# HG changeset patch
# User huffman
# Date 1273518778 25200
# Node ID e05e1283c550f1a3d64269430b1458e9c5ba1862
# Parent  f736a853864fe53dbff8607a1b7c5e2bfbe41268
new construction of real numbers using Cauchy sequences

diff -r f736a853864f -r e05e1283c550 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon May 10 11:47:56 2010 -0700
+++ b/src/HOL/IsaMakefile	Mon May 10 12:12:58 2010 -0700
@@ -354,7 +354,6 @@
   Lubs.thy \
   MacLaurin.thy \
   NthRoot.thy \
-  PReal.thy \
   Parity.thy \
   RComplete.thy \
   Rat.thy \
diff -r f736a853864f -r e05e1283c550 src/HOL/PReal.thy
--- a/src/HOL/PReal.thy	Mon May 10 11:47:56 2010 -0700
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1163 +0,0 @@
-(*  Title:      HOL/PReal.thy
-    Author:     Jacques D. Fleuriot, University of Cambridge
-
-The positive reals as Dedekind sections of positive
-rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
-provides some of the definitions.
-*)
-
-header {* Positive real numbers *}
-
-theory PReal
-imports Rat
-begin
-
-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
-  [code del]: "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::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
-
-
-typedef preal = "{A. cut A}"
-  by (blast intro: cut_of_rat [OF zero_less_one])
-
-definition
-  psup :: "preal set => preal" where
-  [code del]: "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
-  [code del]: "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
-  [code del]: "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 [code del]:
-    "R < S == Rep_preal R < Rep_preal S"
-
-definition
-  preal_le_def [code del]:
-    "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"
-by (unfold preal_def cut_def, blast)
-
-lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"
-by (drule preal_nonempty, fast)
-
-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"
-by (drule preal_imp_psubset_positives, auto)
-
-lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
-by (unfold preal_def cut_def, blast)
-
-lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
-by (unfold preal_def cut_def, 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*}
-
-lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A"
-by blast
-
-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: left_distrib [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 [simp]: "0 < x" "x \<notin> A" by blast
-  from preal_exists_bound [OF B]
-  obtain y where [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 \<in> A" and "v \<in> B" and "x*y = u*v"
-              moreover
-              with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
-              moreover
-              with prems have "0\<le>v"
-                by (blast intro: preal_imp_pos [OF B]  order_less_imp_le prems)
-              moreover
-        from calculation
-              have "u*v < x*y" by (blast intro: mult_strict_mono prems)
-              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 prems)
-      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: right_distrib)
-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 right_distrib 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 right_distrib 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: prems, induct n)
-    case 0
-      from preal_nonempty [OF A]
-      show ?case  by force 
-    case (Suc k)
-      from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
-      hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
-      thus ?case by (force simp add: algebra_simps prems) 
-  qed
-next
-  case (neg n)
-  with prems 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 with 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)
-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 prems
-    by (blast intro: not_in_Rep_preal_ub ) 
-  ultimately show ?thesis using prems
-    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)
-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
-
-end
diff -r f736a853864f -r e05e1283c550 src/HOL/RComplete.thy
--- a/src/HOL/RComplete.thy	Mon May 10 11:47:56 2010 -0700
+++ b/src/HOL/RComplete.thy	Mon May 10 12:12:58 2010 -0700
@@ -30,92 +30,27 @@
   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
 *}
 
+text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}
+
 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
+proof -
+  from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"
+    by (auto intro: less_imp_le)
+  from complete_real [OF not_empty_P this] obtain S
+  where S1: "\<And>x. x \<in> P \<Longrightarrow> x \<le> S" and S2: "\<And>z. \<forall>x\<in>P. x \<le> z \<Longrightarrow> S \<le> z" by fast
+  have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
+  proof
+    fix y show "(\<exists>x\<in>P. y < x) = (y < S)"
+      apply (cases "\<exists>x\<in>P. y < x", simp_all)
+      apply (clarify, drule S1, simp)
+      apply (simp add: not_less S2)
+      done
   qed
+  thus ?thesis ..
 qed
 
 text {*
@@ -130,89 +65,6 @@
 
 
 text {*
-  \medskip Completeness theorem for the positive reals (again).
-*}
-
-lemma posreals_complete:
-  assumes positive_S: "\<forall>x \<in> S. 0 < x"
-    and not_empty_S: "\<exists>x. x \<in> S"
-    and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
-  shows "\<exists>t. isLub (UNIV::real set) S t"
-proof
-  let ?pS = "{w. real_of_preal w \<in> S}"
-
-  obtain u where "isUb UNIV S u" using upper_bound_Ex ..
-  hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
-
-  obtain x where x_in_S: "x \<in> S" using not_empty_S ..
-  hence x_gt_zero: "0 < x" using positive_S by simp
-  have  "x \<le> u" using sup and x_in_S ..
-  hence "0 < u" using x_gt_zero by arith
-
-  then obtain pu where u_is_pu: "u = real_of_preal pu"
-    by (auto simp add: real_gt_zero_preal_Ex)
-
-  have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
-  proof
-    fix pa
-    assume "pa \<in> ?pS"
-    then obtain a where "a \<in> S" and "a = real_of_preal pa"
-      by simp
-    moreover hence "a \<le> u" using sup by simp
-    ultimately show "pa \<le> pu"
-      using sup and u_is_pu by (simp add: real_of_preal_le_iff)
-  qed
-
-  have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
-  proof
-    fix y
-    assume y_in_S: "y \<in> S"
-    hence "0 < y" using positive_S by simp
-    then obtain py where y_is_py: "y = real_of_preal py"
-      by (auto simp add: real_gt_zero_preal_Ex)
-    hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
-    with pS_less_pu have "py \<le> psup ?pS"
-      by (rule preal_psup_le)
-    thus "y \<le> real_of_preal (psup ?pS)"
-      using y_is_py by (simp add: real_of_preal_le_iff)
-  qed
-
-  moreover {
-    fix x
-    assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
-    have "real_of_preal (psup ?pS) \<le> x"
-    proof -
-      obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
-      hence s_pos: "0 < s" using positive_S by simp
-
-      hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
-      then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
-      hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
-
-      from x_ub_S have "s \<le> x" using s_in_S ..
-      hence "0 < x" using s_pos by simp
-      hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
-      then obtain "px" where x_is_px: "x = real_of_preal px" ..
-
-      have "\<forall>pe \<in> ?pS. pe \<le> px"
-      proof
-        fix pe
-        assume "pe \<in> ?pS"
-        hence "real_of_preal pe \<in> S" by simp
-        hence "real_of_preal pe \<le> x" using x_ub_S by simp
-        thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
-      qed
-
-      moreover have "?pS \<noteq> {}" using ps_in_pS by auto
-      ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
-      thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
-    qed
-  }
-  ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
-    by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
-qed
-
-text {*
   \medskip reals Completeness (again!)
 *}
 
