src/HOL/Int.thy
author hoelzl
Thu Nov 13 17:19:52 2014 +0100 (2014-11-13)
changeset 59000 6eb0725503fc
parent 58889 5b7a9633cfa8
child 59536 03bde055a1a0
permissions -rw-r--r--
import general theorems from AFP/Markov_Models
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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 {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
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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 {* Definition of integers as a quotient type *}
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where
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  "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
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  by (simp add: intrel_def)
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quotient_type int = "nat \<times> nat" / "intrel"
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  morphisms Rep_Integ Abs_Integ
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proof (rule equivpI)
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  show "reflp intrel"
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    unfolding reflp_def by auto
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  show "symp intrel"
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    unfolding symp_def by auto
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  show "transp intrel"
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    unfolding transp_def by auto
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
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by (induct z) auto
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subsection {* Integers form a commutative ring *}
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instantiation int :: comm_ring_1
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begin
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lift_definition zero_int :: "int" is "(0, 0)" .
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lift_definition one_int :: "int" is "(1, 0)" .
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
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  by clarsimp
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lift_definition uminus_int :: "int \<Rightarrow> int"
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  is "\<lambda>(x, y). (y, x)"
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  by clarsimp
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
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  by clarsimp
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
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proof (clarsimp)
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  fix s t u v w x y z :: nat
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  assume "s + v = u + t" and "w + z = y + x"
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  hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
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       = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
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    by simp
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  thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
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    by (simp add: algebra_simps)
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qed
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instance
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  by default (transfer, clarsimp simp: algebra_simps)+
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end
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abbreviation int :: "nat \<Rightarrow> int" where
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  "int \<equiv> of_nat"
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lemma int_def: "int n = Abs_Integ (n, 0)"
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  by (induct n, simp add: zero_int.abs_eq,
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    simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]:
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  "(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 {* Integers are totally ordered *}
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instantiation int :: linorder
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begin
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
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  by auto
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v < u + y"
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  by auto
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instance
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  by default (transfer, force)+
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end
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instantiation int :: distrib_lattice
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begin
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definition
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  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (auto simp add: inf_int_def sup_int_def max_min_distrib2)
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end
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subsection {* Ordering properties of arithmetic operations *}
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instance int :: ordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by transfer clarsimp
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qed
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text{*Strict Monotonicity of Multiplication*}
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text{*strict, in 1st argument; proof is by induction on k>0*}
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lemma zmult_zless_mono2_lemma:
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     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
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apply (induct k)
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apply simp
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apply (simp add: distrib_right)
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apply (case_tac "k=0")
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apply (simp_all add: add_strict_mono)
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done
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text{*The integers form an ordered integral domain*}
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instantiation int :: linordered_idom
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begin
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definition
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  zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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instance proof
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  fix i j k :: int
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  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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    by (rule zmult_zless_mono2)
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  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
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    by (simp only: zabs_def)
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  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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    by (simp only: zsgn_def)
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qed
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end
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lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd:
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  "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
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apply transfer
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apply auto
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apply (rename_tac a b c d)
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apply (rule_tac x="c+b - Suc(a+d)" in exI)
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apply arith
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done
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lemmas int_distrib =
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  distrib_right [of z1 z2 w]
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  distrib_left [of w z1 z2]
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  left_diff_distrib [of z1 z2 w]
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  right_diff_distrib [of w z1 z2]
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  for z1 z2 w :: int
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subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
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context ring_1
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begin
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lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
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  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_0 [simp]: "of_int 0 = 0"
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  by transfer simp
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lemma of_int_1 [simp]: "of_int 1 = 1"
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  by transfer simp
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
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  by transfer (clarsimp simp add: algebra_simps)
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
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  by (transfer fixing: uminus) clarsimp
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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  using of_int_add [of w "- z"] by simp
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
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  by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
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text{*Collapse nested embeddings*}
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lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
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by (induct n) auto
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lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
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  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
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lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
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  by simp
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lemma of_int_power:
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  "of_int (z ^ n) = of_int z ^ n"
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  by (induct n) simp_all
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end
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context ring_char_0
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begin
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lemma of_int_eq_iff [simp]:
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   "of_int w = of_int z \<longleftrightarrow> w = z"
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  by transfer (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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text{*Special cases where either operand is zero*}
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lemma of_int_eq_0_iff [simp]:
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  "of_int z = 0 \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of z 0] by simp
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lemma of_int_0_eq_iff [simp]:
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  "0 = of_int z \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of 0 z] by simp
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end
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context linordered_idom
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begin
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text{*Every @{text linordered_idom} has characteristic zero.*}
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subclass ring_char_0 ..
