src/HOL/Int.thy
author haftmann
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(*  Title:      HOL/Int.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
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*)
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header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
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theory Int
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imports Equiv_Relations Wellfounded Quotient FunDef
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begin
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subsection {* Definition of integers as a quotient type *}
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where
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  "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
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  by (simp add: intrel_def)
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quotient_type int = "nat \<times> nat" / "intrel"
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  morphisms Rep_Integ Abs_Integ
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proof (rule equivpI)
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  show "reflp intrel"
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    unfolding reflp_def by auto
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  show "symp intrel"
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    unfolding symp_def by auto
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  show "transp intrel"
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    unfolding transp_def by auto
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
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by (induct z) auto
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subsection {* Integers form a commutative ring *}
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instantiation int :: comm_ring_1
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begin
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lift_definition zero_int :: "int" is "(0, 0)" .
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lift_definition one_int :: "int" is "(1, 0)" .
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
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  by clarsimp
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lift_definition uminus_int :: "int \<Rightarrow> int"
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  is "\<lambda>(x, y). (y, x)"
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  by clarsimp
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
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  by clarsimp
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
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proof (clarsimp)
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  fix s t u v w x y z :: nat
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  assume "s + v = u + t" and "w + z = y + x"
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  hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
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       = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
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    by simp
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  thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
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    by (simp add: algebra_simps)
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qed
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instance
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  by default (transfer, clarsimp simp: algebra_simps)+
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end
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abbreviation int :: "nat \<Rightarrow> int" where
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  "int \<equiv> of_nat"
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lemma int_def: "int n = Abs_Integ (n, 0)"
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  by (induct n, simp add: zero_int.abs_eq,
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    simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]:
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  "(fun_rel (op =) cr_int) (\<lambda>n. (n, 0)) int"
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  unfolding fun_rel_def cr_int_def int_def by simp
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lemma int_diff_cases:
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  obtains (diff) m n where "z = int m - int n"
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  by transfer clarsimp
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subsection {* Integers are totally ordered *}
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instantiation int :: linorder
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begin
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
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  by auto
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v < u + y"
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  by auto
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instance
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  by default (transfer, force)+
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end
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instantiation int :: distrib_lattice
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begin
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definition
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  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
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end
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subsection {* Ordering properties of arithmetic operations *}
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instance int :: ordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by transfer clarsimp
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qed
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text{*Strict Monotonicity of Multiplication*}
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text{*strict, in 1st argument; proof is by induction on k>0*}
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lemma zmult_zless_mono2_lemma:
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     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
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apply (induct k)
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apply simp
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apply (simp add: distrib_right)
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apply (case_tac "k=0")
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apply (simp_all add: add_strict_mono)
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done
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text{*The integers form an ordered integral domain*}
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instantiation int :: linordered_idom
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begin
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definition
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  zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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instance proof
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  fix i j k :: int
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  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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    by (rule zmult_zless_mono2)
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  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
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    by (simp only: zabs_def)
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  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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    by (simp only: zsgn_def)
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qed
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end
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lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd:
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  "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
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apply transfer
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apply auto
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apply (rename_tac a b c d)
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apply (rule_tac x="c+b - Suc(a+d)" in exI)
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apply arith
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done
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lemmas int_distrib =
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  distrib_right [of z1 z2 w]
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  distrib_left [of w z1 z2]
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  left_diff_distrib [of z1 z2 w]
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  right_diff_distrib [of w z1 z2]
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  for z1 z2 w :: int
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subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
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context ring_1
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begin
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lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
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  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_0 [simp]: "of_int 0 = 0"
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  by transfer simp
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lemma of_int_1 [simp]: "of_int 1 = 1"
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  by transfer simp
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
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  by transfer (clarsimp simp add: algebra_simps)
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
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  by (transfer fixing: uminus) clarsimp
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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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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51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   304
instance int :: no_top
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   305
  apply default
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   306
  apply (rule_tac x="x + 1" in exI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   307
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   308
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   309
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   310
instance int :: no_bot
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   311
  apply default
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   312
  apply (rule_tac x="x - 1" in exI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   313
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   314
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   315
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   316
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   317
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   318
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   319
  by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   320
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   321
lemma nat_int [simp]: "nat (int n) = n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   322
