src/HOL/Int.thy
author haftmann
Sat, 08 Aug 2015 10:51:33 +0200
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direct bootstrap of integer division from natural division
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(*  Title:      HOL/Int.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
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*)
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section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>
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theory Int
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imports Equiv_Relations Power Quotient Fun_Def
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begin
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subsection \<open>Definition of integers as a quotient type\<close>
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where
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  "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
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  by (simp add: intrel_def)
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quotient_type int = "nat \<times> nat" / "intrel"
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  morphisms Rep_Integ Abs_Integ
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proof (rule equivpI)
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  show "reflp intrel"
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    unfolding reflp_def by auto
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  show "symp intrel"
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    unfolding symp_def by auto
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  show "transp intrel"
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    unfolding transp_def by auto
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
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by (induct z) auto
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subsection \<open>Integers form a commutative ring\<close>
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instantiation int :: comm_ring_1
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begin
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lift_definition zero_int :: "int" is "(0, 0)" .
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lift_definition one_int :: "int" is "(1, 0)" .
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
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  by clarsimp
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lift_definition uminus_int :: "int \<Rightarrow> int"
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  is "\<lambda>(x, y). (y, x)"
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  by clarsimp
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
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  by clarsimp
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
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proof (clarsimp)
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  fix s t u v w x y z :: nat
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  assume "s + v = u + t" and "w + z = y + x"
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  hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
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       = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
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    by simp
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  thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
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    by (simp add: algebra_simps)
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qed
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instance
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  by 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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  "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
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  unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
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lemma int_diff_cases:
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  obtains (diff) m n where "z = int m - int n"
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  by transfer clarsimp
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subsection \<open>Integers are totally ordered\<close>
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instantiation int :: linorder
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begin
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
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  by auto
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v < u + y"
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  by auto
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instance
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  by 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 max_min_distrib2)
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end
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subsection \<open>Ordering properties of arithmetic operations\<close>
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instance int :: ordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by transfer clarsimp
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qed
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text\<open>Strict Monotonicity of Multiplication\<close>
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text\<open>strict, in 1st argument; proof is by induction on k>0\<close>
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lemma zmult_zless_mono2_lemma:
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     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
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apply (induct k)
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apply simp
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apply (simp add: distrib_right)
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apply (case_tac "k=0")
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apply (simp_all add: add_strict_mono)
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done
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text\<open>The integers form an ordered integral domain\<close>
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instantiation int :: linordered_idom
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begin
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definition
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  zabs_def: "\<bar>i\<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 \<open>Embedding of the Integers into any @{text ring_1}: @{text of_int}\<close>
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context ring_1
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begin
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lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
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  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_0 [simp]: "of_int 0 = 0"
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  by transfer simp
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lemma of_int_1 [simp]: "of_int 1 = 1"
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  by transfer simp
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
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  by transfer (clarsimp simp add: algebra_simps)
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
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  by (transfer fixing: uminus) clarsimp
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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  using of_int_add [of w "- z"] by simp
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
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  by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
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text\<open>Collapse nested embeddings\<close>
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lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
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by (induct n) auto
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lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
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  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
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lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
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  by simp
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lemma of_int_power:
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  "of_int (z ^ n) = of_int z ^ n"
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  by (induct n) simp_all
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end
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context ring_char_0
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begin
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lemma of_int_eq_iff [simp]:
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   "of_int w = of_int z \<longleftrightarrow> w = z"
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  by transfer (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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text\<open>Special cases where either operand is zero\<close>
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lemma of_int_eq_0_iff [simp]:
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  "of_int z = 0 \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of z 0] by simp
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lemma of_int_0_eq_iff [simp]:
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  "0 = of_int z \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of 0 z] by simp
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end
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context linordered_idom
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begin
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text\<open>Every @{text linordered_idom} has characteristic zero.\<close>
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subclass ring_char_0 ..
