src/HOL/Integ/IntDef.thy
 changeset 14378 69c4d5997669 parent 14348 744c868ee0b7 child 14387 e96d5c42c4b0
```     1.1 --- a/src/HOL/Integ/IntDef.thy	Thu Feb 05 10:45:28 2004 +0100
1.2 +++ b/src/HOL/Integ/IntDef.thy	Tue Feb 10 12:02:11 2004 +0100
1.3 @@ -27,13 +27,6 @@
1.4
1.5    int :: "nat => int"
1.6    "int m == Abs_Integ(intrel `` {(m,0)})"
1.7 -
1.8 -  neg   :: "int => bool"
1.9 -  "neg(Z) == \<exists>x y. x<y & (x,y::nat):Rep_Integ(Z)"
1.10 -
1.11 -  (*For simplifying equalities*)
1.12 -  iszero :: "int => bool"
1.13 -  "iszero z == z = (0::int)"
1.14
1.16
1.17 @@ -48,16 +41,17 @@
1.18  		 intrel``{(x1+x2, y1+y2)})"
1.19
1.20    zdiff_def:  "z - (w::int) == z + (-w)"
1.21 -
1.22 -  zless_def:  "z<w == neg(z - w)"
1.23 -
1.24 -  zle_def:    "z <= (w::int) == ~(w < z)"
1.25 -
1.26    zmult_def:
1.27     "z * w ==
1.28         Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).
1.29  		 intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
1.30
1.31 +  zless_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
1.32 +
1.33 +  zle_def:
1.34 +  "z \<le> (w::int) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
1.35 +                            (x1,y1) \<in> Rep_Integ z & (x2,y2) \<in> Rep_Integ w"
1.36 +
1.37  lemma intrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in>  intrel) = (x1+y2 = x2+y1)"
1.38  by (unfold intrel_def, blast)
1.39
1.40 @@ -121,8 +115,8 @@
1.41  done
1.42
1.43  lemma zminus_zminus [simp]: "- (- z) = (z::int)"
1.44 -apply (rule_tac z = z in eq_Abs_Integ)
1.45 -apply (simp (no_asm_simp) add: zminus)
1.46 +apply (rule eq_Abs_Integ [of z])
1.48  done
1.49
1.50  lemma inj_zminus: "inj(%z::int. -z)"
1.51 @@ -134,16 +128,6 @@
1.52  by (simp add: int_def Zero_int_def zminus)
1.53
1.54
1.55 -subsection{*neg: the test for negative integers*}
1.56 -
1.57 -
1.58 -lemma not_neg_int [simp]: "~ neg(int n)"
1.59 -by (simp add: neg_def int_def)
1.60 -
1.61 -lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
1.62 -by (simp add: neg_def int_def zminus)
1.63 -
1.64 -
1.66
1.68 @@ -155,22 +139,22 @@
1.69  done
1.70
1.71  lemma zminus_zadd_distrib [simp]: "- (z + w) = (- z) + (- w::int)"
1.72 -apply (rule_tac z = z in eq_Abs_Integ)
1.73 -apply (rule_tac z = w in eq_Abs_Integ)
1.75 +apply (rule eq_Abs_Integ [of z])
1.76 +apply (rule eq_Abs_Integ [of w])
1.78  done
1.79
1.80  lemma zadd_commute: "(z::int) + w = w + z"
1.81 -apply (rule_tac z = z in eq_Abs_Integ)
1.82 -apply (rule_tac z = w in eq_Abs_Integ)
1.84 +apply (rule eq_Abs_Integ [of z])
1.85 +apply (rule eq_Abs_Integ [of w])
1.87  done
1.88
1.89  lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
1.90 -apply (rule_tac z = z1 in eq_Abs_Integ)
1.91 -apply (rule_tac z = z2 in eq_Abs_Integ)
1.92 -apply (rule_tac z = z3 in eq_Abs_Integ)
1.94 +apply (rule eq_Abs_Integ [of z1])
1.95 +apply (rule eq_Abs_Integ [of z2])
1.96 +apply (rule eq_Abs_Integ [of z3])
1.98  done
1.99
1.100  (*For AC rewriting*)
1.101 @@ -197,8 +181,8 @@
1.102  (*also for the instance declaration int :: plus_ac0*)
1.103  lemma zadd_0 [simp]: "(0::int) + z = z"
1.104  apply (unfold Zero_int_def int_def)
1.105 -apply (rule_tac z = z in eq_Abs_Integ)
1.107 +apply (rule eq_Abs_Integ [of z])
1.109  done
1.110
1.111  lemma zadd_0_right [simp]: "z + (0::int) = z"
1.112 @@ -206,8 +190,8 @@
1.113
1.114  lemma zadd_zminus_inverse [simp]: "z + (- z) = (0::int)"
1.115  apply (unfold int_def Zero_int_def)
1.116 -apply (rule_tac z = z in eq_Abs_Integ)
1.118 +apply (rule eq_Abs_Integ [of z])
1.120  done
1.121
1.122  lemma zadd_zminus_inverse2 [simp]: "(- z) + z = (0::int)"
1.123 @@ -236,57 +220,52 @@
1.124  lemma zadd_assoc_cong: "(z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
1.126
1.127 -lemma zadd_assoc_swap: "(z::int) + (v + w) = v + (z + w)"
1.129 -
1.130
1.131  subsection{*zmult: multiplication on Integ*}
1.132
1.133 -(*Congruence property for multiplication*)
1.134 +text{*Congruence property for multiplication*}
1.135  lemma zmult_congruent2: "congruent2 intrel
1.136          (%p1 p2. (%(x1,y1). (%(x2,y2).