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lemma of_int_le_iff [simp]:
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  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
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  by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_less_iff [simp]:
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  "of_int w < of_int z \<longleftrightarrow> w < z"
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  by (simp add: less_le order_less_le)
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lemma of_int_0_le_iff [simp]:
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  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
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  using of_int_le_iff [of 0 z] by simp
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lemma of_int_le_0_iff [simp]:
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  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
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  using of_int_le_iff [of z 0] by simp
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lemma of_int_0_less_iff [simp]:
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  "0 < of_int z \<longleftrightarrow> 0 < z"
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  using of_int_less_iff [of 0 z] by simp
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lemma of_int_less_0_iff [simp]:
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  "of_int z < 0 \<longleftrightarrow> z < 0"
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  using of_int_less_iff [of z 0] by simp
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end
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lemma of_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"
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proof
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  fix z show "of_int z = id z"
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    by (cases z rule: int_diff_cases, simp)
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qed
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instance int :: no_top
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  apply default
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  apply (rule_tac x="x + 1" in exI)
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  apply simp
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  done
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instance int :: no_bot
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  apply default
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  apply (rule_tac x="x - 1" in exI)
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  apply simp
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  done
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subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
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lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
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  by auto
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lemma nat_int [simp]: "nat (int n) = n"
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  by transfer simp
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lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
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  by transfer clarsimp
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corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
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by simp
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lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
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  by transfer clarsimp
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lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
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  by transfer (clarsimp, arith)
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text{*An alternative condition is @{term "0 \<le> w"} *}
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corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
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by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
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corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919
   344
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   345
haftmann@25919
   346
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
huffman@48045
   347
  by transfer (clarsimp, arith)
haftmann@25919
   348
haftmann@25919
   349
lemma nonneg_eq_int:
haftmann@25919
   350
  fixes z :: int
huffman@44709
   351
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   352
  shows P
haftmann@25919
   353
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   354
haftmann@54223
   355
lemma nat_eq_iff:
haftmann@54223
   356
  "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   357
  by transfer (clarsimp simp add: le_imp_diff_is_add)
haftmann@54223
   358
 
haftmann@54223
   359
corollary nat_eq_iff2:
haftmann@54223
   360
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   361
  using nat_eq_iff [of w m] by auto
haftmann@54223
   362
haftmann@54223
   363
lemma nat_0 [simp]:
haftmann@54223
   364
  "nat 0 = 0"
haftmann@54223
   365
  by (simp add: nat_eq_iff)
haftmann@25919
   366
haftmann@54223
   367
lemma nat_1 [simp]:
haftmann@54223
   368
  "nat 1 = Suc 0"
haftmann@54223
   369
  by (simp add: nat_eq_iff)
haftmann@54223
   370
haftmann@54223
   371
lemma nat_numeral [simp]:
haftmann@54223
   372
  "nat (numeral k) = numeral k"
haftmann@54223
   373
  by (simp add: nat_eq_iff)
haftmann@25919
   374
haftmann@54223
   375
lemma nat_neg_numeral [simp]:
haftmann@54489
   376
  "nat (- numeral k) = 0"
haftmann@54223
   377
  by simp
haftmann@54223
   378
haftmann@54223
   379
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   380
  by simp
haftmann@54223
   381
 
haftmann@25919
   382
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045
   383
  by transfer (clarsimp, arith)
haftmann@25919
   384
huffman@44709
   385
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   386
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   387
huffman@44707
   388
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   389
  by transfer auto
huffman@44707
   390
nipkow@29700
   391
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045
   392
  by transfer clarsimp
nipkow@29700
   393
haftmann@25919
   394
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   395
by (auto simp add: nat_eq_iff2)
haftmann@25919
   396
haftmann@25919
   397
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   398
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   399
haftmann@25919
   400
lemma nat_add_distrib:
haftmann@54223
   401
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   402
  by transfer clarsimp
haftmann@25919
   403
haftmann@54223
   404
lemma nat_diff_distrib':
haftmann@54223
   405
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   406
  by transfer clarsimp
haftmann@54223
   407
 
haftmann@25919
   408
lemma nat_diff_distrib:
haftmann@54223
   409
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   410
  by (rule nat_diff_distrib') auto
haftmann@25919
   411
huffman@44709
   412
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   413
  by transfer simp
haftmann@25919
   414
haftmann@53065
   415
lemma le_nat_iff:
haftmann@53065
   416
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   417
  by transfer auto
haftmann@53065
   418
  
huffman@44709
   419
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045
   420
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   421
haftmann@25919
   422
context ring_1
haftmann@25919
   423
begin
haftmann@25919
   424
haftmann@25919
   425
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   426
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   427
haftmann@25919
   428
end
haftmann@25919
   429
haftmann@54249
   430
lemma diff_nat_numeral [simp]: 
haftmann@54249
   431
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   432
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   433
haftmann@54249
   434
krauss@29779
   435
text {* For termination proofs: *}
krauss@29779
   436
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   437
haftmann@25919
   438
haftmann@25919
   439
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   440
huffman@44709
   441
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   442
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   443
huffman@44709
   444
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   445
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   446
huffman@44709
   447
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   448
by (simp add: minus_le_iff)
haftmann@25919
   449
huffman@44709
   450
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   451
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   452
huffman@44709
   453
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   454
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   455
huffman@44709
   456
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045
   457
  by transfer simp
haftmann@25919
   458
huffman@44709
   459
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   460
by (simp add: linorder_not_less)
haftmann@25919
   461
huffman@44709
   462
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   463
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   464
huffman@44709
   465
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   466
proof -
haftmann@25919
   467
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   468
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   469
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   470
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   471
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   472
    by (simp only: algebra_simps)
haftmann@25919
   473
  finally show ?thesis .
haftmann@25919
   474
qed
haftmann@25919
   475
huffman@44709
   476
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   477
by simp
haftmann@25919
   478
huffman@44709
   479
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919
   480
by simp
haftmann@25919
   481
haftmann@25919
   482
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   483
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   484
      in theory @{text Rings}.
haftmann@25919
   485
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@54147
   486
lemma abs_split [arith_split, no_atp]:
haftmann@35028
   487
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   488
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   489
huffman@44709
   490
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045
   491
apply transfer
huffman@48045
   492
apply clarsimp
huffman@48045
   493
apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919
   494
done
haftmann@25919
   495
haftmann@25919
   496
subsection {* Cases and induction *}
haftmann@25919
   497
haftmann@25919
   498
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   499
whether an integer is negative or not.*}
haftmann@25919
   500
wenzelm@42676
   501
theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709
   502
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   503
apply (cases "z < 0")
wenzelm@42676
   504
apply (blast dest!: negD)
haftmann@25919
   505
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   506
apply auto
haftmann@25919
   507
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   508
done
haftmann@25919
   509
wenzelm@42676
   510
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   511
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   512
  by (cases z) auto
haftmann@25919
   513
huffman@47207
   514
lemma nonneg_int_cases:
huffman@47207
   515
  assumes "0 \<le> k" obtains n where "k = int n"
haftmann@54489
   516
  using assms by (rule nonneg_eq_int)
huffman@47207
   517
huffman@47108
   518
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
huffman@47108
   519
  -- {* Unfold all @{text let}s involving constants *}
haftmann@57514
   520
  by (fact Let_numeral) -- {* FIXME drop *}
haftmann@37767
   521
haftmann@54489
   522
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
haftmann@25919
   523
  -- {* Unfold all @{text let}s involving constants *}
haftmann@57514
   524
  by (fact Let_neg_numeral) -- {* FIXME drop *}
haftmann@25919
   525
huffman@47108
   526
text {* Unfold @{text min} and @{text max} on numerals. *}
huffman@28958
   527
huffman@47108
   528
lemmas max_number_of [simp] =
huffman@47108
   529
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   530
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   531
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   532
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   533
huffman@47108
   534
lemmas min_number_of [simp] =
huffman@47108
   535
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   536
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   537
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   538
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   539
haftmann@25919
   540
huffman@28958
   541
subsubsection {* Binary comparisons *}
huffman@28958
   542
huffman@28958
   543
text {* Preliminaries *}
huffman@28958
   544
huffman@28958
   545
lemma even_less_0_iff:
haftmann@35028
   546
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   547
proof -
webertj@49962
   548
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
huffman@28958
   549
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   550
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   551
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   552
  finally show ?thesis .
