src/HOL/Int.thy
author paulson <lp15@cam.ac.uk>
Tue Jan 03 16:48:49 2017 +0000 (2017-01-03)
changeset 64758 3b33d2fc5fc0
parent 64714 53bab28983f1
child 64849 766db3539859
permissions -rw-r--r--
A few new lemmas and needed adaptations
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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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  then show ?thesis by simp
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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@62347
   360
haftmann@36424
   361
end
haftmann@25919
   362
lp15@61234
   363
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   364
wenzelm@63652
   365
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
lp15@61234
   366
  using of_int_eq_iff by fastforce
lp15@61234
   367
lp15@61649
   368
lemma of_int_le_numeral_iff [simp]:
wenzelm@63652
   369
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   370
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   371
lp15@61649
   372
lemma of_int_numeral_le_iff [simp]:
wenzelm@63652
   373
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   374
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   375
lp15@61649
   376
lemma of_int_less_numeral_iff [simp]:
wenzelm@63652
   377
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
lp15@61234
   378
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   379
lp15@61649
   380
lemma of_int_numeral_less_iff [simp]:
wenzelm@63652
   381
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   382
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   383
wenzelm@63652
   384
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   385
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   386
haftmann@25919
   387
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   388
proof
wenzelm@63652
   389
  show "of_int z = id z" for z
wenzelm@63652
   390
    by (cases z rule: int_diff_cases) simp
haftmann@25919
   391
qed
haftmann@25919
   392
hoelzl@51329
   393
instance int :: no_top
wenzelm@61169
   394
  apply standard
hoelzl@51329
   395
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   396
  apply simp
hoelzl@51329
   397
  done
hoelzl@51329
   398
hoelzl@51329
   399
instance int :: no_bot
wenzelm@61169
   400
  apply standard
hoelzl@51329
   401
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   402
  apply simp
hoelzl@51329
   403
  done
hoelzl@51329
   404
wenzelm@63652
   405
wenzelm@61799
   406
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   407
huffman@48045
   408
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   409
  by auto
haftmann@25919
   410
huffman@44709
   411
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   412
  by transfer simp
haftmann@25919
   413
huffman@44709
   414
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   415
  by transfer clarsimp
haftmann@25919
   416
wenzelm@63652
   417
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
wenzelm@63652
   418
  by simp
haftmann@25919
   419
wenzelm@63652
   420
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
huffman@48045
   421
  by transfer clarsimp
haftmann@25919
   422
wenzelm@63652
   423
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
huffman@48045
   424
  by transfer (clarsimp, arith)
haftmann@25919
   425
wenzelm@63652
   426
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
wenzelm@63652
   427
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   428
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   429
wenzelm@63652
   430
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   431
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   432
wenzelm@63652
   433
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
huffman@48045
   434
  by transfer (clarsimp, arith)
haftmann@25919
   435
haftmann@64714
   436
lemma nonneg_int_cases:
haftmann@64714
   437
  assumes "0 \<le> k"
haftmann@64714
   438
  obtains n where "k = int n"
haftmann@64714
   439
proof -
haftmann@64714
   440
  from assms have "k = int (nat k)"
haftmann@64714
   441
    by simp
haftmann@64714
   442
  then show thesis
haftmann@64714
   443
    by (rule that)
haftmann@64714
   444
qed
haftmann@64714
   445
haftmann@64714
   446
lemma pos_int_cases:
haftmann@64714
   447
  assumes "0 < k"
haftmann@64714
   448
  obtains n where "k = int n" and "n > 0"
haftmann@64714
   449
proof -
haftmann@64714
   450
  from assms have "0 \<le> k"
haftmann@64714
   451
    by simp
haftmann@64714
   452
  then obtain n where "k = int n"
haftmann@64714
   453
    by (rule nonneg_int_cases)
haftmann@64714
   454
  moreover have "n > 0"
haftmann@64714
   455
    using \<open>k = int n\<close> assms by simp
haftmann@64714
   456
  ultimately show thesis
haftmann@64714
   457
    by (rule that)
haftmann@64714
   458
qed
haftmann@64714
   459
haftmann@64714
   460
lemma nonpos_int_cases:
haftmann@64714
   461
  assumes "k \<le> 0"
haftmann@64714
   462
  obtains n where "k = - int n"
haftmann@64714
   463
proof -
haftmann@64714
   464
  from assms have "- k \<ge> 0"
haftmann@64714
   465
    by simp
haftmann@64714
   466
  then obtain n where "- k = int n"
haftmann@64714
   467
    by (rule nonneg_int_cases)
haftmann@64714
   468
  then have "k = - int n"
haftmann@64714
   469
    by simp
haftmann@64714
   470
  then show thesis
haftmann@64714
   471
    by (rule that)
haftmann@64714
   472
qed
haftmann@64714
   473
haftmann@64714
   474
lemma neg_int_cases:
haftmann@64714
   475
  assumes "k < 0"
haftmann@64714
   476
  obtains n where "k = - int n" and "n > 0"
haftmann@64714
   477
proof -
haftmann@64714
   478
  from assms have "- k > 0"
haftmann@64714
   479
    by simp
haftmann@64714
   480
  then obtain n where "- k = int n" and "- k > 0"
haftmann@64714
   481
    by (blast elim: pos_int_cases)
haftmann@64714
   482
  then have "k = - int n" and "n > 0"
haftmann@64714
   483
    by simp_all
haftmann@64714
   484
  then show thesis
haftmann@64714
   485
    by (rule that)
haftmann@64714
   486
qed
haftmann@25919
   487
wenzelm@63652
   488
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   489
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   490
wenzelm@63652
   491
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   492
  using nat_eq_iff [of w m] by auto
haftmann@54223
   493
wenzelm@63652
   494
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@54223
   495
  by (simp add: nat_eq_iff)
haftmann@25919
   496
wenzelm@63652
   497
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@54223
   498
  by (simp add: nat_eq_iff)
haftmann@54223
   499
wenzelm@63652
   500
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
haftmann@54223
   501
  by (simp add: nat_eq_iff)
haftmann@25919
   502
wenzelm@63652
   503
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
haftmann@54223
   504
  by simp
haftmann@54223
   505
haftmann@54223
   506
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   507
  by simp
lp15@60162
   508
wenzelm@63652
   509
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
huffman@48045
   510
  by transfer (clarsimp, arith)
haftmann@25919
   511
huffman@44709
   512
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   513
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   514
huffman@44707
   515
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   516
  by transfer auto
huffman@44707
   517
wenzelm@63652
   518
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
wenzelm@63652
   519
  for i :: int
huffman@48045
   520
  by transfer clarsimp
nipkow@29700
   521
wenzelm@63652
   522
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
wenzelm@63652
   523
  by (auto simp add: nat_eq_iff2)
haftmann@25919
   524
wenzelm@63652
   525
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
wenzelm@63652
   526
  using zless_nat_conj [of 0] by auto
haftmann@25919
   527
wenzelm@63652
   528
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   529
  by transfer clarsimp
haftmann@25919
   530
wenzelm@63652
   531
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   532
  by transfer clarsimp
lp15@60162
   533
wenzelm@63652
   534
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   535
  by (rule nat_diff_distrib') auto
haftmann@25919
   536
huffman@44709
   537
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   538
  by transfer simp
haftmann@25919
   539
wenzelm@63652
   540
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   541
  by transfer auto
lp15@60162
   542
wenzelm@63652
   543
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
huffman@48045
   544
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   545
wenzelm@63652
   546
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   547
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   548
wenzelm@63652
   549
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   550
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   551
haftmann@54249
   552
wenzelm@60758
   553
text \<open>For termination proofs:\<close>
wenzelm@63652
   554
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
krauss@29779
   555
haftmann@25919
   556
wenzelm@63652
   557
subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   558
wenzelm@61076
   559
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
wenzelm@63652
   560
  by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   561
huffman@44709
   562
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
wenzelm@63652
   563
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   564
huffman@44709
   565
lemma negative_zle_0: "- int n \<le> 0"
wenzelm@63652
   566
  by (simp add: minus_le_iff)
haftmann@25919
   567
huffman@44709
   568
lemma negative_zle [iff]: "- int n \<le> int m"
wenzelm@63652
   569
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   570
wenzelm@63652
   571
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
wenzelm@63652
   572
  by (subst le_minus_iff) (simp del: of_nat_Suc)
haftmann@25919
   573
wenzelm@63652
   574
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
huffman@48045
   575
  by transfer simp
haftmann@25919
   576
wenzelm@63652
   577
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
wenzelm@63652
   578
  by (simp add: linorder_not_less)
haftmann@25919
   579
wenzelm@63652
   580
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
wenzelm@63652
   581
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   582
wenzelm@63652
   583
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
wenzelm@63652
   584
  (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@62348
   585
proof
wenzelm@63652
   586
  assume ?rhs
wenzelm@63652
   587
  then show ?lhs by auto
haftmann@62348
   588
next
wenzelm@63652
   589
  assume ?lhs
haftmann@62348
   590
  then have "0 \<le> z - w" by simp
haftmann@62348
   591
  then obtain n where "z - w = int n"
haftmann@62348
   592
    using zero_le_imp_eq_int [of "z - w"] by blast
wenzelm@63652
   593
  then have "z = w + int n" by simp
wenzelm@63652
   594
  then show ?rhs ..
