src/HOL/Int.thy
author haftmann
Mon Oct 09 19:10:47 2017 +0200 (21 months ago)
changeset 66836 4eb431c3f974
parent 66816 212a3334e7da
child 66886 960509bfd47e
permissions -rw-r--r--
tuned imports
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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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lemma of_int_of_bool [simp]:
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  "of_int (of_bool P) = of_bool P"
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  by auto
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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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   331
proof (cases "n = 0")
wenzelm@63652
   332
  case True
wenzelm@63652
   333
  then show ?thesis by simp
haftmann@62347
   334
next
haftmann@62347
   335
  case False
haftmann@62347
   336
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   337
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   338
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   339
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   340
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   341
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   342
  then have "1 < x" using assms by (rule le_less_trans)
haftmann@62347
   343
  then show ?thesis ..
haftmann@62347
   344
qed
haftmann@62347
   345
haftmann@62347
   346
lemma of_int_leD:
haftmann@62347
   347
  assumes "\<bar>of_int n\<bar> \<le> x"
haftmann@62347
   348
  shows "n = 0 \<or> 1 \<le> x"
haftmann@62347
   349
proof (cases "n = 0")
wenzelm@63652
   350
  case True
wenzelm@63652
   351
  then show ?thesis by simp
haftmann@62347
   352
next
haftmann@62347
   353
  case False
haftmann@62347
   354
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   355
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   356
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   357
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   358
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   359
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   360
  then have "1 \<le> x" using assms by (rule order_trans)
haftmann@62347
   361
  then show ?thesis ..
haftmann@62347
   362
qed
haftmann@62347
   363
haftmann@36424
   364
end
haftmann@25919
   365
lp15@61234
   366
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   367
wenzelm@63652
   368
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
lp15@61234
   369
  using of_int_eq_iff by fastforce
lp15@61234
   370
lp15@61649
   371
lemma of_int_le_numeral_iff [simp]:
wenzelm@63652
   372
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   373
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   374
lp15@61649
   375
lemma of_int_numeral_le_iff [simp]:
wenzelm@63652
   376
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   377
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   378
lp15@61649
   379
lemma of_int_less_numeral_iff [simp]:
wenzelm@63652
   380
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
lp15@61234
   381
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   382
lp15@61649
   383
lemma of_int_numeral_less_iff [simp]:
wenzelm@63652
   384
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   385
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   386
wenzelm@63652
   387
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   388
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   389
haftmann@25919
   390
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   391
proof
wenzelm@63652
   392
  show "of_int z = id z" for z
wenzelm@63652
   393
    by (cases z rule: int_diff_cases) simp
haftmann@25919
   394
qed
haftmann@25919
   395
hoelzl@51329
   396
instance int :: no_top
wenzelm@61169
   397
  apply standard
hoelzl@51329
   398
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   399
  apply simp
hoelzl@51329
   400
  done
hoelzl@51329
   401
hoelzl@51329
   402
instance int :: no_bot
wenzelm@61169
   403
  apply standard
hoelzl@51329
   404
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   405
  apply simp
hoelzl@51329
   406
  done
hoelzl@51329
   407
wenzelm@63652
   408
wenzelm@61799
   409
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   410
huffman@48045
   411
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   412
  by auto
haftmann@25919
   413
huffman@44709
   414
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   415
  by transfer simp
haftmann@25919
   416
huffman@44709
   417
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   418
  by transfer clarsimp
haftmann@25919
   419
wenzelm@63652
   420
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
wenzelm@63652
   421
  by simp
haftmann@25919
   422
wenzelm@63652
   423
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
huffman@48045
   424
  by transfer clarsimp
haftmann@25919
   425
wenzelm@63652
   426
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
huffman@48045
   427
  by transfer (clarsimp, arith)
haftmann@25919
   428
wenzelm@63652
   429
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
wenzelm@63652
   430
lemma nat_mono_iff: "0 < z \<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 nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   434
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   435
wenzelm@63652
   436
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
huffman@48045
   437
  by transfer (clarsimp, arith)
haftmann@25919
   438
haftmann@64714
   439
lemma nonneg_int_cases:
haftmann@64714
   440
  assumes "0 \<le> k"
haftmann@64714
   441
  obtains n where "k = int n"
haftmann@64714
   442
proof -
haftmann@64714
   443
  from assms have "k = int (nat k)"
haftmann@64714
   444
    by simp
haftmann@64714
   445
  then show thesis
haftmann@64714
   446
    by (rule that)
haftmann@64714
   447
qed
haftmann@64714
   448
haftmann@64714
   449
lemma pos_int_cases:
haftmann@64714
   450
  assumes "0 < k"
haftmann@64714
   451
  obtains n where "k = int n" and "n > 0"
haftmann@64714
   452
proof -
haftmann@64714
   453
  from assms have "0 \<le> k"
haftmann@64714
   454
    by simp
haftmann@64714
   455
  then obtain n where "k = int n"
haftmann@64714
   456
    by (rule nonneg_int_cases)
haftmann@64714
   457
  moreover have "n > 0"
haftmann@64714
   458
    using \<open>k = int n\<close> assms by simp
haftmann@64714
   459
  ultimately show thesis
haftmann@64714
   460
    by (rule that)
haftmann@64714
   461
qed
haftmann@64714
   462
haftmann@64714
   463
lemma nonpos_int_cases:
haftmann@64714
   464
  assumes "k \<le> 0"
haftmann@64714
   465
  obtains n where "k = - int n"
haftmann@64714
   466
proof -
haftmann@64714
   467
  from assms have "- k \<ge> 0"
haftmann@64714
   468
    by simp
haftmann@64714
   469
  then obtain n where "- k = int n"
haftmann@64714
   470
    by (rule nonneg_int_cases)
haftmann@64714
   471
  then have "k = - int n"
haftmann@64714
   472
    by simp
haftmann@64714
   473
  then show thesis
haftmann@64714
   474
    by (rule that)
haftmann@64714
   475
qed
haftmann@64714
   476
haftmann@64714
   477
lemma neg_int_cases:
haftmann@64714
   478
  assumes "k < 0"
haftmann@64714
   479
  obtains n where "k = - int n" and "n > 0"
haftmann@64714
   480
proof -
haftmann@64714
   481
  from assms have "- k > 0"
haftmann@64714
   482
    by simp
haftmann@64714
   483
  then obtain n where "- k = int n" and "- k > 0"
haftmann@64714
   484
    by (blast elim: pos_int_cases)
haftmann@64714
   485
  then have "k = - int n" and "n > 0"
haftmann@64714
   486
    by simp_all
haftmann@64714
   487
  then show thesis
haftmann@64714
   488
    by (rule that)
haftmann@64714
   489
qed
haftmann@25919
   490
wenzelm@63652
   491
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   492
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   493
wenzelm@63652
   494
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   495
  using nat_eq_iff [of w m] by auto
haftmann@54223
   496
wenzelm@63652
   497
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@54223
   498
  by (simp add: nat_eq_iff)
haftmann@25919
   499
wenzelm@63652
   500
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@54223
   501
  by (simp add: nat_eq_iff)
haftmann@54223
   502
wenzelm@63652
   503
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
haftmann@54223
   504
  by (simp add: nat_eq_iff)
haftmann@25919
   505
wenzelm@63652
   506
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
haftmann@54223
   507
  by simp
haftmann@54223
   508
haftmann@54223
   509
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   510
  by simp
lp15@60162
   511
wenzelm@63652
   512
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
huffman@48045
   513
  by transfer (clarsimp, arith)
haftmann@25919
   514
huffman@44709
   515
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   516
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   517
huffman@44707
   518
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   519
  by transfer auto
huffman@44707
   520
wenzelm@63652
   521
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
wenzelm@63652
   522
  for i :: int
huffman@48045
   523
  by transfer clarsimp
nipkow@29700
   524
wenzelm@63652
   525
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
wenzelm@63652
   526
  by (auto simp add: nat_eq_iff2)
haftmann@25919
   527
wenzelm@63652
   528
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
wenzelm@63652
   529
  using zless_nat_conj [of 0] by auto
haftmann@25919
   530
wenzelm@63652
   531
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   532
  by transfer clarsimp
haftmann@25919
   533
wenzelm@63652
   534
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   535
  by transfer clarsimp