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lemma of_int_le_iff [simp]:
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  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
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  by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_less_iff [simp]:
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  "of_int w < of_int z \<longleftrightarrow> w < z"
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  by (simp add: less_le order_less_le)
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lemma of_int_0_le_iff [simp]:
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  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
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  using of_int_le_iff [of 0 z] by simp
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lemma of_int_le_0_iff [simp]:
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  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
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  using of_int_le_iff [of z 0] by simp
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lemma of_int_0_less_iff [simp]:
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  "0 < of_int z \<longleftrightarrow> 0 < z"
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  using of_int_less_iff [of 0 z] by simp
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lemma of_int_less_0_iff [simp]:
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  "of_int z < 0 \<longleftrightarrow> z < 0"
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  using of_int_less_iff [of z 0] by simp
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end
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lemma of_nat_less_of_int_iff:
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  "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
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  by (metis of_int_of_nat_eq of_int_less_iff)
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lemma of_int_eq_id [simp]: "of_int = id"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff changeset
   301
proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   302
  fix z show "of_int z = id z"
48045
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parents: 48044
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   303
    by (cases z rule: int_diff_cases, simp)
25919
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parents:
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   304
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff changeset
   305
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
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parents:
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   306
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   307
instance int :: no_top
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hoelzl
parents: 51185
diff changeset
   308
  apply default
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   309
  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
   310
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   311
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   312
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   313
instance int :: no_bot
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   314
  apply default
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   315
  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
   316
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   317
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   318
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   319
subsection \<open>Magnitude of an Integer, as a Natural Number: @{text nat}\<close>
25919
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haftmann
parents:
diff changeset
   320
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   321
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
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parents: 48044
diff changeset
   322
  by auto
25919
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parents:
diff changeset
   323
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
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parents: 44707
diff changeset
   324
lemma nat_int [simp]: "nat (int n) = n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   325
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   326
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
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parents: 44707
diff changeset
   327
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
   328
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   329
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   330
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
   331
by simp
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_0 [simp]: "z \<le> 0 ==> nat z = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   334
  by transfer clarsimp
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
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
   337
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   338
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   339
text\<open>An alternative condition is @{term "0 \<le> w"}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   340
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   341
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
25919
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
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   344
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
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 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
   347
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   348
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   349
lemma nonneg_eq_int:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   350
  fixes z :: int
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   351
  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
   352
  shows P
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   353
  using assms by (blast dest: nat_0_le sym)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   354
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   355
lemma nat_eq_iff:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   356
  "nat w = m \<longleftrightarrow> (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
   357
  by transfer (clarsimp simp add: le_imp_diff_is_add)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   358
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   359
corollary nat_eq_iff2:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   360
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   361
  using nat_eq_iff [of w m] by auto
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   362
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   363
lemma nat_0 [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   364
  "nat 0 = 0"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   365
  by (simp add: nat_eq_iff)
25919
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haftmann
parents:
diff changeset
   366
54223
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parents: 54147
diff changeset
   367
lemma nat_1 [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   368
  "nat 1 = Suc 0"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   369
  by (simp add: nat_eq_iff)
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   370
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   371
lemma nat_numeral [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   372
  "nat (numeral k) = numeral k"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   373
  by (simp add: nat_eq_iff)
25919
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haftmann
parents:
diff changeset
   374
54223
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haftmann
parents: 54147
diff changeset
   375
lemma nat_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   376
  "nat (- numeral k) = 0"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   377
  by simp
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   378
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   379
lemma nat_2: "nat 2 = Suc (Suc 0)"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   380
  by simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   381
25919
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haftmann
parents:
diff changeset
   382
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
   383
  by transfer (clarsimp, arith)
25919
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haftmann
parents:
diff changeset
   384
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   385
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
   386
  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
   387
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   388
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
   389
  by transfer auto
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   390
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   391
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
   392
  by transfer clarsimp
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   393
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   394
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
   395
by (auto simp add: nat_eq_iff2)
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 zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   398
by (insert zless_nat_conj [of 0], auto)
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
lemma nat_add_distrib:
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   401
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   402
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   403
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   404
lemma nat_diff_distrib':
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   405
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   406
  by transfer clarsimp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   407
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   408
lemma nat_diff_distrib:
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   409
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   410
  by (rule nat_diff_distrib') auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   411
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   412
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   413
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   414
53065
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   415
lemma le_nat_iff:
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   416
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   417
  by transfer auto
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   418
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   419
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
   420
  by transfer (clarsimp simp add: less_diff_conv)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   421
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   422
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   423
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   424
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   425
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
   426
  by transfer (clarsimp simp add: of_nat_diff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   427
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   428
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   429
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   430
lemma diff_nat_numeral [simp]:
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   431
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   432
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   433
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   434
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   435
text \<open>For termination proofs:\<close>
29779
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   436
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
   437
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   438
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   439
subsection\<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   440
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   441
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   442
by (simp add: order_less_le del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   443
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   444
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   445
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   446
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   447
lemma negative_zle_0: "- int n \<le> 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   448
by (simp add: minus_le_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   449
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   450
lemma negative_zle [iff]: "- int n \<le> int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   451
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
   452
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   453
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
   454
by (subst le_minus_iff, simp del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   455
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   456
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
   457
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   458
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   459
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   460
by (simp add: linorder_not_less)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   461
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   462
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
   463
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
   464
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   465
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
   466
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   467
  have "(w \<le> z) = (0 \<le> z - w)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   468
    by (simp only: le_diff_eq add_0_left)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   469
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   470
    by (auto elim: zero_le_imp_eq_int)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   471
  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
   472
    by (simp only: algebra_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   473
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   474
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   475
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   476
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
   477
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   478
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   479
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   480
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   481
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   482
text\<open>This version is proved for all ordered rings, not just integers!
