src/HOL/Int.thy
author hoelzl
Tue Mar 26 12:20:58 2013 +0100 (2013-03-26)
changeset 51526 155263089e7b
parent 51329 4a3c453f99a1
child 51994 82cc2aeb7d13
permissions -rw-r--r--
move SEQ.thy and Lim.thy to Limits.thy
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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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  by simp
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lift_definition one_int :: "int" is "(1, 0)"
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  by simp
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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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by (simp add: diff_minus Groups.diff_minus)
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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]) 
haftmann@25919
   344
haftmann@25919
   345
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
huffman@48045
   346
  by transfer (clarsimp, arith)
haftmann@25919
   347
haftmann@25919
   348
lemma nonneg_eq_int:
haftmann@25919
   349
  fixes z :: int
huffman@44709
   350
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   351
  shows P
haftmann@25919
   352
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   353
huffman@44709
   354
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
huffman@48045
   355
  by transfer (clarsimp simp add: le_imp_diff_is_add)
haftmann@25919
   356
huffman@44709
   357
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
haftmann@25919
   358
by (simp only: eq_commute [of m] nat_eq_iff)
haftmann@25919
   359
haftmann@25919
   360
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045
   361
  by transfer (clarsimp, arith)
haftmann@25919
   362
huffman@44709
   363
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   364
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   365
huffman@44707
   366
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   367
  by transfer auto
huffman@44707
   368
nipkow@29700
   369
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045
   370
  by transfer clarsimp
nipkow@29700
   371
haftmann@25919
   372
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   373
by (auto simp add: nat_eq_iff2)
haftmann@25919
   374
haftmann@25919
   375
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   376
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   377
haftmann@25919
   378
lemma nat_add_distrib:
haftmann@25919
   379
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
huffman@48045
   380
  by transfer clarsimp
haftmann@25919
   381
haftmann@25919
   382
lemma nat_diff_distrib:
haftmann@25919
   383
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
huffman@48045
   384
  by transfer clarsimp
haftmann@25919
   385
huffman@44709
   386
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   387
  by transfer simp
haftmann@25919
   388
huffman@44709
   389
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045
   390
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   391
haftmann@25919
   392
context ring_1
haftmann@25919
   393
begin
haftmann@25919
   394
haftmann@25919
   395
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   396
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   397
haftmann@25919
   398
end
haftmann@25919
   399
krauss@29779
   400
text {* For termination proofs: *}
krauss@29779
   401
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   402
haftmann@25919
   403
haftmann@25919
   404
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   405
huffman@44709
   406
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   407
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   408
huffman@44709
   409
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   410
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   411
huffman@44709
   412
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   413
by (simp add: minus_le_iff)
haftmann@25919
   414
huffman@44709
   415
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   416
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   417
huffman@44709
   418
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   419
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   420
huffman@44709
   421
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045
   422
  by transfer simp
haftmann@25919
   423
huffman@44709
   424
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   425
by (simp add: linorder_not_less)
haftmann@25919
   426
huffman@44709
   427
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   428
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   429
huffman@44709
   430
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   431
proof -
haftmann@25919
   432
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   433
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   434
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   435
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   436
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   437
    by (simp only: algebra_simps)
haftmann@25919
   438
  finally show ?thesis .
haftmann@25919
   439
qed
haftmann@25919
   440
huffman@44709
   441
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   442
by simp
haftmann@25919
   443
huffman@44709
   444
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919
   445
by simp
haftmann@25919
   446
haftmann@25919
   447
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   448
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   449
      in theory @{text Rings}.
haftmann@25919
   450
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@35828
   451
lemma abs_split [arith_split,no_atp]:
haftmann@35028
   452
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   453
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   454
huffman@44709
   455
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045
   456
apply transfer
huffman@48045
   457
apply clarsimp
huffman@48045
   458
apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919
   459
done
haftmann@25919
   460
haftmann@25919
   461
haftmann@25919
   462
subsection {* Cases and induction *}
haftmann@25919
   463
haftmann@25919
   464
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   465
whether an integer is negative or not.*}
haftmann@25919
   466
wenzelm@42676
   467
theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709
   468
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   469
apply (cases "z < 0")
wenzelm@42676
   470
apply (blast dest!: negD)
haftmann@25919
   471
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   472
apply auto
haftmann@25919
   473
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   474
done
haftmann@25919
   475
wenzelm@42676
   476
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   477
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   478
  by (cases z) auto
haftmann@25919
   479
huffman@47207
   480
lemma nonneg_int_cases:
huffman@47207
   481
  assumes "0 \<le> k" obtains n where "k = int n"
huffman@47207
   482
  using assms by (cases k, simp, simp del: of_nat_Suc)
huffman@47207
   483
huffman@47108
   484
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
huffman@47108
   485
  -- {* Unfold all @{text let}s involving constants *}
huffman@47108
   486
  unfolding Let_def ..
haftmann@37767
   487
huffman@47108
   488
lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
haftmann@25919
   489
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   490
  unfolding Let_def ..
haftmann@25919
   491
huffman@47108
   492
text {* Unfold @{text min} and @{text max} on numerals. *}
huffman@28958
   493
huffman@47108
   494
lemmas max_number_of [simp] =
huffman@47108
   495
  max_def [of "numeral u" "numeral v"]
huffman@47108
   496
  max_def [of "numeral u" "neg_numeral v"]
huffman@47108
   497
  max_def [of "neg_numeral u" "numeral v"]
huffman@47108
   498
  max_def [of "neg_numeral u" "neg_numeral v"] for u v
huffman@28958
   499
huffman@47108
   500
lemmas min_number_of [simp] =
huffman@47108
   501
  min_def [of "numeral u" "numeral v"]
huffman@47108
   502
  min_def [of "numeral u" "neg_numeral v"]
huffman@47108
   503
  min_def [of "neg_numeral u" "numeral v"]
huffman@47108
   504
  min_def [of "neg_numeral u" "neg_numeral v"] for u v
huffman@26075
   505
haftmann@25919
   506
huffman@28958
   507
subsubsection {* Binary comparisons *}
huffman@28958
   508
huffman@28958
   509
text {* Preliminaries *}
huffman@28958
   510
huffman@28958
   511
lemma even_less_0_iff:
haftmann@35028
   512
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   513
proof -
webertj@49962
   514
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
huffman@28958
   515
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   516
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   517
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   518
  finally show ?thesis .
