src/HOL/Int.thy
author haftmann
Tue Nov 19 10:05:53 2013 +0100 (2013-11-19)
changeset 54489 03ff4d1e6784
parent 54249 ce00f2fef556
child 54863 82acc20ded73
permissions -rw-r--r--
eliminiated neg_numeral in favour of - (numeral _)
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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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header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
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theory Int
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imports Equiv_Relations Wellfounded Quotient FunDef
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begin
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subsection {* Definition of integers as a quotient type *}
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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 {* Integers form a commutative ring *}
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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 default (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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  "(fun_rel (op =) cr_int) (\<lambda>n. (n, 0)) int"
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  unfolding fun_rel_def 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 {* Integers are totally ordered *}
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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 default (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 \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> 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 min_max.sup_inf_distrib1)
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end
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subsection {* Ordering properties of arithmetic operations *}
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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{*Strict Monotonicity of Multiplication*}
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text{*strict, in 1st argument; proof is by induction on k>0*}
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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{*The integers form an ordered integral domain*}
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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\<Colon>int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i\<Colon>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\<Colon>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\<Colon>int) \<le> z"
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd:
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  "(w \<Colon> 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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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 {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
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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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text{*Collapse nested embeddings*}
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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:
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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{*Special cases where either operand is zero*}
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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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end
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context linordered_idom
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begin
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text{*Every @{text linordered_idom} has characteristic zero.*}
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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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end
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lemma of_int_eq_id [simp]: "of_int = id"
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proof
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  fix z show "of_int z = id z"
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    by (cases z rule: int_diff_cases, simp)
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qed
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instance int :: no_top
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  apply default
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  apply (rule_tac x="x + 1" in exI)
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  apply simp
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  done
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instance int :: no_bot
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  apply default
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  apply (rule_tac x="x - 1" in exI)
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  apply simp
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  done
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subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
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lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
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  by auto
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lemma nat_int [simp]: "nat (int n) = n"
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  by transfer simp
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lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
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  by transfer clarsimp
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corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
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by simp
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lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
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  by transfer clarsimp
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lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
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  by transfer (clarsimp, arith)
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text{*An alternative condition is @{term "0 \<le> w"} *}
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corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
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by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
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corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
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by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
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lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
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   343
  by transfer (clarsimp, arith)
haftmann@25919
   344
haftmann@25919
   345
lemma nonneg_eq_int:
haftmann@25919
   346
  fixes z :: int
huffman@44709
   347
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   348
  shows P
haftmann@25919
   349
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   350
haftmann@54223
   351
lemma nat_eq_iff:
haftmann@54223
   352
  "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   353
  by transfer (clarsimp simp add: le_imp_diff_is_add)
haftmann@54223
   354
 
haftmann@54223
   355
corollary nat_eq_iff2:
haftmann@54223
   356
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   357
  using nat_eq_iff [of w m] by auto
haftmann@54223
   358
haftmann@54223
   359
lemma nat_0 [simp]:
haftmann@54223
   360
  "nat 0 = 0"
haftmann@54223
   361
  by (simp add: nat_eq_iff)
haftmann@25919
   362
haftmann@54223
   363
lemma nat_1 [simp]:
haftmann@54223
   364
  "nat 1 = Suc 0"
haftmann@54223
   365
  by (simp add: nat_eq_iff)
haftmann@54223
   366
haftmann@54223
   367
lemma nat_numeral [simp]:
haftmann@54223
   368
  "nat (numeral k) = numeral k"
haftmann@54223
   369
  by (simp add: nat_eq_iff)
haftmann@25919
   370
haftmann@54223
   371
lemma nat_neg_numeral [simp]:
haftmann@54489
   372
  "nat (- numeral k) = 0"
haftmann@54223
   373
  by simp
haftmann@54223
   374
haftmann@54223
   375
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   376
  by simp
haftmann@54223
   377
 
haftmann@25919
   378
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045
   379
  by transfer (clarsimp, arith)
haftmann@25919
   380
huffman@44709
   381
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   382
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   383
huffman@44707
   384
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   385
  by transfer auto
huffman@44707
   386
nipkow@29700
   387
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045
   388
  by transfer clarsimp
nipkow@29700
   389
haftmann@25919
   390
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   391
by (auto simp add: nat_eq_iff2)
haftmann@25919
   392
haftmann@25919
   393
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   394
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   395
haftmann@25919
   396
lemma nat_add_distrib:
haftmann@54223
   397
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   398
  by transfer clarsimp
haftmann@25919
   399
haftmann@54223
   400
lemma nat_diff_distrib':
haftmann@54223
   401
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   402
  by transfer clarsimp
haftmann@54223
   403
 
haftmann@25919
   404
lemma nat_diff_distrib:
haftmann@54223
   405
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   406
  by (rule nat_diff_distrib') auto
haftmann@25919
   407
huffman@44709
   408
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   409
  by transfer simp
haftmann@25919
   410
haftmann@53065
   411
lemma le_nat_iff:
haftmann@53065
   412
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   413
  by transfer auto
haftmann@53065
   414
  
huffman@44709
   415
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045
   416
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   417
haftmann@25919
   418
context ring_1
haftmann@25919
   419
begin
haftmann@25919
   420
haftmann@25919
   421
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   422
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   423
haftmann@25919
   424
end
haftmann@25919
   425
haftmann@54249
   426
lemma diff_nat_numeral [simp]: 
haftmann@54249
   427
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   428
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   429
haftmann@54249
   430
krauss@29779
   431
text {* For termination proofs: *}
krauss@29779
   432
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   433
haftmann@25919
   434
haftmann@25919
   435
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   436
huffman@44709
   437
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   438
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   439
huffman@44709
   440
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   441
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   442
huffman@44709
   443
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   444
by (simp add: minus_le_iff)
haftmann@25919
   445
huffman@44709
   446
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   447
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   448
huffman@44709
   449
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   450
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   451
huffman@44709
   452
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045
   453
  by transfer simp
haftmann@25919
   454
huffman@44709
   455
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   456
by (simp add: linorder_not_less)
haftmann@25919
   457
huffman@44709
   458
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   459
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   460
huffman@44709
   461
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   462
proof -
haftmann@25919
   463
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   464
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   465
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   466
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   467
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   468
    by (simp only: algebra_simps)
haftmann@25919
   469
  finally show ?thesis .