@@ -221,87 +73,11 @@
     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
   shows "\<exists>t. isLub (UNIV :: real set) S t"
 proof -
-  obtain X where X_in_S: "X \<in> S" using notempty_S ..
-  obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
-    using exists_Ub ..
-  let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
-
-  {
-    fix x
-    assume "isUb (UNIV::real set) S x"
-    hence S_le_x: "\<forall> y \<in> S. y <= x"
-      by (simp add: isUb_def setle_def)
-    {
-      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 \<in> S" and "s = x1 + (-X) + 1" ..
-      moreover hence "x1 \<le> x" using S_le_x by simp
-      ultimately have "s \<le> x + - X + 1" by arith
-    }
-    then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
-      by (auto simp add: isUb_def setle_def)
-  } note S_Ub_is_SHIFT_Ub = this
-
-  hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
-  hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
-  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: "isLub UNIV ?SHIFT t"
-    using posreals_complete [of ?SHIFT] by blast
-
-  show ?thesis
-  proof
-    show "isLub UNIV S (t + X + (-1))"
-    proof (rule isLubI2)
-      {
-        fix x
-        assume "isUb (UNIV::real set) S x"
-        hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
-          using S_Ub_is_SHIFT_Ub by simp
-        hence "t \<le> (x + (-X) + 1)"
-          using t_is_Lub by (simp add: isLub_le_isUb)
-        hence "t + X + -1 \<le> x" by arith
-      }
-      then show "(t + X + -1) <=* Collect (isUb UNIV S)"
-        by (simp add: setgeI)
-    next
-      show "isUb UNIV S (t + X + -1)"
-      proof -
-        {
-          fix y
-          assume y_in_S: "y \<in> S"
-          have "y \<le> t + X + -1"
-          proof -
-            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" using u_in_shift and t_is_Lub by (simp add: isLubD2)
-
-            show ?thesis
-            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
-              hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
-              thus ?thesis by simp
-            qed
-          qed
-        }
-        then show ?thesis by (simp add: isUb_def setle_def)
-      qed
-    qed
-  qed
+  from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"
+    unfolding isUb_def setle_def by simp_all
+  from complete_real [OF this] show ?thesis
+    unfolding isLub_def leastP_def setle_def setge_def Ball_def
+      Collect_def mem_def isUb_def UNIV_def by simp
 qed
 
 text{*A version of the same theorem without all those predicates!*}
@@ -310,13 +86,7 @@
   assumes "\<exists>y. y\<in>S" and "\<exists>(x::real). \<forall>y\<in>S. y \<le> x"
   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) & 
                (\<forall>z. ((\<forall>y\<in>S. y \<le> z) --> x \<le> z))"
-proof -
-  have "\<exists>x. isLub UNIV S x" 
-    by (rule reals_complete)
-       (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def prems)
-  thus ?thesis
-    by (metis UNIV_I isLub_isUb isLub_le_isUb isUbD isUb_def setleI)
-qed
+using assms by (rule complete_real)
 
 
 subsection {* The Archimedean Property of the Reals *}
@@ -324,88 +94,11 @@
 theorem reals_Archimedean:
   assumes x_pos: "0 < x"
   shows "\<exists>n. inverse (real (Suc n)) < x"
-proof (rule ccontr)
-  assume contr: "\<not> ?thesis"
-  have "\<forall>n. x * real (Suc n) <= 1"
-  proof
-    fix n
-    from contr have "x \<le> inverse (real (Suc n))"
-      by (simp add: linorder_not_less)
-    hence "x \<le> (1 / (real (Suc n)))"
-      by (simp add: inverse_eq_divide)
-    moreover have "0 \<le> real (Suc n)"
-      by (rule real_of_nat_ge_zero)
-    ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
-      by (rule mult_right_mono)
-    thus "x * real (Suc n) \<le> 1" by simp
-  qed
-  hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
-    by (simp add: setle_def, safe, rule spec)
-  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
-    by (simp add: isUbI)
-  hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
-  moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
-  ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
-    by (simp add: reals_complete)
-  then obtain "t" where
-    t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
-
-  have "\<forall>n::nat. x * real n \<le> t + - x"
-  proof
-    fix n
-    from t_is_Lub have "x * real (Suc n) \<le> t"
-      by (simp add: isLubD2)
-    hence  "x * (real n) + x \<le> t"
-      by (simp add: right_distrib real_of_nat_Suc)
-    thus  "x * (real n) \<le> t + - x" by arith
-  qed
-
-  hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
-  hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
-    by (auto simp add: setle_def)
-  hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
-    by (simp add: isUbI)
-  hence "t \<le> t + - x"
-    using t_is_Lub by (simp add: isLub_le_isUb)
-  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"}.
-*}
+  unfolding real_of_nat_def using x_pos
+  by (rule ex_inverse_of_nat_Suc_less)
 
 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
-proof cases
-  assume "x \<le> 0"
-  hence "x < real (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 (real (Suc n)) < inverse x"
-    using reals_Archimedean by blast
-  hence "inverse (real (Suc n)) * x < inverse x * x"
-    using x_greater_zero by (rule mult_strict_right_mono)
-  hence "inverse (real (Suc n)) * x < 1"
-    using x_greater_zero by simp
-  hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
-    by (rule mult_strict_left_mono) simp
-  hence "x < real (Suc n)"
-    by (simp add: algebra_simps)
-  thus "\<exists>(n::nat). x < real n" ..
-qed
-
-instance real :: archimedean_field
-proof
-  fix r :: real
-  obtain n :: nat where "r < real n"
-    using reals_Archimedean2 ..
-  then have "r \<le> of_int (int n)"
-    unfolding real_eq_of_nat by simp
-  then show "\<exists>z. r \<le> of_int z" ..
-qed
+  unfolding real_of_nat_def by (rule ex_less_of_nat)
 
 lemma reals_Archimedean3:
   assumes x_greater_zero: "0 < x"
@@ -458,7 +151,7 @@
   have "x = y-(y-x)" by simp
   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
   also have "\<dots> = real p / real q"
-    by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc 
+    by (simp only: inverse_eq_divide diff_def real_of_nat_Suc 
     minus_divide_left add_divide_distrib[THEN sym]) simp
   finally have "x<r" by (unfold r_def)
   have "p<Suc p" .. also note main[THEN sym]
diff -r f736a853864f -r e05e1283c550 src/HOL/RealDef.thy
--- a/src/HOL/RealDef.thy	Mon May 10 11:47:56 2010 -0700
+++ b/src/HOL/RealDef.thy	Mon May 10 12:12:58 2010 -0700
@@ -1,508 +1,1064 @@
 (*  Title       : HOL/RealDef.thy
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
+    Author      : Jacques D. Fleuriot, 1998
     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
     Additional contributions by Jeremy Avigad
+    Construction of Cauchy Reals by Brian Huffman, 2010
 *)
 
