src/HOL/Int.thy
author haftmann
Wed Feb 17 21:51:57 2016 +0100 (2016-02-17)
changeset 62347 2230b7047376
parent 62128 3201ddb00097
child 62348 9a5f43dac883
permissions -rw-r--r--
generalized some lemmas;
moved some lemmas in more appropriate places;
deleted potentially dangerous simp rule
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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" where
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  "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"
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    unfolding reflp_def by auto
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  show "symp intrel"
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    unfolding symp_def by auto
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  show "transp intrel"
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    unfolding transp_def by auto
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> 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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  hence "(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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  thus "(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" where
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  "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,
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    simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]:
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  "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
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  unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
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lemma int_diff_cases:
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  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
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  "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (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 k>0\<close>
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lemma zmult_zless_mono2_lemma:
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     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
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apply (induct k)
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apply simp
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apply (simp add: distrib_right)
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apply (case_tac "k=0")
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apply (simp_all add: add_strict_mono)
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done
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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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
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  zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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instance 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::int) \<le> z"
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd:
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  "(w :: int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
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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]:
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  fixes z :: int
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  shows "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?Q then show ?P by simp
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next
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  assume ?P
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  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1"
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    by simp
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  then have "\<bar>z\<bar> \<le> 0"
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    by simp
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  then show ?Q
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    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" 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 of_nat_mult)
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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]:
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  "of_int (z ^ n) = of_int z ^ n"
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  by (induct n) simp_all
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end
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context ring_char_0
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begin
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lemma of_int_eq_iff [simp]:
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   "of_int w = of_int z \<longleftrightarrow> w = z"
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  by transfer (clarsimp simp add: algebra_simps
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    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]:
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  "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]:
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  "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]:
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   "of_int z = 1 \<longleftrightarrow> z = 1"
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  using of_int_eq_iff [of z 1] by simp
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end
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context linordered_idom
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begin
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text\<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
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subclass ring_char_0 ..
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lemma of_int_le_iff [simp]:
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  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
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  by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_less_iff [simp]:
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  "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]:
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  "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]:
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  "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]:
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  "0 < of_int z \<longleftrightarrow> 0 < z"
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  using of_int_less_iff [of 0 z] by simp
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lemma of_int_less_0_iff [simp]:
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  "of_int z < 0 \<longleftrightarrow> z < 0"
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  using of_int_less_iff [of z 0] by simp
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lemma of_int_1_le_iff [simp]:
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  "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
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  using of_int_le_iff [of 1 z] by simp
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lemma of_int_le_1_iff [simp]:
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  "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
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  using of_int_le_iff [of z 1] by simp
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lemma of_int_1_less_iff [simp]:
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  "1 < of_int z \<longleftrightarrow> 1 < z"
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  using of_int_less_iff [of 1 z] by simp
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lemma of_int_less_1_iff [simp]:
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  "of_int z < 1 \<longleftrightarrow> z < 1"
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  using of_int_less_iff [of z 1] by simp
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lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
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  by simp
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lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
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  by simp
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lemma of_int_abs [simp]:
haftmann@62347
   339
  "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
haftmann@62347
   340
  by (auto simp add: abs_if)
haftmann@62347
   341
haftmann@62347
   342
lemma of_int_lessD:
haftmann@62347
   343
  assumes "\<bar>of_int n\<bar> < x"
haftmann@62347
   344
  shows "n = 0 \<or> x > 1"
haftmann@62347
   345
proof (cases "n = 0")
haftmann@62347
   346
  case True then show ?thesis by simp
haftmann@62347
   347
next
haftmann@62347
   348
  case False
haftmann@62347
   349
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   350
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   351
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   352
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   353
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   354
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   355
  then have "1 < x" using assms by (rule le_less_trans)
haftmann@62347
   356
  then show ?thesis ..
haftmann@62347
   357
qed
haftmann@62347
   358
haftmann@62347
   359
lemma of_int_leD:
haftmann@62347
   360
  assumes "\<bar>of_int n\<bar> \<le> x"
haftmann@62347
   361
  shows "n = 0 \<or> 1 \<le> x"
haftmann@62347
   362
proof (cases "n = 0")
haftmann@62347
   363
  case True then show ?thesis by simp
haftmann@62347
   364
next
haftmann@62347
   365
  case False
haftmann@62347
   366
