(*  Title:      IntDef.thy

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1996  University of Cambridge

 *)

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

 theory IntDef = Equiv + NatArith:

 constdefs

   intrel :: "((nat * nat) * (nat * nat)) set"

     --{*the equivalence relation underlying the integers*}

     "intrel == {((x,y),(u,v)) | x y u v. x+v = u+y}"

 typedef (Integ)

   int = "UNIV//intrel"

     by (auto simp add: quotient_def)

 instance int :: "{ord, zero, one, plus, times, minus}" ..

 constdefs

   int :: "nat => int"

   "int m == Abs_Integ(intrel `` {(m,0)})"

 defs (overloaded)

   Zero_int_def:  "0 == int 0"

   One_int_def:   "1 == int 1"

   minus_int_def:

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

   add_int_def:

    "z + w ==

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

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

   diff_int_def:  "z - (w::int) == z + (-w)"

   mult_int_def:

    "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)})"

   le_int_def:

    "z \<le> (w::int) == 

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

   less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"

 subsection{*Construction of the Integers*}

 subsubsection{*Preliminary Lemmas about the Equivalence Relation*}

 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_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 = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]

 declare equiv_intrel_iff [simp]

 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)

 lemma inj_on_Abs_Integ: "inj_on Abs_Integ Integ"

 apply (rule inj_on_inverseI)

 apply (erule Abs_Integ_inverse)

 done

 text{*This theorem reduces equality on abstractions to equality on 

       representatives:

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

 declare inj_on_Abs_Integ [THEN inj_on_iff, simp]

 declare Abs_Integ_inverse [simp]

 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 Rep_Integ [of z, unfolded Integ_def, THEN quotientE])

 apply (drule arg_cong [where f=Abs_Integ])

 apply (auto simp add: Rep_Integ_inverse)

 done

 subsubsection{*@{term int}: Embedding the Naturals into the Integers*}

 lemma inj_int: "inj int"

 by (simp add: inj_on_def int_def)

 lemma int_int_eq [iff]: "(int m = int n) = (m = n)"

 by (fast elim!: inj_int [THEN injD])

 subsubsection{*Integer Unary Negation*}

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

 proof -

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

     by (simp add: congruent_def) 

   thus ?thesis

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

 qed

 lemma zminus_zminus: "- (- z) = (z::int)"

 by (cases z, simp add: minus)

 lemma zminus_0: "- 0 = (0::int)"

 by (simp add: int_def Zero_int_def minus)

 subsection{*Integer Addition*}

 lemma add:

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

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

 proof -

   have "congruent2 intrel intrel

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

     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

 lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"

 by (cases z, cases w, simp add: minus add)

 lemma zadd_commute: "(z::int) + w = w + z"

 by (cases z, cases w, simp add: add_ac add)

 lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"

 by (cases z1, cases z2, cases z3, simp add: add add_assoc)

 (*For AC rewriting*)

 lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"

   apply (rule mk_left_commute [of "op +"])

   apply (rule zadd_assoc)

   apply (rule zadd_commute)

   done

 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute

 lemmas zmult_ac = OrderedGroup.mult_ac

 lemma zadd_int: "(int m) + (int n) = int (m + n)"

 by (simp add: int_def add)

 lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"

 by (simp add: zadd_int zadd_assoc [symmetric])

 lemma int_Suc: "int (Suc m) = 1 + (int m)"

 by (simp add: One_int_def zadd_int)

 (*also for the instance declaration int :: comm_monoid_add*)

 lemma zadd_0: "(0::int) + z = z"

 apply (simp add: Zero_int_def int_def)

 apply (cases z, simp add: add)

 done

 lemma zadd_0_right: "z + (0::int) = z"

 by (rule trans [OF zadd_commute zadd_0])

 lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"

 by (cases z, simp add: int_def Zero_int_def minus add)

 subsection{*Integer Multiplication*}

 text{*Congruence property for multiplication*}

 lemma mult_congruent2:

      "congruent2 intrel intrel

         (%p1 p2. (%(x,y). (%(u,v).