haftmann@25919
   595
qed
haftmann@25919
   596
huffman@44709
   597
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
wenzelm@63652
   598
  by simp
haftmann@25919
   599
wenzelm@63652
   600
text \<open>
wenzelm@63652
   601
  This version is proved for all ordered rings, not just integers!
wenzelm@63652
   602
  It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@63652
   603
  in theory \<open>Rings\<close>.
wenzelm@63652
   604
  But is it really better than just rewriting with \<open>abs_if\<close>?
wenzelm@63652
   605
\<close>
wenzelm@63652
   606
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
   607
  for a :: "'a::linordered_idom"
wenzelm@63652
   608
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   609
huffman@44709
   610
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
wenzelm@63652
   611
  apply transfer
wenzelm@63652
   612
  apply clarsimp
wenzelm@63652
   613
  apply (rule_tac x="b - Suc a" in exI)
wenzelm@63652
   614
  apply arith
wenzelm@63652
   615
  done
wenzelm@63652
   616
haftmann@25919
   617
wenzelm@60758
   618
subsection \<open>Cases and induction\<close>
haftmann@25919
   619
wenzelm@63652
   620
text \<open>
wenzelm@63652
   621
  Now we replace the case analysis rule by a more conventional one:
wenzelm@63652
   622
  whether an integer is negative or not.
wenzelm@63652
   623
\<close>
haftmann@25919
   624
wenzelm@63652
   625
text \<open>This version is symmetric in the two subgoals.\<close>
wenzelm@63652
   626
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
wenzelm@63652
   627
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   628
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   629
wenzelm@63652
   630
text \<open>This is the default, with a negative case.\<close>
wenzelm@63652
   631
lemma int_cases [case_names nonneg neg, cases type: int]:
wenzelm@63652
   632
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   633
  apply (cases "z < 0")
wenzelm@63652
   634
   apply (blast dest!: negD)
wenzelm@63652
   635
  apply (simp add: linorder_not_less del: of_nat_Suc)
wenzelm@63652
   636
  apply auto
wenzelm@63652
   637
  apply (blast dest: nat_0_le [THEN sym])
wenzelm@63652
   638
  done
haftmann@25919
   639
haftmann@60868
   640
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   641
  fixes k :: int
haftmann@60868
   642
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   643
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   644
  shows "P"
haftmann@60868
   645
proof (cases k "0::int" rule: linorder_cases)
wenzelm@63652
   646
  case equal
wenzelm@63652
   647
  with assms(1) show P by simp
haftmann@60868
   648
next
haftmann@60868
   649
  case greater
wenzelm@63539
   650
  then have *: "nat k > 0" by simp
wenzelm@63539
   651
  moreover from * have "k = int (nat k)" by auto
haftmann@60868
   652
  ultimately show P using assms(2) by blast
haftmann@60868
   653
next
haftmann@60868
   654
  case less
wenzelm@63539
   655
  then have *: "nat (- k) > 0" by simp
wenzelm@63539
   656
  moreover from * have "k = - int (nat (- k))" by auto
haftmann@60868
   657
  ultimately show P using assms(3) by blast
haftmann@60868
   658
qed
haftmann@60868
   659
wenzelm@63652
   660
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
wenzelm@63652
   661
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
wenzelm@42676
   662
  by (cases z) auto
haftmann@25919
   663
huffman@47108
   664
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   665
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   666
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   667
haftmann@54489
   668
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   669
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   670
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   671
wenzelm@61799
   672
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   673
huffman@47108
   674
lemmas max_number_of [simp] =
huffman@47108
   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"]
haftmann@54489
   678
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   679
huffman@47108
   680
lemmas min_number_of [simp] =
huffman@47108
   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"]
haftmann@54489
   684
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   685
haftmann@25919
   686
wenzelm@60758
   687
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   688
wenzelm@60758
   689
text \<open>Preliminaries\<close>
huffman@28958
   690
lp15@60162
   691
lemma le_imp_0_less:
wenzelm@63652
   692
  fixes z :: int
huffman@28958
   693
  assumes le: "0 \<le> z"
wenzelm@63652
   694
  shows "0 < 1 + z"
huffman@28958
   695
proof -
huffman@28958
   696
  have "0 \<le> z" by fact
wenzelm@63652
   697
  also have "\<dots> < z + 1" by (rule less_add_one)
wenzelm@63652
   698
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
huffman@28958
   699
  finally show "0 < 1 + z" .
huffman@28958
   700
qed
huffman@28958
   701
wenzelm@63652
   702
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
wenzelm@63652
   703
  for z :: int
wenzelm@42676
   704
proof (cases z)
huffman@28958
   705
  case (nonneg n)
wenzelm@63652
   706
  then show ?thesis
wenzelm@63652
   707
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   708
next
huffman@28958
   709
  case (neg n)
wenzelm@63652
   710
  then show ?thesis
wenzelm@63652
   711
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@63652
   712
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   713
qed
huffman@28958
   714
wenzelm@63652
   715
wenzelm@60758
   716
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   717
haftmann@25919
   718
lemmas double_eq_0_iff = double_zero
haftmann@25919
   719
wenzelm@63652
   720
lemma odd_nonzero: "1 + z + z \<noteq> 0"
wenzelm@63652
   721
  for z :: int
wenzelm@42676
   722
proof (cases z)
haftmann@25919
   723
  case (nonneg n)
wenzelm@63652
   724
  have le: "0 \<le> z + z"
wenzelm@63652
   725
    by (simp add: nonneg add_increasing)
wenzelm@63652
   726
  then show ?thesis
wenzelm@63652
   727
    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
haftmann@25919
   728
next
haftmann@25919
   729
  case (neg n)
haftmann@25919
   730
  show ?thesis
haftmann@25919
   731
  proof
haftmann@25919
   732
    assume eq: "1 + z + z = 0"
wenzelm@63652
   733
    have "0 < 1 + (int n + int n)"
lp15@60162
   734
      by (simp add: le_imp_0_less add_increasing)
wenzelm@63652
   735
    also have "\<dots> = - (1 + z + z)"
lp15@60162
   736
      by (simp add: neg add.assoc [symmetric])
wenzelm@63652
   737
    also have "\<dots> = 0" by (simp add: eq)
haftmann@25919
   738
    finally have "0<0" ..
