# Theory RealDef

Up to index of Isabelle/HOL/HOL-Complex

theory RealDef
imports PReal
uses (real_arith.ML)
begin

    (*  Title       : Real/RealDef.thy
        ID          : $Id: RealDef.thy,v 1.58 2005/08/01 17:20:26 wenzelm Exp $
        Author      : Jacques D. Fleuriot
        Copyright   : 1998  University of Cambridge
        Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
        Additional contributions by Jeremy Avigad
    *)

    header{*Defining the Reals from the Positive Reals*}

    theory RealDef
    imports PReal
    uses ("real_arith.ML")
    begin

    constdefs
      realrel   ::  "((preal * preal) * (preal * preal)) set"
      "realrel == {p. ∃x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"

    typedef (Real)  real = "UNIV//realrel"
      by (auto simp add: quotient_def)

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

    constdefs

      (** these don't use the overloaded "real" function: users don't see them **)

      real_of_preal :: "preal => real"
      "real_of_preal m     ==
               Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})"

    consts
       (*Overloaded constant denoting the Real subset of enclosing
         types such as hypreal and complex*)
       Reals :: "'a set"

       (*overloaded constant for injecting other types into "real"*)
       real :: "'a => real"

    syntax (xsymbols)
      Reals     :: "'a set"                   ("\<real>")

    defs (overloaded)

      real_zero_def:
      "0 == Abs_Real(realrel``{(preal_of_rat 1, preal_of_rat 1)})"

      real_one_def:
      "1 == Abs_Real(realrel``
                   {(preal_of_rat 1 + preal_of_rat 1,
                     preal_of_rat 1)})"

      real_minus_def:
      "- r ==  contents (\<Union>(x,y) ∈ Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"

      real_add_def:
       "z + w ==
           contents (\<Union>(x,y) ∈ Rep_Real(z). \<Union>(u,v) ∈ Rep_Real(w).
                     { Abs_Real(realrel``{(x+u, y+v)}) })"

      real_diff_def:
       "r - (s::real) == r + - s"

      real_mult_def:
        "z * w ==
           contents (\<Union>(x,y) ∈ Rep_Real(z). \<Union>(u,v) ∈ Rep_Real(w).
                     { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"

      real_inverse_def:
      "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)"

      real_divide_def:
      "R / (S::real) == R * inverse S"

      real_le_def:
       "z ≤ (w::real) ==
        ∃x y u v. x+v ≤ u+y & (x,y) ∈ Rep_Real z & (u,v) ∈ Rep_Real w"

      real_less_def: "(x < (y::real)) == (x ≤ y & x ≠ y)"

      real_abs_def:  "abs (r::real) == (if 0 ≤ r then r else -r)"

    subsection{*Proving that realrel is an equivalence relation*}

    lemma preal_trans_lemma:
      assumes "x + y1 = x1 + y"
          and "x + y2 = x2 + y"
      shows "x1 + y2 = x2 + (y1::preal)"
    proof -
      have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_add_ac)
      also have "... = (x2 + y) + x1"  by (simp add: prems)
      also have "... = x2 + (x1 + y)"  by (simp add: preal_add_ac)
      also have "... = x2 + (x + y1)"  by (simp add: prems)
      also have "... = (x2 + y1) + x"  by (simp add: preal_add_ac)
      finally have "(x1 + y2) + x = (x2 + y1) + x" .
      thus ?thesis by (simp add: preal_add_right_cancel_iff)
    qed

    lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) ∈ realrel) = (x1 + y2 = x2 + y1)"
    by (simp add: realrel_def)

    lemma equiv_realrel: "equiv UNIV realrel"
    apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def)
    apply (blast dest: preal_trans_lemma)
    done

    text{*Reduces equality of equivalence classes to the @{term realrel} relation:
      @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) ∈ realrel)"} *}
    lemmas equiv_realrel_iff =
           eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]

    declare equiv_realrel_iff [simp]

    lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
    by (simp add: Real_def realrel_def quotient_def, blast)

    lemma inj_on_Abs_Real: "inj_on Abs_Real Real"
    apply (rule inj_on_inverseI)
    apply (erule Abs_Real_inverse)
    done

    declare inj_on_Abs_Real [THEN inj_on_iff, simp]
    declare Abs_Real_inverse [simp]

    text{*Case analysis on the representation of a real number as an equivalence
          class of pairs of positive reals.*}
    lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
         "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
    apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
    apply (drule arg_cong [where f=Abs_Real])
    apply (auto simp add: Rep_Real_inverse)
    done

    subsection{*Congruence property for addition*}

    lemma real_add_congruent2_lemma:
         "[|a + ba = aa + b; ab + bc = ac + bb|]
          ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
    apply (simp add: preal_add_assoc)
    apply (rule preal_add_left_commute [of ab, THEN ssubst])
    apply (simp add: preal_add_assoc [symmetric])
    apply (simp add: preal_add_ac)
    done

    lemma real_add:
         "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
          Abs_Real (realrel``{(x+u, y+v)})"
    proof -
      have "(λz w. (λ(x,y). (λ(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
            respects2 realrel"
        by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma)
      thus ?thesis
        by (simp add: real_add_def UN_UN_split_split_eq
                      UN_equiv_class2 [OF equiv_realrel equiv_realrel])
    qed

    lemma real_add_commute: "(z::real) + w = w + z"
    by (cases z, cases w, simp add: real_add preal_add_ac)

    lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
    by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc)

    lemma real_add_zero_left: "(0::real) + z = z"
    by (cases z, simp add: real_add real_zero_def preal_add_ac)

    instance real :: comm_monoid_add
      by (intro_classes,
          (assumption |
           rule real_add_commute real_add_assoc real_add_zero_left)+)

