src/HOL/Int.thy
author haftmann
Fri Nov 01 18:51:14 2013 +0100 (2013-11-01)
changeset 54230 b1d955791529
parent 54223 85705ba18add
child 54249 ce00f2fef556
permissions -rw-r--r--
more simplification rules on unary and binary minus
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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 [simp, code_post]: "of_int (neg_numeral k) = neg_numeral k"
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  unfolding neg_numeral_def neg_numeral_class.neg_numeral_def
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  by (simp only: of_int_minus of_int_numeral)
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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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   343
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
huffman@48045
   344
  by transfer (clarsimp, arith)
haftmann@25919
   345
haftmann@25919
   346
lemma nonneg_eq_int:
haftmann@25919
   347
  fixes z :: int
huffman@44709
   348
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   349
  shows P
haftmann@25919
   350
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   351
haftmann@54223
   352
lemma nat_eq_iff:
haftmann@54223
   353
  "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   354
  by transfer (clarsimp simp add: le_imp_diff_is_add)
haftmann@54223
   355
 
haftmann@54223
   356
corollary nat_eq_iff2:
haftmann@54223
   357
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   358
  using nat_eq_iff [of w m] by auto
haftmann@54223
   359
haftmann@54223
   360
lemma nat_0 [simp]:
haftmann@54223
   361
  "nat 0 = 0"
haftmann@54223
   362
  by (simp add: nat_eq_iff)
haftmann@25919
   363
haftmann@54223
   364
lemma nat_1 [simp]:
haftmann@54223
   365
  "nat 1 = Suc 0"
haftmann@54223
   366
  by (simp add: nat_eq_iff)
haftmann@54223
   367
haftmann@54223
   368
lemma nat_numeral [simp]:
haftmann@54223
   369
  "nat (numeral k) = numeral k"
haftmann@54223
   370
  by (simp add: nat_eq_iff)
haftmann@25919
   371
haftmann@54223
   372
lemma nat_neg_numeral [simp]:
haftmann@54223
   373
  "nat (neg_numeral k) = 0"
haftmann@54223
   374
  by simp
haftmann@54223
   375
haftmann@54223
   376
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   377
  by simp
haftmann@54223
   378
 
haftmann@25919
   379
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045
   380
  by transfer (clarsimp, arith)
haftmann@25919
   381
huffman@44709
   382
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   383
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   384
huffman@44707
   385
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   386
  by transfer auto
huffman@44707
   387
nipkow@29700
   388
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045
   389
  by transfer clarsimp
nipkow@29700
   390
haftmann@25919
   391
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   392
by (auto simp add: nat_eq_iff2)
haftmann@25919
   393
haftmann@25919
   394
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   395
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   396
haftmann@25919
   397
lemma nat_add_distrib:
haftmann@54223
   398
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   399
  by transfer clarsimp
haftmann@25919
   400
haftmann@54223
   401
lemma nat_diff_distrib':
haftmann@54223
   402
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   403
  by transfer clarsimp
haftmann@54223
   404
 
haftmann@25919
   405
lemma nat_diff_distrib:
haftmann@54223
   406
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   407
  by (rule nat_diff_distrib') auto
haftmann@25919
   408
huffman@44709
   409
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   410
  by transfer simp
haftmann@25919
   411
haftmann@53065
   412
lemma le_nat_iff:
haftmann@53065
   413
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   414
  by transfer auto
haftmann@53065
   415
  
huffman@44709
   416
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045
   417
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   418
haftmann@25919
   419
context ring_1
haftmann@25919
   420
begin
haftmann@25919
   421
haftmann@25919
   422
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   423
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   424
haftmann@25919
   425
end
haftmann@25919
   426
krauss@29779
   427
text {* For termination proofs: *}
krauss@29779
   428
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   429
haftmann@25919
   430
haftmann@25919
   431
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   432
huffman@44709
   433
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   434
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   435
huffman@44709
   436
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   437
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   438
huffman@44709
   439
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   440
by (simp add: minus_le_iff)
haftmann@25919
   441
huffman@44709
   442
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   443
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   444
huffman@44709
   445
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   446
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   447
huffman@44709
   448
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045
   449
  by transfer simp
haftmann@25919
   450
huffman@44709
   451
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   452
by (simp add: linorder_not_less)
haftmann@25919
   453
huffman@44709
   454
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   455
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   456
huffman@44709
   457
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   458
proof -
haftmann@25919
   459
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   460
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   461
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   462
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   463
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   464
    by (simp only: algebra_simps)
haftmann@25919
   465
  finally show ?thesis .
haftmann@25919
   466
qed
haftmann@25919
   467
huffman@44709
   468
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   469
by simp
haftmann@25919
   470
huffman@44709
   471
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919
   472
by simp
haftmann@25919
   473
haftmann@25919
   474
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   475
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   476
      in theory @{text Rings}.
haftmann@25919
   477
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@54147
   478
lemma abs_split [arith_split, no_atp]:
haftmann@35028
   479
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   480
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   481
huffman@44709
   482
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045
   483
apply transfer
huffman@48045
   484
apply clarsimp
huffman@48045
   485
apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919
   486
done
haftmann@25919
   487
haftmann@25919
   488
haftmann@25919
   489
subsection {* Cases and induction *}
haftmann@25919
   490
haftmann@25919
   491
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   492
whether an integer is negative or not.*}
haftmann@25919
   493
wenzelm@42676
   494
theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709
   495
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   496
apply (cases "z < 0")
wenzelm@42676
   497
apply (blast dest!: negD)
haftmann@25919
   498
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   499
apply auto
haftmann@25919
   500
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   501
done
haftmann@25919
   502
wenzelm@42676
   503
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   504
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   505
  by (cases z) auto
haftmann@25919
   506
huffman@47207
   507
lemma nonneg_int_cases:
huffman@47207
   508
  assumes "0 \<le> k" obtains n where "k = int n"
huffman@47207
   509
  using assms by (cases k, simp, simp del: of_nat_Suc)
huffman@47207
   510
huffman@47108
   511
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
huffman@47108
   512
  -- {* Unfold all @{text let}s involving constants *}
huffman@47108
   513
  unfolding Let_def ..
haftmann@37767
   514
huffman@47108
   515
lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
haftmann@25919
   516
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   517
  unfolding Let_def ..