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   367
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   368
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   369
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   370
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   371
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   372
  then have "1 \<le> x" using assms by (rule order_trans)
haftmann@62347
   373
  then show ?thesis ..
haftmann@62347
   374
qed
haftmann@62347
   375
haftmann@62347
   376
haftmann@36424
   377
end
haftmann@25919
   378
lp15@61234
   379
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   380
lp15@61234
   381
lemma of_int_eq_numeral_iff [iff]:
lp15@61649
   382
   "of_int z = (numeral n :: 'a::ring_char_0)
lp15@61234
   383
   \<longleftrightarrow> z = numeral n"
lp15@61234
   384
  using of_int_eq_iff by fastforce
lp15@61234
   385
lp15@61649
   386
lemma of_int_le_numeral_iff [simp]:
lp15@61649
   387
   "of_int z \<le> (numeral n :: 'a::linordered_idom)
lp15@61234
   388
   \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   389
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   390
lp15@61649
   391
lemma of_int_numeral_le_iff [simp]:
lp15@61234
   392
   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   393
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   394
lp15@61649
   395
lemma of_int_less_numeral_iff [simp]:
lp15@61649
   396
   "of_int z < (numeral n :: 'a::linordered_idom)
lp15@61234
   397
   \<longleftrightarrow> z < numeral n"
lp15@61234
   398
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   399
lp15@61649
   400
lemma of_int_numeral_less_iff [simp]:
lp15@61234
   401
   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   402
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   403
hoelzl@56889
   404
lemma of_nat_less_of_int_iff:
hoelzl@56889
   405
  "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   406
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   407
haftmann@25919
   408
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   409
proof
haftmann@25919
   410
  fix z show "of_int z = id z"
huffman@48045
   411
    by (cases z rule: int_diff_cases, simp)
haftmann@25919
   412
qed
haftmann@25919
   413
haftmann@25919
   414
hoelzl@51329
   415
instance int :: no_top
wenzelm@61169
   416
  apply standard
hoelzl@51329
   417
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   418
  apply simp
hoelzl@51329
   419
  done
hoelzl@51329
   420
hoelzl@51329
   421
instance int :: no_bot
wenzelm@61169
   422
  apply standard
hoelzl@51329
   423
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   424
  apply simp
hoelzl@51329
   425
  done
hoelzl@51329
   426
wenzelm@61799
   427
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   428
huffman@48045
   429
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   430
  by auto
haftmann@25919
   431
huffman@44709
   432
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   433
  by transfer simp
haftmann@25919
   434
huffman@44709
   435
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   436
  by transfer clarsimp
haftmann@25919
   437
huffman@44709
   438
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
haftmann@25919
   439
by simp
haftmann@25919
   440
haftmann@25919
   441
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
huffman@48045
   442
  by transfer clarsimp
haftmann@25919
   443
haftmann@25919
   444
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
huffman@48045
   445
  by transfer (clarsimp, arith)
haftmann@25919
   446
wenzelm@60758
   447
text\<open>An alternative condition is @{term "0 \<le> w"}\<close>
haftmann@25919
   448
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
lp15@60162
   449
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   450
haftmann@25919
   451
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
lp15@60162
   452
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   453
haftmann@25919
   454
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
huffman@48045
   455
  by transfer (clarsimp, arith)
haftmann@25919
   456
haftmann@25919
   457
lemma nonneg_eq_int:
haftmann@25919
   458
  fixes z :: int
huffman@44709
   459
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   460
  shows P
haftmann@25919
   461
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   462
haftmann@54223
   463
lemma nat_eq_iff:
haftmann@54223
   464
  "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   465
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   466
haftmann@54223
   467
corollary nat_eq_iff2:
haftmann@54223
   468
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   469
  using nat_eq_iff [of w m] by auto
haftmann@54223
   470
haftmann@54223
   471
lemma nat_0 [simp]:
haftmann@54223
   472
  "nat 0 = 0"
haftmann@54223
   473
  by (simp add: nat_eq_iff)
haftmann@25919
   474
haftmann@54223
   475
lemma nat_1 [simp]:
haftmann@54223
   476
  "nat 1 = Suc 0"
haftmann@54223
   477
  by (simp add: nat_eq_iff)
haftmann@54223
   478
haftmann@54223
   479
lemma nat_numeral [simp]:
haftmann@54223
   480
  "nat (numeral k) = numeral k"
haftmann@54223
   481
  by (simp add: nat_eq_iff)
haftmann@25919
   482
haftmann@54223
   483
lemma nat_neg_numeral [simp]:
haftmann@54489
   484
  "nat (- numeral k) = 0"
haftmann@54223
   485
  by simp
haftmann@54223
   486
haftmann@54223
   487
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   488
  by simp
lp15@60162
   489
haftmann@25919
   490
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045
   491
  by transfer (clarsimp, arith)
haftmann@25919
   492
huffman@44709
   493
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   494
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   495
huffman@44707
   496
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   497
  by transfer auto
huffman@44707
   498
nipkow@29700
   499
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045
   500
  by transfer clarsimp
nipkow@29700
   501
haftmann@25919
   502
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   503
by (auto simp add: nat_eq_iff2)
haftmann@25919
   504
haftmann@25919
   505
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   506
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   507
haftmann@25919
   508
lemma nat_add_distrib:
haftmann@54223
   509
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   510
  by transfer clarsimp
haftmann@25919
   511
haftmann@54223
   512
lemma nat_diff_distrib':
haftmann@54223
   513
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   514
  by transfer clarsimp
lp15@60162
   515
haftmann@25919
   516
lemma nat_diff_distrib:
haftmann@54223
   517
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   518
  by (rule nat_diff_distrib') auto
haftmann@25919
   519
huffman@44709
   520
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   521
  by transfer simp
haftmann@25919
   522
haftmann@53065
   523
lemma le_nat_iff:
haftmann@53065
   524
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   525
  by transfer auto
lp15@60162
   526
huffman@44709
   527
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045
   528
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   529
haftmann@25919
   530
context ring_1
haftmann@25919
   531
begin
haftmann@25919
   532
lp15@61609
   533
lemma of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   534
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   535
haftmann@25919
   536
end
haftmann@25919
   537
lp15@60162
   538
lemma diff_nat_numeral [simp]:
haftmann@54249
   539
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   540
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   541
haftmann@54249
   542
wenzelm@60758
   543
text \<open>For termination proofs:\<close>
krauss@29779
   544
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   545
haftmann@25919
   546
wenzelm@60758
   547
subsection\<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   548
wenzelm@61076
   549
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
haftmann@25919
   550
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   551
huffman@44709
   552
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   553
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   554
huffman@44709
   555
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   556
by (simp add: minus_le_iff)
haftmann@25919
   557
huffman@44709
   558
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   559
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   560
huffman@44709
   561
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   562
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   563
huffman@44709
   564
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045
   565
  by transfer simp
haftmann@25919
   566
huffman@44709
   567
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   568
by (simp add: linorder_not_less)
haftmann@25919
   569
huffman@44709
   570
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   571
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   572
huffman@44709
   573
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   574
proof -
haftmann@25919
   575
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   576
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   577
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   578
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   579
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   580
    by (simp only: algebra_simps)
haftmann@25919
   581
  finally show ?thesis .
haftmann@25919
   582
qed
haftmann@25919
   583
huffman@44709
   584
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   585
by simp
haftmann@25919
   586
wenzelm@60758
   587
text\<open>This version is proved for all ordered rings, not just integers!
wenzelm@61799
   588
      It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@61799
   589
      in theory \<open>Rings\<close>.
