
diff -r 6137d24eef79 -r 9c0b5e081037 src/HOL/Real/PReal.thy
--- a/src/HOL/Real/PReal.thy	Thu Jan 01 10:06:32 2004 +0100
+++ b/src/HOL/Real/PReal.thy	Thu Jan 01 21:47:07 2004 +0100
@@ -3,41 +3,1303 @@
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
     Description : The positive reals as Dedekind sections of positive
-                  rationals. Fundamentals of Abstract Analysis [Gleason- p. 121] 
+         rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
                   provides some of the definitions.
 *)
 
-PReal = PRat +
+theory PReal = PRat:
 
 typedef preal = "{A::prat set. {} < A & A < UNIV &
-                               (!y: A. ((!z. z < y --> z: A) &
-                                        (? u: A. y < u)))}"      (preal_1)
-instance
-   preal :: {ord, plus, times}
+                               (\<forall>y \<in> A. ((\<forall>z. z < y --> z \<in> A) &
+                                        (\<exists>u \<in> A. y < u)))}"
+apply (rule exI) 
+apply (rule preal_1) 
+done
+
+
+instance preal :: ord ..
+instance preal :: plus ..
+instance preal :: times ..
+
 
 constdefs
-  preal_of_prat :: prat => preal             
+  preal_of_prat :: "prat => preal"
    "preal_of_prat q     == Abs_preal({x::prat. x < q})"
 
-  pinv       :: preal => preal
-  "pinv(R)   == Abs_preal({w. ? y. w < y & qinv y ~: Rep_preal(R)})" 
+  pinv       :: "preal => preal"
+  "pinv(R)   == Abs_preal({w. \<exists>y. w < y & qinv y \<notin> Rep_preal(R)})"
 
-  psup       :: preal set => preal
-  "psup(P)   == Abs_preal({w. ? X: P. w: Rep_preal(X)})"
+  psup       :: "preal set => preal"
+  "psup(P)   == Abs_preal({w. \<exists>X \<in> P. w \<in> Rep_preal(X)})"
 
-defs
+defs (overloaded)
 
-  preal_add_def
-    "R + S == Abs_preal({w. ? x: Rep_preal(R). ? y: Rep_preal(S). w = x + y})"
+  preal_add_def:
+    "R + S == Abs_preal({w. \<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). w = x + y})"
 
-  preal_mult_def
-    "R * S == Abs_preal({w. ? x: Rep_preal(R). ? y: Rep_preal(S). w = x * y})"
+  preal_mult_def:
+    "R * S == Abs_preal({w. \<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). w = x * y})"
 
-  preal_less_def
+  preal_less_def:
     "R < (S::preal) == Rep_preal(R) < Rep_preal(S)"
 
-  preal_le_def
-    "R <= (S::preal) == Rep_preal(R) <= Rep_preal(S)"
- 
+  preal_le_def:
+    "R \<le> (S::preal) == Rep_preal(R) \<subseteq> Rep_preal(S)"
+
+
+lemma inj_on_Abs_preal: "inj_on Abs_preal preal"
+apply (rule inj_on_inverseI)
+apply (erule Abs_preal_inverse)
+done
+
+declare inj_on_Abs_preal [THEN inj_on_iff, simp]
+
+lemma inj_Rep_preal: "inj(Rep_preal)"
+apply (rule inj_on_inverseI)
+apply (rule Rep_preal_inverse)
+done
+
+lemma empty_not_mem_preal [simp]: "{} \<notin> preal"
+by (unfold preal_def, fast)
+
+lemma one_set_mem_preal: "{x::prat. x < prat_of_pnat (Abs_pnat (Suc 0))} \<in> preal"
+apply (unfold preal_def)
+apply (rule preal_1)
+done
+
+declare one_set_mem_preal [simp]
+
+lemma preal_psubset_empty: "x \<in> preal ==> {} < x"
+by (unfold preal_def, fast)
+
+lemma Rep_preal_psubset_empty: "{} < Rep_preal x"
+by (rule Rep_preal [THEN preal_psubset_empty])
+
+lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
+apply (cut_tac x = X in Rep_preal_psubset_empty)
+apply (auto intro: equals0I [symmetric] simp add: psubset_def)
+done
+
+lemma prealI1:
+      "[| {} < A; A < UNIV;
+               (\<forall>y \<in> A. ((\<forall>z. z < y --> z \<in> A) &
+                         (\<exists>u \<in> A. y < u))) |] ==> A \<in> preal"
+apply (unfold preal_def, fast)
+done
+
+lemma prealI2:
+      "[| {} < A; A < UNIV;
+               \<forall>y \<in> A. (\<forall>z. z < y --> z \<in> A);
+               \<forall>y \<in> A. (\<exists>u \<in> A. y < u) |] ==> A \<in> preal"
+
+apply (unfold preal_def, best)
+done
+
+lemma prealE_lemma:
+      "A \<in> preal ==> {} < A & A < UNIV &
+                          (\<forall>y \<in> A. ((\<forall>z. z < y --> z \<in> A) &
+                                   (\<exists>u \<in> A. y < u)))"
+apply (unfold preal_def, fast)
+done
+
+declare prealI1 [intro!] prealI2 [intro!]
+
+declare Abs_preal_inverse [simp]
+
+
+lemma prealE_lemma1: "A \<in> preal ==> {} < A"
+by (unfold preal_def, fast)
+
+lemma prealE_lemma2: "A \<in> preal ==> A < UNIV"
+by (unfold preal_def, fast)
+
+lemma prealE_lemma3: "A \<in> preal ==> \<forall>y \<in> A. (\<forall>z. z < y --> z \<in> A)"
+by (unfold preal_def, fast)
+
+lemma prealE_lemma3a: "[| A \<in> preal; y \<in> A |] ==> (\<forall>z. z < y --> z \<in> A)"
+by (fast dest!: prealE_lemma3)
+
+lemma prealE_lemma3b: "[| A \<in> preal; y \<in> A; z < y |] ==> z \<in> A"
+by (fast dest!: prealE_lemma3a)
+
+lemma prealE_lemma4: "A \<in> preal ==> \<forall>y \<in> A. (\<exists>u \<in> A. y < u)"
+by (unfold preal_def, fast)
+
+lemma prealE_lemma4a: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
+by (fast dest!: prealE_lemma4)
+
+lemma not_mem_Rep_preal_Ex: "\<exists>x. x\<notin> Rep_preal X"
+apply (cut_tac x = X in Rep_preal)
+apply (drule prealE_lemma2)
+apply (auto simp add: psubset_def)
+done
+
+
+subsection{*@{term preal_of_prat}: the Injection from prat to preal*}
+
+text{*A few lemmas*}
+
+lemma lemma_prat_less_set_mem_preal: "{u::prat. u < y} \<in> preal"
+apply (cut_tac qless_Ex)
+apply (auto intro: prat_less_trans elim!: prat_less_irrefl)
+apply (blast dest: prat_dense)
+done
+
+lemma lemma_prat_set_eq: "{u::prat. u < x} = {x. x < y} ==> x = y"
+apply (insert prat_linear [of x y], safe)
+apply (drule_tac [2] prat_dense, erule_tac [2] exE)
+apply (drule prat_dense, erule exE)
+apply (blast dest: prat_less_not_sym)
+apply (blast dest: prat_less_not_sym)
+done
+
+lemma inj_preal_of_prat: "inj(preal_of_prat)"
+apply (rule inj_onI)
+apply (unfold preal_of_prat_def)
+apply (drule inj_on_Abs_preal [THEN inj_onD])
+apply (rule lemma_prat_less_set_mem_preal)
+apply (rule lemma_prat_less_set_mem_preal)
+apply (erule lemma_prat_set_eq)
+done
+
+
+subsection{*Theorems for Ordering*}
+
+text{*A positive fraction not in a positive real is an upper bound.
