src/HOL/Int.thy
author boehmes
Thu Jun 08 23:37:01 2017 +0200 (2017-06-08)
changeset 66035 de6cd60b1226
parent 64996 b316cd527a11
child 66816 212a3334e7da
permissions -rw-r--r--
replace non-arithmetic terms by fresh variables before replaying linear-arithmetic proofs: avoid failed proof replays due to an overambitious simpset which may cause proof replay to diverge from the pre-computed proof trace
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(*  Title:      HOL/Int.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
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*)
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section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>
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theory Int
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  imports Equiv_Relations Power Quotient Fun_Def
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begin
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subsection \<open>Definition of integers as a quotient type\<close>
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
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  where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
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  by (simp add: intrel_def)
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quotient_type int = "nat \<times> nat" / "intrel"
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  morphisms Rep_Integ Abs_Integ
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proof (rule equivpI)
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  show "reflp intrel" by (auto simp: reflp_def)
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  show "symp intrel" by (auto simp: symp_def)
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  show "transp intrel" by (auto simp: transp_def)
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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  "(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
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  by (induct z) auto
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subsection \<open>Integers form a commutative ring\<close>
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instantiation int :: comm_ring_1
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begin
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lift_definition zero_int :: "int" is "(0, 0)" .
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lift_definition one_int :: "int" is "(1, 0)" .
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
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  by clarsimp
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lift_definition uminus_int :: "int \<Rightarrow> int"
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  is "\<lambda>(x, y). (y, x)"
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  by clarsimp
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
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  by clarsimp
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
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proof (clarsimp)
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  fix s t u v w x y z :: nat
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  assume "s + v = u + t" and "w + z = y + x"
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  then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
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    (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
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    by simp
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  then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
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    by (simp add: algebra_simps)
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qed
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instance
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  by standard (transfer; clarsimp simp: algebra_simps)+
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end
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abbreviation int :: "nat \<Rightarrow> int"
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  where "int \<equiv> of_nat"
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lemma int_def: "int n = Abs_Integ (n, 0)"
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  by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]: "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
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  by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)
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lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
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  by transfer clarsimp
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subsection \<open>Integers are totally ordered\<close>
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instantiation int :: linorder
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begin
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
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  by auto
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v < u + y"
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  by auto
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instance
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  by standard (transfer, force)+
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end
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instantiation int :: distrib_lattice
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begin
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definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
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definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)
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end
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subsection \<open>Ordering properties of arithmetic operations\<close>
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instance int :: ordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by transfer clarsimp
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qed
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text \<open>Strict Monotonicity of Multiplication.\<close>
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text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close>
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lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j"
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  for i j :: int
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proof (induct k)
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  case 0
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  then show ?case by simp
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next
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  case (Suc k)
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  then show ?case
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    by (cases "k = 0") (simp_all add: distrib_right add_strict_mono)
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qed
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lemma zero_le_imp_eq_int: "0 \<le> k \<Longrightarrow> \<exists>n. k = int n"
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  for k :: int
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  apply transfer
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  apply clarsimp
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  apply (rule_tac x="a - b" in exI)
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  apply simp
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  done
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lemma zero_less_imp_eq_int: "0 < k \<Longrightarrow> \<exists>n>0. k = int n"
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  for k :: int
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  apply transfer
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  apply clarsimp
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  apply (rule_tac x="a - b" in exI)
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  apply simp
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  done
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lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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  for i j k :: int
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  by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)
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text \<open>The integers form an ordered integral domain.\<close>
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instantiation int :: linordered_idom
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begin
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definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
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definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
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instance
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proof
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  fix i j k :: int
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  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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    by (rule zmult_zless_mono2)
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  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
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    by (simp only: zabs_def)
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  show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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    by (simp only: zsgn_def)
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qed
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end
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lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z"
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  for w z :: int
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
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  for w z :: int
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  apply transfer
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  apply auto
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  apply (rename_tac a b c d)
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  apply (rule_tac x="c+b - Suc(a+d)" in exI)
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  apply arith
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  done
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lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs")
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  for z :: int
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proof
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  assume ?rhs
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  then show ?lhs by simp
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next
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  assume ?lhs
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  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp
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  then have "\<bar>z\<bar> \<le> 0" by simp
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  then show ?rhs by simp
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qed
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lemmas int_distrib =
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  distrib_right [of z1 z2 w]
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  distrib_left [of w z1 z2]
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  left_diff_distrib [of z1 z2 w]
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  right_diff_distrib [of w z1 z2]
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  for z1 z2 w :: int
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subsection \<open>Embedding of the Integers into any \<open>ring_1\<close>: \<open>of_int\<close>\<close>
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context ring_1
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begin
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lift_definition of_int :: "int \<Rightarrow> 'a"
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  is "\<lambda>(i, j). of_nat i - of_nat j"
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  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
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      of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_0 [simp]: "of_int 0 = 0"
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  by transfer simp
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lemma of_int_1 [simp]: "of_int 1 = 1"
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  by transfer simp
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lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
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  by transfer (clarsimp simp add: algebra_simps)
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lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
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  by (transfer fixing: uminus) clarsimp
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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  using of_int_add [of w "- z"] by simp
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
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  by (transfer fixing: times) (clarsimp simp add: algebra_simps)
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lemma mult_of_int_commute: "of_int x * y = y * of_int x"
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  by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)
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text \<open>Collapse nested embeddings.\<close>
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lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
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  by (induct n) auto
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lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
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  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
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lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
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  by simp
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lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
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  by (induct n) simp_all
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end
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context ring_char_0
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begin
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lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"
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  by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
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text \<open>Special cases where either operand is zero.\<close>
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lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of z 0] by simp
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lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of 0 z] by simp
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lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
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  using of_int_eq_iff [of z 1] by simp
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end
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context linordered_idom
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begin
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text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
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subclass ring_char_0 ..
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lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
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  by (transfer fixing: less_eq)
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    (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"
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  by (simp add: less_le order_less_le)
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lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
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  using of_int_le_iff [of 0 z] by simp
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lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
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  using of_int_le_iff [of z 0] by simp
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lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
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  using of_int_less_iff [of 0 z] by simp
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lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
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  using of_int_less_iff [of z 0] by simp
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lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
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  using of_int_le_iff [of 1 z] by simp
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lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
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  using of_int_le_iff [of z 1] by simp
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lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
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  using of_int_less_iff [of 1 z] by simp
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lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
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  using of_int_less_iff [of z 1] by simp
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lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
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  by simp
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lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
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  by simp
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lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
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  by (auto simp add: abs_if)
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lemma of_int_lessD:
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  assumes "\<bar>of_int n\<bar> < x"
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  shows "n = 0 \<or> x > 1"
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proof (cases "n = 0")
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  case True
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   329
  then show ?thesis by simp
haftmann@62347
   330
next
haftmann@62347
   331
  case False
haftmann@62347
   332
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   333
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   334
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   335
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   336
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   337
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   338
  then have "1 < x" using assms by (rule le_less_trans)
haftmann@62347
   339
  then show ?thesis ..
haftmann@62347
   340
qed
haftmann@62347
   341
haftmann@62347
   342
lemma of_int_leD:
haftmann@62347
   343
  assumes "\<bar>of_int n\<bar> \<le> x"
haftmann@62347
   344
  shows "n = 0 \<or> 1 \<le> x"
haftmann@62347
   345
proof (cases "n = 0")
wenzelm@63652
   346
  case True
wenzelm@63652
   347
  then show ?thesis by simp
haftmann@62347
   348
next
haftmann@62347
   349
  case False
haftmann@62347
   350
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   351
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   352
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   353
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   354
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   355
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   356
  then have "1 \<le> x" using assms by (rule order_trans)
haftmann@62347
   357
  then show ?thesis ..