lp15@60162
   536
wenzelm@63652
   537
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   538
  by (rule nat_diff_distrib') auto
haftmann@25919
   539
huffman@44709
   540
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   541
  by transfer simp
haftmann@25919
   542
wenzelm@63652
   543
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   544
  by transfer auto
lp15@60162
   545
wenzelm@63652
   546
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
huffman@48045
   547
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   548
wenzelm@63652
   549
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   550
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   551
wenzelm@63652
   552
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   553
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   554
haftmann@66816
   555
lemma nat_of_bool [simp]:
haftmann@66816
   556
  "nat (of_bool P) = of_bool P"
haftmann@66816
   557
  by auto
haftmann@66816
   558
haftmann@66836
   559
lemma split_nat [arith_split]: "P (nat i) \<longleftrightarrow> ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
haftmann@66836
   560
  (is "?P = (?L \<and> ?R)")
haftmann@66836
   561
  for i :: int
haftmann@66836
   562
proof (cases "i < 0")
haftmann@66836
   563
  case True
haftmann@66836
   564
  then show ?thesis
haftmann@66836
   565
    by auto
haftmann@66836
   566
next
haftmann@66836
   567
  case False
haftmann@66836
   568
  have "?P = ?L"
haftmann@66836
   569
  proof
haftmann@66836
   570
    assume ?P
haftmann@66836
   571
    then show ?L using False by auto
haftmann@66836
   572
  next
haftmann@66836
   573
    assume ?L
haftmann@66836
   574
    moreover from False have "int (nat i) = i"
haftmann@66836
   575
      by (simp add: not_less)
haftmann@66836
   576
    ultimately show ?P
haftmann@66836
   577
      by simp
haftmann@66836
   578
  qed
haftmann@66836
   579
  with False show ?thesis by simp
haftmann@66836
   580
qed
haftmann@66836
   581
haftmann@66836
   582
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
haftmann@66836
   583
  by (auto split: split_nat)
haftmann@66836
   584
haftmann@66836
   585
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
haftmann@66836
   586
proof
haftmann@66836
   587
  assume "\<exists>x. P x"
haftmann@66836
   588
  then obtain x where "P x" ..
haftmann@66836
   589
  then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
haftmann@66836
   590
  then show "\<exists>x\<ge>0. P (nat x)" ..
haftmann@66836
   591
next
haftmann@66836
   592
  assume "\<exists>x\<ge>0. P (nat x)"
haftmann@66836
   593
  then show "\<exists>x. P x" by auto
haftmann@66836
   594
qed
haftmann@66836
   595
haftmann@54249
   596
wenzelm@60758
   597
text \<open>For termination proofs:\<close>
wenzelm@63652
   598
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
krauss@29779
   599
haftmann@25919
   600
wenzelm@63652
   601
subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   602
wenzelm@61076
   603
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
wenzelm@63652
   604
  by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   605
huffman@44709
   606
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
wenzelm@63652
   607
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   608
huffman@44709
   609
lemma negative_zle_0: "- int n \<le> 0"
wenzelm@63652
   610
  by (simp add: minus_le_iff)
haftmann@25919
   611
huffman@44709
   612
lemma negative_zle [iff]: "- int n \<le> int m"
wenzelm@63652
   613
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   614
wenzelm@63652
   615
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
wenzelm@63652
   616
  by (subst le_minus_iff) (simp del: of_nat_Suc)
haftmann@25919
   617
wenzelm@63652
   618
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
huffman@48045
   619
  by transfer simp
haftmann@25919
   620
wenzelm@63652
   621
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
wenzelm@63652
   622
  by (simp add: linorder_not_less)
haftmann@25919
   623
wenzelm@63652
   624
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
wenzelm@63652
   625
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   626
wenzelm@63652
   627
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
wenzelm@63652
   628
  (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@62348
   629
proof
wenzelm@63652
   630
  assume ?rhs
wenzelm@63652
   631
  then show ?lhs by auto
haftmann@62348
   632
next
wenzelm@63652
   633
  assume ?lhs
haftmann@62348
   634
  then have "0 \<le> z - w" by simp
haftmann@62348
   635
  then obtain n where "z - w = int n"
haftmann@62348
   636
    using zero_le_imp_eq_int [of "z - w"] by blast
wenzelm@63652
   637
  then have "z = w + int n" by simp
wenzelm@63652
   638
  then show ?rhs ..
haftmann@25919
   639
qed
haftmann@25919
   640
huffman@44709
   641
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
wenzelm@63652
   642
  by simp
haftmann@25919
   643
wenzelm@63652
   644
text \<open>
wenzelm@63652
   645
  This version is proved for all ordered rings, not just integers!
wenzelm@63652
   646
  It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@63652
   647
  in theory \<open>Rings\<close>.
wenzelm@63652
   648
  But is it really better than just rewriting with \<open>abs_if\<close>?
wenzelm@63652
   649
\<close>
wenzelm@63652
   650
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
   651
  for a :: "'a::linordered_idom"
wenzelm@63652
   652
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   653
huffman@44709
   654
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
wenzelm@63652
   655
  apply transfer
wenzelm@63652
   656
  apply clarsimp
wenzelm@63652
   657
  apply (rule_tac x="b - Suc a" in exI)
wenzelm@63652
   658
  apply arith
wenzelm@63652
   659
  done
wenzelm@63652
   660
haftmann@25919
   661
wenzelm@60758
   662
subsection \<open>Cases and induction\<close>
haftmann@25919
   663
wenzelm@63652
   664
text \<open>
wenzelm@63652
   665
  Now we replace the case analysis rule by a more conventional one:
wenzelm@63652
   666
  whether an integer is negative or not.
wenzelm@63652
   667
\<close>
haftmann@25919
   668
wenzelm@63652
   669
text \<open>This version is symmetric in the two subgoals.\<close>
wenzelm@63652
   670
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
wenzelm@63652
   671
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   672
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   673
wenzelm@63652
   674
text \<open>This is the default, with a negative case.\<close>
wenzelm@63652
   675
lemma int_cases [case_names nonneg neg, cases type: int]:
wenzelm@63652
   676
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   677
  apply (cases "z < 0")
wenzelm@63652
   678
   apply (blast dest!: negD)
wenzelm@63652
   679
  apply (simp add: linorder_not_less del: of_nat_Suc)
wenzelm@63652
   680
  apply auto
wenzelm@63652
   681
  apply (blast dest: nat_0_le [THEN sym])
wenzelm@63652
   682
  done
haftmann@25919
   683
haftmann@60868
   684
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   685
  fixes k :: int
haftmann@60868
   686
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   687
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   688
  shows "P"
haftmann@60868
   689
proof (cases k "0::int" rule: linorder_cases)
wenzelm@63652
   690
  case equal
wenzelm@63652
   691
  with assms(1) show P by simp
haftmann@60868
   692
next
haftmann@60868
   693
  case greater
wenzelm@63539
   694
  then have *: "nat k > 0" by simp
wenzelm@63539
   695
  moreover from * have "k = int (nat k)" by auto
haftmann@60868
   696
  ultimately show P using assms(2) by blast
haftmann@60868
   697
next
haftmann@60868
   698
  case less
wenzelm@63539
   699
  then have *: "nat (- k) > 0" by simp
wenzelm@63539
   700
  moreover from * have "k = - int (nat (- k))" by auto
haftmann@60868
   701
  ultimately show P using assms(3) by blast
haftmann@60868
   702
qed
haftmann@60868
   703
wenzelm@63652
   704
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
wenzelm@63652
   705
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
wenzelm@42676
   706
  by (cases z) auto
haftmann@25919
   707
huffman@47108
   708
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   709
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   710
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   711
haftmann@54489
   712
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   713
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   714
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   715
haftmann@66816
   716
lemma sgn_mult_dvd_iff [simp]:
haftmann@66816
   717
  "sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   718
  by (cases r rule: int_cases3) auto
haftmann@66816
   719
haftmann@66816
   720
lemma mult_sgn_dvd_iff [simp]:
haftmann@66816
   721
  "l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   722
  using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)
haftmann@66816
   723
haftmann@66816
   724
lemma dvd_sgn_mult_iff [simp]:
haftmann@66816
   725
  "l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   726
  by (cases r rule: int_cases3) simp_all
haftmann@66816
   727
haftmann@66816
   728
lemma dvd_mult_sgn_iff [simp]:
haftmann@66816
   729
  "l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   730
  using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)
haftmann@66816
   731
haftmann@66816
   732
lemma int_sgnE:
haftmann@66816
   733
  fixes k :: int
haftmann@66816
   734
  obtains n and l where "k = sgn l * int n"
haftmann@66816
   735
proof -
haftmann@66816
   736
  have "k = sgn k * int (nat \<bar>k\<bar>)"
haftmann@66816
   737
    by (simp add: sgn_mult_abs)
haftmann@66816
   738
  then show ?thesis ..
