# Theory IntDef

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theory IntDef
imports Equiv_Relations NatArith
begin

    (*  Title:      IntDef.thy
        ID:         $Id: IntDef.thy,v 1.60 2005/09/27 10:14:39 berghofe Exp $
        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) ∈ Rep_Integ z. intrel``{(y,x)})"

      add_int_def:
       "z + w ==
           Abs_Integ (\<Union>(x,y) ∈ Rep_Integ z. \<Union>(u,v) ∈ 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) ∈ Rep_Integ z. \<Union>(u,v) ∈ Rep_Integ w.
                      intrel``{(x*u + y*v, x*v + y*u)})"

      le_int_def:
       "z ≤ (w::int) ==
        ∃x y u v. x+v ≤ u+y & (x,y) ∈ Rep_Integ z & (u,v) ∈ Rep_Integ w"

      less_int_def: "(z < (w::int)) == (z ≤ w & z ≠ w)"

    subsection{*Construction of the Integers*}

    subsubsection{*Preliminary Lemmas about the Equivalence Relation*}

    lemma intrel_iff [simp]: "(((x,y),(u,v)) ∈ 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) ∈ 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)} ∈ Integ"
    by (auto simp add: Integ_def intrel_def quotient_def)

    text{*Reduces equality on abstractions to equality on representatives:
      @{term "[|x ∈ Integ; y ∈ Integ|] ==> (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 "(λ(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 "(λz w. (λ(x,y). (λ(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)
    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 int_mult: "int (m * n) = (int m) * (int n)"
    by (simp add: int_def mult)

    text{*Compatibility binding*}
    lemmas zmult_int = int_mult [symmetric]

    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 ≠ (1::int)"
        by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
    qed

    subsection{*The @{text "≤"} Ordering*}

    lemma le:
      "(Abs_Integ(intrel``{(x,y)}) ≤ Abs_Integ(intrel``{(u,v)})) = (x+v ≤ u+y)"
    by (force simp add: le_int_def)

    lemma zle_refl: "w ≤ (w::int)"
    by (cases w, simp add: le)

    lemma zle_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::int)"
    by (cases i, cases j, cases k, simp add: le)

    lemma zle_anti_sym: "[| z ≤ w; w ≤ 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 ≤ z & w ≠ 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) ≤ w | w ≤ 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]
    lemmas linorder_neqE_int = linorder_neqE[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 ≠ (1::int)"
    by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)

    lemma zle_int [simp]: "(int m ≤ int n) = (m≤n)"
    by (simp add: linorder_not_less [symmetric])

    lemma zero_zle_int [simp]: "(0 ≤ int n)"
    by (simp add: Zero_int_def)

    lemma int_le_0_conv [simp]: "(int n ≤ 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 ≤ j ==> k + i ≤ 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'≤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 ≤ k ==> ∃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 ≤ j ==> k + i ≤ k + j" by (rule zadd_left_mono)
      show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
      show "¦i¦ = (if i < 0 then -i else i)" by (simp only: zabs_def)
    qed

    lemma zless_imp_add1_zle: "w<z ==> w + (1::int) ≤ 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) ∈ Rep_Integ z. { x-y })"

    lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
    proof -
      have "(λ(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 ≤ z then z else 0)"
    by (cases z, simp add: nat le int_def Zero_int_def)

    corollary nat_0_le: "0 ≤ z ==> int (nat z) = z"
    by simp

    lemma nat_le_0 [simp]: "z ≤ 0 ==> nat z = 0"
    by (cases z, simp add: nat le int_def Zero_int_def)

    lemma nat_le_eq_zle: "0 < w | 0 ≤ z ==> (nat w ≤ nat z) = (w≤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 ≤ 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 ≤ 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 ≤ z;  !!m. z = int m ==> P |] ==> P"
    by (blast dest: nat_0_le sym)

    lemma nat_eq_iff: "(nat w = m) = (if 0 ≤ 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 ≤ w then w = int m else m=0)"
    by (simp only: eq_commute [of m] nat_eq_iff)

    lemma nat_less_iff: "0 ≤ 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 ≤ 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) ≤ z;  0 ≤ 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) ≤ z';  z' ≤ 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 ≤ 0"
    by (simp add: minus_le_iff)

    lemma negative_zle [iff]: "- int n ≤ int m"
    by (rule order_trans [OF negative_zle_0 zero_zle_int])

    lemma not_zle_0_negative [simp]: "~ (0 ≤ - (int (Suc n)))"
    by (subst le_minus_iff, simp)

    lemma int_zle_neg: "(int n ≤ - 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 ≤ z) = (∃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 ≤ c+b) = (∃n. c+b = a+n+d)"
      proof
        assume "a + d ≤ c + b"
        thus "∃n. c + b = a + n + d"
          by (auto intro!: exI [where x="c+b - (a+d)"])
      next
        assume "∃n. c + b = a + n + d"
        then obtain n where "c + b = a + n + d" ..
        thus "a + d ≤ 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 ≤ 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 ≤ 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 ∈ Nats"
    by (simp add: Nats_def)

    lemma Nats_0 [simp]: "0 ∈ Nats"
    apply (simp add: Nats_def)
    apply (rule range_eqI)
    apply (rule of_nat_0 [symmetric])
    done

    lemma Nats_1 [simp]: "1 ∈ Nats"
    apply (simp add: Nats_def)
    apply (rule range_eqI)
    apply (rule of_nat_1 [symmetric])
    done

    lemma Nats_add [simp]: "[|a ∈ Nats; b ∈ Nats|] ==> a+b ∈ Nats"
    apply (auto simp add: Nats_def)
    apply (rule range_eqI)
    apply (rule of_nat_add [symmetric])
    done

    lemma Nats_mult [simp]: "[|a ∈ Nats; b ∈ Nats|] ==> a*b ∈ 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) ∈ 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 "(λ(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 ≤ (of_int z::'a::ordered_idom)) = (w ≤ 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*}
    lemmas of_int_0_le_iff = of_int_le_iff [of 0, simplified]
    lemmas of_int_le_0_iff = of_int_le_iff [of _ 0, simplified]
    declare of_int_0_le_iff [simp]
    declare of_int_le_0_iff [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*}
    lemmas of_int_0_less_iff = of_int_less_iff [of 0, simplified]
    lemmas of_int_less_0_iff = of_int_less_iff [of _ 0, simplified]
    declare of_int_0_less_iff [simp]
    declare of_int_less_0_iff [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*}
    lemmas of_int_0_eq_iff = of_int_eq_iff [of 0, simplified]
    lemmas of_int_eq_0_iff = of_int_eq_iff [of _ 0, simplified]
    declare of_int_0_eq_iff [simp]
    declare of_int_eq_0_iff [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 ∈ Ints"
    apply (simp add: Ints_def)
    apply (rule range_eqI)
    apply (rule of_int_0 [symmetric])
    done

    lemma Ints_1 [simp]: "1 ∈ Ints"
    apply (simp add: Ints_def)
    apply (rule range_eqI)
    apply (rule of_int_1 [symmetric])
    done

    lemma Ints_add [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a+b ∈ Ints"
    apply (auto simp add: Ints_def)
    apply (rule range_eqI)
    apply (rule of_int_add [symmetric])
    done

    lemma Ints_minus [simp]: "a ∈ Ints ==> -a ∈ Ints"
    apply (auto simp add: Ints_def)
    apply (rule range_eqI)
    apply (rule of_int_minus [symmetric])
    done

    lemma Ints_diff [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a-b ∈ Ints"
    apply (auto simp add: Ints_def)
    apply (rule range_eqI)
    apply (rule of_int_diff [symmetric])
    done

    lemma Ints_mult [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a*b ∈ 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 ∈ \<int> ==> (!!z. q = of_int z ==> C) ==> C"
    proof (simp add: Ints_def)
      assume "!!z. q = of_int z ==> C"
      assume "q ∈ range of_int" thus C ..
    qed

    lemma Ints_induct [case_names of_int, induct set: Ints]:
      "q ∈ \<int> ==> (!!z. P (of_int z)) ==> P q"
      by (rule Ints_cases) auto

    (* int (Suc n) = 1 + int n *)
    declare int_Suc [simp]

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

    text{*By Jeremy Avigad*}

    lemma of_nat_setsum: "of_nat (setsum f A) = (∑x∈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) = (∑x∈A. of_int(f x))"
      apply (case_tac "finite A")
      apply (erule finite_induct, auto)
      done

    lemma int_setsum: "int (setsum f A) = (∑x∈A. int(f x))"
      by (simp add: int_eq_of_nat of_nat_setsum)

    lemma of_nat_setprod: "of_nat (setprod f A) = (∏x∈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) = (∏x∈A. of_int(f x))"
      apply (case_tac "finite A")
      apply (erule finite_induct, auto)
      done

    lemma int_setprod: "int (setprod f A) = (∏x∈A. int(f x))"
      by (simp add: int_eq_of_nat of_nat_setprod)

    lemma setprod_nonzero_nat:
        "finite A ==> (∀x ∈ A. f x ≠ (0::nat)) ==> setprod f A ≠ 0"
      by (rule setprod_nonzero, auto)

    lemma setprod_zero_eq_nat:
        "finite A ==> (setprod f A = (0::nat)) = (∃x ∈ A. f x = 0)"
      by (rule setprod_zero_eq, auto)

    lemma setprod_nonzero_int:
        "finite A ==> (∀x ∈ A. f x ≠ (0::int)) ==> setprod f A ≠ 0"
      by (rule setprod_nonzero, auto)

    lemma setprod_zero_eq_int:
        "finite A ==> (setprod f A = (0::int)) = (∃x ∈ 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) = (∃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 ==> ∃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

    text{*Contributed by Brian Huffman*}
    theorem int_diff_cases [case_names diff]:
    assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
     apply (rule_tac z=z in int_cases)
      apply (rule_tac m=n and n=0 in prem, simp)
     apply (rule_tac m=0 and n="Suc n" in prem, simp)
    done

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

    subsection {* Configuration of the code generator *}

    (*FIXME: the IntInf.fromInt below hides a dependence on fixed-precision ints!*)

    types_code
      "int" ("int")
    attach (term_of) {*
    val term_of_int = HOLogic.mk_int o IntInf.fromInt;
    *}
    attach (test) {*
    fun gen_int i = one_of [~1, 1] * random_range 0 i;
    *}

    constdefs
      int_aux :: "int => nat => int"
      "int_aux i n == (i + int n)"
      nat_aux :: "nat => int => nat"
      "nat_aux n i == (n + nat i)"

    lemma [code]:
      "int_aux i 0 = i"
      "int_aux i (Suc n) = int_aux (i + 1) n" -- {* tail recursive *}
      "int n = int_aux 0 n"
      by (simp add: int_aux_def)+

    lemma [code]: "nat_aux n i = (if i <= 0 then n else nat_aux (Suc n) (i - 1))"
      -- {* 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 0 i"
      by (simp add: nat_aux_def)

    consts_code
      "0" :: "int"                  ("0")
      "1" :: "int"                  ("1")
      "uminus" :: "int => int"      ("~")
      "op +" :: "int => int => int" ("(_ +/ _)")
      "op *" :: "int => int => int" ("(_ */ _)")
      "op <" :: "int => int => bool" ("(_ </ _)")
      "op <=" :: "int => int => bool" ("(_ <=/ _)")
      "neg"                         ("(_ < 0)")

