src/HOL/Int.thy
author blanchet
Tue Nov 07 15:16:42 2017 +0100 (19 months ago)
changeset 67022 49309fe530fd
parent 66912 a99a7cbf0fb5
child 67116 7397a6df81d8
permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
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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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lemma numeral_power_eq_of_int_cancel_iff [simp]:
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  "numeral x ^ n = of_int y \<longleftrightarrow> numeral x ^ n = y"
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  using of_int_eq_iff[of "numeral x ^ n" y, unfolded of_int_numeral of_int_power] .
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lemma of_int_eq_numeral_power_cancel_iff [simp]:
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  "of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
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  using numeral_power_eq_of_int_cancel_iff [of x n y] by (metis (mono_tags))
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lemma neg_numeral_power_eq_of_int_cancel_iff [simp]:
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  "(- numeral x) ^ n = of_int y \<longleftrightarrow> (- numeral x) ^ n = y"
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  using of_int_eq_iff[of "(- numeral x) ^ n" y]
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  by simp
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lemma of_int_eq_neg_numeral_power_cancel_iff [simp]:
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  "of_int y = (- numeral x) ^ n \<longleftrightarrow> y = (- numeral x) ^ n"
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  using neg_numeral_power_eq_of_int_cancel_iff[of x n y] by (metis (mono_tags))
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lemma of_int_eq_of_int_power_cancel_iff[simp]: "(of_int b) ^ w = of_int x \<longleftrightarrow> b ^ w = x"
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  by (metis of_int_power of_int_eq_iff)
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lemma of_int_power_eq_of_int_cancel_iff[simp]: "of_int x = (of_int b) ^ w \<longleftrightarrow> x = b ^ w"
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  by (metis of_int_eq_of_int_power_cancel_iff)
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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"
haftmann@36424
   325
  using of_int_less_iff [of 0 z] by simp
haftmann@36424
   326
wenzelm@63652
   327
lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
haftmann@36424
   328
  using of_int_less_iff [of z 0] by simp
haftmann@36424
   329
wenzelm@63652
   330
lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
lp15@61234
   331
  using of_int_le_iff [of 1 z] by simp
lp15@61234
   332
wenzelm@63652
   333
lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
lp15@61234
   334
  using of_int_le_iff [of z 1] by simp
lp15@61234
   335
wenzelm@63652
   336
lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
lp15@61234
   337
  using of_int_less_iff [of 1 z] by simp
lp15@61234
   338
wenzelm@63652
   339
lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
lp15@61234
   340
  using of_int_less_iff [of z 1] by simp
lp15@61234
   341
eberlm@62128
   342
lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
eberlm@62128
   343
  by simp
eberlm@62128
   344
eberlm@62128
   345
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
eberlm@62128
   346
  by simp
eberlm@62128
   347
wenzelm@63652
   348
lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
haftmann@62347
   349
  by (auto simp add: abs_if)
haftmann@62347
   350
haftmann@62347
   351
lemma of_int_lessD:
haftmann@62347
   352
  assumes "\<bar>of_int n\<bar> < x"
haftmann@62347
   353
  shows "n = 0 \<or> x > 1"
haftmann@62347
   354
proof (cases "n = 0")
wenzelm@63652
   355
  case True
wenzelm@63652
   356
  then show ?thesis by simp
haftmann@62347
   357
next
haftmann@62347
   358
  case False
haftmann@62347
   359
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   360
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   361
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   362
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   363
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   364
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   365
  then have "1 < x" using assms by (rule le_less_trans)
haftmann@62347
   366
  then show ?thesis ..
haftmann@62347
   367
qed
haftmann@62347
   368
haftmann@62347
   369
lemma of_int_leD:
haftmann@62347
   370
  assumes "\<bar>of_int n\<bar> \<le> x"
haftmann@62347
   371
  shows "n = 0 \<or> 1 \<le> x"
haftmann@62347
   372
proof (cases "n = 0")
wenzelm@63652
   373
  case True
wenzelm@63652
   374
  then show ?thesis by simp
haftmann@62347
   375
next
haftmann@62347
   376
  case False
haftmann@62347
   377
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   378
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   379
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   380
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   381
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   382
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   383
  then have "1 \<le> x" using assms by (rule order_trans)
haftmann@62347
   384
  then show ?thesis ..
haftmann@62347
   385
qed
haftmann@62347
   386
immler@66912
   387
lemma numeral_power_le_of_int_cancel_iff [simp]:
immler@66912
   388
  "numeral x ^ n \<le> of_int a \<longleftrightarrow> numeral x ^ n \<le> a"
immler@66912
   389
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_le_iff)
immler@66912
   390
immler@66912
   391
lemma of_int_le_numeral_power_cancel_iff [simp]:
immler@66912
   392
  "of_int a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
immler@66912
   393
  by (metis (mono_tags) local.numeral_power_eq_of_int_cancel_iff of_int_le_iff)
immler@66912
   394
immler@66912
   395
lemma numeral_power_less_of_int_cancel_iff [simp]:
immler@66912
   396
  "numeral x ^ n < of_int a \<longleftrightarrow> numeral x ^ n < a"
immler@66912
   397
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
immler@66912
   398
immler@66912
   399
lemma of_int_less_numeral_power_cancel_iff [simp]:
immler@66912
   400
  "of_int a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
immler@66912
   401
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
immler@66912
   402
immler@66912
   403
lemma neg_numeral_power_le_of_int_cancel_iff [simp]:
immler@66912
   404
  "(- numeral x) ^ n \<le> of_int a \<longleftrightarrow> (- numeral x) ^ n \<le> a"
immler@66912
   405
  by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
immler@66912
   406
immler@66912
   407
lemma of_int_le_neg_numeral_power_cancel_iff [simp]:
immler@66912
   408
  "of_int a \<le> (- numeral x) ^ n \<longleftrightarrow> a \<le> (- numeral x) ^ n"
immler@66912
   409
  by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
immler@66912
   410
immler@66912
   411
lemma neg_numeral_power_less_of_int_cancel_iff [simp]:
immler@66912
   412
  "(- numeral x) ^ n < of_int a \<longleftrightarrow> (- numeral x) ^ n < a"
immler@66912
   413
  using of_int_less_iff[of "(- numeral x) ^ n" a]
immler@66912
   414
  by simp
immler@66912
   415
immler@66912
   416
lemma of_int_less_neg_numeral_power_cancel_iff [simp]:
immler@66912
   417
  "of_int a < (- numeral x) ^ n \<longleftrightarrow> a < (- numeral x::int) ^ n"
immler@66912
   418
  using of_int_less_iff[of a "(- numeral x) ^ n"]
immler@66912
   419
  by simp
immler@66912
   420
immler@66912
   421
lemma of_int_le_of_int_power_cancel_iff[simp]: "(of_int b) ^ w \<le> of_int x \<longleftrightarrow> b ^ w \<le> x"
immler@66912
   422
  by (metis (mono_tags) of_int_le_iff of_int_power)
immler@66912
   423
immler@66912
   424
lemma of_int_power_le_of_int_cancel_iff[simp]: "of_int x \<le> (of_int b) ^ w\<longleftrightarrow> x \<le> b ^ w"
immler@66912
   425
  by (metis (mono_tags) of_int_le_iff of_int_power)
immler@66912
   426
immler@66912
   427
lemma of_int_less_of_int_power_cancel_iff[simp]: "(of_int b) ^ w < of_int x \<longleftrightarrow> b ^ w < x"
immler@66912
   428
  by (metis (mono_tags) of_int_less_iff of_int_power)
immler@66912
   429
immler@66912
   430
lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w\<longleftrightarrow> x < b ^ w"
immler@66912
   431
  by (metis (mono_tags) of_int_less_iff of_int_power)
immler@66912
   432
haftmann@36424
   433
end
haftmann@25919
   434
lp15@61234
   435
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   436
wenzelm@63652
   437
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
lp15@61234
   438
  using of_int_eq_iff by fastforce
lp15@61234
   439
lp15@61649
   440
lemma of_int_le_numeral_iff [simp]:
wenzelm@63652
   441
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   442
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   443
lp15@61649
   444
lemma of_int_numeral_le_iff [simp]:
wenzelm@63652
   445
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   446
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   447
lp15@61649
   448
lemma of_int_less_numeral_iff [simp]:
wenzelm@63652
   449
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
lp15@61234
   450
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   451
lp15@61649
   452
lemma of_int_numeral_less_iff [simp]:
wenzelm@63652
   453
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   454
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   455
wenzelm@63652
   456
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   457
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   458
haftmann@25919
   459
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   460
proof
wenzelm@63652
   461
  show "of_int z = id z" for z
wenzelm@63652
   462
    by (cases z rule: int_diff_cases) simp
haftmann@25919
   463
qed
haftmann@25919
   464
hoelzl@51329
   465
instance int :: no_top
wenzelm@61169
   466
  apply standard
hoelzl@51329
   467
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   468
  apply simp
hoelzl@51329
   469
  done
hoelzl@51329
   470
hoelzl@51329
   471
instance int :: no_bot
wenzelm@61169
   472
  apply standard
hoelzl@51329
   473
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   474
  apply simp
hoelzl@51329
   475
  done
hoelzl@51329
   476
wenzelm@63652
   477
wenzelm@61799
   478
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   479
huffman@48045
   480