wenzelm@61799
   590
      But is it really better than just rewriting with \<open>abs_if\<close>?\<close>
blanchet@54147
   591
lemma abs_split [arith_split, no_atp]:
wenzelm@61944
   592
     "P \<bar>a::'a::linordered_idom\<bar> = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   593
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   594
huffman@44709
   595
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045
   596
apply transfer
huffman@48045
   597
apply clarsimp
huffman@48045
   598
apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919
   599
done
haftmann@25919
   600
wenzelm@60758
   601
subsection \<open>Cases and induction\<close>
haftmann@25919
   602
wenzelm@60758
   603
text\<open>Now we replace the case analysis rule by a more conventional one:
wenzelm@60758
   604
whether an integer is negative or not.\<close>
haftmann@25919
   605
wenzelm@60758
   606
text\<open>This version is symmetric in the two subgoals.\<close>
lp15@59613
   607
theorem int_cases2 [case_names nonneg nonpos, cases type: int]:
lp15@59613
   608
  "\<lbrakk>!! n. z = int n \<Longrightarrow> P;  !! n. z = - (int n) \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
lp15@59613
   609
apply (cases "z < 0")
lp15@59613
   610
apply (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   611
done
lp15@59613
   612
wenzelm@60758
   613
text\<open>This is the default, with a negative case.\<close>
wenzelm@42676
   614
theorem int_cases [case_names nonneg neg, cases type: int]:
lp15@59613
   615
  "[|!! n. z = int n ==> P;  !! n. z = - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   616
apply (cases "z < 0")
wenzelm@42676
   617
apply (blast dest!: negD)
haftmann@25919
   618
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   619
apply auto
haftmann@25919
   620
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   621
done
haftmann@25919
   622
haftmann@60868
   623
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   624
  fixes k :: int
haftmann@60868
   625
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   626
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   627
  shows "P"
haftmann@60868
   628
proof (cases k "0::int" rule: linorder_cases)
haftmann@60868
   629
  case equal with assms(1) show P by simp
haftmann@60868
   630
next
haftmann@60868
   631
  case greater
haftmann@60868
   632
  then have "nat k > 0" by simp
haftmann@60868
   633
  moreover from this have "k = int (nat k)" by auto
haftmann@60868
   634
  ultimately show P using assms(2) by blast
haftmann@60868
   635
next
haftmann@60868
   636
  case less
haftmann@60868
   637
  then have "nat (- k) > 0" by simp
haftmann@60868
   638
  moreover from this have "k = - int (nat (- k))" by auto
haftmann@60868
   639
  ultimately show P using assms(3) by blast
haftmann@60868
   640
qed
haftmann@60868
   641
wenzelm@42676
   642
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   643
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   644
  by (cases z) auto
haftmann@25919
   645
huffman@47207
   646
lemma nonneg_int_cases:
huffman@47207
   647
  assumes "0 \<le> k" obtains n where "k = int n"
haftmann@54489
   648
  using assms by (rule nonneg_eq_int)
huffman@47207
   649
huffman@47108
   650
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   651
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   652
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   653
haftmann@54489
   654
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   655
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   656
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   657
wenzelm@61799
   658
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   659
huffman@47108
   660
lemmas max_number_of [simp] =
huffman@47108
   661
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   662
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   663
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   664
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   665
huffman@47108
   666
lemmas min_number_of [simp] =
huffman@47108
   667
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   668
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   669
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   670
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   671
haftmann@25919
   672
wenzelm@60758
   673
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   674
wenzelm@60758
   675
text \<open>Preliminaries\<close>
huffman@28958
   676
lp15@60162
   677
lemma le_imp_0_less:
huffman@28958
   678
  assumes le: "0 \<le> z"
huffman@28958
   679
  shows "(0::int) < 1 + z"
huffman@28958
   680
proof -
huffman@28958
   681
  have "0 \<le> z" by fact
huffman@47108
   682
  also have "... < z + 1" by (rule less_add_one)
haftmann@57514
   683
  also have "... = 1 + z" by (simp add: ac_simps)
huffman@28958
   684
  finally show "0 < 1 + z" .
huffman@28958
   685
qed
huffman@28958
   686
huffman@28958
   687
lemma odd_less_0_iff:
huffman@28958
   688
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   689
proof (cases z)
huffman@28958
   690
  case (nonneg n)
haftmann@57512
   691
  thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
lp15@60162
   692
                             le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   693
next
huffman@28958
   694
  case (neg n)
huffman@30079
   695
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   696
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   697
qed
huffman@28958
   698
wenzelm@60758
   699
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   700
haftmann@25919
   701
lemmas double_eq_0_iff = double_zero
haftmann@25919
   702
haftmann@25919
   703
lemma odd_nonzero:
haftmann@33296
   704
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
   705
proof (cases z)
haftmann@25919
   706
  case (nonneg n)
lp15@60162
   707
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
haftmann@25919
   708
  thus ?thesis using  le_imp_0_less [OF le]
lp15@60162
   709
    by (auto simp add: add.assoc)
haftmann@25919
   710
next
haftmann@25919
   711
  case (neg n)
haftmann@25919
   712
  show ?thesis
haftmann@25919
   713
  proof
haftmann@25919
   714
    assume eq: "1 + z + z = 0"
huffman@44709
   715
    have "(0::int) < 1 + (int n + int n)"
lp15@60162
   716
      by (simp add: le_imp_0_less add_increasing)
lp15@60162
   717
    also have "... = - (1 + z + z)"
lp15@60162
   718
      by (simp add: neg add.assoc [symmetric])
lp15@60162
   719
    also have "... = 0" by (simp add: eq)
haftmann@25919
   720
    finally have "0<0" ..
haftmann@25919
   721
    thus False by blast
haftmann@25919
   722
  qed
haftmann@25919
   723
qed
haftmann@25919
   724
haftmann@30652
   725
wenzelm@60758
   726
subsection \<open>The Set of Integers\<close>
haftmann@25919
   727
haftmann@25919
   728
context ring_1
haftmann@25919
   729
begin
haftmann@25919
   730
wenzelm@61070
   731
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   732
  where "\<int> = range of_int"
haftmann@25919
   733
huffman@35634
   734
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   735
  by (simp add: Ints_def)
huffman@35634
   736
huffman@35634
   737
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   738
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   739
haftmann@25919
   740
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   741
  using Ints_of_int [of "0"] by simp
haftmann@25919
   742
haftmann@25919
   743
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   744
  using Ints_of_int [of "1"] by simp
haftmann@25919
   745
eberlm@61552
   746
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   747
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   748
haftmann@25919
   749
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
   750
apply (auto simp add: Ints_def)
haftmann@25919
   751
apply (rule range_eqI)
haftmann@25919
   752
apply (rule of_int_add [symmetric])
haftmann@25919
   753
done
haftmann@25919
   754
haftmann@25919
   755
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
   756
apply (auto simp add: Ints_def)
haftmann@25919
   757
apply (rule range_eqI)
haftmann@25919
   758
apply (rule of_int_minus [symmetric])
haftmann@25919
   759
done
haftmann@25919
   760
huffman@35634
   761
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
   762
apply (auto simp add: Ints_def)
huffman@35634
   763
apply (rule range_eqI)
huffman@35634
   764
apply (rule of_int_diff [symmetric])
huffman@35634
   765
done
huffman@35634
   766
haftmann@25919
   767
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
   768
apply (auto simp add: Ints_def)
haftmann@25919
   769
apply (rule range_eqI)
haftmann@25919
   770
apply (rule of_int_mult [symmetric])
haftmann@25919
   771
done
haftmann@25919
   772
huffman@35634
   773
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
   774
by (induct n) simp_all
huffman@35634
   775
haftmann@25919
   776
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   777
  assumes "q \<in> \<int>"
haftmann@25919
   778
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   779
  unfolding Ints_def
haftmann@25919
   780
proof -
wenzelm@60758
   781
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   782
  then obtain z where "q = of_int z" ..
haftmann@25919
   783
  then show thesis ..