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   483
      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
   484
      in theory @{text Rings}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   485
      But is it really better than just rewriting with @{text abs_if}?\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53652
diff changeset
   486
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
   487
     "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
   488
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
   489
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   490
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
   491
apply transfer
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   492
apply clarsimp
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   493
apply (rule_tac x="b - Suc a" in exI, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   494
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   495
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   496
subsection \<open>Cases and induction\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   497
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   498
text\<open>Now we replace the case analysis rule by a more conventional one:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   499
whether an integer is negative or not.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   500
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   501
text\<open>This version is symmetric in the two subgoals.\<close>
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   502
theorem int_cases2 [case_names nonneg nonpos, cases type: int]:
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   503
  "\<lbrakk>!! n. z = int n \<Longrightarrow> P;  !! n. z = - (int n) \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   504
apply (cases "z < 0")
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   505
apply (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   506
done
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   507
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   508
text\<open>This is the default, with a negative case.\<close>
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   509
theorem int_cases [case_names nonneg neg, cases type: int]:
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   510
  "[|!! 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
   511
apply (cases "z < 0")
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   512
apply (blast dest!: negD)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   513
apply (simp add: linorder_not_less del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   514
apply auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   515
apply (blast dest: nat_0_le [THEN sym])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   516
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   517
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   518
lemma int_cases3 [case_names zero pos neg]:
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   519
  fixes k :: int
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   520
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   521
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P" 
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   522
  shows "P"
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   523
proof (cases k "0::int" rule: linorder_cases)
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   524
  case equal with assms(1) show P by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   525
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   526
  case greater
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   527
  then have "nat k > 0" by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   528
  moreover from this have "k = int (nat k)" by auto
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   529
  ultimately show P using assms(2) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   530
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   531
  case less
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   532
  then have "nat (- k) > 0" by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   533
  moreover from this have "k = - int (nat (- k))" by auto
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   534
  ultimately show P using assms(3) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   535
qed
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   536
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   537
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
   538
     "[|!! 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
   539
  by (cases z) auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   540
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   541
lemma nonneg_int_cases:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   542
  assumes "0 \<le> k" obtains n where "k = int n"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   543
  using assms by (rule nonneg_eq_int)
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   544
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   545
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   546
  -- \<open>Unfold all @{text let}s involving constants\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   547
  by (fact Let_numeral) -- \<open>FIXME drop\<close>
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36811
diff changeset
   548
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   549
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   550
  -- \<open>Unfold all @{text let}s involving constants\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   551
  by (fact Let_neg_numeral) -- \<open>FIXME drop\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   552
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   553
text \<open>Unfold @{text min} and @{text max} on numerals.\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   554
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   555
lemmas max_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   556
  max_def [of "numeral u" "numeral v"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   557
  max_def [of "numeral u" "- numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   558
  max_def [of "- numeral u" "numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   559
  max_def [of "- numeral u" "- numeral v"] for u v
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   560
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   561
lemmas min_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   562
  min_def [of "numeral u" "numeral v"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   563
  min_def [of "numeral u" "- numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   564
  min_def [of "- numeral u" "numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   565
  min_def [of "- numeral u" "- numeral v"] for u v
26075
815f3ccc0b45 added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents: 26072
diff changeset
   566
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   567
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   568
subsubsection \<open>Binary comparisons\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   569
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   570
text \<open>Preliminaries\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   571
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   572
lemma le_imp_0_less:
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   573
  assumes le: "0 \<le> z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   574
  shows "(0::int) < 1 + z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   575
proof -
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   576
  have "0 \<le> z" by fact
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   577
  also have "... < z + 1" by (rule less_add_one)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   578
  also have "... = 1 + z" by (simp add: ac_simps)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   579
  finally show "0 < 1 + z" .
