(*  Title:      Int.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

                 Tobias Nipkow, Florian Haftmann, TU Muenchen

     Copyright   1994  University of Cambridge

 *)

 header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 

 theory Int

 imports Equiv_Relations Nat Wellfounded

 uses

   ("Tools/numeral.ML")

   ("Tools/numeral_syntax.ML")

   ("Tools/int_arith.ML")

 begin

 subsection {* The equivalence relation underlying the integers *}

 definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where

   [code del]: "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"

 typedef (Integ)

   int = "UNIV//intrel"

   by (auto simp add: quotient_def)

 instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"

 begin

 definition

   Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})"

 definition

   One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})"

 definition

   add_int_def [code del]: "z + w = Abs_Integ

     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.

       intrel `` {(x + u, y + v)})"

 definition

   minus_int_def [code del]:

     "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"

 definition

   diff_int_def [code del]:  "z - w = z + (-w \<Colon> int)"

 definition

   mult_int_def [code del]: "z * w = Abs_Integ

     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.

       intrel `` {(x*u + y*v, x*v + y*u)})"

 definition

   le_int_def [code del]:

    "z \<le> w \<longleftrightarrow> (\<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w)"

 definition

   less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"

 definition

   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"

 definition

   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"

 instance ..

 end

 subsection{*Construction of the Integers*}

 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"

 by (simp add: intrel_def)

 lemma equiv_intrel: "equiv UNIV intrel"

 by (simp add: intrel_def equiv_def refl_on_def sym_def trans_def)

 text{*Reduces equality of equivalence classes to the @{term intrel} relation:

   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}

 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]

 text{*All equivalence classes belong to set of representatives*}

 lemma [simp]: "intrel``{(x,y)} \<in> Integ"

 by (auto simp add: Integ_def intrel_def quotient_def)

 text{*Reduces equality on abstractions to equality on representatives:

   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}

 declare Abs_Integ_inject [simp,no_atp]  Abs_Integ_inverse [simp,no_atp]

 text{*Case analysis on the representation of an integer as an equivalence

       class of pairs of naturals.*}

 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:

      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"

 apply (rule Abs_Integ_cases [of z]) 

 apply (auto simp add: Integ_def quotient_def) 

 done

 subsection {* Arithmetic Operations *}

 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"

 proof -

   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"

     by (simp add: congruent_def) 

   thus ?thesis

     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])

 qed

 lemma add:

      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =

       Abs_Integ (intrel``{(x+u, y+v)})"

 proof -

   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z) 

         respects2 intrel"

     by (simp add: congruent2_def)

   thus ?thesis

     by (simp add: add_int_def UN_UN_split_split_eq

                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])

 qed

 text{*Congruence property for multiplication*}

 lemma mult_congruent2:

      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)

       respects2 intrel"

 apply (rule equiv_intrel [THEN congruent2_commuteI])

  apply (force simp add: mult_ac, clarify) 

 apply (simp add: congruent_def mult_ac)  

 apply (rename_tac u v w x y z)

 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")

 apply (simp add: mult_ac)

 apply (simp add: add_mult_distrib [symmetric])

 done

 lemma mult:

      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =

       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"

 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2

               UN_equiv_class2 [OF equiv_intrel equiv_intrel])

 text{*The integers form a @{text comm_ring_1}*}

 instance int :: comm_ring_1

 proof

   fix i j k :: int

   show "(i + j) + k = i + (j + k)"

     by (cases i, cases j, cases k) (simp add: add add_assoc)

   show "i + j = j + i" 

     by (cases i, cases j) (simp add: add_ac add)

   show "0 + i = i"

     by (cases i) (simp add: Zero_int_def add)

   show "- i + i = 0"

     by (cases i) (simp add: Zero_int_def minus add)

   show "i - j = i + - j"

     by (simp add: diff_int_def)

   show "(i * j) * k = i * (j * k)"

     by (cases i, cases j, cases k) (simp add: mult algebra_simps)

   show "i * j = j * i"

     by (cases i, cases j) (simp add: mult algebra_simps)

   show "1 * i = i"

     by (cases i) (simp add: One_int_def mult)

   show "(i + j) * k = i * k + j * k"

     by (cases i, cases j, cases k) (simp add: add mult algebra_simps)

   show "0 \<noteq> (1::int)"

     by (simp add: Zero_int_def One_int_def)

 qed

 lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"

 by (induct m, simp_all add: Zero_int_def One_int_def add)

 subsection {* The @{text "\<le>"} Ordering *}

 lemma le:

   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"

 by (force simp add: le_int_def)

 lemma less:

   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"

 by (simp add: less_int_def le order_less_le)

 instance int :: linorder

 proof

   fix i j k :: int

   show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"

     by (cases i, cases j) (simp add: le)

   show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"

     by (auto simp add: less_int_def dest: antisym) 

   show "i \<le> i"

     by (cases i) (simp add: le)

   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"

     by (cases i, cases j, cases k) (simp add: le)

   show "i \<le> j \<or> j \<le> i"

     by (cases i, cases j) (simp add: le linorder_linear)

 qed

 instantiation int :: distrib_lattice

 begin

 definition

   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"

 definition

   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"

 instance

   by intro_classes

     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)

 end

 instance int :: ordered_cancel_ab_semigroup_add

 proof

   fix i j k :: int

   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"

     by (cases i, cases j, cases k) (simp add: le add)

 qed

 text{*Strict Monotonicity of Multiplication*}

 text{*strict, in 1st argument; proof is by induction on k>0*}

 lemma zmult_zless_mono2_lemma:

      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"

 apply (induct "k", simp)

 apply (simp add: left_distrib)

 apply (case_tac "k=0")

 apply (simp_all add: add_strict_mono)

 done

 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"

 apply (cases k)

 apply (auto simp add: le add int_def Zero_int_def)

 apply (rule_tac x="x-y" in exI, simp)

 done

 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"

 apply (cases k)

 apply (simp add: less int_def Zero_int_def)

 apply (rule_tac x="x-y" in exI, simp)

 done

 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"

 apply (drule zero_less_imp_eq_int)

 apply (auto simp add: zmult_zless_mono2_lemma)

 done

 text{*The integers form an ordered integral domain*}

 instance int :: linordered_idom

 proof

   fix i j k :: int

   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"

     by (rule zmult_zless_mono2)

   show "\<bar>i\<bar> = (if i < 0 then -i else i)"

     by (simp only: zabs_def)

   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"

     by (simp only: zsgn_def)

 qed

 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"

 apply (cases w, cases z) 

 apply (simp add: less le add One_int_def)

 done

 lemma zless_iff_Suc_zadd:

   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"

 apply (cases z, cases w)

 apply (auto simp add: less add int_def)

 apply (rename_tac a b c d) 

 apply (rule_tac x="a+d - Suc(c+b)" in exI) 

 apply arith

 done

 lemmas int_distrib =

   left_distrib [of "z1::int" "z2" "w", standard]

   right_distrib [of "w::int" "z1" "z2", standard]

   left_diff_distrib [of "z1::int" "z2" "w", standard]

   right_diff_distrib [of "w::int" "z1" "z2", standard]

 subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}

 context ring_1

 begin

 definition of_int :: "int \<Rightarrow> 'a" where

   [code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"

 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"

 proof -

   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"

     by (simp add: congruent_def algebra_simps of_nat_add [symmetric]

             del: of_nat_add) 

   thus ?thesis

     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])

 qed

 lemma of_int_0 [simp]: "of_int 0 = 0"

 by (simp add: of_int Zero_int_def)

 lemma of_int_1 [simp]: "of_int 1 = 1"

 by (simp add: of_int One_int_def)

 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"

 by (cases w, cases z, simp add: algebra_simps of_int add)

 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"

 by (cases z, simp add: algebra_simps of_int minus)

 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"

 by (simp add: diff_minus Groups.diff_minus)

 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"

 apply (cases w, cases z)

 apply (simp add: algebra_simps of_int mult of_nat_mult)

 done

 text{*Collapse nested embeddings*}

 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"

 by (induct n) auto

 lemma of_int_power:

   "of_int (z ^ n) = of_int z ^ n"

   by (induct n) simp_all

 end

 context linordered_idom

 begin

 lemma of_int_le_iff [simp]:

   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"

 by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)

 text{*Special cases where either operand is zero*}

 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]

 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]

 lemma of_int_less_iff [simp]:

   "of_int w < of_int z \<longleftrightarrow> w < z"

   by (simp add: not_le [symmetric] linorder_not_le [symmetric])

 text{*Special cases where either operand is zero*}

 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]

 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]

 end

 text{*Class for unital rings with characteristic zero.