1.137                      intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
1.138  apply (rule equiv_intrel [THEN congruent2_commuteI])
1.139 -apply (rule_tac [2] p=w in PairE)
1.144  apply (rename_tac x1 x2 y1 y2 z1 z2)
1.145  apply (rule equiv_class_eq [OF equiv_intrel intrel_iff [THEN iffD2]])
1.147 -apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2", arith)
1.148 +apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2")
1.149 +apply (simp add: mult_ac, arith)
1.151  done
1.152
1.153  lemma zmult:
1.154     "Abs_Integ((intrel``{(x1,y1)})) * Abs_Integ((intrel``{(x2,y2)})) =
1.155      Abs_Integ(intrel `` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
1.156 -apply (unfold zmult_def)
1.157 -apply (simp (no_asm_simp) add: UN_UN_split_split_eq zmult_congruent2 equiv_intrel [THEN UN_equiv_class2])
1.158 -done
1.159 +by (simp add: zmult_def UN_UN_split_split_eq zmult_congruent2
1.160 +              equiv_intrel [THEN UN_equiv_class2])
1.161
1.162  lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
1.163 -apply (rule_tac z = z in eq_Abs_Integ)
1.164 -apply (rule_tac z = w in eq_Abs_Integ)
1.166 +apply (rule eq_Abs_Integ [of z])
1.167 +apply (rule eq_Abs_Integ [of w])
1.169  done
1.170
1.171  lemma zmult_commute: "(z::int) * w = w * z"
1.172 -apply (rule_tac z = z in eq_Abs_Integ)
1.173 -apply (rule_tac z = w in eq_Abs_Integ)
1.175 +apply (rule eq_Abs_Integ [of z])
1.176 +apply (rule eq_Abs_Integ [of w])
1.178  done
1.179
1.180  lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
1.181 -apply (rule_tac z = z1 in eq_Abs_Integ)
1.182 -apply (rule_tac z = z2 in eq_Abs_Integ)
1.183 -apply (rule_tac z = z3 in eq_Abs_Integ)
1.185 +apply (rule eq_Abs_Integ [of z1])
1.186 +apply (rule eq_Abs_Integ [of z2])
1.187 +apply (rule eq_Abs_Integ [of z3])
1.189  done
1.190
1.191  lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
1.192 -apply (rule_tac z = z1 in eq_Abs_Integ)
1.193 -apply (rule_tac z = z2 in eq_Abs_Integ)
1.194 -apply (rule_tac z = w in eq_Abs_Integ)
1.195 +apply (rule eq_Abs_Integ [of z1])
1.196 +apply (rule eq_Abs_Integ [of z2])
1.197 +apply (rule eq_Abs_Integ [of w])
1.199  done
1.200
1.201 @@ -314,14 +293,14 @@
1.202
1.203  lemma zmult_0 [simp]: "0 * z = (0::int)"
1.204  apply (unfold Zero_int_def int_def)
1.205 -apply (rule_tac z = z in eq_Abs_Integ)
1.206 -apply (simp (no_asm_simp) add: zmult)
1.207 +apply (rule eq_Abs_Integ [of z])
1.209  done
1.210
1.211  lemma zmult_1 [simp]: "(1::int) * z = z"
1.212  apply (unfold One_int_def int_def)
1.213 -apply (rule_tac z = z in eq_Abs_Integ)
1.214 -apply (simp (no_asm_simp) add: zmult)
1.215 +apply (rule eq_Abs_Integ [of z])
1.217  done
1.218
1.219  lemma zmult_0_right [simp]: "z * 0 = (0::int)"
1.220 @@ -352,64 +331,73 @@
1.221  qed
1.222
1.223
1.225 +subsection{*The @{text "\<le>"} Ordering*}
1.226 +
1.227 +lemma zle:
1.228 +  "(Abs_Integ(intrel``{(x1,y1)}) \<le> Abs_Integ(intrel``{(x2,y2)})) =
1.229 +   (x1 + y2 \<le> x2 + y1)"
1.230 +by (force simp add: zle_def)
1.231 +
1.232 +lemma zle_refl: "w \<le> (w::int)"
1.233 +apply (rule eq_Abs_Integ [of w])
1.234 +apply (force simp add: zle)
1.235 +done
1.236 +
1.237 +lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
1.238 +apply (rule eq_Abs_Integ [of i])
1.239 +apply (rule eq_Abs_Integ [of j])
1.240 +apply (rule eq_Abs_Integ [of k])
1.242 +done
1.243 +
1.244 +lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
1.245 +apply (rule eq_Abs_Integ [of w])
1.246 +apply (rule eq_Abs_Integ [of z])
1.248 +done
1.249 +
1.250 +(* Axiom 'order_less_le' of class 'order': *)
1.251 +lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
1.253 +
1.254 +instance int :: order
1.255 +proof qed
1.256 + (assumption |
1.257 +  rule zle_refl zle_trans zle_anti_sym zless_le)+
1.258 +
1.259 +(* Axiom 'linorder_linear' of class 'linorder': *)
1.260 +lemma zle_linear: "(z::int) \<le> w | w \<le> z"
1.261 +apply (rule eq_Abs_Integ [of z])
1.262 +apply (rule eq_Abs_Integ [of w])
1.263 +apply (simp add: zle linorder_linear)
1.264 +done
1.265 +
1.266 +instance int :: plus_ac0
1.268 +