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   580
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   581
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   582
lemma odd_less_0_iff:
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   583
  "(1 + z + z < 0) = (z < (0::int))"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   584
proof (cases z)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   585
  case (nonneg n)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56889
diff changeset
   586
  thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   587
                             le_imp_0_less [THEN order_less_imp_le])
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   588
next
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   589
  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
   590
  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
   591
    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
   592
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   593
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   594
subsubsection \<open>Comparisons, for Ordered Rings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   595
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   596
lemmas double_eq_0_iff = double_zero
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   597
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   598
lemma odd_nonzero:
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
   599
  "1 + z + z \<noteq> (0::int)"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   600
proof (cases z)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   601
  case (nonneg n)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   602
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   603
  thus ?thesis using  le_imp_0_less [OF le]
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   604
    by (auto simp add: add.assoc)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   605
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   606
  case (neg n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   607
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   608
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   609
    assume eq: "1 + z + z = 0"
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   610
    have "(0::int) < 1 + (int n + int n)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   611
      by (simp add: le_imp_0_less add_increasing)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   612
    also have "... = - (1 + z + z)"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   613
      by (simp add: neg add.assoc [symmetric])
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   614
    also have "... = 0" by (simp add: eq)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   615
    finally have "0<0" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   616
    thus False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   617
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   618
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   619
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   620
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   621
subsection \<open>The Set of Integers\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   622
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   623
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   624
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   625
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   626
definition Ints  :: "'a set" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36811
diff changeset
   627
  "Ints = range of_int"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   628
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   629
notation (xsymbols)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   630
  Ints  ("\<int>")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   631
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   632
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   633
  by (simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   635
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   636
  using Ints_of_int [of "of_nat n"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   637
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   638
lemma Ints_0 [simp]: "0 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   639
  using Ints_of_int [of "0"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   640
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   641
lemma Ints_1 [simp]: "1 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   642
  using Ints_of_int [of "1"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   643
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   644
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
   645
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   646
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   647
apply (rule of_int_add [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   648
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   649
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   650
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   651
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   652
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   653
apply (rule of_int_minus [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   654
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   655
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   656
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
   657
apply (auto simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   658
apply (rule range_eqI)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   659
apply (rule of_int_diff [symmetric])
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   660
done
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   661
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   662
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
   663
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   664
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   665
apply (rule of_int_mult [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   666
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   667
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   668
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   669
by (induct n) simp_all
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   670
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   671
lemma Ints_cases [cases set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   672
  assumes "q \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   673
  obtains (of_int) z where "q = of_int z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   674
  unfolding Ints_def
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   675
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   676
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   677
  then obtain z where "q = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   678
  then show thesis ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   679
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   680
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   681
lemma Ints_induct [case_names of_int, induct set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   682
  "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
   683
  by (rule Ints_cases) auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   684
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   685
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   686
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   687
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   688
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   689
lemma Ints_double_eq_0_iff:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   690
  assumes in_Ints: "a \<in> Ints"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   691
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   692
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   693
  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
   694
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   695
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   696
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   697
    assume "a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   698
    thus "a + a = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   699
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   700
    assume eq: "a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   701
    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
   702
    hence "z + z = 0" by (simp only: of_int_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   703
    hence "z = 0" by (simp only: double_eq_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   704
    thus "a = 0" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   705
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   706
qed
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 Ints_odd_nonzero:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   709
  assumes in_Ints: "a \<in> Ints"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   710
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   711
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   712
  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
   713
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   714
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   715
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   716
    assume eq: "1 + a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   717
    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
   718
    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
   719
    with odd_nonzero show False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   720
  qed
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   721
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   722
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   723
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   724
  using of_nat_in_Nats [of "numeral w"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   725
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   726
lemma Ints_odd_less_0:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   727
  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
   728
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   729
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   730
  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
   731
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   732
  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
   733
    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
   734
  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
   735
  also have "... = (a < 0)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   736
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   737
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   738