haftmann@66816
   739
qed
haftmann@66816
   740
wenzelm@61799
   741
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   742
huffman@47108
   743
lemmas max_number_of [simp] =
huffman@47108
   744
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   745
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   746
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   747
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   748
huffman@47108
   749
lemmas min_number_of [simp] =
huffman@47108
   750
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   751
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   752
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   753
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   754
haftmann@25919
   755
wenzelm@60758
   756
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   757
wenzelm@60758
   758
text \<open>Preliminaries\<close>
huffman@28958
   759
lp15@60162
   760
lemma le_imp_0_less:
wenzelm@63652
   761
  fixes z :: int
huffman@28958
   762
  assumes le: "0 \<le> z"
wenzelm@63652
   763
  shows "0 < 1 + z"
huffman@28958
   764
proof -
huffman@28958
   765
  have "0 \<le> z" by fact
wenzelm@63652
   766
  also have "\<dots> < z + 1" by (rule less_add_one)
wenzelm@63652
   767
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
huffman@28958
   768
  finally show "0 < 1 + z" .
huffman@28958
   769
qed
huffman@28958
   770
wenzelm@63652
   771
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
wenzelm@63652
   772
  for z :: int
wenzelm@42676
   773
proof (cases z)
huffman@28958
   774
  case (nonneg n)
wenzelm@63652
   775
  then show ?thesis
wenzelm@63652
   776
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   777
next
huffman@28958
   778
  case (neg n)
wenzelm@63652
   779
  then show ?thesis
wenzelm@63652
   780
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@63652
   781
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   782
qed
huffman@28958
   783
wenzelm@63652
   784
wenzelm@60758
   785
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   786
haftmann@25919
   787
lemmas double_eq_0_iff = double_zero
haftmann@25919
   788
wenzelm@63652
   789
lemma odd_nonzero: "1 + z + z \<noteq> 0"
wenzelm@63652
   790
  for z :: int
wenzelm@42676
   791
proof (cases z)
haftmann@25919
   792
  case (nonneg n)
wenzelm@63652
   793
  have le: "0 \<le> z + z"
wenzelm@63652
   794
    by (simp add: nonneg add_increasing)
wenzelm@63652
   795
  then show ?thesis
wenzelm@63652
   796
    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
haftmann@25919
   797
next
haftmann@25919
   798
  case (neg n)
haftmann@25919
   799
  show ?thesis
haftmann@25919
   800
  proof
haftmann@25919
   801
    assume eq: "1 + z + z = 0"
wenzelm@63652
   802
    have "0 < 1 + (int n + int n)"
lp15@60162
   803
      by (simp add: le_imp_0_less add_increasing)
wenzelm@63652
   804
    also have "\<dots> = - (1 + z + z)"
lp15@60162
   805
      by (simp add: neg add.assoc [symmetric])
wenzelm@63652
   806
    also have "\<dots> = 0" by (simp add: eq)
haftmann@25919
   807
    finally have "0<0" ..
wenzelm@63652
   808
    then show False by blast
haftmann@25919
   809
  qed
haftmann@25919
   810
qed
haftmann@25919
   811
haftmann@30652
   812
wenzelm@60758
   813
subsection \<open>The Set of Integers\<close>
haftmann@25919
   814
haftmann@25919
   815
context ring_1
haftmann@25919
   816
begin
haftmann@25919
   817
wenzelm@61070
   818
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   819
  where "\<int> = range of_int"
haftmann@25919
   820
huffman@35634
   821
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   822
  by (simp add: Ints_def)
huffman@35634
   823
huffman@35634
   824
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   825
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   826
haftmann@25919
   827
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   828
  using Ints_of_int [of "0"] by simp
haftmann@25919
   829
haftmann@25919
   830
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   831
  using Ints_of_int [of "1"] by simp
haftmann@25919
   832
eberlm@61552
   833
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   834
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   835
haftmann@25919
   836
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
wenzelm@63652
   837
  apply (auto simp add: Ints_def)
wenzelm@63652
   838
  apply (rule range_eqI)
wenzelm@63652
   839
  apply (rule of_int_add [symmetric])
wenzelm@63652
   840
  done
haftmann@25919
   841
haftmann@25919
   842
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
wenzelm@63652
   843
  apply (auto simp add: Ints_def)
wenzelm@63652
   844
  apply (rule range_eqI)
wenzelm@63652
   845
  apply (rule of_int_minus [symmetric])
wenzelm@63652
   846
  done
haftmann@25919
   847
huffman@35634
   848
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
wenzelm@63652
   849
  apply (auto simp add: Ints_def)
wenzelm@63652
   850
  apply (rule range_eqI)
wenzelm@63652
   851
  apply (rule of_int_diff [symmetric])
wenzelm@63652
   852
  done
huffman@35634
   853
haftmann@25919
   854
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
wenzelm@63652
   855
  apply (auto simp add: Ints_def)
wenzelm@63652
   856
  apply (rule range_eqI)
wenzelm@63652
   857
  apply (rule of_int_mult [symmetric])
wenzelm@63652
   858
  done
haftmann@25919
   859
huffman@35634
   860
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
wenzelm@63652
   861
  by (induct n) simp_all
huffman@35634
   862
haftmann@25919
   863
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   864
  assumes "q \<in> \<int>"
haftmann@25919
   865
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   866
  unfolding Ints_def
haftmann@25919
   867
proof -
wenzelm@60758
   868
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   869
  then obtain z where "q = of_int z" ..
haftmann@25919
   870
  then show thesis ..
haftmann@25919
   871
qed
haftmann@25919
   872
haftmann@25919
   873
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   874
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   875
  by (rule Ints_cases) auto
haftmann@25919
   876
eberlm@61524
   877
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   878
  unfolding Nats_def Ints_def
eberlm@61524
   879
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   880
eberlm@61524
   881
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   882
proof (intro subsetI equalityI)
wenzelm@63652
   883
  fix x :: 'a
wenzelm@63652
   884
  assume "x \<in> {of_int n |n. n \<ge> 0}"
wenzelm@63652
   885
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   886
    by (auto elim!: Ints_cases)
wenzelm@63652
   887
  then have "x = of_nat (nat n)"
wenzelm@63652
   888
    by (subst of_nat_nat) simp_all
wenzelm@63652
   889
  then show "x \<in> \<nat>"
wenzelm@63652
   890
    by simp
eberlm@61524
   891
next
wenzelm@63652
   892
  fix x :: 'a
wenzelm@63652
   893
  assume "x \<in> \<nat>"
wenzelm@63652
   894
  then obtain n where "x = of_nat n"
wenzelm@63652
   895
    by (auto elim!: Nats_cases)
wenzelm@63652
   896
  then have "x = of_int (int n)" by simp
eberlm@61524
   897
  also have "int n \<ge> 0" by simp
wenzelm@63652
   898
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   899
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   900
qed
eberlm@61524
   901
haftmann@25919
   902
end
haftmann@25919
   903
lp15@64758
   904
lemma (in linordered_idom) Ints_abs [simp]:
lp15@64758
   905
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
lp15@64758
   906
  by (auto simp: abs_if)
lp15@64758
   907
eberlm@61524
   908
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   909
proof (intro subsetI equalityI)
wenzelm@63652
   910
  fix x :: 'a
wenzelm@63652
   911
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
wenzelm@63652
   912
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   913
    by (auto elim!: Ints_cases)
wenzelm@63652
   914
  then have "x = of_nat (nat n)"
wenzelm@63652
   915
    by (subst of_nat_nat) simp_all
wenzelm@63652
   916
  then show "x \<in> \<nat>"
wenzelm@63652
   917
    by simp
eberlm@61524
   918
qed (auto elim!: Nats_cases)
eberlm@61524
   919
haftmann@64849
   920
lemma (in idom_divide) of_int_divide_in_Ints: 
haftmann@64849
   921
  "of_int a div of_int b \<in> \<int>" if "b dvd a"
haftmann@64849
   922
proof -
haftmann@64849
   923
  from that obtain c where "a = b * c" ..