haftmann@25919
   470
qed
haftmann@25919
   471
huffman@44709
   472
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   473
by simp
haftmann@25919
   474
huffman@44709
   475
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919
   476
by simp
haftmann@25919
   477
haftmann@25919
   478
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   479
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   480
      in theory @{text Rings}.
haftmann@25919
   481
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@54147
   482
lemma abs_split [arith_split, no_atp]:
haftmann@35028
   483
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   484
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   485
huffman@44709
   486
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045
   487
apply transfer
huffman@48045
   488
apply clarsimp
huffman@48045
   489
apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919
   490
done
haftmann@25919
   491
haftmann@25919
   492
haftmann@25919
   493
subsection {* Cases and induction *}
haftmann@25919
   494
haftmann@25919
   495
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   496
whether an integer is negative or not.*}
haftmann@25919
   497
wenzelm@42676
   498
theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709
   499
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   500
apply (cases "z < 0")
wenzelm@42676
   501
apply (blast dest!: negD)
haftmann@25919
   502
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   503
apply auto
haftmann@25919
   504
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   505
done
haftmann@25919
   506
wenzelm@42676
   507
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   508
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   509
  by (cases z) auto
haftmann@25919
   510
huffman@47207
   511
lemma nonneg_int_cases:
huffman@47207
   512
  assumes "0 \<le> k" obtains n where "k = int n"
haftmann@54489
   513
  using assms by (rule nonneg_eq_int)
huffman@47207
   514
huffman@47108
   515
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
huffman@47108
   516
  -- {* Unfold all @{text let}s involving constants *}
huffman@47108
   517
  unfolding Let_def ..
haftmann@37767
   518
haftmann@54489
   519
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
haftmann@25919
   520
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   521
  unfolding Let_def ..
haftmann@25919
   522
huffman@47108
   523
text {* Unfold @{text min} and @{text max} on numerals. *}
huffman@28958
   524
huffman@47108
   525
lemmas max_number_of [simp] =
huffman@47108
   526
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   527
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   528
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   529
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   530
huffman@47108
   531
lemmas min_number_of [simp] =
huffman@47108
   532
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   533
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   534
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   535
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   536
haftmann@25919
   537
huffman@28958
   538
subsubsection {* Binary comparisons *}
huffman@28958
   539
huffman@28958
   540
text {* Preliminaries *}
huffman@28958
   541
huffman@28958
   542
lemma even_less_0_iff:
haftmann@35028
   543
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   544
proof -
webertj@49962
   545
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
huffman@28958
   546
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   547
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   548
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   549
  finally show ?thesis .
huffman@28958
   550
qed
huffman@28958
   551
huffman@28958
   552
lemma le_imp_0_less: 
huffman@28958
   553
  assumes le: "0 \<le> z"
huffman@28958
   554
  shows "(0::int) < 1 + z"
huffman@28958
   555
proof -
huffman@28958
   556
  have "0 \<le> z" by fact
huffman@47108
   557
  also have "... < z + 1" by (rule less_add_one)
huffman@28958
   558
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   559
  finally show "0 < 1 + z" .
huffman@28958
   560
qed
huffman@28958
   561
huffman@28958
   562
lemma odd_less_0_iff:
huffman@28958
   563
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   564
proof (cases z)
huffman@28958
   565
  case (nonneg n)
huffman@28958
   566
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   567
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   568
next
huffman@28958
   569
  case (neg n)
huffman@30079
   570
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   571
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   572
qed
huffman@28958
   573
huffman@28958
   574
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
   575
haftmann@25919
   576
lemmas double_eq_0_iff = double_zero
haftmann@25919
   577
haftmann@25919
   578
lemma odd_nonzero:
haftmann@33296
   579
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
   580
proof (cases z)
haftmann@25919
   581
  case (nonneg n)
haftmann@25919
   582
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
   583
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
   584
    by (auto simp add: add_assoc) 
haftmann@25919
   585
next
haftmann@25919
   586
  case (neg n)
haftmann@25919
   587
  show ?thesis
haftmann@25919
   588
  proof
haftmann@25919
   589
    assume eq: "1 + z + z = 0"
huffman@44709
   590
    have "(0::int) < 1 + (int n + int n)"
haftmann@25919
   591
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
   592
    also have "... = - (1 + z + z)" 
haftmann@25919
   593
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
   594
    also have "... = 0" by (simp add: eq) 
haftmann@25919
   595
    finally have "0<0" ..