wenzelm@63652
   739
    then show False by blast
haftmann@25919
   740
  qed
haftmann@25919
   741
qed
haftmann@25919
   742
haftmann@30652
   743
wenzelm@60758
   744
subsection \<open>The Set of Integers\<close>
haftmann@25919
   745
haftmann@25919
   746
context ring_1
haftmann@25919
   747
begin
haftmann@25919
   748
wenzelm@61070
   749
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   750
  where "\<int> = range of_int"
haftmann@25919
   751
huffman@35634
   752
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   753
  by (simp add: Ints_def)
huffman@35634
   754
huffman@35634
   755
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   756
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   757
haftmann@25919
   758
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   759
  using Ints_of_int [of "0"] by simp
haftmann@25919
   760
haftmann@25919
   761
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   762
  using Ints_of_int [of "1"] by simp
haftmann@25919
   763
eberlm@61552
   764
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   765
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   766
haftmann@25919
   767
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
wenzelm@63652
   768
  apply (auto simp add: Ints_def)
wenzelm@63652
   769
  apply (rule range_eqI)
wenzelm@63652
   770
  apply (rule of_int_add [symmetric])
wenzelm@63652
   771
  done
haftmann@25919
   772
haftmann@25919
   773
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
wenzelm@63652
   774
  apply (auto simp add: Ints_def)
wenzelm@63652
   775
  apply (rule range_eqI)
wenzelm@63652
   776
  apply (rule of_int_minus [symmetric])
wenzelm@63652
   777
  done
haftmann@25919
   778
huffman@35634
   779
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
wenzelm@63652
   780
  apply (auto simp add: Ints_def)
wenzelm@63652
   781
  apply (rule range_eqI)
wenzelm@63652
   782
  apply (rule of_int_diff [symmetric])
wenzelm@63652
   783
  done
huffman@35634
   784
haftmann@25919
   785
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
wenzelm@63652
   786
  apply (auto simp add: Ints_def)
wenzelm@63652
   787
  apply (rule range_eqI)
wenzelm@63652
   788
  apply (rule of_int_mult [symmetric])
wenzelm@63652
   789
  done
haftmann@25919
   790
huffman@35634
   791
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
wenzelm@63652
   792
  by (induct n) simp_all
huffman@35634
   793
haftmann@25919
   794
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   795
  assumes "q \<in> \<int>"
haftmann@25919
   796
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   797
  unfolding Ints_def
haftmann@25919
   798
proof -
wenzelm@60758
   799
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   800
  then obtain z where "q = of_int z" ..
haftmann@25919
   801
  then show thesis ..
haftmann@25919
   802
qed
haftmann@25919
   803
haftmann@25919
   804
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   805
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   806
  by (rule Ints_cases) auto
haftmann@25919
   807
eberlm@61524
   808
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   809
  unfolding Nats_def Ints_def
eberlm@61524
   810
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   811
eberlm@61524
   812
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   813
proof (intro subsetI equalityI)
wenzelm@63652
   814
  fix x :: 'a
wenzelm@63652
   815
  assume "x \<in> {of_int n |n. n \<ge> 0}"
wenzelm@63652
   816
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   817
    by (auto elim!: Ints_cases)
wenzelm@63652
   818
  then have "x = of_nat (nat n)"
wenzelm@63652
   819
    by (subst of_nat_nat) simp_all
wenzelm@63652
   820
  then show "x \<in> \<nat>"
wenzelm@63652
   821
    by simp
eberlm@61524
   822
next
wenzelm@63652
   823
  fix x :: 'a
wenzelm@63652
   824
  assume "x \<in> \<nat>"
wenzelm@63652
   825
  then obtain n where "x = of_nat n"
wenzelm@63652
   826
    by (auto elim!: Nats_cases)
wenzelm@63652
   827
  then have "x = of_int (int n)" by simp
eberlm@61524
   828
  also have "int n \<ge> 0" by simp
wenzelm@63652
   829
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   830
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   831
qed
eberlm@61524
   832
haftmann@25919
   833
end
haftmann@25919
   834
lp15@64758
   835
lemma (in linordered_idom) Ints_abs [simp]:
lp15@64758
   836
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
lp15@64758
   837
  by (auto simp: abs_if)
lp15@64758
   838
eberlm@61524
   839
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   840
proof (intro subsetI equalityI)
wenzelm@63652
   841
  fix x :: 'a
wenzelm@63652
   842
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
wenzelm@63652
   843
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   844
    by (auto elim!: Ints_cases)
wenzelm@63652
   845
  then have "x = of_nat (nat n)"
wenzelm@63652
   846
    by (subst of_nat_nat) simp_all
wenzelm@63652
   847
  then show "x \<in> \<nat>"
wenzelm@63652
   848
    by simp
eberlm@61524
   849
qed (auto elim!: Nats_cases)
eberlm@61524
   850
eberlm@61524
   851
wenzelm@60758
   852
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
   853
haftmann@25919
   854
lemma Ints_double_eq_0_iff:
wenzelm@63652
   855
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   856
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   857
  shows "a + a = 0 \<longleftrightarrow> a = 0"
wenzelm@63652
   858
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@25919
   859
proof -
wenzelm@63652
   860
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   861
    unfolding Ints_def [symmetric] .
haftmann@25919
   862
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   863
  show ?thesis
haftmann@25919
   864
  proof
wenzelm@63652
   865
    assume ?rhs
wenzelm@63652
   866
    then show ?lhs by simp
haftmann@25919
   867
  next
wenzelm@63652
   868
    assume ?lhs
wenzelm@63652
   869
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   870
    then have "z + z = 0" by (simp only: of_int_eq_iff)
wenzelm@63652
   871
    then have "z = 0" by (simp only: double_eq_0_iff)
wenzelm@63652
   872
    with a show ?rhs by simp
haftmann@25919
   873
  qed
haftmann@25919
   874
qed
haftmann@25919
   875
haftmann@25919
   876
lemma Ints_odd_nonzero:
wenzelm@63652
   877
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   878
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   879
  shows "1 + a + a \<noteq> 0"
haftmann@25919
   880
proof -
wenzelm@63652
   881
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   882
    unfolding Ints_def [symmetric] .
haftmann@25919
   883
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   884
  show ?thesis
haftmann@25919
   885
  proof
wenzelm@63652
   886
    assume "1 + a + a = 0"
wenzelm@63652
   887
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   888
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   889
    with odd_nonzero show False by blast
haftmann@25919
   890
  qed
lp15@60162
   891
qed
haftmann@25919
   892
wenzelm@61070
   893
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
   894
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   895
lp15@60162
   896
lemma Ints_odd_less_0:
wenzelm@63652
   897
  fixes a :: "'a::linordered_idom"
wenzelm@61070
   898
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   899
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25919
   900
proof -
wenzelm@63652
   901
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   902
    unfolding Ints_def [symmetric] .
haftmann@25919
   903
  then obtain z where a: "a = of_int z" ..