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   481
  by auto
haftmann@25919
   482
huffman@44709
   483
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   484
  by transfer simp
haftmann@25919
   485
huffman@44709
   486
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   487
  by transfer clarsimp
haftmann@25919
   488
wenzelm@63652
   489
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
wenzelm@63652
   490
  by simp
haftmann@25919
   491
wenzelm@63652
   492
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
huffman@48045
   493
  by transfer clarsimp
haftmann@25919
   494
wenzelm@63652
   495
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
huffman@48045
   496
  by transfer (clarsimp, arith)
haftmann@25919
   497
wenzelm@63652
   498
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
wenzelm@63652
   499
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   500
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   501
wenzelm@63652
   502
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   503
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   504
wenzelm@63652
   505
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
huffman@48045
   506
  by transfer (clarsimp, arith)
haftmann@25919
   507
haftmann@64714
   508
lemma nonneg_int_cases:
haftmann@64714
   509
  assumes "0 \<le> k"
haftmann@64714
   510
  obtains n where "k = int n"
haftmann@64714
   511
proof -
haftmann@64714
   512
  from assms have "k = int (nat k)"
haftmann@64714
   513
    by simp
haftmann@64714
   514
  then show thesis
haftmann@64714
   515
    by (rule that)
haftmann@64714
   516
qed
haftmann@64714
   517
haftmann@64714
   518
lemma pos_int_cases:
haftmann@64714
   519
  assumes "0 < k"
haftmann@64714
   520
  obtains n where "k = int n" and "n > 0"
haftmann@64714
   521
proof -
haftmann@64714
   522
  from assms have "0 \<le> k"
haftmann@64714
   523
    by simp
haftmann@64714
   524
  then obtain n where "k = int n"
haftmann@64714
   525
    by (rule nonneg_int_cases)
haftmann@64714
   526
  moreover have "n > 0"
haftmann@64714
   527
    using \<open>k = int n\<close> assms by simp
haftmann@64714
   528
  ultimately show thesis
haftmann@64714
   529
    by (rule that)
haftmann@64714
   530
qed
haftmann@64714
   531
haftmann@64714
   532
lemma nonpos_int_cases:
haftmann@64714
   533
  assumes "k \<le> 0"
haftmann@64714
   534
  obtains n where "k = - int n"
haftmann@64714
   535
proof -
haftmann@64714
   536
  from assms have "- k \<ge> 0"
haftmann@64714
   537
    by simp
haftmann@64714
   538
  then obtain n where "- k = int n"
haftmann@64714
   539
    by (rule nonneg_int_cases)
haftmann@64714
   540
  then have "k = - int n"
haftmann@64714
   541
    by simp
haftmann@64714
   542
  then show thesis
haftmann@64714
   543
    by (rule that)
haftmann@64714
   544
qed
haftmann@64714
   545
haftmann@64714
   546
lemma neg_int_cases:
haftmann@64714
   547
  assumes "k < 0"
haftmann@64714
   548
  obtains n where "k = - int n" and "n > 0"
haftmann@64714
   549
proof -
haftmann@64714
   550
  from assms have "- k > 0"
haftmann@64714
   551
    by simp
haftmann@64714
   552
  then obtain n where "- k = int n" and "- k > 0"
haftmann@64714
   553
    by (blast elim: pos_int_cases)
haftmann@64714
   554
  then have "k = - int n" and "n > 0"
haftmann@64714
   555
    by simp_all
haftmann@64714
   556
  then show thesis
haftmann@64714
   557
    by (rule that)
haftmann@64714
   558
qed
haftmann@25919
   559
wenzelm@63652
   560
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   561
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   562
wenzelm@63652
   563
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   564
  using nat_eq_iff [of w m] by auto
haftmann@54223
   565
wenzelm@63652
   566
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@54223
   567
  by (simp add: nat_eq_iff)
haftmann@25919
   568
wenzelm@63652
   569
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@54223
   570
  by (simp add: nat_eq_iff)
haftmann@54223
   571
wenzelm@63652
   572
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
haftmann@54223
   573
  by (simp add: nat_eq_iff)
haftmann@25919
   574
wenzelm@63652
   575
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
haftmann@54223
   576
  by simp
haftmann@54223
   577
haftmann@54223
   578
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   579
  by simp
lp15@60162
   580
wenzelm@63652
   581
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
huffman@48045
   582
  by transfer (clarsimp, arith)
haftmann@25919
   583
huffman@44709
   584
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   585
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   586
huffman@44707
   587
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   588
  by transfer auto
huffman@44707
   589
wenzelm@63652
   590
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
wenzelm@63652
   591
  for i :: int
huffman@48045
   592
  by transfer clarsimp
nipkow@29700
   593
wenzelm@63652
   594
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
wenzelm@63652
   595
  by (auto simp add: nat_eq_iff2)
haftmann@25919
   596
wenzelm@63652
   597
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
wenzelm@63652
   598
  using zless_nat_conj [of 0] by auto
haftmann@25919
   599
wenzelm@63652
   600
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   601
  by transfer clarsimp
haftmann@25919
   602
wenzelm@63652
   603
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   604
  by transfer clarsimp
lp15@60162
   605
wenzelm@63652
   606
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   607
  by (rule nat_diff_distrib') auto
haftmann@25919
   608
huffman@44709
   609
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   610
  by transfer simp
haftmann@25919
   611
wenzelm@63652
   612
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   613
  by transfer auto
lp15@60162
   614
wenzelm@63652
   615
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
huffman@48045
   616
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   617
wenzelm@63652
   618
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   619
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   620
wenzelm@63652
   621
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   622
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   623
haftmann@66886
   624
lemma nat_abs_triangle_ineq:
haftmann@66886
   625
  "nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>"
haftmann@66886
   626
  by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq)
haftmann@66886
   627
haftmann@66816
   628
lemma nat_of_bool [simp]:
haftmann@66816
   629
  "nat (of_bool P) = of_bool P"
haftmann@66816
   630
  by auto
haftmann@66816
   631
haftmann@66836
   632
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
   633
  (is "?P = (?L \<and> ?R)")
haftmann@66836
   634
  for i :: int
haftmann@66836
   635
proof (cases "i < 0")
haftmann@66836
   636
  case True
haftmann@66836
   637
  then show ?thesis
haftmann@66836
   638
    by auto
haftmann@66836
   639
next
haftmann@66836
   640
  case False
haftmann@66836
   641
  have "?P = ?L"
haftmann@66836
   642
  proof
haftmann@66836
   643
    assume ?P
haftmann@66836
   644
    then show ?L using False by auto
haftmann@66836
   645
  next
haftmann@66836
   646
    assume ?L
haftmann@66836
   647
    moreover from False have "int (nat i) = i"
haftmann@66836
   648
      by (simp add: not_less)
haftmann@66836
   649
    ultimately show ?P
haftmann@66836
   650
      by simp
haftmann@66836
   651
  qed
haftmann@66836
   652
  with False show ?thesis by simp
haftmann@66836
   653
qed
haftmann@66836
   654
haftmann@66836
   655
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
haftmann@66836
   656
  by (auto split: split_nat)
haftmann@66836
   657
haftmann@66836
   658
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
haftmann@66836
   659
proof
haftmann@66836
   660
  assume "\<exists>x. P x"
haftmann@66836
   661
  then obtain x where "P x" ..
haftmann@66836
   662
  then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
haftmann@66836
   663
  then show "\<exists>x\<ge>0. P (nat x)" ..
haftmann@66836
   664
next
haftmann@66836
   665
  assume "\<exists>x\<ge>0. P (nat x)"
haftmann@66836
   666
  then show "\<exists>x. P x" by auto
haftmann@66836
   667
qed
haftmann@66836
   668
haftmann@54249
   669
wenzelm@60758
   670
text \<open>For termination proofs:\<close>
wenzelm@63652
   671
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
krauss@29779
   672
haftmann@25919
   673
wenzelm@63652
   674
subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   675
wenzelm@61076
   676
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
wenzelm@63652
   677
  by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   678
huffman@44709
   679
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
wenzelm@63652
   680
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   681
huffman@44709
   682
lemma negative_zle_0: "- int n \<le> 0"
wenzelm@63652
   683
  by (simp add: minus_le_iff)
haftmann@25919
   684
huffman@44709
   685
lemma negative_zle [iff]: "- int n \<le> int m"
wenzelm@63652
   686
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   687
wenzelm@63652
   688
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
wenzelm@63652
   689
  by (subst le_minus_iff) (simp del: of_nat_Suc)
haftmann@25919
   690
wenzelm@63652
   691
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
huffman@48045
   692
  by transfer simp
haftmann@25919
   693
wenzelm@63652
   694
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
wenzelm@63652
   695
  by (simp add: linorder_not_less)
haftmann@25919
   696
wenzelm@63652
   697
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
wenzelm@63652
   698
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   699
wenzelm@63652
   700
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
wenzelm@63652
   701
  (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@62348
   702
proof
wenzelm@63652
   703
  assume ?rhs
wenzelm@63652
   704
  then show ?lhs by auto
haftmann@62348
   705
next
wenzelm@63652
   706
  assume ?lhs
haftmann@62348
   707
  then have "0 \<le> z - w" by simp
haftmann@62348
   708
  then obtain n where "z - w = int n"
haftmann@62348
   709
    using zero_le_imp_eq_int [of "z - w"] by blast
wenzelm@63652
   710
  then have "z = w + int n" by simp
wenzelm@63652
   711
  then show ?rhs ..
