rat (int(Suc i))), preal_of_rat (rat (int(Suc j))))}) p)"
  34.455 -apply (simp add: congruent_def add_ac del: int_Suc, clarify)
  34.456 -(*OPTION raised if only is changed to add?????????*)  
  34.457 -apply (simp only: add_Suc_right zero_less_rat_of_int_iff zadd_int
  34.458 -          preal_of_rat_add [symmetric] rat_of_int_add_distrib [symmetric], simp) 
  34.459 -done
  34.460 -
  34.461 -lemma real_of_int: 
  34.462 -   "real (Abs_Integ (intrel `` {(i, j)})) =  
  34.463 -      Abs_REAL(realrel ``  
  34.464 -        {(preal_of_rat (rat (int(Suc i))),  
  34.465 -          preal_of_rat (rat (int(Suc j))))})"
  34.466 -apply (unfold real_of_int_def)
  34.467 -apply (rule_tac f = Abs_REAL in arg_cong)
  34.468 -apply (simp del: int_Suc
  34.469 -            add: realrel_in_real [THEN Abs_REAL_inverse] 
  34.470 -             UN_equiv_class [OF equiv_intrel real_of_int_congruent])
  34.471 -done
  34.472 -
  34.473 -lemma inj_real_of_int: "inj(real :: int => real)"
  34.474 -apply (rule inj_onI)
  34.475 -apply (rule_tac z = x in eq_Abs_Integ)
  34.476 -apply (rule_tac z = y in eq_Abs_Integ, clarify) 
  34.477 -apply (simp del: int_Suc 
  34.478 -            add: real_of_int zadd_int preal_of_rat_eq_iff
  34.479 -               preal_of_rat_add [symmetric] rat_of_int_add_distrib [symmetric])
  34.480 -done
  34.481 -
  34.482 -lemma real_of_int_int_zero: "real (int 0) = 0"  
  34.483 -by (simp add: int_def real_zero_def real_of_int preal_add_commute)
  34.484 +defs (overloaded)
  34.485 +  real_of_nat_def: "real z == of_nat z"
  34.486 +  real_of_int_def: "real z == of_int z"
  34.487  
  34.488  lemma real_of_int_zero [simp]: "real (0::int) = 0"  
  34.489 -by (insert real_of_int_int_zero, simp)
  34.490 +by (simp add: real_of_int_def) 
  34.491  
  34.492  lemma real_of_one [simp]: "real (1::int) = (1::real)"
  34.493 -apply (subst int_1 [symmetric])
  34.494 -apply (simp add: int_def real_one_def)
  34.495 -apply (simp add: real_of_int preal_of_rat_add [symmetric])  
  34.496 -done
  34.497 +by (simp add: real_of_int_def) 
  34.498  
  34.499  lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
  34.500 -apply (rule_tac z = x in eq_Abs_Integ)
  34.501 -apply (rule_tac z = y in eq_Abs_Integ, clarify) 
  34.502 -apply (simp del: int_Suc
  34.503 -            add: pos_add_strict real_of_int real_add zadd
  34.504 -                 preal_of_rat_add [symmetric], simp) 
  34.505 -done
  34.506 +by (simp add: real_of_int_def) 
  34.507  declare real_of_int_add [symmetric, simp]
  34.508  
  34.509  lemma real_of_int_minus: "-real (x::int) = real (-x)"
  34.510 -apply (rule_tac z = x in eq_Abs_Integ)
  34.511 -apply (auto simp add: real_of_int real_minus zminus)
  34.512 -done
  34.513 +by (simp add: real_of_int_def) 
  34.514  declare real_of_int_minus [symmetric, simp]
  34.515  
  34.516  lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
  34.517 -by (simp only: zdiff_def real_diff_def real_of_int_add real_of_int_minus)
  34.518 +by (simp add: real_of_int_def) 
  34.519  declare real_of_int_diff [symmetric, simp]
  34.520  
  34.521  lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
  34.522 -apply (rule_tac z = x in eq_Abs_Integ)
  34.523 -apply (rule_tac z = y in eq_Abs_Integ)
  34.524 -apply (rename_tac a b c d) 
  34.525 -apply (simp del: int_Suc
  34.526 -            add: pos_add_strict mult_pos real_of_int real_mult zmult
  34.527 -                 preal_of_rat_add [symmetric] preal_of_rat_mult [symmetric])