    ML {*
    fun number_of_codegen thy defs gr dep module b (Const ("Numeral.number_of",
          Type ("fun", [_, T as Type ("IntDef.int", [])])) $ bin) =
            (SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, T)),
               Pretty.str (IntInf.toString (HOLogic.dest_binum bin))) handle TERM _ => NONE)
      | number_of_codegen thy defs gr s thyname b (Const ("Numeral.number_of",
          Type ("fun", [_, Type ("nat", [])])) $ bin) =
            SOME (Codegen.invoke_codegen thy defs s thyname b (gr,
              Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT) $
                (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) $ bin)))
      | number_of_codegen _ _ _ _ _ _ _ = NONE;
    *}

    setup {* [Codegen.add_codegen "number_of_codegen" number_of_codegen] *}

    quickcheck_params [default_type = int]

    (*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

### Construction of the Integers

#### Preliminary Lemmas about the Equivalence Relation

lemma intrel_iff:

      (((x, y), u, v) ∈ intrel) = (x + v = u + y)

lemma equiv_intrel:

      equiv UNIV intrel

lemmas equiv_intrel_iff:

      (intrel `` {x} = intrel `` {y}) = ((x, y) ∈ intrel)

lemmas equiv_intrel_iff:

      (intrel `` {x} = intrel `` {y}) = ((x, y) ∈ intrel)

lemma

      intrel `` {(x, y)} ∈ Integ

lemma eq_Abs_Integ:

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

#### @{term int}: Embedding the Naturals into the Integers

lemma inj_int:

      inj int

lemma int_int_eq:

      (int m = int n) = (m = n)

#### Integer Unary Negation

lemma minus:

      - Abs_Integ (intrel `` {(x, y)}) = Abs_Integ (intrel `` {(y, x)})

lemma zminus_zminus:

      - (- z) = z

lemma zminus_0:

      - 0 = 0

### Integer Addition

lemma add:

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

lemma zminus_zadd_distrib:

      - (z + w) = - z + - w

lemma zadd_commute:

      z + w = w + z

lemma zadd_assoc:

      z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)

lemma zadd_left_commute:

      x + (y + z) = y + (x + z)

lemmas zadd_ac:

      z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)

      z + w = w + z

      x + (y + z) = y + (x + z)

lemmas zadd_ac:

      z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)

      z + w = w + z

      x + (y + z) = y + (x + z)

lemmas zmult_ac:

      a * b * c = a * (b * c)

      a * b = b * a

      a * (b * c) = b * (a * c)

lemmas zmult_ac:

      a * b * c = a * (b * c)

      a * b = b * a

      a * (b * c) = b * (a * c)

lemma zadd_int:

      int m + int n = int (m + n)

lemma zadd_int_left:

      int m + (int n + z) = int (m + n) + z

lemma int_Suc:

      int (Suc m) = 1 + int m

lemma zadd_0:

      0 + z = z

lemma zadd_0_right:

      z + 0 = z

lemma zadd_zminus_inverse2:

      - z + z = 0

### Integer Multiplication

lemma mult_congruent2:

      congruent2 intrel intrel
       (%p1 p2.
           (%(x, y). (%(u, v). intrel `` {(x * u + y * v, x * v + y * u)}) p2) p1)

lemma mult:

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

lemma zmult_zminus:

      - z * w = - (z * w)

lemma zmult_commute:

      z * w = w * z

lemma zmult_assoc:

      z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)

lemma zadd_zmult_distrib:

      (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w

lemma zadd_zmult_distrib2:

      w * (z1.0 + z2.0) = w * z1.0 + w * z2.0

lemma zdiff_zmult_distrib:

      (z1.0 - z2.0) * w = z1.0 * w - z2.0 * w

lemma zdiff_zmult_distrib2:

      w * (z1.0 - z2.0) = w * z1.0 - w * z2.0

lemmas int_distrib:

      (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w

      w * (z1.0 + z2.0) = w * z1.0 + w * z2.0

      (z1.0 - z2.0) * w = z1.0 * w - z2.0 * w

      w * (z1.0 - z2.0) = w * z1.0 - w * z2.0

lemmas int_distrib:

      (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w

      w * (z1.0 + z2.0) = w * z1.0 + w * z2.0

      (z1.0 - z2.0) * w = z1.0 * w - z2.0 * w

      w * (z1.0 - z2.0) = w * z1.0 - w * z2.0

lemma int_mult:

      int (m * n) = int m * int n

lemmas zmult_int:

      int m1 * int n1 = int (m1 * n1)

lemmas zmult_int:

      int m1 * int n1 = int (m1 * n1)

lemma zmult_1:

      1 * z = z

lemma zmult_1_right:

      z * 1 = z

### The @{text "≤"} Ordering

lemma le:

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

lemma zle_refl:

      w ≤ w

lemma zle_trans:

      [| i ≤ j; j ≤ k |] ==> i ≤ k

lemma zle_anti_sym:

      [| z ≤ w; w ≤ z |] ==> z = w

lemma zless_le:

      (w < z) = (w ≤ z ∧ w ≠ z)

lemma zle_linear:

      z ≤ w ∨ w ≤ z

lemmas zless_linear:

      x < y ∨ x = y ∨ y < x

lemmas zless_linear:

      x < y ∨ x = y ∨ y < x

lemmas linorder_neqE_int:

      [| x ≠ y; x < y ==> R; y < x ==> R |] ==> R

lemmas linorder_neqE_int:

      [| x ≠ y; x < y ==> R; y < x ==> R |] ==> R

lemma int_eq_0_conv:

      (int n = 0) = (n = 0)

lemma zless_int:

      (int m < int n) = (m < n)

lemma int_less_0_conv:

      ¬ int k < 0

lemma zero_less_int_conv:

      (0 < int n) = (0 < n)

lemma int_0_less_1:

      0 < 1

lemma int_0_neq_1:

      0 ≠ 1

lemma zle_int:

      (int m ≤ int n) = (m ≤ n)

lemma zero_zle_int:

      0 ≤ int n

lemma int_le_0_conv:

      (int n ≤ 0) = (n = 0)

lemma int_0:

      int 0 = 0

lemma int_1:

      int 1 = 1

lemma int_Suc0_eq_1:

      int (Suc 0) = 1

### Monotonicity results

lemma zadd_left_mono:

      i ≤ j ==> k + i ≤ k + j

lemma zadd_strict_right_mono:

      i < j ==> i + k < j + k

lemma zadd_zless_mono:

      [| w' < w; z' ≤ z |] ==> w' + z' < w + z

### Strict Monotonicity of Multiplication

lemma zmult_zless_mono2_lemma:

      [| i < j; 0 < k |] ==> int k * i < int k * j

lemma zero_le_imp_eq_int:

      0 ≤ k ==> ∃n. k = int n

lemma zmult_zless_mono2:

      [| i < j; 0 < k |] ==> k * i < k * j

lemma zless_imp_add1_zle:

      w < z ==> w + 1 ≤ z

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

lemma nat:

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

lemma nat_int:

      nat (int n) = n

lemma nat_zero:

      nat 0 = 0

lemma int_nat_eq:

      int (nat z) = (if 0 ≤ z then z else 0)

corollary nat_0_le:

      0 ≤ z ==> int (nat z) = z

lemma nat_le_0:

      z ≤ 0 ==> nat z = 0

lemma nat_le_eq_zle:

      0 < w ∨ 0 ≤ z ==> (nat w ≤ nat z) = (w ≤ z)

corollary nat_mono_iff:

      0 < z ==> (nat w < nat z) = (w < z)

corollary nat_less_eq_zless:

      0 ≤ w ==> (nat w < nat z) = (w < z)

lemma zless_nat_conj:

      (nat w < nat z) = (0 < z ∧ w < z)

lemma nonneg_eq_int:

      [| 0 ≤ z; !!m. z = int m ==> P |] ==> P

lemma nat_eq_iff:

      (nat w = m) = (if 0 ≤ w then w = int m else m = 0)

corollary nat_eq_iff2:

      (m = nat w) = (if 0 ≤ w then w = int m else m = 0)

lemma nat_less_iff:

      0 ≤ w ==> (nat w < m) = (w < int m)

lemma int_eq_iff:

      (int m = z) = (m = nat z ∧ 0 ≤ z)

lemma zero_less_nat_eq:

      (0 < nat z) = (0 < z)

lemma nat_add_distrib:

      [| 0 ≤ z; 0 ≤ z' |] ==> nat (z + z') = nat z + nat z'

lemma nat_diff_distrib:

      [| 0 ≤ z'; z' ≤ z |] ==> nat (z - z') = nat z - nat z'

lemma nat_zminus_int:

      nat (- int n) = 0

lemma zless_nat_eq_int_zless:

      (m < nat z) = (int m < z)

### Lemmas about the Function @{term int} and Orderings

lemma negative_zless_0:

      - int (Suc n) < 0

lemma negative_zless:

      - int (Suc n) < int m

lemma negative_zle_0:

      - int n ≤ 0

lemma negative_zle:

      - int n ≤ int m

lemma not_zle_0_negative:

      ¬ 0 ≤ - int (Suc n)

lemma int_zle_neg:

      (int n ≤ - int m) = (n = 0 ∧ m = 0)

lemma not_int_zless_negative:

      ¬ int n < - int m

lemma negative_eq_positive:

      (- int n = int m) = (n = 0 ∧ m = 0)

lemma zle_iff_zadd:

      (w ≤ z) = (∃n. z = w + int n)

lemma abs_int_eq:

      ¦int m¦ = int m

lemma abs_split:

      P ¦a¦ = (((0::'a) ≤ a --> P a) ∧ (a < (0::'a) --> P (- a)))

### The Constants @{term neg} and @{term iszero}

lemma not_neg_int:

      ¬ neg (int n)

lemma neg_zminus_int:

      neg (- int (Suc n))

lemmas neg_eq_less_0:

      neg Z == Z < (0::'a)

lemmas neg_eq_less_0:

      neg Z == Z < (0::'a)

lemma not_neg_eq_ge_0:

      (¬ neg x) = ((0::'a) ≤ x)

### To simplify inequalities when Numeral1 can get simplified to 1

lemma not_neg_0:

      ¬ neg (0::'a)

lemma not_neg_1:

      ¬ neg (1::'a)

lemma iszero_0:

      iszero (0::'a)

lemma not_iszero_1:

      ¬ iszero (1::'a)

lemma neg_nat:

      neg z ==> nat z = 0

lemma not_neg_nat:

      ¬ neg z ==> int (nat z) = z

### The Set of Natural Numbers

lemma of_nat_in_Nats:

      of_nat n ∈ Nats

lemma Nats_0:

      (0::'a) ∈ Nats

lemma Nats_1:

      (1::'a) ∈ Nats

lemma Nats_add:

      [| a ∈ Nats; b ∈ Nats |] ==> a + b ∈ Nats

lemma Nats_mult:

      [| a ∈ Nats; b ∈ Nats |] ==> a * b ∈ Nats

lemma int_eq_of_nat:

      int = of_nat

lemma of_nat_eq_id:

      of_nat = id

### Embedding of the Integers into any @{text comm_ring_1}: @{term of_int}

lemma of_int:

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

lemma of_int_0:

      of_int 0 = (0::'a)

lemma of_int_1:

      of_int 1 = (1::'a)

lemma of_int_add:

      of_int (w + z) = of_int w + of_int z

lemma of_int_minus:

      of_int (- z) = - of_int z

lemma of_int_diff:

      of_int (w - z) = of_int w - of_int z

lemma of_int_mult:

      of_int (w * z) = of_int w * of_int z

lemma of_int_le_iff:

      (of_int w ≤ of_int z) = (w ≤ z)

lemmas of_int_0_le_iff:

      ((0::'a) ≤ of_int z) = (0 ≤ z)

lemmas of_int_0_le_iff:

      ((0::'a) ≤ of_int z) = (0 ≤ z)

lemmas of_int_le_0_iff:

      (of_int w ≤ (0::'a)) = (w ≤ 0)

lemmas of_int_le_0_iff:

      (of_int w ≤ (0::'a)) = (w ≤ 0)

lemma of_int_less_iff:

      (of_int w < of_int z) = (w < z)

lemmas of_int_0_less_iff:

      ((0::'a) < of_int z) = (0 < z)

lemmas of_int_0_less_iff:

      ((0::'a) < of_int z) = (0 < z)

lemmas of_int_less_0_iff:

      (of_int w < (0::'a)) = (w < 0)

lemmas of_int_less_0_iff:

      (of_int w < (0::'a)) = (w < 0)

lemma of_int_eq_iff:

      (of_int w = of_int z) = (w = z)

lemmas of_int_0_eq_iff:

      ((0::'a) = of_int z) = (0 = z)

lemmas of_int_0_eq_iff:

      ((0::'a) = of_int z) = (0 = z)

lemmas of_int_eq_0_iff:

      (of_int w = (0::'a)) = (w = 0)

lemmas of_int_eq_0_iff:

      (of_int w = (0::'a)) = (w = 0)

lemma of_int_eq_id:

      of_int = id

### The Set of Integers

lemma Ints_0:

      (0::'a) ∈ Ints

lemma Ints_1:

      (1::'a) ∈ Ints

lemma Ints_add:

      [| a ∈ Ints; b ∈ Ints |] ==> a + b ∈ Ints

lemma Ints_minus:

      a ∈ Ints ==> - a ∈ Ints

lemma Ints_diff:

      [| a ∈ Ints; b ∈ Ints |] ==> a - b ∈ Ints

lemma Ints_mult:

      [| a ∈ Ints; b ∈ Ints |] ==> a * b ∈ Ints

lemma of_int_of_nat_eq:

      of_int (of_nat n) = of_nat n

lemma of_int_int_eq:

      of_int (int n) = of_nat n

lemma Ints_cases:

      [| q ∈ Ints; !!z. q = of_int z ==> C |] ==> C

lemma Ints_induct:

      [| q ∈ Ints; !!z. P (of_int z) |] ==> P q

### More Properties of @{term setsum} and @{term setprod}

lemma of_nat_setsum:

      of_nat (setsum f A) = (∑x∈A. of_nat (f x))

lemma of_int_setsum:

      of_int (setsum f A) = (∑x∈A. of_int (f x))

lemma int_setsum:

      int (setsum f A) = (∑x∈A. int (f x))

lemma of_nat_setprod:

      of_nat (setprod f A) = (∏x∈A. of_nat (f x))

lemma of_int_setprod:

      of_int (setprod f A) = (∏x∈A. of_int (f x))

lemma int_setprod:

      int (setprod f A) = (∏x∈A. int (f x))

lemma setprod_nonzero_nat:

      [| finite A; ∀x∈A. f x ≠ 0 |] ==> setprod f A ≠ 0

lemma setprod_zero_eq_nat:

      finite A ==> (setprod f A = 0) = (∃x∈A. f x = 0)

lemma setprod_nonzero_int:

      [| finite A; ∀x∈A. f x ≠ 0 |] ==> setprod f A ≠ 0

lemma setprod_zero_eq_int:

      finite A ==> (setprod f A = 0) = (∃x∈A. f x = 0)

lemma zless_iff_Suc_zadd:

      (w < z) = (∃n. z = w + int (Suc n))

lemma negD:

      x < 0 ==> ∃n. x = - int (Suc n)

theorem int_cases:

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

theorem int_induct:

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

theorem int_diff_cases:

      (!!m n. z = int m - int n ==> P) ==> P

lemma of_nat_nat:

      0 ≤ z ==> of_nat (nat z) = of_int z

### Configuration of the code generator

lemma

      int_aux i 0 = i

      int_aux i (Suc n) = int_aux (i + 1) n

      int n = int_aux 0 n

lemma

      nat_aux n i = (if i ≤ 0 then n else nat_aux (Suc n) (i - 1))

lemma

      nat i = nat_aux 0 i