haftmann@62347
   358
qed
haftmann@62347
   359
haftmann@36424
   360
end
haftmann@25919
   361
lp15@61234
   362
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   363
wenzelm@63652
   364
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
lp15@61234
   365
  using of_int_eq_iff by fastforce
lp15@61234
   366
lp15@61649
   367
lemma of_int_le_numeral_iff [simp]:
wenzelm@63652
   368
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   369
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   370
lp15@61649
   371
lemma of_int_numeral_le_iff [simp]:
wenzelm@63652
   372
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   373
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   374
lp15@61649
   375
lemma of_int_less_numeral_iff [simp]:
wenzelm@63652
   376
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
lp15@61234
   377
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   378
lp15@61649
   379
lemma of_int_numeral_less_iff [simp]:
wenzelm@63652
   380
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   381
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   382
wenzelm@63652
   383
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   384
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   385
haftmann@25919
   386
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   387
proof
wenzelm@63652
   388
  show "of_int z = id z" for z
wenzelm@63652
   389
    by (cases z rule: int_diff_cases) simp
haftmann@25919
   390
qed
haftmann@25919
   391
hoelzl@51329
   392
instance int :: no_top
wenzelm@61169
   393
  apply standard
hoelzl@51329
   394
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   395
  apply simp
hoelzl@51329
   396
  done
hoelzl@51329
   397
hoelzl@51329
   398
instance int :: no_bot
wenzelm@61169
   399
  apply standard
hoelzl@51329
   400
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   401
  apply simp
hoelzl@51329
   402
  done
hoelzl@51329
   403
wenzelm@63652
   404
wenzelm@61799
   405
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   406
huffman@48045
   407
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   408
  by auto
haftmann@25919
   409
huffman@44709
   410
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   411
  by transfer simp
haftmann@25919
   412
huffman@44709
   413
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   414
  by transfer clarsimp
haftmann@25919
   415
wenzelm@63652
   416
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
wenzelm@63652
   417
  by simp
haftmann@25919
   418
wenzelm@63652
   419
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
huffman@48045
   420
  by transfer clarsimp
haftmann@25919
   421
wenzelm@63652
   422
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
huffman@48045
   423
  by transfer (clarsimp, arith)
haftmann@25919
   424
wenzelm@63652
   425
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
wenzelm@63652
   426
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   427
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   428
wenzelm@63652
   429
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   430
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   431
wenzelm@63652
   432
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
huffman@48045
   433
  by transfer (clarsimp, arith)
haftmann@25919
   434
haftmann@64714
   435
lemma nonneg_int_cases:
haftmann@64714
   436
  assumes "0 \<le> k"
haftmann@64714
   437
  obtains n where "k = int n"
haftmann@64714
   438
proof -
haftmann@64714
   439
  from assms have "k = int (nat k)"
haftmann@64714
   440
    by simp
haftmann@64714
   441
  then show thesis
haftmann@64714
   442
    by (rule that)
haftmann@64714
   443
qed
haftmann@64714
   444
haftmann@64714
   445
lemma pos_int_cases:
haftmann@64714
   446
  assumes "0 < k"
haftmann@64714
   447
  obtains n where "k = int n" and "n > 0"
haftmann@64714
   448
proof -
haftmann@64714
   449
  from assms have "0 \<le> k"
haftmann@64714
   450
    by simp
haftmann@64714
   451
  then obtain n where "k = int n"
haftmann@64714
   452
    by (rule nonneg_int_cases)
haftmann@64714
   453
  moreover have "n > 0"
haftmann@64714
   454
    using \<open>k = int n\<close> assms by simp
haftmann@64714
   455
  ultimately show thesis
haftmann@64714
   456
    by (rule that)
haftmann@64714
   457
qed
haftmann@64714
   458
haftmann@64714
   459
lemma nonpos_int_cases:
haftmann@64714
   460
  assumes "k \<le> 0"
haftmann@64714
   461
  obtains n where "k = - int n"
haftmann@64714
   462
proof -
haftmann@64714
   463
  from assms have "- k \<ge> 0"
haftmann@64714
   464
    by simp
haftmann@64714
   465
  then obtain n where "- k = int n"
haftmann@64714
   466
    by (rule nonneg_int_cases)
haftmann@64714
   467
  then have "k = - int n"
haftmann@64714
   468
    by simp
haftmann@64714
   469
  then show thesis
haftmann@64714
   470
    by (rule that)
haftmann@64714
   471
qed
haftmann@64714
   472
haftmann@64714
   473
lemma neg_int_cases:
haftmann@64714
   474
  assumes "k < 0"
haftmann@64714
   475
  obtains n where "k = - int n" and "n > 0"
haftmann@64714
   476
proof -
haftmann@64714
   477
  from assms have "- k > 0"
haftmann@64714
   478
    by simp
haftmann@64714
   479
  then obtain n where "- k = int n" and "- k > 0"
haftmann@64714
   480
    by (blast elim: pos_int_cases)
haftmann@64714
   481
  then have "k = - int n" and "n > 0"
haftmann@64714
   482
    by simp_all
haftmann@64714
   483
  then show thesis
haftmann@64714
   484
    by (rule that)
haftmann@64714
   485
qed
haftmann@25919
   486
wenzelm@63652
   487
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   488
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   489
wenzelm@63652
   490
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   491
  using nat_eq_iff [of w m] by auto
haftmann@54223
   492
wenzelm@63652
   493
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@54223
   494
  by (simp add: nat_eq_iff)
haftmann@25919
   495
wenzelm@63652
   496
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@54223
   497
  by (simp add: nat_eq_iff)
haftmann@54223
   498
wenzelm@63652
   499
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
haftmann@54223
   500
  by (simp add: nat_eq_iff)
haftmann@25919
   501
wenzelm@63652
   502
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
haftmann@54223
   503
  by simp
haftmann@54223
   504
haftmann@54223
   505
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   506
  by simp
lp15@60162
   507
wenzelm@63652
   508
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
huffman@48045
   509
  by transfer (clarsimp, arith)
haftmann@25919
   510
huffman@44709
   511
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   512
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   513
huffman@44707
   514
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   515
  by transfer auto
huffman@44707
   516
wenzelm@63652
   517
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
wenzelm@63652
   518
  for i :: int
huffman@48045
   519
  by transfer clarsimp
nipkow@29700
   520
wenzelm@63652
   521
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
wenzelm@63652
   522
  by (auto simp add: nat_eq_iff2)
haftmann@25919
   523
wenzelm@63652
   524
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
wenzelm@63652
   525
  using zless_nat_conj [of 0] by auto
haftmann@25919
   526
wenzelm@63652
   527
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   528
  by transfer clarsimp
haftmann@25919
   529
wenzelm@63652
   530
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   531
  by transfer clarsimp
lp15@60162
   532
wenzelm@63652
   533
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   534
  by (rule nat_diff_distrib') auto
haftmann@25919
   535
huffman@44709
   536
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   537
  by transfer simp
haftmann@25919
   538
wenzelm@63652
   539
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   540
  by transfer auto
lp15@60162
   541
wenzelm@63652
   542
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
huffman@48045
   543
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   544
wenzelm@63652
   545
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   546
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   547
wenzelm@63652
   548
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   549
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   550
haftmann@54249
   551
wenzelm@60758
   552
text \<open>For termination proofs:\<close>
wenzelm@63652
   553
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
krauss@29779
   554
haftmann@25919
   555
wenzelm@63652
   556
subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   557
wenzelm@61076
   558
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
wenzelm@63652
   559
  by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   560
huffman@44709
   561
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
wenzelm@63652
   562
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   563
huffman@44709
   564
lemma negative_zle_0: "- int n \<le> 0"
wenzelm@63652
   565
  by (simp add: minus_le_iff)
haftmann@25919
   566
huffman@44709
   567
lemma negative_zle [iff]: "- int n \<le> int m"
wenzelm@63652
   568
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   569
wenzelm@63652
   570
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
wenzelm@63652
   571
  by (subst le_minus_iff) (simp del: of_nat_Suc)
haftmann@25919
   572
wenzelm@63652
   573
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
huffman@48045
   574
  by transfer simp
haftmann@25919
   575
wenzelm@63652
   576
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
wenzelm@63652
   577
  by (simp add: linorder_not_less)
haftmann@25919
   578
wenzelm@63652
   579
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
wenzelm@63652
   580
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   581
wenzelm@63652
   582
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
wenzelm@63652
   583
  (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@62348
   584
proof
wenzelm@63652
   585
  assume ?rhs
wenzelm@63652
   586
  then show ?lhs by auto
haftmann@62348
   587
next
wenzelm@63652
   588
  assume ?lhs
haftmann@62348
   589
  then have "0 \<le> z - w" by simp
haftmann@62348
   590
  then obtain n where "z - w = int n"
haftmann@62348
   591
    using zero_le_imp_eq_int [of "z - w"] by blast
wenzelm@63652
   592
  then have "z = w + int n" by simp
wenzelm@63652
   593
  then show ?rhs ..
haftmann@25919
   594
qed
haftmann@25919
   595
huffman@44709
   596
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
wenzelm@63652
   597
  by simp
haftmann@25919
   598
wenzelm@63652
   599
text \<open>
wenzelm@63652
   600
  This version is proved for all ordered rings, not just integers!