haftmann@25919
   596
    thus False by blast
haftmann@25919
   597
  qed
haftmann@25919
   598
qed
haftmann@25919
   599
haftmann@30652
   600
haftmann@25919
   601
subsection {* The Set of Integers *}
haftmann@25919
   602
haftmann@25919
   603
context ring_1
haftmann@25919
   604
begin
haftmann@25919
   605
haftmann@30652
   606
definition Ints  :: "'a set" where
haftmann@37767
   607
  "Ints = range of_int"
haftmann@25919
   608
haftmann@25919
   609
notation (xsymbols)
haftmann@25919
   610
  Ints  ("\<int>")
haftmann@25919
   611
huffman@35634
   612
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   613
  by (simp add: Ints_def)
huffman@35634
   614
huffman@35634
   615
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   616
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   617
haftmann@25919
   618
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   619
  using Ints_of_int [of "0"] by simp
haftmann@25919
   620
haftmann@25919
   621
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   622
  using Ints_of_int [of "1"] by simp
haftmann@25919
   623
haftmann@25919
   624
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
   625
apply (auto simp add: Ints_def)
haftmann@25919
   626
apply (rule range_eqI)
haftmann@25919
   627
apply (rule of_int_add [symmetric])
haftmann@25919
   628
done
haftmann@25919
   629
haftmann@25919
   630
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
   631
apply (auto simp add: Ints_def)
haftmann@25919
   632
apply (rule range_eqI)
haftmann@25919
   633
apply (rule of_int_minus [symmetric])
haftmann@25919
   634
done
haftmann@25919
   635
huffman@35634
   636
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
   637
apply (auto simp add: Ints_def)
huffman@35634
   638
apply (rule range_eqI)
huffman@35634
   639
apply (rule of_int_diff [symmetric])
huffman@35634
   640
done
huffman@35634
   641
haftmann@25919
   642
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
   643
apply (auto simp add: Ints_def)
haftmann@25919
   644
apply (rule range_eqI)
haftmann@25919
   645
apply (rule of_int_mult [symmetric])
haftmann@25919
   646
done
haftmann@25919
   647
huffman@35634
   648
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
   649
by (induct n) simp_all
huffman@35634
   650
haftmann@25919
   651
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   652
  assumes "q \<in> \<int>"
haftmann@25919
   653
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   654
  unfolding Ints_def
haftmann@25919
   655
proof -
haftmann@25919
   656
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   657
  then obtain z where "q = of_int z" ..
haftmann@25919
   658
  then show thesis ..
haftmann@25919
   659
qed
haftmann@25919
   660
haftmann@25919
   661
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   662
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   663
  by (rule Ints_cases) auto
haftmann@25919
   664
haftmann@25919
   665
end
haftmann@25919
   666
haftmann@25919
   667
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
   668
haftmann@25919
   669
lemma Ints_double_eq_0_iff:
haftmann@25919
   670
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   671
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
   672
proof -
haftmann@25919
   673
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   674
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   675
  show ?thesis
haftmann@25919
   676
  proof
haftmann@25919
   677
    assume "a = 0"
haftmann@25919
   678
    thus "a + a = 0" by simp
haftmann@25919
   679
  next
haftmann@25919
   680
    assume eq: "a + a = 0"
haftmann@25919
   681
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   682
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   683
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   684
    thus "a = 0" by (simp add: a)
haftmann@25919
   685
  qed
haftmann@25919
   686
qed
haftmann@25919
   687
haftmann@25919
   688
lemma Ints_odd_nonzero:
haftmann@25919
   689
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   690
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
   691
proof -
haftmann@25919
   692
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   693
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   694
  show ?thesis
haftmann@25919
   695
  proof
haftmann@25919
   696
    assume eq: "1 + a + a = 0"
haftmann@25919
   697
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   698
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   699
    with odd_nonzero show False by blast
haftmann@25919
   700
  qed
haftmann@25919
   701
qed 
haftmann@25919
   702
huffman@47108
   703
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
huffman@47108
   704
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   705
haftmann@25919
   706
lemma Ints_odd_less_0: 
haftmann@25919
   707
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
   708
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
   709
proof -
haftmann@25919
   710
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   711
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   712
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
   713
    by (simp add: a)
huffman@45532
   714
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919
   715
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
   716
  finally show ?thesis .
haftmann@25919
   717
qed
haftmann@25919
   718
haftmann@25919
   719
haftmann@25919
   720
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
   721
haftmann@25919
   722
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
   723
  apply (cases "finite A")
haftmann@25919
   724
  apply (erule finite_induct, auto)
haftmann@25919
   725
  done
haftmann@25919
   726
haftmann@25919
   727
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
   728
  apply (cases "finite A")
haftmann@25919
   729
  apply (erule finite_induct, auto)
haftmann@25919
   730
  done
haftmann@25919
   731
haftmann@25919
   732
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
   733
  apply (cases "finite A")
haftmann@25919
   734
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
   735
  done
haftmann@25919
   736
haftmann@25919
   737
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
   738
  apply (cases "finite A")
haftmann@25919
   739
  apply (erule finite_induct, auto)
haftmann@25919
   740
  done
haftmann@25919
   741
haftmann@25919
   742
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
   743
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
   744
haftmann@25919
   745
haftmann@25919
   746
text {* Legacy theorems *}
haftmann@25919
   747
haftmann@25919
   748
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   749
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
huffman@47108
   750
lemmas numeral_1_eq_1 = numeral_One
haftmann@25919
   751
huffman@30802
   752
subsection {* Setting up simplification procedures *}
huffman@30802
   753
haftmann@54249
   754
lemmas of_int_simps =
haftmann@54249
   755
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   756
huffman@30802
   757
lemmas int_arith_rules =
haftmann@54249
   758
  numeral_One more_arith_simps of_nat_simps of_int_simps
huffman@30802
   759
wenzelm@48891
   760
ML_file "Tools/int_arith.ML"
haftmann@30496
   761
declaration {* K Int_Arith.setup *}
haftmann@25919
   762
huffman@47108
   763
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
huffman@47108
   764
  "(m::'a::linordered_idom) <= n" |
huffman@47108
   765
  "(m::'a::linordered_idom) = n") =
wenzelm@43595
   766
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
   767
haftmann@25919
   768
haftmann@25919
   769
subsection{*Lemmas About Small Numerals*}
haftmann@25919
   770
haftmann@25919
   771
lemma abs_power_minus_one [simp]:
huffman@47108
   772
  "abs(-1 ^ n) = (1::'a::linordered_idom)"
haftmann@25919
   773
by (simp add: power_abs)
haftmann@25919
   774
haftmann@25919
   775
haftmann@25919
   776
subsection{*More Inequality Reasoning*}
haftmann@25919
   777
haftmann@25919
   778
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
   779
by arith
haftmann@25919
   780
haftmann@25919
   781