haftmann@25919
   712
qed
haftmann@25919
   713
huffman@44709
   714
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
wenzelm@63652
   715
  by simp
haftmann@25919
   716
wenzelm@63652
   717
text \<open>
wenzelm@63652
   718
  This version is proved for all ordered rings, not just integers!
wenzelm@63652
   719
  It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@63652
   720
  in theory \<open>Rings\<close>.
wenzelm@63652
   721
  But is it really better than just rewriting with \<open>abs_if\<close>?
wenzelm@63652
   722
\<close>
wenzelm@63652
   723
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
   724
  for a :: "'a::linordered_idom"
wenzelm@63652
   725
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   726
huffman@44709
   727
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
wenzelm@63652
   728
  apply transfer
wenzelm@63652
   729
  apply clarsimp
wenzelm@63652
   730
  apply (rule_tac x="b - Suc a" in exI)
wenzelm@63652
   731
  apply arith
wenzelm@63652
   732
  done
wenzelm@63652
   733
haftmann@25919
   734
wenzelm@60758
   735
subsection \<open>Cases and induction\<close>
haftmann@25919
   736
wenzelm@63652
   737
text \<open>
wenzelm@63652
   738
  Now we replace the case analysis rule by a more conventional one:
wenzelm@63652
   739
  whether an integer is negative or not.
wenzelm@63652
   740
\<close>
haftmann@25919
   741
wenzelm@63652
   742
text \<open>This version is symmetric in the two subgoals.\<close>
wenzelm@63652
   743
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
wenzelm@63652
   744
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   745
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   746
wenzelm@63652
   747
text \<open>This is the default, with a negative case.\<close>
wenzelm@63652
   748
lemma int_cases [case_names nonneg neg, cases type: int]:
wenzelm@63652
   749
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   750
  apply (cases "z < 0")
wenzelm@63652
   751
   apply (blast dest!: negD)
wenzelm@63652
   752
  apply (simp add: linorder_not_less del: of_nat_Suc)
wenzelm@63652
   753
  apply auto
wenzelm@63652
   754
  apply (blast dest: nat_0_le [THEN sym])
wenzelm@63652
   755
  done
haftmann@25919
   756
haftmann@60868
   757
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   758
  fixes k :: int
haftmann@60868
   759
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   760
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   761
  shows "P"
haftmann@60868
   762
proof (cases k "0::int" rule: linorder_cases)
wenzelm@63652
   763
  case equal
wenzelm@63652
   764
  with assms(1) show P by simp
haftmann@60868
   765
next
haftmann@60868
   766
  case greater
wenzelm@63539
   767
  then have *: "nat k > 0" by simp
wenzelm@63539
   768
  moreover from * have "k = int (nat k)" by auto
haftmann@60868
   769
  ultimately show P using assms(2) by blast
haftmann@60868
   770
next
haftmann@60868
   771
  case less
wenzelm@63539
   772
  then have *: "nat (- k) > 0" by simp
wenzelm@63539
   773
  moreover from * have "k = - int (nat (- k))" by auto
haftmann@60868
   774
  ultimately show P using assms(3) by blast
haftmann@60868
   775
qed
haftmann@60868
   776
wenzelm@63652
   777
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
wenzelm@63652
   778
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
wenzelm@42676
   779
  by (cases z) auto
haftmann@25919
   780
huffman@47108
   781
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   782
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   783
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   784
haftmann@54489
   785
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   786
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   787
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   788
haftmann@66816
   789
lemma sgn_mult_dvd_iff [simp]:
haftmann@66816
   790
  "sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   791
  by (cases r rule: int_cases3) auto
haftmann@66816
   792
haftmann@66816
   793
lemma mult_sgn_dvd_iff [simp]:
haftmann@66816
   794
  "l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   795
  using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)
haftmann@66816
   796
haftmann@66816
   797
lemma dvd_sgn_mult_iff [simp]:
haftmann@66816
   798
  "l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   799
  by (cases r rule: int_cases3) simp_all
haftmann@66816
   800
haftmann@66816
   801
lemma dvd_mult_sgn_iff [simp]:
haftmann@66816
   802
  "l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   803
  using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)
haftmann@66816
   804
haftmann@66816
   805
lemma int_sgnE:
haftmann@66816
   806
  fixes k :: int
haftmann@66816
   807
  obtains n and l where "k = sgn l * int n"
haftmann@66816
   808
proof -
haftmann@66816
   809
  have "k = sgn k * int (nat \<bar>k\<bar>)"
haftmann@66816
   810
    by (simp add: sgn_mult_abs)
haftmann@66816
   811
  then show ?thesis ..
haftmann@66816
   812
qed
haftmann@66816
   813
wenzelm@61799
   814
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   815
huffman@47108
   816
lemmas max_number_of [simp] =
huffman@47108
   817
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   818
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   819
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   820
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   821
huffman@47108
   822
lemmas min_number_of [simp] =
huffman@47108
   823
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   824
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   825
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   826
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   827
haftmann@25919
   828
wenzelm@60758
   829
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   830
wenzelm@60758
   831
text \<open>Preliminaries\<close>
huffman@28958
   832
lp15@60162
   833
lemma le_imp_0_less:
wenzelm@63652
   834
  fixes z :: int
huffman@28958
   835
  assumes le: "0 \<le> z"
wenzelm@63652
   836
  shows "0 < 1 + z"
huffman@28958
   837
proof -
huffman@28958
   838
  have "0 \<le> z" by fact
wenzelm@63652
   839
  also have "\<dots> < z + 1" by (rule less_add_one)
wenzelm@63652
   840
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
huffman@28958
   841
  finally show "0 < 1 + z" .
huffman@28958
   842
qed
huffman@28958
   843
wenzelm@63652
   844
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
wenzelm@63652
   845
  for z :: int
wenzelm@42676
   846
proof (cases z)
huffman@28958
   847
  case (nonneg n)
wenzelm@63652
   848
  then show ?thesis
wenzelm@63652
   849
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   850
next
huffman@28958
   851
  case (neg n)
wenzelm@63652
   852
  then show ?thesis
wenzelm@63652
   853
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@63652
   854
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   855
qed
huffman@28958
   856
wenzelm@63652
   857
wenzelm@60758
   858
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   859
haftmann@25919
   860
lemmas double_eq_0_iff = double_zero
haftmann@25919
   861
wenzelm@63652
   862
lemma odd_nonzero: "1 + z + z \<noteq> 0"
wenzelm@63652
   863
  for z :: int
wenzelm@42676
   864
proof (cases z)
haftmann@25919
   865
  case (nonneg n)
wenzelm@63652
   866
  have le: "0 \<le> z + z"
wenzelm@63652
   867
    by (simp add: nonneg add_increasing)
wenzelm@63652
   868
  then show ?thesis
wenzelm@63652
   869
    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
haftmann@25919
   870
next
haftmann@25919
   871
  case (neg n)
haftmann@25919
   872
  show ?thesis
haftmann@25919
   873
  proof
haftmann@25919
   874
    assume eq: "1 + z + z = 0"
wenzelm@63652
   875
    have "0 < 1 + (int n + int n)"
lp15@60162
   876
      by (simp add: le_imp_0_less add_increasing)
wenzelm@63652
   877
    also have "\<dots> = - (1 + z + z)"
lp15@60162
   878
      by (simp add: neg add.assoc [symmetric])
wenzelm@63652
   879
    also have "\<dots> = 0" by (simp add: eq)
haftmann@25919
   880
    finally have "0<0" ..