wenzelm@63652
   601
  It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@63652
   602
  in theory \<open>Rings\<close>.
wenzelm@63652
   603
  But is it really better than just rewriting with \<open>abs_if\<close>?
wenzelm@63652
   604
\<close>
wenzelm@63652
   605
lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
wenzelm@63652
   606
  for a :: "'a::linordered_idom"
wenzelm@63652
   607
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   608
huffman@44709
   609
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
wenzelm@63652
   610
  apply transfer
wenzelm@63652
   611
  apply clarsimp
wenzelm@63652
   612
  apply (rule_tac x="b - Suc a" in exI)
wenzelm@63652
   613
  apply arith
wenzelm@63652
   614
  done
wenzelm@63652
   615
haftmann@25919
   616
wenzelm@60758
   617
subsection \<open>Cases and induction\<close>
haftmann@25919
   618
wenzelm@63652
   619
text \<open>
wenzelm@63652
   620
  Now we replace the case analysis rule by a more conventional one:
wenzelm@63652
   621
  whether an integer is negative or not.
wenzelm@63652
   622
\<close>
haftmann@25919
   623
wenzelm@63652
   624
text \<open>This version is symmetric in the two subgoals.\<close>
wenzelm@63652
   625
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
wenzelm@63652
   626
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   627
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   628
wenzelm@63652
   629
text \<open>This is the default, with a negative case.\<close>
wenzelm@63652
   630
lemma int_cases [case_names nonneg neg, cases type: int]:
wenzelm@63652
   631
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   632
  apply (cases "z < 0")
wenzelm@63652
   633
   apply (blast dest!: negD)
wenzelm@63652
   634
  apply (simp add: linorder_not_less del: of_nat_Suc)
wenzelm@63652
   635
  apply auto
wenzelm@63652
   636
  apply (blast dest: nat_0_le [THEN sym])
wenzelm@63652
   637
  done
haftmann@25919
   638
haftmann@60868
   639
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   640
  fixes k :: int
haftmann@60868
   641
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   642
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   643
  shows "P"
haftmann@60868
   644
proof (cases k "0::int" rule: linorder_cases)
wenzelm@63652
   645
  case equal
wenzelm@63652
   646
  with assms(1) show P by simp
haftmann@60868
   647
next
haftmann@60868
   648
  case greater
wenzelm@63539
   649
  then have *: "nat k > 0" by simp
wenzelm@63539
   650
  moreover from * have "k = int (nat k)" by auto
haftmann@60868
   651
  ultimately show P using assms(2) by blast
haftmann@60868
   652
next
haftmann@60868
   653
  case less
wenzelm@63539
   654
  then have *: "nat (- k) > 0" by simp
wenzelm@63539
   655
  moreover from * have "k = - int (nat (- k))" by auto
haftmann@60868
   656
  ultimately show P using assms(3) by blast
haftmann@60868
   657
qed
haftmann@60868
   658
wenzelm@63652
   659
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
wenzelm@63652
   660
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
wenzelm@42676
   661
  by (cases z) auto
haftmann@25919
   662
huffman@47108
   663
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   664
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   665
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   666
haftmann@54489
   667
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   668
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   669
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   670
wenzelm@61799
   671
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   672
huffman@47108
   673
lemmas max_number_of [simp] =
huffman@47108
   674
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   675
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   676
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   677
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   678
huffman@47108
   679
lemmas min_number_of [simp] =
huffman@47108
   680
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   681
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   682
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   683
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   684
haftmann@25919
   685
wenzelm@60758
   686
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   687
wenzelm@60758
   688
text \<open>Preliminaries\<close>
huffman@28958
   689
lp15@60162
   690
lemma le_imp_0_less:
wenzelm@63652
   691
  fixes z :: int
huffman@28958
   692
  assumes le: "0 \<le> z"
wenzelm@63652
   693
  shows "0 < 1 + z"
huffman@28958
   694
proof -
huffman@28958
   695
  have "0 \<le> z" by fact
wenzelm@63652
   696
  also have "\<dots> < z + 1" by (rule less_add_one)
wenzelm@63652
   697
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
huffman@28958
   698
  finally show "0 < 1 + z" .
huffman@28958
   699
qed
huffman@28958
   700
wenzelm@63652
   701
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
wenzelm@63652
   702
  for z :: int
wenzelm@42676
   703
proof (cases z)
huffman@28958
   704
  case (nonneg n)
wenzelm@63652
   705
  then show ?thesis
wenzelm@63652
   706
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   707
next
huffman@28958
   708
  case (neg n)
wenzelm@63652
   709
  then show ?thesis
wenzelm@63652
   710
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@63652
   711
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   712
qed
huffman@28958
   713
wenzelm@63652
   714
wenzelm@60758
   715
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   716
haftmann@25919
   717
lemmas double_eq_0_iff = double_zero
haftmann@25919
   718
wenzelm@63652
   719
lemma odd_nonzero: "1 + z + z \<noteq> 0"
wenzelm@63652
   720
  for z :: int
wenzelm@42676
   721
proof (cases z)
haftmann@25919
   722
  case (nonneg n)
wenzelm@63652
   723
  have le: "0 \<le> z + z"
wenzelm@63652
   724
    by (simp add: nonneg add_increasing)
wenzelm@63652
   725
  then show ?thesis
wenzelm@63652
   726
    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
haftmann@25919
   727
next
haftmann@25919
   728
  case (neg n)
haftmann@25919
   729
  show ?thesis
haftmann@25919
   730
  proof
haftmann@25919
   731
    assume eq: "1 + z + z = 0"
wenzelm@63652
   732
    have "0 < 1 + (int n + int n)"
lp15@60162
   733
      by (simp add: le_imp_0_less add_increasing)
wenzelm@63652
   734
    also have "\<dots> = - (1 + z + z)"
lp15@60162
   735
      by (simp add: neg add.assoc [symmetric])
wenzelm@63652
   736
    also have "\<dots> = 0" by (simp add: eq)
haftmann@25919
   737
    finally have "0<0" ..
wenzelm@63652
   738
    then show False by blast
haftmann@25919
   739
  qed
haftmann@25919
   740
qed
haftmann@25919
   741
haftmann@30652
   742
wenzelm@60758
   743
subsection \<open>The Set of Integers\<close>
haftmann@25919
   744
haftmann@25919
   745
context ring_1
haftmann@25919
   746
begin
haftmann@25919
   747
wenzelm@61070
   748
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   749
  where "\<int> = range of_int"
haftmann@25919
   750
huffman@35634
   751
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   752
  by (simp add: Ints_def)
huffman@35634
   753
huffman@35634
   754
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   755
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   756
haftmann@25919
   757
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   758
  using Ints_of_int [of "0"] by simp
haftmann@25919
   759
haftmann@25919
   760
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   761
  using Ints_of_int [of "1"] by simp
haftmann@25919
   762
eberlm@61552
   763
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   764
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   765
haftmann@25919
   766
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
wenzelm@63652
   767
  apply (auto simp add: Ints_def)
wenzelm@63652
   768
  apply (rule range_eqI)
wenzelm@63652
   769
  apply (rule of_int_add [symmetric])
wenzelm@63652
   770
  done
haftmann@25919
   771
haftmann@25919
   772
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
wenzelm@63652
   773
  apply (auto simp add: Ints_def)
wenzelm@63652
   774
  apply (rule range_eqI)
wenzelm@63652
   775
  apply (rule of_int_minus [symmetric])
wenzelm@63652
   776
  done
haftmann@25919
   777
huffman@35634
   778
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
wenzelm@63652
   779
  apply (auto simp add: Ints_def)
wenzelm@63652
   780
  apply (rule range_eqI)
wenzelm@63652
   781
  apply (rule of_int_diff [symmetric])
wenzelm@63652
   782
  done
huffman@35634
   783
haftmann@25919
   784
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
wenzelm@63652
   785
  apply (auto simp add: Ints_def)
wenzelm@63652
   786
  apply (rule range_eqI)
wenzelm@63652
   787
  apply (rule of_int_mult [symmetric])
wenzelm@63652
   788
  done
haftmann@25919
   789
huffman@35634
   790
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
wenzelm@63652
   791
  by (induct n) simp_all
huffman@35634
   792
haftmann@25919
   793
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   794
  assumes "q \<in> \<int>"
haftmann@25919
   795
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   796
  unfolding Ints_def
haftmann@25919
   797
proof -
wenzelm@60758
   798
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   799
  then obtain z where "q = of_int z" ..
haftmann@25919
   800
  then show thesis ..