haftmann@25919
   518
huffman@47108
   519
text {* Unfold @{text min} and @{text max} on numerals. *}
huffman@28958
   520
huffman@47108
   521
lemmas max_number_of [simp] =
huffman@47108
   522
  max_def [of "numeral u" "numeral v"]
huffman@47108
   523
  max_def [of "numeral u" "neg_numeral v"]
huffman@47108
   524
  max_def [of "neg_numeral u" "numeral v"]
huffman@47108
   525
  max_def [of "neg_numeral u" "neg_numeral v"] for u v
huffman@28958
   526
huffman@47108
   527
lemmas min_number_of [simp] =
huffman@47108
   528
  min_def [of "numeral u" "numeral v"]
huffman@47108
   529
  min_def [of "numeral u" "neg_numeral v"]
huffman@47108
   530
  min_def [of "neg_numeral u" "numeral v"]
huffman@47108
   531
  min_def [of "neg_numeral u" "neg_numeral v"] for u v
huffman@26075
   532
haftmann@25919
   533
huffman@28958
   534
subsubsection {* Binary comparisons *}
huffman@28958
   535
huffman@28958
   536
text {* Preliminaries *}
huffman@28958
   537
huffman@28958
   538
lemma even_less_0_iff:
haftmann@35028
   539
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   540
proof -
webertj@49962
   541
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
huffman@28958
   542
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   543
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   544
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   545
  finally show ?thesis .
huffman@28958
   546
qed
huffman@28958
   547
huffman@28958
   548
lemma le_imp_0_less: 
huffman@28958
   549
  assumes le: "0 \<le> z"
huffman@28958
   550
  shows "(0::int) < 1 + z"
huffman@28958
   551
proof -
huffman@28958
   552
  have "0 \<le> z" by fact
huffman@47108
   553
  also have "... < z + 1" by (rule less_add_one)
huffman@28958
   554
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   555
  finally show "0 < 1 + z" .
huffman@28958
   556
qed
huffman@28958
   557
huffman@28958
   558
lemma odd_less_0_iff:
huffman@28958
   559
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   560
proof (cases z)
huffman@28958
   561
  case (nonneg n)
huffman@28958
   562
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   563
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   564
next
huffman@28958
   565
  case (neg n)
huffman@30079
   566
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   567
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   568
qed
huffman@28958
   569
huffman@28958
   570
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
   571
haftmann@25919
   572
lemmas double_eq_0_iff = double_zero
haftmann@25919
   573
haftmann@25919
   574
lemma odd_nonzero:
haftmann@33296
   575
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
   576
proof (cases z)
haftmann@25919
   577
  case (nonneg n)
haftmann@25919
   578
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
   579
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
   580
    by (auto simp add: add_assoc) 
haftmann@25919
   581
next
haftmann@25919
   582
  case (neg n)
haftmann@25919
   583
  show ?thesis
haftmann@25919
   584
  proof
haftmann@25919
   585
    assume eq: "1 + z + z = 0"
huffman@44709
   586
    have "(0::int) < 1 + (int n + int n)"
haftmann@25919
   587
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
   588
    also have "... = - (1 + z + z)" 
haftmann@25919
   589
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
   590
    also have "... = 0" by (simp add: eq) 
haftmann@25919
   591
    finally have "0<0" ..
haftmann@25919
   592
    thus False by blast
haftmann@25919
   593
  qed
haftmann@25919
   594
qed
haftmann@25919
   595
haftmann@30652
   596
haftmann@25919
   597
subsection {* The Set of Integers *}
haftmann@25919
   598
haftmann@25919
   599
context ring_1
haftmann@25919
   600
begin
haftmann@25919
   601
haftmann@30652
   602
definition Ints  :: "'a set" where
haftmann@37767
   603
  "Ints = range of_int"
haftmann@25919
   604
haftmann@25919
   605
notation (xsymbols)
haftmann@25919
   606
  Ints  ("\<int>")
haftmann@25919
   607
huffman@35634
   608
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   609
  by (simp add: Ints_def)
huffman@35634
   610
huffman@35634
   611
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   612
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   613
haftmann@25919
   614
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   615
  using Ints_of_int [of "0"] by simp
haftmann@25919
   616
haftmann@25919
   617
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   618
  using Ints_of_int [of "1"] by simp
haftmann@25919
   619
haftmann@25919
   620
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
   621
apply (auto simp add: Ints_def)
haftmann@25919
   622
apply (rule range_eqI)
haftmann@25919
   623
apply (rule of_int_add [symmetric])
haftmann@25919
   624
done
haftmann@25919
   625
haftmann@25919
   626
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
   627
apply (auto simp add: Ints_def)
haftmann@25919
   628
apply (rule range_eqI)
haftmann@25919
   629
apply (rule of_int_minus [symmetric])
haftmann@25919
   630
done
haftmann@25919
   631
huffman@35634
   632
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
   633
apply (auto simp add: Ints_def)
huffman@35634
   634
apply (rule range_eqI)
huffman@35634
   635
apply (rule of_int_diff [symmetric])
huffman@35634
   636
done
huffman@35634
   637
haftmann@25919
   638
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
   639
apply (auto simp add: Ints_def)
haftmann@25919
   640
apply (rule range_eqI)
haftmann@25919
   641
apply (rule of_int_mult [symmetric])
haftmann@25919
   642
done
haftmann@25919
   643
huffman@35634
   644
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
   645
by (induct n) simp_all
huffman@35634
   646
haftmann@25919
   647
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   648
  assumes "q \<in> \<int>"
haftmann@25919
   649
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   650
  unfolding Ints_def
haftmann@25919
   651
proof -
haftmann@25919
   652
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   653
  then obtain z where "q = of_int z" ..
haftmann@25919
   654
  then show thesis ..