  Includes non-ordered rings like the complex numbers.*}

 class ring_char_0 = ring_1 + semiring_char_0

 begin

 lemma of_int_eq_iff [simp]:

    "of_int w = of_int z \<longleftrightarrow> w = z"

 apply (cases w, cases z, simp add: of_int)

 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)

 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)

 done

 text{*Special cases where either operand is zero*}

 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]

 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]

 end

 text{*Every @{text linordered_idom} has characteristic zero.*}

 subclass (in linordered_idom) ring_char_0 by intro_locales

 lemma of_int_eq_id [simp]: "of_int = id"

 proof

   fix z show "of_int z = id z"

     by (cases z) (simp add: of_int add minus int_def diff_minus)

 qed

 subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}

 definition

   nat :: "int \<Rightarrow> nat"

 where

   [code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"

 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"

 proof -

   have "(\<lambda>(x,y). {x-y}) respects intrel"

     by (simp add: congruent_def) arith

   thus ?thesis

     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])

 qed

 lemma nat_int [simp]: "nat (of_nat n) = n"

 by (simp add: nat int_def)

 (* FIXME: duplicates nat_0 *)

 lemma nat_zero [simp]: "nat 0 = 0"

 by (simp add: Zero_int_def nat)

 lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"

 by (cases z, simp add: nat le int_def Zero_int_def)

 corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"

 by simp

 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"

 by (cases z, simp add: nat le Zero_int_def)

 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"

 apply (cases w, cases z) 

 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)

 done

 text{*An alternative condition is @{term "0 \<le> w"} *}

 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"

 by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 

 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"

 by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 

 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"

 apply (cases w, cases z) 

 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)

 done

 lemma nonneg_eq_int:

   fixes z :: int

   assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"

   shows P

   using assms by (blast dest: nat_0_le sym)

 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"

 by (cases w, simp add: nat le int_def Zero_int_def, arith)

 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"

 by (simp only: eq_commute [of m] nat_eq_iff)

 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"

 apply (cases w)

 apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)

 done

 lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"

 by(simp add: nat_eq_iff) arith

 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"

 by (auto simp add: nat_eq_iff2)

 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"

 by (insert zless_nat_conj [of 0], auto)

 lemma nat_add_distrib:

      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"

 by (cases z, cases z', simp add: nat add le Zero_int_def)

 lemma nat_diff_distrib:

      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"

 by (cases z, cases z', 

     simp add: nat add minus diff_minus le Zero_int_def)

 lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"

 by (simp add: int_def minus nat Zero_int_def) 

 lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"

 by (cases z, simp add: nat less int_def, arith)

 context ring_1

 begin

 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"

   by (cases z rule: eq_Abs_Integ)

    (simp add: nat le of_int Zero_int_def of_nat_diff)

 end

 text {* For termination proofs: *}

 lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..

 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}

 lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"

 by (simp add: order_less_le del: of_nat_Suc)

 lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"

 by (rule negative_zless_0 [THEN order_less_le_trans], simp)

 lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"

 by (simp add: minus_le_iff)

 lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"

 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])

 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"

 by (subst le_minus_iff, simp del: of_nat_Suc)

 lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"

 by (simp add: int_def le minus Zero_int_def)

 lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"

 by (simp add: linorder_not_less)

 lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"

 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)

 lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"

 proof -

   have "(w \<le> z) = (0 \<le> z - w)"

     by (simp only: le_diff_eq add_0_left)

   also have "\<dots> = (\<exists>n. z - w = of_nat n)"

     by (auto elim: zero_le_imp_eq_int)

   also have "\<dots> = (\<exists>n. z = w + of_nat n)"

     by (simp only: algebra_simps)

   finally show ?thesis .

 qed

 lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"

 by simp

 lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"

 by simp

 text{*This version is proved for all ordered rings, not just integers!

       It is proved here because attribute @{text arith_split} is not available

       in theory @{text Rings}.

       But is it really better than just rewriting with @{text abs_if}?*}

 lemma abs_split [arith_split,no_atp]:

      "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"

 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)

 lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"

 apply (cases x)

 apply (auto simp add: le minus Zero_int_def int_def order_less_le)

 apply (rule_tac x="y - Suc x" in exI, arith)

 done

 subsection {* Cases and induction *}

 text{*Now we replace the case analysis rule by a more conventional one:

 whether an integer is negative or not.*}

 theorem int_cases [cases type: int, case_names nonneg neg]:

   "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"

 apply (cases "z < 0", blast dest!: negD)

 apply (simp add: linorder_not_less del: of_nat_Suc)

 apply auto

 apply (blast dest: nat_0_le [THEN sym])

 done

 theorem int_induct [induct type: int, case_names nonneg neg]:

      "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"

   by (cases z rule: int_cases) auto

 text{*Contributed by Brian Huffman*}

 theorem int_diff_cases:

   obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"

 apply (cases z rule: eq_Abs_Integ)

 apply (rule_tac m=x and n=y in diff)

 apply (simp add: int_def diff_def minus add)

 done

 subsection {* Binary representation *}

 text {*

   This formalization defines binary arithmetic in terms of the integers

   rather than using a datatype. This avoids multiple representations (leading

   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text

   int_of_binary}, for the numerical interpretation.

   The representation expects that @{text "(m mod 2)"} is 0 or 1,

   even if m is negative;

   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus

   @{text "-5 = (-3)*2 + 1"}.

   This two's complement binary representation derives from the paper 

   "An Efficient Representation of Arithmetic for Term Rewriting" by

   Dave Cohen and Phil Watson, Rewriting Techniques and Applications,

   Springer LNCS 488 (240-251), 1991.

 *}

 subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}

 definition

   Pls :: int where

   [code del]: "Pls = 0"

 definition

   Min :: int where

   [code del]: "Min = - 1"

 definition

   Bit0 :: "int \<Rightarrow> int" where

   [code del]: "Bit0 k = k + k"

 definition

   Bit1 :: "int \<Rightarrow> int" where

   [code del]: "Bit1 k = 1 + k + k"

 class number = -- {* for numeric types: nat, int, real, \dots *}

   fixes number_of :: "int \<Rightarrow> 'a"

 use "Tools/numeral.ML"

 syntax

   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")

 use "Tools/numeral_syntax.ML"

 setup Numeral_Syntax.setup

 abbreviation

   "Numeral0 \<equiv> number_of Pls"

 abbreviation

   "Numeral1 \<equiv> number_of (Bit1 Pls)"

 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"

   -- {* Unfold all @{text let}s involving constants *}

   unfolding Let_def ..

 definition

   succ :: "int \<Rightarrow> int" where

   [code del]: "succ k = k + 1"

 definition

   pred :: "int \<Rightarrow> int" where

   [code del]: "pred k = k - 1"

 lemmas

   max_number_of [simp] = max_def

     [of "number_of u" "number_of v", standard]

 and

   min_number_of [simp] = min_def 

     [of "number_of u" "number_of v", standard]

   -- {* unfolding @{text minx} and @{text max} on numerals *}

 lemmas numeral_simps = 

   succ_def pred_def Pls_def Min_def Bit0_def Bit1_def

 text {* Removal of leading zeroes *}

 lemma Bit0_Pls [simp, code_post]:

   "Bit0 Pls = Pls"

   unfolding numeral_simps by simp

 lemma Bit1_Min [simp, code_post]:

   "Bit1 Min = Min"

   unfolding numeral_simps by simp

 lemmas normalize_bin_simps =

   Bit0_Pls Bit1_Min

 subsubsection {* Successor and predecessor functions *}

 text {* Successor *}

 lemma succ_Pls:

   "succ Pls = Bit1 Pls"

   unfolding numeral_simps by simp

 lemma succ_Min:

   "succ Min = Pls"

   unfolding numeral_simps by simp

 lemma succ_Bit0:

   "succ (Bit0 k) = Bit1 k"

   unfolding numeral_simps by simp

 lemma succ_Bit1:

   "succ (Bit1 k) = Bit0 (succ k)"

   unfolding numeral_simps by simp

 lemmas succ_bin_simps [simp] =

   succ_Pls succ_Min succ_Bit0 succ_Bit1

 text {* Predecessor *}

 lemma pred_Pls:

   "pred Pls = Min"

   unfolding numeral_simps by simp

 lemma pred_Min:

   "pred Min = Bit0 Min"

   unfolding numeral_simps by simp

 lemma pred_Bit0:

   "pred (Bit0 k) = Bit1 (pred k)"

   unfolding numeral_simps by simp 

 lemma pred_Bit1:

   "pred (Bit1 k) = Bit0 k"

   unfolding numeral_simps by simp

 lemmas pred_bin_simps [simp] =

   pred_Pls pred_Min pred_Bit0 pred_Bit1

 subsubsection {* Binary arithmetic *}

 text {* Addition *}

 lemma add_Pls:

   "Pls + k = k"

   unfolding numeral_simps by simp

 lemma add_Min:

   "Min + k = pred k"

   unfolding numeral_simps by simp

 lemma add_Bit0_Bit0:

   "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"

   unfolding numeral_simps by simp

 lemma add_Bit0_Bit1:

   "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"

   unfolding numeral_simps by simp

 lemma add_Bit1_Bit0:

   "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"

   unfolding numeral_simps by simp

 lemma add_Bit1_Bit1:

   "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"

   unfolding numeral_simps by simp

 lemma add_Pls_right:

   "k + Pls = k"

   unfolding numeral_simps by simp

 lemma add_Min_right:

   "k + Min = pred k"

   unfolding numeral_simps by simp

 lemmas add_bin_simps [simp] =

   add_Pls add_Min add_Pls_right add_Min_right

   add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1

 text {* Negation *}

 lemma minus_Pls:

   "- Pls = Pls"

   unfolding numeral_simps by simp

 lemma minus_Min:

   "- Min = Bit1 Pls"

   unfolding numeral_simps by simp

 lemma minus_Bit0:

   "- (Bit0 k) = Bit0 (- k)"

   unfolding numeral_simps by simp

 lemma minus_Bit1:

   "- (Bit1 k) = Bit1 (pred (- k))"

   unfolding numeral_simps by simp

 lemmas minus_bin_simps [simp] =

   minus_Pls minus_Min minus_Bit0 minus_Bit1

 text {* Subtraction *}

 lemma diff_bin_simps [simp]:

   "k - Pls = k"

   "k - Min = succ k"

   "Pls - (Bit0 l) = Bit0 (Pls - l)"

   "Pls - (Bit1 l) = Bit1 (Min - l)"

   "Min - (Bit0 l) = Bit1 (Min - l)"

   "Min - (Bit1 l) = Bit0 (Min - l)"

   "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"

   "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"

   "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"

   "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"

   unfolding numeral_simps by simp_all

 text {* Multiplication *}

 lemma mult_Pls:

   "Pls * w = Pls"

   unfolding numeral_simps by simp

 lemma mult_Min:

   "Min * k = - k"

   unfolding numeral_simps by simp

 lemma mult_Bit0:

   "(Bit0 k) * l = Bit0 (k * l)"

   unfolding numeral_simps int_distrib by simp

 lemma mult_Bit1:

   "(Bit1 k) * l = (Bit0 (k * l)) + l"

   unfolding numeral_simps int_distrib by simp

 lemmas mult_bin_simps [simp] =

   mult_Pls mult_Min mult_Bit0 mult_Bit1

 subsubsection {* Binary comparisons *}

 text {* Preliminaries *}

 lemma even_less_0_iff:

   "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"

 proof -

   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)

   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"

     by (simp add: mult_less_0_iff zero_less_two 

                   order_less_not_sym [OF zero_less_two])

   finally show ?thesis .

 qed

 lemma le_imp_0_less: 

   assumes le: "0 \<le> z"

   shows "(0::int) < 1 + z"

 proof -

   have "0 \<le> z" by fact

   also have "... < z + 1" by (rule less_add_one) 

   also have "... = 1 + z" by (simp add: add_ac)

   finally show "0 < 1 + z" .

 qed

 lemma odd_less_0_iff:

   "(1 + z + z < 0) = (z < (0::int))"

 proof (cases z rule: int_cases)

   case (nonneg n)

   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing

                              le_imp_0_less [THEN order_less_imp_le])  

 next

   case (neg n)

   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1

     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])

 qed

 lemma bin_less_0_simps:

   "Pls < 0 \<longleftrightarrow> False"

   "Min < 0 \<longleftrightarrow> True"

   "Bit0 w < 0 \<longleftrightarrow> w < 0"

   "Bit1 w < 0 \<longleftrightarrow> w < 0"

   unfolding numeral_simps

   by (simp_all add: even_less_0_iff odd_less_0_iff)

 lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"

   by simp

 lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"

   unfolding numeral_simps

   proof

     have "k - 1 < k" by simp

     also assume "k \<le> l"

     finally show "k - 1 < l" .

   next

     assume "k - 1 < l"

     hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)

     thus "k \<le> l" by simp

   qed

 lemma succ_pred: "succ (pred x) = x"

   unfolding numeral_simps by simp

 text {* Less-than *}

 lemma less_bin_simps [simp]:

   "Pls < Pls \<longleftrightarrow> False"

   "Pls < Min \<longleftrightarrow> False"

   "Pls < Bit0 k \<longleftrightarrow> Pls < k"

   "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"

   "Min < Pls \<longleftrightarrow> True"

   "Min < Min \<longleftrightarrow> False"

   "Min < Bit0 k \<longleftrightarrow> Min < k"

   "Min < Bit1 k \<longleftrightarrow> Min < k"

   "Bit0 k < Pls \<longleftrightarrow> k < Pls"

   "Bit0 k < Min \<longleftrightarrow> k \<le> Min"

   "Bit1 k < Pls \<longleftrightarrow> k < Pls"

   "Bit1 k < Min \<longleftrightarrow> k < Min"

   "Bit0 k < Bit0 l \<longleftrightarrow> k < l"

   "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"

   "Bit1 k < Bit0 l \<longleftrightarrow> k < l"

   "Bit1 k < Bit1 l \<longleftrightarrow> k < l"

   unfolding le_iff_pred_less

     less_bin_lemma [of Pls]

     less_bin_lemma [of Min]

     less_bin_lemma [of "k"]

     less_bin_lemma [of "Bit0 k"]

     less_bin_lemma [of "Bit1 k"]

     less_bin_lemma [of "pred Pls"]

     less_bin_lemma [of "pred k"]

   by (simp_all add: bin_less_0_simps succ_pred)

 text {* Less-than-or-equal *}

 lemma le_bin_simps [simp]:

   "Pls \<le> Pls \<longleftrightarrow> True"

   "Pls \<le> Min \<longleftrightarrow> False"

   "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"

   "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"

   "Min \<le> Pls \<longleftrightarrow> True"

   "Min \<le> Min \<longleftrightarrow> True"

   "Min \<le> Bit0 k \<longleftrightarrow> Min < k"

   "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"

   "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"

   "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"

   "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"

   "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"

   "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"

   "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"

   "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"

   "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"

   unfolding not_less [symmetric]

   by (simp_all add: not_le)

 text {* Equality *}

 lemma eq_bin_simps [simp]:

   "Pls = Pls \<longleftrightarrow> True"

   "Pls = Min \<longleftrightarrow> False"

   "Pls = Bit0 l \<longleftrightarrow> Pls = l"

   "Pls = Bit1 l \<longleftrightarrow> False"

   "Min = Pls \<longleftrightarrow> False"

   "Min = Min \<longleftrightarrow> True"

   "Min = Bit0 l \<longleftrightarrow> False"

   "Min = Bit1 l \<longleftrightarrow> Min = l"

   "Bit0 k = Pls \<longleftrightarrow> k = Pls"

   "Bit0 k = Min \<longleftrightarrow> False"

   "Bit1 k = Pls \<longleftrightarrow> False"

   "Bit1 k = Min \<longleftrightarrow> k = Min"

   "Bit0 k = Bit0 l \<longleftrightarrow> k = l"

   "Bit0 k = Bit1 l \<longleftrightarrow> False"

   "Bit1 k = Bit0 l \<longleftrightarrow> False"

   "Bit1 k = Bit1 l \<longleftrightarrow> k = l"

   unfolding order_eq_iff [where 'a=int]

   by (simp_all add: not_less)

 subsection {* Converting Numerals to Rings: @{term number_of} *}

 class number_ring = number + comm_ring_1 +

   assumes number_of_eq: "number_of k = of_int k"

 text {* self-embedding of the integers *}

 instantiation int :: number_ring

 begin

 definition int_number_of_def [code del]:

   "number_of w = (of_int w \<Colon> int)"

 instance proof

 qed (simp only: int_number_of_def)

 end

 lemma number_of_is_id:

   "number_of (k::int) = k"

   unfolding int_number_of_def by simp

 lemma number_of_succ:

   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma number_of_pred:

   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma number_of_minus:

   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"

   unfolding number_of_eq by (rule of_int_minus)

 lemma number_of_add:

   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"

   unfolding number_of_eq by (rule of_int_add)

 lemma number_of_diff:

   "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"

   unfolding number_of_eq by (rule of_int_diff)

 lemma number_of_mult:

   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"

   unfolding number_of_eq by (rule of_int_mult)

 text {*

   The correctness of shifting.

   But it doesn't seem to give a measurable speed-up.

 *}

 lemma double_number_of_Bit0:

   "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"

   unfolding number_of_eq numeral_simps left_distrib by simp

 text {*

   Converting numerals 0 and 1 to their abstract versions.

 *}

 lemma numeral_0_eq_0 [simp, code_post]:

   "Numeral0 = (0::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma numeral_1_eq_1 [simp, code_post]:

   "Numeral1 = (1::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 text {*

   Special-case simplification for small constants.

 *}

 text{*

   Unary minus for the abstract constant 1. Cannot be inserted

   as a simprule until later: it is @{text number_of_Min} re-oriented!

 *}

 lemma numeral_m1_eq_minus_1:

   "(-1::'a::number_ring) = - 1"

   unfolding number_of_eq numeral_simps by simp

 lemma mult_minus1 [simp]:

   "-1 * z = -(z::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma mult_minus1_right [simp]:

   "z * -1 = -(z::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 (*Negation of a coefficient*)

 lemma minus_number_of_mult [simp]:

    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"

    unfolding number_of_eq by simp

 text {* Subtraction *}

 lemma diff_number_of_eq:

   "number_of v - number_of w =

     (number_of (v + uminus w)::'a::number_ring)"

   unfolding number_of_eq by simp

 lemma number_of_Pls:

   "number_of Pls = (0::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma number_of_Min:

   "number_of Min = (- 1::'a::number_ring)"

   unfolding number_of_eq numeral_simps by simp

 lemma number_of_Bit0:

   "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"

   unfolding number_of_eq numeral_simps by simp

 lemma number_of_Bit1:

   "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"

   unfolding number_of_eq numeral_simps by simp

 subsubsection {* Equality of Binary Numbers *}

 text {* First version by Norbert Voelker *}

 definition (*for simplifying equalities*)

   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"

 where

   "iszero z \<longleftrightarrow> z = 0"

 lemma iszero_0: "iszero 0"

 by (simp add: iszero_def)

 lemma not_iszero_1: "~ iszero 1"

 by (simp add: iszero_def eq_commute)

 lemma eq_number_of_eq [simp]:

   "((number_of x::'a::number_ring) = number_of y) =

    iszero (number_of (x + uminus y) :: 'a)"

 unfolding iszero_def number_of_add number_of_minus

 by (simp add: algebra_simps)

 lemma iszero_number_of_Pls:

   "iszero ((number_of Pls)::'a::number_ring)"

 unfolding iszero_def numeral_0_eq_0 ..

 lemma nonzero_number_of_Min:

   "~ iszero ((number_of Min)::'a::number_ring)"

 unfolding iszero_def numeral_m1_eq_minus_1 by simp

 subsubsection {* Comparisons, for Ordered Rings *}

 lemmas double_eq_0_iff = double_zero

 lemma odd_nonzero:

   "1 + z + z \<noteq> (0::int)"

 proof (cases z rule: int_cases)

   case (nonneg n)

   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 

   thus ?thesis using  le_imp_0_less [OF le]

     by (auto simp add: add_assoc) 

 next

   case (neg n)

   show ?thesis

   proof

     assume eq: "1 + z + z = 0"

     have "(0::int) < 1 + (of_nat n + of_nat n)"

       by (simp add: le_imp_0_less add_increasing) 

     also have "... = - (1 + z + z)" 

       by (simp add: neg add_assoc [symmetric]) 

     also have "... = 0" by (simp add: eq) 

     finally have "0<0" ..

     thus False by blast

   qed

 qed

 lemma iszero_number_of_Bit0:

   "iszero (number_of (Bit0 w)::'a) = 

    iszero (number_of w::'a::{ring_char_0,number_ring})"

 proof -

   have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"

   proof -

     assume eq: "of_int w + of_int w = (0::'a)"

     then have "of_int (w + w) = (of_int 0 :: 'a)" by simp

     then have "w + w = 0" by (simp only: of_int_eq_iff)

     then show "w = 0" by (simp only: double_eq_0_iff)

   qed

   thus ?thesis

     by (auto simp add: iszero_def number_of_eq numeral_simps)

 qed

 lemma iszero_number_of_Bit1:

   "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"

 proof -

   have "1 + of_int w + of_int w \<noteq> (0::'a)"

   proof

     assume eq: "1 + of_int w + of_int w = (0::'a)"

     hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 

     hence "1 + w + w = 0" by (simp only: of_int_eq_iff)

     with odd_nonzero show False by blast

   qed

   thus ?thesis

     by (auto simp add: iszero_def number_of_eq numeral_simps)

 qed

 lemmas iszero_simps [simp] =

   iszero_0 not_iszero_1

   iszero_number_of_Pls nonzero_number_of_Min

   iszero_number_of_Bit0 iszero_number_of_Bit1

 (* iszero_number_of_Pls would never normally be used

    because its lhs simplifies to "iszero 0" *)

 subsubsection {* The Less-Than Relation *}

 lemma double_less_0_iff:

   "(a + a < 0) = (a < (0::'a::linordered_idom))"

 proof -

   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)

   also have "... = (a < 0)"

     by (simp add: mult_less_0_iff zero_less_two 

                   order_less_not_sym [OF zero_less_two]) 

   finally show ?thesis .

 qed

 lemma odd_less_0:

   "(1 + z + z < 0) = (z < (0::int))"

 proof (cases z rule: int_cases)

   case (nonneg n)

   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing

                              le_imp_0_less [THEN order_less_imp_le])  

 next

   case (neg n)

   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1

     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])

 qed

 text {* Less-Than or Equals *}

 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}

 lemmas le_number_of_eq_not_less =

   linorder_not_less [of "number_of w" "number_of v", symmetric, 

   standard]

 text {* Absolute value (@{term abs}) *}

 lemma abs_number_of:

   "abs(number_of x::'a::{linordered_idom,number_ring}) =

    (if number_of x < (0::'a) then -number_of x else number_of x)"

   by (simp add: abs_if)

 text {* Re-orientation of the equation nnn=x *}

 lemma number_of_reorient:

   "(number_of w = x) = (x = number_of w)"

   by auto

 subsubsection {* Simplification of arithmetic operations on integer constants. *}

 lemmas arith_extra_simps [standard, simp] =

   number_of_add [symmetric]

   number_of_minus [symmetric]

   numeral_m1_eq_minus_1 [symmetric]

   number_of_mult [symmetric]

   diff_number_of_eq abs_number_of 

 text {*

   For making a minimal simpset, one must include these default simprules.