haftmann@25919
   784
qed
haftmann@25919
   785
haftmann@25919
   786
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   787
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   788
  by (rule Ints_cases) auto
haftmann@25919
   789
eberlm@61524
   790
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   791
  unfolding Nats_def Ints_def
eberlm@61524
   792
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   793
eberlm@61524
   794
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   795
proof (intro subsetI equalityI)
eberlm@61524
   796
  fix x :: 'a assume "x \<in> {of_int n |n. n \<ge> 0}"
eberlm@61524
   797
  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
eberlm@61524
   798
  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
eberlm@61524
   799
  thus "x \<in> \<nat>" by simp
eberlm@61524
   800
next
eberlm@61524
   801
  fix x :: 'a assume "x \<in> \<nat>"
eberlm@61524
   802
  then obtain n where "x = of_nat n" by (auto elim!: Nats_cases)
eberlm@61524
   803
  hence "x = of_int (int n)" by simp
eberlm@61524
   804
  also have "int n \<ge> 0" by simp
eberlm@61524
   805
  hence "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   806
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   807
qed
eberlm@61524
   808
haftmann@25919
   809
end
haftmann@25919
   810
eberlm@61524
   811
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   812
proof (intro subsetI equalityI)
eberlm@61524
   813
  fix x :: 'a assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   814
  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
eberlm@61524
   815
  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
eberlm@61524
   816
  thus "x \<in> \<nat>" by simp
eberlm@61524
   817
qed (auto elim!: Nats_cases)
eberlm@61524
   818
eberlm@61524
   819
wenzelm@60758
   820
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
   821
haftmann@25919
   822
lemma Ints_double_eq_0_iff:
wenzelm@61070
   823
  assumes in_Ints: "a \<in> \<int>"
haftmann@25919
   824
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
   825
proof -
haftmann@25919
   826
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   827
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   828
  show ?thesis
haftmann@25919
   829
  proof
haftmann@25919
   830
    assume "a = 0"
haftmann@25919
   831
    thus "a + a = 0" by simp
haftmann@25919
   832
  next
haftmann@25919
   833
    assume eq: "a + a = 0"
haftmann@25919
   834
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   835
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   836
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   837
    thus "a = 0" by (simp add: a)
haftmann@25919
   838
  qed
haftmann@25919
   839
qed
haftmann@25919
   840
haftmann@25919
   841
lemma Ints_odd_nonzero:
wenzelm@61070
   842
  assumes in_Ints: "a \<in> \<int>"
haftmann@25919
   843
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
   844
proof -
haftmann@25919
   845
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   846
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   847
  show ?thesis
haftmann@25919
   848
  proof
haftmann@25919
   849
    assume eq: "1 + a + a = 0"
haftmann@25919
   850
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   851
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   852
    with odd_nonzero show False by blast
haftmann@25919
   853
  qed
lp15@60162
   854
qed
haftmann@25919
   855
wenzelm@61070
   856
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
   857
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   858
lp15@60162
   859
lemma Ints_odd_less_0:
wenzelm@61070
   860
  assumes in_Ints: "a \<in> \<int>"
haftmann@35028
   861
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
   862
proof -
haftmann@25919
   863
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   864
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   865
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
   866
    by (simp add: a)
huffman@45532
   867
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919
   868
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
   869
  finally show ?thesis .
haftmann@25919
   870
qed
haftmann@25919
   871
haftmann@25919
   872
wenzelm@60758
   873
subsection \<open>@{term setsum} and @{term setprod}\<close>
haftmann@25919
   874
lp15@61649
   875
lemma of_nat_setsum [simp]: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
   876
  apply (cases "finite A")
haftmann@25919
   877
  apply (erule finite_induct, auto)
haftmann@25919
   878
  done
haftmann@25919
   879
lp15@61649
   880
lemma of_int_setsum [simp]: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
   881
  apply (cases "finite A")
haftmann@25919
   882
  apply (erule finite_induct, auto)
haftmann@25919
   883
  done
haftmann@25919
   884
lp15@61649
   885
lemma of_nat_setprod [simp]: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
   886
  apply (cases "finite A")
haftmann@25919
   887
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
   888
  done
haftmann@25919
   889
lp15@61649
   890
lemma of_int_setprod [simp]: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
   891
  apply (cases "finite A")
haftmann@25919
   892
  apply (erule finite_induct, auto)
haftmann@25919
   893
  done
haftmann@25919
   894
haftmann@25919
   895
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
   896
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
   897
haftmann@25919
   898
wenzelm@60758
   899
text \<open>Legacy theorems\<close>
haftmann@25919
   900
haftmann@25919
   901
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   902
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919
   903
wenzelm@60758
   904
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
   905
haftmann@54249
   906
lemmas of_int_simps =
haftmann@54249
   907
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   908
wenzelm@48891
   909
ML_file "Tools/int_arith.ML"
wenzelm@60758
   910
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
   911
huffman@47108
   912
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
wenzelm@61144
   913
  "(m::'a::linordered_idom) \<le> n" |
huffman@47108
   914
  "(m::'a::linordered_idom) = n") =
wenzelm@61144
   915
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
   916
haftmann@25919
   917
wenzelm@60758
   918
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
   919
haftmann@25919
   920
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
   921
by arith
haftmann@25919
   922
haftmann@25919
   923
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
   924
by arith
haftmann@25919
   925
haftmann@25919
   926
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
   927
by arith
haftmann@25919
   928
haftmann@25919
   929
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
   930
by arith
haftmann@25919
   931
haftmann@25919
   932
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
   933
by arith
haftmann@25919
   934
haftmann@25919
   935
wenzelm@60758
   936
subsection\<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
   937
wenzelm@60758
   938
text\<open>Simplify the term @{term "w + - z"}\<close>
haftmann@25919
   939
haftmann@25919
   940
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
lp15@60162
   941
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
   942
wenzelm@60758
   943
text\<open>This simplifies expressions of the form @{term "int n = z"} where
wenzelm@60758
   944
      z is an integer literal.\<close>
huffman@47108
   945
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
   946
haftmann@25919
   947
lemma split_nat [arith_split]:
huffman@44709
   948
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
   949
  (is "?P = (?L & ?R)")
haftmann@25919
   950
proof (cases "i < 0")
haftmann@25919
   951
  case True thus ?thesis by auto
haftmann@25919
   952
next
haftmann@25919
   953
  case False
haftmann@25919
   954
  have "?P = ?L"
haftmann@25919
   955
  proof
haftmann@25919
   956
    assume ?P thus ?L using False by clarsimp
haftmann@25919
   957
  next
haftmann@25919
   958
    assume ?L thus ?P using False by simp
haftmann@25919
   959
  qed
haftmann@25919
   960
  with False show ?thesis by simp
haftmann@25919
   961
qed
haftmann@25919
   962
hoelzl@59000
   963
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
   964
  by auto
hoelzl@59000
   965
hoelzl@59000
   966
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
   967
  by auto
hoelzl@59000
   968
haftmann@25919
   969
context ring_1
haftmann@25919
   970
begin
haftmann@25919
   971
blanchet@33056
   972
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
   973
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
   974
proof (cases "k < 0")
haftmann@25919
   975
  case True then have "0 \<le> - k" by simp
haftmann@25919
   976
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
   977
  with True show ?thesis by simp
haftmann@25919
   978
next
haftmann@25919
   979
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
   980
qed
haftmann@25919