haftmann@25919
   801
qed
haftmann@25919
   802
haftmann@25919
   803
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   804
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   805
  by (rule Ints_cases) auto
haftmann@25919
   806
eberlm@61524
   807
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   808
  unfolding Nats_def Ints_def
eberlm@61524
   809
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   810
eberlm@61524
   811
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   812
proof (intro subsetI equalityI)
wenzelm@63652
   813
  fix x :: 'a
wenzelm@63652
   814
  assume "x \<in> {of_int n |n. n \<ge> 0}"
wenzelm@63652
   815
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   816
    by (auto elim!: Ints_cases)
wenzelm@63652
   817
  then have "x = of_nat (nat n)"
wenzelm@63652
   818
    by (subst of_nat_nat) simp_all
wenzelm@63652
   819
  then show "x \<in> \<nat>"
wenzelm@63652
   820
    by simp
eberlm@61524
   821
next
wenzelm@63652
   822
  fix x :: 'a
wenzelm@63652
   823
  assume "x \<in> \<nat>"
wenzelm@63652
   824
  then obtain n where "x = of_nat n"
wenzelm@63652
   825
    by (auto elim!: Nats_cases)
wenzelm@63652
   826
  then have "x = of_int (int n)" by simp
eberlm@61524
   827
  also have "int n \<ge> 0" by simp
wenzelm@63652
   828
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   829
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   830
qed
eberlm@61524
   831
haftmann@25919
   832
end
haftmann@25919
   833
lp15@64758
   834
lemma (in linordered_idom) Ints_abs [simp]:
lp15@64758
   835
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
lp15@64758
   836
  by (auto simp: abs_if)
lp15@64758
   837
eberlm@61524
   838
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   839
proof (intro subsetI equalityI)
wenzelm@63652
   840
  fix x :: 'a
wenzelm@63652
   841
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
wenzelm@63652
   842
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   843
    by (auto elim!: Ints_cases)
wenzelm@63652
   844
  then have "x = of_nat (nat n)"
wenzelm@63652
   845
    by (subst of_nat_nat) simp_all
wenzelm@63652
   846
  then show "x \<in> \<nat>"
wenzelm@63652
   847
    by simp
eberlm@61524
   848
qed (auto elim!: Nats_cases)
eberlm@61524
   849
haftmann@64849
   850
lemma (in idom_divide) of_int_divide_in_Ints: 
haftmann@64849
   851
  "of_int a div of_int b \<in> \<int>" if "b dvd a"
haftmann@64849
   852
proof -
haftmann@64849
   853
  from that obtain c where "a = b * c" ..
haftmann@64849
   854
  then show ?thesis
haftmann@64849
   855
    by (cases "of_int b = 0") simp_all
haftmann@64849
   856
qed
eberlm@61524
   857
wenzelm@60758
   858
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
   859
haftmann@25919
   860
lemma Ints_double_eq_0_iff:
wenzelm@63652
   861
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   862
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   863
  shows "a + a = 0 \<longleftrightarrow> a = 0"
wenzelm@63652
   864
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@25919
   865
proof -
wenzelm@63652
   866
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   867
    unfolding Ints_def [symmetric] .
haftmann@25919
   868
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   869
  show ?thesis
haftmann@25919
   870
  proof
wenzelm@63652
   871
    assume ?rhs
wenzelm@63652
   872
    then show ?lhs by simp
haftmann@25919
   873
  next
wenzelm@63652
   874
    assume ?lhs
wenzelm@63652
   875
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   876
    then have "z + z = 0" by (simp only: of_int_eq_iff)
wenzelm@63652
   877
    then have "z = 0" by (simp only: double_eq_0_iff)
wenzelm@63652
   878
    with a show ?rhs by simp
haftmann@25919
   879
  qed
haftmann@25919
   880
qed
haftmann@25919
   881
haftmann@25919
   882
lemma Ints_odd_nonzero:
wenzelm@63652
   883
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   884
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   885
  shows "1 + a + a \<noteq> 0"
haftmann@25919
   886
proof -
wenzelm@63652
   887
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   888
    unfolding Ints_def [symmetric] .
haftmann@25919
   889
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   890
  show ?thesis
haftmann@25919
   891
  proof
wenzelm@63652
   892
    assume "1 + a + a = 0"
wenzelm@63652
   893
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   894
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   895
    with odd_nonzero show False by blast
haftmann@25919
   896
  qed
lp15@60162
   897
qed
haftmann@25919
   898
wenzelm@61070
   899
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
   900
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   901
lp15@60162
   902
lemma Ints_odd_less_0:
wenzelm@63652
   903
  fixes a :: "'a::linordered_idom"
wenzelm@61070
   904
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   905
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25919
   906
proof -
wenzelm@63652
   907
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   908
    unfolding Ints_def [symmetric] .
haftmann@25919
   909
  then obtain z where a: "a = of_int z" ..
wenzelm@63652
   910
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
wenzelm@63652
   911
    by simp
wenzelm@63652
   912
  also have "\<dots> \<longleftrightarrow> z < 0"
wenzelm@63652
   913
    by (simp only: of_int_less_iff odd_less_0_iff)
wenzelm@63652
   914
  also have "\<dots> \<longleftrightarrow> a < 0"
haftmann@25919
   915
    by (simp add: a)
haftmann@25919
   916
  finally show ?thesis .
haftmann@25919
   917
qed
haftmann@25919
   918
haftmann@25919
   919
nipkow@64272
   920
subsection \<open>@{term sum} and @{term prod}\<close>
haftmann@25919
   921
nipkow@64267
   922
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
wenzelm@63652
   923
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   924
nipkow@64267
   925
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
wenzelm@63652
   926
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   927
nipkow@64272
   928
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
wenzelm@63652
   929
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   930
nipkow@64272
   931
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
wenzelm@63652
   932
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   933
haftmann@25919
   934
wenzelm@60758
   935
text \<open>Legacy theorems\<close>
haftmann@25919
   936
haftmann@64714
   937
lemmas int_sum = of_nat_sum [where 'a=int]
haftmann@64714
   938
lemmas int_prod = of_nat_prod [where 'a=int]
haftmann@25919
   939
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   940
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@64714
   941
lemmas nonneg_eq_int = nonneg_int_cases
haftmann@25919
   942
wenzelm@63652
   943
wenzelm@60758
   944
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
   945
haftmann@54249
   946
lemmas of_int_simps =
haftmann@54249
   947
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   948
wenzelm@48891
   949
ML_file "Tools/int_arith.ML"
wenzelm@60758
   950
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
   951
wenzelm@63652
   952
simproc_setup fast_arith
wenzelm@63652
   953
  ("(m::'a::linordered_idom) < n" |
wenzelm@63652
   954
    "(m::'a::linordered_idom) \<le> n" |
wenzelm@63652
   955
    "(m::'a::linordered_idom) = n") =
wenzelm@61144
   956
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
   957
haftmann@25919
   958
wenzelm@60758
   959
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
   960
wenzelm@63652
   961
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
wenzelm@63652
   962
  for w z :: int
wenzelm@63652
   963
  by arith
haftmann@25919
   964
wenzelm@63652
   965
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
wenzelm@63652
   966
  for w z :: int
wenzelm@63652
   967
  by arith
haftmann@25919
   968
wenzelm@63652
   969
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
wenzelm@63652
   970
  for w z :: int
wenzelm@63652
   971
  by arith
haftmann@25919
   972
wenzelm@63652
   973
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
wenzelm@63652
   974
  for w z :: int
wenzelm@63652
   975
  by arith
haftmann@25919
   976
wenzelm@63652
   977
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
wenzelm@63652
   978
  for z :: int
wenzelm@63652
   979
  by arith
haftmann@25919
   980
lp15@64758
   981
lemma Ints_nonzero_abs_ge1:
lp15@64758
   982
  fixes x:: "'a :: linordered_idom"
lp15@64758
   983
    assumes "x \<in> Ints" "x \<noteq> 0"
lp15@64758
   984
    shows "1 \<le> abs x"
lp15@64758
   985
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
lp15@64758
   986
  fix z::int
lp15@64758
   987
  assume "x = of_int z"
lp15@64758
   988
    with \<open>x \<noteq> 0\<close> 
lp15@64758
   989
  show "1 \<le> \<bar>x\<bar>"
lp15@64758
   990
    apply (auto simp add: abs_if)
lp15@64758
   991
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
lp15@64758
   992
qed
lp15@64758
   993
  
lp15@64758
   994
lemma Ints_nonzero_abs_less1:
lp15@64758
   995
  fixes x:: "'a :: linordered_idom"
lp15@64758
   996
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
lp15@64758
   997
    using Ints_nonzero_abs_ge1 [of x] by auto
lp15@64758
   998
    
haftmann@25919
   999
wenzelm@63652
  1000
subsection \<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
  1001
wenzelm@63652
  1002
text \<open>Simplify the term @{term "w + - z"}.\<close>
haftmann@25919
  1003
wenzelm@63652
  1004
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
lp15@60162
  1005
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
  1006
wenzelm@63652
  1007
text \<open>
wenzelm@63652
  1008
  This simplifies expressions of the form @{term "int n = z"} where
wenzelm@63652
  1009
  \<open>z\<close> is an integer literal.