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   323
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   324
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   325
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   326
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   327
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   328
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   329
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   330
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   331
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   332
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   333
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   334
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   335
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   336
text{*An alternative condition is @{term "0 \<le> w"} *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   337
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   338
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   339
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   340
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   341
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   342
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   343
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   344
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   345
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   346
lemma nonneg_eq_int:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   347
  fixes z :: int
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   348
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   349
  shows P
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   350
  using assms by (blast dest: nat_0_le sym)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   351
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   352
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   353
  by transfer (clarsimp simp add: le_imp_diff_is_add)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   354
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   355
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   356
by (simp only: eq_commute [of m] nat_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   357
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   358
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   359
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   360
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   361
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   362
  by transfer (clarsimp simp add: le_diff_conv)
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   363
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   364
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   365
  by transfer auto
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   366
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   367
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   368
  by transfer clarsimp
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   369
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   370
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   371
by (auto simp add: nat_eq_iff2)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   372
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   373
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   374
by (insert zless_nat_conj [of 0], auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   375
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   376
lemma nat_add_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   377
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   378
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   379
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   380
lemma nat_diff_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   381
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   382
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   383
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   384
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   385
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   386
53065
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   387
lemma le_nat_iff:
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   388
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   389
  by transfer auto
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   390
  
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   391
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   392
  by transfer (clarsimp simp add: less_diff_conv)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   393
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   394
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   395
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   396
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   397
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   398
  by transfer (clarsimp simp add: of_nat_diff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   399
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   400
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   401
29779
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   402
text {* For termination proofs: *}
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   403
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   404
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   405
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   406
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   407
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   408
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   409
by (simp add: order_less_le del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   410
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   411
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   412
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   413
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   414
lemma negative_zle_0: "- int n \<le> 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   415
by (simp add: minus_le_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   416
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   417
lemma negative_zle [iff]: "- int n \<le> int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   418
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   419
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   420
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   421
by (subst le_minus_iff, simp del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   422
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   423
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   424
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   425
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   426
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   427
by (simp add: linorder_not_less)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   428
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   429
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   430
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   431
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   432
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   433
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   434
  have "(w \<le> z) = (0 \<le> z - w)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   435
    by (simp only: le_diff_eq add_0_left)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   436
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   437
    by (auto elim: zero_le_imp_eq_int)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   438
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29046
diff changeset
   439
    by (simp only: algebra_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   440
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   441
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   442
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   443
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   444
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   445
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   446
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   447
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   448
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   449
text{*This version is proved for all ordered rings, not just integers!
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   450
      It is proved here because attribute @{text arith_split} is not available
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35032
diff changeset
   451
      in theory @{text Rings}.
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   452
      But is it really better than just rewriting with @{text abs_if}?*}
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
   453
lemma abs_split [arith_split,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34055
diff changeset
   454
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   455
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   456
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   457
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   458
apply transfer
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   459
apply clarsimp
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   460
apply (rule_tac x="b - Suc a" in exI, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   461
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   462
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   463
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   464
subsection {* Cases and induction *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   465
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   466
text{*Now we replace the case analysis rule by a more conventional one:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   467
whether an integer is negative or not.*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   468
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   469
theorem int_cases [case_names nonneg neg, cases type: int]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   470
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   471
apply (cases "z < 0")
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   472
apply (blast dest!: negD)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   473
apply (simp add: linorder_not_less del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   474
apply auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   475
apply (blast dest: nat_0_le [THEN sym])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   476
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   477
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   478
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   479
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   480
  by (cases z) auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   481
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   482
lemma nonneg_int_cases:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   483
  assumes "0 \<le> k" obtains n where "k = int n"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   484
  using assms by (cases k, simp, simp del: of_nat_Suc)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   485
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   486
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   487
  -- {* Unfold all @{text let}s involving constants *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   488
  unfolding Let_def ..