   981
haftmann@25919
   982
end
haftmann@25919
   983
haftmann@25919
   984
lemma nat_mult_distrib:
haftmann@25919
   985
  fixes z z' :: int
haftmann@25919
   986
  assumes "0 \<le> z"
haftmann@25919
   987
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
   988
proof (cases "0 \<le> z'")
haftmann@25919
   989
  case False with assms have "z * z' \<le> 0"
haftmann@25919
   990
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
   991
  then have "nat (z * z') = 0" by simp
haftmann@25919
   992
  moreover from False have "nat z' = 0" by simp
haftmann@25919
   993
  ultimately show ?thesis by simp
haftmann@25919
   994
next
haftmann@25919
   995
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
   996
  show ?thesis
haftmann@25919
   997
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
   998
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
   999
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1000
qed
haftmann@25919
  1001
haftmann@25919
  1002
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
  1003
apply (rule trans)
haftmann@25919
  1004
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
  1005
done
haftmann@25919
  1006
wenzelm@61944
  1007
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
haftmann@25919
  1008
apply (cases "z=0 | w=0")
lp15@60162
  1009
apply (auto simp add: abs_if nat_mult_distrib [symmetric]
haftmann@25919
  1010
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1011
done
haftmann@25919
  1012
haftmann@60570
  1013
lemma int_in_range_abs [simp]:
haftmann@60570
  1014
  "int n \<in> range abs"
haftmann@60570
  1015
proof (rule range_eqI)
haftmann@60570
  1016
  show "int n = \<bar>int n\<bar>"
haftmann@60570
  1017
    by simp
haftmann@60570
  1018
qed
haftmann@60570
  1019
haftmann@60570
  1020
lemma range_abs_Nats [simp]:
haftmann@60570
  1021
  "range abs = (\<nat> :: int set)"
haftmann@60570
  1022
proof -
haftmann@60570
  1023
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1024
    by (cases k) simp_all
haftmann@60570
  1025
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1026
    using that by induct simp
haftmann@60570
  1027
  ultimately show ?thesis by blast
paulson@61204
  1028
qed
haftmann@60570
  1029
huffman@47207
  1030
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207
  1031
apply (rule sym)
huffman@47207
  1032
apply (simp add: nat_eq_iff)
huffman@47207
  1033
done
huffman@47207
  1034
huffman@47207
  1035
lemma diff_nat_eq_if:
lp15@60162
  1036
     "nat z - nat z' =
lp15@60162
  1037
        (if z' < 0 then nat z
lp15@60162
  1038
         else let d = z-z' in
huffman@47207
  1039
              if d < 0 then 0 else nat d)"
huffman@47207
  1040
by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1041
huffman@47207
  1042
lemma nat_numeral_diff_1 [simp]:
huffman@47207
  1043
  "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1044
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1045
haftmann@25919
  1046
haftmann@25919
  1047
subsection "Induction principles for int"
haftmann@25919
  1048
wenzelm@60758
  1049
text\<open>Well-founded segments of the integers\<close>
haftmann@25919
  1050
haftmann@25919
  1051
definition
haftmann@25919
  1052
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
  1053
where
haftmann@25919
  1054
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
  1055
haftmann@25919
  1056
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1057
proof -
haftmann@25919
  1058
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
  1059
    by (auto simp add: int_ge_less_than_def)
lp15@60162
  1060
  thus ?thesis
lp15@60162
  1061
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1062
qed
haftmann@25919
  1063
wenzelm@60758
  1064
text\<open>This variant looks odd, but is typical of the relations suggested
wenzelm@60758
  1065
by RankFinder.\<close>
haftmann@25919
  1066
haftmann@25919
  1067
definition
haftmann@25919
  1068
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
  1069
where
haftmann@25919
  1070
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
  1071
haftmann@25919
  1072
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1073
proof -
lp15@60162
  1074
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
haftmann@25919
  1075
    by (auto simp add: int_ge_less_than2_def)
lp15@60162
  1076
  thus ?thesis
lp15@60162
  1077
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1078
qed
haftmann@25919
  1079
haftmann@25919
  1080
(* `set:int': dummy construction *)
haftmann@25919
  1081
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1082
  fixes i :: int
haftmann@25919
  1083
  assumes ge: "k \<le> i" and
haftmann@25919
  1084
    base: "P k" and
haftmann@25919
  1085
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1086
  shows "P i"
haftmann@25919
  1087
proof -
wenzelm@42676
  1088
  { fix n
wenzelm@42676
  1089
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
  1090
    proof (induct n)
haftmann@25919
  1091
      case 0
haftmann@25919
  1092
      hence "i = k" by arith
haftmann@25919
  1093
      thus "P i" using base by simp
haftmann@25919
  1094
    next
haftmann@25919
  1095
      case (Suc n)
haftmann@25919
  1096
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
  1097
      moreover
haftmann@25919
  1098
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
  1099
      ultimately
wenzelm@42676
  1100
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
  1101
      from step [OF ki1 this] show ?case by simp
haftmann@25919
  1102
    qed
haftmann@25919
  1103
  }
haftmann@25919
  1104
  with ge show ?thesis by fast
haftmann@25919
  1105
qed
haftmann@25919
  1106
haftmann@25928
  1107
(* `set:int': dummy construction *)
haftmann@25928
  1108
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
  1109
  assumes gr: "k < (i::int)" and
haftmann@25919
  1110
        base: "P(k+1)" and
haftmann@25919
  1111
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
  1112
  shows "P i"
haftmann@25919
  1113
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
  1114
  using gr apply arith
haftmann@25919
  1115
 apply(rule base)
haftmann@25919
  1116
apply (rule step, simp+)
haftmann@25919
  1117
done
haftmann@25919
  1118
wenzelm@42676
  1119
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
  1120
  assumes le: "i \<le> (k::int)" and
haftmann@25919
  1121
        base: "P(k)" and
haftmann@25919
  1122
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1123
  shows "P i"
haftmann@25919
  1124
proof -
wenzelm@42676
  1125
  { fix n
wenzelm@42676
  1126
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
  1127
    proof (induct n)
haftmann@25919
  1128
      case 0
haftmann@25919
  1129
      hence "i = k" by arith
haftmann@25919
  1130
      thus "P i" using base by simp
haftmann@25919
  1131
    next
haftmann@25919
  1132
      case (Suc n)
wenzelm@42676
  1133
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
  1134
      moreover
haftmann@25919
  1135
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
  1136
      ultimately
wenzelm@42676
  1137
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
  1138
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1139
    qed
haftmann@25919
  1140
  }
haftmann@25919
  1141
  with le show ?thesis by fast
haftmann@25919
  1142
qed
haftmann@25919
  1143
wenzelm@42676
  1144
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
  1145
  assumes less: "(i::int) < k" and
haftmann@25919
  1146
        base: "P(k - 1)" and
haftmann@25919
  1147
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1148
  shows "P i"
haftmann@25919
  1149
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1150
  using less apply arith
haftmann@25919
  1151
 apply(rule base)
haftmann@25919
  1152
apply (rule step, simp+)
haftmann@25919
  1153
done
haftmann@25919
  1154
haftmann@36811
  1155
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1156
  fixes k :: int
haftmann@36801
  1157
  assumes base: "P k"
haftmann@36801
  1158
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1159
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1160
  shows "P i"
haftmann@36801
  1161
proof -
haftmann@36801
  1162
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1163
  then show ?thesis
wenzelm@42676
  1164
  proof
wenzelm@42676
  1165
    assume "i \<ge> k"
wenzelm@42676
  1166
    then show ?thesis using base
haftmann@36801
  1167
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1168
  next
wenzelm@42676
  1169
    assume "i \<le> k"
wenzelm@42676
  1170
    then show ?thesis using base
haftmann@36801
  1171
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1172
  qed
haftmann@36801
  1173
qed
haftmann@36801
  1174
wenzelm@60758
  1175
subsection\<open>Intermediate value theorems\<close>
haftmann@25919
  1176
haftmann@25919
  1177
lemma int_val_lemma:
wenzelm@61944
  1178
     "(\<forall>i<n::nat. \<bar>f(i+1) - f i\<bar> \<le> 1) -->
haftmann@25919
  1179
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1180
unfolding One_nat_def
wenzelm@42676
  1181
apply (induct n)