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

 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, arith)

 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])

 lemma zmult_zminus: "(- z) * w = - (z * (w::int))"

 by (cases z, cases w, simp add: minus mult add_ac)

 lemma zmult_commute: "(z::int) * w = w * z"

 by (cases z, cases w, simp add: mult add_ac mult_ac)

 lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"

 by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)

 lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"

 by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)

 lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"

 by (simp add: zmult_commute [of w] zadd_zmult_distrib)

 lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"

 by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)

 lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"

 by (simp add: zmult_commute [of w] zdiff_zmult_distrib)

 lemmas int_distrib =

   zadd_zmult_distrib zadd_zmult_distrib2

   zdiff_zmult_distrib zdiff_zmult_distrib2

 lemma zmult_int: "(int m) * (int n) = int (m * n)"

 by (simp add: int_def mult)

 lemma zmult_1: "(1::int) * z = z"

 by (cases z, simp add: One_int_def int_def mult)

 lemma zmult_1_right: "z * (1::int) = z"

 by (rule trans [OF zmult_commute zmult_1])

 text{*The Integers Form A comm_ring_1*}

 instance int :: comm_ring_1

 proof

   fix i j k :: int

   show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)

   show "i + j = j + i" by (simp add: zadd_commute)

   show "0 + i = i" by (rule zadd_0)

   show "- i + i = 0" by (rule zadd_zminus_inverse2)

   show "i - j = i + (-j)" by (simp add: diff_int_def)

   show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)

   show "i * j = j * i" by (rule zmult_commute)

   show "1 * i = i" by (rule zmult_1) 

   show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)

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

     by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)

 qed

 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 zle_refl: "w \<le> (w::int)"

 by (cases w, simp add: le)

 lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"

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

 lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"

 by (cases w, cases z, simp add: le)

 (* Axiom 'order_less_le' of class 'order': *)

 lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"

 by (simp add: less_int_def)

 instance int :: order

   by intro_classes

     (assumption |

       rule zle_refl zle_trans zle_anti_sym zless_le)+

 (* Axiom 'linorder_linear' of class 'linorder': *)

 lemma zle_linear: "(z::int) \<le> w | w \<le> z"

 by (cases z, cases w) (simp add: le linorder_linear)

 instance int :: linorder

   by intro_classes (rule zle_linear)

 lemmas zless_linear = linorder_less_linear [where 'a = int]

 lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"

 by (simp add: Zero_int_def)

 lemma zless_int [simp]: "(int m < int n) = (m<n)"

 by (simp add: le add int_def linorder_not_le [symmetric]) 

 lemma int_less_0_conv [simp]: "~ (int k < 0)"

 by (simp add: Zero_int_def)

 lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)"

 by (simp add: Zero_int_def)

 lemma int_0_less_1: "0 < (1::int)"

 by (simp only: Zero_int_def One_int_def One_nat_def zless_int)

 lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"

 by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)

 lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"

 by (simp add: linorder_not_less [symmetric])

 lemma zero_zle_int [simp]: "(0 \<le> int n)"

 by (simp add: Zero_int_def)

 lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"

 by (simp add: Zero_int_def)

 lemma int_0 [simp]: "int 0 = (0::int)"

 by (simp add: Zero_int_def)

 lemma int_1 [simp]: "int 1 = 1"

 by (simp add: One_int_def)

 lemma int_Suc0_eq_1: "int (Suc 0) = 1"

 by (simp add: One_int_def One_nat_def)

 subsection{*Monotonicity results*}

 lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"

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

 lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"

 apply (cases i, cases j, cases k)

 apply (simp add: linorder_not_le [where 'a = int, symmetric]

                  linorder_not_le [where 'a = nat]  le add)

 done

 lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"

 by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])

 subsection{*Strict Monotonicity of Multiplication*}

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

 lemma zmult_zless_mono2_lemma [rule_format]:

      "i<j ==> 0<k --> int k * i < int k * j"

 apply (induct_tac "k", simp)

 apply (simp add: int_Suc)

 apply (case_tac "n=0")

 apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)

 apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)

 done

 lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int 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 zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"

 apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])

 apply (auto simp add: zmult_zless_mono2_lemma)

 done

 defs (overloaded)

     zabs_def:  "abs(i::int) == if i < 0 then -i else i"

 text{*The Integers Form an Ordered comm_ring_1*}

 instance int :: ordered_idom

 proof

   fix i j k :: int

   show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)

   show "i < j ==> 0 < k ==> 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)

 qed

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

 apply (cases w, cases z) 

 apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)

 done

 subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}

 constdefs

    nat  :: "int => nat"