wenzelm@63652
   881
    then show False by blast
haftmann@25919
   882
  qed
haftmann@25919
   883
qed
haftmann@25919
   884
haftmann@30652
   885
wenzelm@60758
   886
subsection \<open>The Set of Integers\<close>
haftmann@25919
   887
haftmann@25919
   888
context ring_1
haftmann@25919
   889
begin
haftmann@25919
   890
wenzelm@61070
   891
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   892
  where "\<int> = range of_int"
haftmann@25919
   893
huffman@35634
   894
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   895
  by (simp add: Ints_def)
huffman@35634
   896
huffman@35634
   897
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   898
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   899
haftmann@25919
   900
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   901
  using Ints_of_int [of "0"] by simp
haftmann@25919
   902
haftmann@25919
   903
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   904
  using Ints_of_int [of "1"] by simp
haftmann@25919
   905
eberlm@61552
   906
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   907
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   908
haftmann@25919
   909
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
wenzelm@63652
   910
  apply (auto simp add: Ints_def)
wenzelm@63652
   911
  apply (rule range_eqI)
wenzelm@63652
   912
  apply (rule of_int_add [symmetric])
wenzelm@63652
   913
  done
haftmann@25919
   914
haftmann@25919
   915
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
wenzelm@63652
   916
  apply (auto simp add: Ints_def)
wenzelm@63652
   917
  apply (rule range_eqI)
wenzelm@63652
   918
  apply (rule of_int_minus [symmetric])
wenzelm@63652
   919
  done
haftmann@25919
   920
huffman@35634
   921
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
wenzelm@63652
   922
  apply (auto simp add: Ints_def)
wenzelm@63652
   923
  apply (rule range_eqI)
wenzelm@63652
   924
  apply (rule of_int_diff [symmetric])
wenzelm@63652
   925
  done
huffman@35634
   926
haftmann@25919
   927
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
wenzelm@63652
   928
  apply (auto simp add: Ints_def)
wenzelm@63652
   929
  apply (rule range_eqI)
wenzelm@63652
   930
  apply (rule of_int_mult [symmetric])
wenzelm@63652
   931
  done
haftmann@25919
   932
huffman@35634
   933
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
wenzelm@63652
   934
  by (induct n) simp_all
huffman@35634
   935
haftmann@25919
   936
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   937
  assumes "q \<in> \<int>"
haftmann@25919
   938
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   939
  unfolding Ints_def
haftmann@25919
   940
proof -
wenzelm@60758
   941
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   942
  then obtain z where "q = of_int z" ..
haftmann@25919
   943
  then show thesis ..
haftmann@25919
   944
qed
haftmann@25919
   945
haftmann@25919
   946
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   947
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   948
  by (rule Ints_cases) auto
haftmann@25919
   949
eberlm@61524
   950
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   951
  unfolding Nats_def Ints_def
eberlm@61524
   952
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   953
eberlm@61524
   954
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   955
proof (intro subsetI equalityI)
wenzelm@63652
   956
  fix x :: 'a
wenzelm@63652
   957
  assume "x \<in> {of_int n |n. n \<ge> 0}"
wenzelm@63652
   958
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   959
    by (auto elim!: Ints_cases)
wenzelm@63652
   960
  then have "x = of_nat (nat n)"
wenzelm@63652
   961
    by (subst of_nat_nat) simp_all
wenzelm@63652
   962
  then show "x \<in> \<nat>"
wenzelm@63652
   963
    by simp
eberlm@61524
   964
next
wenzelm@63652
   965
  fix x :: 'a
wenzelm@63652
   966
  assume "x \<in> \<nat>"
wenzelm@63652
   967
  then obtain n where "x = of_nat n"
wenzelm@63652
   968
    by (auto elim!: Nats_cases)
wenzelm@63652
   969
  then have "x = of_int (int n)" by simp
eberlm@61524
   970
  also have "int n \<ge> 0" by simp
wenzelm@63652
   971
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   972
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   973
qed
eberlm@61524
   974
haftmann@25919
   975
end
haftmann@25919
   976
lp15@64758
   977
lemma (in linordered_idom) Ints_abs [simp]:
lp15@64758
   978
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
lp15@64758
   979
  by (auto simp: abs_if)
lp15@64758
   980
eberlm@61524
   981
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   982
proof (intro subsetI equalityI)
wenzelm@63652
   983
  fix x :: 'a
wenzelm@63652
   984
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
wenzelm@63652
   985
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   986
    by (auto elim!: Ints_cases)
wenzelm@63652
   987
  then have "x = of_nat (nat n)"
wenzelm@63652
   988
    by (subst of_nat_nat) simp_all
wenzelm@63652
   989
  then show "x \<in> \<nat>"
wenzelm@63652
   990
    by simp
eberlm@61524
   991
qed (auto elim!: Nats_cases)
eberlm@61524
   992
haftmann@64849
   993
lemma (in idom_divide) of_int_divide_in_Ints: 
haftmann@64849
   994
  "of_int a div of_int b \<in> \<int>" if "b dvd a"
haftmann@64849
   995
proof -
haftmann@64849
   996
  from that obtain c where "a = b * c" ..
haftmann@64849
   997
  then show ?thesis
haftmann@64849
   998
    by (cases "of_int b = 0") simp_all
haftmann@64849
   999
qed
eberlm@61524
  1000
wenzelm@60758
  1001
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
  1002
haftmann@25919
  1003
lemma Ints_double_eq_0_iff:
wenzelm@63652
  1004
  fixes a :: "'a::ring_char_0"
wenzelm@61070
  1005
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
  1006
  shows "a + a = 0 \<longleftrightarrow> a = 0"
wenzelm@63652
  1007
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@25919
  1008
proof -
wenzelm@63652
  1009
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
  1010
    unfolding Ints_def [symmetric] .
haftmann@25919
  1011
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1012
  show ?thesis
haftmann@25919
  1013
  proof
wenzelm@63652
  1014
    assume ?rhs
wenzelm@63652
  1015
    then show ?lhs by simp
haftmann@25919
  1016
  next
wenzelm@63652
  1017
    assume ?lhs
wenzelm@63652
  1018
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
  1019
    then have "z + z = 0" by (simp only: of_int_eq_iff)
wenzelm@63652
  1020
    then have "z = 0" by (simp only: double_eq_0_iff)
wenzelm@63652
  1021
    with a show ?rhs by simp
haftmann@25919
  1022
  qed
haftmann@25919
  1023
qed
haftmann@25919
  1024
haftmann@25919
  1025
lemma Ints_odd_nonzero:
wenzelm@63652
  1026
  fixes a :: "'a::ring_char_0"
wenzelm@61070
  1027
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
  1028
  shows "1 + a + a \<noteq> 0"
haftmann@25919
  1029
proof -
wenzelm@63652
  1030
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
  1031
    unfolding Ints_def [symmetric] .
haftmann@25919
  1032
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1033
  show ?thesis
haftmann@25919
  1034
  proof
wenzelm@63652
  1035
    assume "1 + a + a = 0"
wenzelm@63652
  1036
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
  1037
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1038
    with odd_nonzero show False by blast
haftmann@25919
  1039
  qed
lp15@60162
  1040
qed
haftmann@25919
  1041
wenzelm@61070
  1042
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
  1043
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
  1044
lp15@60162
  1045
lemma Ints_odd_less_0:
wenzelm@63652
  1046
  fixes a :: "'a::linordered_idom"
wenzelm@61070
  1047
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
  1048
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25919
  1049
proof -
wenzelm@63652
  1050
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
  1051
    unfolding Ints_def [symmetric] .
haftmann@25919
  1052
  then obtain z where a: "a = of_int z" ..
wenzelm@63652
  1053
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
wenzelm@63652
  1054
    by simp
wenzelm@63652
  1055
  also have "\<dots> \<longleftrightarrow> z < 0"
wenzelm@63652
  1056
    by (simp only: of_int_less_iff odd_less_0_iff)
wenzelm@63652
  1057
  also have "\<dots> \<longleftrightarrow> a < 0"
haftmann@25919
  1058
    by (simp add: a)
haftmann@25919
  1059
  finally show ?thesis .
