
wenzelm@41959: (*  Title:      HOL/Int.thy
haftmann@25919:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@41959:     Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
haftmann@25919: *)
haftmann@25919: 
haftmann@25919: header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
haftmann@25919: 
haftmann@25919: theory Int
blanchet@55404: imports Equiv_Relations Power Quotient Fun_Def
haftmann@25919: begin
haftmann@25919: 
huffman@48045: subsection {* Definition of integers as a quotient type *}
haftmann@25919: 
huffman@48045: definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where
huffman@48045:   "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
huffman@48045: 
huffman@48045: lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
huffman@48045:   by (simp add: intrel_def)
haftmann@25919: 
huffman@48045: quotient_type int = "nat \<times> nat" / "intrel"
wenzelm@45694:   morphisms Rep_Integ Abs_Integ
huffman@48045: proof (rule equivpI)
huffman@48045:   show "reflp intrel"
huffman@48045:     unfolding reflp_def by auto
huffman@48045:   show "symp intrel"
huffman@48045:     unfolding symp_def by auto
huffman@48045:   show "transp intrel"
huffman@48045:     unfolding transp_def by auto
huffman@48045: qed
haftmann@25919: 
huffman@48045: lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
huffman@48045:      "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
huffman@48045: by (induct z) auto
huffman@48045: 
huffman@48045: subsection {* Integers form a commutative ring *}
huffman@48045: 
huffman@48045: instantiation int :: comm_ring_1
haftmann@25919: begin
haftmann@25919: 
kuncar@51994: lift_definition zero_int :: "int" is "(0, 0)" .
haftmann@25919: 
kuncar@51994: lift_definition one_int :: "int" is "(1, 0)" .
haftmann@25919: 
huffman@48045: lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045:   is "\<lambda>(x, y) (u, v). (x + u, y + v)"
huffman@48045:   by clarsimp
haftmann@25919: 
huffman@48045: lift_definition uminus_int :: "int \<Rightarrow> int"
huffman@48045:   is "\<lambda>(x, y). (y, x)"
huffman@48045:   by clarsimp
haftmann@25919: 
huffman@48045: lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045:   is "\<lambda>(x, y) (u, v). (x + v, y + u)"
huffman@48045:   by clarsimp
haftmann@25919: 
huffman@48045: lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045:   is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
huffman@48045: proof (clarsimp)
huffman@48045:   fix s t u v w x y z :: nat
huffman@48045:   assume "s + v = u + t" and "w + z = y + x"
huffman@48045:   hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
huffman@48045:        = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
huffman@48045:     by simp
huffman@48045:   thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
huffman@48045:     by (simp add: algebra_simps)
huffman@48045: qed
haftmann@25919: 
huffman@48045: instance
huffman@48045:   by default (transfer, clarsimp simp: algebra_simps)+
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
huffman@44709: abbreviation int :: "nat \<Rightarrow> int" where
huffman@44709:   "int \<equiv> of_nat"
huffman@44709: 
huffman@48045: lemma int_def: "int n = Abs_Integ (n, 0)"
huffman@48045:   by (induct n, simp add: zero_int.abs_eq,
huffman@48045:     simp add: one_int.abs_eq plus_int.abs_eq)
haftmann@25919: 
huffman@48045: lemma int_transfer [transfer_rule]:
kuncar@56525:   "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
kuncar@56525:   unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
haftmann@25919: 
huffman@48045: lemma int_diff_cases:
huffman@48045:   obtains (diff) m n where "z = int m - int n"
huffman@48045:   by transfer clarsimp
huffman@48045: 
huffman@48045: subsection {* Integers are totally ordered *}
haftmann@25919: 
huffman@48045: instantiation int :: linorder
huffman@48045: begin
huffman@48045: 
huffman@48045: lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
huffman@48045:   is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
huffman@48045:   by auto
huffman@48045: 
huffman@48045: lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
huffman@48045:   is "\<lambda>(x, y) (u, v). x + v < u + y"
huffman@48045:   by auto
huffman@48045: 
huffman@48045: instance
huffman@48045:   by default (transfer, force)+
huffman@48045: 
huffman@48045: end
haftmann@25919: 
haftmann@25919: instantiation int :: distrib_lattice
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
haftmann@25919: 
haftmann@25919: instance
haftmann@25919:   by intro_classes
haftmann@54863:     (auto simp add: inf_int_def sup_int_def max_min_distrib2)
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
huffman@48045: subsection {* Ordering properties of arithmetic operations *}
huffman@48045: 
haftmann@35028: instance int :: ordered_cancel_ab_semigroup_add
haftmann@25919: proof
haftmann@25919:   fix i j k :: int
haftmann@25919:   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
huffman@48045:     by transfer clarsimp
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: text{*Strict Monotonicity of Multiplication*}
haftmann@25919: 
haftmann@25919: text{*strict, in 1st argument; proof is by induction on k>0*}
haftmann@25919: lemma zmult_zless_mono2_lemma:
huffman@44709:      "(i::int)<j ==> 0<k ==> int k * i < int k * j"
wenzelm@42676: apply (induct k)
wenzelm@42676: apply simp
webertj@49962: apply (simp add: distrib_right)
haftmann@25919: apply (case_tac "k=0")
haftmann@25919: apply (simp_all add: add_strict_mono)
haftmann@25919: done
haftmann@25919: 
huffman@44709: lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
huffman@48045: apply transfer
huffman@48045: apply clarsimp
huffman@48045: apply (rule_tac x="a - b" in exI, simp)
haftmann@25919: done
haftmann@25919: 
huffman@44709: lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
huffman@48045: apply transfer
huffman@48045: apply clarsimp
huffman@48045: apply (rule_tac x="a - b" in exI, simp)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
haftmann@25919: apply (drule zero_less_imp_eq_int)
haftmann@25919: apply (auto simp add: zmult_zless_mono2_lemma)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*The integers form an ordered integral domain*}
huffman@48045: instantiation int :: linordered_idom
huffman@48045: begin
huffman@48045: 
huffman@48045: definition
huffman@48045:   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
huffman@48045: 
huffman@48045: definition
huffman@48045:   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
huffman@48045: 
huffman@48045: instance proof
haftmann@25919:   fix i j k :: int
haftmann@25919:   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
haftmann@25919:     by (rule zmult_zless_mono2)
haftmann@25919:   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
haftmann@25919:     by (simp only: zabs_def)
haftmann@25919:   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
haftmann@25919:     by (simp only: zsgn_def)
haftmann@25919: qed
haftmann@25919: 
huffman@48045: end
huffman@48045: 
haftmann@25919: lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
huffman@48045:   by transfer clarsimp
haftmann@25919: 
haftmann@25919: lemma zless_iff_Suc_zadd:
huffman@44709:   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
huffman@48045: apply transfer
huffman@48045: apply auto
huffman@48045: apply (rename_tac a b c d)