1.269 +instance int :: linorder
1.270 +proof qed (rule zle_linear)
1.271 +
1.272 +
1.273 +lemmas zless_linear = linorder_less_linear [where 'a = int]
1.274 +
1.275 +
1.276 +lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
1.278
1.279  (*This lemma allows direct proofs of other <-properties*)
1.281      "(w < z) = (\<exists>n. z = w + int(Suc n))"
1.282 -apply (unfold zless_def neg_def zdiff_def int_def)
1.283 -apply (rule_tac z = z in eq_Abs_Integ)
1.284 -apply (rule_tac z = w in eq_Abs_Integ, clarify)
1.286 +apply (rule eq_Abs_Integ [of z])
1.287 +apply (rule eq_Abs_Integ [of w])
1.288 +apply (simp add: linorder_not_le [where 'a = int, symmetric]
1.289 +                 linorder_not_le [where 'a = nat]
1.290 +                 zle int_def zdiff_def zadd zminus)
1.292  apply (rule_tac x = k in exI)
1.294  done
1.295
1.296 -lemma zless_zadd_Suc: "z < z + int (Suc n)"
1.298 -
1.299 -lemma zless_trans: "[| z1<z2; z2<z3 |] ==> z1 < (z3::int)"
1.301 -
1.302 -lemma zless_not_sym: "!!w::int. z<w ==> ~w<z"
1.303 -apply (safe dest!: zless_iff_Suc_zadd [THEN iffD1])
1.304 -apply (rule_tac z = z in eq_Abs_Integ, safe)
1.306 -done
1.307 -
1.308 -(* [| n<m;  ~P ==> m<n |] ==> P *)
1.309 -lemmas zless_asym = zless_not_sym [THEN swap, standard]
1.310 -
1.311 -lemma zless_not_refl: "!!z::int. ~ z<z"
1.312 -apply (rule zless_asym [THEN notI])
1.313 -apply (assumption+)
1.314 -done
1.315 -
1.316 -(* z<z ==> R *)
1.317 -lemmas zless_irrefl = zless_not_refl [THEN notE, standard, elim!]
1.318 -
1.319 -
1.320 -(*"Less than" is a linear ordering*)
1.321 -lemma zless_linear:
1.322 -    "z<w | z=w | w<(z::int)"
1.323 -apply (unfold zless_def neg_def zdiff_def)
1.324 -apply (rule_tac z = z in eq_Abs_Integ)
1.325 -apply (rule_tac z = w in eq_Abs_Integ, safe)
1.327 -apply (rule_tac m1 = "x+ya" and n1 = "xa+y" in less_linear [THEN disjE])
1.329 -done
1.330 -
1.331 -lemma int_neq_iff: "!!w::int. (w ~= z) = (w<z | z<w)"
1.332 -by (cut_tac zless_linear, blast)
1.333 -
1.334 -(*** eliminates ~= in premises ***)
1.335 -lemmas int_neqE = int_neq_iff [THEN iffD1, THEN disjE, standard]
1.336 -
1.337 -lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
1.339 -
1.340  lemma zless_int [simp]: "(int m < int n) = (m<n)"
1.342
1.343 @@ -425,84 +413,553 @@
1.344  lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
1.345  by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
1.346
1.347 -
1.348 -subsection{*Properties of the @{text "\<le>"} Relation*}
1.349 +lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
1.350 +by (simp add: linorder_not_less [symmetric])
1.351
1.352 -lemma zle_int [simp]: "(int m <= int n) = (m<=n)"
1.353 -by (simp add: zle_def le_def)
1.354 +lemma zero_zle_int [simp]: "(0 \<le> int n)"
1.356
1.357 -lemma zero_zle_int [simp]: "(0 <= int n)"
1.358 +lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
1.360 +
1.361 +lemma int_0 [simp]: "int 0 = (0::int)"
1.363
1.364 -lemma int_le_0_conv [simp]: "(int n <= 0) = (n = 0)"
1.366 +lemma int_1 [simp]: "int 1 = 1"
1.368 +
1.369 +lemma int_Suc0_eq_1: "int (Suc 0) = 1"
1.370 +by (simp add: One_int_def One_nat_def)
1.371 +
1.372 +subsection{*Monotonicity results*}
1.373 +
1.374 +lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
1.375 +apply (rule eq_Abs_Integ [of i])
1.376 +apply (rule eq_Abs_Integ [of j])
1.377 +apply (rule eq_Abs_Integ [of k])
1.379 +done
1.380 +
1.381 +lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
1.382 +apply (rule eq_Abs_Integ [of i])
1.383 +apply (rule eq_Abs_Integ [of j])
1.384 +apply (rule eq_Abs_Integ [of k])
1.385 +apply (simp add: linorder_not_le [where 'a = int, symmetric]
1.386 +                 linorder_not_le [where 'a = nat]  zle zadd)
1.387 +done
1.388 +
1.389 +lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
1.391 +
1.392 +
1.393 +subsection{*Strict Monotonicity of Multiplication*}
1.394 +
1.395 +text{*strict, in 1st argument; proof is by induction on k>0*}
1.396 +lemma zmult_zless_mono2_lemma [rule_format]:
1.397 +     "i<j ==> 0<k --> int k * i < int k * j"
1.398 +apply (induct_tac "k", simp)
1.400 +apply (case_tac "n=0")