haftmann@25919
   655
qed
haftmann@25919
   656
haftmann@25919
   657
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   658
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   659
  by (rule Ints_cases) auto
haftmann@25919
   660
haftmann@25919
   661
end
haftmann@25919
   662
haftmann@25919
   663
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
   664
haftmann@25919
   665
lemma Ints_double_eq_0_iff:
haftmann@25919
   666
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   667
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
   668
proof -
haftmann@25919
   669
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   670
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   671
  show ?thesis
haftmann@25919
   672
  proof
haftmann@25919
   673
    assume "a = 0"
haftmann@25919
   674
    thus "a + a = 0" by simp
haftmann@25919
   675
  next
haftmann@25919
   676
    assume eq: "a + a = 0"
haftmann@25919
   677
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   678
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   679
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   680
    thus "a = 0" by (simp add: a)
haftmann@25919
   681
  qed
haftmann@25919
   682
qed
haftmann@25919
   683
haftmann@25919
   684
lemma Ints_odd_nonzero:
haftmann@25919
   685
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   686
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
   687
proof -
haftmann@25919
   688
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   689
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   690
  show ?thesis
haftmann@25919
   691
  proof
haftmann@25919
   692
    assume eq: "1 + a + a = 0"
haftmann@25919
   693
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   694
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   695
    with odd_nonzero show False by blast
haftmann@25919
   696
  qed
haftmann@25919
   697
qed 
haftmann@25919
   698
huffman@47108
   699
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
huffman@47108
   700
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   701
haftmann@25919
   702
lemma Ints_odd_less_0: 
haftmann@25919
   703
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
   704
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
   705
proof -
haftmann@25919
   706
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   707
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   708
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
   709
    by (simp add: a)
huffman@45532
   710
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919
   711
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
   712
  finally show ?thesis .
haftmann@25919
   713
qed
haftmann@25919
   714
haftmann@25919
   715
haftmann@25919
   716
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
   717
haftmann@25919
   718
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
   719
  apply (cases "finite A")
haftmann@25919
   720
  apply (erule finite_induct, auto)
haftmann@25919
   721
  done
haftmann@25919
   722
haftmann@25919
   723
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
   724
  apply (cases "finite A")
haftmann@25919
   725
  apply (erule finite_induct, auto)
haftmann@25919
   726
  done
haftmann@25919
   727
haftmann@25919
   728
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
   729
  apply (cases "finite A")
haftmann@25919
   730
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
   731
  done
haftmann@25919
   732
haftmann@25919
   733
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
   734
  apply (cases "finite A")
haftmann@25919
   735
  apply (erule finite_induct, auto)
haftmann@25919
   736
  done
haftmann@25919
   737
haftmann@25919
   738
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
   739
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
   740
haftmann@25919
   741
haftmann@25919
   742
text {* Legacy theorems *}
haftmann@25919
   743
haftmann@25919
   744
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   745
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
huffman@47108
   746
lemmas numeral_1_eq_1 = numeral_One
haftmann@25919
   747
huffman@30802
   748
subsection {* Setting up simplification procedures *}
huffman@30802
   749
huffman@30802
   750
lemmas int_arith_rules =
huffman@47108
   751
  neg_le_iff_le numeral_One
haftmann@54230
   752
  minus_zero left_minus right_minus
huffman@45219
   753
  mult_zero_left mult_zero_right mult_1_left mult_1_right
huffman@30802
   754
  mult_minus_left mult_minus_right
huffman@30802
   755
  minus_add_distrib minus_minus mult_assoc
huffman@30802
   756
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
huffman@30802
   757
  of_int_0 of_int_1 of_int_add of_int_mult
huffman@30802
   758
wenzelm@48891
   759
ML_file "Tools/int_arith.ML"
haftmann@30496
   760
declaration {* K Int_Arith.setup *}
haftmann@25919
   761
huffman@47108
   762
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
huffman@47108
   763
  "(m::'a::linordered_idom) <= n" |
huffman@47108
   764
  "(m::'a::linordered_idom) = n") =
wenzelm@43595
   765
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
   766
haftmann@25919
   767
haftmann@25919
   768
subsection{*Lemmas About Small Numerals*}
haftmann@25919
   769
haftmann@25919
   770
lemma abs_power_minus_one [simp]:
huffman@47108
   771
  "abs(-1 ^ n) = (1::'a::linordered_idom)"
haftmann@25919
   772
by (simp add: power_abs)
haftmann@25919
   773
haftmann@25919
   774
haftmann@25919
   775
subsection{*More Inequality Reasoning*}
haftmann@25919
   776
haftmann@25919
   777
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
   778
by arith
haftmann@25919
   779
haftmann@25919
   780
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
   781
by arith
haftmann@25919
   782
haftmann@25919
   783
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
   784
by arith
haftmann@25919
   785
haftmann@25919
   786
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
   787
by arith
haftmann@25919
   788
haftmann@25919
   789
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
   790
by arith
haftmann@25919
   791
haftmann@25919
   792
huffman@28958
   793
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
   794
huffman@48044
   795
text{*Simplify the term @{term "w + - z"}*}
haftmann@25919
   796
haftmann@25919
   797
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
   798
apply (insert zless_nat_conj [of 1 z])
huffman@47207
   799
apply auto
haftmann@25919
   800
done
haftmann@25919
   801
haftmann@25919
   802
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
   803
      z is an integer literal.*}
huffman@47108
   804
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
   805
haftmann@25919
   806
lemma split_nat [arith_split]:
huffman@44709
   807
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
   808
  (is "?P = (?L & ?R)")
haftmann@25919
   809
proof (cases "i < 0")
haftmann@25919
   810
  case True thus ?thesis by auto
haftmann@25919
   811
next
haftmann@25919
   812
  case False
haftmann@25919
   813
  have "?P = ?L"
haftmann@25919
   814
  proof
haftmann@25919
   815
    assume ?P thus ?L using False by clarsimp
haftmann@25919
   816
  next
haftmann@25919
   817
    assume ?L thus ?P using False by simp
haftmann@25919
   818
  qed
haftmann@25919
   819
  with False show ?thesis by simp
haftmann@25919
   820
qed
haftmann@25919
   821
haftmann@25919
   822
context ring_1
haftmann@25919
   823
begin
haftmann@25919
   824
blanchet@33056
   825
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
   826
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
   827
proof (cases "k < 0")
haftmann@25919
   828
  case True then have "0 \<le> - k" by simp
haftmann@25919
   829