    subsection{*Additive Inverse on real*}

    lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
    proof -
      have "(λ(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
        by (simp add: congruent_def preal_add_commute)
      thus ?thesis
        by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
    qed

    lemma real_add_minus_left: "(-z) + z = (0::real)"
    by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute)

    subsection{*Congruence property for multiplication*}

    lemma real_mult_congruent2_lemma:
         "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
              x * x1 + y * y1 + (x * y2 + y * x2) =
              x * x2 + y * y2 + (x * y1 + y * x1)"
    apply (simp add: preal_add_left_commute preal_add_assoc [symmetric])
    apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric])
    apply (simp add: preal_add_commute)
    done

    lemma real_mult_congruent2:
        "(%p1 p2.
            (%(x1,y1). (%(x2,y2).
              { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
         respects2 realrel"
    apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
    apply (simp add: preal_mult_commute preal_add_commute)
    apply (auto simp add: real_mult_congruent2_lemma)
    done

    lemma real_mult:
          "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
           Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
    by (simp add: real_mult_def UN_UN_split_split_eq
             UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])

    lemma real_mult_commute: "(z::real) * w = w * z"
    by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac)

    lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
    apply (cases z1, cases z2, cases z3)
    apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac)
    done

    lemma real_mult_1: "(1::real) * z = z"
    apply (cases z)
    apply (simp add: real_mult real_one_def preal_add_mult_distrib2
                     preal_mult_1_right preal_mult_ac preal_add_ac)
    done

    lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
    apply (cases z1, cases z2, cases w)
    apply (simp add: real_add real_mult preal_add_mult_distrib2
                     preal_add_ac preal_mult_ac)
    done

    text{*one and zero are distinct*}
    lemma real_zero_not_eq_one: "0 ≠ (1::real)"
    proof -
      have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1"
        by (simp add: preal_self_less_add_left)
      thus ?thesis
        by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff)
    qed

    subsection{*existence of inverse*}

    lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
    by (simp add: real_zero_def preal_add_commute)

    text{*Instead of using an existential quantifier and constructing the inverse
    within the proof, we could define the inverse explicitly.*}

    lemma real_mult_inverse_left_ex: "x ≠ 0 ==> ∃y. y*x = (1::real)"
    apply (simp add: real_zero_def real_one_def, cases x)
    apply (cut_tac x = xa and y = y in linorder_less_linear)
    apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
    apply (rule_tac
            x = "Abs_Real (realrel `` { (preal_of_rat 1,
                                inverse (D) + preal_of_rat 1)}) "
           in exI)
    apply (rule_tac [2]
            x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1,
                       preal_of_rat 1)})"
           in exI)
    apply (auto simp add: real_mult preal_mult_1_right
                  preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
                  preal_mult_inverse_right preal_add_ac preal_mult_ac)
    done

    lemma real_mult_inverse_left: "x ≠ 0 ==> inverse(x)*x = (1::real)"
    apply (simp add: real_inverse_def)
    apply (frule real_mult_inverse_left_ex, safe)
    apply (rule someI2, auto)
    done

    subsection{*The Real Numbers form a Field*}

    instance real :: field
    proof
      fix x y z :: real
      show "- x + x = 0" by (rule real_add_minus_left)
      show "x - y = x + (-y)" by (simp add: real_diff_def)
      show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
      show "x * y = y * x" by (rule real_mult_commute)
      show "1 * x = x" by (rule real_mult_1)
      show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
      show "0 ≠ (1::real)" by (rule real_zero_not_eq_one)
      show "x ≠ 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
      show "x / y = x * inverse y" by (simp add: real_divide_def)
    qed

    text{*Inverse of zero!  Useful to simplify certain equations*}

    lemma INVERSE_ZERO: "inverse 0 = (0::real)"
    by (simp add: real_inverse_def)

    instance real :: division_by_zero
    proof
      show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
    qed

    (*Pull negations out*)
    declare minus_mult_right [symmetric, simp]
            minus_mult_left [symmetric, simp]

    lemma real_mult_1_right: "z * (1::real) = z"
      by (rule OrderedGroup.mult_1_right)

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

    lemma real_le_refl: "w ≤ (w::real)"
    by (cases w, force simp add: real_le_def)

    text{*The arithmetic decision procedure is not set up for type preal.
      This lemma is currently unused, but it could simplify the proofs of the
      following two lemmas.*}
    lemma preal_eq_le_imp_le:
      assumes eq: "a+b = c+d" and le: "c ≤ a"
      shows "b ≤ (d::preal)"
    proof -
      have "c+d ≤ a+d" by (simp add: prems preal_cancels)
      hence "a+b ≤ a+d" by (simp add: prems)
      thus "b ≤ d" by (simp add: preal_cancels)
    qed

    lemma real_le_lemma:
      assumes l: "u1 + v2 ≤ u2 + v1"
          and "x1 + v1 = u1 + y1"
          and "x2 + v2 = u2 + y2"
      shows "x1 + y2 ≤ x2 + (y1::preal)"
    proof -
      have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
      hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
      also have "... ≤ (x2+y1) + (u2+v1)"
             by (simp add: prems preal_add_le_cancel_left)
      finally show ?thesis by (simp add: preal_add_le_cancel_right)
    qed

    lemma real_le:
         "(Abs_Real(realrel``{(x1,y1)}) ≤ Abs_Real(realrel``{(x2,y2)})) =
          (x1 + y2 ≤ x2 + y1)"
    apply (simp add: real_le_def)
    apply (auto intro: real_le_lemma)
    done

    lemma real_le_anti_sym: "[| z ≤ w; w ≤ z |] ==> z = (w::real)"
    by (cases z, cases w, simp add: real_le)

    lemma real_trans_lemma:
      assumes "x + v ≤ u + y"
          and "u + v' ≤ u' + v"
          and "x2 + v2 = u2 + y2"
      shows "x + v' ≤ u' + (y::preal)"
    proof -
      have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
      also have "... ≤ (u+y) + (u+v')"
        by (simp add: preal_add_le_cancel_right prems)
      also have "... ≤ (u+y) + (u'+v)"
        by (simp add: preal_add_le_cancel_left prems)
      also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
      finally show ?thesis by (simp add: preal_add_le_cancel_right)
    qed

    lemma real_le_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::real)"
    apply (cases i, cases j, cases k)
    apply (simp add: real_le)
    apply (blast intro: real_trans_lemma)
    done

    (* Axiom 'order_less_le' of class 'order': *)
    lemma real_less_le: "((w::real) < z) = (w ≤ z & w ≠ z)"
    by (simp add: real_less_def)

    instance real :: order
    proof qed
     (assumption |
      rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+

    (* Axiom 'linorder_linear' of class 'linorder': *)
    lemma real_le_linear: "(z::real) ≤ w | w ≤ z"
    apply (cases z, cases w)
    apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
    done

    instance real :: linorder
      by (intro_classes, rule real_le_linear)

    lemma real_le_eq_diff: "(x ≤ y) = (x-y ≤ (0::real))"
    apply (cases x, cases y)
    apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
                          preal_add_ac)
    apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
    done

    lemma real_add_left_mono:
      assumes le: "x ≤ y" shows "z + x ≤ z + (y::real)"
    proof -
      have "z + x - (z + y) = (z + -z) + (x - y)"
        by (simp add: diff_minus add_ac)
      with le show ?thesis
        by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
    qed

    lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
    by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)

    lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
    by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)

    lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
    apply (cases x, cases y)
    apply (simp add: linorder_not_le [where 'a = real, symmetric]
                     linorder_not_le [where 'a = preal]
                      real_zero_def real_le real_mult)
      --{*Reduce to the (simpler) @{text "≤"} relation *}
    apply (auto dest!: less_add_left_Ex
         simp add: preal_add_ac preal_mult_ac
              preal_add_mult_distrib2 preal_cancels preal_self_less_add_left)
    done

    lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
    apply (rule real_sum_gt_zero_less)
    apply (drule real_less_sum_gt_zero [of x y])
    apply (drule real_mult_order, assumption)
    apply (simp add: right_distrib)
    done

    text{*lemma for proving @{term "0<(1::real)"}*}
    lemma real_zero_le_one: "0 ≤ (1::real)"
    by (simp add: real_zero_def real_one_def real_le
                     preal_self_less_add_left order_less_imp_le)

    subsection{*The Reals Form an Ordered Field*}

    instance real :: ordered_field
    proof
      fix x y z :: real
      show "x ≤ y ==> z + x ≤ z + y" by (rule real_add_left_mono)
      show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
      show "¦x¦ = (if x < 0 then -x else x)"
        by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
    qed

    text{*The function @{term real_of_preal} requires many proofs, but it seems
    to be essential for proving completeness of the reals from that of the
    positive reals.*}

    lemma real_of_preal_add:
         "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
    by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1
                  preal_add_ac)

    lemma real_of_preal_mult:
         "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
    by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
                  preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)

    text{*Gleason prop 9-4.4 p 127*}
    lemma real_of_preal_trichotomy:
          "∃m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
    apply (simp add: real_of_preal_def real_zero_def, cases x)
    apply (auto simp add: real_minus preal_add_ac)
    apply (cut_tac x = x and y = y in linorder_less_linear)
    apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
    done

    lemma real_of_preal_leD:
          "real_of_preal m1 ≤ real_of_preal m2 ==> m1 ≤ m2"
    by (simp add: real_of_preal_def real_le preal_cancels)

    lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
    by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])

    lemma real_of_preal_lessD:
          "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
    by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]
                  preal_cancels)

    lemma real_of_preal_less_iff [simp]:
         "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
    by (blast intro: real_of_preal_lessI real_of_preal_lessD)

    lemma real_of_preal_le_iff:
         "(real_of_preal m1 ≤ real_of_preal m2) = (m1 ≤ m2)"
    by (simp add: linorder_not_less [symmetric])

    lemma real_of_preal_zero_less: "0 < real_of_preal m"
    apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
                preal_add_ac preal_cancels)
    apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
    apply (blast intro: preal_self_less_add_left order_less_imp_le)
    apply (insert preal_not_eq_self [of "preal_of_rat 1" m])
    apply (simp add: preal_add_ac)
    done

    lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
    by (simp add: real_of_preal_zero_less)

    lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
    proof -
      from real_of_preal_minus_less_zero
      show ?thesis by (blast dest: order_less_trans)
    qed

    subsection{*Theorems About the Ordering*}

    text{*obsolete but used a lot*}

    lemma real_not_refl2: "x < y ==> x ≠ (y::real)"
    by blast

    lemma real_le_imp_less_or_eq: "!!(x::real). x ≤ y ==> x < y | x = y"
    by (simp add: order_le_less)

    lemma real_gt_zero_preal_Ex: "(0 < x) = (∃y. x = real_of_preal y)"
    apply (auto simp add: real_of_preal_zero_less)
    apply (cut_tac x = x in real_of_preal_trichotomy)
    apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
    done

    lemma real_gt_preal_preal_Ex:
         "real_of_preal z < x ==> ∃y. x = real_of_preal y"
    by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
                 intro: real_gt_zero_preal_Ex [THEN iffD1])

    lemma real_ge_preal_preal_Ex:
         "real_of_preal z ≤ x ==> ∃y. x = real_of_preal y"
    by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)

    lemma real_less_all_preal: "y ≤ 0 ==> ∀x. y < real_of_preal x"
    by (auto elim: order_le_imp_less_or_eq [THEN disjE]
                intro: real_of_preal_zero_less [THEN [2] order_less_trans]
                simp add: real_of_preal_zero_less)