haftmann@64849
   924
  then show ?thesis
haftmann@64849
   925
    by (cases "of_int b = 0") simp_all
haftmann@64849
   926
qed
eberlm@61524
   927
wenzelm@60758
   928
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
   929
haftmann@25919
   930
lemma Ints_double_eq_0_iff:
wenzelm@63652
   931
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   932
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   933
  shows "a + a = 0 \<longleftrightarrow> a = 0"
wenzelm@63652
   934
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@25919
   935
proof -
wenzelm@63652
   936
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   937
    unfolding Ints_def [symmetric] .
haftmann@25919
   938
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   939
  show ?thesis
haftmann@25919
   940
  proof
wenzelm@63652
   941
    assume ?rhs
wenzelm@63652
   942
    then show ?lhs by simp
haftmann@25919
   943
  next
wenzelm@63652
   944
    assume ?lhs
wenzelm@63652
   945
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   946
    then have "z + z = 0" by (simp only: of_int_eq_iff)
wenzelm@63652
   947
    then have "z = 0" by (simp only: double_eq_0_iff)
wenzelm@63652
   948
    with a show ?rhs by simp
haftmann@25919
   949
  qed
haftmann@25919
   950
qed
haftmann@25919
   951
haftmann@25919
   952
lemma Ints_odd_nonzero:
wenzelm@63652
   953
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   954
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   955
  shows "1 + a + a \<noteq> 0"
haftmann@25919
   956
proof -
wenzelm@63652
   957
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   958
    unfolding Ints_def [symmetric] .
haftmann@25919
   959
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   960
  show ?thesis
haftmann@25919
   961
  proof
wenzelm@63652
   962
    assume "1 + a + a = 0"
wenzelm@63652
   963
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   964
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   965
    with odd_nonzero show False by blast
haftmann@25919
   966
  qed
lp15@60162
   967
qed
haftmann@25919
   968
wenzelm@61070
   969
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
   970
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   971
lp15@60162
   972
lemma Ints_odd_less_0:
wenzelm@63652
   973
  fixes a :: "'a::linordered_idom"
wenzelm@61070
   974
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   975
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25919
   976
proof -
wenzelm@63652
   977
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   978
    unfolding Ints_def [symmetric] .
haftmann@25919
   979
  then obtain z where a: "a = of_int z" ..
wenzelm@63652
   980
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
wenzelm@63652
   981
    by simp
wenzelm@63652
   982
  also have "\<dots> \<longleftrightarrow> z < 0"
wenzelm@63652
   983
    by (simp only: of_int_less_iff odd_less_0_iff)
wenzelm@63652
   984
  also have "\<dots> \<longleftrightarrow> a < 0"
haftmann@25919
   985
    by (simp add: a)
haftmann@25919
   986
  finally show ?thesis .
haftmann@25919
   987
qed
haftmann@25919
   988
haftmann@25919
   989
nipkow@64272
   990
subsection \<open>@{term sum} and @{term prod}\<close>
haftmann@25919
   991
nipkow@64267
   992
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
wenzelm@63652
   993
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   994
nipkow@64267
   995
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
wenzelm@63652
   996
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   997
nipkow@64272
   998
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
wenzelm@63652
   999
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1000
nipkow@64272
  1001
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
wenzelm@63652
  1002
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1003
haftmann@25919
  1004
wenzelm@60758
  1005
text \<open>Legacy theorems\<close>
haftmann@25919
  1006
haftmann@64714
  1007
lemmas int_sum = of_nat_sum [where 'a=int]
haftmann@64714
  1008
lemmas int_prod = of_nat_prod [where 'a=int]
haftmann@25919
  1009
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1010
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@64714
  1011
lemmas nonneg_eq_int = nonneg_int_cases
haftmann@25919
  1012
wenzelm@63652
  1013
wenzelm@60758
  1014
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
  1015
haftmann@54249
  1016
lemmas of_int_simps =
haftmann@54249
  1017
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
  1018
wenzelm@48891
  1019
ML_file "Tools/int_arith.ML"
wenzelm@60758
  1020
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
  1021
wenzelm@63652
  1022
simproc_setup fast_arith
wenzelm@63652
  1023
  ("(m::'a::linordered_idom) < n" |
wenzelm@63652
  1024
    "(m::'a::linordered_idom) \<le> n" |
wenzelm@63652
  1025
    "(m::'a::linordered_idom) = n") =
wenzelm@61144
  1026
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
  1027
haftmann@25919
  1028
wenzelm@60758
  1029
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
  1030
wenzelm@63652
  1031
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
wenzelm@63652
  1032
  for w z :: int
wenzelm@63652
  1033
  by arith
haftmann@25919
  1034
wenzelm@63652
  1035
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
wenzelm@63652
  1036
  for w z :: int
wenzelm@63652
  1037
  by arith
haftmann@25919
  1038
wenzelm@63652
  1039
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
wenzelm@63652
  1040
  for w z :: int
wenzelm@63652
  1041
  by arith
haftmann@25919
  1042
wenzelm@63652
  1043
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
wenzelm@63652
  1044
  for w z :: int
wenzelm@63652
  1045
  by arith
haftmann@25919
  1046
wenzelm@63652
  1047
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
wenzelm@63652
  1048
  for z :: int
wenzelm@63652
  1049
  by arith
haftmann@25919
  1050
lp15@64758
  1051
lemma Ints_nonzero_abs_ge1:
lp15@64758
  1052
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1053
    assumes "x \<in> Ints" "x \<noteq> 0"
lp15@64758
  1054
    shows "1 \<le> abs x"
lp15@64758
  1055
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
lp15@64758
  1056
  fix z::int
lp15@64758
  1057
  assume "x = of_int z"
lp15@64758
  1058
    with \<open>x \<noteq> 0\<close> 
lp15@64758
  1059
  show "1 \<le> \<bar>x\<bar>"
lp15@64758
  1060
    apply (auto simp add: abs_if)
lp15@64758
  1061
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
lp15@64758
  1062
qed
lp15@64758
  1063
  
lp15@64758
  1064
lemma Ints_nonzero_abs_less1:
lp15@64758
  1065
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1066
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
lp15@64758
  1067
    using Ints_nonzero_abs_ge1 [of x] by auto
lp15@64758
  1068
    
haftmann@25919
  1069
wenzelm@63652
  1070
subsection \<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
  1071
wenzelm@63652
  1072
text \<open>Simplify the term @{term "w + - z"}.\<close>
haftmann@25919
  1073
wenzelm@63652
  1074
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
lp15@60162
  1075
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
  1076
wenzelm@63652
  1077
text \<open>
wenzelm@63652
  1078
  This simplifies expressions of the form @{term "int n = z"} where
wenzelm@63652
  1079
  \<open>z\<close> is an integer literal.