haftmann@25919
  1060
qed
haftmann@25919
  1061
haftmann@25919
  1062
nipkow@64272
  1063
subsection \<open>@{term sum} and @{term prod}\<close>
haftmann@25919
  1064
nipkow@64267
  1065
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
wenzelm@63652
  1066
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1067
nipkow@64267
  1068
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
wenzelm@63652
  1069
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1070
nipkow@64272
  1071
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
wenzelm@63652
  1072
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1073
nipkow@64272
  1074
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
wenzelm@63652
  1075
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
  1076
haftmann@25919
  1077
wenzelm@60758
  1078
text \<open>Legacy theorems\<close>
haftmann@25919
  1079
haftmann@64714
  1080
lemmas int_sum = of_nat_sum [where 'a=int]
haftmann@64714
  1081
lemmas int_prod = of_nat_prod [where 'a=int]
haftmann@25919
  1082
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1083
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@64714
  1084
lemmas nonneg_eq_int = nonneg_int_cases
haftmann@25919
  1085
wenzelm@63652
  1086
wenzelm@60758
  1087
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
  1088
haftmann@54249
  1089
lemmas of_int_simps =
haftmann@54249
  1090
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
  1091
wenzelm@48891
  1092
ML_file "Tools/int_arith.ML"
wenzelm@60758
  1093
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
  1094
wenzelm@63652
  1095
simproc_setup fast_arith
wenzelm@63652
  1096
  ("(m::'a::linordered_idom) < n" |
wenzelm@63652
  1097
    "(m::'a::linordered_idom) \<le> n" |
wenzelm@63652
  1098
    "(m::'a::linordered_idom) = n") =
wenzelm@61144
  1099
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
  1100
haftmann@25919
  1101
wenzelm@60758
  1102
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
  1103
wenzelm@63652
  1104
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
wenzelm@63652
  1105
  for w z :: int
wenzelm@63652
  1106
  by arith
haftmann@25919
  1107
wenzelm@63652
  1108
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
wenzelm@63652
  1109
  for w z :: int
wenzelm@63652
  1110
  by arith
haftmann@25919
  1111
wenzelm@63652
  1112
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
wenzelm@63652
  1113
  for w z :: int
wenzelm@63652
  1114
  by arith
haftmann@25919
  1115
wenzelm@63652
  1116
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
wenzelm@63652
  1117
  for w z :: int
wenzelm@63652
  1118
  by arith
haftmann@25919
  1119
wenzelm@63652
  1120
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
wenzelm@63652
  1121
  for z :: int
wenzelm@63652
  1122
  by arith
haftmann@25919
  1123
lp15@64758
  1124
lemma Ints_nonzero_abs_ge1:
lp15@64758
  1125
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1126
    assumes "x \<in> Ints" "x \<noteq> 0"
lp15@64758
  1127
    shows "1 \<le> abs x"
lp15@64758
  1128
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
lp15@64758
  1129
  fix z::int
lp15@64758
  1130
  assume "x = of_int z"
lp15@64758
  1131
    with \<open>x \<noteq> 0\<close> 
lp15@64758
  1132
  show "1 \<le> \<bar>x\<bar>"
lp15@64758
  1133
    apply (auto simp add: abs_if)
lp15@64758
  1134
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
lp15@64758
  1135
qed
lp15@64758
  1136
  
lp15@64758
  1137
lemma Ints_nonzero_abs_less1:
lp15@64758
  1138
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1139
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
lp15@64758
  1140
    using Ints_nonzero_abs_ge1 [of x] by auto
lp15@64758
  1141
    
haftmann@25919
  1142
wenzelm@63652
  1143
subsection \<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
  1144
wenzelm@63652
  1145
text \<open>Simplify the term @{term "w + - z"}.\<close>
haftmann@25919
  1146
wenzelm@63652
  1147
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
lp15@60162
  1148
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
  1149
wenzelm@63652
  1150
text \<open>
wenzelm@63652
  1151
  This simplifies expressions of the form @{term "int n = z"} where
wenzelm@63652
  1152
  \<open>z\<close> is an integer literal.
wenzelm@63652
  1153
\<close>
huffman@47108
  1154
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
  1155
hoelzl@59000
  1156
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
  1157
  by auto
hoelzl@59000
  1158
hoelzl@59000
  1159
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
  1160
  by auto
hoelzl@59000
  1161
haftmann@25919
  1162
context ring_1
haftmann@25919
  1163
begin
haftmann@25919
  1164
blanchet@33056
  1165
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1166
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1167
proof (cases "k < 0")
wenzelm@63652
  1168
  case True
wenzelm@63652
  1169
  then have "0 \<le> - k" by simp
haftmann@25919
  1170
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1171
  with True show ?thesis by simp
haftmann@25919
  1172
next
wenzelm@63652
  1173
  case False
wenzelm@63652
  1174
  then show ?thesis by (simp add: not_less)
haftmann@25919
  1175
qed
haftmann@25919
  1176
haftmann@25919
  1177
end
haftmann@25919
  1178
haftmann@64014
  1179
lemma transfer_rule_of_int:
haftmann@64014
  1180
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
haftmann@64014
  1181
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64014
  1182
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64014
  1183
    "rel_fun R R uminus uminus"
haftmann@64014
  1184
  shows "rel_fun HOL.eq R of_int of_int"
haftmann@64014
  1185
proof -
haftmann@64014
  1186
  note transfer_rule_of_nat [transfer_rule]
haftmann@64014
  1187
  have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
haftmann@64014
  1188
    by transfer_prover
haftmann@64014
  1189
  show ?thesis
haftmann@64014
  1190
    by (unfold of_int_of_nat [abs_def]) transfer_prover
haftmann@64014
  1191
qed
haftmann@64014
  1192
haftmann@25919
  1193
lemma nat_mult_distrib:
haftmann@25919
  1194
  fixes z z' :: int
haftmann@25919
  1195
  assumes "0 \<le> z"
haftmann@25919
  1196
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1197
proof (cases "0 \<le> z'")
wenzelm@63652
  1198
  case False
wenzelm@63652
  1199
  with assms have "z * z' \<le> 0"
haftmann@25919
  1200
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1201
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1202
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1203
  ultimately show ?thesis by simp
haftmann@25919
  1204
next
wenzelm@63652
  1205
  case True
wenzelm@63652
  1206
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1207
  show ?thesis
haftmann@25919
  1208
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1209
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1210
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1211
qed
haftmann@25919
  1212
wenzelm@63652
  1213
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
wenzelm@63652
  1214
  for z z' :: int
wenzelm@63652
  1215
  apply (rule trans)
wenzelm@63652
  1216
   apply (rule_tac [2] nat_mult_distrib)
wenzelm@63652
  1217
   apply auto
wenzelm@63652
  1218
  done
haftmann@25919
  1219
wenzelm@61944
  1220
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
wenzelm@63652
  1221
  by (cases "z = 0 \<or> w = 0")
wenzelm@63652
  1222
    (auto simp add: abs_if nat_mult_distrib [symmetric]
wenzelm@63652
  1223
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1224
wenzelm@63652
  1225
lemma int_in_range_abs [simp]: "int n \<in> range abs"
haftmann@60570
  1226
proof (rule range_eqI)
wenzelm@63652
  1227
  show "int n = \<bar>int n\<bar>" by simp
haftmann@60570
  1228
qed
haftmann@60570
  1229
wenzelm@63652
  1230
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
haftmann@60570
  1231
proof -
haftmann@60570
  1232
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1233
    by (cases k) simp_all
haftmann@60570
  1234
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1235
    using that by induct simp
haftmann@60570
  1236
  ultimately show ?thesis by blast
paulson@61204
  1237
qed
haftmann@60570
  1238
wenzelm@63652
  1239
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
wenzelm@63652
  1240
  for z :: int
wenzelm@63652
  1241
  by (rule sym) (simp add: nat_eq_iff)
huffman@47207
  1242
huffman@47207
  1243
lemma diff_nat_eq_if:
wenzelm@63652
  1244
  "nat z - nat z' =
wenzelm@63652
  1245
    (if z' < 0 then nat z
wenzelm@63652
  1246
     else
wenzelm@63652
  1247
      let d = z - z'
wenzelm@63652
  1248
      in if d < 0 then 0 else nat d)"
wenzelm@63652
  1249
  by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1250
wenzelm@63652
  1251
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1252
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1253
haftmann@25919
  1254
wenzelm@63652
  1255
subsection \<open>Induction principles for int\<close>
haftmann@25919
  1256
wenzelm@63652
  1257
text \<open>Well-founded segments of the integers.\<close>
haftmann@25919
  1258
wenzelm@63652
  1259
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1260
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
haftmann@25919
  1261
wenzelm@63652
  1262
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1263
proof -
wenzelm@63652
  1264
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
haftmann@25919
  1265
    by (auto simp add: int_ge_less_than_def)
wenzelm@63652
  1266
  then show ?thesis
lp15@60162
  1267
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1268
qed
haftmann@25919
  1269
wenzelm@63652
  1270
text \<open>
wenzelm@63652
  1271
  This variant looks odd, but is typical of the relations suggested
wenzelm@63652
  1272
  by RankFinder.\<close>
haftmann@25919
  1273
wenzelm@63652
  1274
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1275
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
haftmann@25919
  1276
wenzelm@63652
  1277
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1278
proof -
wenzelm@63652
  1279
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
haftmann@25919
  1280
    by (auto simp add: int_ge_less_than2_def)
wenzelm@63652
  1281
  then show ?thesis
lp15@60162
  1282
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1283
qed
haftmann@25919
  1284
haftmann@25919
  1285
(* `set:int': dummy construction *)
haftmann@25919
  1286
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1287
  fixes i :: int
wenzelm@63652
  1288
  assumes ge: "k \<le> i"
wenzelm@63652
  1289
    and base: "P k"
wenzelm@63652
  1290
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1291
  shows "P i"
haftmann@25919
  1292
proof -
wenzelm@63652
  1293