wenzelm@63652
   904
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
wenzelm@63652
   905
    by simp
wenzelm@63652
   906
  also have "\<dots> \<longleftrightarrow> z < 0"
wenzelm@63652
   907
    by (simp only: of_int_less_iff odd_less_0_iff)
wenzelm@63652
   908
  also have "\<dots> \<longleftrightarrow> a < 0"
haftmann@25919
   909
    by (simp add: a)
haftmann@25919
   910
  finally show ?thesis .
haftmann@25919
   911
qed
haftmann@25919
   912
haftmann@25919
   913
nipkow@64272
   914
subsection \<open>@{term sum} and @{term prod}\<close>
haftmann@25919
   915
nipkow@64267
   916
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
wenzelm@63652
   917
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   918
nipkow@64267
   919
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
wenzelm@63652
   920
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   921
nipkow@64272
   922
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
wenzelm@63652
   923
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   924
nipkow@64272
   925
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
wenzelm@63652
   926
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   927
haftmann@25919
   928
wenzelm@60758
   929
text \<open>Legacy theorems\<close>
haftmann@25919
   930
haftmann@64714
   931
lemmas int_sum = of_nat_sum [where 'a=int]
haftmann@64714
   932
lemmas int_prod = of_nat_prod [where 'a=int]
haftmann@25919
   933
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   934
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@64714
   935
lemmas nonneg_eq_int = nonneg_int_cases
haftmann@25919
   936
wenzelm@63652
   937
wenzelm@60758
   938
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
   939
haftmann@54249
   940
lemmas of_int_simps =
haftmann@54249
   941
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   942
wenzelm@48891
   943
ML_file "Tools/int_arith.ML"
wenzelm@60758
   944
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
   945
wenzelm@63652
   946
simproc_setup fast_arith
wenzelm@63652
   947
  ("(m::'a::linordered_idom) < n" |
wenzelm@63652
   948
    "(m::'a::linordered_idom) \<le> n" |
wenzelm@63652
   949
    "(m::'a::linordered_idom) = n") =
wenzelm@61144
   950
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
   951
haftmann@25919
   952
wenzelm@60758
   953
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
   954
wenzelm@63652
   955
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
wenzelm@63652
   956
  for w z :: int
wenzelm@63652
   957
  by arith
haftmann@25919
   958
wenzelm@63652
   959
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
wenzelm@63652
   960
  for w z :: int
wenzelm@63652
   961
  by arith
haftmann@25919
   962
wenzelm@63652
   963
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
wenzelm@63652
   964
  for w z :: int
wenzelm@63652
   965
  by arith
haftmann@25919
   966
wenzelm@63652
   967
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
wenzelm@63652
   968
  for w z :: int
wenzelm@63652
   969
  by arith
haftmann@25919
   970
wenzelm@63652
   971
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
wenzelm@63652
   972
  for z :: int
wenzelm@63652
   973
  by arith
haftmann@25919
   974
lp15@64758
   975
lemma Ints_nonzero_abs_ge1:
lp15@64758
   976
  fixes x:: "'a :: linordered_idom"
lp15@64758
   977
    assumes "x \<in> Ints" "x \<noteq> 0"
lp15@64758
   978
    shows "1 \<le> abs x"
lp15@64758
   979
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
lp15@64758
   980
  fix z::int
lp15@64758
   981
  assume "x = of_int z"
lp15@64758
   982
    with \<open>x \<noteq> 0\<close> 
lp15@64758
   983
  show "1 \<le> \<bar>x\<bar>"
lp15@64758
   984
    apply (auto simp add: abs_if)
lp15@64758
   985
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
lp15@64758
   986
qed
lp15@64758
   987
  
lp15@64758
   988
lemma Ints_nonzero_abs_less1:
lp15@64758
   989
  fixes x:: "'a :: linordered_idom"
lp15@64758
   990
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
lp15@64758
   991
    using Ints_nonzero_abs_ge1 [of x] by auto
lp15@64758
   992
    
haftmann@25919
   993
wenzelm@63652
   994
subsection \<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
   995
wenzelm@63652
   996
text \<open>Simplify the term @{term "w + - z"}.\<close>
haftmann@25919
   997
wenzelm@63652
   998
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
lp15@60162
   999
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
  1000
wenzelm@63652
  1001
text \<open>
wenzelm@63652
  1002
  This simplifies expressions of the form @{term "int n = z"} where
wenzelm@63652
  1003
  \<open>z\<close> is an integer literal.
wenzelm@63652
  1004
\<close>
huffman@47108
  1005
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
  1006
wenzelm@63652
  1007
lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
wenzelm@63652
  1008
  (is "?P = (?L \<and> ?R)")
wenzelm@63652
  1009
  for i :: int
haftmann@25919
  1010
proof (cases "i < 0")
wenzelm@63652
  1011
  case True
wenzelm@63652
  1012
  then show ?thesis by auto
haftmann@25919
  1013
next
haftmann@25919
  1014
  case False
haftmann@25919
  1015
  have "?P = ?L"
haftmann@25919
  1016
  proof
wenzelm@63652
  1017
    assume ?P
wenzelm@63652
  1018
    then show ?L using False by auto
haftmann@25919
  1019
  next
wenzelm@63652
  1020
    assume ?L
wenzelm@63652
  1021
    then show ?P using False by simp
haftmann@25919
  1022
  qed
haftmann@25919
  1023
  with False show ?thesis by simp
haftmann@25919
  1024
qed
haftmann@25919
  1025
hoelzl@59000
  1026
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
  1027
  by auto
hoelzl@59000
  1028
hoelzl@59000
  1029
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
  1030
  by auto
hoelzl@59000
  1031
haftmann@25919
  1032
context ring_1
haftmann@25919
  1033
begin
haftmann@25919
  1034
blanchet@33056
  1035
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1036
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1037
proof (cases "k < 0")
wenzelm@63652
  1038
  case True
wenzelm@63652
  1039
  then have "0 \<le> - k" by simp
haftmann@25919
  1040
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1041
  with True show ?thesis by simp
haftmann@25919
  1042
next
wenzelm@63652
  1043
  case False
wenzelm@63652
  1044
  then show ?thesis by (simp add: not_less)
haftmann@25919
  1045
qed
haftmann@25919
  1046
haftmann@25919
  1047
end
haftmann@25919
  1048
haftmann@64014
  1049
lemma transfer_rule_of_int:
haftmann@64014
  1050
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
haftmann@64014
  1051
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64014
  1052
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64014
  1053
    "rel_fun R R uminus uminus"
haftmann@64014
  1054
  shows "rel_fun HOL.eq R of_int of_int"
haftmann@64014
  1055
proof -
haftmann@64014
  1056
  note transfer_rule_of_nat [transfer_rule]
haftmann@64014
  1057
  have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
haftmann@64014
  1058
    by transfer_prover
haftmann@64014
  1059
  show ?thesis
haftmann@64014
  1060
    by (unfold of_int_of_nat [abs_def]) transfer_prover
haftmann@64014
  1061
qed
haftmann@64014
  1062
haftmann@25919
  1063
lemma nat_mult_distrib:
haftmann@25919
  1064
  fixes z z' :: int
haftmann@25919
  1065
  assumes "0 \<le> z"
haftmann@25919
  1066
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1067
proof (cases "0 \<le> z'")
wenzelm@63652
  1068
  case False
wenzelm@63652
  1069
  with assms have "z * z' \<le> 0"
haftmann@25919
  1070
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1071
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1072
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1073
  ultimately show ?thesis by simp
haftmann@25919
  1074
next
wenzelm@63652
  1075
  case True
wenzelm@63652
  1076
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1077
  show ?thesis
haftmann@25919
  1078
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1079
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1080
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1081
qed
haftmann@25919
  1082
wenzelm@63652
  1083
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
wenzelm@63652
  1084
  for z z' :: int
wenzelm@63652
  1085
  apply (rule trans)
wenzelm@63652
  1086
   apply (rule_tac [2] nat_mult_distrib)
wenzelm@63652
  1087
   apply auto
wenzelm@63652
  1088
  done
haftmann@25919
  1089
wenzelm@61944
  1090
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
wenzelm@63652
  1091
  by (cases "z = 0 \<or> w = 0")
wenzelm@63652
  1092
    (auto simp add: abs_if nat_mult_distrib [symmetric]