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   739
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   740
subsection \<open>@{term setsum} and @{term setprod}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   741
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   742
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
   743
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   744
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   745
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   746
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   747
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
   748
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   749
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   750
  done
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 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
   753
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   754
  apply (erule finite_induct, auto simp add: of_nat_mult)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   755
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   756
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   757
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
   758
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   759
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   760
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   761
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   762
lemmas int_setsum = of_nat_setsum [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   763
lemmas int_setprod = of_nat_setprod [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   764
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   765
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   766
text \<open>Legacy theorems\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   767
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   768
lemmas zle_int = of_nat_le_iff [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   769
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
   770
lemmas numeral_1_eq_1 = numeral_One
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   771
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   772
subsection \<open>Setting up simplification procedures\<close>
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   773
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   774
lemmas of_int_simps =
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   775
  of_int_0 of_int_1 of_int_add of_int_mult
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   776
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48066
diff changeset
   777
ML_file "Tools/int_arith.ML"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   778
declaration \<open>K Int_Arith.setup\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   779
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   780
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   781
  "(m::'a::linordered_idom) <= n" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   782
  "(m::'a::linordered_idom) = n") =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   783
  \<open>fn _ => fn ss => fn ct => Lin_Arith.simproc ss (Thm.term_of ct)\<close>
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43531
diff changeset
   784
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   785
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   786
subsection\<open>More Inequality Reasoning\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   787
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   788
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
   789
by arith
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 add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   792
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   793
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   794
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
   795
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   796
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   797
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
   798
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   799
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   800
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
   801
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   802
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   803
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   804
subsection\<open>The functions @{term nat} and @{term int}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   805
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   806
text\<open>Simplify the term @{term "w + - z"}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   807
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   808
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   809
  using zless_nat_conj [of 1 z] by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   810
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   811
text\<open>This simplifies expressions of the form @{term "int n = z"} where
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   812
      z is an integer literal.\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   813
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
   814
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   815
lemma split_nat [arith_split]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   816
  "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
   817
  (is "?P = (?L & ?R)")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   818
proof (cases "i < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   819
  case True thus ?thesis by auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   820
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   821
  case False
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   822
  have "?P = ?L"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   823
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   824
    assume ?P thus ?L using False by clarsimp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   825
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   826
    assume ?L thus ?P using False by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   827
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   828
  with False show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   829
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   830
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   831
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   832
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   833
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   834
lemma nat_int_add: "nat (int a + int b) = a + b"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   835
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   836
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   837
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   838
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   839
33056
791a4655cae3 renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents: 32437
diff changeset
   840
lemma of_int_of_nat [nitpick_simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   841
  "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
   842
proof (cases "k < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   843
  case True then have "0 \<le> - k" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   844
  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
   845
  with True show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   846
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   847
  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
   848
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   849
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   850
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   851
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   852
lemma nat_mult_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   853
  fixes z z' :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   854
  assumes "0 \<le> z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   855
  shows "nat (z * z') = nat z * nat z'"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   856
proof (cases "0 \<le> z'")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   857
  case False with assms have "z * z' \<le> 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   858
    by (simp add: not_le mult_le_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   859
  then have "nat (z * z') = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   860
  moreover from False have "nat z' = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   861
  ultimately show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   862
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   863
  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
   864
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   865
    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
   866
      (simp only: of_nat_mult of_nat_nat [OF True]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   867
         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
   868
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   869
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   870
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
   871
apply (rule trans)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   872
apply (rule_tac [2] nat_mult_distrib, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   873
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   874
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   875
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
   876
apply (cases "z=0 | w=0")
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   877
apply (auto simp add: abs_if nat_mult_distrib [symmetric]
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   878
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   879
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   880
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   881
lemma int_in_range_abs [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   882
  "int n \<in> range abs"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   883
proof (rule range_eqI)
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   884
  show "int n = \<bar>int n\<bar>"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   885
    by simp
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   886
qed
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   887
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   888
lemma range_abs_Nats [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   889
  "range abs = (\<nat> :: int set)"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   890
proof -
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   891
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   892
    by (cases k) simp_all
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   893