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
wenzelm@63652
  1294
  proof (induct n)
wenzelm@63652
  1295
    case 0
wenzelm@63652
  1296
    then have "i = k" by arith
wenzelm@63652
  1297
    with base show "P i" by simp
wenzelm@63652
  1298
  next
wenzelm@63652
  1299
    case (Suc n)
wenzelm@63652
  1300
    then have "n = nat ((i - 1) - k)" by arith
wenzelm@63652
  1301
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
wenzelm@63652
  1302
    ultimately have "P (i - 1)" by (rule Suc.hyps)
wenzelm@63652
  1303
    from step [OF k this] show ?case by simp
wenzelm@63652
  1304
  qed
haftmann@25919
  1305
  with ge show ?thesis by fast
haftmann@25919
  1306
qed
haftmann@25919
  1307
haftmann@25928
  1308
(* `set:int': dummy construction *)
haftmann@25928
  1309
theorem int_gr_induct [case_names base step, induct set: int]:
wenzelm@63652
  1310
  fixes i k :: int
wenzelm@63652
  1311
  assumes gr: "k < i"
wenzelm@63652
  1312
    and base: "P (k + 1)"
wenzelm@63652
  1313
    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1314
  shows "P i"
wenzelm@63652
  1315
  apply (rule int_ge_induct[of "k + 1"])
haftmann@25919
  1316
  using gr apply arith
wenzelm@63652
  1317
   apply (rule base)
wenzelm@63652
  1318
  apply (rule step)
wenzelm@63652
  1319
   apply simp_all
wenzelm@63652
  1320
  done
haftmann@25919
  1321
wenzelm@42676
  1322
theorem int_le_induct [consumes 1, case_names base step]:
wenzelm@63652
  1323
  fixes i k :: int
wenzelm@63652
  1324
  assumes le: "i \<le> k"
wenzelm@63652
  1325
    and base: "P k"
wenzelm@63652
  1326
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1327
  shows "P i"
haftmann@25919
  1328
proof -
wenzelm@63652
  1329
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
wenzelm@63652
  1330
  proof (induct n)
wenzelm@63652
  1331
    case 0
wenzelm@63652
  1332
    then have "i = k" by arith
wenzelm@63652
  1333
    with base show "P i" by simp
wenzelm@63652
  1334
  next
wenzelm@63652
  1335
    case (Suc n)
wenzelm@63652
  1336
    then have "n = nat (k - (i + 1))" by arith
wenzelm@63652
  1337
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
wenzelm@63652
  1338
    ultimately have "P (i + 1)" by (rule Suc.hyps)
wenzelm@63652
  1339
    from step[OF k this] show ?case by simp
wenzelm@63652
  1340
  qed
haftmann@25919
  1341
  with le show ?thesis by fast
haftmann@25919
  1342
qed
haftmann@25919
  1343
wenzelm@42676
  1344
theorem int_less_induct [consumes 1, case_names base step]:
wenzelm@63652
  1345
  fixes i k :: int
wenzelm@63652
  1346
  assumes less: "i < k"
wenzelm@63652
  1347
    and base: "P (k - 1)"
wenzelm@63652
  1348
    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1349
  shows "P i"
wenzelm@63652
  1350
  apply (rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1351
  using less apply arith
wenzelm@63652
  1352
   apply (rule base)
wenzelm@63652
  1353
  apply (rule step)
wenzelm@63652
  1354
   apply simp_all
wenzelm@63652
  1355
  done
haftmann@25919
  1356
haftmann@36811
  1357
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1358
  fixes k :: int
haftmann@36801
  1359
  assumes base: "P k"
haftmann@36801
  1360
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1361
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1362
  shows "P i"
haftmann@36801
  1363
proof -
haftmann@36801
  1364
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1365
  then show ?thesis
wenzelm@42676
  1366
  proof
wenzelm@42676
  1367
    assume "i \<ge> k"
wenzelm@63652
  1368
    then show ?thesis
wenzelm@63652
  1369
      using base by (rule int_ge_induct) (fact step1)
haftmann@36801
  1370
  next
wenzelm@42676
  1371
    assume "i \<le> k"
wenzelm@63652
  1372
    then show ?thesis
wenzelm@63652
  1373
      using base by (rule int_le_induct) (fact step2)
haftmann@36801
  1374
  qed
haftmann@36801
  1375
qed
haftmann@36801
  1376
wenzelm@63652
  1377
wenzelm@63652
  1378
subsection \<open>Intermediate value theorems\<close>
haftmann@25919
  1379
wenzelm@63652
  1380
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
  1381
  for n :: nat and k :: int
wenzelm@63652
  1382
  unfolding One_nat_def
wenzelm@63652
  1383
  apply (induct n)
wenzelm@63652
  1384
   apply simp
wenzelm@63652
  1385
  apply (intro strip)
wenzelm@63652
  1386
  apply (erule impE)
wenzelm@63652
  1387
   apply simp
wenzelm@63652
  1388
  apply (erule_tac x = n in allE)
wenzelm@63652
  1389
  apply simp
wenzelm@63652
  1390
  apply (case_tac "k = f (Suc n)")
wenzelm@63652
  1391
   apply force
wenzelm@63652
  1392
  apply (erule impE)
wenzelm@63652
  1393
   apply (simp add: abs_if split: if_split_asm)
wenzelm@63652
  1394
  apply (blast intro: le_SucI)
wenzelm@63652
  1395
  done
haftmann@25919
  1396
haftmann@25919
  1397
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1398
haftmann@25919
  1399
lemma nat_intermed_int_val:
wenzelm@63652
  1400
  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
wenzelm@63652
  1401
    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
  1402
    for f :: "nat \<Rightarrow> int" and k :: int
wenzelm@63652
  1403
  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
wenzelm@63652
  1404
  unfolding One_nat_def
wenzelm@63652
  1405
  apply simp
wenzelm@63652
  1406
  apply (erule exE)
wenzelm@63652
  1407
  apply (rule_tac x = "i+m" in exI)
wenzelm@63652
  1408
  apply arith
wenzelm@63652
  1409
  done
haftmann@25919
  1410
haftmann@25919
  1411
wenzelm@63652
  1412
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1413
paulson@34055
  1414
lemma abs_zmult_eq_1:
wenzelm@63652
  1415
  fixes m n :: int
paulson@34055
  1416
  assumes mn: "\<bar>m * n\<bar> = 1"
wenzelm@63652
  1417
  shows "\<bar>m\<bar> = 1"
paulson@34055
  1418
proof -
wenzelm@63652
  1419
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
wenzelm@63652
  1420
  have "\<not> 2 \<le> \<bar>m\<bar>"
paulson@34055
  1421
  proof
paulson@34055
  1422
    assume "2 \<le> \<bar>m\<bar>"
wenzelm@63652
  1423
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
wenzelm@63652
  1424
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
wenzelm@63652
  1425
    also from mn have "\<dots> = 1" by simp
wenzelm@63652
  1426
    finally have "2 * \<bar>n\<bar> \<le> 1" .
wenzelm@63652
  1427
    with 0 show "False" by arith
paulson@34055
  1428
  qed
wenzelm@63652
  1429
  with 0 show ?thesis by auto
paulson@34055
  1430
qed
haftmann@25919
  1431
wenzelm@63652
  1432
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
wenzelm@63652
  1433
  for m n :: int
wenzelm@63652
  1434
  using abs_zmult_eq_1 [of m n] by arith
haftmann@25919
  1435
boehmes@35815
  1436
lemma pos_zmult_eq_1_iff:
wenzelm@63652
  1437
  fixes m n :: int
wenzelm@63652
  1438
  assumes "0 < m"
wenzelm@63652
  1439
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
boehmes@35815
  1440
proof -
wenzelm@63652
  1441
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
wenzelm@63652
  1442
    by (auto dest: pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1443
  then show ?thesis
wenzelm@63652
  1444
    by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1445
qed
haftmann@25919
  1446
wenzelm@63652
  1447
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
wenzelm@63652
  1448
  for m n :: int
wenzelm@63652
  1449
  apply (rule iffI)
wenzelm@63652
  1450
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1451
   apply (simp add: mult.commute [of m])
wenzelm@63652
  1452
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1453
   apply auto
wenzelm@63652
  1454
  done
haftmann@25919
  1455
haftmann@33296
  1456
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1457
proof
haftmann@33296
  1458
  assume "finite (UNIV::int set)"
wenzelm@61076
  1459
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1460
    by (rule injI) simp
wenzelm@61076
  1461
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1462
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1463
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1464
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1465
qed
haftmann@25919
  1466
haftmann@25919
  1467
wenzelm@60758
  1468
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1469
wenzelm@63652
  1470
subsubsection \<open>Special Simplification for Constants\<close>
haftmann@30652
  1471
wenzelm@63652
  1472
text \<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1473
webertj@49962
  1474
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1475
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1476
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1477
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1478
wenzelm@63652
  1479
text \<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1480
huffman@47108
  1481
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1482
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1483
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1484
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1485
haftmann@30652
  1486
wenzelm@61799
  1487
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1488
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1489
huffman@47108
  1490
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1491
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1492
huffman@47108
  1493
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1494
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1495
wenzelm@60758
  1496
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@63652
  1497
  into the literal.\<close>
haftmann@30652
  1498
haftmann@54489
  1499
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1500
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1501
haftmann@54489
  1502
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1503
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1504
haftmann@54489
  1505
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1506
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1507
haftmann@54489
  1508
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1509
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1510
haftmann@54489
  1511
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1512
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1513
haftmann@54489
  1514
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1515
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1516
wenzelm@63652
  1517
(* FIXME maybe simproc *)
haftmann@30652
  1518
haftmann@30652
  1519
wenzelm@61799
  1520
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1521
huffman@47108
  1522
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1523
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1524
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1525