1.403 +done
1.404
1.405 -lemma zle_imp_zless_or_eq: "z <= w ==> z < w | z=(w::int)"
1.406 -apply (unfold zle_def)
1.407 -apply (cut_tac zless_linear)
1.408 -apply (blast elim: zless_asym)
1.409 +lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
1.410 +apply (rule eq_Abs_Integ [of k])
1.412 +apply (rule_tac x="x-y" in exI, simp)
1.413 +done
1.414 +
1.415 +lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
1.416 +apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
1.417 +apply (auto simp add: zmult_zless_mono2_lemma)
1.418 +done
1.419 +
1.420 +
1.422 +    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
1.423 +
1.424 +
1.425 +text{*The Integers Form an Ordered Ring*}
1.426 +instance int :: ordered_ring
1.427 +proof
1.428 +  fix i j k :: int
1.429 +  show "0 < (1::int)" by (rule int_0_less_1)
1.430 +  show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
1.431 +  show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
1.432 +  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
1.433 +qed
1.434 +
1.435 +
1.436 +subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
1.437 +
1.438 +constdefs
1.439 +   nat  :: "int => nat"
1.440 +    "nat(Z) == if Z<0 then 0 else (THE m. Z = int m)"
1.441 +
1.442 +lemma nat_int [simp]: "nat(int n) = n"
1.443 +by (unfold nat_def, auto)
1.444 +
1.445 +lemma nat_zero [simp]: "nat 0 = 0"
1.446 +apply (unfold Zero_int_def)
1.447 +apply (rule nat_int)
1.448 +done
1.449 +
1.450 +lemma nat_0_le [simp]: "0 \<le> z ==> int (nat z) = z"
1.451 +apply (rule eq_Abs_Integ [of z])
1.452 +apply (simp add: nat_def linorder_not_le [symmetric] zle int_def Zero_int_def)
1.453 +apply (subgoal_tac "(THE m. x = m + y) = x-y")
1.454 +apply (auto simp add: the_equality)
1.455  done
1.456
1.457 -lemma zless_or_eq_imp_zle: "z<w | z=w ==> z <= (w::int)"
1.458 -apply (unfold zle_def)
1.459 -apply (cut_tac zless_linear)
1.460 -apply (blast elim: zless_asym)
1.461 +lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
1.462 +by (simp add: nat_def  order_less_le eq_commute [of 0])
1.463 +
1.464 +text{*An alternative condition is @{term "0 \<le> w"} *}
1.465 +lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
1.466 +apply (subst zless_int [symmetric])
1.468 +apply (case_tac "w < 0")
1.469 + apply (simp add: order_less_imp_le)
1.470 + apply (blast intro: order_less_trans)
1.472 +done
1.473 +
1.474 +lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
1.475 +apply (case_tac "0 < z")
1.476 +apply (auto simp add: nat_mono_iff linorder_not_less)
1.477 +done
1.478 +
1.479 +
1.480 +subsection{*Lemmas about the Function @{term int} and Orderings*}
1.481 +
1.482 +lemma negative_zless_0: "- (int (Suc n)) < 0"
1.484 +
1.485 +lemma negative_zless [iff]: "- (int (Suc n)) < int m"
1.486 +by (rule negative_zless_0 [THEN order_less_le_trans], simp)
1.487 +
1.488 +lemma negative_zle_0: "- int n \<le> 0"
1.490 +
1.491 +lemma negative_zle [iff]: "- int n \<le> int m"
1.492 +by (rule order_trans [OF negative_zle_0 zero_zle_int])
1.493 +
1.494 +lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
1.495 +by (subst le_minus_iff, simp)
1.496 +
1.497 +lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
1.498 +apply safe
1.499 +apply (drule_tac [2] le_minus_iff [THEN iffD1])
1.500 +apply (auto dest: zle_trans [OF _ negative_zle_0])
1.501  done
1.502
1.503 -lemma int_le_less: "(x <= (y::int)) = (x < y | x=y)"
1.504 -apply (rule iffI)
1.505 -apply (erule zle_imp_zless_or_eq)
1.506 -apply (erule zless_or_eq_imp_zle)
1.507 +lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
1.509 +
1.510 +lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
1.511 +by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
1.512 +
1.513 +lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
1.514 +by (force intro: exI [of _ "0::nat"]
1.515 +            intro!: not_sym [THEN not0_implies_Suc]
1.517 +
1.518 +
1.519 +text{*This version is proved for all ordered rings, not just integers!
1.520 +      It is proved here because attribute @{text arith_split} is not available
1.521 +      in theory @{text Ring_and_Field}.