-header{*Defining the Reals from the Positive Reals*}
+header {* Development of the Reals using Cauchy Sequences *}
 
 theory RealDef
-imports PReal
+imports Rat
 begin
 
+text {*
+  This theory contains a formalization of the real numbers as
+  equivalence classes of Cauchy sequences of rationals.  See
+  \url{HOL/ex/Dedekind_Real.thy} for an alternative construction
+  using Dedekind cuts.
+*}
+
+subsection {* Preliminary lemmas *}
+
+lemma add_diff_add:
+  fixes a b c d :: "'a::ab_group_add"
+  shows "(a + c) - (b + d) = (a - b) + (c - d)"
+  by simp
+
+lemma minus_diff_minus:
+  fixes a b :: "'a::ab_group_add"
+  shows "- a - - b = - (a - b)"
+  by simp
+
+lemma mult_diff_mult:
+  fixes x y a b :: "'a::ring"
+  shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
+  by (simp add: algebra_simps)
+
+lemma inverse_diff_inverse:
+  fixes a b :: "'a::division_ring"
+  assumes "a \<noteq> 0" and "b \<noteq> 0"
+  shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
+  using assms by (simp add: algebra_simps)
+
+lemma obtain_pos_sum:
+  fixes r :: rat assumes r: "0 < r"
+  obtains s t where "0 < s" and "0 < t" and "r = s + t"
+proof
+    from r show "0 < r/2" by simp
+    from r show "0 < r/2" by simp
+    show "r = r/2 + r/2" by simp
+qed
+
+subsection {* Sequences that converge to zero *}
+
 definition
-  realrel   ::  "((preal * preal) * (preal * preal)) set" where
-  [code del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
+  vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
+where
+  "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
+
+lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
+  unfolding vanishes_def by simp
+
+lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
+  unfolding vanishes_def by simp
+
+lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
+  unfolding vanishes_def
+  apply (cases "c = 0", auto)
+  apply (rule exI [where x="\<bar>c\<bar>"], auto)
+  done
+
+lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
+  unfolding vanishes_def by simp
 
-typedef (Real)  real = "UNIV//realrel"
-  by (auto simp add: quotient_def)
+lemma vanishes_add:
+  assumes X: "vanishes X" and Y: "vanishes Y"
+  shows "vanishes (\<lambda>n. X n + Y n)"
+proof (rule vanishesI)
+  fix r :: rat assume "0 < r"
+  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
+    by (rule obtain_pos_sum)
+  obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
+    using vanishesD [OF X s] ..
+  obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
+    using vanishesD [OF Y t] ..
+  have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
+  proof (clarsimp)
+    fix n assume n: "i \<le> n" "j \<le> n"
+    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
+    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
+    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
+  qed
+  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
+qed
+
+lemma vanishes_diff:
+  assumes X: "vanishes X" and Y: "vanishes Y"
+  shows "vanishes (\<lambda>n. X n - Y n)"
+unfolding diff_minus by (intro vanishes_add vanishes_minus X Y)
+
+lemma vanishes_mult_bounded:
+  assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
+  assumes Y: "vanishes (\<lambda>n. Y n)"
+  shows "vanishes (\<lambda>n. X n * Y n)"
+proof (rule vanishesI)
+  fix r :: rat assume r: "0 < r"
+  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
+    using X by fast
+  obtain b where b: "0 < b" "r = a * b"
+  proof
+    show "0 < r / a" using r a by (simp add: divide_pos_pos)
+    show "r = a * (r / a)" using a by simp
+  qed
+  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
+    using vanishesD [OF Y b(1)] ..
+  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
+    by (simp add: b(2) abs_mult mult_strict_mono' a k)
+  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
+qed
+
+subsection {* Cauchy sequences *}
 