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

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

 proof -

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

     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(int n) = n"

 by (simp add: nat int_def) 

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

 by (simp only: Zero_int_def nat_int)

 lemma int_nat_eq [simp]: "int (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 ==> int (nat z) = z"

 apply simp 

 done

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

 by (cases z, simp add: nat le int_def 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] int_def 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: "(nat w < nat z) = (0 < z & w < z)"

 apply (cases w, cases z) 

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

 done

 lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"

 by (blast dest: nat_0_le sym)

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

 apply (cases w)

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

 done

 lemma int_eq_iff: "(int 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 int_def 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 int_def Zero_int_def)

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

 by (simp add: int_def minus nat Zero_int_def) 

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

 by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)

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

 lemma negative_zless_0: "- (int (Suc n)) < 0"

 by (simp add: order_less_le)

 lemma negative_zless [iff]: "- (int (Suc n)) < int m"

 by (rule negative_zless_0 [THEN order_less_le_trans], simp)

 lemma negative_zle_0: "- int n \<le> 0"

 by (simp add: minus_le_iff)

 lemma negative_zle [iff]: "- int n \<le> int m"

 by (rule order_trans [OF negative_zle_0 zero_zle_int])

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

 by (subst le_minus_iff, simp)

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

 by (simp add: int_def le minus Zero_int_def) 

 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"

 by (simp add: linorder_not_less)

 lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"

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

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

 apply (cases w, cases z)

 apply (auto simp add: le add int_def) 

 apply (rename_tac a b c d) 

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

 apply arith

 done

 lemma abs_int_eq [simp]: "abs (int m) = int m"

 by (simp add: zabs_def)

 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 Ring_and_Field}.

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

 lemma abs_split [arith_split]:

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

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

 subsection{*The Constants @{term neg} and @{term iszero}*}

 constdefs

   neg   :: "'a::ordered_idom => bool"

   "neg(Z) == Z < 0"

   (*For simplifying equalities*)

   iszero :: "'a::comm_semiring_1_cancel => bool"

   "iszero z == z = (0)"

 lemma not_neg_int [simp]: "~ neg(int n)"

 by (simp add: neg_def)

 lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"

 by (simp add: neg_def neg_less_0_iff_less)

 lemmas neg_eq_less_0 = neg_def

 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"

 by (simp add: neg_def linorder_not_less)

 subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}

 lemma not_neg_0: "~ neg 0"

 by (simp add: One_int_def neg_def)

 lemma not_neg_1: "~ neg 1"

 by (simp add: neg_def linorder_not_less zero_le_one)

 lemma iszero_0: "iszero 0"

 by (simp add: iszero_def)

 lemma not_iszero_1: "~ iszero 1"

 by (simp add: iszero_def eq_commute)

 lemma neg_nat: "neg z ==> nat z = 0"

 by (simp add: neg_def order_less_imp_le) 

 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"

 by (simp add: linorder_not_less neg_def)

 subsection{*Embedding of the Naturals into any comm_semiring_1_cancel: @{term of_nat}*}

 consts of_nat :: "nat => 'a::comm_semiring_1_cancel"

 primrec

   of_nat_0:   "of_nat 0 = 0"

   of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"

 lemma of_nat_1 [simp]: "of_nat 1 = 1"

 by simp

 lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"

 apply (induct m)

 apply (simp_all add: add_ac)

 done

 lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"

 apply (induct m)

 apply (simp_all add: mult_ac add_ac right_distrib)

 done

 lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"

 apply (induct m, simp_all)

 apply (erule order_trans)

 apply (rule less_add_one [THEN order_less_imp_le])

 done

 lemma less_imp_of_nat_less:

      "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"

 apply (induct m n rule: diff_induct, simp_all)

 apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)

 done

 lemma of_nat_less_imp_less:

      "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"

 apply (induct m n rule: diff_induct, simp_all)

 apply (insert zero_le_imp_of_nat)

 apply (force simp add: linorder_not_less [symmetric])

 done

 lemma of_nat_less_iff [simp]:

      "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"

 by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)

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

 declare of_nat_less_iff [of 0, simplified, simp]

 declare of_nat_less_iff [of _ 0, simplified, simp]

 lemma of_nat_le_iff [simp]:

      "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"

 by (simp add: linorder_not_less [symmetric])

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

 declare of_nat_le_iff [of 0, simplified, simp]

 declare of_nat_le_iff [of _ 0, simplified, simp]

 text{*The ordering on the comm_semiring_1_cancel is necessary to exclude the possibility of

 a finite field, which indeed wraps back to zero.*}

 lemma of_nat_eq_iff [simp]:

      "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"

 by (simp add: order_eq_iff)

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

 declare of_nat_eq_iff [of 0, simplified, simp]

 declare of_nat_eq_iff [of _ 0, simplified, simp]

 lemma of_nat_diff [simp]:

      "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::comm_ring_1)"

 by (simp del: of_nat_add

 	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)

 subsection{*The Set of Natural Numbers*}

 constdefs

    Nats  :: "'a::comm_semiring_1_cancel set"

     "Nats == range of_nat"

 syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")

 lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"

 by (simp add: Nats_def)

 lemma Nats_0 [simp]: "0 \<in> Nats"

 apply (simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_0 [symmetric])

 done

 lemma Nats_1 [simp]: "1 \<in> Nats"

 apply (simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_1 [symmetric])

 done

 lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"

 apply (auto simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_add [symmetric])

 done

 lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"

 apply (auto simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_mult [symmetric])

 done

 text{*Agreement with the specific embedding for the integers*}

 lemma int_eq_of_nat: "int = (of_nat :: nat => int)"

 proof

   fix n

   show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)

 qed

 lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"

 proof

   fix n

   show "of_nat n = id n"  by (induct n, simp_all)

 qed

 subsection{*Embedding of the Integers into any comm_ring_1: @{term of_int}*}

 constdefs

    of_int :: "int => 'a::comm_ring_1"

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

     by (simp add: congruent_def compare_rls 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 int_def)

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

 by (simp add: of_int One_int_def int_def)

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

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

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

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

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

 by (simp add: 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: compare_rls of_int left_diff_distrib right_diff_distrib

                  mult add_ac)

 done

 lemma of_int_le_iff [simp]:

      "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"

 apply (cases w)

 apply (cases z)

 apply (simp add: compare_rls of_int le diff_int_def add minus

                  of_nat_add [symmetric]   del: of_nat_add)

 done

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

 declare of_int_le_iff [of 0, simplified, simp]

 declare of_int_le_iff [of _ 0, simplified, simp]

 lemma of_int_less_iff [simp]:

      "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"

 by (simp add: linorder_not_le [symmetric])

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

 declare of_int_less_iff [of 0, simplified, simp]

 declare of_int_less_iff [of _ 0, simplified, simp]

 text{*The ordering on the comm_ring_1 is necessary. See @{text of_nat_eq_iff} above.*}

 lemma of_int_eq_iff [simp]:

      "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)"

 by (simp add: order_eq_iff)

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

 declare of_int_eq_iff [of 0, simplified, simp]

 declare of_int_eq_iff [of _ 0, simplified, simp]

 lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"

 proof

  fix z

  show "of_int z = id z"  

   by (cases z,

       simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)

 qed

 subsection{*The Set of Integers*}

 constdefs

    Ints  :: "'a::comm_ring_1 set"

     "Ints == range of_int"

 syntax (xsymbols)

   Ints      :: "'a set"                   ("\<int>")

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

 apply (simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_0 [symmetric])

 done

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

 apply (simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_1 [symmetric])

 done

 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_add [symmetric])

 done

 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_minus [symmetric])

 done

 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_diff [symmetric])

 done

 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"

 apply (auto simp add: Ints_def)

 apply (rule range_eqI)

 apply (rule of_int_mult [symmetric])

 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_int_eq [simp]: "of_int (int n) = int n"

 by (simp add: int_eq_of_nat)

 lemma Ints_cases [case_names of_int, cases set: Ints]:

   "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"

 proof (simp add: Ints_def)

   assume "!!z. q = of_int z ==> C"

   assume "q \<in> range of_int" thus C ..