  34.528 -apply (rule_tac f=preal_of_rat in arg_cong) 
  34.529 -apply (simp only: int_Suc right_distrib add_ac mult_ac zadd_int zmult_int
  34.530 -        rat_of_int_add_distrib [symmetric] rat_of_int_mult_distrib [symmetric]
  34.531 -        rat_inject)
  34.532 -done
  34.533 +by (simp add: real_of_int_def) 
  34.534  declare real_of_int_mult [symmetric, simp]
  34.535  
  34.536 -lemma real_of_int_Suc: "real (int (Suc n)) = real (int n) + (1::real)"
  34.537 -by (simp only: real_of_one [symmetric] zadd_int add_ac int_Suc real_of_int_add)
  34.538 -
  34.539  lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
  34.540 -by (auto intro: inj_real_of_int [THEN injD])
  34.541 -
  34.542 -lemma zero_le_real_of_int: "0 \<le> real y ==> 0 \<le> (y::int)"
  34.543 -apply (rule_tac z = y in eq_Abs_Integ)
  34.544 -apply (simp add: real_le_def, clarify)  
  34.545 -apply (rename_tac a b c d) 
  34.546 -apply (simp del: int_Suc zdiff_def [symmetric]
  34.547 -            add: real_zero_def real_of_int zle_def zless_def zdiff_def zadd
  34.548 -                 zminus neg_def preal_add_ac preal_cancels)
  34.549 -apply (drule sym, drule preal_eq_le_imp_le, assumption) 
  34.550 -apply (simp del: int_Suc add: preal_of_rat_le_iff)
  34.551 -done
  34.552 -
  34.553 -lemma real_of_int_le_cancel:
  34.554 -  assumes le: "real (x::int) \<le> real y"
  34.555 -  shows "x \<le> y"
  34.556 -proof -
  34.557 -  have "real x - real x \<le> real y - real x" using le
  34.558 -    by (simp only: diff_minus add_le_cancel_right) 
  34.559 -  hence "0 \<le> real y - real x" by simp
  34.560 -  hence "0 \<le> y - x" by (simp only: real_of_int_diff zero_le_real_of_int) 
  34.561 -  hence "0 + x \<le> (y - x) + x" by (simp only: add_le_cancel_right) 
  34.562 -  thus  "x \<le> y" by simp 
  34.563 -qed
  34.564 +by (simp add: real_of_int_def) 
  34.565  
  34.566  lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
  34.567 -by (blast dest!: inj_real_of_int [THEN injD])
  34.568 -
  34.569 -lemma real_of_int_less_cancel: "real (x::int) < real y ==> x < y"
  34.570 -by (auto simp add: order_less_le real_of_int_le_cancel)
  34.571 -
  34.572 -lemma real_of_int_less_mono: "x < y ==> (real (x::int) < real y)"
  34.573 -apply (simp add: linorder_not_le [symmetric])
  34.574 -apply (auto dest!: real_of_int_less_cancel simp add: order_le_less)
  34.575 -done
  34.576 +by (simp add: real_of_int_def) 
  34.577  
  34.578  lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
  34.579 -by (blast dest: real_of_int_less_cancel intro: real_of_int_less_mono)
  34.580 +by (simp add: real_of_int_def) 
  34.581  
  34.582  lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
  34.583 -by (simp add: linorder_not_less [symmetric])
  34.584 +by (simp add: real_of_int_def) 
  34.585  
  34.586  
  34.587  subsection{*Embedding the Naturals into the Reals*}
  34.588 @@ -955,73 +767,64 @@
  34.589  by (simp add: real_of_nat_def)
  34.590  
  34.591  lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
  34.592 -by (simp add: real_of_nat_def add_ac)
  34.593 +by (simp add: real_of_nat_def)
  34.594  
  34.595  (*Not for addsimps: often the LHS is used to represent a positive natural*)
  34.596  lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
  34.597 -by (simp add: real_of_nat_def add_ac)
  34.598 +by (simp add: real_of_nat_def)
  34.599  
  34.600  lemma real_of_nat_less_iff [iff]: 
  34.601       "(real (n::nat) < real m) = (n < m)"