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1526
haftmann@30652
  1527
wenzelm@61799
  1528
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1529
lp15@61738
  1530
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1531
lp15@61738
  1532
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1533
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1534
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1535
lp15@61738
  1536
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1537
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1538
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1539
lp15@61738
  1540
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1541
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1542
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1543
lp15@61738
  1544
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1545
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1546
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1547
lp15@61738
  1548
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1549
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1550
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1551
lp15@61738
  1552
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1553
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1554
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1555
haftmann@30652
  1556
wenzelm@63652
  1557
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1558
wenzelm@63652
  1559
text \<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1560
lp15@61738
  1561
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1562
  le_divide_eq [of "numeral w"]
haftmann@54489
  1563
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1564
lp15@61738
  1565
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1566
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1567
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1568
lp15@61738
  1569
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1570
  less_divide_eq [of "numeral w"]
haftmann@54489
  1571
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1572
lp15@61738
  1573
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1574
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1575
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1576
lp15@61738
  1577
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1578
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1579
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1580
lp15@61738
  1581
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1582
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1583
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1584
haftmann@30652
  1585
wenzelm@63652
  1586
text \<open>Not good as automatic simprules because they cause case splits.\<close>
wenzelm@63652
  1587
lemmas [divide_const_simps] =
wenzelm@63652
  1588
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1589
haftmann@30652
  1590
wenzelm@60758
  1591
subsection \<open>The divides relation\<close>
haftmann@33320
  1592
wenzelm@63652
  1593
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
wenzelm@63652
  1594
  for m n :: int
wenzelm@63652
  1595
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1596
wenzelm@63652
  1597
lemma zdvd_antisym_abs:
wenzelm@63652
  1598
  fixes a b :: int
wenzelm@63652
  1599
  assumes "a dvd b" and "b dvd a"
haftmann@33320
  1600
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@63652
  1601
proof (cases "a = 0")
wenzelm@63652
  1602
  case True
wenzelm@63652
  1603
  with assms show ?thesis by simp
nipkow@33657
  1604
next
wenzelm@63652
  1605
  case False
wenzelm@63652
  1606
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
wenzelm@63652
  1607
    unfolding dvd_def by blast
wenzelm@63652
  1608
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
wenzelm@63652
  1609
    unfolding dvd_def by blast
wenzelm@63652
  1610
  from k k' have "a = a * k * k'" by simp
wenzelm@63652
  1611
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
wenzelm@63652
  1612
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
wenzelm@63652
  1613
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
wenzelm@63652
  1614
    by (simp add: zmult_eq_1_iff)
wenzelm@63652
  1615
  with k k' show ?thesis by auto
haftmann@33320
  1616
qed
haftmann@33320
  1617
wenzelm@63652
  1618
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
wenzelm@63652
  1619
  for k m n :: int
lp15@60162
  1620
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1621
wenzelm@63652
  1622
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
wenzelm@63652
  1623
  for k m n :: int
haftmann@58649
  1624
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1625
haftmann@33320
  1626
lemma dvd_imp_le_int:
haftmann@33320
  1627
  fixes d i :: int
haftmann@33320
  1628
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1629
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1630
proof -
wenzelm@60758
  1631
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1632
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1633
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1634
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1635
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1636
qed
haftmann@33320
  1637
haftmann@33320
  1638
lemma zdvd_not_zless:
haftmann@33320
  1639
  fixes m n :: int
haftmann@33320
  1640
  assumes "0 < m" and "m < n"
haftmann@33320
  1641
  shows "\<not> n dvd m"
haftmann@33320
  1642
proof
haftmann@33320
  1643
  from assms have "0 < n" by auto
haftmann@33320
  1644
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1645
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1646
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1647
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1648
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1649
qed
haftmann@33320
  1650
wenzelm@63652
  1651
lemma zdvd_mult_cancel:
wenzelm@63652
  1652
  fixes k m n :: int
wenzelm@63652
  1653
  assumes d: "k * m dvd k * n"
wenzelm@63652
  1654
    and "k \<noteq> 0"
haftmann@33320
  1655
  shows "m dvd n"
wenzelm@63652
  1656
proof -
wenzelm@63652
  1657
  from d obtain h where h: "k * n = k * m * h"
wenzelm@63652
  1658
    unfolding dvd_def by blast
wenzelm@63652
  1659
  have "n = m * h"
wenzelm@63652
  1660
  proof (rule ccontr)
wenzelm@63652
  1661
    assume "\<not> ?thesis"
wenzelm@63652
  1662
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
wenzelm@63652
  1663
    with h show False
wenzelm@63652
  1664
      by (simp add: mult.assoc)
wenzelm@63652
  1665
  qed
wenzelm@63652
  1666
  then show ?thesis by simp
haftmann@33320
  1667
qed
haftmann@33320
  1668
wenzelm@63652
  1669
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
haftmann@33320
  1670
proof -
wenzelm@63652
  1671
  have "x dvd y" if "int y = int x * k" for k
wenzelm@63652
  1672
  proof (cases k)
wenzelm@63652
  1673
    case (nonneg n)
wenzelm@63652
  1674
    with that have "y = x * n"
wenzelm@63652
  1675
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
wenzelm@63652
  1676
    then show ?thesis ..
wenzelm@63652
  1677
  next
wenzelm@63652
  1678
    case (neg n)
wenzelm@63652
  1679
    with that have "int y = int x * (- int (Suc n))"
wenzelm@63652
  1680
      by simp
wenzelm@63652
  1681
    also have "\<dots> = - (int x * int (Suc n))"
wenzelm@63652
  1682
      by (simp only: mult_minus_right)
wenzelm@63652
  1683
    also have "\<dots> = - int (x * Suc n)"
wenzelm@63652
  1684
      by (simp only: of_nat_mult [symmetric])
wenzelm@63652
  1685
    finally have "- int (x * Suc n) = int y" ..
wenzelm@63652
  1686
    then show ?thesis
wenzelm@63652
  1687
      by (simp only: negative_eq_positive) auto
haftmann@33320
  1688
  qed
wenzelm@63652
  1689
  then show ?thesis
wenzelm@63652
  1690
    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1691
qed
haftmann@33320
  1692
wenzelm@63652
  1693
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
wenzelm@63652
  1694
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
  1695
  for x :: int
haftmann@33320
  1696
proof
wenzelm@63652
  1697
  assume ?lhs
wenzelm@63652
  1698
  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@63652
  1699
  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@63652
  1700
  then have "nat \<bar>x\<bar> = 1" by simp
wenzelm@63652
  1701
  then show ?rhs by (cases "x < 0") auto
haftmann@33320
  1702
next
wenzelm@63652
  1703
  assume ?rhs
wenzelm@63652
  1704
  then have "x = 1 \<or> x = - 1" by auto
wenzelm@63652
  1705
  then show ?lhs by (auto intro: dvdI)
haftmann@33320
  1706
qed
haftmann@33320
  1707
lp15@60162
  1708
lemma zdvd_mult_cancel1:
wenzelm@63652
  1709
  fixes m :: int
wenzelm@63652
  1710
  assumes mp: "m \<noteq> 0"
wenzelm@63652
  1711
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
wenzelm@63652
  1712
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@33320
  1713
proof
wenzelm@63652
  1714
  assume ?rhs
wenzelm@63652
  1715
  then show ?lhs
wenzelm@63652
  1716
    by (cases "n > 0") (auto simp add: minus_equation_iff)
haftmann@33320
  1717
next
wenzelm@63652
  1718
  assume ?lhs
wenzelm@63652
  1719
  then have "m * n dvd m * 1" by simp
wenzelm@63652
  1720
  from zdvd_mult_cancel[OF this mp] show ?rhs
wenzelm@63652
  1721
    by (simp only: zdvd1_eq)
haftmann@33320
  1722
qed
haftmann@33320
  1723
wenzelm@63652
  1724
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
wenzelm@63652
  1725
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@33320
  1726
wenzelm@63652
  1727
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
wenzelm@63652
  1728
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@58650
  1729
wenzelm@63652
  1730
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
wenzelm@63652
  1731
  by (simp add: dvd_int_iff [symmetric])
wenzelm@63652
  1732
wenzelm@63652
  1733
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
haftmann@33320
  1734
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1735
wenzelm@63652
  1736
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1737
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1738
wenzelm@63652
  1739
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1740
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1741
immler@66912
  1742
lemma numeral_power_eq_nat_cancel_iff [simp]:
immler@66912
  1743
  "numeral x ^ n = nat y \<longleftrightarrow> numeral x ^ n = y"
immler@66912
  1744
  using nat_eq_iff2 by auto
immler@66912
  1745
immler@66912
  1746
lemma nat_eq_numeral_power_cancel_iff [simp]:
immler@66912
  1747
  "nat y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
immler@66912
  1748
  using numeral_power_eq_nat_cancel_iff[of x n y]