huffman@48045: apply (rule_tac x="c+b - Suc(a+d)" in exI)
haftmann@25919: apply arith
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemmas int_distrib =
webertj@49962:   distrib_right [of z1 z2 w]
webertj@49962:   distrib_left [of w z1 z2]
wenzelm@45607:   left_diff_distrib [of z1 z2 w]
wenzelm@45607:   right_diff_distrib [of w z1 z2]
wenzelm@45607:   for z1 z2 w :: int
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
huffman@48045: lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
huffman@48045:   by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
huffman@48045:     of_nat_add [symmetric] simp del: of_nat_add)
haftmann@25919: 
haftmann@25919: lemma of_int_0 [simp]: "of_int 0 = 0"
huffman@48066:   by transfer simp
haftmann@25919: 
haftmann@25919: lemma of_int_1 [simp]: "of_int 1 = 1"
huffman@48066:   by transfer simp
haftmann@25919: 
haftmann@25919: lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
huffman@48066:   by transfer (clarsimp simp add: algebra_simps)
haftmann@25919: 
haftmann@25919: lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
huffman@48066:   by (transfer fixing: uminus) clarsimp
haftmann@25919: 
haftmann@25919: lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
haftmann@54230:   using of_int_add [of w "- z"] by simp
haftmann@25919: 
haftmann@25919: lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
huffman@48066:   by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
haftmann@25919: 
haftmann@25919: text{*Collapse nested embeddings*}
huffman@44709: lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
nipkow@29667: by (induct n) auto
haftmann@25919: 
huffman@47108: lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
huffman@47108:   by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
huffman@47108: 
haftmann@54489: lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
haftmann@54489:   by simp
huffman@47108: 
haftmann@31015: lemma of_int_power:
haftmann@31015:   "of_int (z ^ n) = of_int z ^ n"
haftmann@31015:   by (induct n) simp_all
haftmann@31015: 
haftmann@25919: end
haftmann@25919: 
huffman@47108: context ring_char_0
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_int_eq_iff [simp]:
haftmann@25919:    "of_int w = of_int z \<longleftrightarrow> w = z"
huffman@48066:   by transfer (clarsimp simp add: algebra_simps
huffman@48066:     of_nat_add [symmetric] simp del: of_nat_add)
haftmann@25919: 
haftmann@25919: text{*Special cases where either operand is zero*}
haftmann@36424: lemma of_int_eq_0_iff [simp]:
haftmann@36424:   "of_int z = 0 \<longleftrightarrow> z = 0"
haftmann@36424:   using of_int_eq_iff [of z 0] by simp
haftmann@36424: 
haftmann@36424: lemma of_int_0_eq_iff [simp]:
haftmann@36424:   "0 = of_int z \<longleftrightarrow> z = 0"
haftmann@36424:   using of_int_eq_iff [of 0 z] by simp
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@36424: context linordered_idom
haftmann@36424: begin
haftmann@36424: 
haftmann@35028: text{*Every @{text linordered_idom} has characteristic zero.*}
haftmann@36424: subclass ring_char_0 ..
haftmann@36424: 
haftmann@36424: lemma of_int_le_iff [simp]:
haftmann@36424:   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
huffman@48066:   by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
huffman@48066:     of_nat_add [symmetric] simp del: of_nat_add)
haftmann@36424: 
haftmann@36424: lemma of_int_less_iff [simp]:
haftmann@36424:   "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@36424:   by (simp add: less_le order_less_le)
haftmann@36424: 
haftmann@36424: lemma of_int_0_le_iff [simp]:
haftmann@36424:   "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
haftmann@36424:   using of_int_le_iff [of 0 z] by simp
haftmann@36424: 
haftmann@36424: lemma of_int_le_0_iff [simp]:
haftmann@36424:   "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
haftmann@36424:   using of_int_le_iff [of z 0] by simp
haftmann@36424: 
haftmann@36424: lemma of_int_0_less_iff [simp]:
haftmann@36424:   "0 < of_int z \<longleftrightarrow> 0 < z"
haftmann@36424:   using of_int_less_iff [of 0 z] by simp
haftmann@36424: 
haftmann@36424: lemma of_int_less_0_iff [simp]:
haftmann@36424:   "of_int z < 0 \<longleftrightarrow> z < 0"
haftmann@36424:   using of_int_less_iff [of z 0] by simp
haftmann@36424: 
haftmann@36424: end
haftmann@25919: 
hoelzl@56889: lemma of_nat_less_of_int_iff:
hoelzl@56889:   "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889:   by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889: 
haftmann@25919: lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919: proof
haftmann@25919:   fix z show "of_int z = id z"
huffman@48045:     by (cases z rule: int_diff_cases, simp)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
hoelzl@51329: instance int :: no_top
hoelzl@51329:   apply default
hoelzl@51329:   apply (rule_tac x="x + 1" in exI)
hoelzl@51329:   apply simp
hoelzl@51329:   done
hoelzl@51329: 
hoelzl@51329: instance int :: no_bot
hoelzl@51329:   apply default
hoelzl@51329:   apply (rule_tac x="x - 1" in exI)
hoelzl@51329:   apply simp
hoelzl@51329:   done
hoelzl@51329: 
haftmann@25919: subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919: 
huffman@48045: lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045:   by auto
haftmann@25919: 
huffman@44709: lemma nat_int [simp]: "nat (int n) = n"
huffman@48045:   by transfer simp
haftmann@25919: 
huffman@44709: lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045:   by transfer clarsimp
haftmann@25919: 
huffman@44709: corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
huffman@48045:   by transfer clarsimp
haftmann@25919: 
haftmann@25919: lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
huffman@48045:   by transfer (clarsimp, arith)
haftmann@25919: 
haftmann@25919: text{*An alternative condition is @{term "0 \<le> w"} *}
haftmann@25919: corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
haftmann@25919: by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919: 
haftmann@25919: corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919: by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919: 
haftmann@25919: lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
huffman@48045:   by transfer (clarsimp, arith)
haftmann@25919: 
haftmann@25919: lemma nonneg_eq_int:
haftmann@25919:   fixes z :: int
huffman@44709:   assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919:   shows P
haftmann@25919:   using assms by (blast dest: nat_0_le sym)
haftmann@25919: 
haftmann@54223: lemma nat_eq_iff:
haftmann@54223:   "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045:   by transfer (clarsimp simp add: le_imp_diff_is_add)
haftmann@54223:  
haftmann@54223: corollary nat_eq_iff2:
haftmann@54223:   "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223:   using nat_eq_iff [of w m] by auto
haftmann@54223: 
haftmann@54223: lemma nat_0 [simp]:
haftmann@54223:   "nat 0 = 0"
haftmann@54223:   by (simp add: nat_eq_iff)
haftmann@25919: 
haftmann@54223: lemma nat_1 [simp]:
haftmann@54223:   "nat 1 = Suc 0"
haftmann@54223:   by (simp add: nat_eq_iff)
haftmann@54223: 
haftmann@54223: lemma nat_numeral [simp]:
haftmann@54223:   "nat (numeral k) = numeral k"
haftmann@54223:   by (simp add: nat_eq_iff)
haftmann@25919: 
haftmann@54223: lemma nat_neg_numeral [simp]:
haftmann@54489:   "nat (- numeral k) = 0"
haftmann@54223:   by simp
haftmann@54223: 
haftmann@54223: lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223:   by simp
haftmann@54223:  
haftmann@25919: lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
huffman@48045:   by transfer (clarsimp, arith)
haftmann@25919: 
huffman@44709: lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045:   by transfer (clarsimp simp add: le_diff_conv)
huffman@44707: 
huffman@44707: lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045:   by transfer auto
huffman@44707: 
nipkow@29700: lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
huffman@48045:   by transfer clarsimp
nipkow@29700: 
haftmann@25919: lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919: by (auto simp add: nat_eq_iff2)
haftmann@25919: 
haftmann@25919: lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919: by (insert zless_nat_conj [of 0], auto)
haftmann@25919: 
haftmann@25919: lemma nat_add_distrib:
haftmann@54223:   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045:   by transfer clarsimp
haftmann@25919: 
haftmann@54223: lemma nat_diff_distrib':
haftmann@54223:   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223:   by transfer clarsimp
haftmann@54223:  
haftmann@25919: lemma nat_diff_distrib:
haftmann@54223:   "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223:   by (rule nat_diff_distrib') auto
haftmann@25919: 
huffman@44709: lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045:   by transfer simp
haftmann@25919: 
haftmann@53065: lemma le_nat_iff:
haftmann@53065:   "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065:   by transfer auto
haftmann@53065:   
huffman@44709: lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
huffman@48045:   by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066:   by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@54249: lemma diff_nat_numeral [simp]: 
haftmann@54249:   "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249:   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249: 
haftmann@54249: 
krauss@29779: text {* For termination proofs: *}
krauss@29779: lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779: 
haftmann@25919: 
haftmann@25919: subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919: 
huffman@44709: lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919: by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919: 
huffman@44709: lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919: by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919: 
huffman@44709: lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919: by (simp add: minus_le_iff)
haftmann@25919: 
huffman@44709: lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919: by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919: 
huffman@44709: lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919: by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919: 
huffman@44709: lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
huffman@48045:   by transfer simp
haftmann@25919: 
huffman@44709: lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919: by (simp add: linorder_not_less)
haftmann@25919: 
huffman@44709: lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919: by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919: 
huffman@44709: lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919: proof -
haftmann@25919:   have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919:     by (simp only: le_diff_eq add_0_left)
haftmann@25919:   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919:     by (auto elim: zero_le_imp_eq_int)
haftmann@25919:   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667:     by (simp only: algebra_simps)
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
huffman@44709: lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919: by simp
haftmann@25919: 
huffman@44709: lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: text{*This version is proved for all ordered rings, not just integers!
haftmann@25919:       It is proved here because attribute @{text arith_split} is not available
haftmann@35050:       in theory @{text Rings}.
haftmann@25919:       But is it really better than just rewriting with @{text abs_if}?*}
blanchet@54147: lemma abs_split [arith_split, no_atp]:
haftmann@35028:      "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919: by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919: 
huffman@44709: lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
huffman@48045: apply transfer
huffman@48045: apply clarsimp
huffman@48045: apply (rule_tac x="b - Suc a" in exI, arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Cases and induction *}
haftmann@25919: 
haftmann@25919: text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919: whether an integer is negative or not.*}
haftmann@25919: 
wenzelm@42676: theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709:   "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676: apply (cases "z < 0")
wenzelm@42676: apply (blast dest!: negD)
haftmann@25919: apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919: apply auto
haftmann@25919: apply (blast dest: nat_0_le [THEN sym])
haftmann@25919: done
haftmann@25919: 
wenzelm@42676: theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709:      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676:   by (cases z) auto
haftmann@25919: 
huffman@47207: lemma nonneg_int_cases:
huffman@47207:   assumes "0 \<le> k" obtains n where "k = int n"
haftmann@54489:   using assms by (rule nonneg_eq_int)
huffman@47207: 
huffman@47108: lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
huffman@47108:   -- {* Unfold all @{text let}s involving constants *}
haftmann@57514:   by (fact Let_numeral) -- {* FIXME drop *}
haftmann@37767: 
haftmann@54489: lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
haftmann@25919:   -- {* Unfold all @{text let}s involving constants *}
haftmann@57514:   by (fact Let_neg_numeral) -- {* FIXME drop *}
haftmann@25919: 
huffman@47108: text {* Unfold @{text min} and @{text max} on numerals. *}
huffman@28958: 
huffman@47108: lemmas max_number_of [simp] =
huffman@47108:   max_def [of "numeral u" "numeral v"]
haftmann@54489:   max_def [of "numeral u" "- numeral v"]
haftmann@54489:   max_def [of "- numeral u" "numeral v"]
haftmann@54489:   max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958: 
huffman@47108: lemmas min_number_of [simp] =
huffman@47108:   min_def [of "numeral u" "numeral v"]
haftmann@54489:   min_def [of "numeral u" "- numeral v"]
haftmann@54489:   min_def [of "- numeral u" "numeral v"]
haftmann@54489:   min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075: 
haftmann@25919: 
huffman@28958: subsubsection {* Binary comparisons *}
huffman@28958: 
huffman@28958: text {* Preliminaries *}
huffman@28958: 
huffman@28958: lemma even_less_0_iff:
haftmann@35028:   "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958: proof -
webertj@49962:   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
huffman@28958:   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958:     by (simp add: mult_less_0_iff zero_less_two 
huffman@28958:                   order_less_not_sym [OF zero_less_two])
huffman@28958:   finally show ?thesis .