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36811
diff changeset
   489
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   490
lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   491
  -- {* Unfold all @{text let}s involving constants *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   492
  unfolding Let_def ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   493
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   494
text {* Unfold @{text min} and @{text max} on numerals. *}
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   495
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   496
lemmas max_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   497
  max_def [of "numeral u" "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   498
  max_def [of "numeral u" "neg_numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   499
  max_def [of "neg_numeral u" "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   500
  max_def [of "neg_numeral u" "neg_numeral v"] for u v
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   501
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   502
lemmas min_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   503
  min_def [of "numeral u" "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   504
  min_def [of "numeral u" "neg_numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   505
  min_def [of "neg_numeral u" "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   506
  min_def [of "neg_numeral u" "neg_numeral v"] for u v
26075
815f3ccc0b45 added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents: 26072
diff changeset
   507
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   508
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   509
subsubsection {* Binary comparisons *}
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   510
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   511
text {* Preliminaries *}
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   512
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   513
lemma even_less_0_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34055
diff changeset
   514
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   515
proof -
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
   516
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   517
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   518
    by (simp add: mult_less_0_iff zero_less_two 
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   519
                  order_less_not_sym [OF zero_less_two])
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   520
  finally show ?thesis .
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   521
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   522
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   523
lemma le_imp_0_less: 
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   524
  assumes le: "0 \<le> z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   525
  shows "(0::int) < 1 + z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   526
proof -
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   527
  have "0 \<le> z" by fact
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   528
  also have "... < z + 1" by (rule less_add_one)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   529
  also have "... = 1 + z" by (simp add: add_ac)
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   530
  finally show "0 < 1 + z" .
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   531
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   532
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   533
lemma odd_less_0_iff:
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   534
  "(1 + z + z < 0) = (z < (0::int))"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   535
proof (cases z)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   536
  case (nonneg n)
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   537
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   538
                             le_imp_0_less [THEN order_less_imp_le])  
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   539
next
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   540
  case (neg n)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
   541
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
   542
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   543
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   544
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   545
subsubsection {* Comparisons, for Ordered Rings *}
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   546
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   547
lemmas double_eq_0_iff = double_zero
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   548
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   549
lemma odd_nonzero:
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
   550
  "1 + z + z \<noteq> (0::int)"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   551
proof (cases z)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   552
  case (nonneg n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   553
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   554
  thus ?thesis using  le_imp_0_less [OF le]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   555
    by (auto simp add: add_assoc) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   556
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   557
  case (neg n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   558
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   559
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   560
    assume eq: "1 + z + z = 0"
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   561
    have "(0::int) < 1 + (int n + int n)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   562
      by (simp add: le_imp_0_less add_increasing) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   563
    also have "... = - (1 + z + z)" 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   564
      by (simp add: neg add_assoc [symmetric]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   565
    also have "... = 0" by (simp add: eq) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   566
    finally have "0<0" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   567
    thus False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   568
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   569
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   570
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   571
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   572
subsection {* The Set of Integers *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   573
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   574
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   575
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   576
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   577
definition Ints  :: "'a set" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36811
diff changeset
   578
  "Ints = range of_int"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   579
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   580
notation (xsymbols)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   581
  Ints  ("\<int>")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   582
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   583
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   584
  by (simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   585
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   586
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   587
  using Ints_of_int [of "of_nat n"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   588
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   589
lemma Ints_0 [simp]: "0 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   590
  using Ints_of_int [of "0"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   591
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   592
lemma Ints_1 [simp]: "1 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   593
  using Ints_of_int [of "1"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   594
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   595
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   596
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   597
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   598
apply (rule of_int_add [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   599
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   600
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   601
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   602
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   603
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   604
apply (rule of_int_minus [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   605
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   606
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   607
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   608
apply (auto simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   609
apply (rule range_eqI)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   610
apply (rule of_int_diff [symmetric])
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   611
done
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   612
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   613
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   614
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   615
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   616
apply (rule of_int_mult [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   617
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   618
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   619
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   620
by (induct n) simp_all
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   621
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   622
lemma Ints_cases [cases set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   623
  assumes "q \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   624
  obtains (of_int) z where "q = of_int z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   625
  unfolding Ints_def
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   626
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   627
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   628
  then obtain z where "q = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   629
  then show thesis ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   630
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   631
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   632
