(*  Title:      HOL/ex/Dedekind_Real.thy

     Author:     Jacques D. Fleuriot, University of Cambridge

     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

 The positive reals as Dedekind sections of positive

 rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]

 provides some of the definitions.

 *)

 theory Dedekind_Real

 imports Complex_Main

 begin

 section {* Positive real numbers *}

 text{*Could be generalized and moved to @{text Groups}*}

 lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"

 by (rule_tac x="b-a" in exI, simp)

 definition

   cut :: "rat set => bool" where

   "cut A = ({} \<subset> A &

             A < {r. 0 < r} &

             (\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"

 lemma interval_empty_iff:

   "{y. (x::'a::unbounded_dense_linorder) < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"

   by (auto dest: dense)

 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 < {r. 0 < r}" by force

   have nonempty: "{} \<subset> ?A"

   proof

     show "{} \<subseteq> ?A" by simp

     show "{} \<noteq> ?A"

       by (force simp only: q eq_commute [of "{}"] interval_empty_iff)

   qed

   show ?thesis

     by (simp add: cut_def pos nonempty,

         blast dest: dense intro: order_less_trans)

 qed

 definition "preal = {A. cut A}"

 typedef preal = preal

   unfolding preal_def by (blast intro: cut_of_rat [OF zero_less_one])

 definition

   psup :: "preal set => preal" where

   "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"

 definition

   add_set :: "[rat set,rat set] => rat set" where

   "add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"

 definition

   diff_set :: "[rat set,rat set] => rat set" where

   "diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"

 definition

   mult_set :: "[rat set,rat set] => rat set" where

   "mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"

 definition

   inverse_set :: "rat set => rat set" where

   "inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"

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

 begin

 definition

   preal_less_def:

     "R < S == Rep_preal R < Rep_preal S"

 definition

   preal_le_def:

     "R \<le> S == Rep_preal R \<subseteq> Rep_preal S"

 definition

   preal_add_def:

     "R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"

 definition

   preal_diff_def:

     "R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"

 definition

   preal_mult_def:

     "R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"

 definition

   preal_inverse_def:

     "inverse R == Abs_preal (inverse_set (Rep_preal R))"

 definition "R / S = R * inverse (S\<Colon>preal)"

 definition

   preal_one_def:

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

 instance ..

 end

 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 ==> {r::rat. 0 < r & r < q} \<in> preal"

 by (simp add: preal_def cut_of_rat)

 lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"

   unfolding preal_def cut_def [abs_def] by blast

 lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"

   apply (drule preal_nonempty)

   apply fast

   done

 lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"

   by (force simp add: preal_def cut_def)

 lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"

   apply (drule preal_imp_psubset_positives)

   apply auto

   done

 lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"

   unfolding preal_def cut_def [abs_def] by blast

 lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"

   unfolding preal_def cut_def [abs_def] by blast

 text{*Relaxing the final premise*}

 lemma preal_downwards_closed':

      "[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"

 apply (simp add: order_le_less)

 apply (blast intro: preal_downwards_closed)

 done

 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: "A \<in> preal"

     and notx: "x \<notin> A"

     and y: "y \<in> 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)

 next

   assume "x=y"

   with notx and y show ?thesis by simp

 next

   assume "y<x"

   thus ?thesis .

 qed

 text {* preal lemmas instantiated to @{term "Rep_preal X"} *}

 lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"

 by (rule preal_Ex_mem [OF Rep_preal])

 lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X"

 by (rule preal_exists_bound [OF Rep_preal])

 lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]

 subsection{*Properties of Ordering*}

 instance preal :: order

 proof

   fix w :: preal

   show "w \<le> w" by (simp add: preal_le_def)

 next

   fix i j k :: preal

   assume "i \<le> j" and "j \<le> k"

   then show "i \<le> k" by (simp add: preal_le_def)

 next

   fix z w :: preal

   assume "z \<le> w" and "w \<le> z"

   then show "z = w" by (simp add: preal_le_def Rep_preal_inject)

 next

   fix z w :: preal

   show "z < w \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z"

   by (auto simp add: preal_le_def preal_less_def Rep_preal_inject)

 qed  

 lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"

 by (insert preal_imp_psubset_positives, blast)

 instance preal :: linorder

 proof

   fix x y :: preal

   show "x <= y | y <= x"

     apply (auto simp add: preal_le_def)

     apply (rule ccontr)

     apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]

              elim: order_less_asym)

     done

 qed

 instantiation preal :: distrib_lattice

 begin

 definition

   "(inf \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = min"

 definition

   "(sup \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = max"

 instance

   by intro_classes

     (auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1)

 end

 subsection{*Properties of Addition*}

 lemma preal_add_commute: "(x::preal) + y = y + x"

 apply (unfold preal_add_def add_set_def)

 apply (rule_tac f = Abs_preal in arg_cong)

 apply (force simp add: add_commute)

 done

 text{*Lemmas for proving that addition of two positive reals gives

  a positive real*}

 text{*Part 1 of Dedekind sections definition*}

 lemma add_set_not_empty:

      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"

 apply (drule preal_nonempty)+

 apply (auto simp add: add_set_def)

 done

 text{*Part 2 of Dedekind sections definition.  A structured version of

 this proof is @{text preal_not_mem_mult_set_Ex} below.*}

 lemma preal_not_mem_add_set_Ex:

      "[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B"

 apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto) 

 apply (rule_tac x = "x+xa" in exI)

 apply (simp add: add_set_def, clarify)

 apply (drule (3) not_in_preal_ub)+

 apply (force dest: add_strict_mono)

 done

 lemma add_set_not_rat_set:

    assumes A: "A \<in> preal" 

        and B: "B \<in> preal"

      shows "add_set A B < {r. 0 < r}"

 proof

   from preal_imp_pos [OF A] preal_imp_pos [OF B]

   show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def) 

 next

   show "add_set A B \<noteq> {r. 0 < r}"

     by (insert preal_not_mem_add_set_Ex [OF A B], blast) 

 qed

 text{*Part 3 of Dedekind sections definition*}

 lemma add_set_lemma3:

      "[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|] 

       ==> z \<in> add_set A B"

 proof (unfold add_set_def, clarify)

   fix x::rat and y::rat

   assume A: "A \<in> preal" 

     and B: "B \<in> preal"

     and [simp]: "0 < z"

     and zless: "z < x + y"

     and x:  "x \<in> A"

     and y:  "y \<in> B"

   have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])

   have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])

   have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)

   let ?f = "z/(x+y)"

   have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)

   show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"

   proof (intro bexI)

     show "z = x*?f + y*?f"

       by (simp add: distrib_right [symmetric] divide_inverse mult_ac

           order_less_imp_not_eq2)

   next

     show "y * ?f \<in> B"

     proof (rule preal_downwards_closed [OF B y])

       show "0 < y * ?f"

         by (simp add: divide_inverse zero_less_mult_iff)

     next

       show "y * ?f < y"

         by (insert mult_strict_left_mono [OF fless ypos], simp)

     qed

   next

     show "x * ?f \<in> A"

     proof (rule preal_downwards_closed [OF A x])

       show "0 < x * ?f"

         by (simp add: divide_inverse zero_less_mult_iff)

     next

       show "x * ?f < x"

         by (insert mult_strict_left_mono [OF fless xpos], simp)

     qed

   qed

 qed

 text{*Part 4 of Dedekind sections definition*}

 lemma add_set_lemma4:

      "[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"

 apply (auto simp add: add_set_def)

 apply (frule preal_exists_greater [of A], auto) 

 apply (rule_tac x="u + y" in exI)

 apply (auto intro: add_strict_left_mono)

 done

 lemma mem_add_set:

      "[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"

 apply (simp (no_asm_simp) add: preal_def cut_def)

 apply (blast intro!: add_set_not_empty add_set_not_rat_set

                      add_set_lemma3 add_set_lemma4)

 done

 lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"

 apply (simp add: preal_add_def mem_add_set Rep_preal)

 apply (force simp add: add_set_def add_ac)

 done

 instance preal :: ab_semigroup_add

 proof

   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)

 qed

 subsection{*Properties of Multiplication*}

 text{*Proofs essentially same as for addition*}

 lemma preal_mult_commute: "(x::preal) * y = y * x"

 apply (unfold preal_mult_def mult_set_def)

 apply (rule_tac f = Abs_preal in arg_cong)

 apply (force simp add: mult_commute)

 done

 text{*Multiplication of two positive reals gives a positive real.*}

 text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}

 text{*Part 1 of Dedekind sections definition*}

 lemma mult_set_not_empty:

      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"

 apply (insert preal_nonempty [of A] preal_nonempty [of B]) 

 apply (auto simp add: mult_set_def)

 done

 text{*Part 2 of Dedekind sections definition*}

 lemma preal_not_mem_mult_set_Ex:

   assumes A: "A \<in> preal" 

     and B: "B \<in> preal"

   shows "\<exists>q. 0 < q & q \<notin> mult_set A B"

 proof -

   from preal_exists_bound [OF A] obtain x where 1 [simp]: "0 < x" "x \<notin> A" by blast

   from preal_exists_bound [OF B] obtain y where 2 [simp]: "0 < y" "y \<notin> B" by blast

   show ?thesis

   proof (intro exI conjI)

     show "0 < x*y" by (simp add: mult_pos_pos)

     show "x * y \<notin> mult_set A B"

     proof -

       {

         fix u::rat and v::rat

         assume u: "u \<in> A" and v: "v \<in> B" and xy: "x*y = u*v"

         moreover from A B 1 2 u v have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+

         moreover

         from A B 1 2 u v have "0\<le>v"

           by (blast intro: preal_imp_pos [OF B] order_less_imp_le)

         moreover

         from A B 1 `u < x` `v < y` `0 \<le> v`

         have "u*v < x*y" by (blast intro: mult_strict_mono)

         ultimately have False by force

       }

       thus ?thesis by (auto simp add: mult_set_def)

     qed

   qed

 qed

 lemma mult_set_not_rat_set:

   assumes A: "A \<in> preal" 

     and B: "B \<in> preal"

   shows "mult_set A B < {r. 0 < r}"

 proof

   show "mult_set A B \<subseteq> {r. 0 < r}"

     by (force simp add: mult_set_def

       intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)

   show "mult_set A B \<noteq> {r. 0 < r}"

     using preal_not_mem_mult_set_Ex [OF A B] by blast

 qed

 text{*Part 3 of Dedekind sections definition*}

 lemma mult_set_lemma3:

      "[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|] 

       ==> z \<in> mult_set A B"

 proof (unfold mult_set_def, clarify)

   fix x::rat and y::rat

   assume A: "A \<in> preal" 

     and B: "B \<in> preal"

     and [simp]: "0 < z"

     and zless: "z < x * y"

     and x:  "x \<in> A"

     and y:  "y \<in> B"

   have [simp]: "0<y" by (rule preal_imp_pos [OF B y])

   show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"

   proof

     show "\<exists>y'\<in>B. z = (z/y) * y'"

     proof

       show "z = (z/y)*y"

         by (simp add: divide_inverse mult_commute [of y] mult_assoc

                       order_less_imp_not_eq2)

       show "y \<in> B" by fact

     qed

   next

     show "z/y \<in> A"

     proof (rule preal_downwards_closed [OF A x])

       show "0 < z/y"

         by (simp add: zero_less_divide_iff)

       show "z/y < x" by (simp add: pos_divide_less_eq zless)

     qed

   qed

 qed

 text{*Part 4 of Dedekind sections definition*}

 lemma mult_set_lemma4:

      "[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"

 apply (auto simp add: mult_set_def)

 apply (frule preal_exists_greater [of A], auto) 

 apply (rule_tac x="u * y" in exI)

 apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B] 

                    mult_strict_right_mono)

 done

 lemma mem_mult_set:

      "[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"

 apply (simp (no_asm_simp) add: preal_def cut_def)

 apply (blast intro!: mult_set_not_empty mult_set_not_rat_set

                      mult_set_lemma3 mult_set_lemma4)

 done

 lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"

 apply (simp add: preal_mult_def mem_mult_set Rep_preal)

 apply (force simp add: mult_set_def mult_ac)

 done

 instance preal :: ab_semigroup_mult

 proof

   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)

 qed

 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: "A \<in> preal"

   have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")

   proof

     show "?lhs \<subseteq> A"

     proof clarify

       fix x::rat and u::rat and v::rat

       assume upos: "0<u" and "u<1" and v: "v \<in> 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 \<in> A"

         by (force intro: preal_downwards_closed [OF A v] mult_pos_pos 

           upos vpos)

     qed

   next

     show "A \<subseteq> ?lhs"

     proof clarify

       fix x::rat

       assume x: "x \<in> 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 \<in> A" and xlessv: "x < v" ..

       have vpos: "0<v" by (rule preal_imp_pos [OF A v])

       show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. 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)

         show "\<exists>v'\<in>A. x = (x / v) * v'"

         proof

           show "x = (x/v)*v"

             by (simp add: divide_inverse mult_assoc vpos

                           order_less_imp_not_eq2)

           show "v \<in> A" by fact

         qed

       qed

     qed

   qed

   thus "1 * Abs_preal A = Abs_preal A"

     by (simp add: preal_one_def preal_mult_def mult_set_def 

                   rat_mem_preal A)

 qed

 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 \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"

 apply (simp add: preal_add_def mem_add_set Rep_preal)

 apply (simp add: add_set_def) 

 done

 lemma mem_Rep_preal_mult_iff:

       "(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"

 apply (simp add: preal_mult_def mem_mult_set Rep_preal)

 apply (simp add: mult_set_def) 

 done

 lemma distrib_subset1:

      "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"

 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)

 apply (force simp add: distrib_left)

 done

 lemma preal_add_mult_distrib_mean:

   assumes a: "a \<in> Rep_preal w"

     and b: "b \<in> Rep_preal w"

     and d: "d \<in> Rep_preal x"

     and e: "e \<in> Rep_preal y"

   shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"

 proof

   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 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 \<in> Rep_preal w"

   proof (cases rule: linorder_le_cases)

     assume "a \<le> b"

     hence "?c \<le> b"

       by (simp add: pos_divide_le_eq distrib_left mult_right_mono

                     order_less_imp_le)

     thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])

   next

     assume "b \<le> a"

     hence "?c \<le> a"

       by (simp add: pos_divide_le_eq distrib_left mult_right_mono

                     order_less_imp_le)

     thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])

   qed

 qed

 lemma distrib_subset2:

      "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"

 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)

 apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)

 done

 lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"

 apply (rule Rep_preal_inject [THEN iffD1])

 apply (rule equalityI [OF distrib_subset1 distrib_subset2])

 done

 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_inv_set_ex:

   assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"

 proof -

   from preal_exists_bound [OF A]

   obtain x where [simp]: "0<x" "x \<notin> 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) \<notin> A"

       by (simp add: order_less_imp_not_eq2)

   qed

 qed

 text{*Part 1 of Dedekind sections definition*}

 lemma inverse_set_not_empty:

      "A \<in> preal ==> {} \<subset> inverse_set A"

 apply (insert mem_inv_set_ex [of A])

 apply (auto simp add: inverse_set_def)

 done

 text{*Part 2 of Dedekind sections definition*}

 lemma preal_not_mem_inverse_set_Ex:

    assumes A: "A \<in> preal"  shows "\<exists>q. 0 < q & q \<notin> inverse_set A"

 proof -

   from preal_nonempty [OF A]

   obtain x where x: "x \<in> A" and  xpos [simp]: "0<x" ..

   show ?thesis

   proof (intro exI conjI)

     show "0 < inverse x" by simp

     show "inverse x \<notin> 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 \<in> A"

           by (simp add: preal_downwards_closed [OF A x] iyless)}

      thus ?thesis by (auto simp add: inverse_set_def)

     qed

   qed

 qed

 lemma inverse_set_not_rat_set:

    assumes A: "A \<in> preal"  shows "inverse_set A < {r. 0 < r}"

 proof

   show "inverse_set A \<subseteq> {r. 0 < r}"  by (force simp add: inverse_set_def)

 next

   show "inverse_set A \<noteq> {r. 0 < r}"

     by (insert preal_not_mem_inverse_set_Ex [OF A], blast)

 qed

 text{*Part 3 of Dedekind sections definition*}

 lemma inverse_set_lemma3:

      "[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|] 

       ==> z \<in> inverse_set A"

 apply (auto simp add: inverse_set_def)

 apply (auto intro: order_less_trans)

 done

 text{*Part 4 of Dedekind sections definition*}

 lemma inverse_set_lemma4:

      "[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"

 apply (auto simp add: inverse_set_def)

 apply (drule dense [of y]) 

 apply (blast intro: order_less_trans)

 done

 lemma mem_inverse_set:

      "A \<in> preal ==> inverse_set A \<in> preal"

 apply (simp (no_asm_simp) add: preal_def cut_def)

 apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set

                      inverse_set_lemma3 inverse_set_lemma4)

 done

 subsection{*Gleason's Lemma 9-3.4, page 122*}

 lemma Gleason9_34_exists:

   assumes A: "A \<in> preal"

     and "\<forall>x\<in>A. x + u \<in> A"

     and "0 \<le> z"

   shows "\<exists>b\<in>A. b + (of_int z) * u \<in> 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 

   next

     case (Suc k)

     then obtain b where b: "b \<in> A" "b + of_nat k * u \<in> A" ..

     hence "b + of_int (int k)*u + u \<in> A" by (simp add: assms)

     thus ?case by (force simp add: algebra_simps b)

   qed

 next

   case (neg n)

   with assms show ?thesis by simp

 qed

 lemma Gleason9_34_contra:

   assumes A: "A \<in> preal"

     shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> 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: "\<forall>x\<in>A. x + (Fract c d) \<in> A"

     and upos: "0 < Fract c d"

     and ypos: "0 < Fract a b"

     and notin: "Fract a b \<notin> 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 \<le> Fract ?k 1 * (Fract c d)" 

   proof -

     have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"

       by (simp add: order_less_imp_not_eq2 mult_ac) 

     moreover

     have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"

       by (rule mult_mono, 

           simp_all add: int_one_le_iff_zero_less zero_less_mult_iff 

                         order_less_imp_le)

     ultimately

     show ?thesis by simp

   qed

   have k: "0 \<le> ?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 \<in> A" 

              and mem: "z + of_int ?k * Fract c d \<in> 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) 

 qed

 lemma Gleason9_34:

   assumes A: "A \<in> preal"

     and upos: "0 < u"

   shows "\<exists>r \<in> A. r + u \<notin> A"

 proof (rule ccontr, simp)

   assume closed: "\<forall>r\<in>A. r + u \<in> A"

   from preal_exists_bound [OF A]

   obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast

   show False

     by (rule Gleason9_34_contra [OF A closed upos ypos y])

 qed

 subsection{*Gleason's Lemma 9-3.6*}

 lemma lemma_gleason9_36:

   assumes A: "A \<in> preal"

     and x: "1 < x"

   shows "\<exists>r \<in> A. r*x \<notin> A"

 proof -

   from preal_nonempty [OF A]

   obtain y where y: "y \<in> A" and  ypos: "0<y" ..

   show ?thesis 

   proof (rule classical)

     assume "~(\<exists>r\<in>A. r * x \<notin> A)"

     with y have ymem: "y * x \<in> 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: "r\<in>A" and notin: "r + ?d \<notin> 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 \<le> y + ?d)"

     proof

       assume le: "r + ?d \<le> y + ?d" 

       from ymem have yd: "y + ?d \<in> A" by (simp add: eq)

       have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])

       with notin show False by simp

     qed

     hence "y < r" by simp

     with ypos have  dless: "?d < (r * ?d)/y"

       by (simp add: pos_less_divide_eq mult_commute [of ?d]

                     mult_strict_right_mono dpos)

     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" .

     qed

     with r notin rdpos

     show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest:  preal_downwards_closed [OF A])

   qed  

 qed

 subsection{*Existence of Inverse: Part 2*}

 lemma mem_Rep_preal_inverse_iff:

       "(z \<in> Rep_preal(inverse R)) = 

        (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"

 apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)

 apply (simp add: inverse_set_def) 

 done

 lemma Rep_preal_one:

      "Rep_preal 1 = {x. 0 < x \<and> 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 "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R & 

     u \<in> Rep_preal R & x = r * u"

 proof -

   from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)

   from lemma_gleason9_36 [OF Rep_preal this]

   obtain r where r: "r \<in> Rep_preal R" 

              and notin: "r * (inverse x) \<notin> Rep_preal R" ..

   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])

   from preal_exists_greater [OF Rep_preal r]

   obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..

   have upos: "0<u" by (rule preal_imp_pos [OF 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/r" using xpos upos rpos

       by (simp add: divide_inverse mult_less_cancel_left rless) 

     show "inverse (x / r) \<notin> Rep_preal R" using notin

       by (simp add: divide_inverse mult_commute) 

     show "u \<in> Rep_preal R" by (rule u) 

     show "x = x / u * u" using upos 

       by (simp add: divide_inverse mult_commute) 

   qed

 qed

 lemma subset_inverse_mult: 

      "Rep_preal 1 \<subseteq> Rep_preal(inverse R * R)"

 apply (auto simp add: Bex_def Rep_preal_one mem_Rep_preal_inverse_iff 

                       mem_Rep_preal_mult_iff)

 apply (blast dest: subset_inverse_mult_lemma) 

 done

 lemma inverse_mult_subset_lemma:

   assumes rpos: "0 < r" 

     and rless: "r < y"

     and notin: "inverse y \<notin> Rep_preal R"

     and q: "q \<in> Rep_preal R"

   shows "r*q < 1"

 proof -

   have "q < inverse y" using rpos rless

     by (simp add: not_in_preal_ub [OF Rep_preal notin] q)

   hence "r * q < r/y" using rpos

     by (simp add: divide_inverse mult_less_cancel_left)

   also have "... \<le> 1" using rpos rless

     by (simp add: pos_divide_le_eq)

   finally show ?thesis .

 qed

 lemma inverse_mult_subset:

      "Rep_preal(inverse R * R) \<subseteq> Rep_preal 1"

 apply (auto simp add: Bex_def Rep_preal_one mem_Rep_preal_inverse_iff

                       mem_Rep_preal_mult_iff)

 apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal]) 

 apply (blast intro: inverse_mult_subset_lemma) 

 done

 lemma preal_mult_inverse: "inverse R * R = (1::preal)"

 apply (rule Rep_preal_inject [THEN iffD1])

 apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult]) 

 done

 lemma preal_mult_inverse_right: "R * inverse R = (1::preal)"

 apply (rule preal_mult_commute [THEN subst])

 apply (rule preal_mult_inverse)

 done

 text{*Theorems needing @{text Gleason9_34}*}

 lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"

 proof 

   fix r

   assume r: "r \<in> Rep_preal R"

   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])

   from mem_Rep_preal_Ex 

   obtain y where y: "y \<in> Rep_preal S" ..

   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])

   have ry: "r+y \<in> Rep_preal(R + S)" using r y

     by (auto simp add: mem_Rep_preal_add_iff)

   show "r \<in> Rep_preal(R + S)" using r ypos rpos 

     by (simp add:  preal_downwards_closed [OF Rep_preal ry]) 

 qed

 lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"

 proof -

   from mem_Rep_preal_Ex 

   obtain y where y: "y \<in> Rep_preal S" ..

   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])

   from  Gleason9_34 [OF Rep_preal ypos]

   obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..

   have "r + y \<in> Rep_preal (R + S)" using r y

     by (auto simp add: mem_Rep_preal_add_iff)

   thus ?thesis using notin by blast

 qed

 lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"

 by (insert Rep_preal_sum_not_subset, blast)

 text{*at last, Gleason prop. 9-3.5(iii) page 123*}

 lemma preal_self_less_add_left: "(R::preal) < R + S"

 apply (unfold preal_less_def less_le)

 apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])

 done

 subsection{*Subtraction for Positive Reals*}

 text{*Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> \<exists>D. A + D =

 B"}. We define the claimed @{term D} and show that it is a positive real*}

 text{*Part 1 of Dedekind sections definition*}

 lemma diff_set_not_empty:

      "R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"

 apply (auto simp add: preal_less_def diff_set_def elim!: equalityE) 

 apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])

 apply (drule preal_imp_pos [OF Rep_preal], clarify)

 apply (cut_tac a=x and b=u in add_eq_exists, force) 

 done

 text{*Part 2 of Dedekind sections definition*}

 lemma diff_set_nonempty:

      "\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"

 apply (cut_tac X = S in Rep_preal_exists_bound)

 apply (erule exE)

 apply (rule_tac x = x in exI, auto)

 apply (simp add: diff_set_def) 

 apply (auto dest: Rep_preal [THEN preal_downwards_closed])

 done

 lemma diff_set_not_rat_set:

   "diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")

 proof

   show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def) 

   show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast

 qed

 text{*Part 3 of Dedekind sections definition*}

 lemma diff_set_lemma3:

      "[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|] 

       ==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"

 apply (auto simp add: diff_set_def) 

 apply (rule_tac x=x in exI) 

 apply (drule Rep_preal [THEN preal_downwards_closed], auto)

 done

 text{*Part 4 of Dedekind sections definition*}

 lemma diff_set_lemma4:

      "[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|] 

       ==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"

 apply (auto simp add: diff_set_def) 

 apply (drule Rep_preal [THEN preal_exists_greater], clarify) 

 apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)  

 apply (rule_tac x="y+xa" in exI) 

 apply (auto simp add: add_ac)

 done

 lemma mem_diff_set:

      "R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"

 apply (unfold preal_def cut_def [abs_def])

 apply (blast intro!: diff_set_not_empty diff_set_not_rat_set

                      diff_set_lemma3 diff_set_lemma4)

 done

 lemma mem_Rep_preal_diff_iff:

       "R < S ==>

        (z \<in> Rep_preal(S-R)) = 

        (\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"

 apply (simp add: preal_diff_def mem_diff_set Rep_preal)

 apply (force simp add: diff_set_def) 

 done

 text{*proving that @{term "R + D \<le> S"}*}

 lemma less_add_left_lemma:

   assumes Rless: "R < S"

     and a: "a \<in> Rep_preal R"

     and cb: "c + b \<in> Rep_preal S"

     and "c \<notin> Rep_preal R"

     and "0 < b"

     and "0 < c"

   shows "a + b \<in> Rep_preal S"

 proof -

   have "0<a" by (rule preal_imp_pos [OF Rep_preal a])

   moreover

   have "a < c" using assms by (blast intro: not_in_Rep_preal_ub ) 

   ultimately show ?thesis

     using assms by (simp add: preal_downwards_closed [OF Rep_preal cb])

 qed

 lemma less_add_left_le1:

        "R < (S::preal) ==> R + (S-R) \<le> S"

 apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff 

                       mem_Rep_preal_diff_iff)

 apply (blast intro: less_add_left_lemma) 

 done

 subsection{*proving that @{term "S \<le> R + D"} --- trickier*}

 lemma lemma_sum_mem_Rep_preal_ex:

      "x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"

 apply (drule Rep_preal [THEN preal_exists_greater], clarify) 

 apply (cut_tac a=x and b=u in add_eq_exists, auto) 

 done

 lemma less_add_left_lemma2:

   assumes Rless: "R < S"

     and x:     "x \<in> Rep_preal S"

     and xnot: "x \<notin>  Rep_preal R"

   shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R & 

                      z + v \<in> Rep_preal S & x = u + v"

 proof -

   have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])

   from lemma_sum_mem_Rep_preal_ex [OF x]

   obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast

   from  Gleason9_34 [OF Rep_preal epos]

   obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..

   with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)

   from add_eq_exists [of r x]

   obtain y where eq: "x = r+y" by auto

   show ?thesis 

   proof (intro exI conjI)

     show "r \<in> Rep_preal R" by (rule r)

     show "r + e \<notin> Rep_preal R" by (rule notin)

     show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)

     show "x = r + y" by (simp add: eq)

     show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]

       by simp

     show "0 < y" using rless eq by arith

   qed

 qed

 lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"

 apply (auto simp add: preal_le_def)

 apply (case_tac "x \<in> Rep_preal R")

 apply (cut_tac Rep_preal_self_subset [of R], force)

 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)

 apply (blast dest: less_add_left_lemma2)

 done

 lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"

 by (blast intro: antisym [OF less_add_left_le1 less_add_left_le2])

 lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"

 by (fast dest: less_add_left)

 lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"

 apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)

 apply (rule_tac y1 = D in preal_add_commute [THEN subst])

 apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])

 done

 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)"

 apply (insert linorder_less_linear [of R S], auto)

 apply (drule_tac R = S and T = T in preal_add_less2_mono1)

 apply (blast dest: order_less_trans) 

 done

 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: "(T + (R::preal) < T + S) = (R < S)"

 by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)

 lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"

 by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left) 

 lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"

 apply (insert linorder_less_linear [of R S], safe)

 apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)

 done

 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_cancel_ab_semigroup_add

 proof

   fix a b c :: preal

   show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel)

   show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left)

 qed

 subsection{*Completeness of type @{typ preal}*}

 text{*Prove that supremum is a cut*}

 text{*Part 1 of Dedekind sections definition*}

 lemma preal_sup_set_not_empty:

      "P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"

 apply auto

 apply (cut_tac X = x in mem_Rep_preal_Ex, auto)

 done

 text{*Part 2 of Dedekind sections definition*}

 lemma preal_sup_not_exists:

      "\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"

 apply (cut_tac X = Y in Rep_preal_exists_bound)

 apply (auto simp add: preal_le_def)

 done

 lemma preal_sup_set_not_rat_set:

      "\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"

 apply (drule preal_sup_not_exists)

 apply (blast intro: preal_imp_pos [OF Rep_preal])  

 done

 text{*Part 3 of Dedekind sections definition*}

 lemma preal_sup_set_lemma3:

      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]

       ==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"

 by (auto elim: Rep_preal [THEN preal_downwards_closed])

 text{*Part 4 of Dedekind sections definition*}

 lemma preal_sup_set_lemma4:

      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]

           ==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"

 by (blast dest: Rep_preal [THEN preal_exists_greater])

 lemma preal_sup:

      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"

 apply (unfold preal_def cut_def [abs_def])

 apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set

                      preal_sup_set_lemma3 preal_sup_set_lemma4)

 done

 lemma preal_psup_le:

      "[| \<forall>X \<in> P. X \<le> Y;  x \<in> P |] ==> x \<le> psup P"

 apply (simp (no_asm_simp) add: preal_le_def) 

 apply (subgoal_tac "P \<noteq> {}") 

 apply (auto simp add: psup_def preal_sup) 

 done

 lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"

 apply (simp (no_asm_simp) add: preal_le_def)

 apply (simp add: psup_def preal_sup) 

 apply (auto simp add: preal_le_def)

 done

 text{*Supremum property*}

 lemma preal_complete:

      "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"

 apply (simp add: preal_less_def psup_def preal_sup)

 apply (auto simp add: preal_le_def)

 apply (rename_tac U) 

 apply (cut_tac x = U and y = Z in linorder_less_linear)

 apply (auto simp add: preal_less_def)

 done

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

 definition

   realrel   ::  "((preal * preal) * (preal * preal)) set" where

   "realrel = {p. \<exists>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 add: quotient_def)

 definition

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

   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}"

 begin

 definition

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

 definition

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

 definition

   real_add_def: "z + w =

        the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).

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

 definition

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

 definition

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

 definition

   real_mult_def:

     "z * w =

        the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).

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

 definition

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

 definition

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

 definition

   real_le_def: "z \<le> (w::real) \<longleftrightarrow>

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

 definition

   real_less_def: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"

 definition

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

 definition

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

 instance ..

 end

 subsection {* Equivalence relation over positive reals *}

 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: add_ac)

   also have "... = (x2 + y) + x1"  by (simp add: assms)

   also have "... = x2 + (x1 + y)"  by (simp add: add_ac)

   also have "... = x2 + (x + y1)"  by (simp add: assms)

   also have "... = (x2 + y1) + x"  by (simp add: add_ac)

   finally have "(x1 + y2) + x = (x2 + y1) + x" .

   thus ?thesis by (rule add_right_imp_eq)

 qed

 lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"

 by (simp add: realrel_def)

 lemma equiv_realrel: "equiv UNIV realrel"

 apply (auto simp add: equiv_def refl_on_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) \<in> 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)

 declare Abs_Real_inject [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 {* Addition and Subtraction *}

 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: add_assoc)

 apply (rule add_left_commute [of ab, THEN ssubst])

 apply (simp add: add_assoc [symmetric])

 apply (simp add: 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 "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)

         respects2 realrel"

     by (auto 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_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"

 proof -

   have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"

     by (auto simp add: congruent_def add_commute) 

   thus ?thesis

     by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])

 qed

 instance real :: ab_group_add

 proof

   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 add_ac)

   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)

 qed

 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)"

 apply (simp add: add_left_commute add_assoc [symmetric])

 apply (simp add: add_assoc distrib_left [symmetric])

 apply (simp add: 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: mult_commute 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 add_ac 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 algebra_simps)

 done

 lemma real_mult_1: "(1::real) * z = z"

 apply (cases z)

 apply (simp add: real_mult real_one_def algebra_simps)

 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 algebra_simps)

 done

 text{*one and zero are distinct*}

 lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"

 proof -

   have "(1::preal) < 1 + 1"

     by (simp add: preal_self_less_add_left)

   thus ?thesis

     by (simp add: real_zero_def real_one_def)

 qed

 instance real :: comm_ring_1

 proof

   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 \<noteq> (1::real)" by (rule real_zero_not_eq_one)

 qed

 subsection {* Inverse and Division *}

 lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"

 by (simp add: real_zero_def 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 \<noteq> 0 ==> \<exists>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``{(1, inverse (D) + 1)})"

        in exI)

 apply (rule_tac [2]

         x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 

        in exI)

 apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)

 done

 lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"

 apply (simp add: real_inverse_def)

 apply (drule real_mult_inverse_left_ex, safe)

 apply (rule theI, assumption, rename_tac z)

 apply (subgoal_tac "(z * x) * y = z * (x * y)")

 apply (simp add: mult_commute)

 apply (rule mult_assoc)

 done

 subsection{*The Real Numbers form a Field*}

 instance real :: field_inverse_zero

 proof

   fix x y z :: real

   show "x \<noteq> 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)

 qed

 subsection{*The @{text "\<le>"} Ordering*}

 lemma real_le_refl: "w \<le> (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 \<le> a"

   shows "b \<le> (d::preal)"

 proof -

   have "c+d \<le> a+d" by (simp add: le)

   hence "a+b \<le> a+d" by (simp add: eq)

   thus "b \<le> d" by simp

 qed

 lemma real_le_lemma:

   assumes l: "u1 + v2 \<le> u2 + v1"

     and "x1 + v1 = u1 + y1"

     and "x2 + v2 = u2 + y2"

   shows "x1 + y2 \<le> 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: add_ac)

   also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: assms)

   finally show ?thesis by simp

 qed

 lemma real_le: 

      "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  

       (x1 + y2 \<le> x2 + y1)"

 apply (simp add: real_le_def)

 apply (auto intro: real_le_lemma)

 done

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

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

 lemma real_trans_lemma:

   assumes "x + v \<le> u + y"

     and "u + v' \<le> u' + v"

     and "x2 + v2 = u2 + y2"

   shows "x + v' \<le> u' + (y::preal)"

 proof -

   have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)

   also have "... \<le> (u+y) + (u+v')" by (simp add: assms)

   also have "... \<le> (u+y) + (u'+v)" by (simp add: assms)

   also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)

   finally show ?thesis by simp

 qed

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

 apply (cases i, cases j, cases k)

 apply (simp add: real_le)

 apply (blast intro: real_trans_lemma)

 done

 instance real :: order

 proof

   fix u v :: real

   show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u" 

     by (auto simp add: real_less_def intro: real_le_antisym)

 qed (assumption | rule real_le_refl real_le_trans real_le_antisym)+

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

 lemma real_le_linear: "(z::real) \<le> w | w \<le> z"

 apply (cases z, cases w)

 apply (auto simp add: real_le real_zero_def add_ac)

 done

 instance real :: linorder

   by (intro_classes, rule real_le_linear)

 lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"

 apply (cases x, cases y) 

 apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus

                       add_ac)

 apply (simp_all add: add_assoc [symmetric])

 done

 lemma real_add_left_mono: 

   assumes le: "x \<le> y" shows "z + x \<le> 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"])

 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])

 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: "[| 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 "\<le>"} relation *}

 apply (auto dest!: less_add_left_Ex

      simp add: algebra_simps 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: algebra_simps)

 done

 instantiation real :: distrib_lattice

 begin

 definition

   "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"

 definition

   "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"

 instance

   by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)

 end

 subsection{*The Reals Form an Ordered Field*}

 instance real :: linordered_field_inverse_zero

 proof

   fix x y z :: real

   show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)

   show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)

   show "\<bar>x\<bar> = (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)

 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 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:

       "\<exists>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 add_ac)

 apply (cut_tac x = x and y = y in linorder_less_linear)

 apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])

 done

 lemma real_of_preal_leD:

       "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"

 by (simp add: real_of_preal_def real_le)

 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])

 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 \<le> real_of_preal m2) = (m1 \<le> m2)"

 by (simp add: linorder_not_less [symmetric])

 lemma real_of_preal_zero_less: "0 < real_of_preal m"

 apply (insert preal_self_less_add_left [of 1 m])

 apply (auto simp add: real_zero_def real_of_preal_def

                       real_less_def real_le_def add_ac)

 apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)

 apply (simp add: 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*}

 lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>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 ==> \<exists>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 \<le> x ==> \<exists>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 \<le> 0 ==> \<forall>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 ==> \<forall>x. y < real_of_preal x"

 by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])

 subsection {* Completeness of Positive Reals *}

 text {*

   Supremum property for the set of positive reals

   Let @{text "P"} be a non-empty set of positive reals, with an upper

   bound @{text "y"}.  Then @{text "P"} has a least upper bound

   (written @{text "S"}).

   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?

 *}

 lemma posreal_complete:

   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"

     and not_empty_P: "\<exists>x. x \<in> P"

     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"

   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

 proof (rule exI, rule allI)

   fix y

   let ?pP = "{w. real_of_preal w \<in> P}"

   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"

   proof (cases "0 < y")

     assume neg_y: "\<not> 0 < y"

     show ?thesis

     proof

       assume "\<exists>x\<in>P. y < x"

       have "\<forall>x. y < real_of_preal x"

         using neg_y by (rule real_less_all_real2)

       thus "y < real_of_preal (psup ?pP)" ..

     next

       assume "y < real_of_preal (psup ?pP)"

       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..

       hence "0 < x" using positive_P by simp

       hence "y < x" using neg_y by simp

       thus "\<exists>x \<in> P. y < x" using x_in_P ..

     qed

   next

     assume pos_y: "0 < y"

     then obtain py where y_is_py: "y = real_of_preal py"

       by (auto simp add: real_gt_zero_preal_Ex)

     obtain a where "a \<in> 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 add: real_gt_zero_preal_Ex)

     hence "pa \<in> ?pP" using `a \<in> P` by auto

     hence pP_not_empty: "?pP \<noteq> {}" by auto

     obtain sup where sup: "\<forall>x \<in> P. x < sup"

       using upper_bound_Ex ..

     from this and `a \<in> P` have "a < sup" ..

     hence "0 < sup" using a_pos by arith

     then obtain possup where "sup = real_of_preal possup"

       by (auto simp add: real_gt_zero_preal_Ex)

     hence "\<forall>X \<in> ?pP. X \<le> possup"

       using sup by (auto simp add: real_of_preal_lessI)

     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"

       by (rule preal_complete)

     show ?thesis

     proof

       assume "\<exists>x \<in> P. y < x"

       then obtain x where x_in_P: "x \<in> 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 add: real_gt_zero_preal_Ex)

       have py_less_X: "\<exists>X \<in> ?pP. py < X"

       proof

         show "py < px" using y_is_py and x_is_px and y_less_x

           by (simp add: real_of_preal_lessI)

         show "px \<in> ?pP" using x_in_P and x_is_px by simp

       qed

       have "(\<exists>X \<in> ?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 \<in> P})"

         using y_is_py and pos_y by (simp add: real_of_preal_lessI)

     next

       assume y_less_psup: "y < real_of_preal (psup ?pP)"

       hence "py < psup ?pP" using y_is_py

         by (simp add: real_of_preal_lessI)

       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?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 add: real_of_preal_lessI)

       moreover have "x \<in> P" using x_is_X and X_in_pP by simp

       ultimately show "\<exists> x \<in> P. y < x" ..

     qed

   qed

 qed

 text {*

   \medskip Completeness

 *}

 lemma reals_complete:

   fixes S :: "real set"

   assumes notempty_S: "\<exists>X. X \<in> S"

     and exists_Ub: "bdd_above S"

   shows "\<exists>x. (\<forall>s\<in>S. s \<le> x) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> x \<le> y)"

 proof -

   obtain X where X_in_S: "X \<in> S" using notempty_S ..

   obtain Y where Y_isUb: "\<forall>s\<in>S. s \<le> Y"

     using exists_Ub by (auto simp: bdd_above_def)

   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"

   {

     fix x

     assume S_le_x: "\<forall>s\<in>S. s \<le> x"

     {

       fix s

       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"

       hence "\<exists> x \<in> S. s = x + -X + 1" ..

       then obtain x1 where x1: "x1 \<in> S" "s = x1 + (-X) + 1" ..

       then have "x1 \<le> x" using S_le_x by simp

       with x1 have "s \<le> x + - X + 1" by arith

     }

     then have "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1"

       by auto

   } note S_Ub_is_SHIFT_Ub = this

   have *: "\<forall>s\<in>?SHIFT. s \<le> Y + (-X) + 1" using Y_isUb by (rule S_Ub_is_SHIFT_Ub)

   have "\<forall>s\<in>?SHIFT. s < Y + (-X) + 2"

   proof

     fix s assume "s\<in>?SHIFT"

     with * have "s \<le> Y + (-X) + 1" by simp

     also have "\<dots> < Y + (-X) + 2" by simp

     finally show "s < Y + (-X) + 2" .

   qed

   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto

   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"

     using X_in_S and Y_isUb by auto

   ultimately obtain t where t_is_Lub: "\<forall>y. (\<exists>x\<in>?SHIFT. y < x) = (y < t)"

     using posreal_complete [of ?SHIFT] unfolding bdd_above_def by blast

   show ?thesis

   proof

     show "(\<forall>s\<in>S. s \<le> (t + X + (-1))) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> (t + X + (-1)) \<le> y)"

     proof safe

       fix x

       assume "\<forall>s\<in>S. s \<le> x"

       hence "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1"

         using S_Ub_is_SHIFT_Ub by simp

       then have "\<not> x + (-X) + 1 < t"

         by (subst t_is_Lub[rule_format, symmetric]) (simp add: not_less)

       thus "t + X + -1 \<le> x" by arith

     next

       fix y

       assume y_in_S: "y \<in> S"

       obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..

       hence "\<exists> x \<in> S. u = x + - X + 1" by simp

       then obtain "x" where x_and_u: "u = x + - X + 1" ..

       have u_le_t: "u \<le> 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 \<le> t"  by simp

       qed

       show "y \<le> t + X + -1"

       proof cases

         assume "y \<le> x"

         moreover have "x = u + X + - 1" using x_and_u by arith

         moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith

         ultimately show "y  \<le> t + X + -1" by arith

       next

         assume "~(y \<le> x)"

         hence x_less_y: "x < y" by arith

         have "x + (-X) + 1 \<in> ?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 \<in> ?SHIFT" using y_in_S by simp

         have "y + (-X) + 1 \<le> t"

         proof (rule dense_le)

           fix x assume "x < y + (-X) + 1" then have "x < t"

             using * t_is_Lub by auto

           then show "x \<le> t"  by simp

         qed

         thus ?thesis by simp

       qed

     qed

   qed

 qed

 subsection {* The Archimedean Property of the Reals *}

 theorem reals_Archimedean:

   fixes x :: real

   assumes x_pos: "0 < x"

   shows "\<exists>n. inverse (of_nat (Suc n)) < x"

 proof (rule ccontr)

   assume contr: "\<not> ?thesis"

   have "\<forall>n. x * of_nat (Suc n) <= 1"

   proof

     fix n

     from contr have "x \<le> inverse (of_nat (Suc n))"

       by (simp add: linorder_not_less)

     hence "x \<le> (1 / (of_nat (Suc n)))"

       by (simp add: inverse_eq_divide)

     moreover have "(0::real) \<le> of_nat (Suc n)"

       by (rule of_nat_0_le_iff)

     ultimately have "x * of_nat (Suc n) \<le> (1 / of_nat (Suc n)) * of_nat (Suc n)"

       by (rule mult_right_mono)

     thus "x * of_nat (Suc n) \<le> 1" by (simp del: of_nat_Suc)

   qed

   hence 2: "bdd_above {z. \<exists>n. z = x * (of_nat (Suc n))}"

     by (auto intro!: bdd_aboveI[of _ 1])

   have 1: "\<exists>X. X \<in> {z. \<exists>n. z = x* (of_nat (Suc n))}" by auto

   obtain t where

     upper: "\<And>z. z \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> z \<le> t" and

     least: "\<And>y. (\<And>a. a \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> a \<le> y) \<Longrightarrow> t \<le> y"

     using reals_complete[OF 1 2] by auto

   have "t \<le> t + - x"

   proof (rule least)

     fix a assume a: "a \<in> {z. \<exists>n. z = x * (of_nat (Suc n))}"

     have "\<forall>n::nat. x * of_nat n \<le> t + - x"

     proof

       fix n

       have "x * of_nat (Suc n) \<le> t"

         by (simp add: upper)

       hence  "x * (of_nat n) + x \<le> t"

         by (simp add: distrib_left)

       thus  "x * (of_nat n) \<le> t + - x" by arith

     qed    hence "\<forall>m. x * of_nat (Suc m) \<le> t + - x" by (simp del: of_nat_Suc)

     with a show "a \<le> t + - x"

       by auto

   qed

   thus False using x_pos by arith

 qed

 text {*

   There must be other proofs, e.g. @{text Suc} of the largest

   integer in the cut representing @{text "x"}.

 *}

 lemma reals_Archimedean2: "\<exists>n. (x::real) < of_nat (n::nat)"

 proof cases

   assume "x \<le> 0"

   hence "x < of_nat (1::nat)" by simp

   thus ?thesis ..

 next

   assume "\<not> x \<le> 0"

   hence x_greater_zero: "0 < x" by simp

   hence "0 < inverse x" by simp

   then obtain n where "inverse (of_nat (Suc n)) < inverse x"

     using reals_Archimedean by blast

   hence "inverse (of_nat (Suc n)) * x < inverse x * x"

     using x_greater_zero by (rule mult_strict_right_mono)

   hence "inverse (of_nat (Suc n)) * x < 1"

     using x_greater_zero by simp

   hence "of_nat (Suc n) * (inverse (of_nat (Suc n)) * x) < of_nat (Suc n) * 1"

     by (rule mult_strict_left_mono) (simp del: of_nat_Suc)

   hence "x < of_nat (Suc n)"

     by (simp add: algebra_simps del: of_nat_Suc)

   thus "\<exists>(n::nat). x < of_nat n" ..

 qed

 instance real :: archimedean_field

 proof

   fix r :: real

   obtain n :: nat where "r < of_nat n"

     using reals_Archimedean2 ..

   then have "r \<le> of_int (int n)"

     by simp

   then show "\<exists>z. r \<le> of_int z" ..

 qed

 end