huffman@28958
   553
qed
huffman@28958
   554
huffman@28958
   555
lemma le_imp_0_less: 
huffman@28958
   556
  assumes le: "0 \<le> z"
huffman@28958
   557
  shows "(0::int) < 1 + z"
huffman@28958
   558
proof -
huffman@28958
   559
  have "0 \<le> z" by fact
huffman@47108
   560
  also have "... < z + 1" by (rule less_add_one)
haftmann@57514
   561
  also have "... = 1 + z" by (simp add: ac_simps)
huffman@28958
   562
  finally show "0 < 1 + z" .
huffman@28958
   563
qed
huffman@28958
   564
huffman@28958
   565
lemma odd_less_0_iff:
huffman@28958
   566
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   567
proof (cases z)
huffman@28958
   568
  case (nonneg n)
haftmann@57512
   569
  thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
huffman@28958
   570
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   571
next
huffman@28958
   572
  case (neg n)
huffman@30079
   573
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   574
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   575
qed
huffman@28958
   576
huffman@28958
   577
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
   578
haftmann@25919
   579
lemmas double_eq_0_iff = double_zero
haftmann@25919
   580
haftmann@25919
   581
lemma odd_nonzero:
haftmann@33296
   582
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
   583
proof (cases z)
haftmann@25919
   584
  case (nonneg n)
haftmann@25919
   585
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
   586
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@57512
   587
    by (auto simp add: add.assoc) 
haftmann@25919
   588
next
haftmann@25919
   589
  case (neg n)
haftmann@25919
   590
  show ?thesis
haftmann@25919
   591
  proof
haftmann@25919
   592
    assume eq: "1 + z + z = 0"
huffman@44709
   593
    have "(0::int) < 1 + (int n + int n)"
haftmann@25919
   594
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
   595
    also have "... = - (1 + z + z)" 
haftmann@57512
   596
      by (simp add: neg add.assoc [symmetric]) 
haftmann@25919
   597
    also have "... = 0" by (simp add: eq) 
haftmann@25919
   598
    finally have "0<0" ..
haftmann@25919
   599
    thus False by blast
haftmann@25919
   600
  qed
haftmann@25919
   601
qed
haftmann@25919
   602
haftmann@30652
   603
haftmann@25919
   604
subsection {* The Set of Integers *}
haftmann@25919
   605
haftmann@25919
   606
context ring_1
haftmann@25919
   607
begin
haftmann@25919
   608
haftmann@30652
   609
definition Ints  :: "'a set" where
haftmann@37767
   610
  "Ints = range of_int"
haftmann@25919
   611
haftmann@25919
   612
notation (xsymbols)
haftmann@25919
   613
  Ints  ("\<int>")
haftmann@25919
   614
huffman@35634
   615
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   616
  by (simp add: Ints_def)
huffman@35634
   617
huffman@35634
   618
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   619
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   620
haftmann@25919
   621
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   622
  using Ints_of_int [of "0"] by simp
haftmann@25919
   623
haftmann@25919
   624
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   625
  using Ints_of_int [of "1"] by simp
haftmann@25919
   626
haftmann@25919
   627
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
   628
apply (auto simp add: Ints_def)
haftmann@25919
   629
apply (rule range_eqI)
haftmann@25919
   630
apply (rule of_int_add [symmetric])
haftmann@25919
   631
done
haftmann@25919
   632
haftmann@25919
   633
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
   634
apply (auto simp add: Ints_def)
haftmann@25919
   635
apply (rule range_eqI)
haftmann@25919
   636
apply (rule of_int_minus [symmetric])
haftmann@25919
   637
done
haftmann@25919
   638
huffman@35634
   639
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
   640
apply (auto simp add: Ints_def)
huffman@35634
   641
apply (rule range_eqI)
huffman@35634
   642
apply (rule of_int_diff [symmetric])
huffman@35634
   643
done
huffman@35634
   644
haftmann@25919
   645
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
   646
apply (auto simp add: Ints_def)
haftmann@25919
   647
apply (rule range_eqI)
haftmann@25919
   648
apply (rule of_int_mult [symmetric])
haftmann@25919
   649
done
haftmann@25919
   650
huffman@35634
   651
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
   652
by (induct n) simp_all
huffman@35634
   653
haftmann@25919
   654
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   655
  assumes "q \<in> \<int>"
haftmann@25919
   656
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   657
  unfolding Ints_def
haftmann@25919
   658
proof -
haftmann@25919
   659
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   660
  then obtain z where "q = of_int z" ..
haftmann@25919
   661
  then show thesis ..
haftmann@25919
   662
qed
haftmann@25919
   663
haftmann@25919
   664
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   665
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   666
  by (rule Ints_cases) auto
haftmann@25919
   667
haftmann@25919
   668
end
haftmann@25919
   669
haftmann@25919
   670
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
   671
haftmann@25919
   672
lemma Ints_double_eq_0_iff:
haftmann@25919
   673
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   674
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
   675
proof -
haftmann@25919
   676
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   677
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   678
  show ?thesis
haftmann@25919
   679
  proof
haftmann@25919
   680
    assume "a = 0"
haftmann@25919
   681
    thus "a + a = 0" by simp
haftmann@25919
   682
  next
haftmann@25919
   683
    assume eq: "a + a = 0"
haftmann@25919
   684
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   685
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   686
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   687
    thus "a = 0" by (simp add: a)
haftmann@25919
   688
  qed
haftmann@25919
   689
qed
haftmann@25919
   690
haftmann@25919
   691
lemma Ints_odd_nonzero:
haftmann@25919
   692
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
   693
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
   694
proof -
haftmann@25919
   695
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   696
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   697
  show ?thesis
haftmann@25919
   698
  proof
haftmann@25919
   699
    assume eq: "1 + a + a = 0"
haftmann@25919
   700
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
   701
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   702
    with odd_nonzero show False by blast
haftmann@25919
   703
  qed
haftmann@25919
   704
qed 
haftmann@25919
   705
huffman@47108
   706
lemma Nats_numeral [simp]: "numeral w \<in> Nats"
huffman@47108
   707
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   708
haftmann@25919
   709
lemma Ints_odd_less_0: 
haftmann@25919
   710
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
   711
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
   712
proof -
haftmann@25919
   713
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
   714
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   715
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
   716
    by (simp add: a)
huffman@45532
   717
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919
   718
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
   719
  finally show ?thesis .