huffman@28958: qed
huffman@28958: 
huffman@28958: lemma le_imp_0_less: 
huffman@28958:   assumes le: "0 \<le> z"
huffman@28958:   shows "(0::int) < 1 + z"
huffman@28958: proof -
huffman@28958:   have "0 \<le> z" by fact
huffman@47108:   also have "... < z + 1" by (rule less_add_one)
haftmann@57514:   also have "... = 1 + z" by (simp add: ac_simps)
huffman@28958:   finally show "0 < 1 + z" .
huffman@28958: qed
huffman@28958: 
huffman@28958: lemma odd_less_0_iff:
huffman@28958:   "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676: proof (cases z)
huffman@28958:   case (nonneg n)
haftmann@57512:   thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
huffman@28958:                              le_imp_0_less [THEN order_less_imp_le])  
huffman@28958: next
huffman@28958:   case (neg n)
huffman@30079:   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079:     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958: qed
huffman@28958: 
huffman@28958: subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919: 
haftmann@25919: lemmas double_eq_0_iff = double_zero
haftmann@25919: 
haftmann@25919: lemma odd_nonzero:
haftmann@33296:   "1 + z + z \<noteq> (0::int)"
wenzelm@42676: proof (cases z)
haftmann@25919:   case (nonneg n)
haftmann@25919:   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919:   thus ?thesis using  le_imp_0_less [OF le]
haftmann@57512:     by (auto simp add: add.assoc) 
haftmann@25919: next
haftmann@25919:   case (neg n)
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume eq: "1 + z + z = 0"
huffman@44709:     have "(0::int) < 1 + (int n + int n)"
haftmann@25919:       by (simp add: le_imp_0_less add_increasing) 
haftmann@25919:     also have "... = - (1 + z + z)" 
haftmann@57512:       by (simp add: neg add.assoc [symmetric]) 
haftmann@25919:     also have "... = 0" by (simp add: eq) 
haftmann@25919:     finally have "0<0" ..
haftmann@25919:     thus False by blast
haftmann@25919:   qed
haftmann@25919: qed
haftmann@25919: 
haftmann@30652: 
haftmann@25919: subsection {* The Set of Integers *}
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@30652: definition Ints  :: "'a set" where
haftmann@37767:   "Ints = range of_int"
haftmann@25919: 
haftmann@25919: notation (xsymbols)
haftmann@25919:   Ints  ("\<int>")
haftmann@25919: 
huffman@35634: lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634:   by (simp add: Ints_def)
huffman@35634: 
huffman@35634: lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533:   using Ints_of_int [of "of_nat n"] by simp
huffman@35634: 
haftmann@25919: lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533:   using Ints_of_int [of "0"] by simp
haftmann@25919: 
haftmann@25919: lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533:   using Ints_of_int [of "1"] by simp
haftmann@25919: 
haftmann@25919: lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_add [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_minus [symmetric])
haftmann@25919: done
haftmann@25919: 
huffman@35634: lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634: apply (auto simp add: Ints_def)
huffman@35634: apply (rule range_eqI)
huffman@35634: apply (rule of_int_diff [symmetric])
huffman@35634: done
huffman@35634: 
haftmann@25919: lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_mult [symmetric])
haftmann@25919: done
haftmann@25919: 
huffman@35634: lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634: by (induct n) simp_all
huffman@35634: 
haftmann@25919: lemma Ints_cases [cases set: Ints]:
haftmann@25919:   assumes "q \<in> \<int>"
haftmann@25919:   obtains (of_int) z where "q = of_int z"
haftmann@25919:   unfolding Ints_def
haftmann@25919: proof -
haftmann@25919:   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919:   then obtain z where "q = of_int z" ..
haftmann@25919:   then show thesis ..
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919:   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919:   by (rule Ints_cases) auto
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919: 
haftmann@25919: lemma Ints_double_eq_0_iff:
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@25919:   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume "a = 0"
haftmann@25919:     thus "a + a = 0" by simp
haftmann@25919:   next
haftmann@25919:     assume eq: "a + a = 0"
haftmann@25919:     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919:     hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919:     thus "a = 0" by (simp add: a)
haftmann@25919:   qed
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma Ints_odd_nonzero:
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@25919:   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume eq: "1 + a + a = 0"
haftmann@25919:     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919:     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     with odd_nonzero show False by blast
haftmann@25919:   qed
haftmann@25919: qed 
haftmann@25919: 
huffman@47108: lemma Nats_numeral [simp]: "numeral w \<in> Nats"
huffman@47108:   using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634: 
haftmann@25919: lemma Ints_odd_less_0: 
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@35028:   shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919:     by (simp add: a)
huffman@45532:   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
haftmann@25919:   also have "... = (a < 0)" by (simp add: a)
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919: 
haftmann@25919: lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919: lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919: 
haftmann@25919: 
haftmann@25919: text {* Legacy theorems *}
haftmann@25919: 
haftmann@25919: lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919: lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
huffman@47108: lemmas numeral_1_eq_1 = numeral_One
haftmann@25919: 
huffman@30802: subsection {* Setting up simplification procedures *}
huffman@30802: 
haftmann@54249: lemmas of_int_simps =
haftmann@54249:   of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249: 
huffman@30802: lemmas int_arith_rules =
haftmann@54249:   numeral_One more_arith_simps of_nat_simps of_int_simps
huffman@30802: 
wenzelm@48891: ML_file "Tools/int_arith.ML"
haftmann@30496: declaration {* K Int_Arith.setup *}
haftmann@25919: 
huffman@47108: simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
huffman@47108:   "(m::'a::linordered_idom) <= n" |
huffman@47108:   "(m::'a::linordered_idom) = n") =
wenzelm@43595:   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595: 
haftmann@25919: 
haftmann@25919: subsection{*More Inequality Reasoning*}
haftmann@25919: 
haftmann@25919: lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919: 
huffman@48044: text{*Simplify the term @{term "w + - z"}*}
haftmann@25919: 
haftmann@25919: lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919: apply (insert zless_nat_conj [of 1 z])
huffman@47207: apply auto
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919:       z is an integer literal.*}
huffman@47108: lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919: 
haftmann@25919: lemma split_nat [arith_split]:
huffman@44709:   "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919:   (is "?P = (?L & ?R)")
haftmann@25919: proof (cases "i < 0")
haftmann@25919:   case True thus ?thesis by auto
haftmann@25919: next
haftmann@25919:   case False
haftmann@25919:   have "?P = ?L"
haftmann@25919:   proof
haftmann@25919:     assume ?P thus ?L using False by clarsimp
haftmann@25919:   next
haftmann@25919:     assume ?L thus ?P using False by simp
haftmann@25919:   qed
haftmann@25919:   with False show ?thesis by simp
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
blanchet@33056: lemma of_int_of_nat [nitpick_simp]:
haftmann@25919:   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919: proof (cases "k < 0")
haftmann@25919:   case True then have "0 \<le> - k" by simp
haftmann@25919:   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919:   with True show ?thesis by simp
haftmann@25919: next
haftmann@25919:   case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: lemma nat_mult_distrib:
haftmann@25919:   fixes z z' :: int
haftmann@25919:   assumes "0 \<le> z"
haftmann@25919:   shows "nat (z * z') = nat z * nat z'"
haftmann@25919: proof (cases "0 \<le> z'")
haftmann@25919:   case False with assms have "z * z' \<le> 0"
haftmann@25919:     by (simp add: not_le mult_le_0_iff)
haftmann@25919:   then have "nat (z * z') = 0" by simp
haftmann@25919:   moreover from False have "nat z' = 0" by simp
haftmann@25919:   ultimately show ?thesis by simp
haftmann@25919: next
haftmann@25919:   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919:   show ?thesis
haftmann@25919:     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919:       (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919:          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919: apply (rule trans)
haftmann@25919: apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919: apply (cases "z=0 | w=0")
haftmann@25919: apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919:                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919: done
haftmann@25919: 
huffman@47207: lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
huffman@47207: apply (rule sym)
huffman@47207: apply (simp add: nat_eq_iff)
huffman@47207: done
huffman@47207: 
huffman@47207: lemma diff_nat_eq_if:
huffman@47207:      "nat z - nat z' =  
huffman@47207:         (if z' < 0 then nat z   
huffman@47207:          else let d = z-z' in     
huffman@47207:               if d < 0 then 0 else nat d)"
huffman@47207: by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207: 
huffman@47207: lemma nat_numeral_diff_1 [simp]:
huffman@47207:   "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207:   using diff_nat_numeral [of v Num.One] by simp
huffman@47207: 
haftmann@25919: 
haftmann@25919: subsection "Induction principles for int"
haftmann@25919: 
haftmann@25919: text{*Well-founded segments of the integers*}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919: where
haftmann@25919:   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919: 
haftmann@25919: theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919: proof -
haftmann@25919:   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919:     by (auto simp add: int_ge_less_than_def)
haftmann@25919:   thus ?thesis 
haftmann@25919:     by (rule wf_subset [OF wf_measure]) 
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919: by RankFinder.*}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919: where
haftmann@25919:   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919: 
haftmann@25919: theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919: proof -
haftmann@25919:   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919:     by (auto simp add: int_ge_less_than2_def)
haftmann@25919:   thus ?thesis 
haftmann@25919:     by (rule wf_subset [OF wf_measure]) 
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: (* `set:int': dummy construction *)
haftmann@25919: theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919:   fixes i :: int
haftmann@25919:   assumes ge: "k \<le> i" and
haftmann@25919:     base: "P k" and
haftmann@25919:     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919:   shows "P i"
haftmann@25919: proof -
wenzelm@42676:   { fix n
wenzelm@42676:     have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919:     proof (induct n)
haftmann@25919:       case 0
haftmann@25919:       hence "i = k" by arith
haftmann@25919:       thus "P i" using base by simp
haftmann@25919:     next
haftmann@25919:       case (Suc n)
haftmann@25919:       then have "n = nat((i - 1) - k)" by arith
haftmann@25919:       moreover
haftmann@25919:       have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919:       ultimately
wenzelm@42676:       have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676:       from step [OF ki1 this] show ?case by simp
haftmann@25919:     qed
haftmann@25919:   }
haftmann@25919:   with ge show ?thesis by fast
haftmann@25919: qed
haftmann@25919: 
haftmann@25928: (* `set:int': dummy construction *)
haftmann@25928: theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919:   assumes gr: "k < (i::int)" and
haftmann@25919:         base: "P(k+1)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919:   shows "P i"
haftmann@25919: apply(rule int_ge_induct[of "k + 1"])
haftmann@25919:   using gr apply arith
haftmann@25919:  apply(rule base)
haftmann@25919: apply (rule step, simp+)
haftmann@25919: done
haftmann@25919: 
wenzelm@42676: theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919:   assumes le: "i \<le> (k::int)" and
haftmann@25919:         base: "P(k)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919:   shows "P i"
haftmann@25919: proof -
wenzelm@42676:   { fix n
wenzelm@42676:     have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919:     proof (induct n)
haftmann@25919:       case 0
haftmann@25919:       hence "i = k" by arith
haftmann@25919:       thus "P i" using base by simp
haftmann@25919:     next
haftmann@25919:       case (Suc n)
wenzelm@42676:       hence "n = nat (k - (i + 1))" by arith
haftmann@25919:       moreover
haftmann@25919:       have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919:       ultimately
wenzelm@42676:       have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919:       from step[OF ki1 this] show ?case by simp
haftmann@25919:     qed
haftmann@25919:   }
haftmann@25919:   with le show ?thesis by fast
haftmann@25919: qed
haftmann@25919: 
wenzelm@42676: theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919:   assumes less: "(i::int) < k" and
haftmann@25919:         base: "P(k - 1)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919:   shows "P i"
haftmann@25919: apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919:   using less apply arith
haftmann@25919:  apply(rule base)
haftmann@25919: apply (rule step, simp+)
haftmann@25919: done
haftmann@25919: 
haftmann@36811: theorem int_induct [case_names base step1 step2]:
haftmann@36801:   fixes k :: int
haftmann@36801:   assumes base: "P k"
haftmann@36801:     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801:     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801:   shows "P i"
haftmann@36801: proof -
haftmann@36801:   have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676:   then show ?thesis
wenzelm@42676:   proof
wenzelm@42676:     assume "i \<ge> k"
wenzelm@42676:     then show ?thesis using base
haftmann@36801:       by (rule int_ge_induct) (fact step1)
haftmann@36801:   next
wenzelm@42676:     assume "i \<le> k"
wenzelm@42676:     then show ?thesis using base
haftmann@36801:       by (rule int_le_induct) (fact step2)
haftmann@36801:   qed
haftmann@36801: qed
haftmann@36801: 
haftmann@25919: subsection{*Intermediate value theorems*}
haftmann@25919: 
haftmann@25919: lemma int_val_lemma:
haftmann@25919:      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919:       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079: unfolding One_nat_def
wenzelm@42676: apply (induct n)
wenzelm@42676: apply simp
haftmann@25919: apply (intro strip)
haftmann@25919: apply (erule impE, simp)
haftmann@25919: apply (erule_tac x = n in allE, simp)
huffman@30079: apply (case_tac "k = f (Suc n)")
haftmann@27106: apply force
haftmann@25919: apply (erule impE)
haftmann@25919:  apply (simp add: abs_if split add: split_if_asm)
haftmann@25919: apply (blast intro: le_SucI)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919: 
haftmann@25919: lemma nat_intermed_int_val:
haftmann@25919:      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919:          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919: apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919:        in int_val_lemma)
huffman@30079: unfolding One_nat_def
haftmann@25919: apply simp
haftmann@25919: apply (erule exE)
haftmann@25919: apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919: 
haftmann@25919: lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919: by arith
haftmann@25919: 
paulson@34055: lemma abs_zmult_eq_1:
paulson@34055:   assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055:   shows "\<bar>m\<bar> = (1::int)"