+ Gleason p. 122 - Remark (1)*}
+
+lemma not_in_preal_ub: "x \<notin> Rep_preal(R) ==> \<forall>y \<in> Rep_preal(R). y < x"
+apply (cut_tac x1 = R in Rep_preal [THEN prealE_lemma]) 
+apply (blast intro: not_less_not_eq_prat_less)
+done
+
+
+text{*@{text preal_less} is a strict order: nonreflexive and transitive *}
+
+lemma preal_less_not_refl: "~ (x::preal) < x"
+apply (unfold preal_less_def)
+apply (simp (no_asm) add: psubset_def)
+done
+
+lemmas preal_less_irrefl = preal_less_not_refl [THEN notE, standard]
+
+lemma preal_not_refl2: "!!(x::preal). x < y ==> x \<noteq> y"
+by (auto simp add: preal_less_not_refl)
+
+lemma preal_less_trans: "!!(x::preal). [| x < y; y < z |] ==> x < z"
+apply (unfold preal_less_def)
+apply (auto dest: subsetD equalityI simp add: psubset_def)
+done
+
+lemma preal_less_not_sym: "!! (q1::preal). q1 < q2 ==> ~ q2 < q1"
+apply (rule notI)
+apply (drule preal_less_trans, assumption)
+apply (simp add: preal_less_not_refl)
+done
+
+(* [| x < y;  ~P ==> y < x |] ==> P *)
+lemmas preal_less_asym = preal_less_not_sym [THEN contrapos_np, standard]
+
+lemma preal_linear:
+      "(x::preal) < y | x = y | y < x"
+apply (unfold preal_less_def)
+apply (auto dest!: inj_Rep_preal [THEN injD] simp add: psubset_def)
+apply (rule prealE_lemma3b, rule Rep_preal, assumption)
+apply (fast dest: not_in_preal_ub)
+done
+
+
+subsection{*Properties of Addition*}
+
+lemma preal_add_commute: "(x::preal) + y = y + x"
+apply (unfold preal_add_def)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (blast intro: prat_add_commute [THEN subst])
+done
+
+text{*Addition of two positive reals gives a positive real*}
+
+text{*Lemmas for proving positive reals addition set in @{typ preal}*}
+
+text{*Part 1 of Dedekind sections definition*}
+lemma preal_add_set_not_empty:
+     "{} < {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y}"
+apply (cut_tac mem_Rep_preal_Ex mem_Rep_preal_Ex)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 2 of Dedekind sections definition*}
+lemma preal_not_mem_add_set_Ex:
+     "\<exists>q. q  \<notin> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y}"
+apply (cut_tac X = R in not_mem_Rep_preal_Ex)
+apply (cut_tac X = S in not_mem_Rep_preal_Ex, clarify) 
+apply (drule not_in_preal_ub)+
+apply (rule_tac x = "x+xa" in exI)
+apply (auto dest!: bspec) 
+apply (drule prat_add_less_mono)
+apply (auto simp add: prat_less_not_refl)
+done
+
+lemma preal_add_set_not_prat_set:
+     "{w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y} < UNIV"
+apply (auto intro!: psubsetI)
+apply (cut_tac R = R and S = S in preal_not_mem_add_set_Ex, auto)
+done
+
+text{*Part 3 of Dedekind sections definition*}
+lemma preal_add_set_lemma3:
+     "\<forall>y \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y}.
+         \<forall>z. z < y --> z \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x+y}"
+apply auto
+apply (frule prat_mult_qinv_less_1)
+apply (frule_tac x = x 
+       in prat_mult_less2_mono1 [of _ "prat_of_pnat (Abs_pnat (Suc 0))"])
+apply (frule_tac x = ya 
+       in prat_mult_less2_mono1 [of _ "prat_of_pnat (Abs_pnat (Suc 0))"])
+apply simp
+apply (drule Rep_preal [THEN prealE_lemma3a])+
+apply (erule allE)+
+apply auto
+apply (rule bexI)+
+apply (auto simp add: prat_add_mult_distrib2 [symmetric] 
+      prat_add_assoc [symmetric] prat_mult_assoc)
+done
+
+lemma preal_add_set_lemma4:
+     "\<forall>y \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y}.
+          \<exists>u \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y}. y < u"
+apply auto
+apply (drule Rep_preal [THEN prealE_lemma4a])
+apply (auto intro: prat_add_less2_mono1)
+done
+
+lemma preal_mem_add_set:
+     "{w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x + y} \<in> preal"
+apply (rule prealI2)
+apply (rule preal_add_set_not_empty)
+apply (rule preal_add_set_not_prat_set)
+apply (rule preal_add_set_lemma3)
+apply (rule preal_add_set_lemma4)
+done
+
+lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
+apply (unfold preal_add_def)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (simp (no_asm) add: preal_mem_add_set [THEN Abs_preal_inverse])
+apply (auto simp add: prat_add_ac)
+apply (rule bexI)
+apply (auto intro!: exI simp add: prat_add_ac)
+done
+
+lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
+  apply (rule mk_left_commute [of "op +"])
+  apply (rule preal_add_assoc)
+  apply (rule preal_add_commute)
+  done
+
+(* Positive Reals addition is an AC operator *)
+lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
+
+
+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)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (blast intro: prat_mult_commute [THEN subst])
+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 preal_mult_set_not_empty:
+     "{} < {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y}"
+apply (cut_tac mem_Rep_preal_Ex mem_Rep_preal_Ex)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 2 of Dedekind sections definition*}
+lemma preal_not_mem_mult_set_Ex:
+     "\<exists>q. q  \<notin> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y}"
+apply (cut_tac X = R in not_mem_Rep_preal_Ex)
+apply (cut_tac X = S in not_mem_Rep_preal_Ex)
+apply (erule exE)+
+apply (drule not_in_preal_ub)+
+apply (rule_tac x = "x*xa" in exI)
+apply (auto, (erule ballE)+, auto)
+apply (drule prat_mult_less_mono)
+apply (auto simp add: prat_less_not_refl)
+done
+
+lemma preal_mult_set_not_prat_set:
+     "{w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y} < UNIV"
+apply (auto intro!: psubsetI)
+apply (cut_tac R = R and S = S in preal_not_mem_mult_set_Ex, auto)
+done
+
+text{*Part 3 of Dedekind sections definition*}
+lemma preal_mult_set_lemma3:
+     "\<forall>y \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y}.
+         \<forall>z. z < y --> z \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x*y}"
+apply auto
+apply (frule_tac x = "qinv (ya)" in prat_mult_left_less2_mono1)
+apply (simp add: prat_mult_ac)
+apply (drule Rep_preal [THEN prealE_lemma3a])
+apply (erule allE)
+apply (rule bexI)+
+apply (auto simp add: prat_mult_assoc)
+done
+
+lemma preal_mult_set_lemma4:
+     "\<forall>y \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y}.