wenzelm@63652
  1080
\<close>
huffman@47108
  1081
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
  1082
hoelzl@59000
  1083
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
  1084
  by auto
hoelzl@59000
  1085
hoelzl@59000
  1086
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
  1087
  by auto
hoelzl@59000
  1088
haftmann@25919
  1089
context ring_1
haftmann@25919
  1090
begin
haftmann@25919
  1091
blanchet@33056
  1092
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1093
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1094
proof (cases "k < 0")
wenzelm@63652
  1095
  case True
wenzelm@63652
  1096
  then have "0 \<le> - k" by simp
haftmann@25919
  1097
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1098
  with True show ?thesis by simp
haftmann@25919
  1099
next
wenzelm@63652
  1100
  case False
wenzelm@63652
  1101
  then show ?thesis by (simp add: not_less)
haftmann@25919
  1102
qed
haftmann@25919
  1103
haftmann@25919
  1104
end
haftmann@25919
  1105
haftmann@64014
  1106
lemma transfer_rule_of_int:
haftmann@64014
  1107
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
haftmann@64014
  1108
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64014
  1109
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64014
  1110
    "rel_fun R R uminus uminus"
haftmann@64014
  1111
  shows "rel_fun HOL.eq R of_int of_int"
haftmann@64014
  1112
proof -
haftmann@64014
  1113
  note transfer_rule_of_nat [transfer_rule]
haftmann@64014
  1114
  have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
haftmann@64014
  1115
    by transfer_prover
haftmann@64014
  1116
  show ?thesis
haftmann@64014
  1117
    by (unfold of_int_of_nat [abs_def]) transfer_prover
haftmann@64014
  1118
qed
haftmann@64014
  1119
haftmann@25919
  1120
lemma nat_mult_distrib:
haftmann@25919
  1121
  fixes z z' :: int
haftmann@25919
  1122
  assumes "0 \<le> z"
haftmann@25919
  1123
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1124
proof (cases "0 \<le> z'")
wenzelm@63652
  1125
  case False
wenzelm@63652
  1126
  with assms have "z * z' \<le> 0"
haftmann@25919
  1127
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1128
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1129
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1130
  ultimately show ?thesis by simp
haftmann@25919
  1131
next
wenzelm@63652
  1132
  case True
wenzelm@63652
  1133
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1134
  show ?thesis
haftmann@25919
  1135
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1136
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1137
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1138
qed
haftmann@25919
  1139
wenzelm@63652
  1140
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
wenzelm@63652
  1141
  for z z' :: int
wenzelm@63652
  1142
  apply (rule trans)
wenzelm@63652
  1143
   apply (rule_tac [2] nat_mult_distrib)
wenzelm@63652
  1144
   apply auto
wenzelm@63652
  1145
  done
haftmann@25919
  1146
wenzelm@61944
  1147
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
wenzelm@63652
  1148
  by (cases "z = 0 \<or> w = 0")
wenzelm@63652
  1149
    (auto simp add: abs_if nat_mult_distrib [symmetric]
wenzelm@63652
  1150
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1151
wenzelm@63652
  1152
lemma int_in_range_abs [simp]: "int n \<in> range abs"
haftmann@60570
  1153
proof (rule range_eqI)
wenzelm@63652
  1154
  show "int n = \<bar>int n\<bar>" by simp
haftmann@60570
  1155
qed
haftmann@60570
  1156
wenzelm@63652
  1157
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
haftmann@60570
  1158
proof -
haftmann@60570
  1159
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1160
    by (cases k) simp_all
haftmann@60570
  1161
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1162
    using that by induct simp
haftmann@60570
  1163
  ultimately show ?thesis by blast
paulson@61204
  1164
qed
haftmann@60570
  1165
wenzelm@63652
  1166
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
wenzelm@63652
  1167
  for z :: int
wenzelm@63652
  1168
  by (rule sym) (simp add: nat_eq_iff)
huffman@47207
  1169
huffman@47207
  1170
lemma diff_nat_eq_if:
wenzelm@63652
  1171
  "nat z - nat z' =
wenzelm@63652
  1172
    (if z' < 0 then nat z
wenzelm@63652
  1173
     else
wenzelm@63652
  1174
      let d = z - z'
wenzelm@63652
  1175
      in if d < 0 then 0 else nat d)"
wenzelm@63652
  1176
  by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1177
wenzelm@63652
  1178
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1179
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1180
haftmann@25919
  1181
wenzelm@63652
  1182
subsection \<open>Induction principles for int\<close>
haftmann@25919
  1183
wenzelm@63652
  1184
text \<open>Well-founded segments of the integers.\<close>
haftmann@25919
  1185
wenzelm@63652
  1186
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1187
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
haftmann@25919
  1188
wenzelm@63652
  1189
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1190
proof -
wenzelm@63652
  1191
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
haftmann@25919
  1192
    by (auto simp add: int_ge_less_than_def)
wenzelm@63652
  1193
  then show ?thesis
lp15@60162
  1194
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1195
qed
haftmann@25919
  1196
wenzelm@63652
  1197
text \<open>
wenzelm@63652
  1198
  This variant looks odd, but is typical of the relations suggested
wenzelm@63652
  1199
  by RankFinder.\<close>
haftmann@25919
  1200
wenzelm@63652
  1201
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1202
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
haftmann@25919
  1203
wenzelm@63652
  1204
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1205
proof -
wenzelm@63652
  1206
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
haftmann@25919
  1207
    by (auto simp add: int_ge_less_than2_def)
wenzelm@63652
  1208
  then show ?thesis
lp15@60162
  1209
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1210
qed
haftmann@25919
  1211
haftmann@25919
  1212
(* `set:int': dummy construction *)
haftmann@25919
  1213
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1214
  fixes i :: int
wenzelm@63652
  1215
  assumes ge: "k \<le> i"
wenzelm@63652
  1216
    and base: "P k"
wenzelm@63652
  1217
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1218
  shows "P i"
haftmann@25919
  1219
proof -
wenzelm@63652
  1220
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
wenzelm@63652
  1221
  proof (induct n)
wenzelm@63652
  1222
    case 0
wenzelm@63652
  1223
    then have "i = k" by arith
wenzelm@63652
  1224
    with base show "P i" by simp
wenzelm@63652
  1225
  next
wenzelm@63652
  1226
    case (Suc n)
wenzelm@63652
  1227
    then have "n = nat ((i - 1) - k)" by arith
wenzelm@63652
  1228
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
wenzelm@63652
  1229
    ultimately have "P (i - 1)" by (rule Suc.hyps)
wenzelm@63652
  1230
    from step [OF k this] show ?case by simp
wenzelm@63652
  1231
  qed
haftmann@25919
  1232
  with ge show ?thesis by fast
haftmann@25919
  1233
qed
haftmann@25919
  1234
haftmann@25928
  1235
(* `set:int': dummy construction *)
haftmann@25928
  1236
theorem int_gr_induct [case_names base step, induct set: int]:
wenzelm@63652
  1237
  fixes i k :: int
wenzelm@63652
  1238
  assumes gr: "k < i"
wenzelm@63652
  1239
    and base: "P (k + 1)"
wenzelm@63652
  1240
    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1241
  shows "P i"
wenzelm@63652
  1242
  apply (rule int_ge_induct[of "k + 1"])
haftmann@25919
  1243
  using gr apply arith
wenzelm@63652
  1244
   apply (rule base)
wenzelm@63652
  1245
  apply (rule step)
wenzelm@63652
  1246
   apply simp_all
wenzelm@63652
  1247
  done
haftmann@25919
  1248
wenzelm@42676
  1249
theorem int_le_induct [consumes 1, case_names base step]:
wenzelm@63652
  1250
  fixes i k :: int
wenzelm@63652
  1251
  assumes le: "i \<le> k"
wenzelm@63652
  1252
    and base: "P k"
wenzelm@63652
  1253
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1254
  shows "P i"
haftmann@25919
  1255
proof -
wenzelm@63652
  1256
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
wenzelm@63652
  1257
  proof (induct n)
wenzelm@63652
  1258