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
   782
by arith
haftmann@25919
   783
haftmann@25919
   784
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
   785
by arith
haftmann@25919
   786
haftmann@25919
   787
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
   788
by arith
haftmann@25919
   789
haftmann@25919
   790
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
   791
by arith
haftmann@25919
   792
haftmann@25919
   793
huffman@28958
   794
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
   795
huffman@48044
   796
text{*Simplify the term @{term "w + - z"}*}
haftmann@25919
   797
haftmann@25919
   798
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
   799
apply (insert zless_nat_conj [of 1 z])
huffman@47207
   800
apply auto
haftmann@25919
   801
done
haftmann@25919
   802
haftmann@25919
   803
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
   804
      z is an integer literal.*}
huffman@47108
   805
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
   806
haftmann@25919
   807
lemma split_nat [arith_split]:
huffman@44709
   808
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
   809
  (is "?P = (?L & ?R)")
haftmann@25919
   810
proof (cases "i < 0")
haftmann@25919
   811
  case True thus ?thesis by auto
haftmann@25919
   812
next
haftmann@25919
   813
  case False
haftmann@25919
   814
  have "?P = ?L"
haftmann@25919
   815
  proof
haftmann@25919
   816
    assume ?P thus ?L using False by clarsimp
haftmann@25919
   817
  next
haftmann@25919
   818
    assume ?L thus ?P using False by simp
haftmann@25919
   819
  qed
haftmann@25919
   820
  with False show ?thesis by simp
haftmann@25919
   821
qed
haftmann@25919
   822
haftmann@25919
   823
context ring_1
haftmann@25919
   824
begin
haftmann@25919
   825
blanchet@33056
   826
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
   827
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
   828
proof (cases "k < 0")
haftmann@25919
   829
  case True then have "0 \<le> - k" by simp
haftmann@25919
   830
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
   831
  with True show ?thesis by simp
haftmann@25919
   832
next
haftmann@25919
   833
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
   834
qed
haftmann@25919
   835
haftmann@25919
   836
end
haftmann@25919
   837
haftmann@25919
   838
lemma nat_mult_distrib:
haftmann@25919
   839
  fixes z z' :: int
haftmann@25919
   840
  assumes "0 \<le> z"
haftmann@25919
   841
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
   842
proof (cases "0 \<le> z'")
haftmann@25919
   843
  case False with assms have "z * z' \<le> 0"
haftmann@25919
   844
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
   845
  then have "nat (z * z') = 0" by simp
haftmann@25919
   846
  moreover from False have "nat z' = 0" by simp
haftmann@25919
   847
  ultimately show ?thesis by simp
haftmann@25919
   848
next
haftmann@25919
   849
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
   850
  show ?thesis
haftmann@25919
   851
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
   852
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
   853
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
   854
qed
haftmann@25919
   855
haftmann@25919
   856
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
   857
apply (rule trans)
haftmann@25919
   858
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
   859
done
haftmann@25919
   860
haftmann@25919
   861
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
   862
apply (cases "z=0 | w=0")
haftmann@25919
   863
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
   864
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
   865
done
haftmann@25919
   866
huffman@47207
   867
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207
   868
apply (rule sym)
huffman@47207
   869
apply (simp add: nat_eq_iff)
huffman@47207
   870
done
huffman@47207
   871
huffman@47207
   872
lemma diff_nat_eq_if:
huffman@47207
   873
     "nat z - nat z' =  
huffman@47207
   874
        (if z' < 0 then nat z   
huffman@47207
   875
         else let d = z-z' in     
huffman@47207
   876
              if d < 0 then 0 else nat d)"
huffman@47207
   877
by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
   878
huffman@47207
   879
lemma nat_numeral_diff_1 [simp]:
huffman@47207
   880
  "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
   881
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
   882
haftmann@25919
   883
haftmann@25919
   884
subsection "Induction principles for int"
haftmann@25919
   885
haftmann@25919
   886
text{*Well-founded segments of the integers*}
haftmann@25919
   887
haftmann@25919
   888
definition
haftmann@25919
   889
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
   890
where
haftmann@25919
   891
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
   892
haftmann@25919
   893
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
   894
proof -
haftmann@25919
   895
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
   896
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
   897
  thus ?thesis 
haftmann@25919
   898
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   899
qed
haftmann@25919
   900
haftmann@25919
   901
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
   902
by RankFinder.*}
haftmann@25919
   903
haftmann@25919
   904
definition
haftmann@25919
   905
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
   906
where
haftmann@25919
   907
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
   908
haftmann@25919
   909
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
   910
proof -
haftmann@25919
   911
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
   912
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
   913
  thus ?thesis 
haftmann@25919
   914
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   915
qed
haftmann@25919
   916
haftmann@25919
   917
(* `set:int': dummy construction *)
haftmann@25919
   918
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
   919
  fixes i :: int
haftmann@25919
   920
  assumes ge: "k \<le> i" and
haftmann@25919
   921
    base: "P k" and
haftmann@25919
   922
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
   923
  shows "P i"
haftmann@25919
   924
proof -
wenzelm@42676
   925
  { fix n
wenzelm@42676
   926
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
   927
    proof (induct n)
haftmann@25919
   928
      case 0
haftmann@25919
   929
      hence "i = k" by arith
haftmann@25919
   930
      thus "P i" using base by simp
haftmann@25919
   931
    next
haftmann@25919
   932
      case (Suc n)
haftmann@25919
   933
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
   934
      moreover
haftmann@25919
   935
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
   936
      ultimately
wenzelm@42676
   937
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
   938
      from step [OF ki1 this] show ?case by simp
haftmann@25919
   939
    qed
haftmann@25919
   940
  }
haftmann@25919
   941
  with ge show ?thesis by fast
haftmann@25919
   942
qed
haftmann@25919
   943
haftmann@25928
   944
(* `set:int': dummy construction *)
haftmann@25928
   945
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
   946
  assumes gr: "k < (i::int)" and
haftmann@25919
   947
        base: "P(k+1)" and
haftmann@25919
   948
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
   949
  shows "P i"
haftmann@25919
   950
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
   951
  using gr apply arith
haftmann@25919
   952
 apply(rule base)
haftmann@25919
   953
apply (rule step, simp+)
haftmann@25919
   954
done
haftmann@25919
   955
wenzelm@42676
   956
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
   957
  assumes le: "i \<le> (k::int)" and
haftmann@25919
   958
        base: "P(k)" and
haftmann@25919
   959
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   960
  shows "P i"
haftmann@25919
   961
proof -
wenzelm@42676
   962
  { fix n
wenzelm@42676
   963
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
   964
    proof (induct n)
haftmann@25919
   965
      case 0
haftmann@25919
   966
      hence "i = k" by arith