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   894
    using that by induct simp
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   895
  ultimately show ?thesis by blast
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   896
qed  
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   897
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   898
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
   899
apply (rule sym)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   900
apply (simp add: nat_eq_iff)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   901
done
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   902
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   903
lemma diff_nat_eq_if:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   904
     "nat z - nat z' =
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   905
        (if z' < 0 then nat z
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   906
         else let d = z-z' in
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   907
              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
   908
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
   909
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   910
lemma nat_numeral_diff_1 [simp]:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   911
  "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
   912
  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
   913
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   914
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   915
subsection "Induction principles for int"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   916
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   917
text\<open>Well-founded segments of the integers\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   918
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   919
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   920
  int_ge_less_than  ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   921
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   922
  "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
   923
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   924
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
   925
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   926
  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
   927
    by (auto simp add: int_ge_less_than_def)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   928
  thus ?thesis
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   929
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   930
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   931
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   932
text\<open>This variant looks odd, but is typical of the relations suggested
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   933
by RankFinder.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   934
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   935
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   936
  int_ge_less_than2 ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   937
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   938
  "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
   939
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   940
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
   941
proof -
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   942
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   943
    by (auto simp add: int_ge_less_than2_def)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   944
  thus ?thesis
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   945
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   946
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   947
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   948
(* `set:int': dummy construction *)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   949
theorem int_ge_induct [case_names base step, induct set: int]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   950
  fixes i :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   951
  assumes ge: "k \<le> i" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   952
    base: "P k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   953
    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
   954
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   955
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   956
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   957
    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
   958
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   959
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   960
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   961
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   962
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   963
      case (Suc n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   964
      then have "n = nat((i - 1) - k)" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   965
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   966
      have ki1: "k \<le> i - 1" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   967
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   968
      have "P (i - 1)" by (rule Suc.hyps)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   969
      from step [OF ki1 this] show ?case by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   970
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   971
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   972
  with ge show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   973
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   974
25928
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
   975
(* `set:int': dummy construction *)
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
   976
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
   977
  assumes gr: "k < (i::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   978
        base: "P(k+1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   979
        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
   980
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   981
apply(rule int_ge_induct[of "k + 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   982
  using gr apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   983
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   984
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   985
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   986
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   987
theorem int_le_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   988
  assumes le: "i \<le> (k::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   989
        base: "P(k)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   990
        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
   991
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   992
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   993
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   994
    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
   995
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   996
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   997
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   998
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   999
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1000
      case (Suc n)
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1001
      hence "n = nat (k - (i + 1))" by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1002
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1003
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1004
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1005
      have "P (i + 1)" by(rule Suc.hyps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1006
      from step[OF ki1 this] show ?case by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1007
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1008
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1009
  with le show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1010
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1011
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1012
theorem int_less_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1013
  assumes less: "(i::int) < k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1014
        base: "P(k - 1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1015
        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
  1016
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1017
apply(rule int_le_induct[of _ "k - 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1018
  using less apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1019
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1020
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1021
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1022
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
  1023
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
  1024
  fixes k :: int
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1025
  assumes base: "P k"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1026
    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
  1027
    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
  1028
  shows "P i"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1029
proof -
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1030
  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
  1031
  then show ?thesis
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1032
  proof
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1033
    assume "i \<ge> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1034
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1035
      by (rule int_ge_induct) (fact step1)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1036
  next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1037
    assume "i \<le> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1038
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1039
      by (rule int_le_induct) (fact step2)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1040
  qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1041
qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1042
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1043
subsection\<open>Intermediate value theorems\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1044
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1045
lemma int_val_lemma:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1046
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1047
      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
  1048
unfolding One_nat_def
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1049
apply (induct n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1050
apply simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1051
apply (intro strip)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1052
