1.1 --- a/src/HOL/Int.thy Sun Sep 04 06:56:10 2011 -0700
1.2 +++ b/src/HOL/Int.thy Sun Sep 04 07:15:13 2011 -0700
1.3 @@ -162,7 +162,10 @@
1.4 by (simp add: Zero_int_def One_int_def)
1.5 qed
1.6
1.7 -lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
1.8 +abbreviation int :: "nat \<Rightarrow> int" where
1.9 + "int \<equiv> of_nat"
1.10 +
1.11 +lemma int_def: "int m = Abs_Integ (intrel `` {(m, 0)})"
1.12 by (induct m) (simp_all add: Zero_int_def One_int_def add)
1.13
1.14
1.15 @@ -218,7 +221,7 @@
1.16
1.17 text{*strict, in 1st argument; proof is by induction on k>0*}
1.18 lemma zmult_zless_mono2_lemma:
1.19 - "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
1.20 + "(i::int)<j ==> 0<k ==> int k * i < int k * j"
1.21 apply (induct k)
1.22 apply simp
1.23 apply (simp add: left_distrib)
1.24 @@ -226,13 +229,13 @@
1.25 apply (simp_all add: add_strict_mono)
1.26 done
1.27
1.28 -lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
1.29 +lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
1.30 apply (cases k)
1.31 apply (auto simp add: le add int_def Zero_int_def)
1.32 apply (rule_tac x="x-y" in exI, simp)
1.33 done
1.34
1.35 -lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
1.36 +lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
1.37 apply (cases k)
1.38 apply (simp add: less int_def Zero_int_def)
1.39 apply (rule_tac x="x-y" in exI, simp)
1.40 @@ -261,7 +264,7 @@
1.41 done
1.42
1.43 lemma zless_iff_Suc_zadd:
1.44 - "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
1.45 + "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
1.46 apply (cases z, cases w)
1.47 apply (auto simp add: less add int_def)
1.48 apply (rename_tac a b c d)
1.49 @@ -314,7 +317,7 @@
1.50 done
1.51
1.52 text{*Collapse nested embeddings*}
1.53 -lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
1.54 +lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
1.55 by (induct n) auto
1.56
1.57 lemma of_int_power:
1.58 @@ -400,13 +403,13 @@
1.59 by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
1.60 qed
1.61
1.62 -lemma nat_int [simp]: "nat (of_nat n) = n"
1.63 +lemma nat_int [simp]: "nat (int n) = n"
1.64 by (simp add: nat int_def)
1.65
1.66 -lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
1.67 +lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
1.68 by (cases z) (simp add: nat le int_def Zero_int_def)
1.69
1.70 -corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
1.71 +corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
1.72 by simp
1.73
1.74 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
1.75 @@ -431,14 +434,14 @@
1.76
1.77 lemma nonneg_eq_int:
1.78 fixes z :: int
1.79 - assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
1.80 + assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
1.81 shows P
1.82 using assms by (blast dest: nat_0_le sym)
1.83
1.84 -lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
1.85 +lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
1.86 by (cases w) (simp add: nat le int_def Zero_int_def, arith)
1.87
1.88 -corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
1.89 +corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
1.90 by (simp only: eq_commute [of m] nat_eq_iff)
1.91
1.92 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
1.93 @@ -446,7 +449,7 @@
1.94 apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)
1.95 done
1.96
1.97 -lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> of_nat n"
1.98 +lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
1.99 by (cases x, simp add: nat le int_def le_diff_conv)
1.100
1.101 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
1.102 @@ -470,10 +473,10 @@
1.103 by (cases z, cases z')
1.104 (simp add: nat add minus diff_minus le Zero_int_def)
1.105
1.106 -lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
1.107 +lemma nat_zminus_int [simp]: "nat (- int n) = 0"
1.108 by (simp add: int_def minus nat Zero_int_def)
1.109
1.110 -lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
1.111 +lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
1.112 by (cases z) (simp add: nat less int_def, arith)
1.113
1.114 context ring_1
1.115 @@ -491,31 +494,31 @@
1.116
1.117 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
1.118
1.119 -lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
1.120 +lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
1.121 by (simp add: order_less_le del: of_nat_Suc)
1.122
1.123 -lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
1.124 +lemma negative_zless [iff]: "- (int (Suc n)) < int m"
1.125 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
1.126
1.127 -lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
1.128 +lemma negative_zle_0: "- int n \<le> 0"
1.129 by (simp add: minus_le_iff)
1.130
1.131 -lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
1.132 +lemma negative_zle [iff]: "- int n \<le> int m"
1.133 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
1.134
1.135 -lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
1.136 +lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
1.137 by (subst le_minus_iff, simp del: of_nat_Suc)
1.138
1.139 -lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
1.140 +lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
1.141 by (simp add: int_def le minus Zero_int_def)
1.142
1.143 -lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
1.144 +lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
1.145 by (simp add: linorder_not_less)
1.146
1.147 -lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
1.148 +lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
1.149 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
1.150
1.151 -lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
1.152 +lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
1.153 proof -
1.154 have "(w \<le> z) = (0 \<le> z - w)"
1.155 by (simp only: le_diff_eq add_0_left)
1.156 @@ -526,10 +529,10 @@
1.157 finally show ?thesis .