paulson@34055: proof -
paulson@34055:   have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055:     by auto
paulson@34055:   have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055:   proof
paulson@34055:     assume "2 \<le> \<bar>m\<bar>"
paulson@34055:     hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055:       by (simp add: mult_mono 0) 
paulson@34055:     also have "... = \<bar>m*n\<bar>" 
paulson@34055:       by (simp add: abs_mult)
paulson@34055:     also have "... = 1"
paulson@34055:       by (simp add: mn)
paulson@34055:     finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055:     thus "False" using 0
huffman@47108:       by arith
paulson@34055:   qed
paulson@34055:   thus ?thesis using 0
paulson@34055:     by auto
paulson@34055: qed
haftmann@25919: 
haftmann@25919: lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919: by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919: 
boehmes@35815: lemma pos_zmult_eq_1_iff:
boehmes@35815:   assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815: proof -
boehmes@35815:   from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815:   thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815: qed
haftmann@25919: 
haftmann@25919: lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919: apply (rule iffI) 
haftmann@25919:  apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@57512:  apply (simp add: mult.commute [of m]) 
haftmann@25919:  apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919: done
haftmann@25919: 
haftmann@33296: lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919: proof
haftmann@33296:   assume "finite (UNIV::int set)"
haftmann@33296:   moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296:     by (rule injI) simp
haftmann@33296:   ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296:     by (rule finite_UNIV_inj_surj)
haftmann@33296:   then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296:   then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@30652: subsection {* Further theorems on numerals *}
haftmann@30652: 
haftmann@30652: subsubsection{*Special Simplification for Constants*}
haftmann@30652: 
haftmann@30652: text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652: 
webertj@49962: lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962: lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108: lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108: lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652: 
haftmann@30652: text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652: 
huffman@47108: lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108: lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108: lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108: lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652: 
haftmann@30652: 
haftmann@30652: text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652:   strange, but then other simprocs simplify the quotient.*}
haftmann@30652: 
huffman@47108: lemmas inverse_eq_divide_numeral [simp] =
huffman@47108:   inverse_eq_divide [of "numeral w"] for w
huffman@47108: 
huffman@47108: lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489:   inverse_eq_divide [of "- numeral w"] for w
haftmann@30652: 
haftmann@30652: text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652: into the literal.*}
haftmann@30652: 
haftmann@54489: lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489:   equation_minus_iff [of "numeral v"] for v
huffman@47108: 
haftmann@54489: lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489:   minus_equation_iff [of _ "numeral v"] for v
huffman@47108: 
haftmann@54489: lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489:   le_minus_iff [of "numeral v"] for v
haftmann@30652: 
haftmann@54489: lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489:   minus_le_iff [of _ "numeral v"] for v
haftmann@30652: 
haftmann@54489: lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489:   less_minus_iff [of "numeral v"] for v
haftmann@30652: 
haftmann@54489: lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489:   minus_less_iff [of _ "numeral v"] for v
haftmann@30652: 
haftmann@54489: -- {* FIXME maybe simproc *}
haftmann@30652: 
haftmann@30652: 
haftmann@30652: text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652: 
huffman@47108: lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108: lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108: lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108: lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652: 
haftmann@30652: 
haftmann@30652: text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652: 
huffman@47108: lemmas le_divide_eq_numeral1 [simp] =
huffman@47108:   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489:   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108: 
huffman@47108: lemmas divide_le_eq_numeral1 [simp] =
huffman@47108:   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489:   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108: 
huffman@47108: lemmas less_divide_eq_numeral1 [simp] =
huffman@47108:   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489:   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652: 
huffman@47108: lemmas divide_less_eq_numeral1 [simp] =
huffman@47108:   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489:   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108: 
huffman@47108: lemmas eq_divide_eq_numeral1 [simp] =
huffman@47108:   eq_divide_eq [of _ _ "numeral w"]
haftmann@54489:   eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas divide_eq_eq_numeral1 [simp] =
huffman@47108:   divide_eq_eq [of _ "numeral w"]
haftmann@54489:   divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489: 
haftmann@30652: 
haftmann@30652: subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652: 
haftmann@30652: text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652: 
huffman@47108: lemmas le_divide_eq_numeral =
huffman@47108:   le_divide_eq [of "numeral w"]
haftmann@54489:   le_divide_eq [of "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas divide_le_eq_numeral =
huffman@47108:   divide_le_eq [of _ _ "numeral w"]
haftmann@54489:   divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas less_divide_eq_numeral =
huffman@47108:   less_divide_eq [of "numeral w"]
haftmann@54489:   less_divide_eq [of "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas divide_less_eq_numeral =
huffman@47108:   divide_less_eq [of _ _ "numeral w"]
haftmann@54489:   divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas eq_divide_eq_numeral =
huffman@47108:   eq_divide_eq [of "numeral w"]
haftmann@54489:   eq_divide_eq [of "- numeral w"] for w
huffman@47108: 
huffman@47108: lemmas divide_eq_eq_numeral =
huffman@47108:   divide_eq_eq [of _ _ "numeral w"]
haftmann@54489:   divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652: 
haftmann@30652: 
haftmann@30652: text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652: lemmas divide_const_simps =
huffman@47108:   le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
huffman@47108:   divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
haftmann@30652:   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652: 
haftmann@30652: text{*Division By @{text "-1"}*}
haftmann@30652: 
huffman@47108: lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
huffman@47108:   unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
huffman@47108:   by simp
haftmann@30652: 
haftmann@30652: lemma half_gt_zero_iff:
haftmann@54489:   "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
haftmann@54489:   by auto
haftmann@30652: 
wenzelm@45607: lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
haftmann@30652: 
huffman@47108: lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
haftmann@54489:   by (fact divide_numeral_1)
haftmann@36719: 
haftmann@30652: 
haftmann@33320: subsection {* The divides relation *}
haftmann@33320: 
nipkow@33657: lemma zdvd_antisym_nonneg:
nipkow@33657:     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320:   apply (simp add: dvd_def, auto)