 definition
-  (** these don't use the overloaded "real" function: users don't see them **)
-  real_of_preal :: "preal => real" where
-  [code del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
+  cauchy :: "(nat \<Rightarrow> rat) set"
+where
+  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
+
+lemma cauchyI:
+  "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r) \<Longrightarrow> cauchy X"
+  unfolding cauchy_def by simp
+
+lemma cauchyD:
+  "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
+  unfolding cauchy_def by simp
+
+lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
+  unfolding cauchy_def by simp
+
+lemma cauchy_add [simp]:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "cauchy (\<lambda>n. X n + Y n)"
+proof (rule cauchyI)
+  fix r :: rat assume "0 < r"
+  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
+    by (rule obtain_pos_sum)
+  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
+    using cauchyD [OF X s] ..
+  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
+    using cauchyD [OF Y t] ..
+  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
+  proof (clarsimp)
+    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
+    have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
+      unfolding add_diff_add by (rule abs_triangle_ineq)
+    also have "\<dots> < s + t"
+      by (rule add_strict_mono, simp_all add: i j *)
+    finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
+  qed
+  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
+qed
+
+lemma cauchy_minus [simp]:
+  assumes X: "cauchy X"
+  shows "cauchy (\<lambda>n. - X n)"
+using assms unfolding cauchy_def
+unfolding minus_diff_minus abs_minus_cancel .
+
+lemma cauchy_diff [simp]:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "cauchy (\<lambda>n. X n - Y n)"
+using assms unfolding diff_minus by simp
+
+lemma cauchy_imp_bounded:
+  assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
+proof -
+  obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
+    using cauchyD [OF assms zero_less_one] ..
+  show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
+  proof (intro exI conjI allI)
+    have "0 \<le> \<bar>X 0\<bar>" by simp
+    also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp
+    finally have "0 \<le> Max (abs ` X ` {..k})" .
+    thus "0 < Max (abs ` X ` {..k}) + 1" by simp
+  next
+    fix n :: nat
+    show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
+    proof (rule linorder_le_cases)
+      assume "n \<le> k"
+      hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
+      thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
+    next
+      assume "k \<le> n"
+      have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
+      also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
+        by (rule abs_triangle_ineq)
+      also have "\<dots> < Max (abs ` X ` {..k}) + 1"
+        by (rule add_le_less_mono, simp, simp add: k `k \<le> n`)
+      finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" .
+    qed
+  qed
+qed
+
+lemma cauchy_mult [simp]:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "cauchy (\<lambda>n. X n * Y n)"
+proof (rule cauchyI)
+  fix r :: rat assume "0 < r"
+  then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
+    by (rule obtain_pos_sum)
+  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
+    using cauchy_imp_bounded [OF X] by fast
+  obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
+    using cauchy_imp_bounded [OF Y] by fast
+  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
+  proof
+    show "0 < v/b" using v b(1) by (rule divide_pos_pos)
+    show "0 < u/a" using u a(1) by (rule divide_pos_pos)
+    show "r = a * (u/a) + (v/b) * b"
+      using a(1) b(1) `r = u + v` by simp
+  qed
+  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
+    using cauchyD [OF X s] ..
+  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
+    using cauchyD [OF Y t] ..
+  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
+  proof (clarsimp)
+    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
+    have "\<bar>X m * Y m - X n * Y n\<bar> = \<bar>X m * (Y m - Y n) + (X m - X n) * Y n\<bar>"
+      unfolding mult_diff_mult ..
+    also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
+      by (rule abs_triangle_ineq)
+    also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
+      unfolding abs_mult ..
+    also have "\<dots> < a * t + s * b"
+      by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
+    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
+  qed
+  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
+qed
+
+lemma cauchy_not_vanishes_cases:
+  assumes X: "cauchy X"
+  assumes nz: "\<not> vanishes X"
+  shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
+proof -
+  obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
+    using nz unfolding vanishes_def by (auto simp add: not_less)
+  obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
+    using `0 < r` by (rule obtain_pos_sum)
+  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
+    using cauchyD [OF X s] ..
+  obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
+    using r by fast
+  have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
+    using i `i \<le> k` by auto
+  have "X k \<le> - r \<or> r \<le> X k"
+    using `r \<le> \<bar>X k\<bar>` by auto
+  hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
+    unfolding `r = s + t` using k by auto
+  hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
+  thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
+    using t by auto
+qed
+
+lemma cauchy_not_vanishes:
+  assumes X: "cauchy X"
+  assumes nz: "\<not> vanishes X"
+  shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
+using cauchy_not_vanishes_cases [OF assms]
+by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
+
+lemma cauchy_inverse [simp]:
+  assumes X: "cauchy X"
+  assumes nz: "\<not> vanishes X"
+  shows "cauchy (\<lambda>n. inverse (X n))"
+proof (rule cauchyI)
+  fix r :: rat assume "0 < r"
+  obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
+    using cauchy_not_vanishes [OF X nz] by fast
+  from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
+  obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
+  proof
+    show "0 < b * r * b"
+      by (simp add: `0 < r` b mult_pos_pos)
+    show "r = inverse b * (b * r * b) * inverse b"
+      using b by simp
+  qed
+  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
+    using cauchyD [OF X s] ..
+  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
+  proof (clarsimp)
+    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
+    have "\<bar>inverse (X m) - inverse (X n)\<bar> =
+          inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
+      by (simp add: inverse_diff_inverse nz * abs_mult)
+    also have "\<dots> < inverse b * s * inverse b"
+      by (simp add: mult_strict_mono less_imp_inverse_less
+                    mult_pos_pos i j b * s)
+    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
+  qed
+  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
+qed
+
+subsection {* Equivalence relation on Cauchy sequences *}
+
+definition
+  realrel :: "((nat \<Rightarrow> rat) \<times> (nat \<Rightarrow> rat)) set"
+where
+  "realrel = {(X, Y). cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n)}"
+
+lemma refl_realrel: "refl_on {X. cauchy X} realrel"
+  unfolding realrel_def by (rule refl_onI, clarsimp, simp)
+
+lemma sym_realrel: "sym realrel"
+  unfolding realrel_def
+  by (rule symI, clarify, drule vanishes_minus, simp)
 
-instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
+lemma trans_realrel: "trans realrel"
+  unfolding realrel_def
+  apply (rule transI, clarify)
+  apply (drule (1) vanishes_add)
+  apply (simp add: algebra_simps)
+  done
+
+lemma equiv_realrel: "equiv {X. cauchy X} realrel"
+  using refl_realrel sym_realrel trans_realrel
+  by (rule equiv.intro)
+
+subsection {* The field of real numbers *}
+
+typedef (open) real = "{X. cauchy X} // realrel"
+  by (fast intro: quotientI cauchy_const)
+
+definition
+  Real :: "(nat \<Rightarrow> rat) \<Rightarrow> real"
+where
+  "Real X = Abs_real (realrel `` {X})"
+
+definition
+  real_case :: "((nat \<Rightarrow> rat) \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
+where
+  "real_case f x = (THE y. \<forall>X\<in>Rep_real x. y = f X)"
+
+lemma Real_induct [induct type: real]:
+  "(\<And>X. cauchy X \<Longrightarrow> P (Real X)) \<Longrightarrow> P x"
+  unfolding Real_def
+  apply (induct x)
+  apply (erule quotientE)
+  apply (simp)
+  done
+
+lemma real_case_1:
+  assumes f: "congruent realrel f"
+  assumes X: "cauchy X"
+  shows "real_case f (Real X) = f X"
+  unfolding real_case_def Real_def
+  apply (subst Abs_real_inverse)
+  apply (simp add: quotientI X)
+  apply (rule the_equality)
+  apply clarsimp
+  apply (erule congruent.congruent [OF f])
+  apply (erule bspec)
+  apply simp
+  apply (rule refl_onD [OF refl_realrel])
+  apply (simp add: X)
+  done
+
+lemma real_case_2:
+  assumes f: "congruent2 realrel realrel f"
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "real_case (\<lambda>X. real_case (\<lambda>Y. f X Y) (Real Y)) (Real X) = f X Y"
+ apply (subst real_case_1 [OF _ X])
+  apply (rule congruent.intro)
+  apply (subst real_case_1 [OF _ Y])
+   apply (rule congruent2_implies_congruent [OF equiv_realrel f])
+   apply (simp add: realrel_def)
+  apply (subst real_case_1 [OF _ Y])
+   apply (rule congruent2_implies_congruent [OF equiv_realrel f])
+   apply (simp add: realrel_def)
+  apply (erule congruent2.congruent2 [OF f])
+  apply (rule refl_onD [OF refl_realrel])
+  apply (simp add: Y)
+  apply (rule real_case_1 [OF _ Y])
+  apply (rule congruent2_implies_congruent [OF equiv_realrel f])
+  apply (simp add: X)
+  done
+
+lemma eq_Real:
+  "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
+  unfolding Real_def
+  apply (subst Abs_real_inject)
+  apply (simp add: quotientI)
+  apply (simp add: quotientI)
+  apply (simp add: eq_equiv_class_iff [OF equiv_realrel])
+  apply (simp add: realrel_def)
+  done
+
+lemma add_respects2_realrel:
+  "(\<lambda>X Y. Real (\<lambda>n. X n + Y n)) respects2 realrel"
+proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq])
+  fix X Y show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. Y n + X n)"
+    by (simp add: add_commute)
+next
+  fix X assume X: "cauchy X"
+  fix Y Z assume "(Y, Z) \<in> realrel"
+  hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n - Z n)"
+    unfolding realrel_def by simp_all
+  show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. X n + Z n)"
+  proof (rule eq_Real [THEN iffD2])
+    show "cauchy (\<lambda>n. X n + Y n)" using X Y by (rule cauchy_add)
+    show "cauchy (\<lambda>n. X n + Z n)" using X Z by (rule cauchy_add)
+    show "vanishes (\<lambda>n. (X n + Y n) - (X n + Z n))"
+      unfolding add_diff_add using YZ by simp
+  qed
+qed
+
+lemma minus_respects_realrel:
+  "(\<lambda>X. Real (\<lambda>n. - X n)) respects realrel"
+proof (rule congruent.intro)
+  fix X Y assume "(X, Y) \<in> realrel"
+  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
+    unfolding realrel_def by simp_all
+  show "Real (\<lambda>n. - X n) = Real (\<lambda>n. - Y n)"
+  proof (rule eq_Real [THEN iffD2])
+    show "cauchy (\<lambda>n. - X n)" using X by (rule cauchy_minus)
+    show "cauchy (\<lambda>n. - Y n)" using Y by (rule cauchy_minus)
+    show "vanishes (\<lambda>n. (- X n) - (- Y n))"
+      unfolding minus_diff_minus using XY by (rule vanishes_minus)
+  qed
+qed
+
+lemma mult_respects2_realrel:
+  "(\<lambda>X Y. Real (\<lambda>n. X n * Y n)) respects2 realrel"
+proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq])
+  fix X Y
+  show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. Y n * X n)"
+    by (simp add: mult_commute)
+next
+  fix X assume X: "cauchy X"
+  fix Y Z assume "(Y, Z) \<in> realrel"
+  hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n - Z n)"
+    unfolding realrel_def by simp_all
+  show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. X n * Z n)"
+  proof (rule eq_Real [THEN iffD2])
+    show "cauchy (\<lambda>n. X n * Y n)" using X Y by (rule cauchy_mult)
+    show "cauchy (\<lambda>n. X n * Z n)" using X Z by (rule cauchy_mult)
+    have "vanishes (\<lambda>n. X n * (Y n - Z n))"
+      by (intro vanishes_mult_bounded cauchy_imp_bounded X YZ)
+    thus "vanishes (\<lambda>n. X n * Y n - X n * Z n)"
+      by (simp add: right_diff_distrib)
+  qed
+qed
+
+lemma vanishes_diff_inverse:
+  assumes X: "cauchy X" "\<not> vanishes X"
+  assumes Y: "cauchy Y" "\<not> vanishes Y"
+  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
+  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
+proof (rule vanishesI)
+  fix r :: rat assume r: "0 < r"
+  obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
+    using cauchy_not_vanishes [OF X] by fast
+  obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
+    using cauchy_not_vanishes [OF Y] by fast
+  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
+  proof
+    show "0 < a * r * b"
+      using a r b by (simp add: mult_pos_pos)
+    show "inverse a * (a * r * b) * inverse b = r"
+      using a r b by simp
+  qed
+  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
+    using vanishesD [OF XY s] ..
+  have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
+  proof (clarsimp)
+    fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
+    have "X n \<noteq> 0" and "Y n \<noteq> 0"
+      using i j a b n by auto
+    hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
+        inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
+      by (simp add: inverse_diff_inverse abs_mult)
+    also have "\<dots> < inverse a * s * inverse b"
+      apply (intro mult_strict_mono' less_imp_inverse_less)
+      apply (simp_all add: a b i j k n mult_nonneg_nonneg)
+      done
+    also note `inverse a * s * inverse b = r`
+    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
+  qed
+  thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
+qed
+
+lemma inverse_respects_realrel:
+  "(\<lambda>X. if vanishes X then c else Real (\<lambda>n. inverse (X n))) respects realrel"
+    (is "?inv respects realrel")
+proof (rule congruent.intro)
+  fix X Y assume "(X, Y) \<in> realrel"
+  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
+    unfolding realrel_def by simp_all
+  have "vanishes X \<longleftrightarrow> vanishes Y"
+  proof
+    assume "vanishes X"
+    from vanishes_diff [OF this XY] show "vanishes Y" by simp
+  next
+    assume "vanishes Y"
+    from vanishes_add [OF this XY] show "vanishes X" by simp
+  qed
+  thus "?inv X = ?inv Y"
+    by (simp add: vanishes_diff_inverse eq_Real X Y XY)
+qed
+
+instantiation real :: field_inverse_zero
 begin
 
 definition
-  real_zero_def [code del]: "0 = Abs_Real(realrel``{(1, 1)})"
+  "0 = Real (\<lambda>n. 0)"
 
 definition
-  real_one_def [code del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
+  "1 = Real (\<lambda>n. 1)"
+
+definition
+  "x + y = real_case (\<lambda>X. real_case (\<lambda>Y. Real (\<lambda>n. X n + Y n)) y) x"
 
 definition
-  real_add_def [code del]: "z + w =
-       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
-                 { Abs_Real(realrel``{(x+u, y+v)}) })"
+  "- x = real_case (\<lambda>X. Real (\<lambda>n. - X n)) x"
+
+definition
+  "x - y = (x::real) + - y"
 
 definition
-  real_minus_def [code del]: "- r =  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
+  "x * y = real_case (\<lambda>X. real_case (\<lambda>Y. Real (\<lambda>n. X n * Y n)) y) x"
 
 definition
-  real_diff_def [code del]: "r - (s::real) = r + - s"
+  "inverse =
+    real_case (\<lambda>X. if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
 
 definition
-  real_mult_def [code del]:
-    "z * w =
-       contents (\<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)}) })"
+  "x / y = (x::real) * inverse y"
+
+lemma add_Real:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
+  unfolding plus_real_def
+  by (rule real_case_2 [OF add_respects2_realrel X Y])
+
+lemma minus_Real:
+  assumes X: "cauchy X"
+  shows "- Real X = Real (\<lambda>n. - X n)"
+  unfolding uminus_real_def
+  by (rule real_case_1 [OF minus_respects_realrel X])
 
-definition
-  real_inverse_def [code del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
+lemma diff_Real:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
+  unfolding minus_real_def diff_minus
+  by (simp add: minus_Real add_Real X Y)
 
-definition
-  real_divide_def [code del]: "R / (S::real) = R * inverse S"
+lemma mult_Real:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
+  unfolding times_real_def
+  by (rule real_case_2 [OF mult_respects2_realrel X Y])
+
+lemma inverse_Real:
+  assumes X: "cauchy X"
+  shows "inverse (Real X) =
+    (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
+  unfolding inverse_real_def
+  by (rule real_case_1 [OF inverse_respects_realrel X])
 