huffman@28958
   519
qed
huffman@28958
   520
huffman@28958
   521
lemma le_imp_0_less: 
huffman@28958
   522
  assumes le: "0 \<le> z"
huffman@28958
   523
  shows "(0::int) < 1 + z"
huffman@28958
   524
proof -
huffman@28958
   525
  have "0 \<le> z" by fact
huffman@47108
   526
  also have "... < z + 1" by (rule less_add_one)
huffman@28958
   527
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   528
  finally show "0 < 1 + z" .
huffman@28958
   529
qed
huffman@28958
   530
huffman@28958
   531
lemma odd_less_0_iff:
huffman@28958
   532
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   533
proof (cases z)
huffman@28958
   534
  case (nonneg n)
huffman@28958
   535
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   536
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   537
next
huffman@28958
   538
  case (neg n)
huffman@30079
   539
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   540
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   541
qed
huffman@28958
   542
huffman@28958
   543
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
   544
haftmann@25919
   545
lemmas double_eq_0_iff = double_zero
haftmann@25919
   546
haftmann@25919
   547
lemma odd_nonzero:
haftmann@33296
   548
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
   549
proof (cases z)
haftmann@25919
   550
  case (nonneg n)
haftmann@25919
   551
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
   552
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
   553
    by (auto simp add: add_assoc) 
haftmann@25919
   554
next
haftmann@25919
   555
  case (neg n)
haftmann@25919
   556
  show ?thesis
haftmann@25919
   557
  proof
haftmann@25919
   558
    assume eq: "1 + z + z = 0"
huffman@44709
   559
    have "(0::int) < 1 + (int n + int n)"
haftmann@25919
   560
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
   561
    also have "... = - (1 + z + z)" 
haftmann@25919
   562
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
   563
    also have "... = 0" by (simp add: eq) 
haftmann@25919
   564
    finally have "0<0" ..
haftmann@25919
   565
    thus False by blast
haftmann@25919
   566
  qed
haftmann@25919
   567
qed
haftmann@25919
   568
haftmann@30652
   569
haftmann@25919
   570
subsection {* The Set of Integers *}
haftmann@25919
   571
haftmann@25919
   572
context ring_1
haftmann@25919
   573
begin
haftmann@25919
   574
haftmann@30652
   575
definition Ints  :: "'a set" where
haftmann@37767
   576
  "Ints = range of_int"
haftmann@25919
   577
haftmann@25919
   578
notation (xsymbols)
haftmann@25919
   579
  Ints  ("\<int>")
haftmann@25919
   580
huffman@35634
   581
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   582
  by (simp add: Ints_def)
huffman@35634
   583
huffman@35634
   584
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   585
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   586
haftmann@25919
   587
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   588
  using Ints_of_int [of "0"] by simp
haftmann@25919
   589
haftmann@25919
   590
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   591
  using Ints_of_int [of "1"] by simp
haftmann@25919
   592
haftmann@25919
   593
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
   594
apply (auto simp add: Ints_def)
haftmann@25919
   595
apply (rule range_eqI)
haftmann@25919
   596
apply (rule of_int_add [symmetric])
haftmann@25919
   597
done
haftmann@25919
   598
haftmann@25919
   599
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
   600
apply (auto simp add: Ints_def)
haftmann@25919
   601
apply (rule range_eqI)
haftmann@25919
   602
apply (rule of_int_minus [symmetric])
haftmann@25919
   603
done
haftmann@25919
   604
huffman@35634
   605
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
   606
apply (auto simp add: Ints_def)
huffman@35634
   607
apply (rule range_eqI)
huffman@35634
   608
apply (rule of_int_diff [symmetric])
huffman@35634
   609
done
huffman@35634
   610
haftmann@25919
   611
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
   612
apply (auto simp add: Ints_def)
haftmann@25919
   613
apply (rule range_eqI)
haftmann@25919
   614
apply (rule of_int_mult [symmetric])
haftmann@25919
   615
done
haftmann@25919
   616
huffman@35634
   617
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
   618
by (induct n) simp_all
huffman@35634
   619
haftmann@25919
   620
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   621
  assumes "q \<in> \<int>"
haftmann@25919
   622
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   623
  unfolding Ints_def
haftmann@25919
   624
proof -
haftmann@25919
   625
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   626
  then obtain z where "q = of_int z" ..
haftmann@25919
   627
  then show thesis ..
haftmann@25919
   628
qed
haftmann@25919
   629
haftmann@25919
   630
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   631
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   632
  by (rule Ints_cases) auto
haftmann@25919
   633
haftmann@25919
   634
end
haftmann@25919
   635
haftmann@25919
   636
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
   637
haftmann@25919
   638
lemma Ints_double_eq_0_iff:
haftmann@25919
   639
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   640
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
   641
proof -
haftmann@25919
   642
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   643
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   644
  show ?thesis
haftmann@25919
   645
  proof
haftmann@25919
   646
    assume "a = 0"
haftmann@25919
   647
    thus "a + a = 0" by simp
haftmann@25919
   648
  next
haftmann@25919
   649
    assume eq: "a + a = 0"
haftmann@25919
   650
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   651
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   652
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   653
    thus "a = 0" by (simp add: a)
haftmann@25919
   654
  qed
haftmann@25919
   655
qed
haftmann@25919
   656
haftmann@25919
   657
lemma Ints_odd_nonzero:
haftmann@25919
   658
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   659
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
   660
proof -
haftmann@25919
   661
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   662
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   663
  show ?thesis
haftmann@25919
   664
  proof
haftmann@25919
   665
    assume eq: "1 + a + a = 0"
haftmann@25919
   666
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   667
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   668
    with odd_nonzero show False by blast
haftmann@25919
   669
  qed
haftmann@25919
   670
qed 
haftmann@25919
   671
huffman@47108
   672
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
huffman@47108
   673
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   674
haftmann@25919
   675
lemma Ints_odd_less_0: 
haftmann@25919
   676
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
   677
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
   678
proof -
haftmann@25919
   679
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   680
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   681
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
   682
    by (simp add: a)
huffman@45532
   683
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919
   684
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
   685
  finally show ?thesis .