1.522 +      But is it really better than just rewriting with @{text abs_if}?*}
1.523 +lemma abs_split [arith_split]:
1.524 +     "P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
1.525 +by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
1.526 +
1.527 +lemma abs_int_eq [simp]: "abs (int m) = int m"
1.529 +
1.530 +
1.531 +subsection{*Misc Results*}
1.532 +
1.533 +lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
1.534 +by (auto simp add: nat_def zero_reorient minus_less_iff)
1.535 +
1.536 +lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
1.537 +apply (case_tac "0 \<le> z")
1.538 +apply (erule nat_0_le [THEN subst], simp)
1.540 +apply (auto dest: order_less_trans simp add: order_less_imp_le)
1.541  done
1.542
1.543 -lemma zle_refl: "w <= (w::int)"
1.545 +
1.546 +
1.547 +subsection{*Monotonicity of Multiplication*}
1.548 +
1.549 +lemma zmult_zle_mono2: "[| i \<le> j;  (0::int) \<le> k |] ==> k*i \<le> k*j"
1.550 +  by (rule Ring_and_Field.mult_left_mono)
1.551 +
1.552 +lemma zmult_zless_cancel2: "(m*k < n*k) = (((0::int) < k & m<n) | (k<0 & n<m))"
1.553 +  by (rule Ring_and_Field.mult_less_cancel_right)
1.554 +
1.555 +lemma zmult_zless_cancel1:
1.556 +     "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))"
1.557 +  by (rule Ring_and_Field.mult_less_cancel_left)
1.558
1.559 -(* Axiom 'linorder_linear' of class 'linorder': *)
1.560 -lemma zle_linear: "(z::int) <= w | w <= z"
1.562 -apply (cut_tac zless_linear, blast)
1.563 +lemma zmult_zle_cancel1:
1.564 +     "(k*m \<le> k*n) = (((0::int) < k --> m\<le>n) & (k < 0 --> n\<le>m))"
1.565 +  by (rule Ring_and_Field.mult_le_cancel_left)
1.566 +
1.567 +
1.568 +
1.569 +text{*A case theorem distinguishing non-negative and negative int*}
1.570 +
1.571 +lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
1.573 +                   diff_eq_eq [symmetric] zdiff_def)
1.574 +
1.575 +lemma int_cases [cases type: int, case_names nonneg neg]:
1.576 +     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
1.577 +apply (case_tac "z < 0", blast dest!: negD)
1.579 +apply (blast dest: nat_0_le [THEN sym])
1.580  done
1.581
1.582 -(* Axiom 'order_trans of class 'order': *)
1.583 -lemma zle_trans: "[| i <= j; j <= k |] ==> i <= (k::int)"
1.584 -apply (drule zle_imp_zless_or_eq)
1.585 -apply (drule zle_imp_zless_or_eq)
1.586 -apply (rule zless_or_eq_imp_zle)
1.587 -apply (blast intro: zless_trans)
1.588 +lemma int_induct [induct type: int, case_names nonneg neg]:
1.589 +     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
1.590 +  by (cases z) auto
1.591 +
1.592 +
1.593 +subsection{*The Constants @{term neg} and @{term iszero}*}
1.594 +
1.595 +constdefs
1.596 +
1.597 +  neg   :: "'a::ordered_ring => bool"
1.598 +  "neg(Z) == Z < 0"
1.599 +
1.600 +  (*For simplifying equalities*)
1.601 +  iszero :: "'a::semiring => bool"
1.602 +  "iszero z == z = (0)"
1.603 +
1.604 +
1.605 +lemma not_neg_int [simp]: "~ neg(int n)"
1.607 +
1.608 +lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
1.609 +by (simp add: neg_def neg_less_0_iff_less)
1.610 +
1.611 +lemmas neg_eq_less_0 = neg_def
1.612 +
1.613 +lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
1.614 +by (simp add: neg_def linorder_not_less)
1.615 +
1.616 +subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
1.617 +
1.618 +lemma not_neg_0: "~ neg 0"
1.619 +by (simp add: One_int_def neg_def)
1.620 +
1.621 +lemma not_neg_1: "~ neg 1"
1.622 +by (simp add: neg_def linorder_not_less zero_le_one)
1.623 +
1.624 +lemma iszero_0: "iszero 0"
1.626 +
1.627 +lemma not_iszero_1: "~ iszero 1"
1.628 +by (simp add: iszero_def eq_commute)
1.629 +
1.630 +lemma neg_nat: "neg z ==> nat z = 0"
1.631 +by (simp add: nat_def neg_def)
1.632 +
1.633 +lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
1.634 +by (simp add: linorder_not_less neg_def)
1.635 +
1.636 +
1.637 +subsection{*Embedding of the Naturals into any Semiring: @{term of_nat}*}
1.638 +
1.639 +consts of_nat :: "nat => 'a::semiring"
1.640 +
1.641 +primrec