haftmann@25919
   967
      thus "P i" using base by simp
haftmann@25919
   968
    next
haftmann@25919
   969
      case (Suc n)
wenzelm@42676
   970
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
   971
      moreover
haftmann@25919
   972
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
   973
      ultimately
wenzelm@42676
   974
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
   975
      from step[OF ki1 this] show ?case by simp
haftmann@25919
   976
    qed
haftmann@25919
   977
  }
haftmann@25919
   978
  with le show ?thesis by fast
haftmann@25919
   979
qed
haftmann@25919
   980
wenzelm@42676
   981
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
   982
  assumes less: "(i::int) < k" and
haftmann@25919
   983
        base: "P(k - 1)" and
haftmann@25919
   984
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   985
  shows "P i"
haftmann@25919
   986
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
   987
  using less apply arith
haftmann@25919
   988
 apply(rule base)
haftmann@25919
   989
apply (rule step, simp+)
haftmann@25919
   990
done
haftmann@25919
   991
haftmann@36811
   992
theorem int_induct [case_names base step1 step2]:
haftmann@36801
   993
  fixes k :: int
haftmann@36801
   994
  assumes base: "P k"
haftmann@36801
   995
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
   996
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
   997
  shows "P i"
haftmann@36801
   998
proof -
haftmann@36801
   999
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1000
  then show ?thesis
wenzelm@42676
  1001
  proof
wenzelm@42676
  1002
    assume "i \<ge> k"
wenzelm@42676
  1003
    then show ?thesis using base
haftmann@36801
  1004
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1005
  next
wenzelm@42676
  1006
    assume "i \<le> k"
wenzelm@42676
  1007
    then show ?thesis using base
haftmann@36801
  1008
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1009
  qed
haftmann@36801
  1010
qed
haftmann@36801
  1011
haftmann@25919
  1012
subsection{*Intermediate value theorems*}
haftmann@25919
  1013
haftmann@25919
  1014
lemma int_val_lemma:
haftmann@25919
  1015
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1016
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1017
unfolding One_nat_def
wenzelm@42676
  1018
apply (induct n)
wenzelm@42676
  1019
apply simp
haftmann@25919
  1020
apply (intro strip)
haftmann@25919
  1021
apply (erule impE, simp)
haftmann@25919
  1022
apply (erule_tac x = n in allE, simp)
huffman@30079
  1023
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1024
apply force
haftmann@25919
  1025
apply (erule impE)
haftmann@25919
  1026
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1027
apply (blast intro: le_SucI)
haftmann@25919
  1028
done
haftmann@25919
  1029
haftmann@25919
  1030
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1031
haftmann@25919
  1032
lemma nat_intermed_int_val:
haftmann@25919
  1033
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1034
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1035
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1036
       in int_val_lemma)
huffman@30079
  1037
unfolding One_nat_def
haftmann@25919
  1038
apply simp
haftmann@25919
  1039
apply (erule exE)
haftmann@25919
  1040
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1041
done
haftmann@25919
  1042
haftmann@25919
  1043
haftmann@25919
  1044
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1045
haftmann@25919
  1046
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1047
by arith
haftmann@25919
  1048
paulson@34055
  1049
lemma abs_zmult_eq_1:
paulson@34055
  1050
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1051
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1052
proof -
paulson@34055
  1053
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1054
    by auto
paulson@34055
  1055
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1056
  proof
paulson@34055
  1057
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1058
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1059
      by (simp add: mult_mono 0) 
paulson@34055
  1060
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1061
      by (simp add: abs_mult)
paulson@34055
  1062
    also have "... = 1"
paulson@34055
  1063
      by (simp add: mn)
paulson@34055
  1064
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1065
    thus "False" using 0
huffman@47108
  1066
      by arith
paulson@34055
  1067
  qed
paulson@34055
  1068
  thus ?thesis using 0
paulson@34055
  1069
    by auto
paulson@34055
  1070
qed
haftmann@25919
  1071
haftmann@25919
  1072
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1073
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1074
boehmes@35815
  1075
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1076
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1077
proof -
boehmes@35815
  1078
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1079
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1080
qed
haftmann@25919
  1081
haftmann@25919
  1082
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1083
apply (rule iffI) 
haftmann@25919
  1084
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1085
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1086
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1087
done
haftmann@25919
  1088
haftmann@33296
  1089
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1090
proof
haftmann@33296
  1091
  assume "finite (UNIV::int set)"
haftmann@33296
  1092
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1093
    by (rule injI) simp
haftmann@33296
  1094
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1095
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1096
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1097
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1098
qed
haftmann@25919
  1099
haftmann@25919
  1100
haftmann@30652
  1101
subsection {* Further theorems on numerals *}
haftmann@30652
  1102
haftmann@30652
  1103
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1104
haftmann@30652
  1105
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1106
webertj@49962
  1107
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1108
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1109
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1110
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1111
haftmann@30652
  1112
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1113
huffman@47108
  1114
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1115
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1116
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1117
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1118
haftmann@30652
  1119
haftmann@30652
  1120
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1121
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1122
huffman@47108
  1123
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1124
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1125
huffman@47108
  1126
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1127
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1128
haftmann@30652
  1129
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1130
into the literal.*}
haftmann@30652
  1131
haftmann@54489
  1132
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1133
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1134
haftmann@54489
  1135
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1136
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1137
haftmann@54489
  1138
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1139
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1140
haftmann@54489
  1141
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1142
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1143
haftmann@54489
  1144
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1145
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1146
haftmann@54489
  1147
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1148
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1149
haftmann@54489
  1150
-- {* FIXME maybe simproc *}