lemma Ints_induct [case_names of_int, induct set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   633
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   634
  by (rule Ints_cases) auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   635
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   636
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   637
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   638
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   639
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   640
lemma Ints_double_eq_0_iff:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   641
  assumes in_Ints: "a \<in> Ints"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   642
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   643
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   644
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   645
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   646
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   647
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   648
    assume "a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   649
    thus "a + a = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   650
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   651
    assume eq: "a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   652
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   653
    hence "z + z = 0" by (simp only: of_int_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   654
    hence "z = 0" by (simp only: double_eq_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   655
    thus "a = 0" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   656
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   657
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   658
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   659
lemma Ints_odd_nonzero:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   660
  assumes in_Ints: "a \<in> Ints"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   661
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   662
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   663
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   664
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   665
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   666
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   667
    assume eq: "1 + a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   668
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   669
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   670
    with odd_nonzero show False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   671
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   672
qed 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   673
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   674
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   675
  using of_nat_in_Nats [of "numeral w"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   676
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   677
lemma Ints_odd_less_0: 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   678
  assumes in_Ints: "a \<in> Ints"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34055
diff changeset
   679
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   680
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   681
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   682
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   683
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   684
    by (simp add: a)
45532
74b17a0881b3 Int.thy: remove duplicate lemmas double_less_0_iff and odd_less_0, use {even,odd}_less_0_iff instead
huffman
parents: 45219
diff changeset
   685
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   686
  also have "... = (a < 0)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   687
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   688
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   689
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   690
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   691
subsection {* @{term setsum} and @{term setprod} *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   692
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   693
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   694
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   695
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   696
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   697
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   698
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   699
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   700
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   701
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   702
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   703
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   704
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   705
  apply (erule finite_induct, auto simp add: of_nat_mult)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   706
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   707
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   708
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   709
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   710
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   711
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   712
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   713
lemmas int_setsum = of_nat_setsum [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   714
lemmas int_setprod = of_nat_setprod [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   715
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   716
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   717
text {* Legacy theorems *}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   718
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   719
lemmas zle_int = of_nat_le_iff [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   720
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   721
lemmas numeral_1_eq_1 = numeral_One
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   722
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   723
subsection {* Setting up simplification procedures *}
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   724
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   725
lemmas int_arith_rules =
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   726
  neg_le_iff_le numeral_One
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   727
  minus_zero diff_minus left_minus right_minus
45219
29f6e990674d removed mult_Bit1 from int_arith_rules (cf. 882403378a41 and 3078fd2eec7b, where mult_num1 erroneously replaced mult_1)
huffman
parents: 45196
diff changeset
   728
  mult_zero_left mult_zero_right mult_1_left mult_1_right
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   729
  mult_minus_left mult_minus_right
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   730
  minus_add_distrib minus_minus mult_assoc
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   731
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   732
  of_int_0 of_int_1 of_int_add of_int_mult
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   733
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48066
diff changeset
   734
ML_file "Tools/int_arith.ML"
30496
7cdcc9dd95cb vague cleanup in arith proof tools setup: deleted dead code, more proper structures, clearer arrangement
haftmann
parents: 30273
diff changeset
   735
declaration {* K Int_Arith.setup *}
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   736
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   737
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   738
  "(m::'a::linordered_idom) <= n" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   739
  "(m::'a::linordered_idom) = n") =
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43531
diff changeset
   740
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43531
diff changeset
   741
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   742
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   743
subsection{*Lemmas About Small Numerals*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   744
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   745
lemma abs_power_minus_one [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   746
  "abs(-1 ^ n) = (1::'a::linordered_idom)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   747
by (simp add: power_abs)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   748
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   749
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   750
subsection{*More Inequality Reasoning*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   751
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   752
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   753
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   754
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   755
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   756
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   757
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   758
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   759
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   760
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   761
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   762
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   763
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   764
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   765
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   766
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   767
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   768
subsection{*The functions @{term nat} and @{term int}*}
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   769
48044
fea6f3060b65 remove unnecessary simp rules involving Abs_Integ
huffman
parents: 47255
diff changeset
   770
text{*Simplify the term @{term "w + - z"}*}
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   771
lemmas diff_int_def_symmetric = diff_def [where 'a=int, symmetric, simp]
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   772
44695
075327b8e841 remove duplicate lemma nat_zero in favor of nat_0
huffman
parents: 43595
diff changeset
   773
lemma nat_0 [simp]: "nat 0 = 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   774
by (simp add: nat_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   775
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   776
lemma nat_1 [simp]: "nat 1 = Suc 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   777
by (subst nat_eq_iff, simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   778