haftmann@25919
   720
qed
haftmann@25919
   721
haftmann@25919
   722
haftmann@25919
   723
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
   724
haftmann@25919
   725
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
   726
  apply (cases "finite A")
haftmann@25919
   727
  apply (erule finite_induct, auto)
haftmann@25919
   728
  done
haftmann@25919
   729
haftmann@25919
   730
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
   731
  apply (cases "finite A")
haftmann@25919
   732
  apply (erule finite_induct, auto)
haftmann@25919
   733
  done
haftmann@25919
   734
haftmann@25919
   735
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
   736
  apply (cases "finite A")
haftmann@25919
   737
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
   738
  done
haftmann@25919
   739
haftmann@25919
   740
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
   741
  apply (cases "finite A")
haftmann@25919
   742
  apply (erule finite_induct, auto)
haftmann@25919
   743
  done
haftmann@25919
   744
haftmann@25919
   745
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
   746
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
   747
haftmann@25919
   748
haftmann@25919
   749
text {* Legacy theorems *}
haftmann@25919
   750
haftmann@25919
   751
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   752
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
huffman@47108
   753
lemmas numeral_1_eq_1 = numeral_One
haftmann@25919
   754
huffman@30802
   755
subsection {* Setting up simplification procedures *}
huffman@30802
   756
haftmann@54249
   757
lemmas of_int_simps =
haftmann@54249
   758
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   759
huffman@30802
   760
lemmas int_arith_rules =
haftmann@54249
   761
  numeral_One more_arith_simps of_nat_simps of_int_simps
huffman@30802
   762
wenzelm@48891
   763
ML_file "Tools/int_arith.ML"
haftmann@30496
   764
declaration {* K Int_Arith.setup *}
haftmann@25919
   765
huffman@47108
   766
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
huffman@47108
   767
  "(m::'a::linordered_idom) <= n" |
huffman@47108
   768
  "(m::'a::linordered_idom) = n") =
wenzelm@43595
   769
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
   770
haftmann@25919
   771
haftmann@25919
   772
subsection{*More Inequality Reasoning*}
haftmann@25919
   773
haftmann@25919
   774
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
   775
by arith
haftmann@25919
   776
haftmann@25919
   777
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
   778
by arith
haftmann@25919
   779
haftmann@25919
   780
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
   781
by arith
haftmann@25919
   782
haftmann@25919
   783
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
   784
by arith
haftmann@25919
   785
haftmann@25919
   786
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
   787
by arith
haftmann@25919
   788
haftmann@25919
   789
huffman@28958
   790
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
   791
huffman@48044
   792
text{*Simplify the term @{term "w + - z"}*}
haftmann@25919
   793
haftmann@25919
   794
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
   795
apply (insert zless_nat_conj [of 1 z])
huffman@47207
   796
apply auto
haftmann@25919
   797
done
haftmann@25919
   798
haftmann@25919
   799
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
   800
      z is an integer literal.*}
huffman@47108
   801
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
   802
haftmann@25919
   803
lemma split_nat [arith_split]:
huffman@44709
   804
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
   805
  (is "?P = (?L & ?R)")
haftmann@25919
   806
proof (cases "i < 0")
haftmann@25919
   807
  case True thus ?thesis by auto
haftmann@25919
   808
next
haftmann@25919
   809
  case False
haftmann@25919
   810
  have "?P = ?L"
haftmann@25919
   811
  proof
haftmann@25919
   812
    assume ?P thus ?L using False by clarsimp
haftmann@25919
   813
  next
haftmann@25919
   814
    assume ?L thus ?P using False by simp
haftmann@25919
   815
  qed
haftmann@25919
   816
  with False show ?thesis by simp
haftmann@25919
   817
qed
haftmann@25919
   818
hoelzl@59000
   819
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
   820
  by auto
hoelzl@59000
   821
hoelzl@59000
   822
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
   823
  by auto
hoelzl@59000
   824
haftmann@25919
   825
context ring_1
haftmann@25919
   826
begin
haftmann@25919
   827
blanchet@33056
   828
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
   829
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
   830
proof (cases "k < 0")
haftmann@25919
   831
  case True then have "0 \<le> - k" by simp
haftmann@25919
   832
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
   833
  with True show ?thesis by simp
haftmann@25919
   834
next
haftmann@25919
   835
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
   836
qed
haftmann@25919
   837
haftmann@25919
   838
end
haftmann@25919
   839
haftmann@25919
   840
lemma nat_mult_distrib:
haftmann@25919
   841
  fixes z z' :: int
haftmann@25919
   842
  assumes "0 \<le> z"
haftmann@25919
   843
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
   844
proof (cases "0 \<le> z'")
haftmann@25919
   845
  case False with assms have "z * z' \<le> 0"
haftmann@25919
   846
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
   847
  then have "nat (z * z') = 0" by simp
haftmann@25919
   848
  moreover from False have "nat z' = 0" by simp
haftmann@25919
   849
  ultimately show ?thesis by simp
haftmann@25919
   850
next
haftmann@25919
   851
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
   852
  show ?thesis
haftmann@25919
   853
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
   854
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
   855
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
   856
qed
haftmann@25919
   857
haftmann@25919
   858
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
   859
apply (rule trans)
haftmann@25919
   860
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
   861
done
haftmann@25919
   862
haftmann@25919
   863
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
   864
apply (cases "z=0 | w=0")
haftmann@25919
   865
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
   866
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
   867
done
haftmann@25919
   868
huffman@47207
   869
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207
   870
apply (rule sym)
huffman@47207
   871
apply (simp add: nat_eq_iff)
huffman@47207
   872
done
huffman@47207
   873
huffman@47207
   874
lemma diff_nat_eq_if:
huffman@47207
   875
     "nat z - nat z' =  
huffman@47207
   876
        (if z' < 0 then nat z   
huffman@47207
   877
         else let d = z-z' in     
huffman@47207
   878
              if d < 0 then 0 else nat d)"
huffman@47207
   879
by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
   880
huffman@47207
   881
lemma nat_numeral_diff_1 [simp]:
huffman@47207
   882
  "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
   883
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
   884
haftmann@25919
   885
haftmann@25919
   886
subsection "Induction principles for int"
haftmann@25919
   887
haftmann@25919
   888
text{*Well-founded segments of the integers*}
haftmann@25919
   889
haftmann@25919
   890
definition
haftmann@25919
   891
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
   892
where
haftmann@25919
   893
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
   894
haftmann@25919
   895
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
   896
proof -
haftmann@25919
   897
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
   898
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
   899
  thus ?thesis 
haftmann@25919
   900
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   901
qed
haftmann@25919
   902
haftmann@25919
   903
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
   904
by RankFinder.*}
haftmann@25919
   905