 qed

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

   "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"

   by (rule Ints_cases) auto

 (* int (Suc n) = 1 + int n *)

 declare int_Suc [simp]

 text{*Simplification of @{term "x-y < 0"}, etc.*}

 declare less_iff_diff_less_0 [symmetric, simp]

 declare eq_iff_diff_eq_0 [symmetric, simp]

 declare le_iff_diff_le_0 [symmetric, simp]

 subsection{*More Properties of @{term setsum} and  @{term setprod}*}

 text{*By Jeremy Avigad*}

 lemma setsum_of_nat: "of_nat (setsum f A) = setsum (of_nat \<circ> f) A"

   apply (case_tac "finite A")

   apply (erule finite_induct, auto)

   apply (simp add: setsum_def)

   done

 lemma setsum_of_int: "of_int (setsum f A) = setsum (of_int \<circ> f) A"

   apply (case_tac "finite A")

   apply (erule finite_induct, auto)

   apply (simp add: setsum_def)

   done

 lemma int_setsum: "int (setsum f A) = setsum (int \<circ> f) A"

   by (subst int_eq_of_nat, rule setsum_of_nat)

 lemma setprod_of_nat: "of_nat (setprod f A) = setprod (of_nat \<circ> f) A"

   apply (case_tac "finite A")

   apply (erule finite_induct, auto)

   apply (simp add: setprod_def)

   done

 lemma setprod_of_int: "of_int (setprod f A) = setprod (of_int \<circ> f) A"

   apply (case_tac "finite A")

   apply (erule finite_induct, auto)

   apply (simp add: setprod_def)

   done

 lemma int_setprod: "int (setprod f A) = setprod (int \<circ> f) A"

   by (subst int_eq_of_nat, rule setprod_of_nat)

 lemma setsum_constant: "finite A ==> (\<Sum>x \<in> A. y) = of_nat(card A) * y"

   apply (erule finite_induct)

   apply (auto simp add: ring_distrib add_ac)

   done

 lemma setprod_nonzero_nat:

     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"

   by (rule setprod_nonzero, auto)

 lemma setprod_zero_eq_nat:

     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"

   by (rule setprod_zero_eq, auto)

 lemma setprod_nonzero_int:

     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"

   by (rule setprod_nonzero, auto)

 lemma setprod_zero_eq_int:

     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"

   by (rule setprod_zero_eq, auto)

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

 whether an integer is negative or not.*}

 lemma zless_iff_Suc_zadd:

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

 apply (cases z, cases w)

 apply (auto simp add: le add int_def linorder_not_le [symmetric]) 

 apply (rename_tac a b c d) 

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

 apply arith

 done

 lemma negD: "x<0 ==> \<exists>n. x = - (int (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

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

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

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

 apply (simp add: linorder_not_less)

 apply (blast dest: nat_0_le [THEN sym])

 done

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

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

   by (cases z) auto

 (*Legacy ML bindings, but no longer the structure Int.*)

 ML

 {*

 val zabs_def = thm "zabs_def"

 val int_0 = thm "int_0";

 val int_1 = thm "int_1";

 val int_Suc0_eq_1 = thm "int_Suc0_eq_1";

 val neg_eq_less_0 = thm "neg_eq_less_0";

 val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";

 val not_neg_0 = thm "not_neg_0";

 val not_neg_1 = thm "not_neg_1";

 val iszero_0 = thm "iszero_0";

 val not_iszero_1 = thm "not_iszero_1";

 val int_0_less_1 = thm "int_0_less_1";

 val int_0_neq_1 = thm "int_0_neq_1";

 val negative_zless = thm "negative_zless";

 val negative_zle = thm "negative_zle";

 val not_zle_0_negative = thm "not_zle_0_negative";

 val not_int_zless_negative = thm "not_int_zless_negative";

 val negative_eq_positive = thm "negative_eq_positive";

 val zle_iff_zadd = thm "zle_iff_zadd";

 val abs_int_eq = thm "abs_int_eq";

 val abs_split = thm"abs_split";

 val nat_int = thm "nat_int";

 val nat_zminus_int = thm "nat_zminus_int";

 val nat_zero = thm "nat_zero";

 val not_neg_nat = thm "not_neg_nat";

 val neg_nat = thm "neg_nat";

 val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";

 val nat_0_le = thm "nat_0_le";

 val nat_le_0 = thm "nat_le_0";

 val zless_nat_conj = thm "zless_nat_conj";

 val int_cases = thm "int_cases";

 val int_def = thm "int_def";

 val Zero_int_def = thm "Zero_int_def";

 val One_int_def = thm "One_int_def";

 val diff_int_def = thm "diff_int_def";

 val inj_int = thm "inj_int";

 val zminus_zminus = thm "zminus_zminus";

 val zminus_0 = thm "zminus_0";

 val zminus_zadd_distrib = thm "zminus_zadd_distrib";

 val zadd_commute = thm "zadd_commute";

 val zadd_assoc = thm "zadd_assoc";

 val zadd_left_commute = thm "zadd_left_commute";

 val zadd_ac = thms "zadd_ac";

 val zmult_ac = thms "zmult_ac";

 val zadd_int = thm "zadd_int";

 val zadd_int_left = thm "zadd_int_left";

 val int_Suc = thm "int_Suc";

 val zadd_0 = thm "zadd_0";

 val zadd_0_right = thm "zadd_0_right";

 val zmult_zminus = thm "zmult_zminus";

 val zmult_commute = thm "zmult_commute";

 val zmult_assoc = thm "zmult_assoc";

 val zadd_zmult_distrib = thm "zadd_zmult_distrib";

 val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";

 val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";

 val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";

 val int_distrib = thms "int_distrib";

 val zmult_int = thm "zmult_int";

 val zmult_1 = thm "zmult_1";

 val zmult_1_right = thm "zmult_1_right";

 val int_int_eq = thm "int_int_eq";

 val int_eq_0_conv = thm "int_eq_0_conv";

 val zless_int = thm "zless_int";

 val int_less_0_conv = thm "int_less_0_conv";

 val zero_less_int_conv = thm "zero_less_int_conv";

 val zle_int = thm "zle_int";

 val zero_zle_int = thm "zero_zle_int";

 val int_le_0_conv = thm "int_le_0_conv";

 val zle_refl = thm "zle_refl";

 val zle_linear = thm "zle_linear";

 val zle_trans = thm "zle_trans";

 val zle_anti_sym = thm "zle_anti_sym";

 val Ints_def = thm "Ints_def";

 val Nats_def = thm "Nats_def";

 val of_nat_0 = thm "of_nat_0";

 val of_nat_Suc = thm "of_nat_Suc";

 val of_nat_1 = thm "of_nat_1";

 val of_nat_add = thm "of_nat_add";

 val of_nat_mult = thm "of_nat_mult";

 val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";

 val less_imp_of_nat_less = thm "less_imp_of_nat_less";

 val of_nat_less_imp_less = thm "of_nat_less_imp_less";

 val of_nat_less_iff = thm "of_nat_less_iff";

 val of_nat_le_iff = thm "of_nat_le_iff";

 val of_nat_eq_iff = thm "of_nat_eq_iff";

 val Nats_0 = thm "Nats_0";

 val Nats_1 = thm "Nats_1";

 val Nats_add = thm "Nats_add";

 val Nats_mult = thm "Nats_mult";

 val int_eq_of_nat = thm"int_eq_of_nat";

 val of_int = thm "of_int";

 val of_int_0 = thm "of_int_0";

 val of_int_1 = thm "of_int_1";

 val of_int_add = thm "of_int_add";

 val of_int_minus = thm "of_int_minus";

 val of_int_diff = thm "of_int_diff";

 val of_int_mult = thm "of_int_mult";

 val of_int_le_iff = thm "of_int_le_iff";

 val of_int_less_iff = thm "of_int_less_iff";

 val of_int_eq_iff = thm "of_int_eq_iff";

 val Ints_0 = thm "Ints_0";

 val Ints_1 = thm "Ints_1";

 val Ints_add = thm "Ints_add";

 val Ints_minus = thm "Ints_minus";

 val Ints_diff = thm "Ints_diff";

 val Ints_mult = thm "Ints_mult";

 val of_int_of_nat_eq = thm"of_int_of_nat_eq";

 val Ints_cases = thm "Ints_cases";

 val Ints_induct = thm "Ints_induct";

 *}

 end