1.642 +  of_nat_0:   "of_nat 0 = 0"
1.643 +  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
1.644 +
1.645 +lemma of_nat_1 [simp]: "of_nat 1 = 1"
1.646 +by simp
1.647 +
1.648 +lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
1.649 +apply (induct m)
1.651 +done
1.652 +
1.653 +lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
1.654 +apply (induct m)
1.656 +done
1.657 +
1.658 +lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semiring)"
1.659 +apply (induct m, simp_all)
1.660 +apply (erule order_trans)
1.661 +apply (rule less_add_one [THEN order_less_imp_le])
1.662  done
1.663
1.664 -lemma zle_anti_sym: "[| z <= w; w <= z |] ==> z = (w::int)"
1.665 -apply (drule zle_imp_zless_or_eq)
1.666 -apply (drule zle_imp_zless_or_eq)
1.667 -apply (blast elim: zless_asym)
1.668 +lemma less_imp_of_nat_less:
1.669 +     "m < n ==> of_nat m < (of_nat n::'a::ordered_semiring)"
1.670 +apply (induct m n rule: diff_induct, simp_all)
1.671 +apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
1.672 +done
1.673 +
1.674 +lemma of_nat_less_imp_less:
1.675 +     "of_nat m < (of_nat n::'a::ordered_semiring) ==> m < n"
1.676 +apply (induct m n rule: diff_induct, simp_all)
1.677 +apply (insert zero_le_imp_of_nat)
1.678 +apply (force simp add: linorder_not_less [symmetric])
1.679  done
1.680
1.681 -(* Axiom 'order_less_le' of class 'order': *)
1.682 -lemma int_less_le: "((w::int) < z) = (w <= z & w ~= z)"
1.683 -apply (simp add: zle_def int_neq_iff)
1.684 -apply (blast elim!: zless_asym)
1.685 +lemma of_nat_less_iff [simp]:
1.686 +     "(of_nat m < (of_nat n::'a::ordered_semiring)) = (m<n)"
1.687 +by (blast intro: of_nat_less_imp_less less_imp_of_nat_less )
1.688 +
1.689 +text{*Special cases where either operand is zero*}
1.690 +declare of_nat_less_iff [of 0, simplified, simp]
1.691 +declare of_nat_less_iff [of _ 0, simplified, simp]
1.692 +
1.693 +lemma of_nat_le_iff [simp]:
1.694 +     "(of_nat m \<le> (of_nat n::'a::ordered_semiring)) = (m \<le> n)"
1.695 +by (simp add: linorder_not_less [symmetric])
1.696 +
1.697 +text{*Special cases where either operand is zero*}
1.698 +declare of_nat_le_iff [of 0, simplified, simp]
1.699 +declare of_nat_le_iff [of _ 0, simplified, simp]
1.700 +
1.701 +lemma of_nat_eq_iff [simp]:
1.702 +     "(of_nat m = (of_nat n::'a::ordered_semiring)) = (m = n)"
1.704 +
1.705 +text{*Special cases where either operand is zero*}
1.706 +declare of_nat_eq_iff [of 0, simplified, simp]
1.707 +declare of_nat_eq_iff [of _ 0, simplified, simp]
1.708 +
1.709 +lemma of_nat_diff [simp]:
1.710 +     "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring)"
1.713 +
1.714 +
1.715 +subsection{*The Set of Natural Numbers*}
1.716 +
1.717 +constdefs
1.718 +   Nats  :: "'a::semiring set"
1.719 +    "Nats == range of_nat"
1.720 +
1.721 +syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
1.722 +
1.723 +lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
1.725 +
1.726 +lemma Nats_0 [simp]: "0 \<in> Nats"
1.728 +apply (rule range_eqI)
1.729 +apply (rule of_nat_0 [symmetric])
1.730 +done
1.731 +
1.732 +lemma Nats_1 [simp]: "1 \<in> Nats"
1.734 +apply (rule range_eqI)
1.735 +apply (rule of_nat_1 [symmetric])
1.736 +done
1.737 +
1.738 +lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
1.739 +apply (auto simp add: Nats_def)
1.740 +apply (rule range_eqI)
1.742 +done
1.743 +
1.744 +lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
1.745 +apply (auto simp add: Nats_def)
1.746 +apply (rule range_eqI)
1.747 +apply (rule of_nat_mult [symmetric])
1.748  done
1.749
1.750 -instance int :: order
1.751 -proof qed (assumption | rule zle_refl zle_trans zle_anti_sym int_less_le)+
1.752 +text{*Agreement with the specific embedding for the integers*}
1.753 +lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
1.754 +proof
1.755 +  fix n
1.756 +  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
1.757 +qed
1.758 +
1.759 +
1.760 +subsection{*Embedding of the Integers into any Ring: @{term of_int}*}
1.761 +
1.762 +constdefs