haftmann@30652
  1151
haftmann@30652
  1152
haftmann@30652
  1153
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  1154
huffman@47108
  1155
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1156
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1157
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1158
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1159
haftmann@30652
  1160
haftmann@30652
  1161
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  1162
huffman@47108
  1163
lemmas le_divide_eq_numeral1 [simp] =
huffman@47108
  1164
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1165
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1166
huffman@47108
  1167
lemmas divide_le_eq_numeral1 [simp] =
huffman@47108
  1168
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1169
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1170
huffman@47108
  1171
lemmas less_divide_eq_numeral1 [simp] =
huffman@47108
  1172
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1173
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1174
huffman@47108
  1175
lemmas divide_less_eq_numeral1 [simp] =
huffman@47108
  1176
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1177
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1178
huffman@47108
  1179
lemmas eq_divide_eq_numeral1 [simp] =
huffman@47108
  1180
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1181
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1182
huffman@47108
  1183
lemmas divide_eq_eq_numeral1 [simp] =
huffman@47108
  1184
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1185
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1186
haftmann@30652
  1187
haftmann@30652
  1188
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  1189
haftmann@30652
  1190
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  1191
huffman@47108
  1192
lemmas le_divide_eq_numeral =
huffman@47108
  1193
  le_divide_eq [of "numeral w"]
haftmann@54489
  1194
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1195
huffman@47108
  1196
lemmas divide_le_eq_numeral =
huffman@47108
  1197
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1198
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1199
huffman@47108
  1200
lemmas less_divide_eq_numeral =
huffman@47108
  1201
  less_divide_eq [of "numeral w"]
haftmann@54489
  1202
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1203
huffman@47108
  1204
lemmas divide_less_eq_numeral =
huffman@47108
  1205
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1206
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1207
huffman@47108
  1208
lemmas eq_divide_eq_numeral =
huffman@47108
  1209
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1210
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1211
huffman@47108
  1212
lemmas divide_eq_eq_numeral =
huffman@47108
  1213
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1214
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1215
haftmann@30652
  1216
haftmann@30652
  1217
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  1218
lemmas divide_const_simps =
huffman@47108
  1219
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
huffman@47108
  1220
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
haftmann@30652
  1221
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1222
haftmann@30652
  1223
text{*Division By @{text "-1"}*}
haftmann@30652
  1224
huffman@47108
  1225
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
huffman@47108
  1226
  unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
huffman@47108
  1227
  by simp
haftmann@30652
  1228
huffman@47108
  1229
lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
haftmann@54489
  1230
  by (fact divide_minus_left)
haftmann@30652
  1231
haftmann@30652
  1232
lemma half_gt_zero_iff:
haftmann@54489
  1233
  "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
haftmann@54489
  1234
  by auto
haftmann@30652
  1235
wenzelm@45607
  1236
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
haftmann@30652
  1237
huffman@47108
  1238
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
haftmann@54489
  1239
  by (fact divide_numeral_1)
haftmann@36719
  1240
haftmann@30652
  1241
haftmann@33320
  1242
subsection {* The divides relation *}
haftmann@33320
  1243
nipkow@33657
  1244
lemma zdvd_antisym_nonneg:
nipkow@33657
  1245
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  1246
  apply (simp add: dvd_def, auto)
nipkow@33657
  1247
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1248
  done
haftmann@33320
  1249
nipkow@33657
  1250
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  1251
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  1252
proof cases
nipkow@33657
  1253
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  1254
next
nipkow@33657
  1255
  assume "a \<noteq> 0"
haftmann@33320
  1256
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  1257
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  1258
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  1259
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@33320
  1260
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
haftmann@33320
  1261
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  1262
  thus ?thesis using k k' by auto
haftmann@33320
  1263
qed
haftmann@33320
  1264
haftmann@33320
  1265
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320
  1266
  apply (subgoal_tac "m = n + (m - n)")
haftmann@33320
  1267
   apply (erule ssubst)
haftmann@33320
  1268
   apply (blast intro: dvd_add, simp)
haftmann@33320
  1269
  done
haftmann@33320
  1270
haftmann@33320
  1271
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320
  1272
apply (rule iffI)
haftmann@33320
  1273
 apply (erule_tac [2] dvd_add)
haftmann@33320
  1274
 apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320
  1275
  apply (erule ssubst)
haftmann@33320
  1276
  apply (erule dvd_diff)
haftmann@33320
  1277
  apply(simp_all)
haftmann@33320
  1278
done
haftmann@33320
  1279
haftmann@33320
  1280
lemma dvd_imp_le_int:
haftmann@33320
  1281
  fixes d i :: int
haftmann@33320
  1282
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1283
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1284
proof -
haftmann@33320
  1285
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  1286
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  1287
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1288
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  1289
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  1290
qed
haftmann@33320
  1291
haftmann@33320
  1292
lemma zdvd_not_zless:
haftmann@33320
  1293
  fixes m n :: int
haftmann@33320
  1294
  assumes "0 < m" and "m < n"
haftmann@33320
  1295
  shows "\<not> n dvd m"
haftmann@33320
  1296
proof
haftmann@33320
  1297
  from assms have "0 < n" by auto
haftmann@33320
  1298
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  1299
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  1300
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  1301
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  1302
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1303
qed
haftmann@33320
  1304
haftmann@33320
  1305
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  1306
  shows "m dvd n"
haftmann@33320
  1307
proof-
haftmann@33320
  1308
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  1309
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@33320
  1310
    with h have False by (simp add: mult_assoc)}
haftmann@33320
  1311
  hence "n = m * h" by blast
haftmann@33320
  1312
  thus ?thesis by simp
haftmann@33320
  1313
qed
haftmann@33320
  1314
haftmann@33320
  1315
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  1316
proof -
haftmann@33320
  1317
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  1318
  proof -
haftmann@33320
  1319
    fix k
haftmann@33320
  1320
    assume A: "int y = int x * k"
wenzelm@42676
  1321
    then show "x dvd y"
wenzelm@42676
  1322
    proof (cases k)
wenzelm@42676
  1323
      case (nonneg n)
wenzelm@42676
  1324
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  1325
      then show ?thesis ..