+          \<exists>u \<in> {w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y}. y < u"
+apply auto
+apply (drule Rep_preal [THEN prealE_lemma4a])
+apply (auto intro: prat_mult_less2_mono1)
+done
+
+lemma preal_mem_mult_set:
+     "{w. \<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. w = x * y} \<in> preal"
+apply (rule prealI2)
+apply (rule preal_mult_set_not_empty)
+apply (rule preal_mult_set_not_prat_set)
+apply (rule preal_mult_set_lemma3)
+apply (rule preal_mult_set_lemma4)
+done
+
+lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
+apply (unfold preal_mult_def)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (simp (no_asm) add: preal_mem_mult_set [THEN Abs_preal_inverse])
+apply (auto simp add: prat_mult_ac)
+apply (rule bexI)
+apply (auto intro!: exI simp add: prat_mult_ac)
+done
+
+lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
+  apply (rule mk_left_commute [of "op *"])
+  apply (rule preal_mult_assoc)
+  apply (rule preal_mult_commute)
+  done
+
+(* Positive Reals multiplication is an AC operator *)
+lemmas preal_mult_ac =
+       preal_mult_assoc preal_mult_commute preal_mult_left_commute
+
+(* Positive Real 1 is the multiplicative identity element *)
+(* long *)
+lemma preal_mult_1:
+      "(preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0)))) * z = z"
+apply (unfold preal_of_prat_def preal_mult_def)
+apply (rule Rep_preal_inverse [THEN subst])
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (rule one_set_mem_preal [THEN Abs_preal_inverse, THEN ssubst])
+apply (auto simp add: Rep_preal_inverse)
+apply (drule Rep_preal [THEN prealE_lemma4a]) 
+apply (erule bexE) 
+apply (drule prat_mult_less_mono)
+apply (auto dest: Rep_preal [THEN prealE_lemma3a])
+apply (frule Rep_preal [THEN prealE_lemma4a]) 
+apply (erule bexE) 
+apply (frule_tac x = "qinv (u)" in prat_mult_less2_mono1)
+apply (rule exI, auto, rule_tac x = u in bexI)
+apply (auto simp add: prat_mult_assoc)
+done
+
+lemma preal_mult_1_right:
+     "z * (preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0)))) = z"
+apply (rule preal_mult_commute [THEN subst])
+apply (rule preal_mult_1)
+done
+
+
+subsection{*Distribution of Multiplication across Addition*}
+
+lemma mem_Rep_preal_addD:
+      "z \<in> Rep_preal(R+S) ==>
+            \<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). z = x + y"
+apply (unfold preal_add_def)
+apply (drule preal_mem_add_set [THEN Abs_preal_inverse, THEN subst], fast)
+done
+
+lemma mem_Rep_preal_addI:
+      "\<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). z = x + y
+       ==> z \<in> Rep_preal(R+S)"
+apply (unfold preal_add_def)
+apply (rule preal_mem_add_set [THEN Abs_preal_inverse, THEN ssubst], fast)
+done
+
+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 (fast intro!: mem_Rep_preal_addD mem_Rep_preal_addI)
+done
+
+lemma mem_Rep_preal_multD:
+      "z \<in> Rep_preal(R*S) ==>
+            \<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). z = x * y"
+apply (unfold preal_mult_def)
+apply (drule preal_mem_mult_set [THEN Abs_preal_inverse, THEN subst], fast)
+done
+
+lemma mem_Rep_preal_multI:
+      "\<exists>x \<in> Rep_preal(R). \<exists>y \<in> Rep_preal(S). z = x * y
+       ==> z \<in> Rep_preal(R*S)"
+apply (unfold preal_mult_def)
+apply (rule preal_mem_mult_set [THEN Abs_preal_inverse, THEN ssubst], fast)
+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)"
+by (fast intro!: mem_Rep_preal_multD mem_Rep_preal_multI)
+
+lemma lemma_add_mult_mem_Rep_preal:
+     "[| xb \<in> Rep_preal z1; xc \<in> Rep_preal z2; ya:
+                   Rep_preal w; yb \<in> Rep_preal w |] ==>
+                   xb * ya + xc * yb \<in> Rep_preal (z1 * w + z2 * w)"
+by (fast intro: mem_Rep_preal_addI mem_Rep_preal_multI)
+
+lemma lemma_add_mult_mem_Rep_preal1:
+     "[| xb \<in> Rep_preal z1; xc \<in> Rep_preal z2; ya:
+                   Rep_preal w; yb \<in> Rep_preal w |] ==>
+                   yb*(xb + xc) \<in> Rep_preal (w*(z1 + z2))"
+by (fast intro: mem_Rep_preal_addI mem_Rep_preal_multI)
+
+lemma lemma_preal_add_mult_distrib:
+     "x \<in> Rep_preal (w * z1 + w * z2) ==>
+               x \<in> Rep_preal (w * (z1 + z2))"
+apply (auto dest!: mem_Rep_preal_addD mem_Rep_preal_multD)
+apply (frule_tac ya = xa and yb = xb and xb = ya and xc = yb in lemma_add_mult_mem_Rep_preal1, auto)
+apply (rule_tac x = xa and y = xb in prat_linear_less2)
+apply (drule_tac b = ya and c = yb in lemma_prat_add_mult_mono)
+apply (rule Rep_preal [THEN prealE_lemma3b])
+apply (auto simp add: prat_add_mult_distrib2)
+apply (drule_tac ya = xb and yb = xa and xc = ya and xb = yb in lemma_add_mult_mem_Rep_preal1, auto)
+apply (drule_tac b = yb and c = ya in lemma_prat_add_mult_mono)
+apply (rule Rep_preal [THEN prealE_lemma3b])
+apply (erule_tac V = "xb * ya + xb * yb \<in> Rep_preal (w * (z1 + z2))" in thin_rl)
+apply (auto simp add: prat_add_mult_distrib prat_add_commute preal_add_ac)
+done
+
+lemma lemma_preal_add_mult_distrib2:
+     "x \<in> Rep_preal (w * (z1 + z2)) ==>
+               x \<in> Rep_preal (w * z1 + w * z2)"
+by (auto dest!: mem_Rep_preal_addD mem_Rep_preal_multD
+         intro!: bexI mem_Rep_preal_addI mem_Rep_preal_multI 
+         simp add: prat_add_mult_distrib2)
+
+lemma preal_add_mult_distrib2: "(w * ((z1::preal) + z2)) = (w * z1) + (w * z2)"
+apply (rule inj_Rep_preal [THEN injD])
+apply (fast intro: lemma_preal_add_mult_distrib lemma_preal_add_mult_distrib2)
+done
+
+lemma preal_add_mult_distrib: "(((z1::preal) + z2) * w) = (z1 * w) + (z2 * w)"
+apply (simp (no_asm) add: preal_mult_commute preal_add_mult_distrib2)
+done
+
+
+subsection{*Existence of Inverse, a Positive Real*}
+
+lemma qinv_not_mem_Rep_preal_Ex: "\<exists>y. qinv(y) \<notin>  Rep_preal X"
+apply (cut_tac X = X in not_mem_Rep_preal_Ex)
+apply (erule exE, cut_tac x = x in prat_as_inverse_ex, auto)
+done
+
+lemma lemma_preal_mem_inv_set_ex:
+     "\<exists>q. q \<in> {x. \<exists>y. x < y & qinv y \<notin>  Rep_preal A}"
+apply (cut_tac X = A in qinv_not_mem_Rep_preal_Ex, auto)
+apply (cut_tac y = y in qless_Ex, fast)
+done
+
+text{*Part 1 of Dedekind sections definition*}
+lemma preal_inv_set_not_empty: "{} < {x. \<exists>y. x < y & qinv y \<notin>  Rep_preal A}"
+apply (cut_tac lemma_preal_mem_inv_set_ex)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 2 of Dedekind sections definition*}
+lemma qinv_mem_Rep_preal_Ex: "\<exists>y. qinv(y) \<in>  Rep_preal X"
+apply (cut_tac X = X in mem_Rep_preal_Ex)
+apply (erule exE, cut_tac x = x in prat_as_inverse_ex, auto)
+done
+
+lemma preal_not_mem_inv_set_Ex:
+     "\<exists>x. x \<notin> {x. \<exists>y. x < y & qinv y \<notin>  Rep_preal A}"
+apply (rule ccontr)
+apply (cut_tac X = A in qinv_mem_Rep_preal_Ex, auto)
+apply (erule allE, clarify) 
+apply (drule qinv_prat_less, drule not_in_preal_ub)
+apply (erule_tac x = "qinv y" in ballE)
+apply (drule prat_less_trans)
+apply (auto simp add: prat_less_not_refl)
+done
+
+lemma preal_inv_set_not_prat_set:
+     "{x. \<exists>y. x < y & qinv y \<notin>  Rep_preal A} < UNIV"
+apply (auto intro!: psubsetI)
+apply (cut_tac A = A in preal_not_mem_inv_set_Ex, auto)
+done
+
+text{*Part 3 of Dedekind sections definition*}
+lemma preal_inv_set_lemma3:
+     "\<forall>y \<in> {x. \<exists>y. x < y & qinv y \<notin> Rep_preal A}.