wenzelm@42676
  1182
apply simp
haftmann@25919
  1183
apply (intro strip)
haftmann@25919
  1184
apply (erule impE, simp)
haftmann@25919
  1185
apply (erule_tac x = n in allE, simp)
huffman@30079
  1186
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1187
apply force
haftmann@25919
  1188
apply (erule impE)
haftmann@25919
  1189
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1190
apply (blast intro: le_SucI)
haftmann@25919
  1191
done
haftmann@25919
  1192
haftmann@25919
  1193
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1194
haftmann@25919
  1195
lemma nat_intermed_int_val:
wenzelm@61944
  1196
     "[| \<forall>i. m \<le> i & i < n --> \<bar>f(i + 1::nat) - f i\<bar> \<le> 1; m < n;
haftmann@25919
  1197
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
lp15@60162
  1198
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
haftmann@25919
  1199
       in int_val_lemma)
huffman@30079
  1200
unfolding One_nat_def
haftmann@25919
  1201
apply simp
haftmann@25919
  1202
apply (erule exE)
haftmann@25919
  1203
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1204
done
haftmann@25919
  1205
haftmann@25919
  1206
wenzelm@60758
  1207
subsection\<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1208
paulson@34055
  1209
lemma abs_zmult_eq_1:
paulson@34055
  1210
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1211
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1212
proof -
paulson@34055
  1213
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1214
    by auto
paulson@34055
  1215
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1216
  proof
paulson@34055
  1217
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1218
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
lp15@60162
  1219
      by (simp add: mult_mono 0)
lp15@60162
  1220
    also have "... = \<bar>m*n\<bar>"
paulson@34055
  1221
      by (simp add: abs_mult)
paulson@34055
  1222
    also have "... = 1"
paulson@34055
  1223
      by (simp add: mn)
paulson@34055
  1224
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1225
    thus "False" using 0
huffman@47108
  1226
      by arith
paulson@34055
  1227
  qed
paulson@34055
  1228
  thus ?thesis using 0
paulson@34055
  1229
    by auto
paulson@34055
  1230
qed
haftmann@25919
  1231
haftmann@25919
  1232
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1233
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1234
boehmes@35815
  1235
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1236
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1237
proof -
boehmes@35815
  1238
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1239
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1240
qed
haftmann@25919
  1241
haftmann@25919
  1242
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
lp15@60162
  1243
apply (rule iffI)
haftmann@25919
  1244
 apply (frule pos_zmult_eq_1_iff_lemma)
lp15@60162
  1245
 apply (simp add: mult.commute [of m])
lp15@60162
  1246
 apply (frule pos_zmult_eq_1_iff_lemma, auto)
haftmann@25919
  1247
done
haftmann@25919
  1248
haftmann@33296
  1249
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1250
proof
haftmann@33296
  1251
  assume "finite (UNIV::int set)"
wenzelm@61076
  1252
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1253
    by (rule injI) simp
wenzelm@61076
  1254
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1255
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1256
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1257
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1258
qed
haftmann@25919
  1259
haftmann@25919
  1260
wenzelm@60758
  1261
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1262
wenzelm@60758
  1263
subsubsection\<open>Special Simplification for Constants\<close>
haftmann@30652
  1264
wenzelm@60758
  1265
text\<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1266
webertj@49962
  1267
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1268
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1269
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1270
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1271
wenzelm@60758
  1272
text\<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1273
huffman@47108
  1274
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1275
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1276
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1277
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1278
haftmann@30652
  1279
wenzelm@61799
  1280
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1281
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1282
huffman@47108
  1283
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1284
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1285
huffman@47108
  1286
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1287
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1288
wenzelm@60758
  1289
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@60758
  1290
into the literal.\<close>
haftmann@30652
  1291
haftmann@54489
  1292
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1293
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1294
haftmann@54489
  1295
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1296
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1297
haftmann@54489
  1298
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1299
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1300
haftmann@54489
  1301
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1302
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1303
haftmann@54489
  1304
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1305
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1306
haftmann@54489
  1307
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1308
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1309
wenzelm@61799
  1310
\<comment> \<open>FIXME maybe simproc\<close>
haftmann@30652
  1311
haftmann@30652
  1312
wenzelm@61799
  1313
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1314
huffman@47108
  1315
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1316
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1317
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1318
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1319
haftmann@30652
  1320
wenzelm@61799
  1321
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1322
lp15@61738
  1323
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1324
lp15@61738
  1325
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1326
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1327
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1328
lp15@61738
  1329
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1330
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1331
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1332
lp15@61738
  1333
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1334
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1335
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1336
lp15@61738
  1337
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1338
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1339
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1340
lp15@61738
  1341
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1342
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1343
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1344
lp15@61738
  1345
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1346
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1347
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1348
haftmann@30652
  1349
wenzelm@60758
  1350
subsubsection\<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1351
wenzelm@60758
  1352
text\<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1353
lp15@61738
  1354
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1355
  le_divide_eq [of "numeral w"]
haftmann@54489
  1356
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1357
lp15@61738
  1358
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1359
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1360
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1361
lp15@61738
  1362
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1363
  less_divide_eq [of "numeral w"]
haftmann@54489
  1364
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1365
lp15@61738
  1366
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1367
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1368
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1369
lp15@61738
  1370
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1371
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1372
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1373
lp15@61738
  1374
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1375
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1376
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1377
haftmann@30652
  1378
wenzelm@60758
  1379