haftmann@57512:   apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320:   done
haftmann@33320: 
nipkow@33657: lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320:   shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657: proof cases
nipkow@33657:   assume "a = 0" with assms show ?thesis by simp
nipkow@33657: next
nipkow@33657:   assume "a \<noteq> 0"
haftmann@33320:   from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320:   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320:   from k k' have "a = a*k*k'" by simp
haftmann@33320:   with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@57512:   have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult.assoc)
haftmann@33320:   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320:   thus ?thesis using k k' by auto
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320:   apply (subgoal_tac "m = n + (m - n)")
haftmann@33320:    apply (erule ssubst)
haftmann@33320:    apply (blast intro: dvd_add, simp)
haftmann@33320:   done
haftmann@33320: 
haftmann@33320: lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320: apply (rule iffI)
haftmann@33320:  apply (erule_tac [2] dvd_add)
haftmann@33320:  apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320:   apply (erule ssubst)
haftmann@33320:   apply (erule dvd_diff)
haftmann@33320:   apply(simp_all)
haftmann@33320: done
haftmann@33320: 
haftmann@33320: lemma dvd_imp_le_int:
haftmann@33320:   fixes d i :: int
haftmann@33320:   assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320:   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320: proof -
haftmann@33320:   from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320:   with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320:   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320:   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320:   with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: lemma zdvd_not_zless:
haftmann@33320:   fixes m n :: int
haftmann@33320:   assumes "0 < m" and "m < n"
haftmann@33320:   shows "\<not> n dvd m"
haftmann@33320: proof
haftmann@33320:   from assms have "0 < n" by auto
haftmann@33320:   assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320:   with `0 < m` have "0 < n * k" by auto
haftmann@33320:   with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320:   with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320:   with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320:   shows "m dvd n"
haftmann@33320: proof-
haftmann@33320:   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320:   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@57512:     with h have False by (simp add: mult.assoc)}
haftmann@33320:   hence "n = m * h" by blast
haftmann@33320:   thus ?thesis by simp
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320: proof -
haftmann@33320:   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320:   proof -
haftmann@33320:     fix k
haftmann@33320:     assume A: "int y = int x * k"
wenzelm@42676:     then show "x dvd y"
wenzelm@42676:     proof (cases k)
wenzelm@42676:       case (nonneg n)
wenzelm@42676:       with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320:       then show ?thesis ..
haftmann@33320:     next
wenzelm@42676:       case (neg n)
wenzelm@42676:       with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320:       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320:       also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320:       finally have "- int (x * Suc n) = int y" ..
haftmann@33320:       then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320:     qed
haftmann@33320:   qed
haftmann@33320:   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320: qed
haftmann@33320: 
wenzelm@42676: lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320: proof
haftmann@33320:   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320:   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320:   hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676:   thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320: next
haftmann@33320:   assume "\<bar>x\<bar>=1"
haftmann@33320:   then have "x = 1 \<or> x = -1" by auto
haftmann@33320:   then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: lemma zdvd_mult_cancel1: 
haftmann@33320:   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320: proof
haftmann@33320:   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676:     by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320: next
haftmann@33320:   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320:   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320: qed
haftmann@33320: 
haftmann@33320: lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320:   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320: 
haftmann@33320: lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320:   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320: 
haftmann@33320: lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320:   by (auto simp add: dvd_int_iff)
haftmann@33320: 
haftmann@33341: lemma eq_nat_nat_iff:
haftmann@33341:   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341:   by (auto elim!: nonneg_eq_int)
haftmann@33341: 
haftmann@33341: lemma nat_power_eq:
haftmann@33341:   "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341:   by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341: 
haftmann@33320: lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676:   apply (cases n)
haftmann@33320:   apply (auto simp add: dvd_int_iff)
wenzelm@42676:   apply (cases z)
haftmann@33320:   apply (auto simp add: dvd_imp_le)
haftmann@33320:   done
haftmann@33320: 
haftmann@36749: lemma zdvd_period:
haftmann@36749:   fixes a d :: int
haftmann@36749:   assumes "a dvd d"
haftmann@36749:   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749: proof -
haftmann@36749:   from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676:   show ?thesis
wenzelm@42676:   proof
haftmann@36749:     assume "a dvd (x + t)"
haftmann@36749:     then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749:     then have "x = a * l - t" by simp
haftmann@36749:     with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749:   next
haftmann@36749:     assume "a dvd x + c * d + t"
haftmann@36749:     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749:     then have "x = a * l - c * d - t" by simp
haftmann@36749:     with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749:   qed
haftmann@36749: qed
haftmann@36749: 
haftmann@33320: 
bulwahn@46756: subsection {* Finiteness of intervals *}
bulwahn@46756: 
bulwahn@46756: lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
bulwahn@46756: proof (cases "a <= b")
bulwahn@46756:   case True
bulwahn@46756:   from this show ?thesis
bulwahn@46756:   proof (induct b rule: int_ge_induct)
bulwahn@46756:     case base
bulwahn@46756:     have "{i. a <= i & i <= a} = {a}" by auto
bulwahn@46756:     from this show ?case by simp
bulwahn@46756:   next
bulwahn@46756:     case (step b)
bulwahn@46756:     from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
bulwahn@46756:     from this step show ?case by simp
bulwahn@46756:   qed
bulwahn@46756: next
bulwahn@46756:   case False from this show ?thesis
bulwahn@46756:     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756: qed
bulwahn@46756: 
bulwahn@46756: lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
bulwahn@46756: by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756: 
bulwahn@46756: lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
bulwahn@46756: by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756: 
bulwahn@46756: lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
bulwahn@46756: by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756: 
bulwahn@46756: 
haftmann@25919: subsection {* Configuration of the code generator *}
haftmann@25919: 
huffman@47108: text {* Constructors *}
huffman@47108: 
huffman@47108: definition Pos :: "num \<Rightarrow> int" where
huffman@47108:   [simp, code_abbrev]: "Pos = numeral"
huffman@47108: 
huffman@47108: definition Neg :: "num \<Rightarrow> int" where