+        \<forall>z. z < y --> z \<in> {x. \<exists>y. x < y & qinv y \<notin> Rep_preal A}"
+apply auto
+apply (rule_tac x = ya in exI)
+apply (auto intro: prat_less_trans)
+done
+
+lemma preal_inv_set_lemma4:
+     "\<forall>y \<in> {x. \<exists>y. x < y & qinv y \<notin> Rep_preal A}.
+        Bex {x. \<exists>y. x < y & qinv y \<notin> Rep_preal A} (op < y)"
+by (blast dest: prat_dense)
+
+lemma preal_mem_inv_set: "{x. \<exists>y. x < y & qinv(y) \<notin> Rep_preal(A)} \<in> preal"
+apply (rule prealI2)
+apply (rule preal_inv_set_not_empty)
+apply (rule preal_inv_set_not_prat_set)
+apply (rule preal_inv_set_lemma3)
+apply (rule preal_inv_set_lemma4)
+done
+
+(*more lemmas for inverse *)
+lemma preal_mem_mult_invD:
+     "x \<in> Rep_preal(pinv(A)*A) ==>
+      x \<in> Rep_preal(preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0))))"
+apply (auto dest!: mem_Rep_preal_multD simp add: pinv_def preal_of_prat_def)
+apply (drule preal_mem_inv_set [THEN Abs_preal_inverse, THEN subst])
+apply (auto dest!: not_in_preal_ub)
+apply (drule prat_mult_less_mono, blast, auto)
+done
+
+subsection{*Gleason's Lemma 9-3.4, page 122*}
+
+lemma lemma1_gleason9_34:
+     "\<forall>xa \<in> Rep_preal(A). xa + x \<in> Rep_preal(A) ==>
+             \<exists>xb \<in> Rep_preal(A). xb + (prat_of_pnat p)*x \<in> Rep_preal(A)"
+apply (cut_tac mem_Rep_preal_Ex)
+apply (induct_tac "p" rule: pnat_induct)
+apply (auto simp add: pnat_one_def pSuc_is_plus_one prat_add_mult_distrib 
+                      prat_of_pnat_add prat_add_assoc [symmetric])
+done
+
+lemma lemma1b_gleason9_34:
+     "Abs_prat (ratrel `` {(y, z)}) < 
+      xb +
+      Abs_prat (ratrel `` {(x*y, Abs_pnat (Suc 0))}) * 
+      Abs_prat (ratrel `` {(w, x)})"
+apply (rule_tac j =
+        "Abs_prat (ratrel `` 
+           { (x * y, Abs_pnat (Suc 0))}) * Abs_prat (ratrel `` {(w, x)})" 
+       in prat_le_less_trans)
+apply (rule_tac [2] prat_self_less_add_right)
+apply (auto intro: lemma_Abs_prat_le3 
+            simp add: prat_mult pre_lemma_gleason9_34b pnat_mult_assoc)
+done
+
+lemma lemma_gleason9_34a:
+     "\<forall>xa \<in> Rep_preal(A). xa + x \<in> Rep_preal(A) ==> False"
+apply (cut_tac X = A in not_mem_Rep_preal_Ex)
+apply (erule exE)
+apply (drule not_in_preal_ub)
+apply (rule_tac z = x in eq_Abs_prat)
+apply (rule_tac z = xa in eq_Abs_prat)
+apply (drule_tac p = "y*xb" in lemma1_gleason9_34)
+apply (erule bexE)
+apply (cut_tac x = y and y = xb and w = xaa and z = ya and xb = xba in lemma1b_gleason9_34)
+apply (drule_tac x = "xba + prat_of_pnat (y * xb) * x" in bspec)
+apply (auto intro: prat_less_asym simp add: prat_of_pnat_def)
+done
+
+lemma lemma_gleason9_34: "\<exists>r \<in> Rep_preal(R). r + x \<notin> Rep_preal(R)"
+apply (rule ccontr)
+apply (blast intro: lemma_gleason9_34a)
+done
+
+
+subsection{*Gleason's Lemma 9-3.6*}
+
+lemma lemma1_gleason9_36: "r + r*qinv(xa)*Q3 = r*qinv(xa)*(xa + Q3)"
+apply (simp (no_asm_use) add: prat_add_mult_distrib2 prat_mult_assoc)
+done
+
+lemma lemma2_gleason9_36: "r*qinv(xa)*(xa*x) = r*x"
+apply (simp (no_asm_use) add: prat_mult_ac)
+done
+
+(*** FIXME: long! ***)
+lemma lemma_gleason9_36:
+     "prat_of_pnat 1 < x ==> \<exists>r \<in> Rep_preal(A). r*x \<notin> Rep_preal(A)"
+apply (rule_tac X1 = A in mem_Rep_preal_Ex [THEN exE])
+apply (rule_tac Q = "xa*x \<in> Rep_preal (A) " in excluded_middle [THEN disjE])
+apply fast
+apply (drule_tac x = xa in prat_self_less_mult_right)
+apply (erule prat_lessE)
+apply (cut_tac R = A and x = Q3 in lemma_gleason9_34)
+apply (drule sym, auto)
+apply (frule not_in_preal_ub)
+apply (drule_tac x = "xa + Q3" in bspec, assumption)
+apply (drule prat_add_right_less_cancel)
+apply (drule_tac x = "qinv (xa) *Q3" in prat_mult_less2_mono1)
+apply (drule_tac x = r in prat_add_less2_mono2)
+apply (simp add: prat_mult_assoc [symmetric] lemma1_gleason9_36)
+apply (drule sym)
+apply (auto simp add: lemma2_gleason9_36)
+apply (rule_tac x = r in bexI)
+apply (rule notI)
+apply (drule_tac y = "r*x" in Rep_preal [THEN prealE_lemma3b], auto)
+done
+
+lemma lemma_gleason9_36a:
+     "prat_of_pnat (Abs_pnat (Suc 0)) < x ==>
+      \<exists>r \<in> Rep_preal(A). r*x \<notin> Rep_preal(A)"
+apply (rule lemma_gleason9_36)
+apply (simp (no_asm_simp) add: pnat_one_def)
+done
+
+
+subsection{*Existence of Inverse: Part 2*}
+lemma preal_mem_mult_invI:
+     "x \<in> Rep_preal(preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0))))
+      ==> x \<in> Rep_preal(pinv(A)*A)"
+apply (auto intro!: mem_Rep_preal_multI simp add: pinv_def preal_of_prat_def)
+apply (rule preal_mem_inv_set [THEN Abs_preal_inverse, THEN ssubst])
+apply (drule prat_qinv_gt_1)
+apply (drule_tac A = A in lemma_gleason9_36a, auto)
+apply (drule Rep_preal [THEN prealE_lemma4a])
+apply (auto, drule qinv_prat_less)
+apply (rule_tac x = "qinv (u) *x" in exI)
+apply (rule conjI)
+apply (rule_tac x = "qinv (r) *x" in exI)
+apply (auto intro: prat_mult_less2_mono1 simp add: qinv_mult_eq qinv_qinv)
+apply (rule_tac x = u in bexI)