haftmann@25919
   686
qed
haftmann@25919
   687
haftmann@25919
   688
haftmann@25919
   689
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
   690
haftmann@25919
   691
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
   692
  apply (cases "finite A")
haftmann@25919
   693
  apply (erule finite_induct, auto)
haftmann@25919
   694
  done
haftmann@25919
   695
haftmann@25919
   696
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
   697
  apply (cases "finite A")
haftmann@25919
   698
  apply (erule finite_induct, auto)
haftmann@25919
   699
  done
haftmann@25919
   700
haftmann@25919
   701
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
   702
  apply (cases "finite A")
haftmann@25919
   703
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
   704
  done
haftmann@25919
   705
haftmann@25919
   706
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
   707
  apply (cases "finite A")
haftmann@25919
   708
  apply (erule finite_induct, auto)
haftmann@25919
   709
  done
haftmann@25919
   710
haftmann@25919
   711
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
   712
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
   713
haftmann@25919
   714
haftmann@25919
   715
text {* Legacy theorems *}
haftmann@25919
   716
haftmann@25919
   717
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   718
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
huffman@47108
   719
lemmas numeral_1_eq_1 = numeral_One
haftmann@25919
   720
huffman@30802
   721
subsection {* Setting up simplification procedures *}
huffman@30802
   722
huffman@30802
   723
lemmas int_arith_rules =
huffman@47108
   724
  neg_le_iff_le numeral_One
huffman@30802
   725
  minus_zero diff_minus left_minus right_minus
huffman@45219
   726
  mult_zero_left mult_zero_right mult_1_left mult_1_right
huffman@30802
   727
  mult_minus_left mult_minus_right
huffman@30802
   728
  minus_add_distrib minus_minus mult_assoc
huffman@30802
   729
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
huffman@30802
   730
  of_int_0 of_int_1 of_int_add of_int_mult
huffman@30802
   731
wenzelm@48891
   732
ML_file "Tools/int_arith.ML"
haftmann@30496
   733
declaration {* K Int_Arith.setup *}
haftmann@25919
   734
huffman@47108
   735
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
huffman@47108
   736
  "(m::'a::linordered_idom) <= n" |
huffman@47108
   737
  "(m::'a::linordered_idom) = n") =
wenzelm@43595
   738
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
   739
haftmann@25919
   740
haftmann@25919
   741
subsection{*Lemmas About Small Numerals*}
haftmann@25919
   742
haftmann@25919
   743
lemma abs_power_minus_one [simp]:
huffman@47108
   744
  "abs(-1 ^ n) = (1::'a::linordered_idom)"
haftmann@25919
   745
by (simp add: power_abs)
haftmann@25919
   746
haftmann@25919
   747
haftmann@25919
   748
subsection{*More Inequality Reasoning*}
haftmann@25919
   749
haftmann@25919
   750
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
   751
by arith
haftmann@25919
   752
haftmann@25919
   753
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
   754
by arith
haftmann@25919
   755
haftmann@25919
   756
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
   757
by arith
haftmann@25919
   758
haftmann@25919
   759
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
   760
by arith
haftmann@25919
   761
haftmann@25919
   762
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
   763
by arith
haftmann@25919
   764
haftmann@25919
   765
huffman@28958
   766
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
   767
huffman@48044
   768
text{*Simplify the term @{term "w + - z"}*}
huffman@48045
   769
lemmas diff_int_def_symmetric = diff_def [where 'a=int, symmetric, simp]
haftmann@25919
   770
huffman@44695
   771
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@25919
   772
by (simp add: nat_eq_iff)
haftmann@25919
   773
huffman@47207
   774
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@25919
   775
by (subst nat_eq_iff, simp)
haftmann@25919
   776
haftmann@25919
   777
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919
   778
by (subst nat_eq_iff, simp)
haftmann@25919
   779
haftmann@25919
   780
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
   781
apply (insert zless_nat_conj [of 1 z])
huffman@47207
   782
apply auto
haftmann@25919
   783
done
haftmann@25919
   784
haftmann@25919
   785
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
   786
      z is an integer literal.*}
huffman@47108
   787
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
   788
haftmann@25919
   789
lemma split_nat [arith_split]:
huffman@44709
   790
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
   791
  (is "?P = (?L & ?R)")
haftmann@25919
   792
proof (cases "i < 0")
haftmann@25919
   793
  case True thus ?thesis by auto
haftmann@25919
   794
next
haftmann@25919
   795
  case False
haftmann@25919
   796
  have "?P = ?L"
haftmann@25919
   797
  proof
haftmann@25919
   798
    assume ?P thus ?L using False by clarsimp
haftmann@25919
   799
  next
haftmann@25919
   800
    assume ?L thus ?P using False by simp
haftmann@25919
   801
  qed
haftmann@25919
   802
  with False show ?thesis by simp
haftmann@25919
   803
qed
haftmann@25919
   804
haftmann@25919
   805
context ring_1
haftmann@25919
   806
begin
haftmann@25919
   807
blanchet@33056
   808
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
   809
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
   810
proof (cases "k < 0")
haftmann@25919
   811
  case True then have "0 \<le> - k" by simp
haftmann@25919
   812
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
   813
  with True show ?thesis by simp
haftmann@25919
   814
next
haftmann@25919
   815
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
   816
qed
haftmann@25919
   817
haftmann@25919
   818
end
haftmann@25919
   819
haftmann@25919
   820
lemma nat_mult_distrib:
haftmann@25919
   821
  fixes z z' :: int
haftmann@25919
   822
  assumes "0 \<le> z"
haftmann@25919
   823
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
   824
proof (cases "0 \<le> z'")
haftmann@25919
   825
  case False with assms have "z * z' \<le> 0"
haftmann@25919
   826
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
   827
  then have "nat (z * z') = 0" by simp
haftmann@25919
   828
  moreover from False have "nat z' = 0" by simp
haftmann@25919
   829
  ultimately show ?thesis by simp
haftmann@25919
   830
next
haftmann@25919
   831
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
   832
  show ?thesis
haftmann@25919
   833
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
   834
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
   835
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
   836
qed
haftmann@25919
   837
haftmann@25919
   838
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
   839
apply (rule trans)
haftmann@25919
   840
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
   841
done
haftmann@25919
   842
haftmann@25919
   843
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
   844
apply (cases "z=0 | w=0")
haftmann@25919
   845
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
   846
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
   847
done
haftmann@25919
   848
huffman@47207
   849
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207
   850
apply (rule sym)
huffman@47207
   851
apply (simp add: nat_eq_iff)
huffman@47207
   852
done
huffman@47207
   853
huffman@47207
   854
lemma diff_nat_eq_if:
huffman@47207
   855
     "nat z - nat z' =  
huffman@47207
   856
        (if z' < 0 then nat z   
huffman@47207
   857
         else let d = z-z' in     
huffman@47207
   858
              if d < 0 then 0 else nat d)"
huffman@47207
   859