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
   830
  with True show ?thesis by simp
haftmann@25919
   831
next
haftmann@25919
   832
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
   833
qed
haftmann@25919
   834
haftmann@25919
   835
end
haftmann@25919
   836
haftmann@25919
   837
lemma nat_mult_distrib:
haftmann@25919
   838
  fixes z z' :: int
haftmann@25919
   839
  assumes "0 \<le> z"
haftmann@25919
   840
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
   841
proof (cases "0 \<le> z'")
haftmann@25919
   842
  case False with assms have "z * z' \<le> 0"
haftmann@25919
   843
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
   844
  then have "nat (z * z') = 0" by simp
haftmann@25919
   845
  moreover from False have "nat z' = 0" by simp
haftmann@25919
   846
  ultimately show ?thesis by simp
haftmann@25919
   847
next
haftmann@25919
   848
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
   849
  show ?thesis
haftmann@25919
   850
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
   851
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
   852
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
   853
qed
haftmann@25919
   854
haftmann@25919
   855
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
   856
apply (rule trans)
haftmann@25919
   857
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
   858
done
haftmann@25919
   859
haftmann@25919
   860
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
   861
apply (cases "z=0 | w=0")
haftmann@25919
   862
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
   863
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
   864
done
haftmann@25919
   865
huffman@47207
   866
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207
   867
apply (rule sym)
huffman@47207
   868
apply (simp add: nat_eq_iff)
huffman@47207
   869
done
huffman@47207
   870
huffman@47207
   871
lemma diff_nat_eq_if:
huffman@47207
   872
     "nat z - nat z' =  
huffman@47207
   873
        (if z' < 0 then nat z   
huffman@47207
   874
         else let d = z-z' in     
huffman@47207
   875
              if d < 0 then 0 else nat d)"
huffman@47207
   876
by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
   877
huffman@47207
   878
lemma diff_nat_numeral [simp]: 
huffman@47207
   879
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
huffman@47207
   880
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
huffman@47207
   881
huffman@47207
   882
lemma nat_numeral_diff_1 [simp]:
huffman@47207
   883
  "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
   884
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
   885
huffman@47255
   886
lemmas nat_arith = diff_nat_numeral
huffman@47255
   887
haftmann@25919
   888
haftmann@25919
   889
subsection "Induction principles for int"
haftmann@25919
   890
haftmann@25919
   891
text{*Well-founded segments of the integers*}
haftmann@25919
   892
haftmann@25919
   893
definition
haftmann@25919
   894
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
   895
where
haftmann@25919
   896
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
   897
haftmann@25919
   898
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
   899
proof -
haftmann@25919
   900
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
   901
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
   902
  thus ?thesis 
haftmann@25919
   903
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   904
qed
haftmann@25919
   905
haftmann@25919
   906
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
   907
by RankFinder.*}
haftmann@25919
   908
haftmann@25919
   909
definition
haftmann@25919
   910
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
   911
where
haftmann@25919
   912
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
   913
haftmann@25919
   914
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
   915
proof -
haftmann@25919
   916
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
   917
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
   918
  thus ?thesis 
haftmann@25919
   919
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   920
qed
haftmann@25919
   921
haftmann@25919
   922
(* `set:int': dummy construction *)
haftmann@25919
   923
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
   924
  fixes i :: int
haftmann@25919
   925
  assumes ge: "k \<le> i" and
haftmann@25919
   926
    base: "P k" and
haftmann@25919
   927
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
   928
  shows "P i"
haftmann@25919
   929
proof -
wenzelm@42676
   930
  { fix n
wenzelm@42676
   931
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
   932
    proof (induct n)
haftmann@25919
   933
      case 0
haftmann@25919
   934
      hence "i = k" by arith
haftmann@25919
   935
      thus "P i" using base by simp
haftmann@25919
   936
    next
haftmann@25919
   937
      case (Suc n)
haftmann@25919
   938
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
   939
      moreover
haftmann@25919
   940
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
   941
      ultimately
wenzelm@42676
   942
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
   943
      from step [OF ki1 this] show ?case by simp
haftmann@25919
   944
    qed
haftmann@25919
   945
  }
haftmann@25919
   946
  with ge show ?thesis by fast
haftmann@25919
   947
qed
haftmann@25919
   948
haftmann@25928
   949
(* `set:int': dummy construction *)
haftmann@25928
   950
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
   951
  assumes gr: "k < (i::int)" and
haftmann@25919
   952
        base: "P(k+1)" and
haftmann@25919
   953
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
   954
  shows "P i"
haftmann@25919
   955
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
   956
  using gr apply arith
haftmann@25919
   957
 apply(rule base)
haftmann@25919
   958
apply (rule step, simp+)
haftmann@25919
   959
done
haftmann@25919
   960
wenzelm@42676
   961
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
   962
  assumes le: "i \<le> (k::int)" and
haftmann@25919
   963
        base: "P(k)" and
haftmann@25919
   964
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   965
  shows "P i"
haftmann@25919
   966
proof -
wenzelm@42676
   967
  { fix n
wenzelm@42676
   968
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
   969
    proof (induct n)
haftmann@25919
   970
      case 0
haftmann@25919
   971
      hence "i = k" by arith
haftmann@25919
   972
      thus "P i" using base by simp
haftmann@25919
   973
    next
haftmann@25919
   974
      case (Suc n)
wenzelm@42676
   975
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
   976
      moreover
haftmann@25919
   977
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
   978
      ultimately
wenzelm@42676
   979
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
   980
      from step[OF ki1 this] show ?case by simp
haftmann@25919
   981
    qed
haftmann@25919
   982
  }
haftmann@25919
   983
  with le show ?thesis by fast
haftmann@25919
   984
qed
haftmann@25919
   985
wenzelm@42676
   986
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
   987
  assumes less: "(i::int) < k" and
haftmann@25919
   988
        base: "P(k - 1)" and
haftmann@25919
   989
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   990
  shows "P i"
haftmann@25919
   991
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
   992
  using less apply arith
haftmann@25919
   993
 apply(rule base)
haftmann@25919
   994
apply (rule step, simp+)
haftmann@25919
   995
done
haftmann@25919
   996
haftmann@36811
   997
theorem int_induct [case_names base step1 step2]:
haftmann@36801
   998
  fixes k :: int
haftmann@36801
   999
  assumes base: "P k"
haftmann@36801
  1000