+apply (auto simp add: prat_mult_assoc prat_mult_left_commute)
+done
+
+lemma preal_mult_inv:
+     "pinv(A)*A = (preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0))))"
+apply (rule inj_Rep_preal [THEN injD])
+apply (fast dest: preal_mem_mult_invD preal_mem_mult_invI)
+done
+
+lemma preal_mult_inv_right:
+     "A*pinv(A) = (preal_of_prat (prat_of_pnat (Abs_pnat (Suc 0))))"
+apply (rule preal_mult_commute [THEN subst])
+apply (rule preal_mult_inv)
+done
+
+
+text{*Theorems needing @{text lemma_gleason9_34}*}
+
+lemma Rep_preal_self_subset: "Rep_preal (R1) \<subseteq> Rep_preal(R1 + R2)"
+apply (cut_tac X = R2 in mem_Rep_preal_Ex)
+apply (auto intro!: bexI 
+            intro: Rep_preal [THEN prealE_lemma3b] prat_self_less_add_left 
+                   mem_Rep_preal_addI)
+done
+
+lemma Rep_preal_sum_not_subset: "~ Rep_preal (R1 + R2) \<subseteq> Rep_preal(R1)"
+apply (cut_tac X = R2 in mem_Rep_preal_Ex)
+apply (erule exE)
+apply (cut_tac R = R1 in lemma_gleason9_34)
+apply (auto intro: mem_Rep_preal_addI)
+done
+
+lemma Rep_preal_sum_not_eq: "Rep_preal (R1 + R2) \<noteq> Rep_preal(R1)"
+apply (rule notI)
+apply (erule equalityE)
+apply (simp add: Rep_preal_sum_not_subset)
+done
+
+text{*at last, Gleason prop. 9-3.5(iii) page 123*}
+lemma preal_self_less_add_left: "(R1::preal) < R1 + R2"
+apply (unfold preal_less_def psubset_def)
+apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
+done
+
+lemma preal_self_less_add_right: "(R1::preal) < R2 + R1"
+apply (simp add: preal_add_commute preal_self_less_add_left)
+done
+
+
+subsection{*The @{text "\<le>"} Ordering*}
+
+lemma preal_less_le_iff: "(~(w < z)) = (z \<le> (w::preal))"
+apply (unfold preal_le_def psubset_def preal_less_def)
+apply (insert preal_linear [of w z])
+apply (auto simp add: preal_less_def psubset_def)
+done
+
+lemma preal_le_iff_less_or_eq:
+     "((x::preal) \<le> y) = (x < y | x = y)"
+apply (unfold preal_le_def preal_less_def psubset_def)
+apply (auto intro: inj_Rep_preal [THEN injD])
+done
+
+lemma preal_le_refl: "w \<le> (w::preal)"
+apply (simp add: preal_le_def)
+done
+
+lemma preal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::preal)"
+apply (simp add: preal_le_iff_less_or_eq) 
+apply (blast intro: preal_less_trans)
+done
+
+lemma preal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::preal)"
+apply (simp add: preal_le_iff_less_or_eq) 
+apply (blast intro: preal_less_asym)
+done
+
+lemma preal_neq_iff: "(w \<noteq> z) = (w<z | z < (w::preal))"
+apply (insert preal_linear [of w z])
+apply (auto elim: preal_less_irrefl)
+done
+
+(* Axiom 'order_less_le' of class 'order': *)
+lemma preal_less_le: "((w::preal) < z) = (w \<le> z & w \<noteq> z)"
+apply (simp (no_asm) add: preal_less_le_iff [symmetric] preal_neq_iff)
+apply (blast elim!: preal_less_asym)
+done
+
+instance preal :: order
+proof qed
+ (assumption |
+  rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+
+
+lemma preal_le_linear: "x <= y | y <= (x::preal)"
+apply (insert preal_linear [of x y]) 
+apply (auto simp add: order_less_le) 
+done
+
+instance preal :: linorder
+  by (intro_classes, rule preal_le_linear)
+
+
+subsection{*Gleason prop. 9-3.5(iv), page 123*}
+
+text{*Proving @{term "A < B ==> \<exists>D. A + D = B"}*}
+
+text{*Define the claimed D and show that it is a positive real*}
+
+text{*Part 1 of Dedekind sections definition*}
+lemma lemma_ex_mem_less_left_add1:
+     "A < B ==>
+      \<exists>q. q \<in> {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}"
+apply (unfold preal_less_def psubset_def)
+apply (clarify) 
+apply (drule_tac x1 = B in Rep_preal [THEN prealE_lemma4a])
+apply (auto simp add: prat_less_def)
+done
+
+lemma preal_less_set_not_empty:
+     "A < B ==> {} < {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}"
+apply (drule lemma_ex_mem_less_left_add1)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 2 of Dedekind sections definition*}
+lemma lemma_ex_not_mem_less_left_add1:
+     "\<exists>q. q \<notin> {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}"
+apply (cut_tac X = B in not_mem_Rep_preal_Ex)
+apply (erule exE)
+apply (rule_tac x = x in exI, auto)
+apply (cut_tac x = x and y = n in prat_self_less_add_right)
+apply (auto dest: Rep_preal [THEN prealE_lemma3b])
+done
+
+lemma preal_less_set_not_prat_set:
+     "{d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)} < UNIV"
+apply (auto intro!: psubsetI)
+apply (cut_tac A = A and B = B in lemma_ex_not_mem_less_left_add1, auto)
+done
+
+text{*Part 3 of Dedekind sections definition*}
+lemma preal_less_set_lemma3:
+     "A < B ==> \<forall>y \<in> {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}.
+     \<forall>z. z < y --> z \<in> {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}"
+apply auto
+apply (drule_tac x = n in prat_add_less2_mono2)
+apply (drule Rep_preal [THEN prealE_lemma3b], auto)
+done
+
+lemma preal_less_set_lemma4:
+     "A < B ==> \<forall>y \<in> {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}.