immler@66912
  1749
  by (metis (mono_tags))
immler@66912
  1750
immler@66912
  1751
lemma numeral_power_le_nat_cancel_iff [simp]:
immler@66912
  1752
  "numeral x ^ n \<le> nat a \<longleftrightarrow> numeral x ^ n \<le> a"
immler@66912
  1753
  using nat_le_eq_zle[of "numeral x ^ n" a]
immler@66912
  1754
  by (auto simp: nat_power_eq)
immler@66912
  1755
immler@66912
  1756
lemma nat_le_numeral_power_cancel_iff [simp]:
immler@66912
  1757
  "nat a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
immler@66912
  1758
  by (simp add: nat_le_iff)
immler@66912
  1759
immler@66912
  1760
lemma numeral_power_less_nat_cancel_iff [simp]:
immler@66912
  1761
  "numeral x ^ n < nat a \<longleftrightarrow> numeral x ^ n < a"
immler@66912
  1762
  using nat_less_eq_zless[of "numeral x ^ n" a]
immler@66912
  1763
  by (auto simp: nat_power_eq)
immler@66912
  1764
immler@66912
  1765
lemma nat_less_numeral_power_cancel_iff [simp]:
immler@66912
  1766
  "nat a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
immler@66912
  1767
  using nat_less_eq_zless[of a "numeral x ^ n"]
immler@66912
  1768
  by (cases "a < 0") (auto simp: nat_power_eq less_le_trans[where y=0])
immler@66912
  1769
wenzelm@63652
  1770
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
wenzelm@63652
  1771
  for n z :: int
wenzelm@42676
  1772
  apply (cases n)
wenzelm@63652
  1773
   apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1774
  apply (cases z)
wenzelm@63652
  1775
   apply (auto simp add: dvd_imp_le)
haftmann@33320
  1776
  done
haftmann@33320
  1777
haftmann@36749
  1778
lemma zdvd_period:
haftmann@36749
  1779
  fixes a d :: int
haftmann@36749
  1780
  assumes "a dvd d"
haftmann@36749
  1781
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
wenzelm@63652
  1782
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@36749
  1783
proof -
haftmann@66816
  1784
  from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)"
haftmann@66816
  1785
    by (simp add: dvd_add_left_iff)
haftmann@66816
  1786
  then show ?thesis
haftmann@66816
  1787
    by (simp add: ac_simps)
haftmann@36749
  1788
qed
haftmann@36749
  1789
haftmann@33320
  1790
wenzelm@60758
  1791
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1792
wenzelm@63652
  1793
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
wenzelm@63652
  1794
proof (cases "a \<le> b")
bulwahn@46756
  1795
  case True
wenzelm@63652
  1796
  then show ?thesis
bulwahn@46756
  1797
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1798
    case base
wenzelm@63652
  1799
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
wenzelm@63652
  1800
    then show ?case by simp
bulwahn@46756
  1801
  next
bulwahn@46756
  1802
    case (step b)
wenzelm@63652
  1803
    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
  1804
    with step show ?case by simp
bulwahn@46756
  1805
  qed
bulwahn@46756
  1806
next
wenzelm@63652
  1807
  case False
wenzelm@63652
  1808
  then show ?thesis
bulwahn@46756
  1809
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1810
qed
bulwahn@46756
  1811
wenzelm@63652
  1812
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
wenzelm@63652
  1813
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1814
wenzelm@63652
  1815
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
wenzelm@63652
  1816
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1817
wenzelm@63652
  1818
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
wenzelm@63652
  1819
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1820
bulwahn@46756
  1821
wenzelm@60758
  1822
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1823
wenzelm@60758
  1824
text \<open>Constructors\<close>
huffman@47108
  1825
wenzelm@63652
  1826
definition Pos :: "num \<Rightarrow> int"
wenzelm@63652
  1827
  where [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1828
wenzelm@63652
  1829
definition Neg :: "num \<Rightarrow> int"
wenzelm@63652
  1830
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1831
huffman@47108
  1832
code_datatype "0::int" Pos Neg
huffman@47108
  1833
huffman@47108
  1834
wenzelm@63652
  1835
text \<open>Auxiliary operations.\<close>
huffman@47108
  1836
wenzelm@63652
  1837
definition dup :: "int \<Rightarrow> int"
wenzelm@63652
  1838
  where [simp]: "dup k = k + k"
haftmann@26507
  1839
huffman@47108
  1840
lemma dup_code [code]:
huffman@47108
  1841
  "dup 0 = 0"
huffman@47108
  1842
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1843
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1844
  by (simp_all add: numeral_Bit0)
huffman@47108
  1845
wenzelm@63652
  1846
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
wenzelm@63652
  1847
  where [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1848
huffman@47108
  1849
lemma sub_code [code]:
huffman@47108
  1850
  "sub Num.One Num.One = 0"
huffman@47108
  1851
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1852
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1853
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1854
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1855
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1856
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1857
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1858
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
boehmes@66035
  1859
  by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
huffman@47108
  1860
wenzelm@63652
  1861
text \<open>Implementations.\<close>
huffman@47108
  1862
haftmann@64996
  1863
lemma one_int_code [code]: "1 = Pos Num.One"
huffman@47108
  1864
  by simp
huffman@47108
  1865
huffman@47108
  1866
lemma plus_int_code [code]:
wenzelm@63652
  1867
  "k + 0 = k"
wenzelm@63652
  1868
  "0 + l = l"
huffman@47108
  1869
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1870
  "Pos m + Neg n = sub m n"
huffman@47108
  1871
  "Neg m + Pos n = sub n m"
huffman@47108
  1872
  "Neg m + Neg n = Neg (m + n)"
wenzelm@63652
  1873
  for k l :: int
huffman@47108
  1874
  by simp_all
haftmann@26507
  1875
huffman@47108
  1876
lemma uminus_int_code [code]:
huffman@47108
  1877
  "uminus 0 = (0::int)"
huffman@47108
  1878
  "uminus (Pos m) = Neg m"
huffman@47108
  1879
  "uminus (Neg m) = Pos m"
huffman@47108
  1880
  by simp_all
huffman@47108
  1881
huffman@47108
  1882
lemma minus_int_code [code]:
wenzelm@63652
  1883
  "k - 0 = k"
wenzelm@63652
  1884
  "0 - l = uminus l"
huffman@47108
  1885
  "Pos m - Pos n = sub m n"
huffman@47108
  1886
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1887
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1888
  "Neg m - Neg n = sub n m"
wenzelm@63652
  1889
  for k l :: int
huffman@47108
  1890
  by simp_all
huffman@47108
  1891
huffman@47108
  1892
lemma times_int_code [code]:
wenzelm@63652
  1893
  "k * 0 = 0"
wenzelm@63652
  1894
  "0 * l = 0"
huffman@47108
  1895
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1896
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1897
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1898
  "Neg m * Neg n = Pos (m * n)"
wenzelm@63652
  1899
  for k l :: int
huffman@47108
  1900
  by simp_all
haftmann@26507
  1901
haftmann@38857
  1902
instantiation int :: equal
haftmann@26507
  1903
begin
haftmann@26507
  1904
wenzelm@63652
  1905
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1906
wenzelm@61169
  1907
instance
wenzelm@61169
  1908
  by standard (rule equal_int_def)
haftmann@26507
  1909
haftmann@26507
  1910
end
haftmann@26507
  1911
huffman@47108
  1912
lemma equal_int_code [code]:
huffman@47108
  1913
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1914
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1915
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1916
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1917
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1918
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1919
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1920
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1921
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1922
  by (auto simp add: equal)
haftmann@26507
  1923
wenzelm@63652
  1924
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
wenzelm@63652
  1925
  for k :: int
huffman@47108
  1926
  by (fact equal_refl)
haftmann@26507
  1927
haftmann@28562
  1928
lemma less_eq_int_code [code]:
huffman@47108
  1929
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1930
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1931
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1932
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1933
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1934
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1935
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1936
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1937
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1938
  by simp_all
haftmann@26507
  1939
haftmann@28562
  1940
lemma less_int_code [code]:
huffman@47108
  1941
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1942
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1943
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1944
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1945
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1946
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1947
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1948
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1949
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1950
  by simp_all
haftmann@25919
  1951
huffman@47108
  1952
lemma nat_code [code]:
huffman@47108
  1953
  "nat (Int.Neg k) = 0"
huffman@47108
  1954
  "nat 0 = 0"
huffman@47108
  1955
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1956
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1957
huffman@47108
  1958
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1959
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1960
  "of_int 0 = 0"
huffman@47108
  1961
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1962
  by simp_all
haftmann@25919
  1963
huffman@47108
  1964
wenzelm@63652
  1965
text \<open>Serializer setup.\<close>
haftmann@25919
  1966
haftmann@52435
  1967
code_identifier
haftmann@52435
  1968
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1969
haftmann@25919
  1970
quickcheck_params [default_type = int]
haftmann@25919
  1971
huffman@47108
  1972
hide_const (open) Pos Neg sub dup
haftmann@25919
  1973
haftmann@25919
  1974
wenzelm@61799
  1975
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1976
kuncar@53652
  1977
lifting_update int.lifting
kuncar@53652
  1978
lifting_forget int.lifting
huffman@48045
  1979
haftmann@25919
  1980
end