apply (erule impE, simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1053
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
  1054
apply (case_tac "k = f (Suc n)")
27106
ff27dc6e7d05 removed some dubious code lemmas
haftmann
parents: 26975
diff changeset
  1055
apply force
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1056
apply (erule impE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1057
 apply (simp add: abs_if split add: split_if_asm)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1058
apply (blast intro: le_SucI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1059
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1060
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1061
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
  1062
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1063
lemma nat_intermed_int_val:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1064
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1065
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1066
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1067
       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
  1068
unfolding One_nat_def
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1069
apply simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1070
apply (erule exE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1071
apply (rule_tac x = "i+m" in exI, arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1072
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1073
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1074
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1075
subsection\<open>Products and 1, by T. M. Rasmussen\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1076
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1077
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
  1078
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1079
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1080
lemma abs_zmult_eq_1:
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1081
  assumes mn: "\<bar>m * n\<bar> = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1082
  shows "\<bar>m\<bar> = (1::int)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1083
proof -
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1084
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1085
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1086
  have "~ (2 \<le> \<bar>m\<bar>)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1087
  proof
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1088
    assume "2 \<le> \<bar>m\<bar>"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1089
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1090
      by (simp add: mult_mono 0)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1091
    also have "... = \<bar>m*n\<bar>"
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1092
      by (simp add: abs_mult)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1093
    also have "... = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1094
      by (simp add: mn)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1095
    finally have "2*\<bar>n\<bar> \<le> 1" .
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1096
    thus "False" using 0
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1097
      by arith
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1098
  qed
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1099
  thus ?thesis using 0
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1100
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1101
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1102
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1103
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
  1104
by (insert abs_zmult_eq_1 [of m n], arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1105
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1106
lemma pos_zmult_eq_1_iff:
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1107
  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
  1108
proof -
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1109
  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
  1110
  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
  1111
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1112
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1113
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1114
apply (rule iffI)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1115
 apply (frule pos_zmult_eq_1_iff_lemma)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1116
 apply (simp add: mult.commute [of m])
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1117
 apply (frule pos_zmult_eq_1_iff_lemma, auto)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1118
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1119
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1120
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1121
proof
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1122
  assume "finite (UNIV::int set)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1123
  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
  1124
    by (rule injI) simp
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1125
  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
  1126
    by (rule finite_UNIV_inj_surj)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1127
  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
  1128
  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
  1129
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1130
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1131
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1132
subsection \<open>Further theorems on numerals\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1133
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1134
subsubsection\<open>Special Simplification for Constants\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1135
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1136
text\<open>These distributive laws move literals inside sums and differences.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1137
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1138
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
  1139
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
  1140
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
  1141
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
  1142
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1143
text\<open>These are actually for fields, like real: but where else to put them?\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1144
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1145
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
  1146
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
  1147
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
  1148
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
  1149
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1150
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1151
text \<open>Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1152
  strange, but then other simprocs simplify the quotient.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1153
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1154
lemmas inverse_eq_divide_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1155
  inverse_eq_divide [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1156
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1157
lemmas inverse_eq_divide_neg_numeral [simp] =
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1158
  inverse_eq_divide [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
  1159
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1160
text \<open>These laws simplify inequalities, moving unary minus from a term
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1161
into the literal.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1162
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1163
lemmas equation_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1164
  equation_minus_iff [of "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1165
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1166
lemmas minus_equation_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1167
  minus_equation_iff [of _ "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1168
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1169
lemmas le_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1170
  le_minus_iff [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
  1171
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1172
lemmas minus_le_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1173
  minus_le_iff [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
  1174
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1175
lemmas less_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1176
  less_minus_iff [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
  1177
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1178
lemmas minus_less_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1179
  minus_less_iff [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
  1180
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1181
-- \<open>FIXME maybe simproc\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1182
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1183
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1184
text \<open>Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"})\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1185
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1186
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
  1187
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
  1188
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
  1189
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
  1190
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1191
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1192
text \<open>Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="})\<close>
30652
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 le_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1195
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1196
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1197
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1198
lemmas divide_le_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1199
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1200
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1201
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1202
lemmas less_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1203
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1204
  neg_less_divide_eq [of "- 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
  1205
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1206
lemmas divide_less_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1207
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1208
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