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   779
lemma nat_2: "nat 2 = Suc (Suc 0)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   780
by (subst nat_eq_iff, simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   781
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   782
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   783
apply (insert zless_nat_conj [of 1 z])
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   784
apply auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   785
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   786
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   787
text{*This simplifies expressions of the form @{term "int n = z"} where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   788
      z is an integer literal.*}
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   789
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   790
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   791
lemma split_nat [arith_split]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   792
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   793
  (is "?P = (?L & ?R)")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   794
proof (cases "i < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   795
  case True thus ?thesis by auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   796
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   797
  case False
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   798
  have "?P = ?L"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   799
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   800
    assume ?P thus ?L using False by clarsimp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   801
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   802
    assume ?L thus ?P using False by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   803
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   804
  with False show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   805
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   806
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   807
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   808
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   809
33056
791a4655cae3 renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents: 32437
diff changeset
   810
lemma of_int_of_nat [nitpick_simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   811
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   812
proof (cases "k < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   813
  case True then have "0 \<le> - k" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   814
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   815
  with True show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   816
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   817
  case False then show ?thesis by (simp add: not_less of_nat_nat)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   818
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   819
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   820
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   821
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   822
lemma nat_mult_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   823
  fixes z z' :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   824
  assumes "0 \<le> z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   825
  shows "nat (z * z') = nat z * nat z'"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   826
proof (cases "0 \<le> z'")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   827
  case False with assms have "z * z' \<le> 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   828
    by (simp add: not_le mult_le_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   829
  then have "nat (z * z') = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   830
  moreover from False have "nat z' = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   831
  ultimately show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   832
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   833
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   834
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   835
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   836
      (simp only: of_nat_mult of_nat_nat [OF True]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   837
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   838
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   839
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   840
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   841
apply (rule trans)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   842
apply (rule_tac [2] nat_mult_distrib, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   843
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   844
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   845
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   846
apply (cases "z=0 | w=0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   847
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   848
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   849
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   850
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   851
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   852
apply (rule sym)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   853
apply (simp add: nat_eq_iff)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   854
done
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   855
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   856
lemma diff_nat_eq_if:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   857
     "nat z - nat z' =  
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   858
        (if z' < 0 then nat z   
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   859
         else let d = z-z' in     
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   860
              if d < 0 then 0 else nat d)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   861
by (simp add: Let_def nat_diff_distrib [symmetric])
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   862
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   863
(* nat_diff_distrib has too-strong premises *)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   864
lemma nat_diff_distrib': "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x - y) = nat x - nat y"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   865
apply (rule int_int_eq [THEN iffD1], clarsimp)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   866
apply (subst of_nat_diff)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   867
apply (rule nat_mono, simp_all)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   868
done
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   869
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51112
diff changeset
   870
lemma nat_numeral [simp]:
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   871
  "nat (numeral k) = numeral k"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   872
  by (simp add: nat_eq_iff)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   873
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   874
lemma nat_neg_numeral [simp]:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   875
  "nat (neg_numeral k) = 0"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   876
  by simp
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   877
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   878
lemma diff_nat_numeral [simp]: 
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   879
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   880
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   881
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   882
lemma nat_numeral_diff_1 [simp]:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   883
  "numeral v - (1::nat) = nat (numeral v - 1)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   884
  using diff_nat_numeral [of v Num.One] by simp
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   885
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47228
diff changeset
   886
lemmas nat_arith = diff_nat_numeral
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47228
diff changeset
   887
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   888
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   889
subsection "Induction principles for int"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   890
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   891
text{*Well-founded segments of the integers*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   892
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   893
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   894
  int_ge_less_than  ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   895
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   896
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   897
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   898
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   899
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   900
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   901
    by (auto simp add: int_ge_less_than_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   902
  thus ?thesis 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   903
    by (rule wf_subset [OF wf_measure]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   904
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   905
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   906
text{*This variant looks odd, but is typical of the relations suggested
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   907
by RankFinder.*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   908
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   909
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   910
  int_ge_less_than2 ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   911
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   912
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   913
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   914
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   915
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   916
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   917
    by (auto simp add: int_ge_less_than2_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   918
  thus ?thesis 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   919
    by (rule wf_subset [OF wf_measure]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   920
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   921