+        Bex {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)} (op < y)"
+apply auto
+apply (drule Rep_preal [THEN prealE_lemma4a])
+apply (auto simp add: prat_less_def prat_add_assoc)
+done
+
+lemma preal_mem_less_set:
+     "!! (A::preal). A < B ==>
+      {d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}: preal"
+apply (rule prealI2)
+apply (rule preal_less_set_not_empty)
+apply (rule_tac [2] preal_less_set_not_prat_set)
+apply (rule_tac [2] preal_less_set_lemma3)
+apply (rule_tac [3] preal_less_set_lemma4, auto)
+done
+
+text{*proving that @{term "A + D \<le> B"}*}
+lemma preal_less_add_left_subsetI:
+       "!! (A::preal). A < B ==>
+          A + Abs_preal({d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}) \<le> B"
+apply (unfold preal_le_def)
+apply (rule subsetI)
+apply (drule mem_Rep_preal_addD)
+apply (auto simp add: preal_mem_less_set [THEN Abs_preal_inverse])
+apply (drule not_in_preal_ub)
+apply (drule bspec, assumption)
+apply (drule_tac x = y in prat_add_less2_mono1)
+apply (drule_tac x1 = B in Rep_preal [THEN prealE_lemma3b], auto)
+done
+
+subsection{*proving that @{term "B \<le> A + D"} --- trickier*}
+
+lemma lemma_sum_mem_Rep_preal_ex:
+     "x \<in> Rep_preal(B) ==> \<exists>e. x + e \<in> Rep_preal(B)"
+apply (drule Rep_preal [THEN prealE_lemma4a])
+apply (auto simp add: prat_less_def)
+done
+
+lemma preal_less_add_left_subsetI2:
+       "!! (A::preal). A < B ==>
+          B \<le> A + Abs_preal({d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)})"
+apply (unfold preal_le_def)
+apply (rule subsetI)
+apply (rule_tac Q = "x \<in> Rep_preal (A) " in excluded_middle [THEN disjE])
+apply (rule mem_Rep_preal_addI)
+apply (drule lemma_sum_mem_Rep_preal_ex)
+apply (erule exE)
+apply (cut_tac R = A and x = e in lemma_gleason9_34, erule bexE)
+apply (drule not_in_preal_ub, drule bspec, assumption)
+apply (erule prat_lessE)
+apply (rule_tac x = r in bexI)
+apply (rule_tac x = Q3 in bexI)
+apply (cut_tac [4] Rep_preal_self_subset)
+apply (auto simp add: preal_mem_less_set [THEN Abs_preal_inverse])
+apply (rule_tac x = "r+e" in exI)
+apply (simp add: prat_add_ac)
+done
+
+(*** required proof ***)
+lemma preal_less_add_left:
+     "!! (A::preal). A < B ==>
+          A + Abs_preal({d. \<exists>n. n \<notin> Rep_preal(A) & n + d \<in> Rep_preal(B)}) = B"
+apply (blast intro: preal_le_anti_sym preal_less_add_left_subsetI preal_less_add_left_subsetI2)
+done
+
+lemma preal_less_add_left_Ex: "!! (A::preal). A < B ==> \<exists>D. A + D = B"
+by (fast dest: preal_less_add_left)
+
+lemma preal_add_less2_mono1: "!!(A::preal). A < B ==> A + C < B + C"
+apply (auto dest!: preal_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: "!!(A::preal). A < B ==> C + A < C + B"
+by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute)
+
+lemma preal_mult_less_mono1:
+      "!!(q1::preal). q1 < q2 ==> q1 * x < q2 * x"
+apply (drule preal_less_add_left_Ex)
+apply (auto simp add: preal_add_mult_distrib preal_self_less_add_left)
+done
+
+lemma preal_mult_left_less_mono1: "!!(q1::preal). q1 < q2  ==> x * q1 < x * q2"
+by (auto dest: preal_mult_less_mono1 simp add: preal_mult_commute)
+
+lemma preal_mult_left_le_mono1: "!!(q1::preal). q1 \<le> q2  ==> x * q1 \<le> x * q2"
+apply (simp add: preal_le_iff_less_or_eq) 
+apply (blast intro!: preal_mult_left_less_mono1)
+done
+
+lemma preal_mult_le_mono1: "!!(q1::preal). q1 \<le> q2  ==> q1 * x \<le> q2 * x"
+by (auto dest: preal_mult_left_le_mono1 simp add: preal_mult_commute)
+
+lemma preal_add_left_le_mono1: "!!(q1::preal). q1 \<le> q2  ==> x + q1 \<le> x + q2"
+apply (simp add: preal_le_iff_less_or_eq) 
+apply (auto intro!: preal_add_less2_mono1 simp add: preal_add_commute)
+done
+
+lemma preal_add_le_mono1: "!!(q1::preal). q1 \<le> q2  ==> q1 + x \<le> q2 + x"
+by (auto dest: preal_add_left_le_mono1 simp add: preal_add_commute)
+
+lemma preal_add_right_less_cancel: "!!(A::preal). A + C < B + C ==> A < B"
+apply (cut_tac preal_linear)
+apply (auto elim: preal_less_irrefl)
+apply (drule_tac A = B and C = C in preal_add_less2_mono1)
+apply (fast dest: preal_less_trans elim: preal_less_irrefl)
+done
+
+lemma preal_add_left_less_cancel: "!!(A::preal). C + A < C + B ==> A < B"
+by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute)
+
+lemma preal_add_less_iff1 [simp]: "((A::preal) + C < B + C) = (A < B)"
+by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
+
+lemma preal_add_less_iff2 [simp]: "(C + (A::preal) < C + B) = (A < B)"
+by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
+
+lemma preal_add_less_mono:
+     "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
+apply (auto dest!: preal_less_add_left_Ex simp add: preal_add_ac)
+apply (rule preal_add_assoc [THEN subst])
+apply (rule preal_self_less_add_right)
+done
+
+lemma preal_mult_less_mono:
+     "[| x1 < y1; x2 < y2 |] ==> x1 * x2 < y1 * (y2::preal)"
+apply (auto dest!: preal_less_add_left_Ex simp add: preal_add_mult_distrib preal_add_mult_distrib2 preal_self_less_add_left preal_add_assoc preal_mult_ac)
+done
+
+lemma preal_add_right_cancel: "(A::preal) + C = B + C ==> A = B"
+apply (cut_tac preal_linear [of A B], safe)
+apply (drule_tac [!] C = C in preal_add_less2_mono1)
+apply (auto elim: preal_less_irrefl)
+done
+
+lemma preal_add_left_cancel: "!!(A::preal). C + A = C + B ==> A = B"
+by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
+
+lemma preal_add_left_cancel_iff [simp]: "(C + A = C + B) = ((A::preal) = B)"
+by (fast intro: preal_add_left_cancel)
+
+lemma preal_add_right_cancel_iff [simp]: "(A + C = B + C) = ((A::preal) = B)"
+by (fast intro: preal_add_right_cancel)
+
+
+
+subsection{*Completeness of type @{typ preal}*}
+
+text{*Prove that supremum is a cut*}