by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
   860
huffman@47207
   861
(* nat_diff_distrib has too-strong premises *)
huffman@47207
   862
lemma nat_diff_distrib': "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x - y) = nat x - nat y"
huffman@47207
   863
apply (rule int_int_eq [THEN iffD1], clarsimp)
huffman@47207
   864
apply (subst of_nat_diff)
huffman@47207
   865
apply (rule nat_mono, simp_all)
huffman@47207
   866
done
huffman@47207
   867
haftmann@51143
   868
lemma nat_numeral [simp]:
huffman@47207
   869
  "nat (numeral k) = numeral k"
huffman@47207
   870
  by (simp add: nat_eq_iff)
huffman@47207
   871
huffman@47207
   872
lemma nat_neg_numeral [simp]:
huffman@47207
   873
  "nat (neg_numeral k) = 0"
huffman@47207
   874
  by simp
huffman@47207
   875
huffman@47207
   876
lemma diff_nat_numeral [simp]: 
huffman@47207
   877
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
huffman@47207
   878
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
huffman@47207
   879
huffman@47207
   880
lemma nat_numeral_diff_1 [simp]:
huffman@47207
   881
  "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
   882
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
   883
huffman@47255
   884
lemmas nat_arith = diff_nat_numeral
huffman@47255
   885
haftmann@25919
   886
haftmann@25919
   887
subsection "Induction principles for int"
haftmann@25919
   888
haftmann@25919
   889
text{*Well-founded segments of the integers*}
haftmann@25919
   890
haftmann@25919
   891
definition
haftmann@25919
   892
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
   893
where
haftmann@25919
   894
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
   895
haftmann@25919
   896
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
   897
proof -
haftmann@25919
   898
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
   899
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
   900
  thus ?thesis 
haftmann@25919
   901
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   902
qed
haftmann@25919
   903
haftmann@25919
   904
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
   905
by RankFinder.*}
haftmann@25919
   906
haftmann@25919
   907
definition
haftmann@25919
   908
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
   909
where
haftmann@25919
   910
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
   911
haftmann@25919
   912
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
   913
proof -
haftmann@25919
   914
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
   915
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
   916
  thus ?thesis 
haftmann@25919
   917
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   918
qed
haftmann@25919
   919
haftmann@25919
   920
(* `set:int': dummy construction *)
haftmann@25919
   921
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
   922
  fixes i :: int
haftmann@25919
   923
  assumes ge: "k \<le> i" and
haftmann@25919
   924
    base: "P k" and
haftmann@25919
   925
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
   926
  shows "P i"
haftmann@25919
   927
proof -
wenzelm@42676
   928
  { fix n
wenzelm@42676
   929
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
   930
    proof (induct n)
haftmann@25919
   931
      case 0
haftmann@25919
   932
      hence "i = k" by arith
haftmann@25919
   933
      thus "P i" using base by simp
haftmann@25919
   934
    next
haftmann@25919
   935
      case (Suc n)
haftmann@25919
   936
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
   937
      moreover
haftmann@25919
   938
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
   939
      ultimately
wenzelm@42676
   940
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
   941
      from step [OF ki1 this] show ?case by simp
haftmann@25919
   942
    qed
haftmann@25919
   943
  }
haftmann@25919
   944
  with ge show ?thesis by fast
haftmann@25919
   945
qed
haftmann@25919
   946
haftmann@25928
   947
(* `set:int': dummy construction *)
haftmann@25928
   948
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
   949
  assumes gr: "k < (i::int)" and
haftmann@25919
   950
        base: "P(k+1)" and
haftmann@25919
   951
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
   952
  shows "P i"
haftmann@25919
   953
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
   954
  using gr apply arith
haftmann@25919
   955
 apply(rule base)
haftmann@25919
   956
apply (rule step, simp+)
haftmann@25919
   957
done
haftmann@25919
   958
wenzelm@42676
   959
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
   960
  assumes le: "i \<le> (k::int)" and
haftmann@25919
   961
        base: "P(k)" and
haftmann@25919
   962
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   963
  shows "P i"
haftmann@25919
   964
proof -
wenzelm@42676
   965
  { fix n
wenzelm@42676
   966
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
   967
    proof (induct n)
haftmann@25919
   968
      case 0
haftmann@25919
   969
      hence "i = k" by arith
haftmann@25919
   970
      thus "P i" using base by simp
haftmann@25919
   971
    next
haftmann@25919
   972
      case (Suc n)
wenzelm@42676
   973
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
   974
      moreover
haftmann@25919
   975
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
   976
      ultimately
wenzelm@42676
   977
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
   978
      from step[OF ki1 this] show ?case by simp
haftmann@25919
   979
    qed
haftmann@25919
   980
  }
haftmann@25919
   981
  with le show ?thesis by fast
haftmann@25919
   982
qed
haftmann@25919
   983
wenzelm@42676
   984
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
   985
  assumes less: "(i::int) < k" and
haftmann@25919
   986
        base: "P(k - 1)" and
haftmann@25919
   987
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   988
  shows "P i"
haftmann@25919
   989
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
   990
  using less apply arith
haftmann@25919
   991
 apply(rule base)
haftmann@25919
   992
apply (rule step, simp+)
haftmann@25919
   993
done
haftmann@25919
   994
haftmann@36811
   995
theorem int_induct [case_names base step1 step2]:
haftmann@36801
   996
  fixes k :: int
haftmann@36801
   997
  assumes base: "P k"
haftmann@36801
   998
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
   999
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1000
  shows "P i"
haftmann@36801
  1001
proof -
haftmann@36801
  1002
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1003
  then show ?thesis
wenzelm@42676
  1004
  proof
wenzelm@42676
  1005
    assume "i \<ge> k"
wenzelm@42676
  1006
    then show ?thesis using base
haftmann@36801
  1007
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1008
  next
wenzelm@42676
  1009
    assume "i \<le> k"
wenzelm@42676
  1010
    then show ?thesis using base
haftmann@36801
  1011
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1012
  qed
haftmann@36801
  1013
qed
haftmann@36801
  1014
haftmann@25919
  1015
subsection{*Intermediate value theorems*}
haftmann@25919
  1016
haftmann@25919
  1017
lemma int_val_lemma:
haftmann@25919
  1018
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1019
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1020
unfolding One_nat_def
wenzelm@42676
  1021
apply (induct n)
wenzelm@42676
  1022
apply simp
haftmann@25919
  1023
apply (intro strip)
haftmann@25919
  1024
apply (erule impE, simp)
haftmann@25919
  1025
apply (erule_tac x = n in allE, simp)
huffman@30079
  1026
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1027
apply force
haftmann@25919
  1028
apply (erule impE)
haftmann@25919
  1029
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1030
apply (blast intro: le_SucI)