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 eq_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1211
  eq_divide_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1212
  eq_divide_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1213
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1214
lemmas divide_eq_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1215
  divide_eq_eq [of _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1216
  divide_eq_eq [of _ "- numeral w"] for w
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1217
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1218
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1219
subsubsection\<open>Optional Simplification Rules Involving Constants\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1220
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1221
text\<open>Simplify quotients that are compared with a literal constant.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1222
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1223
lemmas le_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1224
  le_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1225
  le_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1226
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1227
lemmas divide_le_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1228
  divide_le_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1229
  divide_le_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1230
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1231
lemmas less_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1232
  less_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1233
  less_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1234
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1235
lemmas divide_less_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1236
  divide_less_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1237
  divide_less_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1238
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1239
lemmas eq_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1240
  eq_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1241
  eq_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1242
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1243
lemmas divide_eq_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1244
  divide_eq_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1245
  divide_eq_eq [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
  1246
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1247
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1248
text\<open>Not good as automatic simprules because they cause case splits.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1249
lemmas divide_const_simps =
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1250
  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
  1251
  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
  1252
  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
  1253
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1254
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1255
subsection \<open>The divides relation\<close>
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1256
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1257
lemma zdvd_antisym_nonneg:
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1258
    "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
  1259
  apply (simp add: dvd_def, auto)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56889
diff changeset
  1260
  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
  1261
  done
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1262
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1263
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
  1264
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1265
proof cases
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1266
  assume "a = 0" with assms show ?thesis by simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1267
next
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1268
  assume "a \<noteq> 0"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1269
  from \<open>a dvd b\<close> obtain k where k:"b = a*k" unfolding dvd_def by blast
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1270
  from \<open>b dvd a\<close> obtain k' where k':"a = b*k'" unfolding dvd_def by blast
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1271
  from k k' have "a = a*k*k'" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1272
  with mult_cancel_left1[where c="a" and b="k*k'"]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1273
  have kk':"k*k' = 1" using \<open>a\<noteq>0\<close> by (simp add: mult.assoc)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1274
  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
  1275
  thus ?thesis using k k' by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1276
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1277
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1278
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1279
  using dvd_add_right_iff [of k "- n" m] by simp
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1280
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1281
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
58649
a62065b5e1e2 generalized and consolidated some theorems concerning divisibility
haftmann
parents: 58512
diff changeset
  1282
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1283
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1284
lemma dvd_imp_le_int:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1285
  fixes d i :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1286
  assumes "i \<noteq> 0" and "d dvd i"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1287
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1288
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1289
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1290
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1291
  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
  1292
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1293
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1294
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1295
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1296
lemma zdvd_not_zless:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1297
  fixes m n :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1298
  assumes "0 < m" and "m < n"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1299
  shows "\<not> n dvd m"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1300
proof
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1301
  from assms have "0 < n" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1302
  assume "n dvd m" then obtain k where k: "m = n * k" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1303
  with \<open>0 < m\<close> have "0 < n * k" by auto
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1304
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1305
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1306
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1307
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1308
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1309
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
  1310
  shows "m dvd n"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1311
proof-
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1312
  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
  1313
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56889
diff changeset
  1314
    with h have False by (simp add: mult.assoc)}
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1315
  hence "n = m * h" by blast
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1316
  thus ?thesis by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1317
qed
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
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1320
proof -
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1321
  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
  1322
  proof -
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1323
    fix k
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1324
    assume A: "int y = int x * k"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1325
    then show "x dvd y"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1326
    proof (cases k)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1327
      case (nonneg n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1328
      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
  1329
      then show ?thesis ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1330
    next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1331
      case (neg n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1332
      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
  1333
      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
  1334
      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
  1335
      finally have "- int (x * Suc n) = int y" ..
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1336
      then show ?thesis by (simp only: negative_eq_positive) auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1337
    qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1338
  qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1339
  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
  1340
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1341
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1342
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
  1343
proof
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1344
  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
  1345
  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
  1346
  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
  1347
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1348
next
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1349
  assume "\<bar>x\<bar>=1"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1350
  then have "x = 1 \<or> x = -1" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1351
  then show "x dvd 1" by (auto intro: dvdI)
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
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1354
lemma zdvd_mult_cancel1:
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1355
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1356
proof
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1357
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1358
    by (cases "n >0") (auto simp add: minus_equation_iff)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1359
next
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1360
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1361
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1362
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1363
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1364
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: