src/HOL/Integ/IntDef.thy

integrated Jeremy's FiniteLib

(*  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
imports Equiv_Relations NatArith
begin

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)

text{*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 Abs_Integ_inject [simp]  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 Abs_Integ_cases [of z]) 
apply (auto simp add: Integ_def quotient_def) 
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 "(\<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 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 "(\<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

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:
     "(%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, 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 @{text 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:
     "i<j ==> 0<k ==> int k * i < int k * j"
apply (induct "k", simp)
apply (simp add: int_Suc)
apply (case_tac "k=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 @{text 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 "(\<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(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"
by simp

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)"
proof (cases w, cases z, simp add: le add int_def)
  fix a b c d
  assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})"
  show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)"
  proof
    assume "a + d \<le> c + b" 
    thus "\<exists>n. c + b = a + n + d" 
      by (auto intro!: exI [where x="c+b - (a+d)"])
  next    
    assume "\<exists>n. c + b = a + n + d" 
    then obtain n where "c + b = a + n + d" ..
    thus "a + d \<le> c + b" by arith
  qed
qed

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

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{*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 @{text 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 "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
    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 @{text 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) = of_nat 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 of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
  apply (case_tac "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 (case_tac "finite A")
  apply (erule finite_induct, auto)
  done

lemma int_setsum: "int (setsum f A) = (\<Sum>x\<in>A. int(f x))"
  by (simp add: int_eq_of_nat of_nat_setsum)

lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
  apply (case_tac "finite A")
  apply (erule finite_induct, auto)
  done

lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
  apply (case_tac "finite A")
  apply (erule finite_induct, auto)
  done

lemma int_setprod: "int (setprod f A) = (\<Prod>x\<in>A. int(f x))"
  by (simp add: int_eq_of_nat of_nat_setprod)

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


lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
apply (cases z)
apply (simp_all add: not_zle_0_negative del: int_Suc)
done


(*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