haftmann@33320
  1326
    next
wenzelm@42676
  1327
      case (neg n)
wenzelm@42676
  1328
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  1329
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  1330
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  1331
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  1332
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  1333
    qed
haftmann@33320
  1334
  qed
haftmann@33320
  1335
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1336
qed
haftmann@33320
  1337
wenzelm@42676
  1338
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  1339
proof
haftmann@33320
  1340
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  1341
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  1342
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  1343
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  1344
next
haftmann@33320
  1345
  assume "\<bar>x\<bar>=1"
haftmann@33320
  1346
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  1347
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  1348
qed
haftmann@33320
  1349
haftmann@33320
  1350
lemma zdvd_mult_cancel1: 
haftmann@33320
  1351
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  1352
proof
haftmann@33320
  1353
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  1354
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  1355
next
haftmann@33320
  1356
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  1357
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  1358
qed
haftmann@33320
  1359
haftmann@33320
  1360
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  1361
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1362
haftmann@33320
  1363
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  1364
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1365
haftmann@33320
  1366
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  1367
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1368
haftmann@33341
  1369
lemma eq_nat_nat_iff:
haftmann@33341
  1370
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1371
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1372
haftmann@33341
  1373
lemma nat_power_eq:
haftmann@33341
  1374
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1375
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1376
haftmann@33320
  1377
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  1378
  apply (cases n)
haftmann@33320
  1379
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1380
  apply (cases z)
haftmann@33320
  1381
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  1382
  done
haftmann@33320
  1383
haftmann@36749
  1384
lemma zdvd_period:
haftmann@36749
  1385
  fixes a d :: int
haftmann@36749
  1386
  assumes "a dvd d"
haftmann@36749
  1387
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  1388
proof -
haftmann@36749
  1389
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1390
  show ?thesis
wenzelm@42676
  1391
  proof
haftmann@36749
  1392
    assume "a dvd (x + t)"
haftmann@36749
  1393
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1394
    then have "x = a * l - t" by simp
haftmann@36749
  1395
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  1396
  next
haftmann@36749
  1397
    assume "a dvd x + c * d + t"
haftmann@36749
  1398
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1399
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  1400
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  1401
  qed
haftmann@36749
  1402
qed
haftmann@36749
  1403
haftmann@33320
  1404
bulwahn@46756
  1405
subsection {* Finiteness of intervals *}
bulwahn@46756
  1406
bulwahn@46756
  1407
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756
  1408
proof (cases "a <= b")
bulwahn@46756
  1409
  case True
bulwahn@46756
  1410
  from this show ?thesis
bulwahn@46756
  1411
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1412
    case base
bulwahn@46756
  1413
    have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756
  1414
    from this show ?case by simp
bulwahn@46756
  1415
  next
bulwahn@46756
  1416
    case (step b)
bulwahn@46756
  1417
    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756
  1418
    from this step show ?case by simp
bulwahn@46756
  1419
  qed
bulwahn@46756
  1420
next
bulwahn@46756
  1421
  case False from this show ?thesis
bulwahn@46756
  1422
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1423
qed
bulwahn@46756
  1424
bulwahn@46756
  1425
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756
  1426
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1427
bulwahn@46756
  1428
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756
  1429
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1430
bulwahn@46756
  1431
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756
  1432
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1433
bulwahn@46756
  1434
haftmann@25919
  1435
subsection {* Configuration of the code generator *}
haftmann@25919
  1436
huffman@47108
  1437
text {* Constructors *}
huffman@47108
  1438
huffman@47108
  1439
definition Pos :: "num \<Rightarrow> int" where
huffman@47108
  1440
  [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1441
huffman@47108
  1442
definition Neg :: "num \<Rightarrow> int" where
haftmann@54489
  1443
  [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1444
huffman@47108
  1445
code_datatype "0::int" Pos Neg
huffman@47108
  1446
huffman@47108
  1447
huffman@47108
  1448
text {* Auxiliary operations *}
huffman@47108
  1449
huffman@47108
  1450
definition dup :: "int \<Rightarrow> int" where
huffman@47108
  1451
  [simp]: "dup k = k + k"
haftmann@26507
  1452
huffman@47108
  1453
lemma dup_code [code]:
huffman@47108
  1454
  "dup 0 = 0"
huffman@47108
  1455
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1456
  "dup (Neg n) = Neg (Num.Bit0 n)"
haftmann@54489
  1457
  unfolding Pos_def Neg_def
huffman@47108
  1458
  by (simp_all add: numeral_Bit0)
huffman@47108
  1459
huffman@47108
  1460
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108
  1461
  [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1462
huffman@47108
  1463
lemma sub_code [code]:
huffman@47108
  1464
  "sub Num.One Num.One = 0"
huffman@47108
  1465
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1466
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1467
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1468
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1469
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1470
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1471
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1472
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@54489
  1473
  apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
haftmann@54230
  1474
  apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230
  1475
  apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230
  1476
  apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230
  1477