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1001
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1002
  shows "P i"
haftmann@36801
  1003
proof -
haftmann@36801
  1004
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1005
  then show ?thesis
wenzelm@42676
  1006
  proof
wenzelm@42676
  1007
    assume "i \<ge> k"
wenzelm@42676
  1008
    then show ?thesis using base
haftmann@36801
  1009
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1010
  next
wenzelm@42676
  1011
    assume "i \<le> k"
wenzelm@42676
  1012
    then show ?thesis using base
haftmann@36801
  1013
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1014
  qed
haftmann@36801
  1015
qed
haftmann@36801
  1016
haftmann@25919
  1017
subsection{*Intermediate value theorems*}
haftmann@25919
  1018
haftmann@25919
  1019
lemma int_val_lemma:
haftmann@25919
  1020
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1021
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1022
unfolding One_nat_def
wenzelm@42676
  1023
apply (induct n)
wenzelm@42676
  1024
apply simp
haftmann@25919
  1025
apply (intro strip)
haftmann@25919
  1026
apply (erule impE, simp)
haftmann@25919
  1027
apply (erule_tac x = n in allE, simp)
huffman@30079
  1028
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1029
apply force
haftmann@25919
  1030
apply (erule impE)
haftmann@25919
  1031
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1032
apply (blast intro: le_SucI)
haftmann@25919
  1033
done
haftmann@25919
  1034
haftmann@25919
  1035
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1036
haftmann@25919
  1037
lemma nat_intermed_int_val:
haftmann@25919
  1038
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1039
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1040
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1041
       in int_val_lemma)
huffman@30079
  1042
unfolding One_nat_def
haftmann@25919
  1043
apply simp
haftmann@25919
  1044
apply (erule exE)
haftmann@25919
  1045
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1046
done
haftmann@25919
  1047
haftmann@25919
  1048
haftmann@25919
  1049
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1050
haftmann@25919
  1051
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1052
by arith
haftmann@25919
  1053
paulson@34055
  1054
lemma abs_zmult_eq_1:
paulson@34055
  1055
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1056
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1057
proof -
paulson@34055
  1058
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1059
    by auto
paulson@34055
  1060
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1061
  proof
paulson@34055
  1062
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1063
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1064
      by (simp add: mult_mono 0) 
paulson@34055
  1065
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1066
      by (simp add: abs_mult)
paulson@34055
  1067
    also have "... = 1"
paulson@34055
  1068
      by (simp add: mn)
paulson@34055
  1069
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1070
    thus "False" using 0
huffman@47108
  1071
      by arith
paulson@34055
  1072
  qed
paulson@34055
  1073
  thus ?thesis using 0
paulson@34055
  1074
    by auto
paulson@34055
  1075
qed
haftmann@25919
  1076
huffman@47108
  1077
ML_val {* @{const_name neg_numeral} *}
huffman@47108
  1078
haftmann@25919
  1079
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1080
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1081
boehmes@35815
  1082
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1083
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1084
proof -
boehmes@35815
  1085
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1086
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1087
qed
haftmann@25919
  1088
haftmann@25919
  1089
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1090
apply (rule iffI) 
haftmann@25919
  1091
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1092
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1093
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1094
done
haftmann@25919
  1095
haftmann@33296
  1096
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1097
proof
haftmann@33296
  1098
  assume "finite (UNIV::int set)"
haftmann@33296
  1099
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1100
    by (rule injI) simp
haftmann@33296
  1101
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1102
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1103
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1104
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1105
qed
haftmann@25919
  1106
haftmann@25919
  1107
haftmann@30652
  1108
subsection {* Further theorems on numerals *}
haftmann@30652
  1109
haftmann@30652
  1110
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1111
haftmann@30652
  1112
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1113
webertj@49962
  1114
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1115
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1116
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1117
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1118
haftmann@30652
  1119
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1120
huffman@47108
  1121
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1122
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1123
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1124
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1125
haftmann@30652
  1126
haftmann@30652
  1127
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1128
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1129
huffman@47108
  1130
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1131
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1132
huffman@47108
  1133
lemmas inverse_eq_divide_neg_numeral [simp] =
huffman@47108
  1134
  inverse_eq_divide [of "neg_numeral w"] for w
haftmann@30652
  1135
haftmann@30652
  1136
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1137
into the literal.*}
haftmann@30652
  1138
huffman@47108
  1139
lemmas le_minus_iff_numeral [simp, no_atp] =
huffman@47108
  1140
  le_minus_iff [of "numeral v"]
huffman@47108
  1141
  le_minus_iff [of "neg_numeral v"] for v
huffman@47108
  1142
huffman@47108
  1143
lemmas equation_minus_iff_numeral [simp, no_atp] =
huffman@47108
  1144
  equation_minus_iff [of "numeral v"]
huffman@47108
  1145
  equation_minus_iff [of "neg_numeral v"] for v
huffman@47108
  1146
huffman@47108
  1147
lemmas minus_less_iff_numeral [simp, no_atp] =
huffman@47108
  1148
  minus_less_iff [of _ "numeral v"]
huffman@47108
  1149
  minus_less_iff [of _ "neg_numeral v"] for v
huffman@47108
  1150
huffman@47108
  1151
lemmas minus_le_iff_numeral [simp, no_atp] =
huffman@47108
  1152
  minus_le_iff [of _ "numeral v"]
huffman@47108
  1153
  minus_le_iff [of _ "neg_numeral v"] for v
huffman@47108
  1154
huffman@47108
  1155
lemmas minus_equation_iff_numeral [simp, no_atp] =
huffman@47108
  1156
  minus_equation_iff [of _ "numeral v"]
huffman@47108
  1157
  minus_equation_iff [of _ "neg_numeral v"] for v
haftmann@30652
  1158
haftmann@30652
  1159
text{*To Simplify Inequalities Where One Side is the Constant 1*}
haftmann@30652
  1160
blanchet@54147
  1161
lemma less_minus_iff_1 [simp]:
huffman@47108
  1162
  fixes b::"'b::linordered_idom"
haftmann@30652
  1163
  shows "(1 < - b) = (b < -1)"
haftmann@30652
  1164
by auto
haftmann@30652
  1165
blanchet@54147
  1166
lemma le_minus_iff_1 [simp]:
huffman@47108
  1167
  fixes b::"'b::linordered_idom"
haftmann@30652
  1168
  shows "(1 \<le> - b) = (b \<le> -1)"
haftmann@30652
  1169
by auto
haftmann@30652
  1170
blanchet@54147
  1171
lemma equation_minus_iff_1 [simp]:
huffman@47108
  1172
  fixes b::"'b::ring_1"
haftmann@30652
  1173
  shows "(1 = - b) = (b = -1)"
haftmann@30652
  1174
by (subst equation_minus_iff, auto)
haftmann@30652
  1175
blanchet@54147
  1176
lemma minus_less_iff_1 [simp]:
huffman@47108
  1177
  fixes a::"'b::linordered_idom"
haftmann@30652
  1178
  shows "(- a < 1) = (-1 < a)"
haftmann@30652
  1179
by auto
haftmann@30652
  1180
blanchet@54147
  1181
lemma minus_le_iff_1 [simp]:
huffman@47108
  1182
  fixes a::"'b::linordered_idom"
haftmann@30652
  1183
  shows "(- a \<le> 1) = (-1 \<le> a)"
haftmann@30652
  1184
by auto
haftmann@30652
  1185
blanchet@54147
  1186
lemma minus_equation_iff_1 [simp]:
huffman@47108
  1187
  fixes a::"'b::ring_1"
haftmann@30652
  1188
  shows "(- a = 1) = (a = -1)"
haftmann@30652
  1189
by (subst minus_equation_iff, auto)
haftmann@30652
  1190
haftmann@30652
  1191
haftmann@30652
  1192
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  1193
huffman@47108
  1194
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1195
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1196
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1197
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1198
haftmann@30652
  1199
haftmann@30652
  1200
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  1201
huffman@47108
  1202
lemmas le_divide_eq_numeral1 [simp] =
huffman@47108
  1203
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1204
  neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1205
huffman@47108
  1206
lemmas divide_le_eq_numeral1 [simp] =
huffman@47108
  1207
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1208
  neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1209
huffman@47108
  1210
lemmas less_divide_eq_numeral1 [simp] =
huffman@47108
  1211
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1212
  neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1213
huffman@47108
  1214
lemmas divide_less_eq_numeral1 [simp] =
huffman@47108
  1215
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1216
  neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1217
huffman@47108
  1218
lemmas eq_divide_eq_numeral1 [simp] =
huffman@47108
  1219
  eq_divide_eq [of _ _ "numeral w"]
huffman@47108
  1220
  eq_divide_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1221
huffman@47108
  1222
lemmas divide_eq_eq_numeral1 [simp] =
huffman@47108
  1223
  divide_eq_eq [of _ "numeral w"]
huffman@47108
  1224
  divide_eq_eq [of _ "neg_numeral w"] for w
haftmann@30652
  1225
haftmann@30652
  1226
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  1227
haftmann@30652
  1228
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  1229
huffman@47108
  1230
lemmas le_divide_eq_numeral =
huffman@47108
  1231
  le_divide_eq [of "numeral w"]
huffman@47108
  1232
  le_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1233
huffman@47108
  1234
lemmas divide_le_eq_numeral =
huffman@47108
  1235
  divide_le_eq [of _ _ "numeral w"]
huffman@47108
  1236
  divide_le_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1237
huffman@47108
  1238
lemmas less_divide_eq_numeral =
huffman@47108
  1239
  less_divide_eq [of "numeral w"]
huffman@47108
  1240
  less_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1241
huffman@47108
  1242
lemmas divide_less_eq_numeral =
huffman@47108
  1243
  divide_less_eq [of _ _ "numeral w"]
huffman@47108
  1244
  divide_less_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1245
huffman@47108
  1246
lemmas eq_divide_eq_numeral =
huffman@47108
  1247
  eq_divide_eq [of "numeral w"]
huffman@47108
  1248
  eq_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1249
huffman@47108
  1250
lemmas divide_eq_eq_numeral =
huffman@47108
  1251
  divide_eq_eq [of _ _ "numeral w"]
huffman@47108
  1252
  divide_eq_eq [of _ _ "neg_numeral w"] for w
haftmann@30652
  1253
haftmann@30652
  1254
haftmann@30652
  1255
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  1256
lemmas divide_const_simps =
huffman@47108
  1257
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
huffman@47108
  1258
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
haftmann@30652
  1259
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1260
haftmann@30652
  1261
text{*Division By @{text "-1"}*}
haftmann@30652
  1262
huffman@47108
  1263
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
huffman@47108
  1264
  unfolding minus_one [symmetric]
huffman@47108
  1265
  unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
huffman@47108
  1266
  by simp
haftmann@30652
  1267
huffman@47108
  1268
lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
huffman@47108
  1269
  unfolding minus_one [symmetric] by (rule divide_minus_left)
haftmann@30652
  1270
haftmann@30652
  1271
lemma half_gt_zero_iff:
huffman@47108
  1272
     "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
haftmann@30652
  1273
by auto
haftmann@30652
  1274
wenzelm@45607
  1275
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
haftmann@30652
  1276
huffman@47108
  1277
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
haftmann@36719
  1278
  by simp
haftmann@36719
  1279
haftmann@30652
  1280
haftmann@33320
  1281
subsection {* The divides relation *}
haftmann@33320
  1282
nipkow@33657
  1283
lemma zdvd_antisym_nonneg:
nipkow@33657
  1284
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  1285
  apply (simp add: dvd_def, auto)
nipkow@33657
  1286
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1287
  done
haftmann@33320
  1288
nipkow@33657
  1289
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  1290
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  1291
proof cases
nipkow@33657
  1292
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  1293
next
nipkow@33657
  1294
  assume "a \<noteq> 0"
haftmann@33320
  1295
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  1296
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  1297
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  1298
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@33320
  1299
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
haftmann@33320
  1300
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  1301
  thus ?thesis using k k' by auto
haftmann@33320
  1302
qed
haftmann@33320
  1303
haftmann@33320
  1304
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320
  1305
  apply (subgoal_tac "m = n + (m - n)")
haftmann@33320
  1306
   apply (erule ssubst)
haftmann@33320
  1307
   apply (blast intro: dvd_add, simp)
haftmann@33320
  1308
  done
haftmann@33320
  1309
haftmann@33320
  1310
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320
  1311
apply (rule iffI)
haftmann@33320
  1312
 apply (erule_tac [2] dvd_add)
haftmann@33320
  1313
 apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320
  1314
  apply (erule ssubst)
haftmann@33320
  1315
  apply (erule dvd_diff)
haftmann@33320
  1316
  apply(simp_all)
haftmann@33320
  1317
done
haftmann@33320
  1318
haftmann@33320
  1319
lemma dvd_imp_le_int:
haftmann@33320
  1320
  fixes d i :: int
haftmann@33320
  1321
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1322
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1323
proof -
haftmann@33320
  1324
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  1325
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  1326
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1327
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  1328
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  1329
qed
haftmann@33320
  1330
haftmann@33320
  1331
lemma zdvd_not_zless:
haftmann@33320
  1332
  fixes m n :: int
haftmann@33320
  1333
  assumes "0 < m" and "m < n"
haftmann@33320
  1334
  shows "\<not> n dvd m"
haftmann@33320
  1335
proof
haftmann@33320
  1336
  from assms have "0 < n" by auto
haftmann@33320
  1337
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  1338
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  1339
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  1340
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  1341
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1342
qed
haftmann@33320
  1343
haftmann@33320
  1344
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  1345
  shows "m dvd n"
haftmann@33320
  1346
proof-
haftmann@33320
  1347
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  1348
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@33320
  1349
    with h have False by (simp add: mult_assoc)}
haftmann@33320
  1350
  hence "n = m * h" by blast
haftmann@33320
  1351
  thus ?thesis by simp
haftmann@33320
  1352
qed
haftmann@33320
  1353
haftmann@33320
  1354
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  1355
proof -
haftmann@33320
  1356
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  1357
  proof -
haftmann@33320
  1358
    fix k
haftmann@33320
  1359
    assume A: "int y = int x * k"
wenzelm@42676
  1360
    then show "x dvd y"
wenzelm@42676
  1361
    proof (cases k)
wenzelm@42676
  1362
      case (nonneg n)
wenzelm@42676
  1363
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  1364
      then show ?thesis ..
haftmann@33320
  1365
    next
wenzelm@42676
  1366
      case (neg n)
wenzelm@42676
  1367
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  1368
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  1369
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  1370
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  1371
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  1372
    qed
haftmann@33320
  1373
  qed
haftmann@33320
  1374
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1375
qed
haftmann@33320
  1376
wenzelm@42676
  1377
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  1378
proof
haftmann@33320
  1379
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  1380
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  1381
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  1382
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  1383
next
haftmann@33320
  1384
  assume "\<bar>x\<bar>=1"
haftmann@33320
  1385
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  1386
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  1387
qed
haftmann@33320
  1388
haftmann@33320
  1389
lemma zdvd_mult_cancel1: 
haftmann@33320
  1390
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  1391
proof
haftmann@33320
  1392
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  1393
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  1394
next
haftmann@33320
  1395
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  1396
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  1397
qed
haftmann@33320
  1398
haftmann@33320
  1399
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  1400
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1401
haftmann@33320
  1402
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  1403
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1404
haftmann@33320
  1405
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  1406
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1407
haftmann@33341
  1408
lemma eq_nat_nat_iff:
haftmann@33341
  1409
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1410
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1411
haftmann@33341
  1412
lemma nat_power_eq:
haftmann@33341
  1413
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1414
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1415
haftmann@33320
  1416
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  1417
  apply (cases n)
haftmann@33320
  1418
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1419
  apply (cases z)
haftmann@33320
  1420
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  1421
  done
haftmann@33320
  1422
haftmann@36749
  1423
lemma zdvd_period:
haftmann@36749
  1424
  fixes a d :: int
haftmann@36749
  1425
  assumes "a dvd d"
haftmann@36749
  1426
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  1427
proof -
haftmann@36749
  1428
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1429
  show ?thesis
wenzelm@42676
  1430
  proof
haftmann@36749
  1431
    assume "a dvd (x + t)"
haftmann@36749
  1432
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1433
    then have "x = a * l - t" by simp
haftmann@36749
  1434
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  1435
  next
haftmann@36749
  1436
    assume "a dvd x + c * d + t"
haftmann@36749
  1437
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1438
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  1439
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  1440
  qed
haftmann@36749
  1441
qed
haftmann@36749
  1442
haftmann@33320
  1443
bulwahn@46756
  1444
subsection {* Finiteness of intervals *}
bulwahn@46756
  1445
bulwahn@46756
  1446
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756
  1447
proof (cases "a <= b")
bulwahn@46756
  1448
  case True
bulwahn@46756
  1449
  from this show ?thesis
bulwahn@46756
  1450
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1451
    case base
bulwahn@46756
  1452
    have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756
  1453
    from this show ?case by simp
bulwahn@46756
  1454
  next
bulwahn@46756
  1455
    case (step b)
bulwahn@46756
  1456
    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756
  1457
    from this step show ?case by simp
bulwahn@46756
  1458
  qed
bulwahn@46756
  1459
next
bulwahn@46756
  1460
  case False from this show ?thesis
bulwahn@46756
  1461
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1462
qed
bulwahn@46756
  1463
bulwahn@46756
  1464
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756
  1465
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1466
bulwahn@46756
  1467
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756
  1468
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1469
bulwahn@46756
  1470
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756
  1471
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1472
bulwahn@46756
  1473
haftmann@25919
  1474
subsection {* Configuration of the code generator *}
haftmann@25919
  1475
huffman@47108
  1476
text {* Constructors *}
huffman@47108
  1477
huffman@47108
  1478
definition Pos :: "num \<Rightarrow> int" where
huffman@47108
  1479
  [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1480
huffman@47108
  1481
definition Neg :: "num \<Rightarrow> int" where
huffman@47108
  1482
  [simp, code_abbrev]: "Neg = neg_numeral"
huffman@47108
  1483
huffman@47108
  1484
code_datatype "0::int" Pos Neg
huffman@47108
  1485
huffman@47108
  1486
huffman@47108
  1487
text {* Auxiliary operations *}
huffman@47108
  1488
huffman@47108
  1489
definition dup :: "int \<Rightarrow> int" where
huffman@47108
  1490
  [simp]: "dup k = k + k"
haftmann@26507
  1491
huffman@47108
  1492
lemma dup_code [code]:
huffman@47108
  1493
  "dup 0 = 0"
huffman@47108
  1494
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1495
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1496
  unfolding Pos_def Neg_def neg_numeral_def
huffman@47108
  1497
  by (simp_all add: numeral_Bit0)
huffman@47108
  1498
huffman@47108
  1499
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108
  1500
  [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1501
huffman@47108
  1502
lemma sub_code [code]:
huffman@47108
  1503
  "sub Num.One Num.One = 0"
huffman@47108
  1504
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1505
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1506
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1507
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1508
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1509
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1510
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1511
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@54230
  1512
  apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def
haftmann@54230
  1513
    neg_numeral_def numeral_BitM)
haftmann@54230
  1514
  apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230
  1515
  apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230
  1516
  apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230
  1517
  apply (simp_all only: algebra_simps right_minus)
haftmann@54230
  1518
  done
huffman@47108
  1519
huffman@47108
  1520
text {* Implementations *}
huffman@47108
  1521
huffman@47108
  1522
lemma one_int_code [code, code_unfold]:
huffman@47108
  1523
  "1 = Pos Num.One"
huffman@47108
  1524
  by simp
huffman@47108
  1525
huffman@47108
  1526
lemma plus_int_code [code]:
huffman@47108
  1527
  "k + 0 = (k::int)"
huffman@47108
  1528
  "0 + l = (l::int)"
huffman@47108
  1529
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1530
  "Pos m + Neg n = sub m n"
huffman@47108
  1531
  "Neg m + Pos n = sub n m"
huffman@47108
  1532
  "Neg m + Neg n = Neg (m + n)"
huffman@47108
  1533
  by simp_all
haftmann@26507
  1534
huffman@47108
  1535
lemma uminus_int_code [code]:
huffman@47108
  1536
  "uminus 0 = (0::int)"
huffman@47108
  1537
  "uminus (Pos m) = Neg m"
huffman@47108
  1538
  "uminus (Neg m) = Pos m"
huffman@47108
  1539
  by simp_all
huffman@47108
  1540
huffman@47108
  1541
lemma minus_int_code [code]:
huffman@47108
  1542
  "k - 0 = (k::int)"
huffman@47108
  1543
  "0 - l = uminus (l::int)"
huffman@47108
  1544
  "Pos m - Pos n = sub m n"
huffman@47108
  1545
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1546
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1547
  "Neg m - Neg n = sub n m"
huffman@47108
  1548
  by simp_all
huffman@47108
  1549
huffman@47108
  1550
lemma times_int_code [code]:
huffman@47108
  1551
  "k * 0 = (0::int)"
huffman@47108
  1552
  "0 * l = (0::int)"
huffman@47108
  1553
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1554
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1555
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1556
  "Neg m * Neg n = Pos (m * n)"
huffman@47108
  1557
  by simp_all
haftmann@26507
  1558
haftmann@38857
  1559
instantiation int :: equal
haftmann@26507
  1560
begin
haftmann@26507
  1561
haftmann@37767
  1562
definition
huffman@47108
  1563
  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1564
huffman@47108
  1565
instance by default (rule equal_int_def)
haftmann@26507
  1566
haftmann@26507
  1567
end
haftmann@26507
  1568
huffman@47108
  1569
lemma equal_int_code [code]:
huffman@47108
  1570
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1571
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1572
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1573
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1574
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1575
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1576
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1577
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1578
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1579
  by (auto simp add: equal)
haftmann@26507
  1580
huffman@47108
  1581
lemma equal_int_refl [code nbe]:
haftmann@38857
  1582
  "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108
  1583
  by (fact equal_refl)
haftmann@26507
  1584
haftmann@28562
  1585
lemma less_eq_int_code [code]:
huffman@47108
  1586
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1587
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1588
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1589
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1590
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1591
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1592
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1593
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1594
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1595
  by simp_all
haftmann@26507
  1596
haftmann@28562
  1597
lemma less_int_code [code]:
huffman@47108
  1598
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1599
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1600
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1601
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1602
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1603
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1604
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1605
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1606
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1607
  by simp_all
haftmann@25919
  1608
huffman@47108
  1609
lemma nat_code [code]:
huffman@47108
  1610
  "nat (Int.Neg k) = 0"
huffman@47108
  1611
  "nat 0 = 0"
huffman@47108
  1612
  "nat (Int.Pos k) = nat_of_num k"
huffman@47108
  1613
  by (simp_all add: nat_of_num_numeral nat_numeral)
haftmann@25928
  1614
huffman@47108
  1615
lemma (in ring_1) of_int_code [code]:
huffman@47108
  1616
  "of_int (Int.Neg k) = neg_numeral k"
huffman@47108
  1617
  "of_int 0 = 0"
huffman@47108
  1618
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1619
  by simp_all
haftmann@25919
  1620
huffman@47108
  1621
huffman@47108
  1622
text {* Serializer setup *}
haftmann@25919
  1623
haftmann@52435
  1624
code_identifier
haftmann@52435
  1625
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1626
haftmann@25919
  1627
quickcheck_params [default_type = int]
haftmann@25919
  1628
huffman@47108
  1629
hide_const (open) Pos Neg sub dup
haftmann@25919
  1630
haftmann@25919
  1631
haftmann@25919
  1632
subsection {* Legacy theorems *}
haftmann@25919
  1633
haftmann@25919
  1634
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1635
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1636
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  1637
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
wenzelm@45607
  1638
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1639
lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607
  1640
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919
  1641
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1642
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607
  1643
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1644
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1645
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1646
lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207
  1647
lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607
  1648
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919
  1649
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1650
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255
  1651
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255
  1652
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960
  1653
haftmann@31015
  1654
lemma zpower_zpower:
haftmann@31015
  1655
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  1656
  by (rule power_mult [symmetric])
haftmann@31015
  1657
haftmann@31015
  1658
lemma int_power:
haftmann@31015
  1659
  "int (m ^ n) = int m ^ n"
haftmann@31015
  1660
  by (rule of_nat_power)
haftmann@31015
  1661
haftmann@31015
  1662
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  1663
huffman@48045
  1664
text {* De-register @{text "int"} as a quotient type: *}
huffman@48045
  1665
kuncar@53652
  1666
lifting_update int.lifting
kuncar@53652
  1667
lifting_forget int.lifting
huffman@48045
  1668
haftmann@25919
  1669
end