   Also include @{text simp_thms}.

 *}

 lemmas arith_simps = 

   normalize_bin_simps pred_bin_simps succ_bin_simps

   add_bin_simps minus_bin_simps mult_bin_simps

   abs_zero abs_one arith_extra_simps

 text {* Simplification of relational operations *}

 lemma less_number_of [simp]:

   "(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"

   unfolding number_of_eq by (rule of_int_less_iff)

 lemma le_number_of [simp]:

   "(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"

   unfolding number_of_eq by (rule of_int_le_iff)

 lemma eq_number_of [simp]:

   "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"

   unfolding number_of_eq by (rule of_int_eq_iff)

 lemmas rel_simps =

   less_number_of less_bin_simps

   le_number_of le_bin_simps

   eq_number_of_eq eq_bin_simps

   iszero_simps

 subsubsection {* Simplification of arithmetic when nested to the right. *}

 lemma add_number_of_left [simp]:

   "number_of v + (number_of w + z) =

    (number_of(v + w) + z::'a::number_ring)"

   by (simp add: add_assoc [symmetric])

 lemma mult_number_of_left [simp]:

   "number_of v * (number_of w * z) =

    (number_of(v * w) * z::'a::number_ring)"

   by (simp add: mult_assoc [symmetric])

 lemma add_number_of_diff1:

   "number_of v + (number_of w - c) = 

   number_of(v + w) - (c::'a::number_ring)"

   by (simp add: diff_minus)

 lemma add_number_of_diff2 [simp]:

   "number_of v + (c - number_of w) =

    number_of (v + uminus w) + (c::'a::number_ring)"

 by (simp add: algebra_simps diff_number_of_eq [symmetric])

 subsection {* The Set of Integers *}

 context ring_1

 begin

 definition Ints  :: "'a set" where

   [code del]: "Ints = range of_int"

 notation (xsymbols)

   Ints  ("\<int>")

 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"

   by (simp add: Ints_def)

 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"

 apply (simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_of_nat_eq [symmetric])

 done

 lemma Ints_0 [simp]: "0 \<in> \<int>"

 apply (simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_0 [symmetric])

 done

 lemma Ints_1 [simp]: "1 \<in> \<int>"

 apply (simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_1 [symmetric])

 done

 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_add [symmetric])

 done

 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_minus [symmetric])

 done

 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_diff [symmetric])

 done

 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_mult [symmetric])

 done

 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"

 by (induct n) simp_all

 lemma Ints_cases [cases set: Ints]:

   assumes "q \<in> \<int>"

   obtains (of_int) z where "q = of_int z"

   unfolding Ints_def

 proof -

   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .

   then obtain z where "q = of_int z" ..

   then show thesis ..

 qed

 lemma Ints_induct [case_names of_int, induct set: Ints]:

   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"

   by (rule Ints_cases) auto

 end

 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}

 lemma Ints_double_eq_0_iff:

   assumes in_Ints: "a \<in> Ints"

   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"

 proof -

   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .

   then obtain z where a: "a = of_int z" ..

   show ?thesis

   proof

     assume "a = 0"

     thus "a + a = 0" by simp

   next

     assume eq: "a + a = 0"

     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)

     hence "z + z = 0" by (simp only: of_int_eq_iff)

     hence "z = 0" by (simp only: double_eq_0_iff)

     thus "a = 0" by (simp add: a)

   qed

 qed

 lemma Ints_odd_nonzero:

   assumes in_Ints: "a \<in> Ints"

   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"

 proof -

   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .

   then obtain z where a: "a = of_int z" ..

   show ?thesis

   proof

     assume eq: "1 + a + a = 0"

     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)

     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)

     with odd_nonzero show False by blast

   qed

 qed 

 lemma Ints_number_of [simp]:

   "(number_of w :: 'a::number_ring) \<in> Ints"

   unfolding number_of_eq Ints_def by simp

 lemma Nats_number_of [simp]:

   "Int.Pls \<le> w \<Longrightarrow> (number_of w :: 'a::number_ring) \<in> Nats"

 unfolding Int.Pls_def number_of_eq

 by (simp only: of_nat_nat [symmetric] of_nat_in_Nats)

 lemma Ints_odd_less_0: 

   assumes in_Ints: "a \<in> Ints"

   shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"

 proof -

   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .

   then obtain z where a: "a = of_int z" ..

   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"

     by (simp add: a)

   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)

   also have "... = (a < 0)" by (simp add: a)

   finally show ?thesis .

 qed

 subsection {* @{term setsum} and @{term setprod} *}

 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"

   apply (cases "finite A")

   apply (erule finite_induct, auto)

   done

 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"

   apply (cases "finite A")

   apply (erule finite_induct, auto)

   done

 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"

   apply (cases "finite A")

   apply (erule finite_induct, auto simp add: of_nat_mult)

   done

 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"

   apply (cases "finite A")

   apply (erule finite_induct, auto)

   done

 lemmas int_setsum = of_nat_setsum [where 'a=int]

 lemmas int_setprod = of_nat_setprod [where 'a=int]

 subsection{*Inequality Reasoning for the Arithmetic Simproc*}

 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"

 by simp 

 lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"

 by simp

 lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"

 by simp 

 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"

 by simp

 lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"

 by simp

 lemma inverse_numeral_1:

   "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"

 by simp

 text{*Theorem lists for the cancellation simprocs. The use of binary numerals

 for 0 and 1 reduces the number of special cases.*}

 lemmas add_0s = add_numeral_0 add_numeral_0_right

 lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 

                  mult_minus1 mult_minus1_right

 subsection{*Special Arithmetic Rules for Abstract 0 and 1*}

 text{*Arithmetic computations are defined for binary literals, which leaves 0

 and 1 as special cases. Addition already has rules for 0, but not 1.

 Multiplication and unary minus already have rules for both 0 and 1.*}

 lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"

 by simp

 lemmas add_number_of_eq = number_of_add [symmetric]

 text{*Allow 1 on either or both sides*}

 lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"

 by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric])

 lemmas add_special =

     one_add_one_is_two

     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]

     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]

 text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}

 lemmas diff_special =

     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]

     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]

 text{*Allow 0 or 1 on either side with a binary numeral on the other*}

 lemmas eq_special =

     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]

     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]

     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]

     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]

 text{*Allow 0 or 1 on either side with a binary numeral on the other*}

 lemmas less_special =

   binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]

   binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]

   binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]

   binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]

 text{*Allow 0 or 1 on either side with a binary numeral on the other*}

 lemmas le_special =

     binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]

     binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]

     binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]

     binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]

 lemmas arith_special[simp] = 

        add_special diff_special eq_special less_special le_special

 text {* Legacy theorems *}

 lemmas zle_int = of_nat_le_iff [where 'a=int]

 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]

 subsection {* Setting up simplification procedures *}

 lemmas int_arith_rules =

   neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1

   minus_zero diff_minus left_minus right_minus

   mult_zero_left mult_zero_right mult_Bit1 mult_1_left mult_1_right

   mult_minus_left mult_minus_right

   minus_add_distrib minus_minus mult_assoc

   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult

   of_int_0 of_int_1 of_int_add of_int_mult

 use "Tools/int_arith.ML"

 setup {* Int_Arith.global_setup *}

 declaration {* K Int_Arith.setup *}

 setup {*

   Reorient_Proc.add

     (fn Const (@{const_name number_of}, _) $ _ => true | _ => false)