  done
huffman@47108
  1478
huffman@47108
  1479
text {* Implementations *}
huffman@47108
  1480
huffman@47108
  1481
lemma one_int_code [code, code_unfold]:
huffman@47108
  1482
  "1 = Pos Num.One"
huffman@47108
  1483
  by simp
huffman@47108
  1484
huffman@47108
  1485
lemma plus_int_code [code]:
huffman@47108
  1486
  "k + 0 = (k::int)"
huffman@47108
  1487
  "0 + l = (l::int)"
huffman@47108
  1488
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1489
  "Pos m + Neg n = sub m n"
huffman@47108
  1490
  "Neg m + Pos n = sub n m"
huffman@47108
  1491
  "Neg m + Neg n = Neg (m + n)"
huffman@47108
  1492
  by simp_all
haftmann@26507
  1493
huffman@47108
  1494
lemma uminus_int_code [code]:
huffman@47108
  1495
  "uminus 0 = (0::int)"
huffman@47108
  1496
  "uminus (Pos m) = Neg m"
huffman@47108
  1497
  "uminus (Neg m) = Pos m"
huffman@47108
  1498
  by simp_all
huffman@47108
  1499
huffman@47108
  1500
lemma minus_int_code [code]:
huffman@47108
  1501
  "k - 0 = (k::int)"
huffman@47108
  1502
  "0 - l = uminus (l::int)"
huffman@47108
  1503
  "Pos m - Pos n = sub m n"
huffman@47108
  1504
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1505
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1506
  "Neg m - Neg n = sub n m"
huffman@47108
  1507
  by simp_all
huffman@47108
  1508
huffman@47108
  1509
lemma times_int_code [code]:
huffman@47108
  1510
  "k * 0 = (0::int)"
huffman@47108
  1511
  "0 * l = (0::int)"
huffman@47108
  1512
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1513
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1514
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1515
  "Neg m * Neg n = Pos (m * n)"
huffman@47108
  1516
  by simp_all
haftmann@26507
  1517
haftmann@38857
  1518
instantiation int :: equal
haftmann@26507
  1519
begin
haftmann@26507
  1520
haftmann@37767
  1521
definition
huffman@47108
  1522
  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1523
huffman@47108
  1524
instance by default (rule equal_int_def)
haftmann@26507
  1525
haftmann@26507
  1526
end
haftmann@26507
  1527
huffman@47108
  1528
lemma equal_int_code [code]:
huffman@47108
  1529
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1530
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1531
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1532
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1533
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1534
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1535
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1536
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1537
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1538
  by (auto simp add: equal)
haftmann@26507
  1539
huffman@47108
  1540
lemma equal_int_refl [code nbe]:
haftmann@38857
  1541
  "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108
  1542
  by (fact equal_refl)
haftmann@26507
  1543
haftmann@28562
  1544
lemma less_eq_int_code [code]:
huffman@47108
  1545
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1546
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1547
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1548
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1549
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1550
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1551
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1552
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1553
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1554
  by simp_all
haftmann@26507
  1555
haftmann@28562
  1556
lemma less_int_code [code]:
huffman@47108
  1557
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1558
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1559
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1560
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1561
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1562
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1563
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1564
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1565
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1566
  by simp_all
haftmann@25919
  1567
huffman@47108
  1568
lemma nat_code [code]:
huffman@47108
  1569
  "nat (Int.Neg k) = 0"
huffman@47108
  1570
  "nat 0 = 0"
huffman@47108
  1571
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1572
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1573
huffman@47108
  1574
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1575
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1576
  "of_int 0 = 0"
huffman@47108
  1577
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1578
  by simp_all
haftmann@25919
  1579
huffman@47108
  1580
huffman@47108
  1581
text {* Serializer setup *}
haftmann@25919
  1582
haftmann@52435
  1583
code_identifier
haftmann@52435
  1584
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1585
haftmann@25919
  1586
quickcheck_params [default_type = int]
haftmann@25919
  1587
huffman@47108
  1588
hide_const (open) Pos Neg sub dup
haftmann@25919
  1589
haftmann@25919
  1590
haftmann@25919
  1591
subsection {* Legacy theorems *}
haftmann@25919
  1592
haftmann@25919
  1593
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1594
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1595
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  1596
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
wenzelm@45607
  1597
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1598
lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607
  1599
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919
  1600
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1601
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607
  1602
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1603
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1604
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1605
lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207
  1606
lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607
  1607
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919
  1608
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1609
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255
  1610
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255
  1611
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960
  1612
haftmann@31015
  1613
lemma zpower_zpower:
haftmann@31015
  1614
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  1615
  by (rule power_mult [symmetric])
haftmann@31015
  1616
haftmann@31015
  1617
lemma int_power:
haftmann@31015
  1618
  "int (m ^ n) = int m ^ n"
haftmann@54489
  1619
  by (fact of_nat_power)
haftmann@31015
  1620
haftmann@31015
  1621
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  1622
huffman@48045
  1623
text {* De-register @{text "int"} as a quotient type: *}
huffman@48045
  1624
kuncar@53652
  1625
lifting_update int.lifting
kuncar@53652
  1626
lifting_forget int.lifting
huffman@48045
  1627
haftmann@25919
  1628
end