-definition
-  real_le_def [code del]: "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 [code del]: "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 ..
+instance proof
+  fix a b c :: real
+  show "a + b = b + a"
+    by (induct a, induct b) (simp add: add_Real add_ac)
+  show "(a + b) + c = a + (b + c)"
+    by (induct a, induct b, induct c) (simp add: add_Real add_ac)
+  show "0 + a = a"
+    unfolding zero_real_def
+    by (induct a) (simp add: add_Real)
+  show "- a + a = 0"
+    unfolding zero_real_def
+    by (induct a) (simp add: minus_Real add_Real)
+  show "a - b = a + - b"
+    by (rule minus_real_def)
+  show "(a * b) * c = a * (b * c)"
+    by (induct a, induct b, induct c) (simp add: mult_Real mult_ac)
+  show "a * b = b * a"
+    by (induct a, induct b) (simp add: mult_Real mult_ac)
+  show "1 * a = a"
+    unfolding one_real_def
+    by (induct a) (simp add: mult_Real)
+  show "(a + b) * c = a * c + b * c"
+    by (induct a, induct b, induct c)
+       (simp add: mult_Real add_Real algebra_simps)
+  show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
+    unfolding zero_real_def one_real_def
+    by (simp add: eq_Real)
+  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
+    unfolding zero_real_def one_real_def
+    apply (induct a)
+    apply (simp add: eq_Real inverse_Real mult_Real)
+    apply (rule vanishesI)
+    apply (frule (1) cauchy_not_vanishes, clarify)
+    apply (rule_tac x=k in exI, clarify)
+    apply (drule_tac x=n in spec, simp)
+    done
+  show "a / b = a * inverse b"
+    by (rule divide_real_def)
+  show "inverse (0::real) = 0"
+    by (simp add: zero_real_def inverse_Real)
+qed
 
 end
 
-subsection {* Equivalence relation over positive reals *}
+subsection {* 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: prems)
-  also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
-  also have "... = x2 + (x + y1)"  by (simp add: prems)
-  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]
+definition
+  positive :: "real \<Rightarrow> bool"
+where
+  "positive = real_case (\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
 
-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)
+lemma bool_congruentI:
+  assumes sym: "sym r"
+  assumes P: "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x \<Longrightarrow> P y"
+  shows "P respects r"
+apply (rule congruent.intro)
+apply (rule iffI)
+apply (erule (1) P)
+apply (erule (1) P [OF symD [OF sym]])
 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 (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 (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 right_distrib [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)
+lemma positive_respects_realrel:
+  "(\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n) respects realrel"
+proof (rule bool_congruentI)
+  show "sym realrel" by (rule sym_realrel)
+next
+  fix X Y assume "(X, Y) \<in> realrel"
+  hence XY: "vanishes (\<lambda>n. X n - Y n)"
+    unfolding realrel_def by simp_all
+  assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
+  then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
+    by fast
+  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
+    using `0 < r` by (rule obtain_pos_sum)
+  obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
+    using vanishesD [OF XY s] ..
+  have "\<forall>n\<ge>max i j. t < Y n"
+  proof (clarsimp)
+    fix n assume n: "i \<le> n" "j \<le> n"
+    have "\<bar>X n - Y n\<bar> < s" and "r < X n"
+      using i j n by simp_all
+    thus "t < Y n" unfolding r by simp
+  qed
+  thus "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
 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
+lemma positive_Real:
+  assumes X: "cauchy X"
+  shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
+unfolding positive_def
+by (rule real_case_1 [OF positive_respects_realrel X])
 
-subsection {* Inverse and Division *}
-
-lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
-by (simp add: real_zero_def add_commute)
+lemma positive_zero: "\<not> positive 0"
+unfolding zero_real_def by (auto simp add: positive_Real)
 
-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)
+lemma positive_add:
+  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
+apply (induct x, induct y, rename_tac Y X)
+apply (simp add: add_Real positive_Real)
+apply (clarify, rename_tac a b i j)
+apply (rule_tac x="a + b" in exI, simp)
+apply (rule_tac x="max i j" in exI, clarsimp)
+apply (simp add: add_strict_mono)
 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)
+lemma positive_mult:
+  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
+apply (induct x, induct y, rename_tac Y X)
+apply (simp add: mult_Real positive_Real)
+apply (clarify, rename_tac a b i j)
+apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)
+apply (rule_tac x="max i j" in exI, clarsimp)
+apply (rule mult_strict_mono, auto)
+done
+
+lemma positive_minus:
+  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
+apply (induct x, rename_tac X)
+apply (simp add: zero_real_def eq_Real minus_Real positive_Real)
+apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
 done
 
+instantiation real :: linordered_field_inverse_zero
+begin
 
-subsection{*The Real Numbers form a Field*}
+definition
+  "x < y \<longleftrightarrow> positive (y - x)"
 
-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
+definition
+  "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
 
-lemma INVERSE_ZERO: "inverse 0 = (0::real)"
-  by (fact inverse_zero)
-
+definition
+  "abs (a::real) = (if a < 0 then - a else a)"
 
-subsection{*The @{text "\<le>"} Ordering*}
-
-lemma real_le_refl: "w \<le> (w::real)"
-by (cases w, force simp add: real_le_def)
+definition
+  "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
 
-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: prems)
-  hence "a+b \<le> a+d" by (simp add: prems)
-  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: prems)
-  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
-  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
-  finally show ?thesis by simp
+instance proof
+  fix a b c :: real
+  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
+    by (rule abs_real_def)
+  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
+    unfolding less_eq_real_def less_real_def
+    by (auto, drule (1) positive_add, simp_all add: positive_zero)
+  show "a \<le> a"
+    unfolding less_eq_real_def by simp
+  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
+    unfolding less_eq_real_def less_real_def
+    by (auto, drule (1) positive_add, simp add: algebra_simps)
+  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
+    unfolding less_eq_real_def less_real_def
+    by (auto, drule (1) positive_add, simp add: positive_zero)
+  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
+    unfolding less_eq_real_def less_real_def by auto
+  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
+    by (rule sgn_real_def)
+  show "a \<le> b \<or> b \<le> a"
+    unfolding less_eq_real_def less_real_def
+    by (auto dest!: positive_minus)
+  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
+    unfolding less_real_def
+    by (drule (1) positive_mult, simp add: algebra_simps)
 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: prems)
-  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
-  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"] diff_minus)
-qed
-
-lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
-by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
-
-lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
-by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
-
-lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
-apply (cases x, cases y)
-apply (simp add: linorder_not_le [where 'a = real, symmetric] 
-                 linorder_not_le [where 'a = preal] 
-                  real_zero_def real_le real_mult)
-  --{*Reduce to the (simpler) @{text "\<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: right_distrib)
-done
+end
 
 instantiation real :: distrib_lattice
 begin
 
 definition
-  "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
+  "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
 
 definition
-  "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
+  "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
+
+instance proof
+qed (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
+
+end
 