text\<open>Not good as automatic simprules because they cause case splits.\<close>
lp15@61738
  1380
lemmas [divide_const_simps] = le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 
haftmann@30652
  1381
haftmann@30652
  1382
wenzelm@60758
  1383
subsection \<open>The divides relation\<close>
haftmann@33320
  1384
nipkow@33657
  1385
lemma zdvd_antisym_nonneg:
nipkow@33657
  1386
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  1387
  apply (simp add: dvd_def, auto)
haftmann@57512
  1388
  apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1389
  done
haftmann@33320
  1390
lp15@60162
  1391
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
haftmann@33320
  1392
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  1393
proof cases
nipkow@33657
  1394
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  1395
next
nipkow@33657
  1396
  assume "a \<noteq> 0"
wenzelm@60758
  1397
  from \<open>a dvd b\<close> obtain k where k:"b = a*k" unfolding dvd_def by blast
wenzelm@60758
  1398
  from \<open>b dvd a\<close> obtain k' where k':"a = b*k'" unfolding dvd_def by blast
haftmann@33320
  1399
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  1400
  with mult_cancel_left1[where c="a" and b="k*k'"]
wenzelm@60758
  1401
  have kk':"k*k' = 1" using \<open>a\<noteq>0\<close> by (simp add: mult.assoc)
haftmann@33320
  1402
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  1403
  thus ?thesis using k k' by auto
haftmann@33320
  1404
qed
haftmann@33320
  1405
haftmann@33320
  1406
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
lp15@60162
  1407
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1408
haftmann@33320
  1409
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@58649
  1410
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1411
haftmann@33320
  1412
lemma dvd_imp_le_int:
haftmann@33320
  1413
  fixes d i :: int
haftmann@33320
  1414
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1415
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1416
proof -
wenzelm@60758
  1417
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1418
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1419
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1420
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1421
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1422
qed
haftmann@33320
  1423
haftmann@33320
  1424
lemma zdvd_not_zless:
haftmann@33320
  1425
  fixes m n :: int
haftmann@33320
  1426
  assumes "0 < m" and "m < n"
haftmann@33320
  1427
  shows "\<not> n dvd m"
haftmann@33320
  1428
proof
haftmann@33320
  1429
  from assms have "0 < n" by auto
haftmann@33320
  1430
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1431
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1432
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1433
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1434
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1435
qed
haftmann@33320
  1436
haftmann@33320
  1437
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  1438
  shows "m dvd n"
haftmann@33320
  1439
proof-
haftmann@33320
  1440
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  1441
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@57512
  1442
    with h have False by (simp add: mult.assoc)}
haftmann@33320
  1443
  hence "n = m * h" by blast
haftmann@33320
  1444
  thus ?thesis by simp
haftmann@33320
  1445
qed
haftmann@33320
  1446
haftmann@33320
  1447
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  1448
proof -
haftmann@33320
  1449
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  1450
  proof -
haftmann@33320
  1451
    fix k
haftmann@33320
  1452
    assume A: "int y = int x * k"
wenzelm@42676
  1453
    then show "x dvd y"
wenzelm@42676
  1454
    proof (cases k)
wenzelm@42676
  1455
      case (nonneg n)
lp15@61649
  1456
      with A have "y = x * n" by (simp del: of_nat_mult add: of_nat_mult [symmetric])
haftmann@33320
  1457
      then show ?thesis ..
haftmann@33320
  1458
    next
wenzelm@42676
  1459
      case (neg n)
wenzelm@42676
  1460
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  1461
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  1462
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  1463
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  1464
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  1465
    qed
haftmann@33320
  1466
  qed
haftmann@33320
  1467
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1468
qed
haftmann@33320
  1469
wenzelm@42676
  1470
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  1471
proof
haftmann@33320
  1472
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  1473
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  1474
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  1475
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  1476
next
haftmann@33320
  1477
  assume "\<bar>x\<bar>=1"
haftmann@33320
  1478
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  1479
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  1480
qed
haftmann@33320
  1481
lp15@60162
  1482
lemma zdvd_mult_cancel1:
haftmann@33320
  1483
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  1484
proof
lp15@60162
  1485
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
wenzelm@42676
  1486
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  1487
next
haftmann@33320
  1488
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  1489
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  1490
qed
haftmann@33320
  1491
wenzelm@61944
  1492
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat \<bar>z\<bar>)"
haftmann@33320
  1493
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1494
wenzelm@61944
  1495
lemma dvd_int_iff: "(z dvd int m) = (nat \<bar>z\<bar> dvd m)"
haftmann@33320
  1496
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1497
haftmann@58650
  1498
lemma dvd_int_unfold_dvd_nat:
haftmann@58650
  1499
  "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
haftmann@58650
  1500
  unfolding dvd_int_iff [symmetric] by simp
haftmann@58650
  1501
haftmann@33320
  1502
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  1503
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1504
haftmann@33341
  1505
lemma eq_nat_nat_iff:
haftmann@33341
  1506
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1507
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1508
haftmann@33341
  1509
lemma nat_power_eq:
haftmann@33341
  1510
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1511
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1512
haftmann@33320
  1513
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  1514
  apply (cases n)
haftmann@33320
  1515
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1516
  apply (cases z)
haftmann@33320
  1517
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  1518
  done
haftmann@33320
  1519
haftmann@36749
  1520
lemma zdvd_period:
haftmann@36749
  1521
  fixes a d :: int
haftmann@36749
  1522
  assumes "a dvd d"
haftmann@36749
  1523
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  1524
proof -
haftmann@36749
  1525
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1526
  show ?thesis
wenzelm@42676
  1527
  proof
haftmann@36749
  1528
    assume "a dvd (x + t)"
haftmann@36749
  1529
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1530
    then have "x = a * l - t" by simp
wenzelm@60758
  1531
    with \<open>d = a * k\<close> show "a dvd x + c * d + t" by simp
haftmann@36749
  1532
  next
haftmann@36749
  1533
    assume "a dvd x + c * d + t"
haftmann@36749
  1534
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1535
    then have "x = a * l - c * d - t" by simp
wenzelm@60758
  1536
    with \<open>d = a * k\<close> show "a dvd (x + t)" by simp
haftmann@36749
  1537
  qed
haftmann@36749
  1538
qed
haftmann@36749
  1539
haftmann@33320
  1540
wenzelm@60758
  1541
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1542
bulwahn@46756
  1543
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756
  1544
proof (cases "a <= b")
bulwahn@46756
  1545
  case True
bulwahn@46756
  1546
  from this show ?thesis
bulwahn@46756
  1547
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1548
    case base
bulwahn@46756
  1549
    have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756
  1550
    from this show ?case by simp
bulwahn@46756
  1551
  next
bulwahn@46756
  1552
    case (step b)
bulwahn@46756
  1553
    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756
  1554
    from this step show ?case by simp
bulwahn@46756
  1555
  qed
bulwahn@46756
  1556
next
bulwahn@46756
  1557
  case False from this show ?thesis
bulwahn@46756
  1558
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1559
qed
bulwahn@46756
  1560
bulwahn@46756
  1561
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756
  1562
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1563
bulwahn@46756
  1564
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756
  1565
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1566