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   922
(* `set:int': dummy construction *)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   923
theorem int_ge_induct [case_names base step, induct set: int]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   924
  fixes i :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   925
  assumes ge: "k \<le> i" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   926
    base: "P k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   927
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   928
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   929
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   930
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   931
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   932
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   933
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   934
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   935
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   936
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   937
      case (Suc n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   938
      then have "n = nat((i - 1) - k)" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   939
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   940
      have ki1: "k \<le> i - 1" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   941
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   942
      have "P (i - 1)" by (rule Suc.hyps)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   943
      from step [OF ki1 this] show ?case by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   944
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   945
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   946
  with ge show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   947
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   948
25928
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
   949
(* `set:int': dummy construction *)
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
   950
theorem int_gr_induct [case_names base step, induct set: int]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   951
  assumes gr: "k < (i::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   952
        base: "P(k+1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   953
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   954
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   955
apply(rule int_ge_induct[of "k + 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   956
  using gr apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   957
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   958
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   959
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   960
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   961
theorem int_le_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   962
  assumes le: "i \<le> (k::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   963
        base: "P(k)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   964
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   965
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   966
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   967
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   968
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   969
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   970
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   971
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   972
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   973
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   974
      case (Suc n)
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   975
      hence "n = nat (k - (i + 1))" by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   976
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   977
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   978
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   979
      have "P (i + 1)" by(rule Suc.hyps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   980
      from step[OF ki1 this] show ?case by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   981
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   982
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   983
  with le show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   984
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   985
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   986
theorem int_less_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   987
  assumes less: "(i::int) < k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   988
        base: "P(k - 1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   989
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   990
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   991
apply(rule int_le_induct[of _ "k - 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   992
  using less apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   993
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   994
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   995
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   996
36811
4ab4aa5bee1c renamed former Int.int_induct to Int.int_of_nat_induct, former Presburger.int_induct to Int.int_induct: is more conservative and more natural than the intermediate solution
haftmann
parents: 36801
diff changeset
   997
theorem int_induct [case_names base step1 step2]:
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
   998
  fixes k :: int
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
   999
  assumes base: "P k"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1000
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1001
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1002
  shows "P i"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1003
proof -
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1004
  have "i \<le> k \<or> i \<ge> k" by arith
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1005
  then show ?thesis
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1006
  proof
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1007
    assume "i \<ge> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1008
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1009
      by (rule int_ge_induct) (fact step1)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1010
  next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1011
    assume "i \<le> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1012
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1013
      by (rule int_le_induct) (fact step2)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1014
  qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1015
qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1016
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1017
subsection{*Intermediate value theorems*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1018
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1019
lemma int_val_lemma:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1020
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1021
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1022
unfolding One_nat_def
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1023
apply (induct n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1024
apply simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1025
apply (intro strip)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1026
apply (erule impE, simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1027
apply (erule_tac x = n in allE, simp)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1028
apply (case_tac "k = f (Suc n)")
27106
ff27dc6e7d05 removed some dubious code lemmas
haftmann
parents: 26975
diff changeset
  1029
apply force
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1030
apply (erule impE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1031
 apply (simp add: abs_if split add: split_if_asm)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1032
apply (blast intro: le_SucI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1033
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1034
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1035
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1036
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1037
lemma nat_intermed_int_val:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1038
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1039
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1040
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1041
       in int_val_lemma)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1042
unfolding One_nat_def
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1043
apply simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1044
apply (erule exE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1045
apply (rule_tac x = "i+m" in exI, arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1046
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1047
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1048
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1049
subsection{*Products and 1, by T. M. Rasmussen*}
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1050
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1051
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1052
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1053
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1054
lemma abs_zmult_eq_1:
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1055
  assumes mn: "\<bar>m * n\<bar> = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1056
  shows "\<bar>m\<bar> = (1::int)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1057
proof -
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1058
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1059
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1060
  have "~ (2 \<le> \<bar>m\<bar>)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1061
  proof
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1062
    assume "2 \<le> \<bar>m\<bar>"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1063
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1064
      by (simp add: mult_mono 0) 
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1065
    also have "... = \<bar>m*n\<bar>" 
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1066
      by (simp add: abs_mult)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1067
    also have "... = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1068
      by (simp add: mn)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1069
    finally have "2*\<bar>n\<bar> \<le> 1" .