+
+lemma preal_sup_mem_Ex:
+     "\<exists>X. X \<in> P ==> \<exists>q.  q \<in> {w. \<exists>X. X \<in> P & w \<in> Rep_preal X}"
+apply safe
+apply (cut_tac X = X in mem_Rep_preal_Ex, auto)
+done
+
+text{*Part 1 of Dedekind sections definition*}
+lemma preal_sup_set_not_empty:
+     "\<exists>(X::preal). X \<in> P ==>
+          {} < {w. \<exists>X \<in> P. w \<in> Rep_preal X}"
+apply (drule preal_sup_mem_Ex)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 2 of Dedekind sections definition*}
+lemma preal_sup_not_mem_Ex:
+     "\<exists>Y. (\<forall>X \<in> P. X < Y)
+          ==> \<exists>q. q \<notin> {w. \<exists>X. X \<in> P & w \<in> Rep_preal(X)}"
+apply (unfold preal_less_def)
+apply (auto simp add: psubset_def)
+apply (cut_tac X = Y in not_mem_Rep_preal_Ex)
+apply (erule exE)
+apply (rule_tac x = x in exI)
+apply (auto dest!: bspec)
+done
+
+lemma preal_sup_not_mem_Ex1:
+     "\<exists>Y. (\<forall>X \<in> P. X \<le> Y)
+          ==> \<exists>q. q \<notin> {w. \<exists>X. X \<in> P & w \<in> Rep_preal(X)}"
+apply (unfold preal_le_def, safe)
+apply (cut_tac X = Y in not_mem_Rep_preal_Ex)
+apply (erule exE)
+apply (rule_tac x = x in exI)
+apply (auto dest!: bspec)
+done
+
+lemma preal_sup_set_not_prat_set:
+     "\<exists>Y. (\<forall>X \<in> P. X < Y) ==> {w. \<exists>X \<in> P. w \<in> Rep_preal(X)} < UNIV"
+apply (drule preal_sup_not_mem_Ex)
+apply (auto intro!: psubsetI)
+done
+
+lemma preal_sup_set_not_prat_set1:
+     "\<exists>Y. (\<forall>X \<in> P. X \<le> Y) ==> {w. \<exists>X \<in> P. w \<in> Rep_preal(X)} < UNIV"
+apply (drule preal_sup_not_mem_Ex1)
+apply (auto intro!: psubsetI)
+done
+
+text{*Part 3 of Dedekind sections definition*}
+lemma preal_sup_set_lemma3:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X < Y) |]
+          ==> \<forall>y \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}.
+              \<forall>z. z < y --> z \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}"
+apply (auto elim: Rep_preal [THEN prealE_lemma3b])
+done
+
+lemma preal_sup_set_lemma3_1:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X \<le> Y) |]
+          ==> \<forall>y \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}.
+              \<forall>z. z < y --> z \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}"
+apply (auto elim: Rep_preal [THEN prealE_lemma3b])
+done
+
+lemma preal_sup_set_lemma4:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X < Y) |]
+          ==>  \<forall>y \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}.
+              Bex {w. \<exists>X \<in> P. w \<in> Rep_preal X} (op < y)"
+apply (blast dest: Rep_preal [THEN prealE_lemma4a])
+done
+
+lemma preal_sup_set_lemma4_1:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X \<le> Y) |]
+          ==>  \<forall>y \<in> {w. \<exists>X \<in> P. w \<in> Rep_preal X}.
+              Bex {w. \<exists>X \<in> P. w \<in> Rep_preal X} (op < y)"
+apply (blast dest: Rep_preal [THEN prealE_lemma4a])
+done
+
+lemma preal_sup:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X < Y) |]
+          ==> {w. \<exists>X \<in> P. w \<in> Rep_preal(X)}: preal"
+apply (rule prealI2)
+apply (rule preal_sup_set_not_empty)
+apply (rule_tac [2] preal_sup_set_not_prat_set)
+apply (rule_tac [3] preal_sup_set_lemma3)
+apply (rule_tac [5] preal_sup_set_lemma4, auto)
+done
+
+lemma preal_sup1:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X \<le> Y) |]
+          ==> {w. \<exists>X \<in> P. w \<in> Rep_preal(X)}: preal"
+apply (rule prealI2)
+apply (rule preal_sup_set_not_empty)
+apply (rule_tac [2] preal_sup_set_not_prat_set1)
+apply (rule_tac [3] preal_sup_set_lemma3_1)
+apply (rule_tac [5] preal_sup_set_lemma4_1, auto)
+done
+
+lemma preal_psup_leI: "\<exists>Y. (\<forall>X \<in> P. X < Y) ==> \<forall>x \<in> P. x \<le> psup P"
+apply (unfold psup_def)
+apply (auto simp add: preal_le_def)
+apply (rule preal_sup [THEN Abs_preal_inverse, THEN ssubst], auto)
+done
+
+lemma preal_psup_leI2: "\<exists>Y. (\<forall>X \<in> P. X \<le> Y) ==> \<forall>x \<in> P. x \<le> psup P"
+apply (unfold psup_def)
+apply (auto simp add: preal_le_def)
+apply (rule preal_sup1 [THEN Abs_preal_inverse, THEN ssubst])
+apply (auto simp add: preal_le_def)
+done
+
+lemma preal_psup_leI2b:
+     "[| \<exists>Y. (\<forall>X \<in> P. X < Y); x \<in> P |] ==> x \<le> psup P"
+apply (blast dest!: preal_psup_leI)
+done
+
+lemma preal_psup_leI2a:
+     "[| \<exists>Y. (\<forall>X \<in> P. X \<le> Y); x \<in> P |] ==> x \<le> psup P"
+apply (blast dest!: preal_psup_leI2)
+done
+
+lemma psup_le_ub: "[| \<exists>X. X \<in> P; \<forall>X \<in> P. X < Y |] ==> psup P \<le> Y"
+apply (unfold psup_def)
+apply (auto simp add: preal_le_def)
+apply (drule preal_sup [OF exI exI, THEN Abs_preal_inverse, THEN subst])
+apply (rotate_tac [2] 1)
+prefer 2 apply assumption
+apply (auto dest!: bspec simp add: preal_less_def psubset_def)
+done
+
+lemma psup_le_ub1: "[| \<exists>X. X \<in> P; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
+apply (unfold psup_def)
+apply (auto simp add: preal_le_def)
+apply (drule preal_sup1 [OF exI exI, THEN Abs_preal_inverse, THEN subst])
+apply (rotate_tac [2] 1)
+prefer 2 apply assumption
+apply (auto dest!: bspec simp add: preal_less_def psubset_def preal_le_def)
+done
+
+text{*Supremum property*}
+lemma preal_complete:
+     "[|\<exists>(X::preal). X \<in> P; \<exists>Y. (\<forall>X \<in> P. X < Y) |]
+          ==> (\<forall>Y. (\<exists>X \<in> P. Y < X) = (Y < psup P))"
+apply (frule preal_sup [THEN Abs_preal_inverse], fast)
+apply (auto simp add: psup_def preal_less_def)
+apply (cut_tac x = Xa and y = Ya in preal_linear)
+apply (auto dest: psubsetD simp add: preal_less_def)
+done
+
+