 *}

 simproc_setup reorient_numeral ("number_of w = x") = Reorient_Proc.proc

 subsection{*Lemmas About Small Numerals*}

 lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"

 proof -

   have "(of_int -1 :: 'a) = of_int (- 1)" by simp

   also have "... = - of_int 1" by (simp only: of_int_minus)

   also have "... = -1" by simp

   finally show ?thesis .

 qed

 lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})"

 by (simp add: abs_if)

 lemma abs_power_minus_one [simp]:

   "abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})"

 by (simp add: power_abs)

 lemma of_int_number_of_eq [simp]:

      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"

 by (simp add: number_of_eq) 

 text{*Lemmas for specialist use, NOT as default simprules*}

 lemma mult_2: "2 * z = (z+z::'a::number_ring)"

 unfolding one_add_one_is_two [symmetric] left_distrib by simp

 lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"

 by (subst mult_commute, rule mult_2)

 subsection{*More Inequality Reasoning*}

 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"

 by arith

 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"

 by arith

 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"

 by arith

 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"

 by arith

 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"

 by arith

 subsection{*The functions @{term nat} and @{term int}*}

 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and

   @{term "w + - z"}*}

 declare Zero_int_def [symmetric, simp]

 declare One_int_def [symmetric, simp]

 lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]

 (* FIXME: duplicates nat_zero *)

 lemma nat_0: "nat 0 = 0"

 by (simp add: nat_eq_iff)

 lemma nat_1: "nat 1 = Suc 0"

 by (subst nat_eq_iff, simp)

 lemma nat_2: "nat 2 = Suc (Suc 0)"

 by (subst nat_eq_iff, simp)

 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"

 apply (insert zless_nat_conj [of 1 z])

 apply (auto simp add: nat_1)

 done

 text{*This simplifies expressions of the form @{term "int n = z"} where

       z is an integer literal.*}

 lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]

 lemma split_nat [arith_split]:

   "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"

   (is "?P = (?L & ?R)")

 proof (cases "i < 0")

   case True thus ?thesis by auto

 next

   case False

   have "?P = ?L"

   proof

     assume ?P thus ?L using False by clarsimp

   next

     assume ?L thus ?P using False by simp

   qed

   with False show ?thesis by simp

 qed

 context ring_1

 begin

 lemma of_int_of_nat [nitpick_simp]:

   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"

 proof (cases "k < 0")

   case True then have "0 \<le> - k" by simp

   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)

   with True show ?thesis by simp

 next

   case False then show ?thesis by (simp add: not_less of_nat_nat)

 qed

 end

 lemma nat_mult_distrib:

   fixes z z' :: int

   assumes "0 \<le> z"

   shows "nat (z * z') = nat z * nat z'"

 proof (cases "0 \<le> z'")

   case False with assms have "z * z' \<le> 0"

     by (simp add: not_le mult_le_0_iff)

   then have "nat (z * z') = 0" by simp

   moreover from False have "nat z' = 0" by simp

   ultimately show ?thesis by simp

 next

   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)

   show ?thesis

     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])

       (simp only: of_nat_mult of_nat_nat [OF True]

          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)

 qed

 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"

 apply (rule trans)

 apply (rule_tac [2] nat_mult_distrib, auto)

 done

 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"

 apply (cases "z=0 | w=0")

 apply (auto simp add: abs_if nat_mult_distrib [symmetric] 

                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)

 done

 subsection "Induction principles for int"

 text{*Well-founded segments of the integers*}

 definition

   int_ge_less_than  ::  "int => (int * int) set"

 where

   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"

 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"

 proof -

   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"

     by (auto simp add: int_ge_less_than_def)

   thus ?thesis 

     by (rule wf_subset [OF wf_measure]) 

 qed

 text{*This variant looks odd, but is typical of the relations suggested

 by RankFinder.*}

 definition

   int_ge_less_than2 ::  "int => (int * int) set"

 where

   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"

 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"

 proof -

   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 

     by (auto simp add: int_ge_less_than2_def)

   thus ?thesis 

     by (rule wf_subset [OF wf_measure]) 

 qed

 abbreviation

   int :: "nat \<Rightarrow> int"

 where

   "int \<equiv> of_nat"

 (* `set:int': dummy construction *)

 theorem int_ge_induct [case_names base step, induct set: int]:

   fixes i :: int

   assumes ge: "k \<le> i" and

     base: "P k" and

     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"

   shows "P i"

 proof -

   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"

     proof (induct n)

       case 0

       hence "i = k" by arith

       thus "P i" using base by simp

     next

       case (Suc n)

       then have "n = nat((i - 1) - k)" by arith

       moreover

       have ki1: "k \<le> i - 1" using Suc.prems by arith

       ultimately

       have "P(i - 1)" by(rule Suc.hyps)

       from step[OF ki1 this] show ?case by simp

     qed

   }

   with ge show ?thesis by fast

 qed

 (* `set:int': dummy construction *)

 theorem int_gr_induct [case_names base step, induct set: int]:

   assumes gr: "k < (i::int)" and

         base: "P(k+1)" and

         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"

   shows "P i"

 apply(rule int_ge_induct[of "k + 1"])

   using gr apply arith

  apply(rule base)

 apply (rule step, simp+)

 done

 theorem int_le_induct[consumes 1,case_names base step]:

   assumes le: "i \<le> (k::int)" and

         base: "P(k)" and

         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"

   shows "P i"

 proof -

   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"

     proof (induct n)

       case 0

       hence "i = k" by arith

       thus "P i" using base by simp

     next

       case (Suc n)

       hence "n = nat(k - (i+1))" by arith

       moreover

       have ki1: "i + 1 \<le> k" using Suc.prems by arith

       ultimately

       have "P(i+1)" by(rule Suc.hyps)

       from step[OF ki1 this] show ?case by simp

     qed

   }

   with le show ?thesis by fast

 qed

 theorem int_less_induct [consumes 1,case_names base step]:

   assumes less: "(i::int) < k" and

         base: "P(k - 1)" and

         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"

   shows "P i"

 apply(rule int_le_induct[of _ "k - 1"])

   using less apply arith

  apply(rule base)

 apply (rule step, simp+)

 done

 subsection{*Intermediate value theorems*}

 lemma int_val_lemma:

      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  

       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"

 unfolding One_nat_def

 apply (induct n, simp)

 apply (intro strip)

 apply (erule impE, simp)

 apply (erule_tac x = n in allE, simp)

 apply (case_tac "k = f (Suc n)")

 apply force

 apply (erule impE)

  apply (simp add: abs_if split add: split_if_asm)

 apply (blast intro: le_SucI)

 done

 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]

 lemma nat_intermed_int_val:

      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  

          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"

 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 

        in int_val_lemma)

 unfolding One_nat_def

 apply simp

 apply (erule exE)

 apply (rule_tac x = "i+m" in exI, arith)

 done

 subsection{*Products and 1, by T. M. Rasmussen*}

 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"

 by arith

 lemma abs_zmult_eq_1:

   assumes mn: "\<bar>m * n\<bar> = 1"

   shows "\<bar>m\<bar> = (1::int)"

 proof -

   have 0: "m \<noteq> 0 & n \<noteq> 0" using mn

     by auto

   have "~ (2 \<le> \<bar>m\<bar>)"

   proof

     assume "2 \<le> \<bar>m\<bar>"

     hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"

       by (simp add: mult_mono 0) 

     also have "... = \<bar>m*n\<bar>" 

       by (simp add: abs_mult)

     also have "... = 1"

       by (simp add: mn)

     finally have "2*\<bar>n\<bar> \<le> 1" .