1.763 +   of_int :: "int => 'a::ring"
1.764 +   "of_int z ==
1.765 +      (THE a. \<exists>i j. (i,j) \<in> Rep_Integ z & a = (of_nat i) - (of_nat j))"
1.766 +
1.767 +
1.768 +lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
1.770 +apply (rule the_equality, auto)
1.773 +done
1.774 +
1.775 +lemma of_int_0 [simp]: "of_int 0 = 0"
1.776 +by (simp add: of_int Zero_int_def int_def)
1.777 +
1.778 +lemma of_int_1 [simp]: "of_int 1 = 1"
1.779 +by (simp add: of_int One_int_def int_def)
1.780 +
1.781 +lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
1.782 +apply (rule eq_Abs_Integ [of w])
1.783 +apply (rule eq_Abs_Integ [of z])
1.785 +done
1.786 +
1.787 +lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
1.788 +apply (rule eq_Abs_Integ [of z])
1.789 +apply (simp add: compare_rls of_int zminus)
1.790 +done
1.791
1.792 -instance int :: plus_ac0
1.794 +lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
1.796 +
1.797 +lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
1.798 +apply (rule eq_Abs_Integ [of w])
1.799 +apply (rule eq_Abs_Integ [of z])
1.800 +apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
1.802 +done
1.803 +
1.804 +lemma of_int_le_iff [simp]:
1.805 +     "(of_int w \<le> (of_int z::'a::ordered_ring)) = (w \<le> z)"
1.806 +apply (rule eq_Abs_Integ [of w])
1.807 +apply (rule eq_Abs_Integ [of z])
1.810 +done
1.811 +
1.812 +text{*Special cases where either operand is zero*}
1.813 +declare of_int_le_iff [of 0, simplified, simp]
1.814 +declare of_int_le_iff [of _ 0, simplified, simp]
1.815
1.816 -instance int :: linorder
1.817 -proof qed (rule zle_linear)
1.818 +lemma of_int_less_iff [simp]:
1.819 +     "(of_int w < (of_int z::'a::ordered_ring)) = (w < z)"
1.820 +by (simp add: linorder_not_le [symmetric])
1.821 +
1.822 +text{*Special cases where either operand is zero*}
1.823 +declare of_int_less_iff [of 0, simplified, simp]
1.824 +declare of_int_less_iff [of _ 0, simplified, simp]
1.825 +
1.826 +lemma of_int_eq_iff [simp]:
1.827 +     "(of_int w = (of_int z::'a::ordered_ring)) = (w = z)"
1.829 +
1.830 +text{*Special cases where either operand is zero*}
1.831 +declare of_int_eq_iff [of 0, simplified, simp]
1.832 +declare of_int_eq_iff [of _ 0, simplified, simp]
1.833 +
1.834 +
1.835 +subsection{*The Set of Integers*}
1.836 +
1.837 +constdefs
1.838 +   Ints  :: "'a::ring set"
1.839 +    "Ints == range of_int"
1.840
1.841
1.842 -lemma zadd_left_cancel [simp]: "!!w::int. (z + w' = z + w) = (w' = w)"
1.844 +syntax (xsymbols)
1.845 +  Ints      :: "'a set"                   ("\<int>")
1.846 +
1.847 +lemma Ints_0 [simp]: "0 \<in> Ints"
1.849 +apply (rule range_eqI)
1.850 +apply (rule of_int_0 [symmetric])
1.851 +done
1.852 +
1.853 +lemma Ints_1 [simp]: "1 \<in> Ints"
1.855 +apply (rule range_eqI)
1.856 +apply (rule of_int_1 [symmetric])
1.857 +done
1.858 +
1.859 +lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
1.860 +apply (auto simp add: Ints_def)
1.861 +apply (rule range_eqI)
1.863 +done
1.864 +
1.865 +lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
1.866 +apply (auto simp add: Ints_def)
1.867 +apply (rule range_eqI)
1.868 +apply (rule of_int_minus [symmetric])
1.869 +done
1.870 +
1.871 +lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
1.872 +apply (auto simp add: Ints_def)
1.873 +apply (rule range_eqI)
1.874 +apply (rule of_int_diff [symmetric])
1.875 +done
1.876 +
1.877 +lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
1.878 +apply (auto simp add: Ints_def)
1.879 +apply (rule range_eqI)
1.880 +apply (rule of_int_mult [symmetric])
1.881 +done
1.882 +
1.883 +text{*Collapse nested embeddings*}
1.884 +lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
1.885 +by (induct n, auto)
1.886 +
1.887 +lemma of_int_int_eq [simp]: "of_int (int n) = int n"
1.889
1.890
1.891 +lemma Ints_cases [case_names of_int, cases set: Ints]:
1.892 +  "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
1.893 +proof (unfold Ints_def)
1.894 +  assume "!!z. q = of_int z ==> C"
1.895 +  assume "q \<in> range of_int" thus C ..
1.896 +qed
1.897 +
1.898 +lemma Ints_induct [case_names of_int, induct set: Ints]:
1.899 +  "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
1.900 +  by (rule Ints_cases) auto
1.901 +
1.902 +
1.903 +
1.904 +(*Legacy ML bindings, but no longer the structure Int.*)
1.905  ML
1.906  {*
1.907 +val zabs_def = thm "zabs_def"
1.908 +val nat_def  = thm "nat_def"
1.909 +
1.910 +val int_0 = thm "int_0";
1.911 +val int_1 = thm "int_1";
1.912 +val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
1.913 +val neg_eq_less_0 = thm "neg_eq_less_0";
1.914 +val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
1.915 +val not_neg_0 = thm "not_neg_0";
1.916 +val not_neg_1 = thm "not_neg_1";
1.917 +val iszero_0 = thm "iszero_0";
1.918 +val not_iszero_1 = thm "not_iszero_1";
1.919 +val int_0_less_1 = thm "int_0_less_1";
1.920 +val int_0_neq_1 = thm "int_0_neq_1";
1.921 +val negative_zless = thm "negative_zless";
1.922 +val negative_zle = thm "negative_zle";
1.923 +val not_zle_0_negative = thm "not_zle_0_negative";
1.924 +val not_int_zless_negative = thm "not_int_zless_negative";
1.925 +val negative_eq_positive = thm "negative_eq_positive";
1.927 +val abs_int_eq = thm "abs_int_eq";
1.928 +val abs_split = thm"abs_split";
1.929 +val nat_int = thm "nat_int";
1.930 +val nat_zminus_int = thm "nat_zminus_int";
1.931 +val nat_zero = thm "nat_zero";
1.932 +val not_neg_nat = thm "not_neg_nat";
1.933 +val neg_nat = thm "neg_nat";
1.934 +val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
1.935 +val nat_0_le = thm "nat_0_le";
1.936 +val nat_le_0 = thm "nat_le_0";
1.937 +val zless_nat_conj = thm "zless_nat_conj";
1.938 +val int_cases = thm "int_cases";
1.939 +
1.940  val int_def = thm "int_def";
1.941 -val neg_def = thm "neg_def";
1.942 -val iszero_def = thm "iszero_def";
1.943  val Zero_int_def = thm "Zero_int_def";
1.944  val One_int_def = thm "One_int_def";
1.946 @@ -524,8 +981,6 @@
1.947  val zminus_zminus = thm "zminus_zminus";
1.948  val inj_zminus = thm "inj_zminus";
1.949  val zminus_0 = thm "zminus_0";
1.950 -val not_neg_int = thm "not_neg_int";
1.951 -val neg_zminus_int = thm "neg_zminus_int";
1.955 @@ -545,8 +1000,6 @@
1.956  val zdiff0 = thm "zdiff0";
1.957  val zdiff0_right = thm "zdiff0_right";
1.958  val zdiff_self = thm "zdiff_self";
1.961  val zmult_congruent2 = thm "zmult_congruent2";
1.962  val zmult = thm "zmult";
1.963  val zmult_zminus = thm "zmult_zminus";
1.964 @@ -564,15 +1017,6 @@
1.965  val zmult_0_right = thm "zmult_0_right";
1.966  val zmult_1_right = thm "zmult_1_right";
1.969 -val zless_trans = thm "zless_trans";
1.970 -val zless_not_sym = thm "zless_not_sym";
1.971 -val zless_asym = thm "zless_asym";
1.972 -val zless_not_refl = thm "zless_not_refl";
1.973 -val zless_irrefl = thm "zless_irrefl";
1.974 -val zless_linear = thm "zless_linear";
1.975 -val int_neq_iff = thm "int_neq_iff";
1.976 -val int_neqE = thm "int_neqE";
1.977  val int_int_eq = thm "int_int_eq";
1.978  val int_eq_0_conv = thm "int_eq_0_conv";
1.979  val zless_int = thm "zless_int";
1.980 @@ -581,15 +1025,48 @@
1.981  val zle_int = thm "zle_int";
1.982  val zero_zle_int = thm "zero_zle_int";
1.983  val int_le_0_conv = thm "int_le_0_conv";
1.984 -val zle_imp_zless_or_eq = thm "zle_imp_zless_or_eq";
1.985 -val zless_or_eq_imp_zle = thm "zless_or_eq_imp_zle";
1.986 -val int_le_less = thm "int_le_less";
1.987  val zle_refl = thm "zle_refl";
1.988  val zle_linear = thm "zle_linear";
1.989  val zle_trans = thm "zle_trans";
1.990  val zle_anti_sym = thm "zle_anti_sym";
1.991 -val int_less_le = thm "int_less_le";
1.993 +
1.994 +val Ints_def = thm "Ints_def";
1.995 +val Nats_def = thm "Nats_def";
1.996 +
1.997 +val of_nat_0 = thm "of_nat_0";
1.998 +val of_nat_Suc = thm "of_nat_Suc";
1.999 +val of_nat_1 = thm "of_nat_1";
1.1001 +val of_nat_mult = thm "of_nat_mult";
1.1002 +val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
1.1003 +val less_imp_of_nat_less = thm "less_imp_of_nat_less";
1.1004 +val of_nat_less_imp_less = thm "of_nat_less_imp_less";
1.1005 +val of_nat_less_iff = thm "of_nat_less_iff";
1.1006 +val of_nat_le_iff = thm "of_nat_le_iff";
1.1007 +val of_nat_eq_iff = thm "of_nat_eq_iff";
1.1008 +val Nats_0 = thm "Nats_0";
1.1009 +val Nats_1 = thm "Nats_1";
1.1011 +val Nats_mult = thm "Nats_mult";
1.1012 +val of_int = thm "of_int";
1.1013 +val of_int_0 = thm "of_int_0";
1.1014 +val of_int_1 = thm "of_int_1";
1.1016 +val of_int_minus = thm "of_int_minus";
1.1017 +val of_int_diff = thm "of_int_diff";
1.1018 +val of_int_mult = thm "of_int_mult";
1.1019 +val of_int_le_iff = thm "of_int_le_iff";
1.1020 +val of_int_less_iff = thm "of_int_less_iff";
1.1021 +val of_int_eq_iff = thm "of_int_eq_iff";
1.1022 +val Ints_0 = thm "Ints_0";
1.1023 +val Ints_1 = thm "Ints_1";