haftmann@25919
   906
definition
haftmann@25919
   907
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
   908
where
haftmann@25919
   909
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
   910
haftmann@25919
   911
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
   912
proof -
haftmann@25919
   913
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
   914
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
   915
  thus ?thesis 
haftmann@25919
   916
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
   917
qed
haftmann@25919
   918
haftmann@25919
   919
(* `set:int': dummy construction *)
haftmann@25919
   920
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
   921
  fixes i :: int
haftmann@25919
   922
  assumes ge: "k \<le> i" and
haftmann@25919
   923
    base: "P k" and
haftmann@25919
   924
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
   925
  shows "P i"
haftmann@25919
   926
proof -
wenzelm@42676
   927
  { fix n
wenzelm@42676
   928
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
   929
    proof (induct n)
haftmann@25919
   930
      case 0
haftmann@25919
   931
      hence "i = k" by arith
haftmann@25919
   932
      thus "P i" using base by simp
haftmann@25919
   933
    next
haftmann@25919
   934
      case (Suc n)
haftmann@25919
   935
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
   936
      moreover
haftmann@25919
   937
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
   938
      ultimately
wenzelm@42676
   939
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
   940
      from step [OF ki1 this] show ?case by simp
haftmann@25919
   941
    qed
haftmann@25919
   942
  }
haftmann@25919
   943
  with ge show ?thesis by fast
haftmann@25919
   944
qed
haftmann@25919
   945
haftmann@25928
   946
(* `set:int': dummy construction *)
haftmann@25928
   947
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
   948
  assumes gr: "k < (i::int)" and
haftmann@25919
   949
        base: "P(k+1)" and
haftmann@25919
   950
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
   951
  shows "P i"
haftmann@25919
   952
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
   953
  using gr apply arith
haftmann@25919
   954
 apply(rule base)
haftmann@25919
   955
apply (rule step, simp+)
haftmann@25919
   956
done
haftmann@25919
   957
wenzelm@42676
   958
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
   959
  assumes le: "i \<le> (k::int)" and
haftmann@25919
   960
        base: "P(k)" and
haftmann@25919
   961
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   962
  shows "P i"
haftmann@25919
   963
proof -
wenzelm@42676
   964
  { fix n
wenzelm@42676
   965
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
   966
    proof (induct n)
haftmann@25919
   967
      case 0
haftmann@25919
   968
      hence "i = k" by arith
haftmann@25919
   969
      thus "P i" using base by simp
haftmann@25919
   970
    next
haftmann@25919
   971
      case (Suc n)
wenzelm@42676
   972
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
   973
      moreover
haftmann@25919
   974
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
   975
      ultimately
wenzelm@42676
   976
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
   977
      from step[OF ki1 this] show ?case by simp
haftmann@25919
   978
    qed
haftmann@25919
   979
  }
haftmann@25919
   980
  with le show ?thesis by fast
haftmann@25919
   981
qed
haftmann@25919
   982
wenzelm@42676
   983
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
   984
  assumes less: "(i::int) < k" and
haftmann@25919
   985
        base: "P(k - 1)" and
haftmann@25919
   986
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
   987
  shows "P i"
haftmann@25919
   988
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
   989
  using less apply arith
haftmann@25919
   990
 apply(rule base)
haftmann@25919
   991
apply (rule step, simp+)
haftmann@25919
   992
done
haftmann@25919
   993
haftmann@36811
   994
theorem int_induct [case_names base step1 step2]:
haftmann@36801
   995
  fixes k :: int
haftmann@36801
   996
  assumes base: "P k"
haftmann@36801
   997
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
   998
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
   999
  shows "P i"
haftmann@36801
  1000
proof -
haftmann@36801
  1001
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1002
  then show ?thesis
wenzelm@42676
  1003
  proof
wenzelm@42676
  1004
    assume "i \<ge> k"
wenzelm@42676
  1005
    then show ?thesis using base
haftmann@36801
  1006
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1007
  next
wenzelm@42676
  1008
    assume "i \<le> k"
wenzelm@42676
  1009
    then show ?thesis using base
haftmann@36801
  1010
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1011
  qed
haftmann@36801
  1012
qed
haftmann@36801
  1013
haftmann@25919
  1014
subsection{*Intermediate value theorems*}
haftmann@25919
  1015
haftmann@25919
  1016
lemma int_val_lemma:
haftmann@25919
  1017
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1018
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1019
unfolding One_nat_def
wenzelm@42676
  1020
apply (induct n)
wenzelm@42676
  1021
apply simp
haftmann@25919
  1022
apply (intro strip)
haftmann@25919
  1023
apply (erule impE, simp)
haftmann@25919
  1024
apply (erule_tac x = n in allE, simp)
huffman@30079
  1025
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1026
apply force
haftmann@25919
  1027
apply (erule impE)
haftmann@25919
  1028
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1029
apply (blast intro: le_SucI)
haftmann@25919
  1030
done
haftmann@25919
  1031
haftmann@25919
  1032
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1033
haftmann@25919
  1034
lemma nat_intermed_int_val:
haftmann@25919
  1035
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1036
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1037
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1038
       in int_val_lemma)
huffman@30079
  1039
unfolding One_nat_def
haftmann@25919
  1040
apply simp
haftmann@25919
  1041
apply (erule exE)
haftmann@25919
  1042
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1043
done
haftmann@25919
  1044
haftmann@25919
  1045
haftmann@25919
  1046
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1047
haftmann@25919
  1048
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1049
by arith
haftmann@25919
  1050
paulson@34055
  1051
lemma abs_zmult_eq_1:
paulson@34055
  1052
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1053
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1054
proof -
paulson@34055
  1055
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1056
    by auto
paulson@34055
  1057
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1058
  proof
paulson@34055
  1059
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1060
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1061
      by (simp add: mult_mono 0) 
paulson@34055
  1062
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1063
      by (simp add: abs_mult)
paulson@34055
  1064
    also have "... = 1"
paulson@34055
  1065
      by (simp add: mn)
paulson@34055
  1066
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1067
    thus "False" using 0
huffman@47108
  1068
      by arith
paulson@34055
  1069
  qed
paulson@34055
  1070
  thus ?thesis using 0
paulson@34055
  1071
    by auto
paulson@34055
  1072
qed
haftmann@25919
  1073
haftmann@25919
  1074
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1075
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1076
boehmes@35815
  1077
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1078
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1079
proof -
boehmes@35815
  1080
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1081
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1082
qed
haftmann@25919
  1083
haftmann@25919
  1084
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1085
apply (rule iffI) 
haftmann@25919
  1086
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@57512