haftmann@25919
  1031
done
haftmann@25919
  1032
haftmann@25919
  1033
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1034
haftmann@25919
  1035
lemma nat_intermed_int_val:
haftmann@25919
  1036
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1037
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1038
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1039
       in int_val_lemma)
huffman@30079
  1040
unfolding One_nat_def
haftmann@25919
  1041
apply simp
haftmann@25919
  1042
apply (erule exE)
haftmann@25919
  1043
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1044
done
haftmann@25919
  1045
haftmann@25919
  1046
haftmann@25919
  1047
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1048
haftmann@25919
  1049
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1050
by arith
haftmann@25919
  1051
paulson@34055
  1052
lemma abs_zmult_eq_1:
paulson@34055
  1053
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1054
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1055
proof -
paulson@34055
  1056
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1057
    by auto
paulson@34055
  1058
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1059
  proof
paulson@34055
  1060
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1061
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1062
      by (simp add: mult_mono 0) 
paulson@34055
  1063
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1064
      by (simp add: abs_mult)
paulson@34055
  1065
    also have "... = 1"
paulson@34055
  1066
      by (simp add: mn)
paulson@34055
  1067
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1068
    thus "False" using 0
huffman@47108
  1069
      by arith
paulson@34055
  1070
  qed
paulson@34055
  1071
  thus ?thesis using 0
paulson@34055
  1072
    by auto
paulson@34055
  1073
qed
haftmann@25919
  1074
huffman@47108
  1075
ML_val {* @{const_name neg_numeral} *}
huffman@47108
  1076
haftmann@25919
  1077
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1078
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1079
boehmes@35815
  1080
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1081
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1082
proof -
boehmes@35815
  1083
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1084
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1085
qed
haftmann@25919
  1086
haftmann@25919
  1087
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1088
apply (rule iffI) 
haftmann@25919
  1089
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1090
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1091
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1092
done
haftmann@25919
  1093
haftmann@33296
  1094
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1095
proof
haftmann@33296
  1096
  assume "finite (UNIV::int set)"
haftmann@33296
  1097
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1098
    by (rule injI) simp
haftmann@33296
  1099
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1100
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1101
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1102
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1103
qed
haftmann@25919
  1104
haftmann@25919
  1105
haftmann@30652
  1106
subsection {* Further theorems on numerals *}
haftmann@30652
  1107
haftmann@30652
  1108
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1109
haftmann@30652
  1110
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1111
webertj@49962
  1112
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1113
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1114
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1115
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1116
haftmann@30652
  1117
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1118
huffman@47108
  1119
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1120
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1121
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1122
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1123
haftmann@30652
  1124
haftmann@30652
  1125
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1126
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1127
huffman@47108
  1128
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1129
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1130
huffman@47108
  1131
lemmas inverse_eq_divide_neg_numeral [simp] =
huffman@47108
  1132
  inverse_eq_divide [of "neg_numeral w"] for w
haftmann@30652
  1133
haftmann@30652
  1134
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1135
into the literal.*}
haftmann@30652
  1136
huffman@47108
  1137
lemmas le_minus_iff_numeral [simp, no_atp] =
huffman@47108
  1138
  le_minus_iff [of "numeral v"]
huffman@47108
  1139
  le_minus_iff [of "neg_numeral v"] for v
huffman@47108
  1140
huffman@47108
  1141
lemmas equation_minus_iff_numeral [simp, no_atp] =
huffman@47108
  1142
  equation_minus_iff [of "numeral v"]
huffman@47108
  1143
  equation_minus_iff [of "neg_numeral v"] for v
huffman@47108
  1144
huffman@47108
  1145
lemmas minus_less_iff_numeral [simp, no_atp] =
huffman@47108
  1146
  minus_less_iff [of _ "numeral v"]
huffman@47108
  1147
  minus_less_iff [of _ "neg_numeral v"] for v
huffman@47108
  1148
huffman@47108
  1149
lemmas minus_le_iff_numeral [simp, no_atp] =
huffman@47108
  1150
  minus_le_iff [of _ "numeral v"]
huffman@47108
  1151
  minus_le_iff [of _ "neg_numeral v"] for v
huffman@47108
  1152
huffman@47108
  1153
lemmas minus_equation_iff_numeral [simp, no_atp] =
huffman@47108
  1154
  minus_equation_iff [of _ "numeral v"]
huffman@47108
  1155
  minus_equation_iff [of _ "neg_numeral v"] for v
haftmann@30652
  1156
haftmann@30652
  1157
text{*To Simplify Inequalities Where One Side is the Constant 1*}
haftmann@30652
  1158
blanchet@35828
  1159
lemma less_minus_iff_1 [simp,no_atp]:
huffman@47108
  1160
  fixes b::"'b::linordered_idom"
haftmann@30652
  1161
  shows "(1 < - b) = (b < -1)"
haftmann@30652
  1162
by auto
haftmann@30652
  1163
blanchet@35828
  1164
lemma le_minus_iff_1 [simp,no_atp]:
huffman@47108
  1165
  fixes b::"'b::linordered_idom"
haftmann@30652
  1166
  shows "(1 \<le> - b) = (b \<le> -1)"
haftmann@30652
  1167
by auto
haftmann@30652
  1168
blanchet@35828
  1169
lemma equation_minus_iff_1 [simp,no_atp]:
huffman@47108
  1170
  fixes b::"'b::ring_1"
haftmann@30652
  1171
  shows "(1 = - b) = (b = -1)"
haftmann@30652
  1172
by (subst equation_minus_iff, auto)
haftmann@30652
  1173
blanchet@35828
  1174
lemma minus_less_iff_1 [simp,no_atp]:
huffman@47108
  1175
  fixes a::"'b::linordered_idom"
haftmann@30652
  1176
  shows "(- a < 1) = (-1 < a)"
haftmann@30652
  1177
by auto
haftmann@30652
  1178
blanchet@35828
  1179
lemma minus_le_iff_1 [simp,no_atp]:
huffman@47108
  1180
  fixes a::"'b::linordered_idom"
haftmann@30652
  1181
  shows "(- a \<le> 1) = (-1 \<le> a)"
haftmann@30652
  1182
by auto
haftmann@30652
  1183
blanchet@35828
  1184
lemma minus_equation_iff_1 [simp,no_atp]:
huffman@47108
  1185
  fixes a::"'b::ring_1"
haftmann@30652
  1186
  shows "(- a = 1) = (a = -1)"
haftmann@30652
  1187
by (subst minus_equation_iff, auto)
haftmann@30652
  1188
haftmann@30652
  1189
haftmann@30652
  1190
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  1191
huffman@47108
  1192
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1193
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1194
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1195
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1196
haftmann@30652
  1197
haftmann@30652
  1198
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  1199
huffman@47108
  1200
lemmas le_divide_eq_numeral1 [simp] =
huffman@47108
  1201
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1202
  neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1203
huffman@47108
  1204
lemmas divide_le_eq_numeral1 [simp] =
huffman@47108
  1205
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1206
  neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1207
huffman@47108
  1208
lemmas less_divide_eq_numeral1 [simp] =
huffman@47108
  1209
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1210
  neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1211
huffman@47108
  1212
lemmas divide_less_eq_numeral1 [simp] =
huffman@47108
  1213
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
huffman@47108
  1214
  neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1215
huffman@47108
  1216
lemmas eq_divide_eq_numeral1 [simp] =
huffman@47108
  1217
  eq_divide_eq [of _ _ "numeral w"]
huffman@47108
  1218
  eq_divide_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1219
huffman@47108
  1220
lemmas divide_eq_eq_numeral1 [simp] =
huffman@47108
  1221
  divide_eq_eq [of _ "numeral w"]
huffman@47108
  1222
  divide_eq_eq [of _ "neg_numeral w"] for w
haftmann@30652
  1223
haftmann@30652
  1224
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  1225
haftmann@30652
  1226
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  1227
huffman@47108
  1228
lemmas le_divide_eq_numeral =
huffman@47108
  1229
  le_divide_eq [of "numeral w"]
huffman@47108
  1230
  le_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1231
huffman@47108
  1232
lemmas divide_le_eq_numeral =
huffman@47108
  1233
  divide_le_eq [of _ _ "numeral w"]
huffman@47108
  1234
  divide_le_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1235
huffman@47108
  1236
lemmas less_divide_eq_numeral =
huffman@47108
  1237
  less_divide_eq [of "numeral w"]
huffman@47108
  1238
  less_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1239
huffman@47108
  1240
lemmas divide_less_eq_numeral =
huffman@47108
  1241
  divide_less_eq [of _ _ "numeral w"]
huffman@47108
  1242
  divide_less_eq [of _ _ "neg_numeral w"] for w
huffman@47108
  1243
huffman@47108
  1244
lemmas eq_divide_eq_numeral =
huffman@47108
  1245
  eq_divide_eq [of "numeral w"]
huffman@47108
  1246
  eq_divide_eq [of "neg_numeral w"] for w
huffman@47108
  1247
huffman@47108
  1248
lemmas divide_eq_eq_numeral =
huffman@47108
  1249
  divide_eq_eq [of _ _ "numeral w"]
huffman@47108
  1250
  divide_eq_eq [of _ _ "neg_numeral w"] for w
haftmann@30652
  1251
haftmann@30652
  1252
haftmann@30652
  1253
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  1254
lemmas divide_const_simps =
huffman@47108
  1255
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
huffman@47108
  1256
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
haftmann@30652
  1257
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1258
haftmann@30652
  1259
text{*Division By @{text "-1"}*}
haftmann@30652
  1260
huffman@47108
  1261
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
huffman@47108
  1262
  unfolding minus_one [symmetric]
huffman@47108
  1263
  unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
huffman@47108
  1264
  by simp
haftmann@30652
  1265
huffman@47108
  1266
lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
huffman@47108
  1267
  unfolding minus_one [symmetric] by (rule divide_minus_left)
haftmann@30652
  1268
haftmann@30652
  1269
lemma half_gt_zero_iff:
huffman@47108
  1270
     "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
haftmann@30652
  1271
by auto
haftmann@30652
  1272
wenzelm@45607
  1273
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
haftmann@30652
  1274
huffman@47108
  1275
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
haftmann@36719
  1276
  by simp
haftmann@36719
  1277
haftmann@30652
  1278
haftmann@33320
  1279
subsection {* The divides relation *}
haftmann@33320
  1280
nipkow@33657
  1281
lemma zdvd_antisym_nonneg:
nipkow@33657
  1282
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  1283
  apply (simp add: dvd_def, auto)
nipkow@33657
  1284
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1285
  done
haftmann@33320
  1286
nipkow@33657
  1287
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  1288
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  1289
proof cases
nipkow@33657
  1290
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  1291
next
nipkow@33657
  1292
  assume "a \<noteq> 0"
haftmann@33320
  1293
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  1294
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  1295
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  1296
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@33320
  1297
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
haftmann@33320
  1298
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  1299
  thus ?thesis using k k' by auto
haftmann@33320
  1300
qed
haftmann@33320
  1301
haftmann@33320
  1302
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320
  1303
  apply (subgoal_tac "m = n + (m - n)")
haftmann@33320
  1304
   apply (erule ssubst)
haftmann@33320
  1305
   apply (blast intro: dvd_add, simp)
haftmann@33320
  1306
  done
haftmann@33320
  1307
haftmann@33320
  1308
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320
  1309
apply (rule iffI)
haftmann@33320
  1310
 apply (erule_tac [2] dvd_add)
haftmann@33320
  1311
 apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320
  1312
  apply (erule ssubst)
haftmann@33320
  1313
  apply (erule dvd_diff)
haftmann@33320
  1314
  apply(simp_all)
haftmann@33320
  1315
done
haftmann@33320
  1316
haftmann@33320
  1317
lemma dvd_imp_le_int:
haftmann@33320
  1318
  fixes d i :: int
haftmann@33320
  1319
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1320
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1321
proof -
haftmann@33320
  1322
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  1323
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  1324
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1325
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  1326
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  1327
qed
haftmann@33320
  1328
haftmann@33320
  1329
lemma zdvd_not_zless:
haftmann@33320
  1330
  fixes m n :: int
haftmann@33320
  1331
  assumes "0 < m" and "m < n"
haftmann@33320
  1332
  shows "\<not> n dvd m"
haftmann@33320
  1333
proof
haftmann@33320
  1334
  from assms have "0 < n" by auto
haftmann@33320
  1335
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  1336
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  1337
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  1338
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  1339
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1340
qed
haftmann@33320
  1341
haftmann@33320
  1342
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  1343
  shows "m dvd n"
haftmann@33320
  1344
proof-
haftmann@33320
  1345
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  1346
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@33320
  1347
    with h have False by (simp add: mult_assoc)}
haftmann@33320
  1348
  hence "n = m * h" by blast
haftmann@33320
  1349
  thus ?thesis by simp
haftmann@33320
  1350
qed
haftmann@33320
  1351
haftmann@33320
  1352
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  1353
proof -
haftmann@33320
  1354
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  1355
  proof -
haftmann@33320
  1356
    fix k
haftmann@33320
  1357
    assume A: "int y = int x * k"
wenzelm@42676
  1358
    then show "x dvd y"
wenzelm@42676
  1359
    proof (cases k)
wenzelm@42676
  1360
      case (nonneg n)
wenzelm@42676
  1361
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  1362
      then show ?thesis ..
haftmann@33320
  1363
    next
wenzelm@42676
  1364
      case (neg n)
wenzelm@42676
  1365
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  1366
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  1367
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  1368
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  1369
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  1370
    qed
haftmann@33320
  1371
  qed
haftmann@33320
  1372
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1373
qed
haftmann@33320
  1374
wenzelm@42676
  1375
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  1376
proof
haftmann@33320
  1377
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  1378
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  1379
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  1380
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  1381
next
haftmann@33320
  1382
  assume "\<bar>x\<bar>=1"
haftmann@33320
  1383
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  1384
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  1385
qed
haftmann@33320
  1386
haftmann@33320
  1387
lemma zdvd_mult_cancel1: 
haftmann@33320
  1388
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  1389
proof
haftmann@33320
  1390
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  1391
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  1392
next
haftmann@33320
  1393
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  1394
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  1395
qed
haftmann@33320
  1396
haftmann@33320
  1397
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  1398
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1399
haftmann@33320
  1400
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  1401
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1402
haftmann@33320
  1403
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  1404
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1405
haftmann@33341
  1406
lemma eq_nat_nat_iff:
haftmann@33341
  1407
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1408
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1409
haftmann@33341
  1410
lemma nat_power_eq:
haftmann@33341
  1411
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1412
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1413
haftmann@33320
  1414
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  1415
  apply (cases n)
haftmann@33320
  1416
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1417
  apply (cases z)
haftmann@33320
  1418
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  1419
  done
haftmann@33320
  1420
haftmann@36749
  1421
lemma zdvd_period:
haftmann@36749
  1422
  fixes a d :: int
haftmann@36749
  1423
  assumes "a dvd d"
haftmann@36749
  1424
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  1425
proof -
haftmann@36749
  1426
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1427
  show ?thesis
wenzelm@42676
  1428
  proof
haftmann@36749
  1429
    assume "a dvd (x + t)"
haftmann@36749
  1430
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1431
    then have "x = a * l - t" by simp
haftmann@36749
  1432
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  1433
  next
haftmann@36749
  1434
    assume "a dvd x + c * d + t"
haftmann@36749
  1435
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1436
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  1437
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  1438
  qed
haftmann@36749
  1439
qed
haftmann@36749
  1440
haftmann@33320
  1441
bulwahn@46756
  1442
subsection {* Finiteness of intervals *}
bulwahn@46756
  1443
bulwahn@46756
  1444
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756
  1445
proof (cases "a <= b")
bulwahn@46756
  1446
  case True
bulwahn@46756
  1447
  from this show ?thesis
bulwahn@46756
  1448
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1449
    case base
bulwahn@46756
  1450
    have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756
  1451
    from this show ?case by simp
bulwahn@46756
  1452
  next
bulwahn@46756
  1453
    case (step b)
bulwahn@46756
  1454
    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756
  1455
    from this step show ?case by simp
bulwahn@46756
  1456
  qed
bulwahn@46756
  1457
next
bulwahn@46756
  1458
  case False from this show ?thesis
bulwahn@46756
  1459
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1460
qed
bulwahn@46756
  1461
bulwahn@46756
  1462
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756
  1463
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1464
bulwahn@46756
  1465
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756
  1466
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1467
bulwahn@46756
  1468
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756
  1469
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1470
bulwahn@46756
  1471
haftmann@25919
  1472
subsection {* Configuration of the code generator *}
haftmann@25919
  1473
huffman@47108
  1474
text {* Constructors *}
huffman@47108
  1475
huffman@47108
  1476
definition Pos :: "num \<Rightarrow> int" where
huffman@47108
  1477
  [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1478
huffman@47108
  1479
definition Neg :: "num \<Rightarrow> int" where
huffman@47108
  1480
  [simp, code_abbrev]: "Neg = neg_numeral"
huffman@47108
  1481
huffman@47108
  1482
code_datatype "0::int" Pos Neg
huffman@47108
  1483
huffman@47108
  1484
huffman@47108
  1485
text {* Auxiliary operations *}
huffman@47108
  1486
huffman@47108
  1487
definition dup :: "int \<Rightarrow> int" where
huffman@47108
  1488
  [simp]: "dup k = k + k"
haftmann@26507
  1489
huffman@47108
  1490
lemma dup_code [code]:
huffman@47108
  1491
  "dup 0 = 0"
huffman@47108
  1492
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1493
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1494
  unfolding Pos_def Neg_def neg_numeral_def
huffman@47108
  1495
  by (simp_all add: numeral_Bit0)
huffman@47108
  1496
huffman@47108
  1497
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108
  1498
  [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1499
huffman@47108
  1500
lemma sub_code [code]:
huffman@47108
  1501
  "sub Num.One Num.One = 0"
huffman@47108
  1502
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1503
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1504
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1505
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1506
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1507
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1508
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1509
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
huffman@47108
  1510
  unfolding sub_def dup_def numeral.simps Pos_def Neg_def
huffman@47108
  1511
    neg_numeral_def numeral_BitM
huffman@47108
  1512
  by (simp_all only: algebra_simps)
haftmann@26507
  1513
huffman@47108
  1514
huffman@47108
  1515
text {* Implementations *}
huffman@47108
  1516
huffman@47108
  1517
lemma one_int_code [code, code_unfold]:
huffman@47108
  1518
  "1 = Pos Num.One"
huffman@47108
  1519
  by simp
huffman@47108
  1520
huffman@47108
  1521
lemma plus_int_code [code]:
huffman@47108
  1522
  "k + 0 = (k::int)"
huffman@47108
  1523
  "0 + l = (l::int)"
huffman@47108
  1524
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1525
  "Pos m + Neg n = sub m n"
huffman@47108
  1526
  "Neg m + Pos n = sub n m"
huffman@47108
  1527
  "Neg m + Neg n = Neg (m + n)"
huffman@47108
  1528
  by simp_all
haftmann@26507
  1529
huffman@47108
  1530
lemma uminus_int_code [code]:
huffman@47108
  1531
  "uminus 0 = (0::int)"
huffman@47108
  1532
  "uminus (Pos m) = Neg m"
huffman@47108
  1533
  "uminus (Neg m) = Pos m"
huffman@47108
  1534
  by simp_all
huffman@47108
  1535
huffman@47108
  1536
lemma minus_int_code [code]:
huffman@47108
  1537
  "k - 0 = (k::int)"
huffman@47108
  1538
  "0 - l = uminus (l::int)"
huffman@47108
  1539
  "Pos m - Pos n = sub m n"
huffman@47108
  1540
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1541
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1542
  "Neg m - Neg n = sub n m"
huffman@47108
  1543
  by simp_all
huffman@47108
  1544
huffman@47108
  1545
lemma times_int_code [code]:
huffman@47108
  1546
  "k * 0 = (0::int)"
huffman@47108
  1547
  "0 * l = (0::int)"
huffman@47108
  1548
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1549
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1550
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1551
  "Neg m * Neg n = Pos (m * n)"
huffman@47108
  1552
  by simp_all
haftmann@26507
  1553
haftmann@38857
  1554
instantiation int :: equal
haftmann@26507
  1555
begin
haftmann@26507
  1556
haftmann@37767
  1557
definition
huffman@47108
  1558
  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1559
huffman@47108
  1560
instance by default (rule equal_int_def)
haftmann@26507
  1561
haftmann@26507
  1562
end
haftmann@26507
  1563
huffman@47108
  1564
lemma equal_int_code [code]:
huffman@47108
  1565
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1566
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1567
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1568
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1569
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1570
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1571
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1572
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1573
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1574
  by (auto simp add: equal)
haftmann@26507
  1575
huffman@47108
  1576
lemma equal_int_refl [code nbe]:
haftmann@38857
  1577
  "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108
  1578
  by (fact equal_refl)
haftmann@26507
  1579
haftmann@28562
  1580
lemma less_eq_int_code [code]:
huffman@47108
  1581
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1582
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1583
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1584
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1585
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1586
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1587
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1588
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1589
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1590
  by simp_all
haftmann@26507
  1591
haftmann@28562
  1592
lemma less_int_code [code]:
huffman@47108
  1593
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1594
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1595
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1596
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1597
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1598
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1599
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1600
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1601
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1602
  by simp_all
haftmann@25919
  1603
huffman@47108
  1604
lemma nat_code [code]:
huffman@47108
  1605
  "nat (Int.Neg k) = 0"
huffman@47108
  1606
  "nat 0 = 0"
huffman@47108
  1607
  "nat (Int.Pos k) = nat_of_num k"
huffman@47108
  1608
  by (simp_all add: nat_of_num_numeral nat_numeral)
haftmann@25928
  1609
huffman@47108
  1610
lemma (in ring_1) of_int_code [code]:
huffman@47108
  1611
  "of_int (Int.Neg k) = neg_numeral k"
huffman@47108
  1612
  "of_int 0 = 0"
huffman@47108
  1613
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1614
  by simp_all
haftmann@25919
  1615
huffman@47108
  1616
huffman@47108
  1617
text {* Serializer setup *}
haftmann@25919
  1618
haftmann@25919
  1619
code_modulename SML
haftmann@33364
  1620
  Int Arith
haftmann@25919
  1621
haftmann@25919
  1622
code_modulename OCaml
haftmann@33364
  1623
  Int Arith
haftmann@25919
  1624
haftmann@25919
  1625
code_modulename Haskell
haftmann@33364
  1626
  Int Arith
haftmann@25919
  1627
haftmann@25919
  1628
quickcheck_params [default_type = int]
haftmann@25919
  1629
huffman@47108
  1630
hide_const (open) Pos Neg sub dup
haftmann@25919
  1631
haftmann@25919
  1632
haftmann@25919
  1633
subsection {* Legacy theorems *}
haftmann@25919
  1634
haftmann@25919
  1635
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1636
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1637
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  1638
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
wenzelm@45607
  1639
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1640
lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607
  1641
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919
  1642
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1643
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607
  1644
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1645
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1646
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1647
lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207
  1648
lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607
  1649
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919
  1650
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1651
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255
  1652
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255
  1653
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960
  1654
haftmann@31015
  1655
lemma zpower_zpower:
haftmann@31015
  1656
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  1657
  by (rule power_mult [symmetric])
haftmann@31015
  1658
haftmann@31015
  1659
lemma int_power:
haftmann@31015
  1660
  "int (m ^ n) = int m ^ n"
haftmann@31015
  1661
  by (rule of_nat_power)
haftmann@31015
  1662
haftmann@31015
  1663
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  1664
huffman@48045
  1665
text {* De-register @{text "int"} as a quotient type: *}
huffman@48045
  1666
huffman@48045
  1667
lemmas [transfer_rule del] =
huffman@48045
  1668
  int.id_abs_transfer int.rel_eq_transfer zero_int.transfer one_int.transfer
huffman@48045
  1669
  plus_int.transfer uminus_int.transfer minus_int.transfer times_int.transfer
huffman@48045
  1670
  int_transfer less_eq_int.transfer less_int.transfer of_int.transfer
kuncar@51185
  1671
  nat.transfer int.right_unique int.right_total int.bi_total
huffman@48045
  1672
huffman@48045
  1673
declare Quotient_int [quot_del]
huffman@48045
  1674
haftmann@25919
  1675
end