    lemma real_less_all_real2: "~ 0 < y ==> ∀x. y < real_of_preal x"
    by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])

    lemma real_add_less_le_mono: "[| w'<w; z'≤z |] ==> w' + z' < w + (z::real)"
      by (rule OrderedGroup.add_less_le_mono)

    lemma real_add_le_less_mono:
         "!!z z'::real. [| w'≤w; z'<z |] ==> w' + z' < w + z"
      by (rule OrderedGroup.add_le_less_mono)

    lemma real_le_square [simp]: "(0::real) ≤ x*x"
     by (rule Ring_and_Field.zero_le_square)

    subsection{*More Lemmas*}

    lemma real_mult_left_cancel: "(c::real) ≠ 0 ==> (c*a=c*b) = (a=b)"
    by auto

    lemma real_mult_right_cancel: "(c::real) ≠ 0 ==> (a*c=b*c) = (a=b)"
    by auto

    text{*The precondition could be weakened to @{term "0≤x"}*}
    lemma real_mult_less_mono:
         "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
     by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)

    lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
      by (force elim: order_less_asym
                simp add: Ring_and_Field.mult_less_cancel_right)

    lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z ≤ y*z) = (x≤y)"
    apply (simp add: mult_le_cancel_right)
    apply (blast intro: elim: order_less_asym)
    done

    lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x ≤ z*y) = (x≤y)"
    by(simp add:mult_commute)

    text{*Only two uses?*}
    lemma real_mult_less_mono':
         "[| x < y;  r1 < r2;  (0::real) ≤ r1;  0 ≤ x|] ==> r1 * x < r2 * y"
     by (rule Ring_and_Field.mult_strict_mono')

    text{*FIXME: delete or at least combine the next two lemmas*}
    lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
    apply (drule OrderedGroup.equals_zero_I [THEN sym])
    apply (cut_tac x = y in real_le_square)
    apply (auto, drule order_antisym, auto)
    done

    lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
    apply (rule_tac y = x in real_sum_squares_cancel)
    apply (simp add: add_commute)
    done

    lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
    by (drule add_strict_mono [of concl: 0 0], assumption, simp)

    lemma real_le_add_order: "[| 0 ≤ x; 0 ≤ y |] ==> (0::real) ≤ x + y"
    apply (drule order_le_imp_less_or_eq)+
    apply (auto intro: real_add_order order_less_imp_le)
    done

    lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
    apply (case_tac "x ≠ 0")
    apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
    done

    lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
    by (auto dest: less_imp_inverse_less)

    lemma real_mult_self_sum_ge_zero: "(0::real) ≤ x*x + y*y"
    proof -
      have "0 + 0 ≤ x*x + y*y" by (blast intro: add_mono zero_le_square)
      thus ?thesis by simp
    qed

    subsection{*Embedding the Integers into the Reals*}

    defs (overloaded)
      real_of_nat_def: "real z == of_nat z"
      real_of_int_def: "real z == of_int z"

    lemma real_eq_of_nat: "real = of_nat"
      apply (rule ext)
      apply (unfold real_of_nat_def)
      apply (rule refl)
      done

    lemma real_eq_of_int: "real = of_int"
      apply (rule ext)
      apply (unfold real_of_int_def)
      apply (rule refl)
      done

    lemma real_of_int_zero [simp]: "real (0::int) = 0"
    by (simp add: real_of_int_def)

    lemma real_of_one [simp]: "real (1::int) = (1::real)"
    by (simp add: real_of_int_def)

    lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
    by (simp add: real_of_int_def)

    lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
    by (simp add: real_of_int_def)

    lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
    by (simp add: real_of_int_def)

    lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
    by (simp add: real_of_int_def)

    lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
      apply (subst real_eq_of_int)+
      apply (rule of_int_setsum)
    done

    lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
        (PROD x:A. real(f x))"
      apply (subst real_eq_of_int)+
      apply (rule of_int_setprod)
    done

    lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
    by (simp add: real_of_int_def)

    lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
    by (simp add: real_of_int_def)

    lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
    by (simp add: real_of_int_def)

    lemma real_of_int_le_iff [simp]: "(real (x::int) ≤ real y) = (x ≤ y)"
    by (simp add: real_of_int_def)

    lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
    by (simp add: real_of_int_def)

    lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
    by (simp add: real_of_int_def)

    lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
    by (simp add: real_of_int_def)

    lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
    by (simp add: real_of_int_def)

    lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
    by (auto simp add: abs_if)

    lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
      apply (subgoal_tac "real n + 1 = real (n + 1)")
      apply (simp del: real_of_int_add)
      apply auto
    done

    lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
      apply (subgoal_tac "real m + 1 = real (m + 1)")
      apply (simp del: real_of_int_add)
      apply simp
    done

    lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) =
        real (x div d) + (real (x mod d)) / (real d)"
    proof -
      assume "d ~= 0"
      have "x = (x div d) * d + x mod d"
        by auto
      then have "real x = real (x div d) * real d + real(x mod d)"
        by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
      then have "real x / real d = ... / real d"
        by simp
      then show ?thesis
        by (auto simp add: add_divide_distrib ring_eq_simps prems)
    qed

    lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
        real(n div d) = real n / real d"
      apply (frule real_of_int_div_aux [of d n])
      apply simp
      apply (simp add: zdvd_iff_zmod_eq_0)
    done

    lemma real_of_int_div2:
      "0 <= real (n::int) / real (x) - real (n div x)"
      apply (case_tac "x = 0")
      apply simp
      apply (case_tac "0 < x")
      apply (simp add: compare_rls)
      apply (subst real_of_int_div_aux)
      apply simp
      apply simp
      apply (subst zero_le_divide_iff)
      apply auto
      apply (simp add: compare_rls)
      apply (subst real_of_int_div_aux)
      apply simp
      apply simp
      apply (subst zero_le_divide_iff)
      apply auto
    done