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1070
    thus "False" using 0
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1071
      by arith
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1072
  qed
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1073
  thus ?thesis using 0
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1074
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1075
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1076
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1077
ML_val {* @{const_name neg_numeral} *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1078
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1079
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1080
by (insert abs_zmult_eq_1 [of m n], arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1081
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1082
lemma pos_zmult_eq_1_iff:
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1083
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1084
proof -
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1085
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1086
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1087
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1088
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1089
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1090
apply (rule iffI) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1091
 apply (frule pos_zmult_eq_1_iff_lemma)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1092
 apply (simp add: mult_commute [of m]) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1093
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1094
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1095
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1096
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1097
proof
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1098
  assume "finite (UNIV::int set)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1099
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1100
    by (rule injI) simp
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1101
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1102
    by (rule finite_UNIV_inj_surj)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1103
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1104
  then show False by (simp add: pos_zmult_eq_1_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1105
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1106
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1107
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1108
subsection {* Further theorems on numerals *}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1109
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1110
subsubsection{*Special Simplification for Constants*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1111
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1112
text{*These distributive laws move literals inside sums and differences.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1113
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1114
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1115
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1116
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1117
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1118
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1119
text{*These are actually for fields, like real: but where else to put them?*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1120
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1121
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1122
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1123
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1124
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1125
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1126
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1127
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1128
  strange, but then other simprocs simplify the quotient.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1129
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1130
lemmas inverse_eq_divide_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1131
  inverse_eq_divide [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1132
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1133
lemmas inverse_eq_divide_neg_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1134
  inverse_eq_divide [of "neg_numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1135
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1136
text {*These laws simplify inequalities, moving unary minus from a term
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1137
into the literal.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1138
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1139
lemmas le_minus_iff_numeral [simp, no_atp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1140
  le_minus_iff [of "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1141
  le_minus_iff [of "neg_numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1142
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1143
lemmas equation_minus_iff_numeral [simp, no_atp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1144
  equation_minus_iff [of "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1145
  equation_minus_iff [of "neg_numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1146
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1147
lemmas minus_less_iff_numeral [simp, no_atp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1148
  minus_less_iff [of _ "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1149
  minus_less_iff [of _ "neg_numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1150
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1151
lemmas minus_le_iff_numeral [simp, no_atp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1152
  minus_le_iff [of _ "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1153
  minus_le_iff [of _ "neg_numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1154
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1155
lemmas minus_equation_iff_numeral [simp, no_atp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1156
  minus_equation_iff [of _ "numeral v"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1157
  minus_equation_iff [of _ "neg_numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1158
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1159
text{*To Simplify Inequalities Where One Side is the Constant 1*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1160
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1161
lemma less_minus_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1162
  fixes b::"'b::linordered_idom"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1163
  shows "(1 < - b) = (b < -1)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1164
by auto
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1165
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1166
lemma le_minus_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1167
  fixes b::"'b::linordered_idom"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1168
  shows "(1 \<le> - b) = (b \<le> -1)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1169
by auto
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1170
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1171
lemma equation_minus_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1172
  fixes b::"'b::ring_1"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1173
  shows "(1 = - b) = (b = -1)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1174
by (subst equation_minus_iff, auto)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1175
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1176
lemma minus_less_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1177
  fixes a::"'b::linordered_idom"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1178
  shows "(- a < 1) = (-1 < a)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1179
by auto
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1180
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1181
lemma minus_le_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1182
  fixes a::"'b::linordered_idom"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1183
  shows "(- a \<le> 1) = (-1 \<le> a)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1184
by auto
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1185
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35634
diff changeset
  1186
lemma minus_equation_iff_1 [simp,no_atp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1187
  fixes a::"'b::ring_1"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1188
  shows "(- a = 1) = (a = -1)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1189
by (subst minus_equation_iff, auto)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1190
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1191
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1192
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1193
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1194
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1195
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1196
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1197
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1198
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1199
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1200
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1201
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1202
lemmas le_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1203
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1204
  neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1205
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1206
lemmas divide_le_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1207
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1208
  neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1209
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1210
lemmas less_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1211
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1212
  neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1213
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1214
lemmas divide_less_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1215
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1216
  neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1217
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1218
lemmas eq_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1219
  eq_divide_eq [of _ _ "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1220
  eq_divide_eq [of _ _ "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1221
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1222
lemmas divide_eq_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1223
  divide_eq_eq [of _ "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1224
  divide_eq_eq [of _ "neg_numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1225
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1226
subsubsection{*Optional Simplification Rules Involving Constants*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1227
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1228
text{*Simplify quotients that are compared with a literal constant.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1229