-instance
-  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
+instantiation real :: number_ring
+begin
+
+definition
+  "(number_of x :: real) = of_int x"
+
+instance proof
+qed (rule number_of_real_def)
 
 end
 
+lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
+apply (induct x)
+apply (simp add: zero_real_def)
+apply (simp add: one_real_def add_Real)
+done
 
-subsection{*The Reals Form an Ordered Field*}
+lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
+apply (cases x rule: int_diff_cases)
+apply (simp add: of_nat_Real diff_Real)
+done
 
-instance real :: linordered_field_inverse_zero
+lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
+apply (induct x)
+apply (simp add: Fract_of_int_quotient of_rat_divide)
+apply (simp add: of_int_Real divide_inverse)
+apply (simp add: inverse_Real mult_Real)
+done
+
+instance real :: archimedean_field
 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)
+  fix x :: real
+  show "\<exists>z. x \<le> of_int z"
+    apply (induct x)
+    apply (frule cauchy_imp_bounded, clarify)
+    apply (rule_tac x="ceiling b + 1" in exI)
+    apply (rule less_imp_le)
+    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
+    apply (rule_tac x=1 in exI, simp add: algebra_simps)
+    apply (rule_tac x=0 in exI, clarsimp)
+    apply (rule le_less_trans [OF abs_ge_self])
+    apply (rule less_le_trans [OF _ le_of_int_ceiling])
+    apply simp
+    done
 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.*}
+subsection {* Completeness *}
 
-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 not_positive_Real:
+  assumes X: "cauchy X"
+  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
+unfolding positive_Real [OF X]
+apply (auto, unfold not_less)
+apply (erule obtain_pos_sum)
+apply (drule_tac x=s in spec, simp)
+apply (drule_tac r=t in cauchyD [OF X], clarify)
+apply (drule_tac x=k in spec, clarsimp)
+apply (rule_tac x=n in exI, clarify, rename_tac m)
+apply (drule_tac x=m in spec, simp)
+apply (drule_tac x=n in spec, simp)
+apply (drule spec, drule (1) mp, clarify, rename_tac i)
+apply (rule_tac x="max i k" in exI, simp)
+done
+
+lemma le_Real:
+  assumes X: "cauchy X" and Y: "cauchy Y"
+  shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
+unfolding not_less [symmetric, where 'a=real] less_real_def
+apply (simp add: diff_Real not_positive_Real X Y)
+apply (simp add: diff_le_eq add_ac)
+done
 
-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)
-
+lemma le_RealI:
+  assumes Y: "cauchy Y"
+  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
+proof (induct x)
+  fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
+  hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
+    by (simp add: of_rat_Real le_Real)
+  {
+    fix r :: rat assume "0 < r"
+    then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
+      by (rule obtain_pos_sum)
+    obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
+      using cauchyD [OF Y s] ..
+    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
+      using le [OF t] ..
+    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
+    proof (clarsimp)
+      fix n assume n: "i \<le> n" "j \<le> n"
+      have "X n \<le> Y i + t" using n j by simp
+      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
+      ultimately show "X n \<le> Y n + r" unfolding r by simp
+    qed
+    hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
+  }
+  thus "Real X \<le> Real Y"
+    by (simp add: of_rat_Real le_Real X Y)
+qed
 
-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])
+lemma Real_leI:
+  assumes X: "cauchy X"
+  assumes le: "\<forall>n. of_rat (X n) \<le> y"
+  shows "Real X \<le> y"
+proof -
+  have "- y \<le> - Real X"
+    by (simp add: minus_Real X le_RealI of_rat_minus le)
+  thus ?thesis by simp
+qed
+
+lemma less_RealD:
+  assumes Y: "cauchy Y"
+  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
+by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
+
+lemma of_nat_less_two_power:
+  "of_nat n < (2::'a::{linordered_idom,number_ring}) ^ n"
+apply (induct n)
+apply simp
+apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
+apply (drule (1) add_le_less_mono, simp)
+apply simp
 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 complete_real:
+  fixes S :: "real set"
+  assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
+  shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
+proof -
+  obtain x where x: "x \<in> S" using assms(1) ..
+  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
 
-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])
+  def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
+  obtain a where a: "\<not> P a"
+  proof
+    have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
+    also have "x - 1 < x" by simp
+    finally have "of_int (floor (x - 1)) < x" .
+    hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
+    then show "\<not> P (of_int (floor (x - 1)))"
+      unfolding P_def of_rat_of_int_eq using x by fast
+  qed
+  obtain b where b: "P b"
+  proof
+    show "P (of_int (ceiling z))"
+    unfolding P_def of_rat_of_int_eq
+    proof
+      fix y assume "y \<in> S"
+      hence "y \<le> z" using z by simp
+      also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
+      finally show "y \<le> of_int (ceiling z)" .
+    qed
+  qed
 
-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])
+  def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
+  def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
+  def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
+  def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
+  def C \<equiv> "\<lambda>n. avg (A n) (B n)"
+  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
+  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
+  have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
+    unfolding A_def B_def C_def bisect_def split_def by simp
+  have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
+    unfolding A_def B_def C_def bisect_def split_def by simp
 
-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)
+  have width: "\<And>n. B n - A n = (b - a) / 2^n"
+    apply (simp add: eq_divide_eq)
+    apply (induct_tac n, simp)
+    apply (simp add: C_def avg_def algebra_simps)
+    done
+
+  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
+    apply (simp add: divide_less_eq)
+    apply (subst mult_commute)
+    apply (frule_tac y=y in ex_less_of_nat_mult)
+    apply clarify
+    apply (rule_tac x=n in exI)
+    apply (erule less_trans)
+    apply (rule mult_strict_right_mono)
+    apply (rule le_less_trans [OF _ of_nat_less_two_power])
+    apply simp
+    apply assumption
+    done
 
-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)
+  have PA: "\<And>n. \<not> P (A n)"
+    by (induct_tac n, simp_all add: a)
+  have PB: "\<And>n. P (B n)"
+    by (induct_tac n, simp_all add: b)
+  have ab: "a < b"
+    using a b unfolding P_def
+    apply (clarsimp simp add: not_le)
+    apply (drule (1) bspec)
+    apply (drule (1) less_le_trans)
+    apply (simp add: of_rat_less)
+    done
+  have AB: "\<And>n. A n < B n"
+    by (induct_tac n, simp add: ab, simp add: C_def avg_def)
+  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
+    apply (auto simp add: le_less [where 'a=nat])
+    apply (erule less_Suc_induct)
+    apply (clarsimp simp add: C_def avg_def)
+    apply (simp add: add_divide_distrib [symmetric])
+    apply (rule AB [THEN less_imp_le])
+    apply simp
+    done
+  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
+    apply (auto simp add: le_less [where 'a=nat])
+    apply (erule less_Suc_induct)
+    apply (clarsimp simp add: C_def avg_def)
+    apply (simp add: add_divide_distrib [symmetric])
+    apply (rule AB [THEN less_imp_le])
+    apply simp
+    done
+  have cauchy_lemma:
+    "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
+    apply (rule cauchyI)
+    apply (drule twos [where y="b - a"])
+    apply (erule exE)
+    apply (rule_tac x=n in exI, clarify, rename_tac i j)
+    apply (rule_tac y="B n - A n" in le_less_trans) defer
+    apply (simp add: width)
+    apply (drule_tac x=n in spec)
+    apply (frule_tac x=i in spec, drule (1) mp)
+    apply (frule_tac x=j in spec, drule (1) mp)
+    apply (frule A_mono, drule B_mono)
+    apply (frule A_mono, drule B_mono)
+    apply arith
+    done
+  have "cauchy A"
+    apply (rule cauchy_lemma [rule_format])
+    apply (simp add: A_mono)
+    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
+    done
+  have "cauchy B"
+    apply (rule cauchy_lemma [rule_format])
+    apply (simp add: B_mono)
+    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
+    done
+  have 1: "\<forall>x\<in>S. x \<le> Real B"
+  proof
+    fix x assume "x \<in> S"
+    then show "x \<le> Real B"
+      using PB [unfolded P_def] `cauchy B`
+      by (simp add: le_RealI)
+  qed
+  have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
+    apply clarify
+    apply (erule contrapos_pp)
+    apply (simp add: not_le)
+    apply (drule less_RealD [OF `cauchy A`], clarify)
+    apply (subgoal_tac "\<not> P (A n)")
+    apply (simp add: P_def not_le, clarify)
+    apply (erule rev_bexI)
+    apply (erule (1) less_trans)
+    apply (simp add: PA)
+    done
+  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
+  proof (rule vanishesI)
+    fix r :: rat assume "0 < r"
+    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
+      using twos by fast
+    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
+    proof (clarify)
+      fix n assume n: "k \<le> n"
+      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
+        by simp
+      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
+        using n by (simp add: divide_left_mono mult_pos_pos)
+      also note k
+      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
+    qed
+    thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
+  qed
+  hence 3: "Real B = Real A"
+    by (simp add: eq_Real `cauchy A` `cauchy B` width)
+  show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
+    using 1 2 3 by (rule_tac x="Real B" in exI, simp)
 qed
 
+subsection {* Hiding implementation details *}
 
-subsection{*Theorems About the Ordering*}
+hide_const (open) vanishes cauchy positive Real real_case
 
-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
+declare Real_induct [induct del]
+declare Abs_real_induct [induct del]
+declare Abs_real_cases [cases del]
+
+subsection {* Legacy theorem names *}
+
+text {* TODO: Could we have a way to mark theorem names as deprecated,
+and have Isabelle issue a warning and display the preferred name? *}
 
-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])
-
+lemmas real_diff_def = minus_real_def [of "r" "s", standard]
+lemmas real_divide_def = divide_real_def [of "R" "S", standard]
+lemmas real_less_def = less_le [of "x::real" "y", standard]
+lemmas real_abs_def = abs_real_def [of "r", standard]
+lemmas real_sgn_def = sgn_real_def [of "x", standard]
+lemmas real_mult_assoc = mult_assoc [of "z1::real" "z2" "z3", standard]
+lemmas real_mult_1 = mult_1_left [of "z::real", standard]
+lemmas real_add_mult_distrib = left_distrib [of "z1::real" "z2" "w", standard]
+lemmas real_zero_not_eq_one = zero_neq_one [where 'a=real]
+lemmas real_mult_inverse_left = left_inverse [of "x::real", standard]
+lemmas INVERSE_ZERO = inverse_zero [where 'a=real]
+lemmas real_le_refl = order_refl [of "w::real", standard]
+lemmas real_le_antisym = order_antisym [of "z::real" "w", standard]
+lemmas real_le_trans = order_trans [of "i::real" "j" "k", standard]
+lemmas real_le_eq_diff = le_iff_diff_le_0 [of "x::real" "y", standard]
+lemmas real_add_left_mono = add_left_mono [of "x::real" "y" "z", standard]
+lemmas real_mult_order = mult_pos_pos [of "x::real" "y", standard]
+lemmas real_mult_less_mono2 =
+  mult_strict_left_mono [of "x::real" "y" "z", COMP swap_prems_rl, standard]
 
 subsection{*More Lemmas*}
 
@@ -940,20 +1496,11 @@
 
 subsection{*Numerals and Arithmetic*}
 
-instantiation real :: number_ring
-begin
-
-definition
-  real_number_of_def [code del]: "number_of w = real_of_int w"
-
-instance
-  by intro_classes (simp add: real_number_of_def)
-
-end
+declare number_of_real_def [code del]
 
 lemma [code_unfold_post]:
   "number_of k = real_of_int (number_of k)"
-  unfolding number_of_is_id real_number_of_def ..
+  unfolding number_of_is_id number_of_real_def ..
 
 
 text{*Collapse applications of @{term real} to @{term number_of}*}
@@ -976,8 +1523,8 @@
       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
       @{thm real_of_nat_number_of}, @{thm real_number_of}]
-  #> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.natT --> HOLogic.realT)
-  #> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.intT --> HOLogic.realT))
+  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
+  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
 *}
 
 
@@ -986,7 +1533,7 @@
 text{*Needed in this non-standard form by Hyperreal/Transcendental*}
 lemma real_0_le_divide_iff:
      "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
-by (simp add: real_divide_def zero_le_mult_iff, auto)
+by (auto simp add: zero_le_divide_iff)
 
 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
 by arith
diff -r f736a853864f -r e05e1283c550 src/HOL/RealVector.thy
--- a/src/HOL/RealVector.thy	Mon May 10 11:47:56 2010 -0700
+++ b/src/HOL/RealVector.thy	Mon May 10 12:12:58 2010 -0700
@@ -793,7 +793,7 @@
 apply (intro_classes, unfold real_norm_def real_scaleR_def)
 apply (rule dist_real_def)
 apply (rule open_real_def)
-apply (simp add: real_sgn_def)
+apply (simp add: sgn_real_def)
 apply (rule abs_ge_zero)
 apply (rule abs_eq_0)
 apply (rule abs_triangle_ineq)