wenzelm@63652
  1093
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1094
wenzelm@63652
  1095
lemma int_in_range_abs [simp]: "int n \<in> range abs"
haftmann@60570
  1096
proof (rule range_eqI)
wenzelm@63652
  1097
  show "int n = \<bar>int n\<bar>" by simp
haftmann@60570
  1098
qed
haftmann@60570
  1099
wenzelm@63652
  1100
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
haftmann@60570
  1101
proof -
haftmann@60570
  1102
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1103
    by (cases k) simp_all
haftmann@60570
  1104
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1105
    using that by induct simp
haftmann@60570
  1106
  ultimately show ?thesis by blast
paulson@61204
  1107
qed
haftmann@60570
  1108
wenzelm@63652
  1109
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
wenzelm@63652
  1110
  for z :: int
wenzelm@63652
  1111
  by (rule sym) (simp add: nat_eq_iff)
huffman@47207
  1112
huffman@47207
  1113
lemma diff_nat_eq_if:
wenzelm@63652
  1114
  "nat z - nat z' =
wenzelm@63652
  1115
    (if z' < 0 then nat z
wenzelm@63652
  1116
     else
wenzelm@63652
  1117
      let d = z - z'
wenzelm@63652
  1118
      in if d < 0 then 0 else nat d)"
wenzelm@63652
  1119
  by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1120
wenzelm@63652
  1121
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1122
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1123
haftmann@25919
  1124
wenzelm@63652
  1125
subsection \<open>Induction principles for int\<close>
haftmann@25919
  1126
wenzelm@63652
  1127
text \<open>Well-founded segments of the integers.\<close>
haftmann@25919
  1128
wenzelm@63652
  1129
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1130
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
haftmann@25919
  1131
wenzelm@63652
  1132
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1133
proof -
wenzelm@63652
  1134
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
haftmann@25919
  1135
    by (auto simp add: int_ge_less_than_def)
wenzelm@63652
  1136
  then show ?thesis
lp15@60162
  1137
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1138
qed
haftmann@25919
  1139
wenzelm@63652
  1140
text \<open>
wenzelm@63652
  1141
  This variant looks odd, but is typical of the relations suggested
wenzelm@63652
  1142
  by RankFinder.\<close>
haftmann@25919
  1143
wenzelm@63652
  1144
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1145
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
haftmann@25919
  1146
wenzelm@63652
  1147
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1148
proof -
wenzelm@63652
  1149
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
haftmann@25919
  1150
    by (auto simp add: int_ge_less_than2_def)
wenzelm@63652
  1151
  then show ?thesis
lp15@60162
  1152
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1153
qed
haftmann@25919
  1154
haftmann@25919
  1155
(* `set:int': dummy construction *)
haftmann@25919
  1156
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1157
  fixes i :: int
wenzelm@63652
  1158
  assumes ge: "k \<le> i"
wenzelm@63652
  1159
    and base: "P k"
wenzelm@63652
  1160
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1161
  shows "P i"
haftmann@25919
  1162
proof -
wenzelm@63652
  1163
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
wenzelm@63652
  1164
  proof (induct n)
wenzelm@63652
  1165
    case 0
wenzelm@63652
  1166
    then have "i = k" by arith
wenzelm@63652
  1167
    with base show "P i" by simp
wenzelm@63652
  1168
  next
wenzelm@63652
  1169
    case (Suc n)
wenzelm@63652
  1170
    then have "n = nat ((i - 1) - k)" by arith
wenzelm@63652
  1171
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
wenzelm@63652
  1172
    ultimately have "P (i - 1)" by (rule Suc.hyps)
wenzelm@63652
  1173
    from step [OF k this] show ?case by simp
wenzelm@63652
  1174
  qed
haftmann@25919
  1175
  with ge show ?thesis by fast
haftmann@25919
  1176
qed
haftmann@25919
  1177
haftmann@25928
  1178
(* `set:int': dummy construction *)
haftmann@25928
  1179
theorem int_gr_induct [case_names base step, induct set: int]:
wenzelm@63652
  1180
  fixes i k :: int
wenzelm@63652
  1181
  assumes gr: "k < i"
wenzelm@63652
  1182
    and base: "P (k + 1)"
wenzelm@63652
  1183
    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1184
  shows "P i"
wenzelm@63652
  1185
  apply (rule int_ge_induct[of "k + 1"])
haftmann@25919
  1186
  using gr apply arith
wenzelm@63652
  1187
   apply (rule base)
wenzelm@63652
  1188
  apply (rule step)
wenzelm@63652
  1189
   apply simp_all
wenzelm@63652
  1190
  done
haftmann@25919
  1191
wenzelm@42676
  1192
theorem int_le_induct [consumes 1, case_names base step]:
wenzelm@63652
  1193
  fixes i k :: int
wenzelm@63652
  1194
  assumes le: "i \<le> k"
wenzelm@63652
  1195
    and base: "P k"
wenzelm@63652
  1196
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1197
  shows "P i"
haftmann@25919
  1198
proof -
wenzelm@63652
  1199
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
wenzelm@63652
  1200
  proof (induct n)
wenzelm@63652
  1201
    case 0
wenzelm@63652
  1202
    then have "i = k" by arith
wenzelm@63652
  1203
    with base show "P i" by simp
wenzelm@63652
  1204
  next
wenzelm@63652
  1205
    case (Suc n)
wenzelm@63652
  1206
    then have "n = nat (k - (i + 1))" by arith
wenzelm@63652
  1207
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
wenzelm@63652
  1208
    ultimately have "P (i + 1)" by (rule Suc.hyps)
wenzelm@63652
  1209
    from step[OF k this] show ?case by simp
wenzelm@63652
  1210
  qed
haftmann@25919
  1211
  with le show ?thesis by fast
haftmann@25919
  1212
qed
haftmann@25919
  1213
wenzelm@42676
  1214
theorem int_less_induct [consumes 1, case_names base step]:
wenzelm@63652
  1215
  fixes i k :: int
wenzelm@63652
  1216
  assumes less: "i < k"
wenzelm@63652
  1217
    and base: "P (k - 1)"
wenzelm@63652
  1218
    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1219
  shows "P i"
wenzelm@63652
  1220
  apply (rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1221
  using less apply arith
wenzelm@63652
  1222
   apply (rule base)
wenzelm@63652
  1223
  apply (rule step)
wenzelm@63652
  1224
   apply simp_all
wenzelm@63652
  1225
  done
haftmann@25919
  1226
haftmann@36811
  1227
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1228
  fixes k :: int
haftmann@36801
  1229
  assumes base: "P k"
haftmann@36801
  1230
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1231
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1232
  shows "P i"
haftmann@36801
  1233
proof -
haftmann@36801
  1234
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1235
  then show ?thesis
wenzelm@42676
  1236
  proof
wenzelm@42676
  1237
    assume "i \<ge> k"
wenzelm@63652
  1238
    then show ?thesis
wenzelm@63652
  1239
      using base by (rule int_ge_induct) (fact step1)
haftmann@36801
  1240
  next
wenzelm@42676
  1241
    assume "i \<le> k"
wenzelm@63652
  1242
    then show ?thesis
wenzelm@63652
  1243
      using base by (rule int_le_induct) (fact step2)
haftmann@36801
  1244
  qed
haftmann@36801
  1245
qed
haftmann@36801
  1246
wenzelm@63652
  1247
wenzelm@63652
  1248
subsection \<open>Intermediate value theorems\<close>
haftmann@25919
  1249
wenzelm@63652
  1250
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
  1251
  for n :: nat and k :: int
wenzelm@63652
  1252
  unfolding One_nat_def
wenzelm@63652
  1253
  apply (induct n)
wenzelm@63652
  1254
   apply simp
wenzelm@63652
  1255
  apply (intro strip)
wenzelm@63652
  1256
  apply (erule impE)
wenzelm@63652
  1257
   apply simp
wenzelm@63652
  1258
  apply (erule_tac x = n in allE)
wenzelm@63652
  1259
  apply simp
wenzelm@63652
  1260
  apply (case_tac "k = f (Suc n)")
wenzelm@63652
  1261
   apply force
wenzelm@63652
  1262
  apply (erule impE)
wenzelm@63652
  1263
   apply (simp add: abs_if split: if_split_asm)
wenzelm@63652
  1264
  apply (blast intro: le_SucI)
wenzelm@63652
  1265
  done
haftmann@25919
  1266
haftmann@25919
  1267
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1268
haftmann@25919
  1269
lemma nat_intermed_int_val:
wenzelm@63652
  1270
  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
wenzelm@63652
  1271
    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
  1272
    for f :: "nat \<Rightarrow> int" and k :: int
wenzelm@63652
  1273
  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
wenzelm@63652
  1274
  unfolding One_nat_def
wenzelm@63652
  1275
  apply simp
wenzelm@63652
  1276
  apply (erule exE)
wenzelm@63652
  1277
  apply (rule_tac x = "i+m" in exI)
wenzelm@63652
  1278