  34.602  by (simp add: real_of_nat_def)
  34.603  
  34.604  lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
  34.605 -by (simp add: linorder_not_less [symmetric])
  34.606 -
  34.607 -lemma inj_real_of_nat: "inj (real :: nat => real)"
  34.608 -apply (rule inj_onI)
  34.609 -apply (simp add: real_of_nat_def)
  34.610 -done
  34.611 +by (simp add: real_of_nat_def)
  34.612  
  34.613  lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
  34.614 -apply (insert real_of_int_le_iff [of 0 "int n"]) 
  34.615 -apply (simp add: real_of_nat_def) 
  34.616 -done
  34.617 +by (simp add: real_of_nat_def zero_le_imp_of_nat)
  34.618  
  34.619  lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
  34.620 -by (insert real_of_nat_less_iff [of 0 "Suc n"], simp) 
  34.621 +by (simp add: real_of_nat_def del: of_nat_Suc)
  34.622  
  34.623  lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
  34.624 -by (simp add: real_of_nat_def zmult_int [symmetric]) 
  34.625 +by (simp add: real_of_nat_def)
  34.626  
  34.627  lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
  34.628 -by (auto dest: inj_real_of_nat [THEN injD])
  34.629 +by (simp add: real_of_nat_def)
  34.630  
  34.631  lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)"
  34.632 -  proof 
  34.633 -    assume "real n = 0"
  34.634 -    have "real n = real (0::nat)" by simp
  34.635 -    then show "n = 0" by (simp only: real_of_nat_inject)
  34.636 -  next
  34.637 -    show "n = 0 \<Longrightarrow> real n = 0" by simp
  34.638 -  qed
  34.639 +by (simp add: real_of_nat_def)
  34.640  
  34.641  lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
  34.642 -by (simp add: real_of_nat_def zdiff_int [symmetric])
  34.643 -
  34.644 -lemma real_of_nat_neg_int [simp]: "neg z ==> real (nat z) = 0"
  34.645 -by (simp add: neg_nat)
  34.646 +by (simp add: add: real_of_nat_def) 
  34.647  
  34.648  lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
  34.649 -by (rule real_of_nat_less_iff [THEN subst], auto)
  34.650 +by (simp add: add: real_of_nat_def) 
  34.651  
  34.652  lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
  34.653 -apply (rule real_of_nat_zero [THEN subst])
  34.654 -apply (simp only: real_of_nat_le_iff, simp) 
  34.655 -done
  34.656 -
  34.657 +by (simp add: add: real_of_nat_def)
  34.658  
  34.659  lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
  34.660 -by (simp add: linorder_not_less real_of_nat_ge_zero)
  34.661 +by (simp add: add: real_of_nat_def)
  34.662  
  34.663  lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
  34.664 -by (simp add: linorder_not_less)
  34.665 +by (simp add: add: real_of_nat_def)
  34.666  
  34.667  lemma real_of_int_real_of_nat: "real (int n) = real n"
  34.668 -by (simp add: real_of_nat_def)
  34.669 +by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
  34.670 +
  34.671 +
  34.672  
  34.673 +text{*Still needed for binary arith*}
  34.674  lemma real_of_nat_real_of_int: "~neg z ==> real (nat z) = real z"
  34.675 -by (simp add: not_neg_eq_ge_0 real_of_nat_def)
  34.676 +proof (simp add: not_neg_eq_ge_0 real_of_nat_def real_of_int_def)
  34.677 +  assume "0 \<le> z"
  34.678 +  hence eq: "of_nat (nat z) = z" 
  34.679 +    by (simp add: nat_0_le int_eq_of_nat[symmetric]) 
  34.680 +  have "of_nat (nat z) = of_int (of_nat (nat z))" by simp
  34.681 +  also have "... = of_int z" by (simp add: eq)
  34.682 +  finally show "of_nat (nat z) = of_int z" .
  34.683 +qed
  34.684  
  34.685  ML
  34.686  {*
  34.687 @@ -1031,7 +834,6 @@
  34.688  val real_diff_def = thm "real_diff_def";
  34.689  val real_divide_def = thm "real_divide_def";
  34.690  
  34.691 -val preal_trans_lemma = thm"preal_trans_lemma";
  34.692  val realrel_iff = thm"realrel_iff";
  34.693  val realrel_refl = thm"realrel_refl";
  34.694  val equiv_realrel = thm"equiv_realrel";
  34.695 @@ -1099,20 +901,14 @@
  34.696  val real_inverse_unique = thm "real_inverse_unique";
  34.697  val real_inverse_gt_one = thm "real_inverse_gt_one";
  34.698  
  34.699 -val real_of_int = thm"real_of_int";
  34.700 -val inj_real_of_int = thm"inj_real_of_int";
  34.701  val real_of_int_zero = thm"real_of_int_zero";
  34.702  val real_of_one = thm"real_of_one";
  34.703  val real_of_int_add = thm"real_of_int_add";
  34.704  val real_of_int_minus = thm"real_of_int_minus";
  34.705  val real_of_int_diff = thm"real_of_int_diff";
  34.706  val real_of_int_mult = thm"real_of_int_mult";
  34.707 -val real_of_int_Suc = thm"real_of_int_Suc";
  34.708  val real_of_int_real_of_nat = thm"real_of_int_real_of_nat";
  34.709 -val real_of_nat_real_of_int = thm"real_of_nat_real_of_int";
  34.710 -val real_of_int_less_cancel = thm"real_of_int_less_cancel";
  34.711  val real_of_int_inject = thm"real_of_int_inject";
  34.712 -val real_of_int_less_mono = thm"real_of_int_less_mono";
  34.713  val real_of_int_less_iff = thm"real_of_int_less_iff";
  34.714  val real_of_int_le_iff = thm"real_of_int_le_iff";
  34.715  val real_of_nat_zero = thm "real_of_nat_zero";
  34.716 @@ -1121,14 +917,12 @@
  34.717  val real_of_nat_Suc = thm "real_of_nat_Suc";
  34.718  val real_of_nat_less_iff = thm "real_of_nat_less_iff";
  34.719  val real_of_nat_le_iff = thm "real_of_nat_le_iff";
  34.720 -val inj_real_of_nat = thm "inj_real_of_nat";
  34.721  val real_of_nat_ge_zero = thm "real_of_nat_ge_zero";
  34.722  val real_of_nat_Suc_gt_zero = thm "real_of_nat_Suc_gt_zero";
  34.723  val real_of_nat_mult = thm "real_of_nat_mult";
  34.724  val real_of_nat_inject = thm "real_of_nat_inject";
  34.725  val real_of_nat_diff = thm "real_of_nat_diff";
  34.726  val real_of_nat_zero_iff = thm "real_of_nat_zero_iff";
  34.727 -val real_of_nat_neg_int = thm "real_of_nat_neg_int";
  34.728  val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff";
  34.729  val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff";
  34.730  val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero";
    35.1 --- a/src/HOL/Real/rat_arith.ML	Thu Feb 05 10:45:28 2004 +0100
    35.2 +++ b/src/HOL/Real/rat_arith.ML	Tue Feb 10 12:02:11 2004 +0100
    35.3 @@ -19,18 +19,6 @@
    35.4  val rat_number_of_def = thm "rat_number_of_def";
    35.5  val diff_rat_def = thm "diff_rat_def";
    35.6  
    35.7 -val rat_of_int_zero = thm "rat_of_int_zero";
    35.8 -val rat_of_int_one = thm "rat_of_int_one";
    35.9 -val rat_of_int_add_distrib = thm "rat_of_int_add_distrib";
   35.10 -val rat_of_int_minus_distrib = thm "rat_of_int_minus_distrib";
   35.11 -val rat_of_int_diff_distrib = thm "rat_of_int_diff_distrib";
   35.12 -val rat_of_int_mult_distrib = thm "rat_of_int_mult_distrib";
   35.13 -val rat_inject = thm "rat_inject";
   35.14 -val rat_of_int_zero_cancel = thm "rat_of_int_zero_cancel";
   35.15 -val rat_of_int_less_iff = thm "rat_of_int_less_iff";
   35.16 -val rat_of_int_le_iff = thm "rat_of_int_le_iff";
   35.17 -
   35.18 -val number_of_rat = thm "number_of_rat";
   35.19  val rat_numeral_0_eq_0 = thm "rat_numeral_0_eq_0";
   35.20  val rat_numeral_1_eq_1 = thm "rat_numeral_1_eq_1";
   35.21  val add_rat_number_of = thm "add_rat_number_of";
   35.22 @@ -615,9 +603,8 @@
   35.23  val simps = [True_implies_equals,
   35.24               inst "a" "(number_of ?v)" right_distrib,
   35.25               divide_1, times_divide_eq_right, times_divide_eq_left,
   35.26 -	     rat_of_int_zero, rat_of_int_one, rat_of_int_add_distrib,
   35.27 -	     rat_of_int_minus_distrib, rat_of_int_diff_distrib,
   35.28 -	     rat_of_int_mult_distrib, number_of_rat RS sym];
   35.29 +	     of_int_0, of_int_1, of_int_add, of_int_minus, of_int_diff,
   35.30 +	     of_int_mult, of_int_of_nat_eq, rat_number_of_def];
   35.31  
   35.32  in
   35.33  
   35.34 @@ -625,8 +612,11 @@
   35.35    "fast_rat_arith" ["(m::rat) < n","(m::rat) <= n", "(m::rat) = n"]
   35.36    Fast_Arith.lin_arith_prover;
   35.37  
   35.38 -val int_inj_thms = [rat_of_int_le_iff RS iffD2, rat_of_int_less_iff RS iffD2,
   35.39 -                    rat_inject RS iffD2];
   35.40 +val nat_inj_thms = [of_nat_le_iff RS iffD2, of_nat_less_iff RS iffD2,
   35.41 +                    of_nat_eq_iff RS iffD2];
   35.42 +
   35.43 +val int_inj_thms = [of_int_le_iff RS iffD2, of_int_less_iff RS iffD2,
   35.44 +                    of_int_eq_iff RS iffD2];
   35.45  
   35.46  val rat_arith_setup =
   35.47   [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
   35.48 @@ -637,7 +627,8 @@
   35.49      simpset = simpset addsimps add_rules
   35.50                        addsimps simps
   35.51                        addsimprocs simprocs}),
   35.52 -  arith_inj_const ("Rational.rat", HOLogic.intT --> Rat_Numeral_Simprocs.ratT),
   35.53 +  arith_inj_const("IntDef.of_nat", HOLogic.natT --> Rat_Numeral_Simprocs.ratT),
   35.54 +  arith_inj_const("IntDef.of_int", HOLogic.intT --> Rat_Numeral_Simprocs.ratT),
   35.55    arith_discrete ("Rational.rat",false),
   35.56    Simplifier.change_simpset_of (op addsimprocs) [fast_rat_arith_simproc]];
   35.57  
    36.1 --- a/src/HOL/UNITY/Simple/Lift.thy	Thu Feb 05 10:45:28 2004 +0100
    36.2 +++ b/src/HOL/UNITY/Simple/Lift.thy	Tue Feb 10 12:02:11 2004 +0100
    36.3 @@ -215,13 +215,13 @@
    36.4  lemma moving_up: "Lift \<in> Always moving_up"
    36.5  apply (rule AlwaysI, force) 
    36.6  apply (unfold Lift_def, constrains)
    36.7 -apply (auto dest: zle_imp_zless_or_eq simp add: add1_zle_eq)
    36.8 +apply (auto dest: order_le_imp_less_or_eq simp add: add1_zle_eq)
    36.9  done
   36.10  
   36.11  lemma moving_down: "Lift \<in> Always moving_down"
   36.12  apply (rule AlwaysI, force) 
   36.13  apply (unfold Lift_def, constrains)
   36.14 -apply (blast dest: zle_imp_zless_or_eq)
   36.15 +apply (blast dest: order_le_imp_less_or_eq)
   36.16  done
   36.17  
   36.18  lemma bounded: "Lift \<in> Always bounded"
   36.19 @@ -290,7 +290,7 @@
   36.20      "Lift \<in> (Req n \<inter> closed - (atFloor n - queueing))    
   36.21               LeadsTo ((closed \<inter> goingup \<inter> Req n)  \<union> 
   36.22                        (closed \<inter> goingdown \<inter> Req n))"
   36.23 -by (auto intro!: subset_imp_LeadsTo elim!: int_neqE)
   36.24 +by (auto intro!: subset_imp_LeadsTo simp add: linorder_neq_iff)
   36.25  
   36.26  (*lift_2*)
   36.27  lemma (in Floor) lift_2: "Lift \<in> (Req n \<inter> closed - (atFloor n - queueing))    
    37.1 --- a/src/HOL/ex/BinEx.thy	Thu Feb 05 10:45:28 2004 +0100
    37.2 +++ b/src/HOL/ex/BinEx.thy	Tue Feb 10 12:02:11 2004 +0100
    37.3 @@ -470,7 +470,7 @@
    37.4     apply assumption
    37.5    apply (simp add: normal_Pls_eq_0)
    37.6    apply (simp only: number_of_minus zminus_0 iszero_def
    37.7 -                    zminus_equation [of _ "0"])
    37.8 +                    minus_equation_iff [of _ "0"])
    37.9    apply (simp add: eq_commute)
   37.10    done
   37.11