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1230
lemmas le_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1231
  le_divide_eq [of "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1232
  le_divide_eq [of "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1233
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1234
lemmas divide_le_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1235
  divide_le_eq [of _ _ "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1236
  divide_le_eq [of _ _ "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1237
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1238
lemmas less_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1239
  less_divide_eq [of "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1240
  less_divide_eq [of "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1241
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1242
lemmas divide_less_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1243
  divide_less_eq [of _ _ "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1244
  divide_less_eq [of _ _ "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1245
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1246
lemmas eq_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1247
  eq_divide_eq [of "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1248
  eq_divide_eq [of "neg_numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1249
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1250
lemmas divide_eq_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1251
  divide_eq_eq [of _ _ "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1252
  divide_eq_eq [of _ _ "neg_numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1253
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1254
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1255
text{*Not good as automatic simprules because they cause case splits.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1256
lemmas divide_const_simps =
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1257
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1258
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1259
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1260
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1261
text{*Division By @{text "-1"}*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1262
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1263
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1264
  unfolding minus_one [symmetric]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1265
  unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1266
  by simp
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1267
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1268
lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1269
  unfolding minus_one [symmetric] by (rule divide_minus_left)
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1270
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1271
lemma half_gt_zero_iff:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1272
     "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1273
by auto
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1274
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45533
diff changeset
  1275
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1276
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1277
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
36719
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36716
diff changeset
  1278
  by simp
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36716
diff changeset
  1279
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1280
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1281
subsection {* The divides relation *}
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1282
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1283
lemma zdvd_antisym_nonneg:
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1284
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1285
  apply (simp add: dvd_def, auto)
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1286
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1287
  done
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1288
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1289
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1290
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1291
proof cases
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1292
  assume "a = 0" with assms show ?thesis by simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1293
next
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1294
  assume "a \<noteq> 0"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1295
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1296
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1297
  from k k' have "a = a*k*k'" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1298
  with mult_cancel_left1[where c="a" and b="k*k'"]
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1299
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1300
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1301
  thus ?thesis using k k' by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1302
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1303
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1304
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1305
  apply (subgoal_tac "m = n + (m - n)")
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1306
   apply (erule ssubst)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1307
   apply (blast intro: dvd_add, simp)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1308
  done
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1309
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1310
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1311
apply (rule iffI)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1312
 apply (erule_tac [2] dvd_add)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1313
 apply (subgoal_tac "n = (n + k * m) - k * m")
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1314
  apply (erule ssubst)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1315
  apply (erule dvd_diff)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1316
  apply(simp_all)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1317
done
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1318
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1319
lemma dvd_imp_le_int:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1320
  fixes d i :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1321
  assumes "i \<noteq> 0" and "d dvd i"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1322
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1323
proof -
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1324
  from `d dvd i` obtain k where "i = d * k" ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1325
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1326
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1327
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1328
  with `i = d * k` show ?thesis by (simp add: abs_mult)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1329
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1330
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1331
lemma zdvd_not_zless:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1332
  fixes m n :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1333
  assumes "0 < m" and "m < n"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1334
  shows "\<not> n dvd m"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1335
proof
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1336
  from assms have "0 < n" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1337
  assume "n dvd m" then obtain k where k: "m = n * k" ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1338
  with `0 < m` have "0 < n * k" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1339
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1340
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1341
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1342
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1343
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1344
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1345
  shows "m dvd n"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1346
proof-
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1347
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1348
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1349
    with h have False by (simp add: mult_assoc)}
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1350
  hence "n = m * h" by blast
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1351
  thus ?thesis by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1352
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1353
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1354
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1355
proof -
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1356
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1357
  proof -
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1358
    fix k
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1359
    assume A: "int y = int x * k"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1360
    then show "x dvd y"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1361
    proof (cases k)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1362
      case (nonneg n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1363
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1364
      then show ?thesis ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1365
    next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1366
      case (neg n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1367
      with A have "int y = int x * (- int (Suc n))" by simp
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1368
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1369
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1370
      finally have "- int (x * Suc n) = int y" ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1371
      then show ?thesis by (simp only: negative_eq_positive) auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1372
    qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1373
  qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1374
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1375
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1376
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1377
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1378
proof
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1379
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1380
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1381
  hence "nat \<bar>x\<bar> = 1"  by simp
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1382
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1383
next
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1384
  assume "\<bar>x\<bar>=1"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1385
  then have "x = 1 \<or> x = -1" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1386
  then show "x dvd 1" by (auto intro: dvdI)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1387
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1388
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296