wenzelm@63652
  1010
\<close>
huffman@47108
  1011
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
  1012
wenzelm@63652
  1013
lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
wenzelm@63652
  1014
  (is "?P = (?L \<and> ?R)")
wenzelm@63652
  1015
  for i :: int
haftmann@25919
  1016
proof (cases "i < 0")
wenzelm@63652
  1017
  case True
wenzelm@63652
  1018
  then show ?thesis by auto
haftmann@25919
  1019
next
haftmann@25919
  1020
  case False
haftmann@25919
  1021
  have "?P = ?L"
haftmann@25919
  1022
  proof
wenzelm@63652
  1023
    assume ?P
wenzelm@63652
  1024
    then show ?L using False by auto
haftmann@25919
  1025
  next
wenzelm@63652
  1026
    assume ?L
wenzelm@63652
  1027
    then show ?P using False by simp
haftmann@25919
  1028
  qed
haftmann@25919
  1029
  with False show ?thesis by simp
haftmann@25919
  1030
qed
haftmann@25919
  1031
hoelzl@59000
  1032
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
  1033
  by auto
hoelzl@59000
  1034
hoelzl@59000
  1035
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
  1036
  by auto
hoelzl@59000
  1037
haftmann@25919
  1038
context ring_1
haftmann@25919
  1039
begin
haftmann@25919
  1040
blanchet@33056
  1041
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1042
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1043
proof (cases "k < 0")
wenzelm@63652
  1044
  case True
wenzelm@63652
  1045
  then have "0 \<le> - k" by simp
haftmann@25919
  1046
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1047
  with True show ?thesis by simp
haftmann@25919
  1048
next
wenzelm@63652
  1049
  case False
wenzelm@63652
  1050
  then show ?thesis by (simp add: not_less)
haftmann@25919
  1051
qed
haftmann@25919
  1052
haftmann@25919
  1053
end
haftmann@25919
  1054
haftmann@64014
  1055
lemma transfer_rule_of_int:
haftmann@64014
  1056
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
haftmann@64014
  1057
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64014
  1058
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64014
  1059
    "rel_fun R R uminus uminus"
haftmann@64014
  1060
  shows "rel_fun HOL.eq R of_int of_int"
haftmann@64014
  1061
proof -
haftmann@64014
  1062
  note transfer_rule_of_nat [transfer_rule]
haftmann@64014
  1063
  have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
haftmann@64014
  1064
    by transfer_prover
haftmann@64014
  1065
  show ?thesis
haftmann@64014
  1066
    by (unfold of_int_of_nat [abs_def]) transfer_prover
haftmann@64014
  1067
qed
haftmann@64014
  1068
haftmann@25919
  1069
lemma nat_mult_distrib:
haftmann@25919
  1070
  fixes z z' :: int
haftmann@25919
  1071
  assumes "0 \<le> z"
haftmann@25919
  1072
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1073
proof (cases "0 \<le> z'")
wenzelm@63652
  1074
  case False
wenzelm@63652
  1075
  with assms have "z * z' \<le> 0"
haftmann@25919
  1076
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1077
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1078
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1079
  ultimately show ?thesis by simp
haftmann@25919
  1080
next
wenzelm@63652
  1081
  case True
wenzelm@63652
  1082
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1083
  show ?thesis
haftmann@25919
  1084
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1085
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1086
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1087
qed
haftmann@25919
  1088
wenzelm@63652
  1089
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
wenzelm@63652
  1090
  for z z' :: int
wenzelm@63652
  1091
  apply (rule trans)
wenzelm@63652
  1092
   apply (rule_tac [2] nat_mult_distrib)
wenzelm@63652
  1093
   apply auto
wenzelm@63652
  1094
  done
haftmann@25919
  1095
wenzelm@61944
  1096
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
wenzelm@63652
  1097
  by (cases "z = 0 \<or> w = 0")
wenzelm@63652
  1098
    (auto simp add: abs_if nat_mult_distrib [symmetric]
wenzelm@63652
  1099
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1100
wenzelm@63652
  1101
lemma int_in_range_abs [simp]: "int n \<in> range abs"
haftmann@60570
  1102
proof (rule range_eqI)
wenzelm@63652
  1103
  show "int n = \<bar>int n\<bar>" by simp
haftmann@60570
  1104
qed
haftmann@60570
  1105
wenzelm@63652
  1106
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
haftmann@60570
  1107
proof -
haftmann@60570
  1108
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1109
    by (cases k) simp_all
haftmann@60570
  1110
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1111
    using that by induct simp
haftmann@60570
  1112
  ultimately show ?thesis by blast
paulson@61204
  1113
qed
haftmann@60570
  1114
wenzelm@63652
  1115
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
wenzelm@63652
  1116
  for z :: int
wenzelm@63652
  1117
  by (rule sym) (simp add: nat_eq_iff)
huffman@47207
  1118
huffman@47207
  1119
lemma diff_nat_eq_if:
wenzelm@63652
  1120
  "nat z - nat z' =
wenzelm@63652
  1121
    (if z' < 0 then nat z
wenzelm@63652
  1122
     else
wenzelm@63652
  1123
      let d = z - z'
wenzelm@63652
  1124
      in if d < 0 then 0 else nat d)"
wenzelm@63652
  1125
  by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1126
wenzelm@63652
  1127
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1128
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1129
haftmann@25919
  1130
wenzelm@63652
  1131
subsection \<open>Induction principles for int\<close>
haftmann@25919
  1132
wenzelm@63652
  1133
text \<open>Well-founded segments of the integers.\<close>
haftmann@25919
  1134
wenzelm@63652
  1135
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1136
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
haftmann@25919
  1137
wenzelm@63652
  1138
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1139
proof -
wenzelm@63652
  1140
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
haftmann@25919
  1141
    by (auto simp add: int_ge_less_than_def)
wenzelm@63652
  1142
  then show ?thesis
lp15@60162
  1143
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1144
qed
haftmann@25919
  1145
wenzelm@63652
  1146
text \<open>
wenzelm@63652
  1147
  This variant looks odd, but is typical of the relations suggested
wenzelm@63652
  1148
  by RankFinder.\<close>
haftmann@25919
  1149
wenzelm@63652
  1150
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1151
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
haftmann@25919
  1152
wenzelm@63652
  1153
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1154
proof -
wenzelm@63652
  1155
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
haftmann@25919
  1156
    by (auto simp add: int_ge_less_than2_def)
wenzelm@63652
  1157
  then show ?thesis
lp15@60162
  1158
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1159
qed
haftmann@25919
  1160
haftmann@25919
  1161
(* `set:int': dummy construction *)
haftmann@25919
  1162
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1163
  fixes i :: int
wenzelm@63652
  1164
  assumes ge: "k \<le> i"
wenzelm@63652
  1165
    and base: "P k"
wenzelm@63652
  1166
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1167
  shows "P i"
haftmann@25919
  1168
proof -
wenzelm@63652
  1169
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
wenzelm@63652
  1170
  proof (induct n)
wenzelm@63652
  1171
    case 0
wenzelm@63652
  1172
    then have "i = k" by arith
wenzelm@63652
  1173
    with base show "P i" by simp
wenzelm@63652
  1174
  next
wenzelm@63652
  1175
    case (Suc n)
wenzelm@63652
  1176
    then have "n = nat ((i - 1) - k)" by arith
wenzelm@63652
  1177
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
wenzelm@63652
  1178
    ultimately have "P (i - 1)" by (rule Suc.hyps)
wenzelm@63652
  1179
    from step [OF k this] show ?case by simp
wenzelm@63652
  1180
  qed
haftmann@25919
  1181
  with ge show ?thesis by fast
haftmann@25919
  1182
qed
haftmann@25919
  1183
haftmann@25928
  1184
(* `set:int': dummy construction *)
haftmann@25928
  1185
theorem int_gr_induct [case_names base step, induct set: int]:
wenzelm@63652
  1186
  fixes i k :: int
wenzelm@63652
  1187
  assumes gr: "k < i"
wenzelm@63652
  1188
    and base: "P (k + 1)"
wenzelm@63652
  1189
    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1190
  shows "P i"
wenzelm@63652
  1191
  apply (rule int_ge_induct[of "k + 1"])
haftmann@25919
  1192
  using gr apply arith
wenzelm@63652
  1193
   apply (rule base)
wenzelm@63652
  1194
  apply (rule step)
wenzelm@63652
  1195
   apply simp_all
wenzelm@63652
  1196
  done
haftmann@25919
  1197
wenzelm@42676
  1198
theorem int_le_induct [consumes 1, case_names base step]:
wenzelm@63652
  1199
  fixes i k :: int
wenzelm@63652
  1200
  assumes le: "i \<le> k"
wenzelm@63652
  1201
    and base: "P k"
wenzelm@63652
  1202
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1203
  shows "P i"
haftmann@25919
  1204
proof -
wenzelm@63652
  1205
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
wenzelm@63652
  1206
  proof (induct n)
wenzelm@63652
  1207
    case 0
wenzelm@63652
  1208
    then have "i = k" by arith
wenzelm@63652
  1209
    with base show "P i" by simp
wenzelm@63652
  1210
  next
wenzelm@63652
  1211
    case (Suc n)
wenzelm@63652
  1212
    then have "n = nat (k - (i + 1))" by arith
wenzelm@63652
  1213
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
wenzelm@63652
  1214
    ultimately have "P (i + 1)" by (rule Suc.hyps)
wenzelm@63652
  1215
    from step[OF k this] show ?case by simp
wenzelm@63652
  1216
  qed
haftmann@25919
  1217
  with le show ?thesis by fast
haftmann@25919
  1218
qed
haftmann@25919
  1219
wenzelm@42676
  1220
theorem int_less_induct [consumes 1, case_names base step]:
wenzelm@63652
  1221
  fixes i k :: int
wenzelm@63652
  1222
  assumes less: "i < k"
wenzelm@63652
  1223
    and base: "P (k - 1)"
wenzelm@63652
  1224
    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1225
  shows "P i"
wenzelm@63652
  1226
  apply (rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1227
  using less apply arith
wenzelm@63652
  1228
   apply (rule base)
wenzelm@63652
  1229
  apply (rule step)
wenzelm@63652
  1230
   apply simp_all
wenzelm@63652
  1231
  done
haftmann@25919
  1232
haftmann@36811
  1233
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1234
  fixes k :: int
haftmann@36801
  1235
  assumes base: "P k"
haftmann@36801
  1236
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1237
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1238
  shows "P i"
haftmann@36801
  1239
proof -
haftmann@36801
  1240
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1241
  then show ?thesis
wenzelm@42676
  1242
  proof
wenzelm@42676
  1243
    assume "i \<ge> k"
wenzelm@63652
  1244
    then show ?thesis
wenzelm@63652
  1245
      using base by (rule int_ge_induct) (fact step1)
haftmann@36801
  1246
  next
wenzelm@42676
  1247
    assume "i \<le> k"
wenzelm@63652
  1248
    then show ?thesis
wenzelm@63652
  1249
      using base by (rule int_le_induct) (fact step2)
haftmann@36801
  1250
  qed
haftmann@36801
  1251
qed
haftmann@36801
  1252
wenzelm@63652
  1253
wenzelm@63652
  1254
subsection \<open>Intermediate value theorems\<close>
haftmann@25919
  1255
wenzelm@63652
  1256
lemma int_val_lemma: "(\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1) \<longrightarrow> f 0 \<le> k \<longrightarrow> k \<le> f n \<longrightarrow> (\<exists>i \<le> n. f i = k)"
wenzelm@63652
  1257
  for n :: nat and k :: int
wenzelm@63652
  1258
  unfolding One_nat_def
wenzelm@63652
  1259
  apply (induct n)
wenzelm@63652
  1260
   apply simp
wenzelm@63652
  1261
  apply (intro strip)
wenzelm@63652
  1262
  apply (erule impE)
wenzelm@63652
  1263
   apply simp
wenzelm@63652
  1264
  apply (erule_tac x = n in allE)
wenzelm@63652
  1265
  apply simp
wenzelm@63652
  1266
  apply (case_tac "k = f (Suc n)")
wenzelm@63652
  1267
   apply force
wenzelm@63652
  1268
  apply (erule impE)
wenzelm@63652
  1269
   apply (simp add: abs_if split: if_split_asm)
wenzelm@63652
  1270
  apply (blast intro: le_SucI)
wenzelm@63652
  1271
  done
haftmann@25919
  1272
haftmann@25919
  1273
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1274
haftmann@25919
  1275
lemma nat_intermed_int_val:
wenzelm@63652
  1276
  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
wenzelm@63652
  1277
    f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
wenzelm@63652
  1278
    for f :: "nat \<Rightarrow> int" and k :: int
wenzelm@63652
  1279
  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
wenzelm@63652
  1280
  unfolding One_nat_def
wenzelm@63652
  1281
  apply simp
wenzelm@63652
  1282
  apply (erule exE)
wenzelm@63652
  1283
  apply (rule_tac x = "i+m" in exI)
wenzelm@63652
  1284
  apply arith
wenzelm@63652
  1285
  done
haftmann@25919
  1286
haftmann@25919
  1287
wenzelm@63652
  1288
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1289
paulson@34055
  1290
lemma abs_zmult_eq_1:
wenzelm@63652
  1291
  fixes m n :: int
paulson@34055
  1292
  assumes mn: "\<bar>m * n\<bar> = 1"
wenzelm@63652
  1293
  shows "\<bar>m\<bar> = 1"
paulson@34055
  1294
proof -
wenzelm@63652
  1295
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
wenzelm@63652
  1296
  have "\<not> 2 \<le> \<bar>m\<bar>"
paulson@34055
  1297
  proof
paulson@34055
  1298
    assume "2 \<le> \<bar>m\<bar>"
wenzelm@63652
  1299
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
wenzelm@63652
  1300
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
wenzelm@63652
  1301
    also from mn have "\<dots> = 1" by simp
wenzelm@63652
  1302
    finally have "2 * \<bar>n\<bar> \<le> 1" .
wenzelm@63652
  1303
    with 0 show "False" by arith
paulson@34055
  1304
  qed
wenzelm@63652
  1305
  with 0 show ?thesis by auto
paulson@34055
  1306
qed
haftmann@25919
  1307
wenzelm@63652
  1308
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
wenzelm@63652
  1309
  for m n :: int
wenzelm@63652
  1310
  using abs_zmult_eq_1 [of m n] by arith
haftmann@25919
  1311
boehmes@35815
  1312
lemma pos_zmult_eq_1_iff:
wenzelm@63652
  1313
  fixes m n :: int
wenzelm@63652
  1314
  assumes "0 < m"
wenzelm@63652
  1315
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
boehmes@35815
  1316
proof -
wenzelm@63652
  1317
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
wenzelm@63652
  1318
    by (auto dest: pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1319
  then show ?thesis
wenzelm@63652
  1320
    by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1321
qed
haftmann@25919
  1322
wenzelm@63652
  1323
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
wenzelm@63652
  1324
  for m n :: int
wenzelm@63652
  1325
  apply (rule iffI)
wenzelm@63652
  1326
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1327
   apply (simp add: mult.commute [of m])
wenzelm@63652
  1328
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1329
   apply auto
wenzelm@63652
  1330
  done
haftmann@25919
  1331
haftmann@33296
  1332
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1333
proof
haftmann@33296
  1334
  assume "finite (UNIV::int set)"
wenzelm@61076
  1335
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1336
    by (rule injI) simp
wenzelm@61076
  1337
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1338
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1339
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1340
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1341
qed
haftmann@25919
  1342
haftmann@25919
  1343
wenzelm@60758
  1344
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1345
wenzelm@63652
  1346
subsubsection \<open>Special Simplification for Constants\<close>
haftmann@30652
  1347
wenzelm@63652
  1348
text \<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1349
webertj@49962
  1350
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1351
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1352
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1353
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1354
wenzelm@63652
  1355
text \<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1356
huffman@47108
  1357
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1358
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1359
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1360
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1361
haftmann@30652
  1362
wenzelm@61799
  1363
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1364
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1365
huffman@47108
  1366
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1367
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1368
huffman@47108
  1369
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1370
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1371
wenzelm@60758
  1372
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@63652
  1373
  into the literal.\<close>
haftmann@30652
  1374
haftmann@54489
  1375
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1376
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1377
haftmann@54489
  1378
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1379
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1380
haftmann@54489
  1381
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1382
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1383
haftmann@54489
  1384
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1385
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1386
haftmann@54489
  1387
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1388
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1389
haftmann@54489
  1390
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1391
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1392
wenzelm@63652
  1393
(* FIXME maybe simproc *)
haftmann@30652
  1394
haftmann@30652
  1395
wenzelm@61799
  1396
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1397
huffman@47108
  1398
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1399
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1400
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1401
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1402
haftmann@30652
  1403
wenzelm@61799
  1404
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1405
lp15@61738
  1406
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1407
lp15@61738
  1408
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1409
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1410
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1411
lp15@61738
  1412
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1413
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1414
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1415
lp15@61738
  1416
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1417
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1418
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1419
lp15@61738
  1420
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1421
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1422
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1423
lp15@61738
  1424
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1425
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1426
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1427
lp15@61738
  1428
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1429
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1430
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1431
haftmann@30652
  1432
wenzelm@63652
  1433
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1434
wenzelm@63652
  1435
text \<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1436
lp15@61738
  1437
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1438
  le_divide_eq [of "numeral w"]
haftmann@54489
  1439
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1440
lp15@61738
  1441
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1442
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1443
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1444
lp15@61738
  1445
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1446
  less_divide_eq [of "numeral w"]
haftmann@54489
  1447
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1448
lp15@61738
  1449
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1450
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1451
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1452
lp15@61738
  1453
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1454
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1455
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1456
lp15@61738
  1457
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1458
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1459
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1460
haftmann@30652
  1461
wenzelm@63652
  1462
text \<open>Not good as automatic simprules because they cause case splits.\<close>
wenzelm@63652
  1463
lemmas [divide_const_simps] =
wenzelm@63652
  1464
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1465
haftmann@30652
  1466
wenzelm@60758
  1467
subsection \<open>The divides relation\<close>
haftmann@33320
  1468
wenzelm@63652
  1469
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
wenzelm@63652
  1470
  for m n :: int
wenzelm@63652
  1471
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1472
wenzelm@63652
  1473
lemma zdvd_antisym_abs:
wenzelm@63652
  1474
  fixes a b :: int
wenzelm@63652
  1475
  assumes "a dvd b" and "b dvd a"
haftmann@33320
  1476
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@63652
  1477
proof (cases "a = 0")
wenzelm@63652
  1478
  case True
wenzelm@63652
  1479
  with assms show ?thesis by simp
nipkow@33657
  1480
next
wenzelm@63652
  1481
  case False
wenzelm@63652
  1482
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
wenzelm@63652
  1483
    unfolding dvd_def by blast
wenzelm@63652
  1484
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
wenzelm@63652
  1485
    unfolding dvd_def by blast
wenzelm@63652
  1486
  from k k' have "a = a * k * k'" by simp
wenzelm@63652
  1487
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
wenzelm@63652
  1488
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
wenzelm@63652
  1489
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
wenzelm@63652
  1490
    by (simp add: zmult_eq_1_iff)
wenzelm@63652
  1491
  with k k' show ?thesis by auto
haftmann@33320
  1492
qed
haftmann@33320
  1493
wenzelm@63652
  1494
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
wenzelm@63652
  1495
  for k m n :: int
lp15@60162
  1496
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1497
wenzelm@63652
  1498
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
wenzelm@63652
  1499
  for k m n :: int
haftmann@58649
  1500
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1501
haftmann@33320
  1502
lemma dvd_imp_le_int:
haftmann@33320
  1503
  fixes d i :: int
haftmann@33320
  1504
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1505
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1506
proof -
wenzelm@60758
  1507
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1508
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1509
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1510
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1511
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1512
qed
haftmann@33320
  1513
haftmann@33320
  1514
lemma zdvd_not_zless:
haftmann@33320
  1515
  fixes m n :: int
haftmann@33320
  1516
  assumes "0 < m" and "m < n"
haftmann@33320
  1517
  shows "\<not> n dvd m"
haftmann@33320
  1518
proof
haftmann@33320
  1519
  from assms have "0 < n" by auto
haftmann@33320
  1520
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1521
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1522
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1523
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1524
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1525
qed
haftmann@33320
  1526
wenzelm@63652
  1527
lemma zdvd_mult_cancel:
wenzelm@63652
  1528
  fixes k m n :: int
wenzelm@63652
  1529
  assumes d: "k * m dvd k * n"
wenzelm@63652
  1530
    and "k \<noteq> 0"
haftmann@33320
  1531
  shows "m dvd n"
wenzelm@63652
  1532
proof -
wenzelm@63652
  1533
  from d obtain h where h: "k * n = k * m * h"
wenzelm@63652
  1534
    unfolding dvd_def by blast
wenzelm@63652
  1535
  have "n = m * h"
wenzelm@63652
  1536
  proof (rule ccontr)
wenzelm@63652
  1537
    assume "\<not> ?thesis"
wenzelm@63652
  1538
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
wenzelm@63652
  1539
    with h show False
wenzelm@63652
  1540
      by (simp add: mult.assoc)
wenzelm@63652
  1541
  qed
wenzelm@63652
  1542
  then show ?thesis by simp
haftmann@33320
  1543
qed
haftmann@33320
  1544
wenzelm@63652
  1545
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
haftmann@33320
  1546
proof -
wenzelm@63652
  1547
  have "x dvd y" if "int y = int x * k" for k
wenzelm@63652
  1548
  proof (cases k)
wenzelm@63652
  1549
    case (nonneg n)
wenzelm@63652
  1550
    with that have "y = x * n"
wenzelm@63652
  1551
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
wenzelm@63652
  1552
    then show ?thesis ..
wenzelm@63652
  1553
  next
wenzelm@63652
  1554
    case (neg n)
wenzelm@63652
  1555
    with that have "int y = int x * (- int (Suc n))"
wenzelm@63652
  1556
      by simp
wenzelm@63652
  1557
    also have "\<dots> = - (int x * int (Suc n))"
wenzelm@63652
  1558
      by (simp only: mult_minus_right)
wenzelm@63652
  1559
    also have "\<dots> = - int (x * Suc n)"
wenzelm@63652
  1560
      by (simp only: of_nat_mult [symmetric])
wenzelm@63652
  1561
    finally have "- int (x * Suc n) = int y" ..
wenzelm@63652
  1562
    then show ?thesis
wenzelm@63652
  1563
      by (simp only: negative_eq_positive) auto
haftmann@33320
  1564
  qed
wenzelm@63652
  1565
  then show ?thesis
wenzelm@63652
  1566
    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1567
qed
haftmann@33320
  1568
wenzelm@63652
  1569
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
wenzelm@63652
  1570
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
  1571
  for x :: int
haftmann@33320
  1572
proof
wenzelm@63652
  1573
  assume ?lhs
wenzelm@63652
  1574
  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@63652
  1575
  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@63652
  1576
  then have "nat \<bar>x\<bar> = 1" by simp
wenzelm@63652
  1577
  then show ?rhs by (cases "x < 0") auto
haftmann@33320
  1578
next
wenzelm@63652
  1579
  assume ?rhs
wenzelm@63652
  1580
  then have "x = 1 \<or> x = - 1" by auto
wenzelm@63652
  1581
  then show ?lhs by (auto intro: dvdI)
haftmann@33320
  1582
qed
haftmann@33320
  1583
lp15@60162
  1584
lemma zdvd_mult_cancel1:
wenzelm@63652
  1585
  fixes m :: int
wenzelm@63652
  1586
  assumes mp: "m \<noteq> 0"
wenzelm@63652
  1587
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
wenzelm@63652
  1588
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@33320
  1589
proof
wenzelm@63652
  1590
  assume ?rhs
wenzelm@63652
  1591
  then show ?lhs
wenzelm@63652
  1592
    by (cases "n > 0") (auto simp add: minus_equation_iff)
haftmann@33320
  1593
next
wenzelm@63652
  1594
  assume ?lhs
wenzelm@63652
  1595
  then have "m * n dvd m * 1" by simp
wenzelm@63652
  1596
  from zdvd_mult_cancel[OF this mp] show ?rhs
wenzelm@63652
  1597
    by (simp only: zdvd1_eq)
haftmann@33320
  1598
qed
haftmann@33320
  1599
wenzelm@63652
  1600
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
wenzelm@63652
  1601
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@33320
  1602
wenzelm@63652
  1603
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
wenzelm@63652
  1604
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@58650
  1605
wenzelm@63652
  1606
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
wenzelm@63652
  1607
  by (simp add: dvd_int_iff [symmetric])
wenzelm@63652
  1608
wenzelm@63652
  1609
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
haftmann@33320
  1610
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1611
wenzelm@63652
  1612
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1613
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1614
wenzelm@63652
  1615
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1616
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1617
wenzelm@63652
  1618
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
wenzelm@63652
  1619
  for n z :: int
wenzelm@42676
  1620
  apply (cases n)
wenzelm@63652
  1621
   apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1622
  apply (cases z)
wenzelm@63652
  1623
   apply (auto simp add: dvd_imp_le)
haftmann@33320
  1624
  done
haftmann@33320
  1625
haftmann@36749
  1626
lemma zdvd_period:
haftmann@36749
  1627
  fixes a d :: int
haftmann@36749
  1628
  assumes "a dvd d"
haftmann@36749
  1629
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
wenzelm@63652
  1630
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@36749
  1631
proof -
haftmann@36749
  1632
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1633
  show ?thesis
wenzelm@42676
  1634
  proof
wenzelm@63652
  1635
    assume ?lhs
haftmann@36749
  1636
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1637
    then have "x = a * l - t" by simp
wenzelm@63652
  1638
    with \<open>d = a * k\<close> show ?rhs by simp
haftmann@36749
  1639
  next
wenzelm@63652
  1640
    assume ?rhs
haftmann@36749
  1641
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1642
    then have "x = a * l - c * d - t" by simp
wenzelm@63652
  1643
    with \<open>d = a * k\<close> show ?lhs by simp
haftmann@36749
  1644
  qed
haftmann@36749
  1645
qed
haftmann@36749
  1646
haftmann@33320
  1647
wenzelm@60758
  1648
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1649
wenzelm@63652
  1650
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
wenzelm@63652
  1651
proof (cases "a \<le> b")
bulwahn@46756
  1652
  case True
wenzelm@63652
  1653
  then show ?thesis
bulwahn@46756
  1654
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1655
    case base
wenzelm@63652
  1656
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
wenzelm@63652
  1657
    then show ?case by simp
bulwahn@46756
  1658
  next
bulwahn@46756
  1659
    case (step b)
wenzelm@63652
  1660
    then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
wenzelm@63652
  1661
    with step show ?case by simp
bulwahn@46756
  1662
  qed
bulwahn@46756
  1663
next
wenzelm@63652
  1664
  case False
wenzelm@63652
  1665
  then show ?thesis
bulwahn@46756
  1666
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1667
qed
bulwahn@46756
  1668
wenzelm@63652
  1669
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
wenzelm@63652
  1670
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1671
wenzelm@63652
  1672
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
wenzelm@63652
  1673
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1674
wenzelm@63652
  1675
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
wenzelm@63652
  1676
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1677
bulwahn@46756
  1678
wenzelm@60758
  1679
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1680
wenzelm@60758
  1681
text \<open>Constructors\<close>
huffman@47108
  1682
wenzelm@63652
  1683
definition Pos :: "num \<Rightarrow> int"
wenzelm@63652
  1684
  where [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1685
wenzelm@63652
  1686
definition Neg :: "num \<Rightarrow> int"
wenzelm@63652
  1687
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1688
huffman@47108
  1689
code_datatype "0::int" Pos Neg
huffman@47108
  1690
huffman@47108
  1691
wenzelm@63652
  1692
text \<open>Auxiliary operations.\<close>
huffman@47108
  1693
wenzelm@63652
  1694
definition dup :: "int \<Rightarrow> int"
wenzelm@63652
  1695
  where [simp]: "dup k = k + k"
haftmann@26507
  1696
huffman@47108
  1697
lemma dup_code [code]:
huffman@47108
  1698
  "dup 0 = 0"
huffman@47108
  1699
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1700
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1701
  by (simp_all add: numeral_Bit0)
huffman@47108
  1702
wenzelm@63652
  1703
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
wenzelm@63652
  1704
  where [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1705
huffman@47108
  1706
lemma sub_code [code]:
huffman@47108
  1707
  "sub Num.One Num.One = 0"
huffman@47108
  1708
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1709
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1710
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1711
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1712
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1713
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1714
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1715
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
boehmes@66035
  1716
  by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
huffman@47108
  1717
wenzelm@63652
  1718
text \<open>Implementations.\<close>
huffman@47108
  1719
haftmann@64996
  1720
lemma one_int_code [code]: "1 = Pos Num.One"
huffman@47108
  1721
  by simp
huffman@47108
  1722
huffman@47108
  1723
lemma plus_int_code [code]:
wenzelm@63652
  1724
  "k + 0 = k"
wenzelm@63652
  1725
  "0 + l = l"
huffman@47108
  1726
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1727
  "Pos m + Neg n = sub m n"
huffman@47108
  1728
  "Neg m + Pos n = sub n m"
huffman@47108
  1729
  "Neg m + Neg n = Neg (m + n)"
wenzelm@63652
  1730
  for k l :: int
huffman@47108
  1731
  by simp_all
haftmann@26507
  1732
huffman@47108
  1733
lemma uminus_int_code [code]:
huffman@47108
  1734
  "uminus 0 = (0::int)"
huffman@47108
  1735
  "uminus (Pos m) = Neg m"
huffman@47108
  1736
  "uminus (Neg m) = Pos m"
huffman@47108
  1737
  by simp_all
huffman@47108
  1738
huffman@47108
  1739
lemma minus_int_code [code]:
wenzelm@63652
  1740
  "k - 0 = k"
wenzelm@63652
  1741
  "0 - l = uminus l"
huffman@47108
  1742
  "Pos m - Pos n = sub m n"
huffman@47108
  1743
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1744
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1745
  "Neg m - Neg n = sub n m"
wenzelm@63652
  1746
  for k l :: int
huffman@47108
  1747
  by simp_all
huffman@47108
  1748
huffman@47108
  1749
lemma times_int_code [code]:
wenzelm@63652
  1750
  "k * 0 = 0"
wenzelm@63652
  1751
  "0 * l = 0"
huffman@47108
  1752
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1753
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1754
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1755
  "Neg m * Neg n = Pos (m * n)"
wenzelm@63652
  1756
  for k l :: int
huffman@47108
  1757
  by simp_all
haftmann@26507
  1758
haftmann@38857
  1759
instantiation int :: equal
haftmann@26507
  1760
begin
haftmann@26507
  1761
wenzelm@63652
  1762
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1763
wenzelm@61169
  1764
instance
wenzelm@61169
  1765
  by standard (rule equal_int_def)
haftmann@26507
  1766
haftmann@26507
  1767
end
haftmann@26507
  1768
huffman@47108
  1769
lemma equal_int_code [code]:
huffman@47108
  1770
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1771
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1772
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1773
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1774
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1775
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1776
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1777
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1778
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1779
  by (auto simp add: equal)
haftmann@26507
  1780
wenzelm@63652
  1781
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
wenzelm@63652
  1782
  for k :: int
huffman@47108
  1783
  by (fact equal_refl)
haftmann@26507
  1784
haftmann@28562
  1785
lemma less_eq_int_code [code]:
huffman@47108
  1786
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1787
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1788
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1789
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1790
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1791
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1792
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1793
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1794
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1795
  by simp_all
haftmann@26507
  1796
haftmann@28562
  1797
lemma less_int_code [code]:
huffman@47108
  1798
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1799
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1800
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1801
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1802
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1803
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1804
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1805
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1806
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1807
  by simp_all
haftmann@25919
  1808
huffman@47108
  1809
lemma nat_code [code]:
huffman@47108
  1810
  "nat (Int.Neg k) = 0"
huffman@47108
  1811
  "nat 0 = 0"
huffman@47108
  1812
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1813
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1814
huffman@47108
  1815
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1816
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1817
  "of_int 0 = 0"
huffman@47108
  1818
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1819
  by simp_all
haftmann@25919
  1820
huffman@47108
  1821
wenzelm@63652
  1822
text \<open>Serializer setup.\<close>
haftmann@25919
  1823
haftmann@52435
  1824
code_identifier
haftmann@52435
  1825
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1826
haftmann@25919
  1827
quickcheck_params [default_type = int]
haftmann@25919
  1828
huffman@47108
  1829
hide_const (open) Pos Neg sub dup
haftmann@25919
  1830
haftmann@25919
  1831
wenzelm@61799
  1832
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1833
kuncar@53652
  1834
lifting_update int.lifting
kuncar@53652
  1835
lifting_forget int.lifting
huffman@48045
  1836
haftmann@25919
  1837
end