  1087
 apply (simp add: mult.commute [of m]) 
haftmann@25919
  1088
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1089
done
haftmann@25919
  1090
haftmann@33296
  1091
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1092
proof
haftmann@33296
  1093
  assume "finite (UNIV::int set)"
haftmann@33296
  1094
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1095
    by (rule injI) simp
haftmann@33296
  1096
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1097
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1098
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1099
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1100
qed
haftmann@25919
  1101
haftmann@25919
  1102
haftmann@30652
  1103
subsection {* Further theorems on numerals *}
haftmann@30652
  1104
haftmann@30652
  1105
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1106
haftmann@30652
  1107
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1108
webertj@49962
  1109
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1110
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1111
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1112
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1113
haftmann@30652
  1114
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1115
huffman@47108
  1116
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1117
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1118
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1119
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1120
haftmann@30652
  1121
haftmann@30652
  1122
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1123
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1124
huffman@47108
  1125
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1126
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1127
huffman@47108
  1128
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1129
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1130
haftmann@30652
  1131
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1132
into the literal.*}
haftmann@30652
  1133
haftmann@54489
  1134
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1135
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1136
haftmann@54489
  1137
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1138
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1139
haftmann@54489
  1140
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1141
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1142
haftmann@54489
  1143
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1144
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1145
haftmann@54489
  1146
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1147
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1148
haftmann@54489
  1149
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1150
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1151
haftmann@54489
  1152
-- {* FIXME maybe simproc *}
haftmann@30652
  1153
haftmann@30652
  1154
haftmann@30652
  1155
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  1156
huffman@47108
  1157
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1158
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1159
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1160
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1161
haftmann@30652
  1162
haftmann@30652
  1163
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  1164
huffman@47108
  1165
lemmas le_divide_eq_numeral1 [simp] =
huffman@47108
  1166
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1167
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1168
huffman@47108
  1169
lemmas divide_le_eq_numeral1 [simp] =
huffman@47108
  1170
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1171
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1172
huffman@47108
  1173
lemmas less_divide_eq_numeral1 [simp] =
huffman@47108
  1174
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1175
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1176
huffman@47108
  1177
lemmas divide_less_eq_numeral1 [simp] =
huffman@47108
  1178
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1179
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1180
huffman@47108
  1181
lemmas eq_divide_eq_numeral1 [simp] =
huffman@47108
  1182
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1183
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1184
huffman@47108
  1185
lemmas divide_eq_eq_numeral1 [simp] =
huffman@47108
  1186
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1187
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1188
haftmann@30652
  1189
haftmann@30652
  1190
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  1191
haftmann@30652
  1192
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  1193
huffman@47108
  1194
lemmas le_divide_eq_numeral =
huffman@47108
  1195
  le_divide_eq [of "numeral w"]
haftmann@54489
  1196
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1197
huffman@47108
  1198
lemmas divide_le_eq_numeral =
huffman@47108
  1199
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1200
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1201
huffman@47108
  1202
lemmas less_divide_eq_numeral =
huffman@47108
  1203
  less_divide_eq [of "numeral w"]
haftmann@54489
  1204
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1205
huffman@47108
  1206
lemmas divide_less_eq_numeral =
huffman@47108
  1207
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1208
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1209
huffman@47108
  1210
lemmas eq_divide_eq_numeral =
huffman@47108
  1211
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1212
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1213
huffman@47108
  1214
lemmas divide_eq_eq_numeral =
huffman@47108
  1215
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1216
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1217
haftmann@30652
  1218
haftmann@30652
  1219
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  1220
lemmas divide_const_simps =
huffman@47108
  1221
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
huffman@47108
  1222
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
haftmann@30652
  1223
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1224
haftmann@30652
  1225
haftmann@33320
  1226
subsection {* The divides relation *}
haftmann@33320
  1227
nipkow@33657
  1228
lemma zdvd_antisym_nonneg:
nipkow@33657
  1229
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  1230
  apply (simp add: dvd_def, auto)
haftmann@57512
  1231
  apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1232
  done
haftmann@33320
  1233
nipkow@33657
  1234
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  1235
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  1236
proof cases
nipkow@33657
  1237
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  1238
next
nipkow@33657
  1239
  assume "a \<noteq> 0"
haftmann@33320
  1240
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  1241
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  1242
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  1243
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@57512
  1244
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult.assoc)
haftmann@33320
  1245
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  1246
  thus ?thesis using k k' by auto
haftmann@33320
  1247
qed
haftmann@33320
  1248
haftmann@33320
  1249
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@58649
  1250
  using dvd_add_right_iff [of k "- n" m] by simp 
haftmann@33320
  1251
haftmann@33320
  1252
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@58649
  1253
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1254
haftmann@33320
  1255
lemma dvd_imp_le_int:
haftmann@33320
  1256
  fixes d i :: int
haftmann@33320
  1257
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1258
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1259
proof -
haftmann@33320
  1260
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  1261