     thus "False" using 0

       by auto

   qed

   thus ?thesis using 0

     by auto

 qed

 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"

 by (insert abs_zmult_eq_1 [of m n], arith)

 lemma pos_zmult_eq_1_iff:

   assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"

 proof -

   from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)

   thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)

 qed

 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"

 apply (rule iffI) 

  apply (frule pos_zmult_eq_1_iff_lemma)

  apply (simp add: mult_commute [of m]) 

  apply (frule pos_zmult_eq_1_iff_lemma, auto) 

 done

 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"

 proof

   assume "finite (UNIV::int set)"

   moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"

     by (rule injI) simp

   ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"

     by (rule finite_UNIV_inj_surj)

   then obtain i :: int where "1 = 2 * i" by (rule surjE)

   then show False by (simp add: pos_zmult_eq_1_iff)

 qed

 subsection {* Further theorems on numerals *}

 subsubsection{*Special Simplification for Constants*}

 text{*These distributive laws move literals inside sums and differences.*}

 lemmas left_distrib_number_of [simp] =

   left_distrib [of _ _ "number_of v", standard]

 lemmas right_distrib_number_of [simp] =

   right_distrib [of "number_of v", standard]

 lemmas left_diff_distrib_number_of [simp] =

   left_diff_distrib [of _ _ "number_of v", standard]

 lemmas right_diff_distrib_number_of [simp] =

   right_diff_distrib [of "number_of v", standard]

 text{*These are actually for fields, like real: but where else to put them?*}

 lemmas zero_less_divide_iff_number_of [simp, no_atp] =

   zero_less_divide_iff [of "number_of w", standard]

 lemmas divide_less_0_iff_number_of [simp, no_atp] =

   divide_less_0_iff [of "number_of w", standard]

 lemmas zero_le_divide_iff_number_of [simp, no_atp] =

   zero_le_divide_iff [of "number_of w", standard]

 lemmas divide_le_0_iff_number_of [simp, no_atp] =

   divide_le_0_iff [of "number_of w", standard]

 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks

   strange, but then other simprocs simplify the quotient.*}

 lemmas inverse_eq_divide_number_of [simp] =

   inverse_eq_divide [of "number_of w", standard]

 text {*These laws simplify inequalities, moving unary minus from a term

 into the literal.*}

 lemmas less_minus_iff_number_of [simp, no_atp] =

   less_minus_iff [of "number_of v", standard]

 lemmas le_minus_iff_number_of [simp, no_atp] =

   le_minus_iff [of "number_of v", standard]

 lemmas equation_minus_iff_number_of [simp, no_atp] =

   equation_minus_iff [of "number_of v", standard]

 lemmas minus_less_iff_number_of [simp, no_atp] =

   minus_less_iff [of _ "number_of v", standard]

 lemmas minus_le_iff_number_of [simp, no_atp] =

   minus_le_iff [of _ "number_of v", standard]

 lemmas minus_equation_iff_number_of [simp, no_atp] =

   minus_equation_iff [of _ "number_of v", standard]

 text{*To Simplify Inequalities Where One Side is the Constant 1*}

 lemma less_minus_iff_1 [simp,no_atp]:

   fixes b::"'b::{linordered_idom,number_ring}"

   shows "(1 < - b) = (b < -1)"

 by auto

 lemma le_minus_iff_1 [simp,no_atp]:

   fixes b::"'b::{linordered_idom,number_ring}"

   shows "(1 \<le> - b) = (b \<le> -1)"

 by auto

 lemma equation_minus_iff_1 [simp,no_atp]:

   fixes b::"'b::number_ring"

   shows "(1 = - b) = (b = -1)"

 by (subst equation_minus_iff, auto)

 lemma minus_less_iff_1 [simp,no_atp]:

   fixes a::"'b::{linordered_idom,number_ring}"

   shows "(- a < 1) = (-1 < a)"

 by auto

 lemma minus_le_iff_1 [simp,no_atp]:

   fixes a::"'b::{linordered_idom,number_ring}"

   shows "(- a \<le> 1) = (-1 \<le> a)"

 by auto

 lemma minus_equation_iff_1 [simp,no_atp]:

   fixes a::"'b::number_ring"

   shows "(- a = 1) = (a = -1)"

 by (subst minus_equation_iff, auto)

 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}

 lemmas mult_less_cancel_left_number_of [simp, no_atp] =

   mult_less_cancel_left [of "number_of v", standard]

 lemmas mult_less_cancel_right_number_of [simp, no_atp] =

   mult_less_cancel_right [of _ "number_of v", standard]

 lemmas mult_le_cancel_left_number_of [simp, no_atp] =

   mult_le_cancel_left [of "number_of v", standard]

 lemmas mult_le_cancel_right_number_of [simp, no_atp] =

   mult_le_cancel_right [of _ "number_of v", standard]

 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}

 lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]

 lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]

 lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]

 lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]

 lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]

 lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]

 subsubsection{*Optional Simplification Rules Involving Constants*}

 text{*Simplify quotients that are compared with a literal constant.*}

 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]

 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]

 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]

 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]

 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]

 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]

 text{*Not good as automatic simprules because they cause case splits.*}

 lemmas divide_const_simps =

   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of

   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of

   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1

 text{*Division By @{text "-1"}*}

 lemma divide_minus1 [simp]:

      "x/-1 = -(x::'a::{field_inverse_zero, number_ring})"

 by simp

 lemma minus1_divide [simp]:

      "-1 / (x::'a::{field_inverse_zero, number_ring}) = - (1/x)"

 by (simp add: divide_inverse)

 lemma half_gt_zero_iff:

      "(0 < r/2) = (0 < (r::'a::{linordered_field_inverse_zero,number_ring}))"

 by auto

 lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]

 subsection {* The divides relation *}

 lemma zdvd_antisym_nonneg:

     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"

   apply (simp add: dvd_def, auto)

   apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)

   done

 lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 

   shows "\<bar>a\<bar> = \<bar>b\<bar>"

 proof cases

   assume "a = 0" with assms show ?thesis by simp

 next

   assume "a \<noteq> 0"

   from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 

   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 

   from k k' have "a = a*k*k'" by simp

   with mult_cancel_left1[where c="a" and b="k*k'"]

   have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)

   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)

   thus ?thesis using k k' by auto

 qed

 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"

   apply (subgoal_tac "m = n + (m - n)")

    apply (erule ssubst)

    apply (blast intro: dvd_add, simp)

   done

 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

 apply (rule iffI)

  apply (erule_tac [2] dvd_add)

  apply (subgoal_tac "n = (n + k * m) - k * m")

   apply (erule ssubst)

   apply (erule dvd_diff)

   apply(simp_all)

 done

 lemma dvd_imp_le_int:

   fixes d i :: int

   assumes "i \<noteq> 0" and "d dvd i"

   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"

 proof -

   from `d dvd i` obtain k where "i = d * k" ..

   with `i \<noteq> 0` have "k \<noteq> 0" by auto

   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto

   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)

   with `i = d * k` show ?thesis by (simp add: abs_mult)

 qed

 lemma zdvd_not_zless:

   fixes m n :: int

   assumes "0 < m" and "m < n"

   shows "\<not> n dvd m"

 proof

   from assms have "0 < n" by auto

   assume "n dvd m" then obtain k where k: "m = n * k" ..

   with `0 < m` have "0 < n * k" by auto

   with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)

   with k `0 < n` `m < n` have "n * k < n * 1" by simp

   with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto

 qed

 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"

   shows "m dvd n"

 proof-

   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast

   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp

     with h have False by (simp add: mult_assoc)}

   hence "n = m * h" by blast

   thus ?thesis by simp

 qed

 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"

 proof -

   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"

   proof -

     fix k

     assume A: "int y = int x * k"

     then show "x dvd y" proof (cases k)

       case (1 n) with A have "y = x * n" by (simp add: of_nat_mult [symmetric])

       then show ?thesis ..

     next

       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp

       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)

       also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])

       finally have "- int (x * Suc n) = int y" ..