    case 0
wenzelm@63652
  1259
    then have "i = k" by arith
wenzelm@63652
  1260
    with base show "P i" by simp
wenzelm@63652
  1261
  next
wenzelm@63652
  1262
    case (Suc n)
wenzelm@63652
  1263
    then have "n = nat (k - (i + 1))" by arith
wenzelm@63652
  1264
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
wenzelm@63652
  1265
    ultimately have "P (i + 1)" by (rule Suc.hyps)
wenzelm@63652
  1266
    from step[OF k this] show ?case by simp
wenzelm@63652
  1267
  qed
haftmann@25919
  1268
  with le show ?thesis by fast
haftmann@25919
  1269
qed
haftmann@25919
  1270
wenzelm@42676
  1271
theorem int_less_induct [consumes 1, case_names base step]:
wenzelm@63652
  1272
  fixes i k :: int
wenzelm@63652
  1273
  assumes less: "i < k"
wenzelm@63652
  1274
    and base: "P (k - 1)"
wenzelm@63652
  1275
    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1276
  shows "P i"
wenzelm@63652
  1277
  apply (rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1278
  using less apply arith
wenzelm@63652
  1279
   apply (rule base)
wenzelm@63652
  1280
  apply (rule step)
wenzelm@63652
  1281
   apply simp_all
wenzelm@63652
  1282
  done
haftmann@25919
  1283
haftmann@36811
  1284
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1285
  fixes k :: int
haftmann@36801
  1286
  assumes base: "P k"
haftmann@36801
  1287
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1288
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1289
  shows "P i"
haftmann@36801
  1290
proof -
haftmann@36801
  1291
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1292
  then show ?thesis
wenzelm@42676
  1293
  proof
wenzelm@42676
  1294
    assume "i \<ge> k"
wenzelm@63652
  1295
    then show ?thesis
wenzelm@63652
  1296
      using base by (rule int_ge_induct) (fact step1)
haftmann@36801
  1297
  next
wenzelm@42676
  1298
    assume "i \<le> k"
wenzelm@63652
  1299
    then show ?thesis
wenzelm@63652
  1300
      using base by (rule int_le_induct) (fact step2)
haftmann@36801
  1301
  qed
haftmann@36801
  1302
qed
haftmann@36801
  1303
wenzelm@63652
  1304
wenzelm@63652
  1305
subsection \<open>Intermediate value theorems\<close>
haftmann@25919
  1306
wenzelm@63652
  1307
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
  1308
  for n :: nat and k :: int
wenzelm@63652
  1309
  unfolding One_nat_def
wenzelm@63652
  1310
  apply (induct n)
wenzelm@63652
  1311
   apply simp
wenzelm@63652
  1312
  apply (intro strip)
wenzelm@63652
  1313
  apply (erule impE)
wenzelm@63652
  1314
   apply simp
wenzelm@63652
  1315
  apply (erule_tac x = n in allE)
wenzelm@63652
  1316
  apply simp
wenzelm@63652
  1317
  apply (case_tac "k = f (Suc n)")
wenzelm@63652
  1318
   apply force
wenzelm@63652
  1319
  apply (erule impE)
wenzelm@63652
  1320
   apply (simp add: abs_if split: if_split_asm)
wenzelm@63652
  1321
  apply (blast intro: le_SucI)
wenzelm@63652
  1322
  done
haftmann@25919
  1323
haftmann@25919
  1324
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1325
haftmann@25919
  1326
lemma nat_intermed_int_val:
wenzelm@63652
  1327
  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
wenzelm@63652
  1328
    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
  1329
    for f :: "nat \<Rightarrow> int" and k :: int
wenzelm@63652
  1330
  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
wenzelm@63652
  1331
  unfolding One_nat_def
wenzelm@63652
  1332
  apply simp
wenzelm@63652
  1333
  apply (erule exE)
wenzelm@63652
  1334
  apply (rule_tac x = "i+m" in exI)
wenzelm@63652
  1335
  apply arith
wenzelm@63652
  1336
  done
haftmann@25919
  1337
haftmann@25919
  1338
wenzelm@63652
  1339
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1340
paulson@34055
  1341
lemma abs_zmult_eq_1:
wenzelm@63652
  1342
  fixes m n :: int
paulson@34055
  1343
  assumes mn: "\<bar>m * n\<bar> = 1"
wenzelm@63652
  1344
  shows "\<bar>m\<bar> = 1"
paulson@34055
  1345
proof -
wenzelm@63652
  1346
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
wenzelm@63652
  1347
  have "\<not> 2 \<le> \<bar>m\<bar>"
paulson@34055
  1348
  proof
paulson@34055
  1349
    assume "2 \<le> \<bar>m\<bar>"
wenzelm@63652
  1350
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
wenzelm@63652
  1351
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
wenzelm@63652
  1352
    also from mn have "\<dots> = 1" by simp
wenzelm@63652
  1353
    finally have "2 * \<bar>n\<bar> \<le> 1" .
wenzelm@63652
  1354
    with 0 show "False" by arith
paulson@34055
  1355
  qed
wenzelm@63652
  1356
  with 0 show ?thesis by auto
paulson@34055
  1357
qed
haftmann@25919
  1358
wenzelm@63652
  1359
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
wenzelm@63652
  1360
  for m n :: int
wenzelm@63652
  1361
  using abs_zmult_eq_1 [of m n] by arith
haftmann@25919
  1362
boehmes@35815
  1363
lemma pos_zmult_eq_1_iff:
wenzelm@63652
  1364
  fixes m n :: int
wenzelm@63652
  1365
  assumes "0 < m"
wenzelm@63652
  1366
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
boehmes@35815
  1367
proof -
wenzelm@63652
  1368
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
wenzelm@63652
  1369
    by (auto dest: pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1370
  then show ?thesis
wenzelm@63652
  1371
    by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1372
qed
haftmann@25919
  1373
wenzelm@63652
  1374
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
wenzelm@63652
  1375
  for m n :: int
wenzelm@63652
  1376
  apply (rule iffI)
wenzelm@63652
  1377
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1378
   apply (simp add: mult.commute [of m])
wenzelm@63652
  1379
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1380
   apply auto
wenzelm@63652
  1381
  done
haftmann@25919
  1382
haftmann@33296
  1383
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1384
proof
haftmann@33296
  1385
  assume "finite (UNIV::int set)"
wenzelm@61076
  1386
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1387
    by (rule injI) simp
wenzelm@61076
  1388
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1389
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1390
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1391
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1392
qed
haftmann@25919
  1393
haftmann@25919
  1394
wenzelm@60758
  1395
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1396
wenzelm@63652
  1397
subsubsection \<open>Special Simplification for Constants\<close>
haftmann@30652
  1398
wenzelm@63652
  1399
text \<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1400
webertj@49962
  1401
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1402
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1403
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1404
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1405
wenzelm@63652
  1406
text \<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1407
huffman@47108
  1408
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1409
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1410
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1411
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1412
haftmann@30652
  1413
wenzelm@61799
  1414
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1415
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1416
huffman@47108
  1417
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1418
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1419
huffman@47108
  1420
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1421
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1422
wenzelm@60758
  1423
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@63652
  1424
  into the literal.\<close>
haftmann@30652
  1425
haftmann@54489
  1426
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1427
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1428
haftmann@54489
  1429
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1430
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1431
haftmann@54489
  1432
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1433
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1434
haftmann@54489
  1435
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1436
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1437
haftmann@54489
  1438
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1439
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1440
haftmann@54489
  1441
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1442
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1443
wenzelm@63652
  1444
(* FIXME maybe simproc *)
haftmann@30652
  1445
haftmann@30652
  1446
wenzelm@61799
  1447
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1448
huffman@47108
  1449
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1450
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1451
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1452
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1453
haftmann@30652
  1454
wenzelm@61799
  1455
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1456
lp15@61738
  1457
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1458
lp15@61738
  1459
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1460
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1461
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1462
lp15@61738
  1463
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1464
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1465
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1466
lp15@61738
  1467
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1468
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1469
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1470
lp15@61738
  1471
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1472
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1473
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1474
lp15@61738
  1475
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1476
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1477
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1478
lp15@61738
  1479
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1480
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1481
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1482
haftmann@30652
  1483
wenzelm@63652
  1484
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1485
wenzelm@63652
  1486
text \<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1487
lp15@61738
  1488
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1489
  le_divide_eq [of "numeral w"]
haftmann@54489
  1490
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1491
lp15@61738
  1492
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1493
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1494
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1495
lp15@61738
  1496
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1497
  less_divide_eq [of "numeral w"]
haftmann@54489
  1498
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1499
lp15@61738
  1500
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1501
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1502
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1503
lp15@61738
  1504
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1505
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1506
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1507
lp15@61738
  1508
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1509
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1510
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1511
haftmann@30652
  1512
wenzelm@63652
  1513
text \<open>Not good as automatic simprules because they cause case splits.\<close>
wenzelm@63652
  1514
lemmas [divide_const_simps] =
wenzelm@63652
  1515
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1516
haftmann@30652
  1517
wenzelm@60758
  1518
subsection \<open>The divides relation\<close>
haftmann@33320
  1519
wenzelm@63652
  1520
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
wenzelm@63652
  1521
  for m n :: int
wenzelm@63652
  1522
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1523
wenzelm@63652
  1524
lemma zdvd_antisym_abs:
wenzelm@63652
  1525
  fixes a b :: int
wenzelm@63652
  1526
  assumes "a dvd b" and "b dvd a"
haftmann@33320
  1527
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@63652
  1528
proof (cases "a = 0")
wenzelm@63652
  1529
  case True
wenzelm@63652
  1530
  with assms show ?thesis by simp
nipkow@33657
  1531
next
wenzelm@63652
  1532
  case False
wenzelm@63652
  1533
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
wenzelm@63652
  1534
    unfolding dvd_def by blast
wenzelm@63652
  1535
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
wenzelm@63652
  1536
    unfolding dvd_def by blast
wenzelm@63652
  1537
  from k k' have "a = a * k * k'" by simp
wenzelm@63652
  1538
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
wenzelm@63652
  1539
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
wenzelm@63652
  1540
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
wenzelm@63652
  1541
    by (simp add: zmult_eq_1_iff)
wenzelm@63652
  1542
  with k k' show ?thesis by auto
haftmann@33320
  1543
qed
haftmann@33320
  1544
wenzelm@63652
  1545
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
wenzelm@63652
  1546
  for k m n :: int
lp15@60162
  1547
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1548
wenzelm@63652
  1549
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
wenzelm@63652
  1550
  for k m n :: int
haftmann@58649
  1551
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1552
haftmann@33320
  1553
lemma dvd_imp_le_int:
haftmann@33320
  1554
  fixes d i :: int
haftmann@33320
  1555
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1556
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1557
proof -
wenzelm@60758
  1558
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1559
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1560
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1561
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1562
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1563
qed
haftmann@33320
  1564
haftmann@33320
  1565
lemma zdvd_not_zless:
haftmann@33320
  1566
  fixes m n :: int
haftmann@33320
  1567
  assumes "0 < m" and "m < n"
haftmann@33320
  1568
  shows "\<not> n dvd m"
haftmann@33320
  1569
proof
haftmann@33320
  1570
  from assms have "0 < n" by auto
haftmann@33320
  1571
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1572
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1573
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1574
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1575
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1576
qed
haftmann@33320
  1577
wenzelm@63652
  1578
lemma zdvd_mult_cancel:
wenzelm@63652
  1579
  fixes k m n :: int
wenzelm@63652
  1580
  assumes d: "k * m dvd k * n"
wenzelm@63652
  1581
    and "k \<noteq> 0"
haftmann@33320
  1582
  shows "m dvd n"
wenzelm@63652
  1583
proof -
wenzelm@63652
  1584
  from d obtain h where h: "k * n = k * m * h"
wenzelm@63652
  1585
    unfolding dvd_def by blast
wenzelm@63652
  1586
  have "n = m * h"
wenzelm@63652
  1587
  proof (rule ccontr)
wenzelm@63652
  1588
    assume "\<not> ?thesis"
wenzelm@63652
  1589
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
wenzelm@63652
  1590
    with h show False
wenzelm@63652
  1591
      by (simp add: mult.assoc)
wenzelm@63652
  1592
  qed
wenzelm@63652
  1593
  then show ?thesis by simp
haftmann@33320
  1594
qed
haftmann@33320
  1595
wenzelm@63652
  1596
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
haftmann@33320
  1597
proof -
wenzelm@63652
  1598
  have "x dvd y" if "int y = int x * k" for k
wenzelm@63652
  1599
  proof (cases k)
wenzelm@63652
  1600
    case (nonneg n)
wenzelm@63652
  1601
    with that have "y = x * n"
wenzelm@63652
  1602
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
wenzelm@63652
  1603
    then show ?thesis ..
wenzelm@63652
  1604
  next
wenzelm@63652
  1605
    case (neg n)
wenzelm@63652
  1606
    with that have "int y = int x * (- int (Suc n))"
wenzelm@63652
  1607
      by simp
wenzelm@63652
  1608
    also have "\<dots> = - (int x * int (Suc n))"
wenzelm@63652
  1609
      by (simp only: mult_minus_right)
wenzelm@63652
  1610
    also have "\<dots> = - int (x * Suc n)"
wenzelm@63652
  1611
      by (simp only: of_nat_mult [symmetric])
wenzelm@63652
  1612
    finally have "- int (x * Suc n) = int y" ..
wenzelm@63652
  1613
    then show ?thesis
wenzelm@63652
  1614
      by (simp only: negative_eq_positive) auto
haftmann@33320
  1615
  qed
wenzelm@63652
  1616
  then show ?thesis
wenzelm@63652
  1617
    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1618
qed
haftmann@33320
  1619
wenzelm@63652
  1620
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
wenzelm@63652
  1621
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
  1622
  for x :: int
haftmann@33320
  1623
proof
wenzelm@63652
  1624
  assume ?lhs
wenzelm@63652
  1625
  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@63652
  1626
  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@63652
  1627
  then have "nat \<bar>x\<bar> = 1" by simp
wenzelm@63652
  1628
  then show ?rhs by (cases "x < 0") auto
haftmann@33320
  1629
next
wenzelm@63652
  1630
  assume ?rhs
wenzelm@63652
  1631
  then have "x = 1 \<or> x = - 1" by auto
wenzelm@63652
  1632
  then show ?lhs by (auto intro: dvdI)
haftmann@33320
  1633
qed
haftmann@33320
  1634
lp15@60162
  1635
lemma zdvd_mult_cancel1:
wenzelm@63652
  1636
  fixes m :: int
wenzelm@63652
  1637
  assumes mp: "m \<noteq> 0"
wenzelm@63652
  1638
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
wenzelm@63652
  1639
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@33320
  1640
proof
wenzelm@63652
  1641
  assume ?rhs
wenzelm@63652
  1642
  then show ?lhs
wenzelm@63652
  1643
    by (cases "n > 0") (auto simp add: minus_equation_iff)
haftmann@33320
  1644
next
wenzelm@63652
  1645
  assume ?lhs
wenzelm@63652
  1646
  then have "m * n dvd m * 1" by simp
wenzelm@63652
  1647
  from zdvd_mult_cancel[OF this mp] show ?rhs
wenzelm@63652
  1648
    by (simp only: zdvd1_eq)
haftmann@33320
  1649
qed
haftmann@33320
  1650
wenzelm@63652
  1651
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
wenzelm@63652
  1652
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@33320
  1653
wenzelm@63652
  1654
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
wenzelm@63652
  1655
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@58650
  1656
wenzelm@63652
  1657
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
wenzelm@63652
  1658
  by (simp add: dvd_int_iff [symmetric])
wenzelm@63652
  1659
wenzelm@63652
  1660
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
haftmann@33320
  1661
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1662
wenzelm@63652
  1663
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1664
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1665
wenzelm@63652
  1666
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1667
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1668
wenzelm@63652
  1669
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
wenzelm@63652
  1670
  for n z :: int
wenzelm@42676
  1671
  apply (cases n)
wenzelm@63652
  1672
   apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1673
  apply (cases z)
wenzelm@63652
  1674
   apply (auto simp add: dvd_imp_le)
haftmann@33320
  1675
  done
haftmann@33320
  1676
haftmann@36749
  1677
lemma zdvd_period:
haftmann@36749
  1678
  fixes a d :: int
haftmann@36749
  1679
  assumes "a dvd d"
haftmann@36749
  1680
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
wenzelm@63652
  1681
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@36749
  1682
proof -
haftmann@66816
  1683
  from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)"
haftmann@66816
  1684
    by (simp add: dvd_add_left_iff)
haftmann@66816
  1685
  then show ?thesis
haftmann@66816
  1686
    by (simp add: ac_simps)
haftmann@36749
  1687
qed
haftmann@36749
  1688
haftmann@33320
  1689
wenzelm@60758
  1690
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1691
wenzelm@63652
  1692
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
wenzelm@63652
  1693
proof (cases "a \<le> b")
bulwahn@46756
  1694
  case True
wenzelm@63652
  1695
  then show ?thesis
bulwahn@46756
  1696
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1697
    case base
wenzelm@63652
  1698
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
wenzelm@63652
  1699
    then show ?case by simp
bulwahn@46756
  1700
  next
bulwahn@46756
  1701
    case (step b)
wenzelm@63652
  1702
    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
  1703
    with step show ?case by simp
bulwahn@46756
  1704
  qed
bulwahn@46756
  1705
next
wenzelm@63652
  1706
  case False
wenzelm@63652
  1707
  then show ?thesis
bulwahn@46756
  1708
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1709
qed
bulwahn@46756
  1710
wenzelm@63652
  1711
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
wenzelm@63652
  1712
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1713
wenzelm@63652
  1714
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
wenzelm@63652
  1715
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1716
wenzelm@63652
  1717
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
wenzelm@63652
  1718
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1719
bulwahn@46756
  1720
wenzelm@60758
  1721
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1722
wenzelm@60758
  1723
text \<open>Constructors\<close>
huffman@47108
  1724
wenzelm@63652
  1725
definition Pos :: "num \<Rightarrow> int"
wenzelm@63652
  1726
  where [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1727
wenzelm@63652
  1728
definition Neg :: "num \<Rightarrow> int"
wenzelm@63652
  1729
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1730
huffman@47108
  1731
code_datatype "0::int" Pos Neg
huffman@47108
  1732
huffman@47108
  1733
wenzelm@63652
  1734
text \<open>Auxiliary operations.\<close>
huffman@47108
  1735
wenzelm@63652
  1736
definition dup :: "int \<Rightarrow> int"
wenzelm@63652
  1737
  where [simp]: "dup k = k + k"
haftmann@26507
  1738
huffman@47108
  1739
lemma dup_code [code]:
huffman@47108
  1740
  "dup 0 = 0"
huffman@47108
  1741
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1742
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1743
  by (simp_all add: numeral_Bit0)
huffman@47108
  1744
wenzelm@63652
  1745
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
wenzelm@63652
  1746
  where [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1747
huffman@47108
  1748
lemma sub_code [code]:
huffman@47108
  1749
  "sub Num.One Num.One = 0"
huffman@47108
  1750
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1751
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1752
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1753
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1754
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1755
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1756
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1757
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
boehmes@66035
  1758
  by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
huffman@47108
  1759
wenzelm@63652
  1760
text \<open>Implementations.\<close>
huffman@47108
  1761
haftmann@64996
  1762
lemma one_int_code [code]: "1 = Pos Num.One"
huffman@47108
  1763
  by simp
huffman@47108
  1764
huffman@47108
  1765
lemma plus_int_code [code]:
wenzelm@63652
  1766
  "k + 0 = k"
wenzelm@63652
  1767
  "0 + l = l"
huffman@47108
  1768
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1769
  "Pos m + Neg n = sub m n"
huffman@47108
  1770
  "Neg m + Pos n = sub n m"
huffman@47108
  1771
  "Neg m + Neg n = Neg (m + n)"
wenzelm@63652
  1772
  for k l :: int
huffman@47108
  1773
  by simp_all
haftmann@26507
  1774
huffman@47108
  1775
lemma uminus_int_code [code]:
huffman@47108
  1776
  "uminus 0 = (0::int)"
huffman@47108
  1777
  "uminus (Pos m) = Neg m"
huffman@47108
  1778
  "uminus (Neg m) = Pos m"
huffman@47108
  1779
  by simp_all
huffman@47108
  1780
huffman@47108
  1781
lemma minus_int_code [code]:
wenzelm@63652
  1782
  "k - 0 = k"
wenzelm@63652
  1783
  "0 - l = uminus l"
huffman@47108
  1784
  "Pos m - Pos n = sub m n"
huffman@47108
  1785
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1786
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1787
  "Neg m - Neg n = sub n m"
wenzelm@63652
  1788
  for k l :: int
huffman@47108
  1789
  by simp_all
huffman@47108
  1790
huffman@47108
  1791
lemma times_int_code [code]:
wenzelm@63652
  1792
  "k * 0 = 0"
wenzelm@63652
  1793
  "0 * l = 0"
huffman@47108
  1794
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1795
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1796
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1797
  "Neg m * Neg n = Pos (m * n)"
wenzelm@63652
  1798
  for k l :: int
huffman@47108
  1799
  by simp_all
haftmann@26507
  1800
haftmann@38857
  1801
instantiation int :: equal
haftmann@26507
  1802
begin
haftmann@26507
  1803
wenzelm@63652
  1804
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1805
wenzelm@61169
  1806
instance
wenzelm@61169
  1807
  by standard (rule equal_int_def)
haftmann@26507
  1808
haftmann@26507
  1809
end
haftmann@26507
  1810
huffman@47108
  1811
lemma equal_int_code [code]:
huffman@47108
  1812
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1813
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1814
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1815
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1816
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1817
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1818
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1819
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1820
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1821
  by (auto simp add: equal)
haftmann@26507
  1822
wenzelm@63652
  1823
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
wenzelm@63652
  1824
  for k :: int
huffman@47108
  1825
  by (fact equal_refl)
haftmann@26507
  1826
haftmann@28562
  1827
lemma less_eq_int_code [code]:
huffman@47108
  1828
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1829
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1830
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1831
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1832
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1833
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1834
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1835
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1836
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1837
  by simp_all
haftmann@26507
  1838
haftmann@28562
  1839
lemma less_int_code [code]:
huffman@47108
  1840
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1841
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1842
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1843
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1844
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1845
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1846
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1847
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1848
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1849
  by simp_all
haftmann@25919
  1850
huffman@47108
  1851
lemma nat_code [code]:
huffman@47108
  1852
  "nat (Int.Neg k) = 0"
huffman@47108
  1853
  "nat 0 = 0"
huffman@47108
  1854
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1855
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1856
huffman@47108
  1857
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1858
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1859
  "of_int 0 = 0"
huffman@47108
  1860
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1861
  by simp_all
haftmann@25919
  1862
huffman@47108
  1863
wenzelm@63652
  1864
text \<open>Serializer setup.\<close>
haftmann@25919
  1865
haftmann@52435
  1866
code_identifier
haftmann@52435
  1867
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1868
haftmann@25919
  1869
quickcheck_params [default_type = int]
haftmann@25919
  1870
huffman@47108
  1871
hide_const (open) Pos Neg sub dup
haftmann@25919
  1872
haftmann@25919
  1873
wenzelm@61799
  1874
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1875
kuncar@53652
  1876
lifting_update int.lifting
kuncar@53652
  1877
lifting_forget int.lifting
huffman@48045
  1878
haftmann@25919
  1879
end