haftmann@54489:   [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108: 
huffman@47108: code_datatype "0::int" Pos Neg
huffman@47108: 
huffman@47108: 
huffman@47108: text {* Auxiliary operations *}
huffman@47108: 
huffman@47108: definition dup :: "int \<Rightarrow> int" where
huffman@47108:   [simp]: "dup k = k + k"
haftmann@26507: 
huffman@47108: lemma dup_code [code]:
huffman@47108:   "dup 0 = 0"
huffman@47108:   "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108:   "dup (Neg n) = Neg (Num.Bit0 n)"
haftmann@54489:   unfolding Pos_def Neg_def
huffman@47108:   by (simp_all add: numeral_Bit0)
huffman@47108: 
huffman@47108: definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
huffman@47108:   [simp]: "sub m n = numeral m - numeral n"
haftmann@26507: 
huffman@47108: lemma sub_code [code]:
huffman@47108:   "sub Num.One Num.One = 0"
huffman@47108:   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108:   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108:   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108:   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108:   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108:   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108:   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108:   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@54489:   apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
haftmann@54230:   apply (simp_all only: algebra_simps minus_diff_eq)
haftmann@54230:   apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
haftmann@54230:   apply (simp_all only: minus_add add.assoc left_minus)
haftmann@54230:   done
huffman@47108: 
huffman@47108: text {* Implementations *}
huffman@47108: 
huffman@47108: lemma one_int_code [code, code_unfold]:
huffman@47108:   "1 = Pos Num.One"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma plus_int_code [code]:
huffman@47108:   "k + 0 = (k::int)"
huffman@47108:   "0 + l = (l::int)"
huffman@47108:   "Pos m + Pos n = Pos (m + n)"
huffman@47108:   "Pos m + Neg n = sub m n"
huffman@47108:   "Neg m + Pos n = sub n m"
huffman@47108:   "Neg m + Neg n = Neg (m + n)"
huffman@47108:   by simp_all
haftmann@26507: 
huffman@47108: lemma uminus_int_code [code]:
huffman@47108:   "uminus 0 = (0::int)"
huffman@47108:   "uminus (Pos m) = Neg m"
huffman@47108:   "uminus (Neg m) = Pos m"
huffman@47108:   by simp_all
huffman@47108: 
huffman@47108: lemma minus_int_code [code]:
huffman@47108:   "k - 0 = (k::int)"
huffman@47108:   "0 - l = uminus (l::int)"
huffman@47108:   "Pos m - Pos n = sub m n"
huffman@47108:   "Pos m - Neg n = Pos (m + n)"
huffman@47108:   "Neg m - Pos n = Neg (m + n)"
huffman@47108:   "Neg m - Neg n = sub n m"
huffman@47108:   by simp_all
huffman@47108: 
huffman@47108: lemma times_int_code [code]:
huffman@47108:   "k * 0 = (0::int)"
huffman@47108:   "0 * l = (0::int)"
huffman@47108:   "Pos m * Pos n = Pos (m * n)"
huffman@47108:   "Pos m * Neg n = Neg (m * n)"
huffman@47108:   "Neg m * Pos n = Neg (m * n)"
huffman@47108:   "Neg m * Neg n = Pos (m * n)"
huffman@47108:   by simp_all
haftmann@26507: 
haftmann@38857: instantiation int :: equal
haftmann@26507: begin
haftmann@26507: 
haftmann@37767: definition
huffman@47108:   "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857: 
huffman@47108: instance by default (rule equal_int_def)
haftmann@26507: 
haftmann@26507: end
haftmann@26507: 
huffman@47108: lemma equal_int_code [code]:
huffman@47108:   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108:   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108:   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108:   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108:   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108:   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108:   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108:   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108:   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108:   by (auto simp add: equal)
haftmann@26507: 
huffman@47108: lemma equal_int_refl [code nbe]:
haftmann@38857:   "HOL.equal (k::int) k \<longleftrightarrow> True"
huffman@47108:   by (fact equal_refl)
haftmann@26507: 
haftmann@28562: lemma less_eq_int_code [code]:
huffman@47108:   "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108:   "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108:   "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108:   "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108:   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108:   "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108:   "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108:   "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108:   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958:   by simp_all
haftmann@26507: 
haftmann@28562: lemma less_int_code [code]:
huffman@47108:   "0 < (0::int) \<longleftrightarrow> False"
huffman@47108:   "0 < Pos l \<longleftrightarrow> True"
huffman@47108:   "0 < Neg l \<longleftrightarrow> False"
huffman@47108:   "Pos k < 0 \<longleftrightarrow> False"
huffman@47108:   "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108:   "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108:   "Neg k < 0 \<longleftrightarrow> True"
huffman@47108:   "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108:   "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958:   by simp_all
haftmann@25919: 
huffman@47108: lemma nat_code [code]:
huffman@47108:   "nat (Int.Neg k) = 0"
huffman@47108:   "nat 0 = 0"
huffman@47108:   "nat (Int.Pos k) = nat_of_num k"
haftmann@54489:   by (simp_all add: nat_of_num_numeral)
haftmann@25928: 
huffman@47108: lemma (in ring_1) of_int_code [code]:
haftmann@54489:   "of_int (Int.Neg k) = - numeral k"
huffman@47108:   "of_int 0 = 0"
huffman@47108:   "of_int (Int.Pos k) = numeral k"
huffman@47108:   by simp_all
haftmann@25919: 
huffman@47108: 
huffman@47108: text {* Serializer setup *}
haftmann@25919: 
haftmann@52435: code_identifier
haftmann@52435:   code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919: 
haftmann@25919: quickcheck_params [default_type = int]
haftmann@25919: 
huffman@47108: hide_const (open) Pos Neg sub dup
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Legacy theorems *}
haftmann@25919: 
haftmann@25919: lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919: lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919: lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919: lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
wenzelm@45607: lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
haftmann@25919: lemmas zless_int = of_nat_less_iff [where 'a=int]
wenzelm@45607: lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
haftmann@25919: lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919: lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
wenzelm@45607: lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
haftmann@25919: lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919: lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919: lemmas int_Suc = of_nat_Suc [where 'a=int]
huffman@47207: lemmas int_numeral = of_nat_numeral [where 'a=int]
wenzelm@45607: lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
haftmann@25919: lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919: lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
huffman@47255: lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
huffman@47255: lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
haftmann@30960: 
haftmann@31015: lemma zpower_zpower:
haftmann@31015:   "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015:   by (rule power_mult [symmetric])
haftmann@31015: 
haftmann@31015: lemma int_power:
haftmann@31015:   "int (m ^ n) = int m ^ n"
haftmann@54489:   by (fact of_nat_power)
haftmann@31015: 
haftmann@31015: lemmas zpower_int = int_power [symmetric]
haftmann@31015: 
huffman@48045: text {* De-register @{text "int"} as a quotient type: *}
huffman@48045: 
kuncar@53652: lifting_update int.lifting
kuncar@53652: lifting_forget int.lifting
huffman@48045: 
haftmann@25919: end