       then show ?thesis by (simp only: negative_eq_positive) auto

     qed

   qed

   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)

 qed

 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"

 proof

   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp

   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)

   hence "nat \<bar>x\<bar> = 1"  by simp

   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)

 next

   assume "\<bar>x\<bar>=1"

   then have "x = 1 \<or> x = -1" by auto

   then show "x dvd 1" by (auto intro: dvdI)

 qed

 lemma zdvd_mult_cancel1: 

   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"

 proof

   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 

     by (cases "n >0", auto simp add: minus_equation_iff)

 next

   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp

   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)

 qed

 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

   unfolding zdvd_int by (cases "z \<ge> 0") simp_all

 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

   unfolding zdvd_int by (cases "z \<ge> 0") simp_all

 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

   by (auto simp add: dvd_int_iff)

 lemma eq_nat_nat_iff:

   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"

   by (auto elim!: nonneg_eq_int)

 lemma nat_power_eq:

   "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"

   by (induct n) (simp_all add: nat_mult_distrib)

 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"

   apply (rule_tac z=n in int_cases)

   apply (auto simp add: dvd_int_iff)

   apply (rule_tac z=z in int_cases)

   apply (auto simp add: dvd_imp_le)

   done

 subsection {* Configuration of the code generator *}

 code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"

 lemmas pred_succ_numeral_code [code] =

   pred_bin_simps succ_bin_simps

 lemmas plus_numeral_code [code] =

   add_bin_simps

   arith_extra_simps(1) [where 'a = int]

 lemmas minus_numeral_code [code] =

   minus_bin_simps

   arith_extra_simps(2) [where 'a = int]

   arith_extra_simps(5) [where 'a = int]

 lemmas times_numeral_code [code] =

   mult_bin_simps

   arith_extra_simps(4) [where 'a = int]

 instantiation int :: eq

 begin

 definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)"

 instance by default (simp add: eq_int_def)

 end

 lemma eq_number_of_int_code [code]:

   "eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l"

   unfolding eq_int_def number_of_is_id ..

 lemma eq_int_code [code]:

   "eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True"

   "eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False"

   "eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2"

   "eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"

   "eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False"

   "eq_class.eq Int.Min Int.Min \<longleftrightarrow> True"

   "eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"

   "eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2"

   "eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls"

   "eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"

   "eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"

   "eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min"

   "eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2"

   "eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"

   "eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"

   "eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2"

   unfolding eq_equals by simp_all

 lemma eq_int_refl [code nbe]:

   "eq_class.eq (k::int) k \<longleftrightarrow> True"

   by (rule HOL.eq_refl)

 lemma less_eq_number_of_int_code [code]:

   "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"

   unfolding number_of_is_id ..

 lemma less_eq_int_code [code]:

   "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"

   "Int.Pls \<le> Int.Min \<longleftrightarrow> False"

   "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"

   "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"

   "Int.Min \<le> Int.Pls \<longleftrightarrow> True"

   "Int.Min \<le> Int.Min \<longleftrightarrow> True"

   "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"

   "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"

   "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"

   "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"

   "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"

   "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"

   "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"

   "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"

   "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"

   "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"

   by simp_all

 lemma less_number_of_int_code [code]:

   "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"

   unfolding number_of_is_id ..

 lemma less_int_code [code]:

   "Int.Pls < Int.Pls \<longleftrightarrow> False"

   "Int.Pls < Int.Min \<longleftrightarrow> False"

   "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"

   "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"

   "Int.Min < Int.Pls \<longleftrightarrow> True"

   "Int.Min < Int.Min \<longleftrightarrow> False"

   "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"

   "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"

   "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"

   "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"

   "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"

   "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"

   "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"

   "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"

   "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"

   "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"

   by simp_all

 definition

   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where

   "nat_aux i n = nat i + n"

 lemma [code]:

   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}

   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le

     dest: zless_imp_add1_zle)

 lemma [code]: "nat i = nat_aux i 0"

   by (simp add: nat_aux_def)

 hide_const (open) nat_aux

 lemma zero_is_num_zero [code, code_unfold_post]:

   "(0\<Colon>int) = Numeral0" 

   by simp

 lemma one_is_num_one [code, code_unfold_post]:

   "(1\<Colon>int) = Numeral1" 

   by simp

 code_modulename SML

   Int Arith

 code_modulename OCaml

   Int Arith

 code_modulename Haskell

   Int Arith

 types_code

   "int" ("int")

 attach (term_of) {*

 val term_of_int = HOLogic.mk_number HOLogic.intT;

 *}

 attach (test) {*

 fun gen_int i =

   let val j = one_of [~1, 1] * random_range 0 i

   in (j, fn () => term_of_int j) end;

 *}

 setup {*

 let

 fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t

   | strip_number_of t = t;

 fun numeral_codegen thy defs dep module b t gr =

   let val i = HOLogic.dest_numeral (strip_number_of t)

   in

     SOME (Codegen.str (string_of_int i),

       snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr))

   end handle TERM _ => NONE;

 in

 Codegen.add_codegen "numeral_codegen" numeral_codegen

 end

 *}

 consts_code

   "number_of :: int \<Rightarrow> int"    ("(_)")

   "0 :: int"                   ("0")

   "1 :: int"                   ("1")

   "uminus :: int => int"       ("~")

   "op + :: int => int => int"  ("(_ +/ _)")

   "op * :: int => int => int"  ("(_ */ _)")

   "op \<le> :: int => int => bool" ("(_ <=/ _)")

   "op < :: int => int => bool" ("(_ </ _)")

 quickcheck_params [default_type = int]

 hide_const (open) Pls Min Bit0 Bit1 succ pred

 subsection {* Legacy theorems *}

 lemmas zminus_zminus = minus_minus [of "z::int", standard]

 lemmas zminus_0 = minus_zero [where 'a=int]

 lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]

 lemmas zadd_commute = add_commute [of "z::int" "w", standard]

 lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]

 lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]

 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute

 lemmas zmult_ac = mult_ac

 lemmas zadd_0 = add_0_left [of "z::int", standard]

 lemmas zadd_0_right = add_0_right [of "z::int", standard]

 lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]

 lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]

 lemmas zmult_commute = mult_commute [of "z::int" "w", standard]

 lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]

 lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]

 lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]

 lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]

 lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]

 lemmas zmult_1 = mult_1_left [of "z::int", standard]

 lemmas zmult_1_right = mult_1_right [of "z::int", standard]

 lemmas zle_refl = order_refl [of "w::int", standard]

 lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]

 lemmas zle_antisym = order_antisym [of "z::int" "w", standard]

 lemmas zle_linear = linorder_linear [of "z::int" "w", standard]

 lemmas zless_linear = linorder_less_linear [where 'a = int]

 lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]

 lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]

 lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]

 lemmas int_0_less_1 = zero_less_one [where 'a=int]

 lemmas int_0_neq_1 = zero_neq_one [where 'a=int]

 lemmas inj_int = inj_of_nat [where 'a=int]

 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]

 lemmas int_mult = of_nat_mult [where 'a=int]

 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]

 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]

 lemmas zless_int = of_nat_less_iff [where 'a=int]

 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]

 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]

 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]

 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]

 lemmas int_0 = of_nat_0 [where 'a=int]

 lemmas int_1 = of_nat_1 [where 'a=int]

 lemmas int_Suc = of_nat_Suc [where 'a=int]

 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]

 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]

 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]

 lemmas zless_le = less_int_def

 lemmas int_eq_of_nat = TrueI

 lemma zpower_zadd_distrib:

   "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"

   by (rule power_add)

 lemma zero_less_zpower_abs_iff:

   "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"

   by (rule zero_less_power_abs_iff)

 lemma zero_le_zpower_abs: "(0::int) \<le> abs x ^ n"

   by (rule zero_le_power_abs)

 lemma zpower_zpower:

   "(x ^ y) ^ z = (x ^ (y * z)::int)"

   by (rule power_mult [symmetric])

 lemma int_power:

   "int (m ^ n) = int m ^ n"

   by (rule of_nat_power)

 lemmas zpower_int = int_power [symmetric]

 end