Theory Dedekind_Real

section ‹The Reals as Dedekind Sections of Positive Rationals›

text ‹Fundamentals of Abstract Analysis [Gleason, p. 121] provides some of the definitions.›

theory Dedekind_Real
imports Complex_Main 

lemma add_eq_exists: "x. a+x = (b::'a::ab_group_add)"
  by (rule_tac x="b-a" in exI, simp)

subsection ‹Dedekind cuts or sections›

  cut :: "rat set  bool" where
  "cut A  {}  A  A  {0<..} 
            (y  A. ((z. 0<z  z < y  z  A)  (u  A. y < u)))"

lemma cut_of_rat: 
  assumes q: "0 < q" shows "cut {r::rat. 0 < r  r < q}" (is "cut ?A")
proof -
  from q have pos: "?A  {0<..}" by force
  have nonempty: "{}  ?A"
    show "{}  ?A" by simp
    show "{}  ?A"
      using field_lbound_gt_zero q by auto
  show ?thesis
    by (simp add: cut_def pos nonempty,
        blast dest: dense intro: order_less_trans)

typedef preal = "Collect cut"
  by (blast intro: cut_of_rat [OF zero_less_one])

lemma Abs_preal_induct [induct type: preal]:
  "(x. cut x  P (Abs_preal x))  P x"
  using Abs_preal_induct [of P x] by simp

lemma cut_Rep_preal [simp]: "cut (Rep_preal x)"
  using Rep_preal [of x] by simp

  psup :: "preal set  preal" where
  "psup P = Abs_preal (X  P. Rep_preal X)"

  add_set :: "[rat set,rat set]  rat set" where
  "add_set A B = {w. x  A. y  B. w = x + y}"

  diff_set :: "[rat set,rat set]  rat set" where
  "diff_set A B = {w. x. 0 < w  0 < x  x  B  x + w  A}"

  mult_set :: "[rat set,rat set]  rat set" where
  "mult_set A B = {w. x  A. y  B. w = x * y}"

  inverse_set :: "rat set  rat set" where
  "inverse_set A  {x. y. 0 < x  x < y  inverse y  A}"

instantiation preal :: "{ord, plus, minus, times, inverse, one}"

    "r < s  Rep_preal r < Rep_preal s"

    "r  s  Rep_preal r  Rep_preal s"

    "r + s  Abs_preal (add_set (Rep_preal r) (Rep_preal s))"

    "r - s  Abs_preal (diff_set (Rep_preal r) (Rep_preal s))"

    "r * s  Abs_preal (mult_set (Rep_preal r) (Rep_preal s))"

    "inverse r  Abs_preal (inverse_set (Rep_preal r))"

definition "r div s = r * inverse (s::preal)"

    "1  Abs_preal {x. 0 < x  x < 1}"

instance ..


text‹Reduces equality on abstractions to equality on representatives›
declare Abs_preal_inject [simp]
declare Abs_preal_inverse [simp]

lemma rat_mem_preal: "0 < q  cut {r::rat. 0 < r  r < q}"
by (simp add: cut_of_rat)

lemma preal_nonempty: "cut A  xA. 0 < x"
  unfolding cut_def [abs_def] by blast

lemma preal_Ex_mem: "cut A  x. x  A"
  using preal_nonempty by blast

lemma preal_exists_bound: "cut A  x. 0 < x  x  A"
  using Dedekind_Real.cut_def by fastforce

lemma preal_exists_greater: "cut A; y  A  u  A. y < u"
  unfolding cut_def [abs_def] by blast

lemma preal_downwards_closed: "cut A; y  A; 0 < z; z < y  z  A"
  unfolding cut_def [abs_def] by blast

text‹Relaxing the final premise›
lemma preal_downwards_closed': "cut A; y  A; 0 < z; z  y  z  A"
  using less_eq_rat_def preal_downwards_closed by blast

text‹A positive fraction not in a positive real is an upper bound.
 Gleason p. 122 - Remark (1)›

lemma not_in_preal_ub:
  assumes A: "cut A"
    and notx: "x  A"
    and y: "y  A"
    and pos: "0 < x"
  shows "y < x"
proof (cases rule: linorder_cases)
  assume "x<y"
  with notx show ?thesis
    by (simp add:  preal_downwards_closed [OF A y] pos)
  assume "x=y"
  with notx and y show ?thesis by simp
  assume "y<x"
  thus ?thesis .

text ‹preal lemmas instantiated to termRep_preal X

lemma mem_Rep_preal_Ex: "x. x  Rep_preal X"
thm preal_Ex_mem
by (rule preal_Ex_mem [OF cut_Rep_preal])

lemma Rep_preal_exists_bound: "x>0. x  Rep_preal X"
by (rule preal_exists_bound [OF cut_Rep_preal])

lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF cut_Rep_preal]

subsection‹Properties of Ordering›

instance preal :: order
  fix w :: preal
  show "w  w" by (simp add: preal_le_def)
  fix i j k :: preal
  assume "i  j" and "j  k"
  then show "i  k" by (simp add: preal_le_def)
  fix z w :: preal
  assume "z  w" and "w  z"
  then show "z = w" by (simp add: preal_le_def Rep_preal_inject)
  fix z w :: preal
  show "z < w  z  w  ¬ w  z"
  by (auto simp: preal_le_def preal_less_def Rep_preal_inject)

lemma preal_imp_pos: "cut A; r  A  0 < r"
  by (auto simp: cut_def)

instance preal :: linorder
  fix x y :: preal
  show "x  y  y  x"
    unfolding preal_le_def
    by (meson cut_Rep_preal not_in_preal_ub preal_downwards_closed preal_imp_pos subsetI)

instantiation preal :: distrib_lattice

  "(inf :: preal  preal  preal) = min"

  "(sup :: preal  preal  preal) = max"

  by intro_classes
    (auto simp: inf_preal_def sup_preal_def max_min_distrib2)


subsection‹Properties of Addition›

lemma preal_add_commute: "(x::preal) + y = y + x"
  unfolding preal_add_def add_set_def
  by (metis (no_types, opaque_lifting) add.commute)

text‹Lemmas for proving that addition of two positive reals gives
 a positive real›

lemma mem_add_set:
  assumes "cut A" "cut B"
  shows "cut (add_set A B)"
proof -
  have "{}  add_set A B"
    using assms by (force simp: add_set_def dest: preal_nonempty)
  obtain q where "q > 0" "q  add_set A B"
  proof -
    obtain a b where "a > 0" "a  A" "b > 0" "b  B" "x. x  A  x < a" "y. y  B  y < b"
      by (meson assms preal_exists_bound not_in_preal_ub)
    with assms have "a+b  add_set A B"
      by (fastforce simp add: add_set_def)
    then show thesis
      using 0 < a 0 < b add_pos_pos that by blast
  then have "add_set A B  {0<..}"
    unfolding add_set_def
    using preal_imp_pos [OF cut A] preal_imp_pos [OF cut B]  by fastforce
  moreover have "z  add_set A B" 
    if u: "u  add_set A B" and "0 < z" "z < u" for u z
    using u unfolding add_set_def
  proof (clarify)
    fix x::rat and y::rat
    assume ueq: "u = x + y" and x: "x  A" and y:"y  B"
    have xpos [simp]: "x > 0" and ypos [simp]: "y > 0"
      using assms preal_imp_pos x y by blast+
    have xypos [simp]: "x+y > 0" by (simp add: pos_add_strict)
    let ?f = "z/(x+y)"
    have fless: "?f < 1"
      using divide_less_eq_1_pos z < u ueq xypos by blast
    show "x'  A. y'B. z = x' + y'"
    proof (intro bexI)
      show "z = x*?f + y*?f"
        by (simp add: distrib_right [symmetric] divide_inverse ac_simps order_less_imp_not_eq2)
      show "y * ?f  B"
      proof (rule preal_downwards_closed [OF cut B y])
        show "0 < y * ?f"
          by (simp add: 0 < z)
        show "y * ?f < y"
          by (insert mult_strict_left_mono [OF fless ypos], simp)
      show "x * ?f  A"
      proof (rule preal_downwards_closed [OF cut A x])
        show "0 < x * ?f"
          by (simp add: 0 < z)
        show "x * ?f < x"
          by (insert mult_strict_left_mono [OF fless xpos], simp)
  have "y. y  add_set A B  u  add_set A B. y < u"
    unfolding add_set_def using preal_exists_greater assms by fastforce
  ultimately show ?thesis
    by (simp add: Dedekind_Real.cut_def)

lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
  apply (simp add: preal_add_def mem_add_set)
  apply (force simp: add_set_def ac_simps)

instance preal :: ab_semigroup_add
  fix a b c :: preal
  show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc)
  show "a + b = b + a" by (rule preal_add_commute)

subsection‹Properties of Multiplication›

text‹Proofs essentially same as for addition›

lemma preal_mult_commute: "(x::preal) * y = y * x"
  unfolding preal_mult_def mult_set_def
  by (metis (no_types, opaque_lifting) mult.commute)

text‹Multiplication of two positive reals gives a positive real.›

lemma mem_mult_set:
  assumes "cut A" "cut B"
  shows "cut (mult_set A B)"
proof -
  have "{}  mult_set A B"
    using assms
      by (force simp: mult_set_def dest: preal_nonempty)
    obtain q where "q > 0" "q  mult_set A B"
    proof -
      obtain x y where x [simp]: "0 < x" "x  A" and y [simp]: "0 < y" "y  B"
        using preal_exists_bound assms by blast
      show thesis
        show "0 < x*y" by simp
        show "x * y  mult_set A B"
        proof -
            fix u::rat and v::rat
            assume u: "u  A" and v: "v  B" and xy: "x*y = u*v"
            moreover have "u<x" and "v<y" using assms x y u v by (blast dest: not_in_preal_ub)+
            moreover have "0v"
              using less_imp_le preal_imp_pos assms x y u v by blast
            moreover have "u*v < x*y"
                using assms x u < x v < y 0  v by (blast intro: mult_strict_mono)
            ultimately have False by force
          thus ?thesis by (auto simp: mult_set_def)
  then have "mult_set A B  {0<..}"
    unfolding mult_set_def
    using preal_imp_pos [OF cut A] preal_imp_pos [OF cut B]  by fastforce
  moreover have "z  mult_set A B"
    if u: "u  mult_set A B" and "0 < z" "z < u" for u z
    using u unfolding mult_set_def
  proof (clarify)
    fix x::rat and y::rat
    assume ueq: "u = x * y" and x: "x  A" and y: "y  B"  
    have [simp]: "y > 0"
      using cut B preal_imp_pos y by blast
    show "x'  A. y'  B. z = x' * y'"
      have "z = (z/y)*y"
          by (simp add: divide_inverse mult.commute [of y] mult.assoc order_less_imp_not_eq2)
      then show "y'B. z = (z/y) * y'"
        using y by blast
      show "z/y  A"
      proof (rule preal_downwards_closed [OF cut A x])
        show "0 < z/y"
          by (simp add: 0 < z)
        show "z/y < x"
          using 0 < y pos_divide_less_eq z < u ueq by blast  
  moreover have "y. y  mult_set A B  u  mult_set A B. y < u"
    apply (simp add: mult_set_def)
    by (metis preal_exists_greater mult_strict_right_mono preal_imp_pos assms)
  ultimately show ?thesis
    by (simp add: Dedekind_Real.cut_def)

lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
  apply (simp add: preal_mult_def mem_mult_set Rep_preal)
  apply (simp add: mult_set_def)
  apply (metis (no_types, opaque_lifting) ab_semigroup_mult_class.mult_ac(1))

instance preal :: ab_semigroup_mult
  fix a b c :: preal
  show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc)
  show "a * b = b * a" by (rule preal_mult_commute)

text‹Positive real 1 is the multiplicative identity element›

lemma preal_mult_1: "(1::preal) * z = z"
proof (induct z)
  fix A :: "rat set"
  assume A: "cut A"
  have "{w. u. 0 < u  u < 1  (v  A. w = u * v)} = A" (is "?lhs = A")
    show "?lhs  A"
    proof clarify
      fix x::rat and u::rat and v::rat
      assume upos: "0<u" and "u<1" and v: "v  A"
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
      hence "u*v < 1*v" by (simp only: mult_strict_right_mono upos u < 1 v)
      thus "u * v  A"
        by (force intro: preal_downwards_closed [OF A v] mult_pos_pos  upos vpos)
    show "A  ?lhs"
    proof clarify
      fix x::rat
      assume x: "x  A"
      have xpos: "0<x" by (rule preal_imp_pos [OF A x])
      from preal_exists_greater [OF A x]
      obtain v where v: "v  A" and xlessv: "x < v" ..
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
      show "u. 0 < u  u < 1  (vA. x = u * v)"
      proof (intro exI conjI)
        show "0 < x/v"
          by (simp add: zero_less_divide_iff xpos vpos)
        show "x / v < 1"
          by (simp add: pos_divide_less_eq vpos xlessv)
        have "x = (x/v)*v"
            by (simp add: divide_inverse mult.assoc vpos order_less_imp_not_eq2)
        then show "v'A. x = (x / v) * v'"
          using v by blast
  thus "1 * Abs_preal A = Abs_preal A"
    by (simp add: preal_one_def preal_mult_def mult_set_def rat_mem_preal A)

instance preal :: comm_monoid_mult
  by intro_classes (rule preal_mult_1)

subsection‹Distribution of Multiplication across Addition›

lemma mem_Rep_preal_add_iff:
  "(z  Rep_preal(r+s)) = (x  Rep_preal r. y  Rep_preal s. z = x + y)"
  apply (simp add: preal_add_def mem_add_set Rep_preal)
  apply (simp add: add_set_def) 

lemma mem_Rep_preal_mult_iff:
  "(z  Rep_preal(r*s)) = (x  Rep_preal r. y  Rep_preal s. z = x * y)"
  apply (simp add: preal_mult_def mem_mult_set Rep_preal)
  apply (simp add: mult_set_def) 

lemma distrib_subset1:
  "Rep_preal (w * (x + y))  Rep_preal (w * x + w * y)"
  by (force simp: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff distrib_left)

lemma preal_add_mult_distrib_mean:
  assumes a: "a  Rep_preal w"
    and b: "b  Rep_preal w"
    and d: "d  Rep_preal x"
    and e: "e  Rep_preal y"
  shows "c  Rep_preal w. a * d + b * e = c * (d + e)"
  let ?c = "(a*d + b*e)/(d+e)"
  have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
    by (blast intro: preal_imp_pos [OF cut_Rep_preal] a b d e pos_add_strict)+
  have cpos: "0 < ?c"
    by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
  show "a * d + b * e = ?c * (d + e)"
    by (simp add: divide_inverse mult.assoc order_less_imp_not_eq2)
  show "?c  Rep_preal w"
  proof (cases rule: linorder_le_cases)
    assume "a  b"
    hence "?c  b"
      by (simp add: pos_divide_le_eq distrib_left mult_right_mono
    thus ?thesis by (rule preal_downwards_closed' [OF cut_Rep_preal b cpos])
    assume "b  a"
    hence "?c  a"
      by (simp add: pos_divide_le_eq distrib_left mult_right_mono
    thus ?thesis by (rule preal_downwards_closed' [OF cut_Rep_preal a cpos])

lemma distrib_subset2:
  "Rep_preal (w * x + w * y)  Rep_preal (w * (x + y))"
  apply (clarsimp simp: mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
  using mem_Rep_preal_add_iff preal_add_mult_distrib_mean by blast

lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
  by (metis Rep_preal_inverse distrib_subset1 distrib_subset2 subset_antisym)

lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
  by (simp add: preal_mult_commute preal_add_mult_distrib2)

instance preal :: comm_semiring
  by intro_classes (rule preal_add_mult_distrib)

subsection‹Existence of Inverse, a Positive Real›

lemma mem_inverse_set:
  assumes "cut A" shows "cut (inverse_set A)"
proof -
  have "x y. 0 < x  x < y  inverse y  A"
  proof -
    from preal_exists_bound [OF cut A]
    obtain x where [simp]: "0<x" "x  A" by blast
    show ?thesis
    proof (intro exI conjI)
      show "0 < inverse (x+1)"
        by (simp add: order_less_trans [OF _ less_add_one]) 
      show "inverse(x+1) < inverse x"
        by (simp add: less_imp_inverse_less less_add_one)
      show "inverse (inverse x)  A"
        by (simp add: order_less_imp_not_eq2)
  then have "{}  inverse_set A"
    using inverse_set_def by fastforce
  moreover obtain q where "q > 0" "q  inverse_set A"
  proof -
    from preal_nonempty [OF cut A]
    obtain x where x: "x  A" and  xpos [simp]: "0<x" ..
    show ?thesis
      show "0 < inverse x" by simp
      show "inverse x  inverse_set A"
      proof -
        { fix y::rat 
          assume ygt: "inverse x < y"
          have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
          have iyless: "inverse y < x" 
            by (simp add: inverse_less_imp_less [of x] ygt)
          have "inverse y  A"
            by (simp add: preal_downwards_closed [OF cut A x] iyless)}
        thus ?thesis by (auto simp: inverse_set_def)
  moreover have "inverse_set A  {0<..}"
    using calculation inverse_set_def by blast
  moreover have "z  inverse_set A"
    if u: "u  inverse_set A" and "0 < z" "z < u" for u z
    using u that less_trans unfolding inverse_set_def by auto
  moreover have "y. y  inverse_set A  u  inverse_set A. y < u"
    by (simp add: inverse_set_def) (meson dense less_trans)
  ultimately show ?thesis
    by (simp add: Dedekind_Real.cut_def)

subsection‹Gleason's Lemma 9-3.4, page 122›

lemma Gleason9_34_exists:
  assumes A: "cut A"
    and "xA. x + u  A"
    and "0  z"
  shows "bA. b + (of_int z) * u  A"
proof (cases z rule: int_cases)
  case (nonneg n)
  show ?thesis
  proof (simp add: nonneg, induct n)
    case 0
    from preal_nonempty [OF A]
    show ?case  by force 
    case (Suc k)
    then obtain b where b: "b  A" "b + of_nat k * u  A" ..
    hence "b + of_int (int k)*u + u  A" by (simp add: assms)
    thus ?case by (force simp: algebra_simps b)
  case (neg n)
  with assms show ?thesis by simp

lemma Gleason9_34_contra:
  assumes A: "cut A"
    shows "xA. x + u  A; 0 < u; 0 < y; y  A  False"
proof (induct u, induct y)
  fix a::int and b::int
  fix c::int and d::int
  assume bpos [simp]: "0 < b"
    and dpos [simp]: "0 < d"
    and closed: "xA. x + (Fract c d)  A"
    and upos: "0 < Fract c d"
    and ypos: "0 < Fract a b"
    and notin: "Fract a b  A"
  have cpos [simp]: "0 < c" 
    by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos) 
  have apos [simp]: "0 < a" 
    by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) 
  let ?k = "a*d"
  have frle: "Fract a b  Fract ?k 1 * (Fract c d)" 
  proof -
    have "?thesis = ((a * d * b * d)  c * b * (a * d * b * d))"
      by (simp add: order_less_imp_not_eq2 ac_simps) 
    have "(1 * (a * d * b * d))  c * b * (a * d * b * d)"
      by (rule mult_mono, 
          simp_all add: int_one_le_iff_zero_less zero_less_mult_iff 
    show ?thesis by simp
  have k: "0  ?k" by (simp add: order_less_imp_le zero_less_mult_iff)  
  from Gleason9_34_exists [OF A closed k]
  obtain z where z: "z  A" 
             and mem: "z + of_int ?k * Fract c d  A" ..
  have less: "z + of_int ?k * Fract c d < Fract a b"
    by (rule not_in_preal_ub [OF A notin mem ypos])
  have "0<z" by (rule preal_imp_pos [OF A z])
  with frle and less show False by (simp add: Fract_of_int_eq) 

lemma Gleason9_34:
  assumes "cut A" "0 < u"
  shows "r  A. r + u  A"
  using assms Gleason9_34_contra preal_exists_bound by blast

subsection‹Gleason's Lemma 9-3.6›

lemma lemma_gleason9_36:
  assumes A: "cut A"
    and x: "1 < x"
  shows "r  A. r*x  A"
proof -
  from preal_nonempty [OF A]
  obtain y where y: "y  A" and  ypos: "0<y" ..
  show ?thesis 
  proof (rule classical)
    assume "~(rA. r * x  A)"
    with y have ymem: "y * x  A" by blast 
    from ypos mult_strict_left_mono [OF x]
    have yless: "y < y*x" by simp 
    let ?d = "y*x - y"
    from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
    from Gleason9_34 [OF A dpos]
    obtain r where r: "rA" and notin: "r + ?d  A" ..
    have rpos: "0<r" by (rule preal_imp_pos [OF A r])
    with dpos have rdpos: "0 < r + ?d" by arith
    have "~ (r + ?d  y + ?d)"
      assume le: "r + ?d  y + ?d" 
      from ymem have yd: "y + ?d  A" by (simp add: eq)
      have "r + ?d  A" by (rule preal_downwards_closed' [OF A yd rdpos le])
      with notin show False by simp
    hence "y < r" by simp
    with ypos have  dless: "?d < (r * ?d)/y"
      using dpos less_divide_eq_1 by fastforce
    have "r + ?d < r*x"
    proof -
      have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
      also from ypos have " = (r/y) * (y + ?d)"
        by (simp only: algebra_simps divide_inverse, simp)
      also have " = r*x" using ypos
        by simp
      finally show "r + ?d < r*x" .
    with r notin rdpos
    show "rA. r * x  A" by (blast dest:  preal_downwards_closed [OF A])

subsection‹Existence of Inverse: Part 2›

lemma mem_Rep_preal_inverse_iff:
  "(z  Rep_preal(inverse r))  (0 < z  (y. z < y  inverse y  Rep_preal r))"
  apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
  apply (simp add: inverse_set_def) 

lemma Rep_preal_one:
     "Rep_preal 1 = {x. 0 < x  x < 1}"
by (simp add: preal_one_def rat_mem_preal)

lemma subset_inverse_mult_lemma:
  assumes xpos: "0 < x" and xless: "x < 1"
  shows "v u y. 0 < v  v < y  inverse y  Rep_preal R  
    u  Rep_preal R  x = v * u"
proof -
  from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
  from lemma_gleason9_36 [OF cut_Rep_preal this]
  obtain t where t: "t  Rep_preal R" 
             and notin: "t * (inverse x)  Rep_preal R" ..
  have rpos: "0<t" by (rule preal_imp_pos [OF cut_Rep_preal t])
  from preal_exists_greater [OF cut_Rep_preal t]
  obtain u where u: "u  Rep_preal R" and rless: "t < u" ..
  have upos: "0<u" by (rule preal_imp_pos [OF cut_Rep_preal u])
  show ?thesis
  proof (intro exI conjI)
    show "0 < x/u" using xpos upos
      by (simp add: zero_less_divide_iff)  
    show "x/u < x/t" using xpos upos rpos
      by (simp add: divide_inverse mult_less_cancel_left rless) 
    show "inverse (x / t)  Rep_preal R" using notin
      by (simp add: divide_inverse mult.commute) 
    show "u  Rep_preal R" by (rule u) 
    show "x = x / u * u" using upos 
      by (simp add: divide_inverse mult.commute) 

lemma subset_inverse_mult: 
     "Rep_preal 1  Rep_preal(inverse r * r)"
  by (force simp: Rep_preal_one mem_Rep_preal_inverse_iff mem_Rep_preal_mult_iff dest: subset_inverse_mult_lemma)

lemma inverse_mult_subset: "Rep_preal(inverse r * r)  Rep_preal 1"
  proof -
  have "0 < u * v" if "v  Rep_preal r" "0 < u" "u < t" for u v t :: rat
    using that by (simp add: zero_less_mult_iff preal_imp_pos [OF cut_Rep_preal]) 
  moreover have "t * q < 1"
    if "q  Rep_preal r" "0 < t" "t < y" "inverse y  Rep_preal r"
    for t q y :: rat
  proof -
    have "q < inverse y"
      using not_in_Rep_preal_ub that by auto 
    hence "t * q < t/y" 
      using that by (simp add: divide_inverse mult_less_cancel_left)
    also have "  1" 
      using that by (simp add: pos_divide_le_eq)
    finally show ?thesis .
  ultimately show ?thesis
    by (auto simp: Rep_preal_one mem_Rep_preal_inverse_iff mem_Rep_preal_mult_iff)

lemma preal_mult_inverse: "inverse r * r = (1::preal)"
  by (meson Rep_preal_inject inverse_mult_subset subset_antisym subset_inverse_mult)

lemma preal_mult_inverse_right: "r * inverse r = (1::preal)"
  using preal_mult_commute preal_mult_inverse by auto

text‹Theorems needing Gleason9_34›

lemma Rep_preal_self_subset: "Rep_preal (r)  Rep_preal(r + s)"
  fix x
  assume x: "x  Rep_preal r"
  obtain y where y: "y  Rep_preal s" and "y > 0"
    using Rep_preal preal_nonempty by blast
  have ry: "x+y  Rep_preal(r + s)" using x y
    by (auto simp: mem_Rep_preal_add_iff)
  then show "x  Rep_preal(r + s)"
    by (meson 0 < y add_less_same_cancel1 not_in_Rep_preal_ub order.asym preal_imp_pos [OF cut_Rep_preal x])

lemma Rep_preal_sum_not_subset: "~ Rep_preal (r + s)  Rep_preal(r)"
proof -
  obtain y where y: "y  Rep_preal s" and "y > 0"
    using Rep_preal preal_nonempty by blast
  obtain x where "x  Rep_preal r" and notin: "x + y  Rep_preal r"
    using Dedekind_Real.Rep_preal Gleason9_34 0 < y by blast 
  then have "x + y  Rep_preal (r + s)" using y
    by (auto simp: mem_Rep_preal_add_iff)
  thus ?thesis using notin by blast

text‹at last, Gleason prop. 9-3.5(iii) page 123›
proposition preal_self_less_add_left: "(r::preal) < r + s"
  by (meson Rep_preal_sum_not_subset not_less preal_le_def)

subsection‹Subtraction for Positive Reals›

text‹gleason prop. 9-3.5(iv), page 123: proving propa < b  d. a + d = b. 
We define the claimed termD and show that it is a positive real›

lemma mem_diff_set:
  assumes "r < s"
  shows "cut (diff_set (Rep_preal s) (Rep_preal r))"
proof -
  obtain p where "Rep_preal r  Rep_preal s" "p  Rep_preal s" "p  Rep_preal r"
    using assms unfolding preal_less_def by auto
  then have "{}  diff_set (Rep_preal s) (Rep_preal r)"
    apply (simp add: diff_set_def psubset_eq)
    by (metis cut_Rep_preal add_eq_exists less_add_same_cancel1 preal_exists_greater preal_imp_pos)
  obtain q where "q > 0" "q  Rep_preal s"
    using Rep_preal_exists_bound by blast
  then have qnot: "q  diff_set (Rep_preal s) (Rep_preal r)"
    by (auto simp: diff_set_def dest: cut_Rep_preal [THEN preal_downwards_closed])
  moreover have "diff_set (Rep_preal s) (Rep_preal r)  {0<..}" (is "?lhs < ?rhs")
    using 0 < q diff_set_def qnot by blast
  moreover have "z  diff_set (Rep_preal s) (Rep_preal r)"
    if u: "u  diff_set (Rep_preal s) (Rep_preal r)" and "0 < z" "z < u" for u z
    using u that less_trans Rep_preal unfolding diff_set_def Dedekind_Real.cut_def by auto
  moreover have "u  diff_set (Rep_preal s) (Rep_preal r). y < u"
    if y: "y  diff_set (Rep_preal s) (Rep_preal r)" for y
  proof -
    obtain a b where "0 < a" "0 < b" "a  Rep_preal r" "a + y + b  Rep_preal s"
      using y
      by (simp add: diff_set_def) (metis cut_Rep_preal add_eq_exists less_add_same_cancel1 preal_exists_greater) 
    then have "a + (y + b)  Rep_preal s"
      by (simp add: add.assoc)
    then have "y + b  diff_set (Rep_preal s) (Rep_preal r)"
      using 0 < a 0 < b a  Rep_preal r y
      by (auto simp: diff_set_def)
    then show ?thesis
      using 0 < b less_add_same_cancel1 by blast
  ultimately show ?thesis
    by (simp add: Dedekind_Real.cut_def)

lemma mem_Rep_preal_diff_iff:
  "r < s 
       (z  Rep_preal (s - r))  
       (x. 0 < x  0 < z  x  Rep_preal r  x + z  Rep_preal s)"
  apply (simp add: preal_diff_def mem_diff_set Rep_preal)
  apply (force simp: diff_set_def) 

proposition less_add_left:
  fixes r::preal 
  assumes "r < s"
  shows "r + (s-r) = s"
proof -
  have "a + b  Rep_preal s"
    if "a  Rep_preal r" "c + b  Rep_preal s" "c  Rep_preal r"
    and "0 < b" "0 < c" for a b c
    by (meson cut_Rep_preal add_less_imp_less_right add_pos_pos not_in_Rep_preal_ub preal_downwards_closed preal_imp_pos that)
  then have "r + (s-r)  s"
    using assms mem_Rep_preal_add_iff mem_Rep_preal_diff_iff preal_le_def by auto
  have "x  Rep_preal (r + (s - r))" if "x  Rep_preal s" for x
  proof (cases "x  Rep_preal r")
    case True
    then show ?thesis
      using Rep_preal_self_subset by blast
    case False
    have "u v z. 0 < v  0 < z  u  Rep_preal r  z  Rep_preal r  z + v  Rep_preal s  x = u + v"
      if x: "x  Rep_preal s"
    proof -
      have xpos: "x > 0"
        using Rep_preal preal_imp_pos that by blast 
      obtain e where epos: "0 < e" and xe: "x + e  Rep_preal s"
        by (metis cut_Rep_preal x add_eq_exists less_add_same_cancel1 preal_exists_greater)
      from  Gleason9_34 [OF cut_Rep_preal epos]
      obtain u where r: "u  Rep_preal r" and notin: "u + e  Rep_preal r" ..
      with x False xpos have rless: "u < x" by (blast intro: not_in_Rep_preal_ub)
      from add_eq_exists [of u x]
      obtain y where eq: "x = u+y" by auto
      show ?thesis 
      proof (intro exI conjI)
        show "u + e  Rep_preal r" by (rule notin)
        show "u + e + y  Rep_preal s" using xe eq by (simp add: ac_simps)
        show "0 < u + e" 
          using epos preal_imp_pos [OF cut_Rep_preal r] by simp
      qed (use r rless eq in auto)
    then show ?thesis
      using assms mem_Rep_preal_add_iff mem_Rep_preal_diff_iff that by blast
  then have "s  r + (s-r)"
    by (auto simp: preal_le_def)
  then show ?thesis
    by (simp add: r + (s - r)  s antisym)

lemma preal_add_less2_mono1: "r < (s::preal)  r + t < s + t"
  by (metis add.assoc add.commute less_add_left preal_self_less_add_left)

lemma preal_add_less2_mono2: "r < (s::preal)  t + r < t + s"
  by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of t])

lemma preal_add_right_less_cancel: "r + t < s + t  r < (s::preal)"
  by (metis linorder_cases order.asym preal_add_less2_mono1)

lemma preal_add_left_less_cancel: "t + r < t + s  r < (s::preal)"
  by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of t])

lemma preal_add_less_cancel_left [simp]: "(t + (r::preal) < t + s)  (r < s)"
  by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)

lemma preal_add_less_cancel_right [simp]: "((r::preal) + t < s + t) = (r < s)"
  using preal_add_less_cancel_left [symmetric, of r s t] by (simp add: ac_simps)

lemma preal_add_le_cancel_left [simp]: "(t + (r::preal)  t + s) = (r  s)"
  by (simp add: linorder_not_less [symmetric]) 

lemma preal_add_le_cancel_right [simp]: "((r::preal) + t  s + t) = (r  s)"
  using preal_add_le_cancel_left [symmetric, of r s t] by (simp add: ac_simps)

lemma preal_add_right_cancel: "(r::preal) + t = s + t  r = s"
  by (metis less_irrefl linorder_cases preal_add_less_cancel_right)

lemma preal_add_left_cancel: "c + a = c + b  a = (b::preal)"
  by (auto intro: preal_add_right_cancel simp add: preal_add_commute)

instance preal :: linordered_ab_semigroup_add
  fix a b c :: preal
  show "a  b  c + a  c + b" by (simp only: preal_add_le_cancel_left)

subsection‹Completeness of type typpreal

text‹Prove that supremum is a cut›

text‹Part 1 of Dedekind sections definition›

lemma preal_sup:
  assumes le: "X. X  P  X  Y" and "P  {}" 
  shows "cut (X  P. Rep_preal(X))"
proof -
  have "{}  (X  P. Rep_preal(X))"
    using P  {} mem_Rep_preal_Ex by fastforce
  obtain q where "q > 0" and "q  (X  P. Rep_preal(X))"
    using Rep_preal_exists_bound [of Y] le by (auto simp: preal_le_def)
  then have "(X  P. Rep_preal(X))  {0<..}"
    using cut_Rep_preal preal_imp_pos by force
  have "u z. u  (X  P. Rep_preal(X)); 0 < z; z < u  z  (X  P. Rep_preal(X))"
    by (auto elim: cut_Rep_preal [THEN preal_downwards_closed])
  have "y. y  (X  P. Rep_preal(X))  u  (X  P. Rep_preal(X)). y < u"
    by (blast dest: cut_Rep_preal [THEN preal_exists_greater])
  ultimately show ?thesis
    by (simp add: Dedekind_Real.cut_def)

lemma preal_psup_le:
     "X. X  P  X  Y;  x  P  x  psup P"
  using preal_sup [of P Y] unfolding preal_le_def psup_def by fastforce 

lemma psup_le_ub: "X. X  P  X  Y; P  {}  psup P  Y"
  using preal_sup [of P Y] by (simp add: SUP_least preal_le_def psup_def) 

text‹Supremum property›
proposition preal_complete:
  assumes le: "X. X  P  X  Y" and "P  {}" 
  shows "(X  P. Z < X)  (Z < psup P)" (is "?lhs = ?rhs")
  assume ?lhs
  then show ?rhs
    using preal_sup [OF assms] preal_less_def psup_def by auto
  assume ?rhs
  then show ?lhs
    by (meson P  {} not_less psup_le_ub) 

subsection ‹Defining the Reals from the Positive Reals›

text ‹Here we do quotients the old-fashioned way›

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

definition "Real = UNIV//realrel"

typedef real = Real
  morphisms Rep_Real Abs_Real
  unfolding Real_def by (auto simp: quotient_def)

text ‹This doesn't involve the overloaded "real" function: users don't see it›
  real_of_preal :: "preal  real" where
  "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"

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

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

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

  real_add_def: "z + w =
       the_elem ((x,y)  Rep_Real z. (u,v)  Rep_Real w.
                 { Abs_Real(realrel``{(x+u, y+v)}) })"

  real_minus_def: "- r =  the_elem ((x,y)  Rep_Real r. { Abs_Real(realrel``{(y,x)}) })"

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

    "z * w =
       the_elem ((x,y)  Rep_Real z. (u,v)  Rep_Real w.
                 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"

  real_inverse_def: "inverse (r::real)  (THE s. (r = 0  s = 0)  s * r = 1)"

  real_divide_def: "r div (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: "¦r::real¦ = (if r < 0 then - r else r)"

  real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"

instance ..


subsection ‹Equivalence relation over positive reals›

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

lemma preal_trans_lemma:
  assumes "x + y1 = x1 + y" and "x + y2 = x2 + y"
  shows "x1 + y2 = x2 + (y1::preal)"
  by (metis add.left_commute assms preal_add_left_cancel)

lemma equiv_realrel: "equiv UNIV realrel"
  by (auto simp: equiv_def refl_on_def sym_def trans_def realrel_def intro: dest: preal_trans_lemma)

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

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

declare Abs_Real_inject [simp] 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"
  by (metis Rep_Real_inverse prod.exhaust  Rep_Real [of z, unfolded Real_def, THEN quotientE])

subsection ‹Addition and Subtraction›

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 (clarsimp simp: congruent2_def) (metis add.left_commute preal_add_assoc)
  thus ?thesis
    by (simp add: real_add_def UN_UN_split_split_eq UN_equiv_class2 [OF equiv_realrel equiv_realrel])

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 (auto simp: congruent_def add.commute) 
  thus ?thesis
    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])

instance real :: ab_group_add
  fix x y z :: real
  show "(x + y) + z = x + (y + z)"
    by (cases x, cases y, cases z, simp add: real_add add.assoc)
  show "x + y = y + x"
    by (cases x, cases y, simp add: real_add add.commute)
  show "0 + x = x"
    by (cases x, simp add: real_add real_zero_def ac_simps)
  show "- x + x = 0"
    by (cases x, simp add: real_minus real_add real_zero_def add.commute)
  show "x - y = x + - y"
    by (simp add: real_diff_def)

subsection ‹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)"
  by (metis (no_types, opaque_lifting) add.left_commute preal_add_commute preal_add_mult_distrib2)

lemma real_mult_congruent2:
  "(λp1 p2.
        (λ(x1,y1). (λ(x2,y2). 
          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
     respects2 realrel"
  apply (rule congruent2_commuteI [OF equiv_realrel])
  by (auto simp: mult.commute add.commute combine_common_factor preal_add_assoc preal_add_commute)

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 ac_simps)

lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
  by (cases z1, cases z2, cases z3) (simp add: real_mult algebra_simps)

lemma real_mult_1: "(1::real) * z = z"
  by (cases z) (simp add: real_mult real_one_def algebra_simps)

lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
  by (cases z1, cases z2, cases w) (simp add: real_add real_mult algebra_simps)

text‹one and zero are distinct›
lemma real_zero_not_eq_one: "0  (1::real)"
proof -
  have "(1::preal) < 1 + 1"
    by (simp add: preal_self_less_add_left)
  then show ?thesis
    by (simp add: real_zero_def real_one_def neq_iff)

instance real :: comm_ring_1
  fix x y z :: real
  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 (rule real_add_mult_distrib)
  show "0  (1::real)" by (rule real_zero_not_eq_one)

subsection ‹Inverse and Division›

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

lemma real_mult_inverse_left_ex:
  assumes "x  0" obtains y::real where "y*x = 1"
proof (cases x)
  case (Abs_Real u v)
  show ?thesis
  proof (cases u v rule: linorder_cases)
    case less
    then have "v * inverse (v - u) = 1 + u * inverse (v - u)"
      using less_add_left [of u v]
      by (metis preal_add_commute preal_add_mult_distrib preal_mult_inverse_right)
    then have "Abs_Real (realrel``{(1, inverse (v-u) + 1)}) * x - 1 = 0"
      by (simp add: Abs_Real real_mult preal_mult_inverse_right real_one_def) (simp add: algebra_simps)
    with that show thesis by auto
    case equal
    then show ?thesis
      using Abs_Real assms real_zero_iff by blast
    case greater
    then have "u * inverse (u - v) = 1 + v * inverse (u - v)"
      using less_add_left [of v u] by (metis add.commute distrib_right preal_mult_inverse_right)
    then have "Abs_Real (realrel``{(inverse (u-v) + 1, 1)}) * x - 1 = 0"
      by (simp add: Abs_Real real_mult preal_mult_inverse_right real_one_def) (simp add: algebra_simps)
    with that show thesis by auto

lemma real_mult_inverse_left:
  fixes x :: real
  assumes "x  0" shows "inverse x * x = 1"
proof -
  obtain y where "y*x = 1"
    using assms real_mult_inverse_left_ex by blast
  then have "(THE s. s * x = 1) * x = 1"
  proof (rule theI)
    show "y' = y" if "y' * x = 1" for y'
      by (metis y * x = 1 mult.left_commute mult.right_neutral that) 
  then show ?thesis
    using assms real_inverse_def by auto

subsection‹The Real Numbers form a Field›

instance real :: field
  fix x y z :: real
  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)
  show "inverse 0 = (0::real)" by (simp add: real_inverse_def)

subsection‹The ≤› Ordering›

lemma real_le_refl: "w  (w::real)"
  by (cases w, force simp: 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 -
  from le have "c+d  a+d" by simp
  hence "a+b  a+d" by (simp add: eq)
  thus "b  d" by simp

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: assms)
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: ac_simps)
  also have "  (x2+y1) + (u2+v1)" by (simp add: assms)
  finally show ?thesis by simp

lemma real_le: 
  "Abs_Real(realrel``{(x1,y1)})  Abs_Real(realrel``{(x2,y2)})    x1 + y2  x2 + y1"
  unfolding real_le_def by (auto intro: real_le_lemma)

lemma real_le_antisym: "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: ac_simps)
  also have "  (u+y) + (u+v')" by (simp add: assms)
  also have "  (u+y) + (u'+v)" by (simp add: assms)
  also have " = (u'+y) + (u+v)"  by (simp add: ac_simps)
  finally show ?thesis by simp

lemma real_le_trans: "i  j; j  k  i  (k::real)"
  by (cases i, cases j, cases k) (auto simp: real_le intro: real_trans_lemma)

instance real :: order
  show "u < v  u  v  ¬ v  u" for u v::real
    by (auto simp: real_less_def intro: real_le_antisym)
qed (auto intro: real_le_refl real_le_trans real_le_antisym)

instance real :: linorder
  show "x  y  y  x" for x y :: real
    by (meson eq_refl le_cases real_le_def)

instantiation real :: distrib_lattice

  "(inf :: real  real  real) = min"

  "(sup :: real  real  real) = max"

  by standard (auto simp: inf_real_def sup_real_def max_min_distrib2)


subsection‹The Reals Form an Ordered Field›

lemma real_le_eq_diff: "(x  y)  (x-y  (0::real))"
  by (cases x, cases y) (simp add: real_le real_zero_def real_diff_def real_add real_minus preal_add_commute)

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: algebra_simps) 
  with le show ?thesis 
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"])

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])

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])

lemma real_mult_order: 
  fixes x y::real
  assumes "0 < x" "0 < y"
  shows "0 < x * y"
  proof (cases x, cases y)
  show "0 < x * y"
    if x: "x = Abs_Real (Dedekind_Real.realrel `` {(x1, x2)})"
      and y: "y = Abs_Real (Dedekind_Real.realrel `` {(y1, y2)})"
    for x1 x2 y1 y2
  proof -
    have "x2 < x1" "y2 < y1"
      using assms not_le real_zero_def real_le x y
      by (metis preal_add_le_cancel_left real_zero_iff)+
    then obtain xd yd where "x1 = x2 + xd" "y1 = y2 + yd"
      using less_add_left by metis
    then have "¬ (x * y  0)"
      apply (simp add: x y real_mult real_zero_def real_le)
      apply (simp add: not_le algebra_simps preal_self_less_add_left)
    then show ?thesis
      by auto

lemma real_mult_less_mono2: "(0::real) < z; x < y  z * x < z * y"
  by (metis add_uminus_conv_diff real_less_sum_gt_zero real_mult_order real_sum_gt_zero_less right_diff_distrib')

instance real :: linordered_field
  fix x y z :: real
  show "x  y  z + x  z + y" by (rule real_add_left_mono)
  show "¦x¦ = (if x < 0 then -x else x)" by (simp only: real_abs_def)
  show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
    by (simp only: real_sgn_def)
  show "z * x < z * y" if "x < y" "0 < z"
    by (simp add: real_mult_less_mono2 that)

subsection ‹Completeness of the reals›

text‹The function termreal_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 algebra_simps)

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 algebra_simps)

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)"
proof (cases x)
  case (Abs_Real u v)
  show ?thesis
  proof (cases u v rule: linorder_cases)
    case less
    then show ?thesis
      using less_add_left
      apply (simp add: Abs_Real real_of_preal_def real_minus real_zero_def)
      by (metis preal_add_assoc preal_add_commute)      
    case equal
    then show ?thesis
      using Abs_Real real_zero_iff by blast
    case greater
    then show ?thesis
      using less_add_left
      apply (simp add: Abs_Real real_of_preal_def real_minus real_zero_def)
      by (metis preal_add_assoc preal_add_commute)      

lemma real_of_preal_less_iff [simp]:
  "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
  by (metis not_less preal_add_less_cancel_right real_le real_of_preal_def)

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

lemma real_of_preal_zero_less [simp]: "0 < real_of_preal m"
  by (metis less_add_same_cancel2 preal_self_less_add_left real_of_preal_add real_of_preal_less_iff)

subsection‹Theorems About the Ordering›

lemma real_gt_zero_preal_Ex: "(0 < x)  (y. x = real_of_preal y)"
  using order.asym real_of_preal_trichotomy by fastforce

subsection ‹Completeness of Positive Reals›

text ‹
  Supremum property for the set of positive reals

  Let P› be a non-empty set of positive reals, with an upper
  bound y›.  Then P› has a least upper bound
  (written S›).

  FIXME: Can the premise be weakened to ∀x ∈ P. x≤ y›?

lemma posreal_complete:
  assumes positive_P: "x  P. (0::real) < x"
    and not_empty_P: "x. x  P"
    and upper_bound_Ex: "y. x  P. x<y"
  shows "s. y. (x  P. y < x) = (y < s)"
proof (rule exI, rule allI)
  fix y
  let ?pP = "{w. real_of_preal w  P}"

  show "(xP. y < x) = (y < real_of_preal (psup ?pP))"
  proof (cases "0 < y")
    assume neg_y: "¬ 0 < y"
    show ?thesis
      assume "xP. y < x"
      thus "y < real_of_preal (psup ?pP)"
        by (metis dual_order.strict_trans neg_y not_less_iff_gr_or_eq real_of_preal_zero_less) 
      assume "y < real_of_preal (psup ?pP)"
      obtain "x" where x_in_P: "x  P" using not_empty_P ..
      thus "x  P. y < x" using x_in_P
        using neg_y not_less_iff_gr_or_eq positive_P by fastforce 
    assume pos_y: "0 < y"
    then obtain py where y_is_py: "y = real_of_preal py"
      by (auto simp: real_gt_zero_preal_Ex)

    obtain a where "a  P" using not_empty_P ..
    with positive_P have a_pos: "0 < a" ..
    then obtain pa where "a = real_of_preal pa"
      by (auto simp: real_gt_zero_preal_Ex)
    hence "pa  ?pP" using a  P by auto
    hence pP_not_empty: "?pP  {}" by auto

    obtain sup where sup: "x  P. x < sup"
      using upper_bound_Ex ..
    from this and a  P have "a < sup" ..
    hence "0 < sup" using a_pos by arith
    then obtain possup where "sup = real_of_preal possup"
      by (auto simp: real_gt_zero_preal_Ex)
    hence "X  ?pP. X  possup"
      using sup by auto
    with pP_not_empty have psup: "Z. (X  ?pP. Z < X) = (Z < psup ?pP)"
      by (meson preal_complete)
    show ?thesis
      assume "x  P. y < x"
      then obtain x where x_in_P: "x  P" and y_less_x: "y < x" ..
      hence "0 < x" using pos_y by arith
      then obtain px where x_is_px: "x = real_of_preal px"
        by (auto simp: real_gt_zero_preal_Ex)

      have py_less_X: "X  ?pP. py < X"
        show "py < px" using y_is_py and x_is_px and y_less_x
          by simp
        show "px  ?pP" using x_in_P and x_is_px by simp

      have "(X  ?pP. py < X)  (py < psup ?pP)"
        using psup by simp
      hence "py < psup ?pP" using py_less_X by simp
      thus "y < real_of_preal (psup {w. real_of_preal w  P})"
        using y_is_py and pos_y by simp
      assume y_less_psup: "y < real_of_preal (psup ?pP)"

      hence "py < psup ?pP" using y_is_py
        by simp
      then obtain "X" where py_less_X: "py < X" and X_in_pP: "X  ?pP"
        using psup by auto
      then obtain x where x_is_X: "x = real_of_preal X"
        by (simp add: real_gt_zero_preal_Ex)
      hence "y < x" using py_less_X and y_is_py
        by simp
      moreover have "x  P" 
        using x_is_X and X_in_pP by simp
      ultimately show " x  P. y < x" ..

subsection ‹Completeness›

lemma reals_complete:
  fixes S :: "real set"
  assumes notempty_S: "X. X  S"
    and exists_Ub: "bdd_above S"
  shows "x. (sS. s  x)  (y. (sS. s  y)  x  y)"
proof -
  obtain X where X_in_S: "X  S" using notempty_S ..
  obtain Y where Y_isUb: "sS. s  Y"
    using exists_Ub by (auto simp: bdd_above_def)
  let ?SHIFT = "{z. x S. z = x + (-X) + 1}  {x. 0 < x}"

    fix x
    assume S_le_x: "sS. s  x"
      fix s
      assume "s  {z. xS. z = x + - X + 1}"
      hence " x  S. s = x + -X + 1" ..
      then obtain x1 where x1: "x1  S" "s = x1 + (-X) + 1" ..
      then have "x1  x" using S_le_x by simp
      with x1 have "s  x + - X + 1" by arith
    then have "s?SHIFT. s  x + (-X) + 1"
      by auto
  } note S_Ub_is_SHIFT_Ub = this

  have *: "s?SHIFT. s  Y + (-X) + 1" using Y_isUb by (rule S_Ub_is_SHIFT_Ub)
  have "s?SHIFT. s < Y + (-X) + 2"
    fix s assume "s?SHIFT"
    with * have "s  Y + (-X) + 1" by simp
    also have " < Y + (-X) + 2" by simp
    finally show "s < Y + (-X) + 2" .
  moreover have "y  ?SHIFT. 0 < y" by auto
  moreover have shifted_not_empty: "u. u  ?SHIFT"
    using X_in_S and Y_isUb by auto
  ultimately obtain t where t_is_Lub: "y. (x?SHIFT. y < x) = (y < t)"
    using posreal_complete [of ?SHIFT] unfolding bdd_above_def by blast

  show ?thesis
    show "(sS. s  (t + X + (-1)))  (y. (sS. s  y)  (t + X + (-1))  y)"
    proof safe
      fix x
      assume "sS. s  x"
      hence "s?SHIFT. s  x + (-X) + 1"
        using S_Ub_is_SHIFT_Ub by simp
      then have "¬ x + (-X) + 1 < t"
        by (subst t_is_Lub[rule_format, symmetric]) (simp add: not_less)
      thus "t + X + -1  x" by arith
      fix y
      assume y_in_S: "y  S"
      obtain "u" where u_in_shift: "u  ?SHIFT" using shifted_not_empty ..
      hence " x  S. u = x + - X + 1" by simp
      then obtain "x" where x_and_u: "u = x + - X + 1" ..
      have u_le_t: "u  t"
      proof (rule dense_le)
        fix x assume "x < u" then have "x < t"
          using u_in_shift t_is_Lub by auto
        then show "x  t"  by simp

      show "y  t + X + -1"
      proof cases
        assume "y  x"
        moreover have "x = u + X + - 1" using x_and_u by arith
        moreover have "u + X + - 1   t + X + -1" using u_le_t by arith
        ultimately show "y   t + X + -1" by arith
        assume "~(y  x)"
        hence x_less_y: "x < y" by arith

        have "x + (-X) + 1  ?SHIFT" using x_and_u and u_in_shift by simp
        hence "0 < x + (-X) + 1" by simp
        hence "0 < y + (-X) + 1" using x_less_y by arith
        hence *: "y + (-X) + 1  ?SHIFT" using y_in_S by simp
        have "y + (-X) + 1  t"
        proof (rule dense_le)
          fix x assume "x < y + (-X) + 1" then have