  apply arith
wenzelm@63652
  1279
  done
haftmann@25919
  1280
haftmann@25919
  1281
wenzelm@63652
  1282
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1283
paulson@34055
  1284
lemma abs_zmult_eq_1:
wenzelm@63652
  1285
  fixes m n :: int
paulson@34055
  1286
  assumes mn: "\<bar>m * n\<bar> = 1"
wenzelm@63652
  1287
  shows "\<bar>m\<bar> = 1"
paulson@34055
  1288
proof -
wenzelm@63652
  1289
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
wenzelm@63652
  1290
  have "\<not> 2 \<le> \<bar>m\<bar>"
paulson@34055
  1291
  proof
paulson@34055
  1292
    assume "2 \<le> \<bar>m\<bar>"
wenzelm@63652
  1293
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
wenzelm@63652
  1294
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
wenzelm@63652
  1295
    also from mn have "\<dots> = 1" by simp
wenzelm@63652
  1296
    finally have "2 * \<bar>n\<bar> \<le> 1" .
wenzelm@63652
  1297
    with 0 show "False" by arith
paulson@34055
  1298
  qed
wenzelm@63652
  1299
  with 0 show ?thesis by auto
paulson@34055
  1300
qed
haftmann@25919
  1301
wenzelm@63652
  1302
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
wenzelm@63652
  1303
  for m n :: int
wenzelm@63652
  1304
  using abs_zmult_eq_1 [of m n] by arith
haftmann@25919
  1305
boehmes@35815
  1306
lemma pos_zmult_eq_1_iff:
wenzelm@63652
  1307
  fixes m n :: int
wenzelm@63652
  1308
  assumes "0 < m"
wenzelm@63652
  1309
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
boehmes@35815
  1310
proof -
wenzelm@63652
  1311
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
wenzelm@63652
  1312
    by (auto dest: pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1313
  then show ?thesis
wenzelm@63652
  1314
    by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1315
qed
haftmann@25919
  1316
wenzelm@63652
  1317
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
wenzelm@63652
  1318
  for m n :: int
wenzelm@63652
  1319
  apply (rule iffI)
wenzelm@63652
  1320
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1321
   apply (simp add: mult.commute [of m])
wenzelm@63652
  1322
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1323
   apply auto
wenzelm@63652
  1324
  done
haftmann@25919
  1325
haftmann@33296
  1326
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1327
proof
haftmann@33296
  1328
  assume "finite (UNIV::int set)"
wenzelm@61076
  1329
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1330
    by (rule injI) simp
wenzelm@61076
  1331
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1332
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1333
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1334
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1335
qed
haftmann@25919
  1336
haftmann@25919
  1337
wenzelm@60758
  1338
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1339
wenzelm@63652
  1340
subsubsection \<open>Special Simplification for Constants\<close>
haftmann@30652
  1341
wenzelm@63652
  1342
text \<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1343
webertj@49962
  1344
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1345
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1346
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1347
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1348
wenzelm@63652
  1349
text \<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1350
huffman@47108
  1351
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1352
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1353
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1354
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1355
haftmann@30652
  1356
wenzelm@61799
  1357
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1358
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1359
huffman@47108
  1360
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1361
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1362
huffman@47108
  1363
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1364
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1365
wenzelm@60758
  1366
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@63652
  1367
  into the literal.\<close>
haftmann@30652
  1368
haftmann@54489
  1369
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1370
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1371
haftmann@54489
  1372
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1373
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1374
haftmann@54489
  1375
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1376
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1377
haftmann@54489
  1378
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1379
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1380
haftmann@54489
  1381
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1382
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1383
haftmann@54489
  1384
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1385
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1386
wenzelm@63652
  1387
(* FIXME maybe simproc *)
haftmann@30652
  1388
haftmann@30652
  1389
wenzelm@61799
  1390
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1391
huffman@47108
  1392
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1393
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1394
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1395
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1396
haftmann@30652
  1397
wenzelm@61799
  1398
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1399
lp15@61738
  1400
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1401
lp15@61738
  1402
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1403
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1404
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1405
lp15@61738
  1406
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1407
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1408
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1409
lp15@61738
  1410
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1411
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1412
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1413
lp15@61738
  1414
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1415
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1416
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1417
lp15@61738
  1418
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1419
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1420
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1421
lp15@61738
  1422
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1423
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1424
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1425
haftmann@30652
  1426
wenzelm@63652
  1427
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1428
wenzelm@63652
  1429
text \<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1430
lp15@61738
  1431
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1432
  le_divide_eq [of "numeral w"]
haftmann@54489
  1433
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1434
lp15@61738
  1435
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1436
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1437
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1438
lp15@61738
  1439
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1440
  less_divide_eq [of "numeral w"]
haftmann@54489
  1441
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1442
lp15@61738
  1443
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1444
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1445
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1446
lp15@61738
  1447
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1448
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1449
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1450
lp15@61738
  1451
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1452
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1453
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1454
haftmann@30652
  1455
wenzelm@63652
  1456
text \<open>Not good as automatic simprules because they cause case splits.\<close>
wenzelm@63652
  1457
lemmas [divide_const_simps] =
wenzelm@63652
  1458
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1459
haftmann@30652
  1460
wenzelm@60758
  1461
subsection \<open>The divides relation\<close>
haftmann@33320
  1462
wenzelm@63652
  1463
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
wenzelm@63652
  1464
  for m n :: int
wenzelm@63652
  1465
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1466
wenzelm@63652
  1467
lemma zdvd_antisym_abs:
wenzelm@63652
  1468
  fixes a b :: int
wenzelm@63652
  1469
  assumes "a dvd b" and "b dvd a"
haftmann@33320
  1470
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@63652
  1471
proof (cases "a = 0")
wenzelm@63652
  1472
  case True
wenzelm@63652
  1473
  with assms show ?thesis by simp
nipkow@33657
  1474
next
wenzelm@63652
  1475
  case False
wenzelm@63652
  1476
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
wenzelm@63652
  1477
    unfolding dvd_def by blast
wenzelm@63652
  1478
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
wenzelm@63652
  1479
    unfolding dvd_def by blast
wenzelm@63652
  1480
  from k k' have "a = a * k * k'" by simp
wenzelm@63652
  1481
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
wenzelm@63652
  1482
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
wenzelm@63652
  1483
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
wenzelm@63652
  1484
    by (simp add: zmult_eq_1_iff)
wenzelm@63652
  1485
  with k k' show ?thesis by auto
haftmann@33320
  1486
qed
haftmann@33320
  1487
wenzelm@63652
  1488
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
wenzelm@63652
  1489
  for k m n :: int
lp15@60162
  1490
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1491
wenzelm@63652
  1492
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
wenzelm@63652
  1493
  for k m n :: int
haftmann@58649
  1494
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1495
haftmann@33320
  1496
lemma dvd_imp_le_int:
haftmann@33320
  1497
  fixes d i :: int
haftmann@33320
  1498
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1499
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1500
proof -
wenzelm@60758
  1501
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1502
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1503
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1504
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1505
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1506
qed
haftmann@33320
  1507
haftmann@33320
  1508
lemma zdvd_not_zless:
haftmann@33320
  1509
  fixes m n :: int
haftmann@33320
  1510
  assumes "0 < m" and "m < n"
haftmann@33320
  1511
  shows "\<not> n dvd m"
haftmann@33320
  1512
proof
haftmann@33320
  1513
  from assms have "0 < n" by auto
haftmann@33320
  1514
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1515
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1516
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1517
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1518
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1519
qed
haftmann@33320
  1520
wenzelm@63652
  1521
lemma zdvd_mult_cancel:
wenzelm@63652
  1522
  fixes k m n :: int
wenzelm@63652
  1523
  assumes d: "k * m dvd k * n"
wenzelm@63652
  1524
    and "k \<noteq> 0"
haftmann@33320
  1525
  shows "m dvd n"
wenzelm@63652
  1526
proof -
wenzelm@63652
  1527
  from d obtain h where h: "k * n = k * m * h"
wenzelm@63652
  1528
    unfolding dvd_def by blast
wenzelm@63652
  1529
  have "n = m * h"
wenzelm@63652
  1530
  proof (rule ccontr)
wenzelm@63652
  1531
    assume "\<not> ?thesis"
wenzelm@63652
  1532
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
wenzelm@63652
  1533
    with h show False
wenzelm@63652
  1534
      by (simp add: mult.assoc)
wenzelm@63652
  1535
  qed
wenzelm@63652
  1536
  then show ?thesis by simp
haftmann@33320
  1537
qed
haftmann@33320
  1538
wenzelm@63652
  1539
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
haftmann@33320
  1540
proof -
wenzelm@63652
  1541
  have "x dvd y" if "int y = int x * k" for k
wenzelm@63652
  1542
  proof (cases k)
wenzelm@63652
  1543
    case (nonneg n)
wenzelm@63652
  1544
    with that have "y = x * n"
wenzelm@63652
  1545
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
wenzelm@63652
  1546
    then show ?thesis ..
wenzelm@63652
  1547
  next
wenzelm@63652
  1548
    case (neg n)
wenzelm@63652
  1549
    with that have "int y = int x * (- int (Suc n))"
wenzelm@63652
  1550
      by simp
wenzelm@63652
  1551
    also have "\<dots> = - (int x * int (Suc n))"
wenzelm@63652
  1552
      by (simp only: mult_minus_right)
wenzelm@63652
  1553
    also have "\<dots> = - int (x * Suc n)"
wenzelm@63652
  1554
      by (simp only: of_nat_mult [symmetric])
wenzelm@63652
  1555
    finally have "- int (x * Suc n) = int y" ..
wenzelm@63652
  1556
    then show ?thesis
wenzelm@63652
  1557
      by (simp only: negative_eq_positive) auto
haftmann@33320
  1558
  qed
wenzelm@63652
  1559
  then show ?thesis
wenzelm@63652
  1560
    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1561
qed
haftmann@33320
  1562
wenzelm@63652
  1563
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
wenzelm@63652
  1564
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
  1565
  for x :: int
haftmann@33320
  1566
proof
wenzelm@63652
  1567
  assume ?lhs
wenzelm@63652
  1568
  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@63652
  1569
  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@63652
  1570
  then have "nat \<bar>x\<bar> = 1" by simp
wenzelm@63652
  1571
  then show ?rhs by (cases "x < 0") auto
haftmann@33320
  1572
next
wenzelm@63652
  1573
  assume ?rhs
wenzelm@63652
  1574
  then have "x = 1 \<or> x = - 1" by auto
wenzelm@63652
  1575
  then show ?lhs by (auto intro: dvdI)
haftmann@33320
  1576
qed
haftmann@33320
  1577
lp15@60162
  1578
lemma zdvd_mult_cancel1:
wenzelm@63652
  1579
  fixes m :: int
wenzelm@63652
  1580
  assumes mp: "m \<noteq> 0"
wenzelm@63652
  1581
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
wenzelm@63652
  1582
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@33320
  1583
proof
wenzelm@63652
  1584
  assume ?rhs
wenzelm@63652
  1585
  then show ?lhs
wenzelm@63652
  1586
    by (cases "n > 0") (auto simp add: minus_equation_iff)
haftmann@33320
  1587
next
wenzelm@63652
  1588
  assume ?lhs
wenzelm@63652
  1589
  then have "m * n dvd m * 1" by simp
wenzelm@63652
  1590
  from zdvd_mult_cancel[OF this mp] show ?rhs
wenzelm@63652
  1591
    by (simp only: zdvd1_eq)
haftmann@33320
  1592
qed
haftmann@33320
  1593
wenzelm@63652
  1594
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
wenzelm@63652
  1595
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@33320
  1596
wenzelm@63652
  1597
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
wenzelm@63652
  1598
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@58650
  1599
wenzelm@63652
  1600
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
wenzelm@63652
  1601
  by (simp add: dvd_int_iff [symmetric])
wenzelm@63652
  1602
wenzelm@63652
  1603
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
haftmann@33320
  1604
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1605
wenzelm@63652
  1606
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1607
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1608
wenzelm@63652
  1609
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1610
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1611
wenzelm@63652
  1612
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
wenzelm@63652
  1613
  for n z :: int
wenzelm@42676
  1614
  apply (cases n)
wenzelm@63652
  1615
   apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1616
  apply (cases z)
wenzelm@63652
  1617
   apply (auto simp add: dvd_imp_le)
haftmann@33320
  1618
  done
haftmann@33320
  1619
haftmann@36749
  1620
lemma zdvd_period:
haftmann@36749
  1621
  fixes a d :: int
haftmann@36749
  1622
  assumes "a dvd d"
haftmann@36749
  1623
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
wenzelm@63652
  1624
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@36749
  1625
proof -
haftmann@36749
  1626
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1627
  show ?thesis
wenzelm@42676
  1628
  proof
wenzelm@63652
  1629
    assume ?lhs
haftmann@36749
  1630
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1631
    then have "x = a * l - t" by simp
wenzelm@63652
  1632
    with \<open>d = a * k\<close> show ?rhs by simp
haftmann@36749
  1633
  next
wenzelm@63652
  1634
    assume ?rhs
haftmann@36749
  1635
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1636
    then have "x = a * l - c * d - t" by simp
wenzelm@63652
  1637
    with \<open>d = a * k\<close> show ?lhs by simp
haftmann@36749
  1638
  qed
haftmann@36749
  1639
qed
haftmann@36749
  1640
haftmann@33320
  1641
wenzelm@60758
  1642
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1643
wenzelm@63652
  1644
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
wenzelm@63652
  1645
proof (cases "a \<le> b")
bulwahn@46756
  1646
  case True
wenzelm@63652
  1647
  then show ?thesis
bulwahn@46756
  1648
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1649
    case base
wenzelm@63652
  1650
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
wenzelm@63652
  1651
    then show ?case by simp
bulwahn@46756
  1652
  next
bulwahn@46756
  1653
    case (step b)
wenzelm@63652
  1654
    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
  1655
    with step show ?case by simp
bulwahn@46756
  1656
  qed
bulwahn@46756
  1657
next
wenzelm@63652
  1658
  case False
wenzelm@63652
  1659
  then show ?thesis
bulwahn@46756
  1660
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1661
qed
bulwahn@46756
  1662
wenzelm@63652
  1663
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
wenzelm@63652
  1664
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1665
wenzelm@63652
  1666
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
wenzelm@63652
  1667
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1668
wenzelm@63652
  1669
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
wenzelm@63652
  1670
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1671
bulwahn@46756
  1672
wenzelm@60758
  1673
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1674
wenzelm@60758
  1675
text \<open>Constructors\<close>
huffman@47108
  1676
wenzelm@63652
  1677
definition Pos :: "num \<Rightarrow> int"
wenzelm@63652
  1678
  where [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1679
wenzelm@63652
  1680
definition Neg :: "num \<Rightarrow> int"
wenzelm@63652
  1681
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1682
huffman@47108
  1683
code_datatype "0::int" Pos Neg
huffman@47108
  1684
huffman@47108
  1685
wenzelm@63652
  1686
text \<open>Auxiliary operations.\<close>
huffman@47108
  1687
wenzelm@63652
  1688
definition dup :: "int \<Rightarrow> int"
wenzelm@63652
  1689
  where [simp]: "dup k = k + k"
haftmann@26507
  1690
huffman@47108
  1691
lemma dup_code [code]:
huffman@47108
  1692
  "dup 0 = 0"
huffman@47108
  1693
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1694
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1695
  by (simp_all add: numeral_Bit0)
huffman@47108
  1696
wenzelm@63652
  1697
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
wenzelm@63652
  1698
  where [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1699
huffman@47108
  1700
lemma sub_code [code]:
huffman@47108
  1701
  "sub Num.One Num.One = 0"
huffman@47108
  1702
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1703
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1704
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1705
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1706
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1707
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1708
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1709
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
wenzelm@63652
  1710
          apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
wenzelm@63652
  1711
        apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230
  1712
  apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230
  1713
  apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230
  1714
  done
huffman@47108
  1715
wenzelm@63652
  1716
text \<open>Implementations.\<close>
huffman@47108
  1717
wenzelm@63652
  1718
lemma one_int_code [code, code_unfold]: "1 = Pos Num.One"
huffman@47108
  1719
  by simp
huffman@47108
  1720
huffman@47108
  1721
lemma plus_int_code [code]:
wenzelm@63652
  1722
  "k + 0 = k"
wenzelm@63652
  1723
  "0 + l = l"
huffman@47108
  1724
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1725
  "Pos m + Neg n = sub m n"
huffman@47108
  1726
  "Neg m + Pos n = sub n m"
huffman@47108
  1727
  "Neg m + Neg n = Neg (m + n)"
wenzelm@63652
  1728
  for k l :: int
huffman@47108
  1729
  by simp_all
haftmann@26507
  1730
huffman@47108
  1731
lemma uminus_int_code [code]:
huffman@47108
  1732
  "uminus 0 = (0::int)"
huffman@47108
  1733
  "uminus (Pos m) = Neg m"
huffman@47108
  1734
  "uminus (Neg m) = Pos m"
huffman@47108
  1735
  by simp_all
huffman@47108
  1736
huffman@47108
  1737
lemma minus_int_code [code]:
wenzelm@63652
  1738
  "k - 0 = k"
wenzelm@63652
  1739
  "0 - l = uminus l"
huffman@47108
  1740
  "Pos m - Pos n = sub m n"
huffman@47108
  1741
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1742
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1743
  "Neg m - Neg n = sub n m"
wenzelm@63652
  1744
  for k l :: int
huffman@47108
  1745
  by simp_all
huffman@47108
  1746
huffman@47108
  1747
lemma times_int_code [code]:
wenzelm@63652
  1748
  "k * 0 = 0"
wenzelm@63652
  1749
  "0 * l = 0"
huffman@47108
  1750
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1751
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1752
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1753
  "Neg m * Neg n = Pos (m * n)"
wenzelm@63652
  1754
  for k l :: int
huffman@47108
  1755
  by simp_all
haftmann@26507
  1756
haftmann@38857
  1757
instantiation int :: equal
haftmann@26507
  1758
begin
haftmann@26507
  1759
wenzelm@63652
  1760
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1761
wenzelm@61169
  1762
instance
wenzelm@61169
  1763
  by standard (rule equal_int_def)
haftmann@26507
  1764
haftmann@26507
  1765
end
haftmann@26507
  1766
huffman@47108
  1767
lemma equal_int_code [code]:
huffman@47108
  1768
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1769
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1770
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1771
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1772
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1773
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1774
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1775
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1776
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1777
  by (auto simp add: equal)
haftmann@26507
  1778
wenzelm@63652
  1779
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
wenzelm@63652
  1780
  for k :: int
huffman@47108
  1781
  by (fact equal_refl)
haftmann@26507
  1782
haftmann@28562
  1783
lemma less_eq_int_code [code]:
huffman@47108
  1784
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1785
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1786
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1787
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1788
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1789
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1790
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1791
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1792
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1793
  by simp_all
haftmann@26507
  1794
haftmann@28562
  1795
lemma less_int_code [code]:
huffman@47108
  1796
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1797
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1798
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1799
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1800
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1801
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1802
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1803
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1804
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1805
  by simp_all
haftmann@25919
  1806
huffman@47108
  1807
lemma nat_code [code]:
huffman@47108
  1808
  "nat (Int.Neg k) = 0"
huffman@47108
  1809
  "nat 0 = 0"
huffman@47108
  1810
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1811
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1812
huffman@47108
  1813
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1814
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1815
  "of_int 0 = 0"
huffman@47108
  1816
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1817
  by simp_all
haftmann@25919
  1818
huffman@47108
  1819
wenzelm@63652
  1820
text \<open>Serializer setup.\<close>
haftmann@25919
  1821
haftmann@52435
  1822
code_identifier
haftmann@52435
  1823
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1824
haftmann@25919
  1825
quickcheck_params [default_type = int]
haftmann@25919
  1826
huffman@47108
  1827
hide_const (open) Pos Neg sub dup
haftmann@25919
  1828
haftmann@25919
  1829
wenzelm@61799
  1830
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1831
kuncar@53652
  1832
lifting_update int.lifting
kuncar@53652
  1833
lifting_forget int.lifting
huffman@48045
  1834
haftmann@25919
  1835
end