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  1262
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1263
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  1264
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  1265
qed
haftmann@33320
  1266
haftmann@33320
  1267
lemma zdvd_not_zless:
haftmann@33320
  1268
  fixes m n :: int
haftmann@33320
  1269
  assumes "0 < m" and "m < n"
haftmann@33320
  1270
  shows "\<not> n dvd m"
haftmann@33320
  1271
proof
haftmann@33320
  1272
  from assms have "0 < n" by auto
haftmann@33320
  1273
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  1274
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  1275
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  1276
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  1277
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1278
qed
haftmann@33320
  1279
haftmann@33320
  1280
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  1281
  shows "m dvd n"
haftmann@33320
  1282
proof-
haftmann@33320
  1283
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  1284
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@57512
  1285
    with h have False by (simp add: mult.assoc)}
haftmann@33320
  1286
  hence "n = m * h" by blast
haftmann@33320
  1287
  thus ?thesis by simp
haftmann@33320
  1288
qed
haftmann@33320
  1289
haftmann@33320
  1290
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  1291
proof -
haftmann@33320
  1292
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  1293
  proof -
haftmann@33320
  1294
    fix k
haftmann@33320
  1295
    assume A: "int y = int x * k"
wenzelm@42676
  1296
    then show "x dvd y"
wenzelm@42676
  1297
    proof (cases k)
wenzelm@42676
  1298
      case (nonneg n)
wenzelm@42676
  1299
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  1300
      then show ?thesis ..
haftmann@33320
  1301
    next
wenzelm@42676
  1302
      case (neg n)
wenzelm@42676
  1303
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  1304
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  1305
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  1306
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  1307
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  1308
    qed
haftmann@33320
  1309
  qed
haftmann@33320
  1310
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1311
qed
haftmann@33320
  1312
wenzelm@42676
  1313
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  1314
proof
haftmann@33320
  1315
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  1316
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  1317
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  1318
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  1319
next
haftmann@33320
  1320
  assume "\<bar>x\<bar>=1"
haftmann@33320
  1321
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  1322
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  1323
qed
haftmann@33320
  1324
haftmann@33320
  1325
lemma zdvd_mult_cancel1: 
haftmann@33320
  1326
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  1327
proof
haftmann@33320
  1328
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  1329
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  1330
next
haftmann@33320
  1331
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  1332
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  1333
qed
haftmann@33320
  1334
haftmann@33320
  1335
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  1336
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1337
haftmann@33320
  1338
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  1339
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  1340
haftmann@58650
  1341
lemma dvd_int_unfold_dvd_nat:
haftmann@58650
  1342
  "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
haftmann@58650
  1343
  unfolding dvd_int_iff [symmetric] by simp
haftmann@58650
  1344
haftmann@33320
  1345
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  1346
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1347
haftmann@33341
  1348
lemma eq_nat_nat_iff:
haftmann@33341
  1349
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1350
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1351
haftmann@33341
  1352
lemma nat_power_eq:
haftmann@33341
  1353
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1354
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1355
haftmann@33320
  1356
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  1357
  apply (cases n)
haftmann@33320
  1358
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1359
  apply (cases z)
haftmann@33320
  1360
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  1361
  done
haftmann@33320
  1362
haftmann@36749
  1363
lemma zdvd_period:
haftmann@36749
  1364
  fixes a d :: int
haftmann@36749
  1365
  assumes "a dvd d"
haftmann@36749
  1366
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  1367
proof -
haftmann@36749
  1368
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  1369
  show ?thesis
wenzelm@42676
  1370
  proof
haftmann@36749
  1371
    assume "a dvd (x + t)"
haftmann@36749
  1372
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  1373
    then have "x = a * l - t" by simp
haftmann@36749
  1374
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  1375
  next
haftmann@36749
  1376
    assume "a dvd x + c * d + t"
haftmann@36749
  1377
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  1378
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  1379
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  1380
  qed
haftmann@36749
  1381
qed
haftmann@36749
  1382
haftmann@33320
  1383
bulwahn@46756
  1384
subsection {* Finiteness of intervals *}
bulwahn@46756
  1385
bulwahn@46756
  1386
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756
  1387
proof (cases "a <= b")
bulwahn@46756
  1388
  case True
bulwahn@46756
  1389
  from this show ?thesis
bulwahn@46756
  1390
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1391
    case base
bulwahn@46756
  1392
    have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756
  1393
    from this show ?case by simp
bulwahn@46756
  1394
  next
bulwahn@46756
  1395
    case (step b)
bulwahn@46756
  1396
    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756
  1397
    from this step show ?case by simp
bulwahn@46756
  1398
  qed
bulwahn@46756
  1399
next
bulwahn@46756
  1400
  case False from this show ?thesis
bulwahn@46756
  1401
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1402
qed
bulwahn@46756
  1403
bulwahn@46756
  1404
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756
  1405
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1406
bulwahn@46756
  1407
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756
  1408
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1409
bulwahn@46756
  1410
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756
  1411
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1412
bulwahn@46756
  1413
haftmann@25919
  1414
subsection {* Configuration of the code generator *}
haftmann@25919
  1415
huffman@47108
  1416
text {* Constructors *}
huffman@47108
  1417
huffman@47108
  1418
definition Pos :: "num \<Rightarrow> int" where
huffman@47108
  1419
  [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1420
huffman@47108
  1421
definition Neg :: "num \<Rightarrow> int" where
haftmann@54489
  1422
  [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1423
huffman@47108
  1424
code_datatype "0::int" Pos Neg
huffman@47108
  1425
huffman@47108
  1426
huffman@47108
  1427
text {* Auxiliary operations *}
huffman@47108
  1428
huffman@47108
  1429
definition dup :: "int \<Rightarrow> int" where
huffman@47108
  1430
  [simp]: "dup k = k + k"
haftmann@26507
  1431
huffman@47108
  1432
lemma dup_code [code]:
huffman@47108
  1433
  "dup 0 = 0"
huffman@47108
  1434
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1435
  "dup (Neg n) = Neg (Num.Bit0 n)"
haftmann@54489
  1436
  unfolding Pos_def Neg_def
huffman@47108
  1437
  by (simp_all add: numeral_Bit0)