bulwahn@46756
  1567
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756
  1568
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1569
bulwahn@46756
  1570
wenzelm@60758
  1571
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1572
wenzelm@60758
  1573
text \<open>Constructors\<close>
huffman@47108
  1574
huffman@47108
  1575
definition Pos :: "num \<Rightarrow> int" where
huffman@47108
  1576
  [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1577
huffman@47108
  1578
definition Neg :: "num \<Rightarrow> int" where
haftmann@54489
  1579
  [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1580
huffman@47108
  1581
code_datatype "0::int" Pos Neg
huffman@47108
  1582
huffman@47108
  1583
wenzelm@60758
  1584
text \<open>Auxiliary operations\<close>
huffman@47108
  1585
huffman@47108
  1586
definition dup :: "int \<Rightarrow> int" where
huffman@47108
  1587
  [simp]: "dup k = k + k"
haftmann@26507
  1588
huffman@47108
  1589
lemma dup_code [code]:
huffman@47108
  1590
  "dup 0 = 0"
huffman@47108
  1591
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1592
  "dup (Neg n) = Neg (Num.Bit0 n)"
haftmann@54489
  1593
  unfolding Pos_def Neg_def
huffman@47108
  1594
  by (simp_all add: numeral_Bit0)
huffman@47108
  1595
huffman@47108
  1596
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108
  1597
  [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1598
huffman@47108
  1599
lemma sub_code [code]:
huffman@47108
  1600
  "sub Num.One Num.One = 0"
huffman@47108
  1601
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1602
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1603
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1604
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1605
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1606
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1607
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1608
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@54489
  1609
  apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
haftmann@54230
  1610
  apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230
  1611
  apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230
  1612
  apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230
  1613
  done
huffman@47108
  1614
wenzelm@60758
  1615
text \<open>Implementations\<close>
huffman@47108
  1616
huffman@47108
  1617
lemma one_int_code [code, code_unfold]:
huffman@47108
  1618
  "1 = Pos Num.One"
huffman@47108
  1619
  by simp
huffman@47108
  1620
huffman@47108
  1621
lemma plus_int_code [code]:
huffman@47108
  1622
  "k + 0 = (k::int)"
huffman@47108
  1623
  "0 + l = (l::int)"
huffman@47108
  1624
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1625
  "Pos m + Neg n = sub m n"
huffman@47108
  1626
  "Neg m + Pos n = sub n m"
huffman@47108
  1627
  "Neg m + Neg n = Neg (m + n)"
huffman@47108
  1628
  by simp_all
haftmann@26507
  1629
huffman@47108
  1630
lemma uminus_int_code [code]:
huffman@47108
  1631
  "uminus 0 = (0::int)"
huffman@47108
  1632
  "uminus (Pos m) = Neg m"
huffman@47108
  1633
  "uminus (Neg m) = Pos m"
huffman@47108
  1634
  by simp_all
huffman@47108
  1635
huffman@47108
  1636
lemma minus_int_code [code]:
huffman@47108
  1637
  "k - 0 = (k::int)"
huffman@47108
  1638
  "0 - l = uminus (l::int)"
huffman@47108
  1639
  "Pos m - Pos n = sub m n"
huffman@47108
  1640
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1641
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1642
  "Neg m - Neg n = sub n m"
huffman@47108
  1643
  by simp_all
huffman@47108
  1644
huffman@47108
  1645
lemma times_int_code [code]:
huffman@47108
  1646
  "k * 0 = (0::int)"
huffman@47108
  1647
  "0 * l = (0::int)"
huffman@47108
  1648
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1649
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1650
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1651
  "Neg m * Neg n = Pos (m * n)"
huffman@47108
  1652
  by simp_all
haftmann@26507
  1653
haftmann@38857
  1654
instantiation int :: equal
haftmann@26507
  1655
begin
haftmann@26507
  1656
haftmann@37767
  1657
definition
huffman@47108
  1658
  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1659
wenzelm@61169
  1660
instance
wenzelm@61169
  1661
  by standard (rule equal_int_def)
haftmann@26507
  1662
haftmann@26507
  1663
end
haftmann@26507
  1664
huffman@47108
  1665
lemma equal_int_code [code]:
huffman@47108
  1666
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1667
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1668
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1669
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1670
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1671
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1672
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1673
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1674
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1675
  by (auto simp add: equal)
haftmann@26507
  1676
huffman@47108
  1677
lemma equal_int_refl [code nbe]:
haftmann@38857
  1678
  "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108
  1679
  by (fact equal_refl)
haftmann@26507
  1680
haftmann@28562
  1681
lemma less_eq_int_code [code]:
huffman@47108
  1682
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1683
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1684
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1685
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1686
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1687
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1688
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1689
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1690
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1691
  by simp_all
haftmann@26507
  1692
haftmann@28562
  1693
lemma less_int_code [code]:
huffman@47108
  1694
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1695
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1696
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1697
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1698
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1699
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1700
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1701
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1702
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1703
  by simp_all
haftmann@25919
  1704
huffman@47108
  1705
lemma nat_code [code]:
huffman@47108
  1706
  "nat (Int.Neg k) = 0"
huffman@47108
  1707
  "nat 0 = 0"
huffman@47108
  1708
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1709
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1710
huffman@47108
  1711
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1712
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1713
  "of_int 0 = 0"
huffman@47108
  1714
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1715
  by simp_all
haftmann@25919
  1716
huffman@47108
  1717
wenzelm@60758
  1718
text \<open>Serializer setup\<close>
haftmann@25919
  1719
haftmann@52435
  1720
code_identifier
haftmann@52435
  1721
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1722
haftmann@25919
  1723
quickcheck_params [default_type = int]
haftmann@25919
  1724
huffman@47108
  1725
hide_const (open) Pos Neg sub dup
haftmann@25919
  1726
haftmann@25919
  1727
wenzelm@60758
  1728
subsection \<open>Legacy theorems\<close>
haftmann@25919
  1729
haftmann@25919
  1730
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1731
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1732
lemmas int_mult = of_nat_mult [where 'a=int]
wenzelm@45607
  1733
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1734
lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607
  1735
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919
  1736
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1737
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607
  1738
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1739
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1740
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1741
lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207
  1742
lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607
  1743
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919
  1744
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1745
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255
  1746
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255
  1747
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960
  1748
wenzelm@61799
  1749
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1750
kuncar@53652
  1751
lifting_update int.lifting
kuncar@53652
  1752
lifting_forget int.lifting
huffman@48045
  1753
wenzelm@60758
  1754
text\<open>Also the class for fields with characteristic zero.\<close>
lp15@59667
  1755
class field_char_0 = field + ring_char_0
lp15@59667
  1756
subclass (in linordered_field) field_char_0 ..
lp15@59667
  1757
haftmann@25919
  1758
end