1.158 qed
1.159
1.160 -lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
1.161 +lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
1.162 by simp
1.163
1.164 -lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
1.165 +lemma int_Suc0_eq_1: "int (Suc 0) = 1"
1.166 by simp
1.167
1.168 text{*This version is proved for all ordered rings, not just integers!
1.169 @@ -540,7 +543,7 @@
1.170 "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
1.171 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
1.172
1.173 -lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
1.174 +lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
1.175 apply (cases x)
1.176 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
1.177 apply (rule_tac x="y - Suc x" in exI, arith)
1.178 @@ -553,7 +556,7 @@
1.179 whether an integer is negative or not.*}
1.180
1.181 theorem int_cases [case_names nonneg neg, cases type: int]:
1.182 - "[|!! n. (z \<Colon> int) = of_nat n ==> P; !! n. z = - (of_nat (Suc n)) ==> P |] ==> P"
1.183 + "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P"
1.184 apply (cases "z < 0")
1.185 apply (blast dest!: negD)
1.186 apply (simp add: linorder_not_less del: of_nat_Suc)
1.187 @@ -562,12 +565,12 @@
1.188 done
1.189
1.190 theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
1.191 - "[|!! n. P (of_nat n \<Colon> int); !!n. P (- (of_nat (Suc n))) |] ==> P z"
1.192 + "[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z"
1.193 by (cases z) auto
1.194
1.195 text{*Contributed by Brian Huffman*}
1.196 theorem int_diff_cases:
1.197 - obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
1.198 + obtains (diff) m n where "z = int m - int n"
1.199 apply (cases z rule: eq_Abs_Integ)
1.200 apply (rule_tac m=x and n=y in diff)
1.201 apply (simp add: int_def minus add diff_minus)
1.202 @@ -944,11 +947,11 @@
1.203 assumes number_of_eq: "number_of k = of_int k"
1.204
1.205 class number_semiring = number + comm_semiring_1 +
1.206 - assumes number_of_int: "number_of (of_nat n) = of_nat n"
1.207 + assumes number_of_int: "number_of (int n) = of_nat n"
1.208
1.209 instance number_ring \<subseteq> number_semiring
1.210 proof
1.211 - fix n show "number_of (of_nat n) = (of_nat n :: 'a)"
1.212 + fix n show "number_of (int n) = (of_nat n :: 'a)"
1.213 unfolding number_of_eq by (rule of_int_of_nat_eq)
1.214 qed
1.215
1.216 @@ -1124,7 +1127,7 @@
1.217 show ?thesis
1.218 proof
1.219 assume eq: "1 + z + z = 0"
1.220 - have "(0::int) < 1 + (of_nat n + of_nat n)"
1.221 + have "(0::int) < 1 + (int n + int n)"
1.222 by (simp add: le_imp_0_less add_increasing)
1.223 also have "... = - (1 + z + z)"
1.224 by (simp add: neg add_assoc [symmetric])
1.225 @@ -1644,7 +1647,7 @@
1.226 lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
1.227
1.228 lemma split_nat [arith_split]:
1.229 - "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
1.230 + "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
1.231 (is "?P = (?L & ?R)")
1.232 proof (cases "i < 0")
1.233 case True thus ?thesis by auto
1.234 @@ -1737,11 +1740,6 @@
1.235 by (rule wf_subset [OF wf_measure])
1.236 qed
1.237
1.238 -abbreviation
1.239 - int :: "nat \<Rightarrow> int"
1.240 -where
1.241 - "int \<equiv> of_nat"
1.242 -
1.243 (* `set:int': dummy construction *)
1.244 theorem int_ge_induct [case_names base step, induct set: int]:
1.245 fixes i :: int