huffman@47108
  1438
huffman@47108
  1439
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108
  1440
  [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1441
huffman@47108
  1442
lemma sub_code [code]:
huffman@47108
  1443
  "sub Num.One Num.One = 0"
huffman@47108
  1444
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1445
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1446
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1447
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1448
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1449
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1450
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1451
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@54489
  1452
  apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
haftmann@54230
  1453
  apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230
  1454
  apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230
  1455
  apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230
  1456
  done
huffman@47108
  1457
huffman@47108
  1458
text {* Implementations *}
huffman@47108
  1459
huffman@47108
  1460
lemma one_int_code [code, code_unfold]:
huffman@47108
  1461
  "1 = Pos Num.One"
huffman@47108
  1462
  by simp
huffman@47108
  1463
huffman@47108
  1464
lemma plus_int_code [code]:
huffman@47108
  1465
  "k + 0 = (k::int)"
huffman@47108
  1466
  "0 + l = (l::int)"
huffman@47108
  1467
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1468
  "Pos m + Neg n = sub m n"
huffman@47108
  1469
  "Neg m + Pos n = sub n m"
huffman@47108
  1470
  "Neg m + Neg n = Neg (m + n)"
huffman@47108
  1471
  by simp_all
haftmann@26507
  1472
huffman@47108
  1473
lemma uminus_int_code [code]:
huffman@47108
  1474
  "uminus 0 = (0::int)"
huffman@47108
  1475
  "uminus (Pos m) = Neg m"
huffman@47108
  1476
  "uminus (Neg m) = Pos m"
huffman@47108
  1477
  by simp_all
huffman@47108
  1478
huffman@47108
  1479
lemma minus_int_code [code]:
huffman@47108
  1480
  "k - 0 = (k::int)"
huffman@47108
  1481
  "0 - l = uminus (l::int)"
huffman@47108
  1482
  "Pos m - Pos n = sub m n"
huffman@47108
  1483
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1484
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1485
  "Neg m - Neg n = sub n m"
huffman@47108
  1486
  by simp_all
huffman@47108
  1487
huffman@47108
  1488
lemma times_int_code [code]:
huffman@47108
  1489
  "k * 0 = (0::int)"
huffman@47108
  1490
  "0 * l = (0::int)"
huffman@47108
  1491
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1492
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1493
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1494
  "Neg m * Neg n = Pos (m * n)"
huffman@47108
  1495
  by simp_all
haftmann@26507
  1496
haftmann@38857
  1497
instantiation int :: equal
haftmann@26507
  1498
begin
haftmann@26507
  1499
haftmann@37767
  1500
definition
huffman@47108
  1501
  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1502
huffman@47108
  1503
instance by default (rule equal_int_def)
haftmann@26507
  1504
haftmann@26507
  1505
end
haftmann@26507
  1506
huffman@47108
  1507
lemma equal_int_code [code]:
huffman@47108
  1508
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1509
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1510
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1511
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1512
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1513
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1514
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1515
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1516
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1517
  by (auto simp add: equal)
haftmann@26507
  1518
huffman@47108
  1519
lemma equal_int_refl [code nbe]:
haftmann@38857
  1520
  "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108
  1521
  by (fact equal_refl)
haftmann@26507
  1522
haftmann@28562
  1523
lemma less_eq_int_code [code]:
huffman@47108
  1524
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1525
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1526
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1527
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1528
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1529
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1530
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1531
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1532
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1533
  by simp_all
haftmann@26507
  1534
haftmann@28562
  1535
lemma less_int_code [code]:
huffman@47108
  1536
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1537
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1538
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1539
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1540
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1541
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1542
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1543
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1544
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1545
  by simp_all
haftmann@25919
  1546
huffman@47108
  1547
lemma nat_code [code]:
huffman@47108
  1548
  "nat (Int.Neg k) = 0"
huffman@47108
  1549
  "nat 0 = 0"
huffman@47108
  1550
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1551
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1552
huffman@47108
  1553
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1554
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1555
  "of_int 0 = 0"
huffman@47108
  1556
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1557
  by simp_all
haftmann@25919
  1558
huffman@47108
  1559
huffman@47108
  1560
text {* Serializer setup *}
haftmann@25919
  1561
haftmann@52435
  1562
code_identifier
haftmann@52435
  1563
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1564
haftmann@25919
  1565
quickcheck_params [default_type = int]
haftmann@25919
  1566
huffman@47108
  1567
hide_const (open) Pos Neg sub dup
haftmann@25919
  1568
haftmann@25919
  1569
haftmann@25919
  1570
subsection {* Legacy theorems *}
haftmann@25919
  1571
haftmann@25919
  1572
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1573
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1574
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  1575
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
wenzelm@45607
  1576
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1577
lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607
  1578
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919
  1579
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1580
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607
  1581
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919
  1582
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1583
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1584
lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207
  1585
lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607
  1586
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919
  1587
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1588
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255
  1589
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255
  1590
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960
  1591
haftmann@31015
  1592
lemma zpower_zpower:
haftmann@31015
  1593
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  1594
  by (rule power_mult [symmetric])
haftmann@31015
  1595
haftmann@31015
  1596
lemma int_power:
haftmann@31015
  1597
  "int (m ^ n) = int m ^ n"
haftmann@54489
  1598
  by (fact of_nat_power)
haftmann@31015
  1599
haftmann@31015
  1600
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  1601
huffman@48045
  1602
text {* De-register @{text "int"} as a quotient type: *}
huffman@48045
  1603
kuncar@53652
  1604
lifting_update int.lifting
kuncar@53652
  1605
lifting_forget int.lifting
huffman@48045
  1606
haftmann@25919
  1607
end