    lemma real_of_int_div3:
      "real (n::int) / real (x) - real (n div x) <= 1"
      apply(case_tac "x = 0")
      apply simp
      apply (simp add: compare_rls)
      apply (subst real_of_int_div_aux)
      apply assumption
      apply simp
      apply (subst divide_le_eq)
      apply clarsimp
      apply (rule conjI)
      apply (rule impI)
      apply (rule order_less_imp_le)
      apply simp
      apply (rule impI)
      apply (rule order_less_imp_le)
      apply simp
    done

    lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
      by (insert real_of_int_div2 [of n x], simp)

    subsection{*Embedding the Naturals into the Reals*}

    lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
    by (simp add: real_of_nat_def)

    (*Not for addsimps: often the LHS is used to represent a positive natural*)
    lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_less_iff [iff]:
         "(real (n::nat) < real m) = (n < m)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_le_iff [iff]: "(real (n::nat) ≤ real m) = (n ≤ m)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_ge_zero [iff]: "0 ≤ real (n::nat)"
    by (simp add: real_of_nat_def zero_le_imp_of_nat)

    lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
    by (simp add: real_of_nat_def del: of_nat_Suc)

    lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
        (SUM x:A. real(f x))"
      apply (subst real_eq_of_nat)+
      apply (rule of_nat_setsum)
    done

    lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
        (PROD x:A. real(f x))"
      apply (subst real_eq_of_nat)+
      apply (rule of_nat_setprod)
    done

    lemma real_of_card: "real (card A) = setsum (%x.1) A"
      apply (subst card_eq_setsum)
      apply (subst real_of_nat_setsum)
      apply simp
    done

    lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
    by (simp add: real_of_nat_def)

    lemma real_of_nat_diff: "n ≤ m ==> real (m - n) = real (m::nat) - real n"
    by (simp add: add: real_of_nat_def)

    lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
    by (simp add: add: real_of_nat_def)

    lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) ≤ 0) = (n = 0)"
    by (simp add: add: real_of_nat_def)

    lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
    by (simp add: add: real_of_nat_def)

    lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 ≤ real (n::nat)) = (0 ≤ n)"
    by (simp add: add: real_of_nat_def)

    lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
      apply (subgoal_tac "real n + 1 = real (Suc n)")
      apply simp
      apply (auto simp add: real_of_nat_Suc)
    done

    lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
      apply (subgoal_tac "real m + 1 = real (Suc m)")
      apply (simp add: less_Suc_eq_le)
      apply (simp add: real_of_nat_Suc)
    done

    lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) =
        real (x div d) + (real (x mod d)) / (real d)"
    proof -
      assume "0 < d"
      have "x = (x div d) * d + x mod d"
        by auto
      then have "real x = real (x div d) * real d + real(x mod d)"
        by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
      then have "real x / real d = … / real d"
        by simp
      then show ?thesis
        by (auto simp add: add_divide_distrib ring_eq_simps prems)
    qed

    lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
        real(n div d) = real n / real d"
      apply (frule real_of_nat_div_aux [of d n])
      apply simp
      apply (subst dvd_eq_mod_eq_0 [THEN sym])
      apply assumption
    done

    lemma real_of_nat_div2:
      "0 <= real (n::nat) / real (x) - real (n div x)"
      apply(case_tac "x = 0")
      apply simp
      apply (simp add: compare_rls)
      apply (subst real_of_nat_div_aux)
      apply assumption
      apply simp
      apply (subst zero_le_divide_iff)
      apply simp
    done

    lemma real_of_nat_div3:
      "real (n::nat) / real (x) - real (n div x) <= 1"
      apply(case_tac "x = 0")
      apply simp
      apply (simp add: compare_rls)
      apply (subst real_of_nat_div_aux)
      apply assumption
      apply simp
    done

    lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
      by (insert real_of_nat_div2 [of n x], simp)

    lemma real_of_int_real_of_nat: "real (int n) = real n"
    by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)

    lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
    by (simp add: real_of_int_def real_of_nat_def)

    lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
      apply (subgoal_tac "real(int(nat x)) = real(nat x)")
      apply force
      apply (simp only: real_of_int_real_of_nat)
    done

    subsection{*Numerals and Arithmetic*}

    instance real :: number ..

    defs (overloaded)
      real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)"
        --{*the type constraint is essential!*}

    instance real :: number_ring
    by (intro_classes, simp add: real_number_of_def)

    text{*Collapse applications of @{term real} to @{term number_of}*}
    lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
    by (simp add:  real_of_int_def of_int_number_of_eq)

    lemma real_of_nat_number_of [simp]:
         "real (number_of v :: nat) =
            (if neg (number_of v :: int) then 0
             else (number_of v :: real))"
    by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)

    use "real_arith.ML"

    setup real_arith_setup

    subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}

    text{*Needed in this non-standard form by Hyperreal/Transcendental*}
    lemma real_0_le_divide_iff:
         "((0::real) ≤ x/y) = ((x ≤ 0 | 0 ≤ y) & (0 ≤ x | y ≤ 0))"
    by (simp add: real_divide_def zero_le_mult_iff, auto)

    lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
    by arith

    lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
    by auto

    lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
    by auto

    lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
    by auto

    lemma real_add_le_0_iff: "(x+y ≤ (0::real)) = (y ≤ -x)"
    by auto

    lemma real_0_le_add_iff: "((0::real) ≤ x+y) = (-x ≤ y)"
    by auto

    (*
    FIXME: we should have this, as for type int, but many proofs would break.
    It replaces x+-y by x-y.
    declare real_diff_def [symmetric, simp]
    *)

    subsubsection{*Density of the Reals*}

    lemma real_lbound_gt_zero:
         "[| (0::real) < d1; 0 < d2 |] ==> ∃e. 0 < e & e < d1 & e < d2"
    apply (rule_tac x = " (min d1 d2) /2" in exI)
    apply (simp add: min_def)
    done

    text{*Similar results are proved in @{text Ring_and_Field}*}
    lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
      by auto

    lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
      by auto

    subsection{*Absolute Value Function for the Reals*}

    lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
    by (simp add: abs_if)

    lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
    by (force simp add: Ring_and_Field.abs_less_iff)

    lemma abs_le_interval_iff: "(abs x ≤ r) = (-r≤x & x≤(r::real))"
    by (force simp add: OrderedGroup.abs_le_iff)

    (*FIXME: used only once, in SEQ.ML*)
    lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
    by (simp add: abs_if)

    lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
    by (simp add: real_of_nat_ge_zero)

    lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
    apply (simp add: linorder_not_less)
    apply (auto intro: abs_ge_self [THEN order_trans])
    done

    text{*Used only in Hyperreal/Lim.ML*}
    lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) ≤ abs(x + -l) + abs(y + -m)"
    apply (simp add: real_add_assoc)
    apply (rule_tac a1 = y in add_left_commute [THEN ssubst])
    apply (rule real_add_assoc [THEN subst])
    apply (rule abs_triangle_ineq)
    done

    ML
    {*
    val real_lbound_gt_zero = thm"real_lbound_gt_zero";
    val real_less_half_sum = thm"real_less_half_sum";
    val real_gt_half_sum = thm"real_gt_half_sum";

    val abs_interval_iff = thm"abs_interval_iff";
    val abs_le_interval_iff = thm"abs_le_interval_iff";
    val abs_add_one_gt_zero = thm"abs_add_one_gt_zero";
    val abs_add_one_not_less_self = thm"abs_add_one_not_less_self";
    val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq";
    *}

    end

### Proving that realrel is an equivalence relation

lemma preal_trans_lemma:

      [| x + y1.0 = x1.0 + y; x + y2.0 = x2.0 + y |] ==> x1.0 + y2.0 = x2.0 + y1.0

lemma realrel_iff:

      (((x1.0, y1.0), x2.0, y2.0) ∈ realrel) = (x1.0 + y2.0 = x2.0 + y1.0)

lemma equiv_realrel:

      equiv UNIV realrel

lemmas equiv_realrel_iff:

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

lemmas equiv_realrel_iff:

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

lemma realrel_in_real:

      realrel `` {(x, y)} ∈ Real

lemma inj_on_Abs_Real:

      inj_on Abs_Real Real

lemma eq_Abs_Real:

      (!!x y. z = Abs_Real (realrel `` {(x, y)}) ==> P) ==> P

### Congruence property for addition

lemma real_add_congruent2_lemma:

      [| a + ba = aa + b; ab + bc = ac + bb |]
      ==> a + ab + (ba + bc) = aa + ac + (b + bb)

lemma real_add:

      Abs_Real (realrel `` {(x, y)}) + Abs_Real (realrel `` {(u, v)}) =
      Abs_Real (realrel `` {(x + u, y + v)})

lemma real_add_commute:

      z + w = w + z

lemma real_add_assoc:

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

lemma real_add_zero_left:

      0 + z = z

### Additive Inverse on real

lemma real_minus:

      - Abs_Real (realrel `` {(x, y)}) = Abs_Real (realrel `` {(y, x)})

lemma real_add_minus_left:

      - z + z = 0

### Congruence property for multiplication

lemma real_mult_congruent2_lemma:

      x1.0 + y2.0 = x2.0 + y1.0
      ==> x * x1.0 + y * y1.0 + (x * y2.0 + y * x2.0) =
          x * x2.0 + y * y2.0 + (x * y1.0 + y * x1.0)

lemma real_mult_congruent2:

      congruent2 realrel realrel
       (%p1 p2.
           (%(x1, y1).
               (%(x2, y2).
                   {Abs_Real (realrel `` {(x1 * x2 + y1 * y2, x1 * y2 + y1 * x2)})})
                p2)
            p1)

lemma real_mult:

      Abs_Real (realrel `` {(x1.0, y1.0)}) * Abs_Real (realrel `` {(x2.0, y2.0)}) =
      Abs_Real (realrel `` {(x1.0 * x2.0 + y1.0 * y2.0, x1.0 * y2.0 + y1.0 * x2.0)})

lemma real_mult_commute:

      z * w = w * z

lemma real_mult_assoc:

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

lemma real_mult_1:

      1 * z = z

lemma real_add_mult_distrib:

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

lemma real_zero_not_eq_one:

      0 ≠ 1

### existence of inverse

lemma real_zero_iff:

      Abs_Real (realrel `` {(x, x)}) = 0

lemma real_mult_inverse_left_ex:

      x ≠ 0 ==> ∃y. y * x = 1

lemma real_mult_inverse_left:

      x ≠ 0 ==> inverse x * x = 1

### The Real Numbers form a Field

lemma INVERSE_ZERO:

      inverse 0 = 0

lemma real_mult_1_right:

      z * 1 = z

### The @{text "≤"} Ordering

lemma real_le_refl:

      w ≤ w

lemma preal_eq_le_imp_le:

      [| a + b = c + d; c ≤ a |] ==> b ≤ d

lemma real_le_lemma:

      [| u1.0 + v2.0 ≤ u2.0 + v1.0; x1.0 + v1.0 = u1.0 + y1.0;
         x2.0 + v2.0 = u2.0 + y2.0 |]
      ==> x1.0 + y2.0 ≤ x2.0 + y1.0

lemma real_le:

      (Abs_Real (realrel `` {(x1.0, y1.0)}) ≤ Abs_Real (realrel `` {(x2.0, y2.0)})) =
      (x1.0 + y2.0 ≤ x2.0 + y1.0)

lemma real_le_anti_sym:

      [| z ≤ w; w ≤ z |] ==> z = w

lemma real_trans_lemma:

      [| x + v ≤ u + y; u + v' ≤ u' + v; x2.0 + v2.0 = u2.0 + y2.0 |]
      ==> x + v' ≤ u' + y

lemma real_le_trans:

      [| i ≤ j; j ≤ k |] ==> i ≤ k

lemma real_less_le:

      (w < z) = (w ≤ z ∧ w ≠ z)

lemma real_le_linear:

      z ≤ w ∨ w ≤ z

lemma real_le_eq_diff:

      (x ≤ y) = (x - y ≤ 0)

lemma real_add_left_mono:

      x ≤ y ==> z + x ≤ z + y

lemma real_sum_gt_zero_less:

      0 < S + - W ==> W < S

lemma real_less_sum_gt_zero:

      W < S ==> 0 < S + - W

lemma real_mult_order:

      [| 0 < x; 0 < y |] ==> 0 < x * y

lemma real_mult_less_mono2:

      [| 0 < z; x < y |] ==> z * x < z * y

lemma real_zero_le_one:

      0 ≤ 1

### The Reals Form an Ordered Field

lemma real_of_preal_add:

      real_of_preal (x + y) = real_of_preal x + real_of_preal y

lemma real_of_preal_mult:

      real_of_preal (x * y) = real_of_preal x * real_of_preal y

lemma real_of_preal_trichotomy:

      ∃m. x = real_of_preal m ∨ x = 0 ∨ x = - real_of_preal m

lemma real_of_preal_leD:

      real_of_preal m1.0 ≤ real_of_preal m2.0 ==> m1.0 ≤ m2.0

lemma real_of_preal_lessI:

      m1.0 < m2.0 ==> real_of_preal m1.0 < real_of_preal m2.0

lemma real_of_preal_lessD:

      real_of_preal m1.0 < real_of_preal m2.0 ==> m1.0 < m2.0

lemma real_of_preal_less_iff:

      (real_of_preal m1.0 < real_of_preal m2.0) = (m1.0 < m2.0)

lemma real_of_preal_le_iff:

      (real_of_preal m1.0 ≤ real_of_preal m2.0) = (m1.0 ≤ m2.0)

lemma real_of_preal_zero_less:

      0 < real_of_preal m

lemma real_of_preal_minus_less_zero:

      - real_of_preal m < 0

lemma real_of_preal_not_minus_gt_zero:

      ¬ 0 < - real_of_preal m

### Theorems About the Ordering

lemma real_not_refl2:

      x < y ==> x ≠ y

lemma real_le_imp_less_or_eq:

      x ≤ y ==> x < y ∨ x = y

lemma real_gt_zero_preal_Ex:

      (0 < x) = (∃y. x = real_of_preal y)

lemma real_gt_preal_preal_Ex:

      real_of_preal z < x ==> ∃y. x = real_of_preal y

lemma real_ge_preal_preal_Ex:

      real_of_preal z ≤ x ==> ∃y. x = real_of_preal y

lemma real_less_all_preal:

      y ≤ 0 ==> ∀x. y < real_of_preal x

lemma real_less_all_real2:

      ¬ 0 < y ==> ∀x. y < real_of_preal x

lemma real_add_less_le_mono:

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

lemma real_add_le_less_mono:

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

lemma real_le_square:

      0 ≤ x * x

### More Lemmas

lemma real_mult_left_cancel:

      c ≠ 0 ==> (c * a = c * b) = (a = b)

lemma real_mult_right_cancel:

      c ≠ 0 ==> (a * c = b * c) = (a = b)

lemma real_mult_less_mono:

      [| u < v; x < y; 0 < v; 0 < x |] ==> u * x < v * y

lemma real_mult_less_iff1:

      0 < z ==> (x * z < y * z) = (x < y)

lemma real_mult_le_cancel_iff1:

      0 < z ==> (x * z ≤ y * z) = (x ≤ y)

lemma real_mult_le_cancel_iff2:

      0 < z ==> (z * x ≤ z * y) = (x ≤ y)

lemma real_mult_less_mono':

      [| x < y; r1.0 < r2.0; 0 ≤ r1.0; 0 ≤ x |] ==> r1.0 * x < r2.0 * y

lemma real_sum_squares_cancel:

      x * x + y * y = 0 ==> x = 0

lemma real_sum_squares_cancel2:

      x * x + y * y = 0 ==> y = 0

lemma real_add_order:

      [| 0 < x; 0 < y |] ==> 0 < x + y

lemma real_le_add_order:

      [| 0 ≤ x; 0 ≤ y |] ==> 0 ≤ x + y

lemma real_inverse_unique:

      x * y = 1 ==> y = inverse x

lemma real_inverse_gt_one:

      [| 0 < x; x < 1 |] ==> 1 < inverse x

lemma real_mult_self_sum_ge_zero:

      0 ≤ x * x + y * y

### Embedding the Integers into the Reals

lemma real_eq_of_nat:

      real = of_nat

lemma real_eq_of_int:

      real = of_int

lemma real_of_int_zero:

      real 0 = 0

lemma real_of_one:

      real 1 = 1

lemma real_of_int_add:

      real (x + y) = real x + real y

lemma real_of_int_minus:

      real (- x) = - real x

lemma real_of_int_diff:

      real (x - y) = real x - real y

lemma real_of_int_mult:

      real (x * y) = real x * real y

lemma real_of_int_setsum:

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

lemma real_of_int_setprod:

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

lemma real_of_int_zero_cancel:

      (real x = 0) = (x = 0)

lemma real_of_int_inject:

      (real x = real y) = (x = y)

lemma real_of_int_less_iff:

      (real x < real y) = (x < y)

lemma real_of_int_le_iff:

      (real x ≤ real y) = (x ≤ y)

lemma real_of_int_gt_zero_cancel_iff:

      (0 < real n) = (0 < n)

lemma real_of_int_ge_zero_cancel_iff:

      (0 ≤ real n) = (0 ≤ n)

lemma real_of_int_lt_zero_cancel_iff:

      (real n < 0) = (n < 0)

lemma real_of_int_le_zero_cancel_iff:

      (real n ≤ 0) = (n ≤ 0)

lemma real_of_int_abs:

      real ¦x¦ = ¦real x¦

lemma int_less_real_le:

      (n < m) = (real n + 1 ≤ real m)

lemma int_le_real_less:

      (n ≤ m) = (real n < real m + 1)

lemma real_of_int_div_aux:

      d ≠ 0 ==> real x / real d = real (x div d) + real (x mod d) / real d

lemma real_of_int_div:

      [| d ≠ 0; d dvd n |] ==> real (n div d) = real n / real d

lemma real_of_int_div2:

      0 ≤ real n / real x - real (n div x)

lemma real_of_int_div3:

      real n / real x - real (n div x) ≤ 1

lemma real_of_int_div4:

      real (n div x) ≤ real n / real x

### Embedding the Naturals into the Reals

lemma real_of_nat_zero:

      real 0 = 0

lemma real_of_nat_one:

      real (Suc 0) = 1

lemma real_of_nat_add:

      real (m + n) = real m + real n

lemma real_of_nat_Suc:

      real (Suc n) = real n + 1

lemma real_of_nat_less_iff:

      (real n < real m) = (n < m)

lemma real_of_nat_le_iff:

      (real n ≤ real m) = (n ≤ m)

lemma real_of_nat_ge_zero:

      0 ≤ real n

lemma real_of_nat_Suc_gt_zero:

      0 < real (Suc n)

lemma real_of_nat_mult:

      real (m * n) = real m * real n

lemma real_of_nat_setsum:

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

lemma real_of_nat_setprod:

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

lemma real_of_card:

      real (card A) = (∑x∈A. 1)

lemma real_of_nat_inject:

      (real n = real m) = (n = m)

lemma real_of_nat_zero_iff:

      (real n = 0) = (n = 0)

lemma real_of_nat_diff:

      n ≤ m ==> real (m - n) = real m - real n

lemma real_of_nat_gt_zero_cancel_iff:

      (0 < real n) = (0 < n)

lemma real_of_nat_le_zero_cancel_iff:

      (real n ≤ 0) = (n = 0)

lemma not_real_of_nat_less_zero:

      ¬ real n < 0

lemma real_of_nat_ge_zero_cancel_iff:

      (0 ≤ real n) = (0 ≤ n)

lemma nat_less_real_le:

      (n < m) = (real n + 1 ≤ real m)

lemma nat_le_real_less:

      (n ≤ m) = (real n < real m + 1)

lemma real_of_nat_div_aux:

      0 < d ==> real x / real d = real (x div d) + real (x mod d) / real d

lemma real_of_nat_div:

      [| 0 < d; d dvd n |] ==> real (n div d) = real n / real d

lemma real_of_nat_div2:

      0 ≤ real n / real x - real (n div x)

lemma real_of_nat_div3:

      real n / real x - real (n div x) ≤ 1

lemma real_of_nat_div4:

      real (n div x) ≤ real n / real x

lemma real_of_int_real_of_nat:

      real (int n) = real n

lemma real_of_int_of_nat_eq:

      real (of_nat n) = real n

lemma real_nat_eq_real:

      0 ≤ x ==> real (nat x) = real x

### Numerals and Arithmetic

lemma real_number_of:

      real (number_of v) = number_of v

lemma real_of_nat_number_of:

      real (number_of v) = (if neg (number_of v) then 0 else number_of v)

### Simprules combining x+y and 0: ARE THEY NEEDED?

lemma real_0_le_divide_iff:

      (0 ≤ x / y) = ((x ≤ 0 ∨ 0 ≤ y) ∧ (0 ≤ x ∨ y ≤ 0))

lemma real_add_minus_iff:

      (x + - a = 0) = (x = a)

lemma real_add_eq_0_iff:

      (x + y = 0) = (y = - x)

lemma real_add_less_0_iff:

      (x + y < 0) = (y < - x)

lemma real_0_less_add_iff:

      (0 < x + y) = (- x < y)

lemma real_add_le_0_iff:

      (x + y ≤ 0) = (y ≤ - x)

lemma real_0_le_add_iff:

      (0 ≤ x + y) = (- x ≤ y)

#### Density of the Reals

lemma real_lbound_gt_zero:

      [| 0 < d1.0; 0 < d2.0 |] ==> ∃e>0. e < d1.0 ∧ e < d2.0

lemma real_less_half_sum:

      x < y ==> x < (x + y) / 2

lemma real_gt_half_sum:

      x < y ==> (x + y) / 2 < y

### Absolute Value Function for the Reals

lemma abs_minus_add_cancel:

      ¦x + - y¦ = ¦y + - x¦

lemma abs_interval_iff:

      (¦x¦ < r) = (- r < x ∧ x < r)

lemma abs_le_interval_iff:

      (¦x¦ ≤ r) = (- r ≤ x ∧ x ≤ r)

lemma abs_add_one_gt_zero:

      0 < 1 + ¦x¦

lemma abs_real_of_nat_cancel:

      ¦real x¦ = real x

lemma abs_add_one_not_less_self:

      ¬ ¦x¦ + 1 < x

lemma abs_sum_triangle_ineq:

      ¦x + y + (- l + - m)¦ ≤ ¦x + - l¦ + ¦y + - m¦