+subsection{*The Embadding from @{typ prat} into @{typ preal}*}
+
+lemma lemma_preal_rat_less: "x < z1 + z2 ==> x * z1 * qinv (z1 + z2) < z1"
+apply (drule_tac x = "z1 * qinv (z1 + z2) " in prat_mult_less2_mono1)
+apply (simp add: prat_mult_ac)
+done
+
+lemma lemma_preal_rat_less2: "x < z1 + z2 ==> x * z2 * qinv (z1 + z2) < z2"
+apply (subst prat_add_commute)
+apply (drule prat_add_commute [THEN subst])
+apply (erule lemma_preal_rat_less)
+done
+
+lemma preal_of_prat_add:
+      "preal_of_prat ((z1::prat) + z2) =
+       preal_of_prat z1 + preal_of_prat z2"
+apply (unfold preal_of_prat_def preal_add_def)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (auto intro: prat_add_less_mono 
+            simp add: lemma_prat_less_set_mem_preal [THEN Abs_preal_inverse])
+apply (rule_tac x = "x*z1*qinv (z1+z2) " in exI, rule conjI)
+apply (erule lemma_preal_rat_less)
+apply (rule_tac x = "x*z2*qinv (z1+z2) " in exI, rule conjI)
+apply (erule lemma_preal_rat_less2)
+apply (simp add: prat_add_mult_distrib [symmetric] 
+                 prat_add_mult_distrib2 [symmetric] prat_mult_ac)
+done
+
+lemma lemma_preal_rat_less3: "x < xa ==> x*z1*qinv(xa) < z1"
+apply (drule_tac x = "z1 * qinv xa" in prat_mult_less2_mono1)
+apply (drule prat_mult_left_commute [THEN subst])
+apply (simp add: prat_mult_ac)
+done
+
+lemma lemma_preal_rat_less4: "xa < z1 * z2 ==> xa*z2*qinv(z1*z2) < z2"
+apply (drule_tac x = "z2 * qinv (z1*z2) " in prat_mult_less2_mono1)
+apply (drule prat_mult_left_commute [THEN subst])
+apply (simp add: prat_mult_ac)
+done
+
+lemma preal_of_prat_mult:
+      "preal_of_prat ((z1::prat) * z2) =
+       preal_of_prat z1 * preal_of_prat z2"
+apply (unfold preal_of_prat_def preal_mult_def)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (auto intro: prat_mult_less_mono 
+            simp add: lemma_prat_less_set_mem_preal [THEN Abs_preal_inverse])
+apply (drule prat_dense, safe)
+apply (rule_tac x = "x*z1*qinv (xa) " in exI, rule conjI)
+apply (erule lemma_preal_rat_less3)
+apply (rule_tac x = " xa*z2*qinv (z1*z2) " in exI, rule conjI)
+apply (erule lemma_preal_rat_less4)
+apply (simp add: qinv_mult_eq [symmetric] prat_mult_ac)
+apply (simp add: prat_mult_assoc [symmetric])
+done
+
+lemma preal_of_prat_less_iff [simp]:
+      "(preal_of_prat p < preal_of_prat q) = (p < q)"
+apply (unfold preal_of_prat_def preal_less_def)
+apply (auto dest!: lemma_prat_set_eq elim: prat_less_trans 
+        simp add: lemma_prat_less_set_mem_preal psubset_def prat_less_not_refl)
+apply (rule_tac x = p and y = q in prat_linear_less2)
+apply (auto intro: prat_less_irrefl)
+done
+
+
+ML
+{*
+val inj_on_Abs_preal = thm"inj_on_Abs_preal";
+val inj_Rep_preal = thm"inj_Rep_preal";
+val empty_not_mem_preal = thm"empty_not_mem_preal";
+val one_set_mem_preal = thm"one_set_mem_preal";
+val preal_psubset_empty = thm"preal_psubset_empty";
+val mem_Rep_preal_Ex = thm"mem_Rep_preal_Ex";
+val inj_preal_of_prat = thm"inj_preal_of_prat";
+val not_in_preal_ub = thm"not_in_preal_ub";
+val preal_less_not_refl = thm"preal_less_not_refl";
+val preal_less_trans = thm"preal_less_trans";
+val preal_less_not_sym = thm"preal_less_not_sym";
+val preal_linear = thm"preal_linear";
+val preal_add_commute = thm"preal_add_commute";
+val preal_add_set_not_empty = thm"preal_add_set_not_empty";
+val preal_not_mem_add_set_Ex = thm"preal_not_mem_add_set_Ex";
+val preal_add_set_not_prat_set = thm"preal_add_set_not_prat_set";
+val preal_mem_add_set = thm"preal_mem_add_set";
+val preal_add_assoc = thm"preal_add_assoc";
+val preal_add_left_commute = thm"preal_add_left_commute";
+val preal_mult_commute = thm"preal_mult_commute";
+val preal_mult_set_not_empty = thm"preal_mult_set_not_empty";
+val preal_not_mem_mult_set_Ex = thm"preal_not_mem_mult_set_Ex";
+val preal_mult_set_not_prat_set = thm"preal_mult_set_not_prat_set";
+val preal_mem_mult_set = thm"preal_mem_mult_set";
+val preal_mult_assoc = thm"preal_mult_assoc";
+val preal_mult_left_commute = thm"preal_mult_left_commute";
+val preal_mult_1 = thm"preal_mult_1";
+val preal_mult_1_right = thm"preal_mult_1_right";
+val mem_Rep_preal_addD = thm"mem_Rep_preal_addD";
+val mem_Rep_preal_addI = thm"mem_Rep_preal_addI";
+val mem_Rep_preal_add_iff = thm"mem_Rep_preal_add_iff";
+val mem_Rep_preal_multD = thm"mem_Rep_preal_multD";
+val mem_Rep_preal_multI = thm"mem_Rep_preal_multI";
+val mem_Rep_preal_mult_iff = thm"mem_Rep_preal_mult_iff";
+val preal_add_mult_distrib2 = thm"preal_add_mult_distrib2";
+val preal_add_mult_distrib = thm"preal_add_mult_distrib";
+val qinv_not_mem_Rep_preal_Ex = thm"qinv_not_mem_Rep_preal_Ex";
+val preal_inv_set_not_empty = thm"preal_inv_set_not_empty";
+val qinv_mem_Rep_preal_Ex = thm"qinv_mem_Rep_preal_Ex";
+val preal_not_mem_inv_set_Ex = thm"preal_not_mem_inv_set_Ex";
+val preal_inv_set_not_prat_set = thm"preal_inv_set_not_prat_set";
+val preal_mem_inv_set = thm"preal_mem_inv_set";
+val preal_mem_mult_invD = thm"preal_mem_mult_invD";
+val preal_mem_mult_invI = thm"preal_mem_mult_invI";
+val preal_mult_inv = thm"preal_mult_inv";
+val preal_mult_inv_right = thm"preal_mult_inv_right";
+val Rep_preal_self_subset = thm"Rep_preal_self_subset";
+val Rep_preal_sum_not_subset = thm"Rep_preal_sum_not_subset";
+val Rep_preal_sum_not_eq = thm"Rep_preal_sum_not_eq";
+val preal_self_less_add_left = thm"preal_self_less_add_left";
+val preal_self_less_add_right = thm"preal_self_less_add_right";
+val preal_less_le_iff = thm"preal_less_le_iff";
+val preal_le_refl = thm"preal_le_refl";
+val preal_le_trans = thm"preal_le_trans";
+val preal_le_anti_sym = thm"preal_le_anti_sym";
+val preal_neq_iff = thm"preal_neq_iff";
+val preal_less_le = thm"preal_less_le";
+val psubset_trans = thm"psubset_trans";
+val preal_less_set_not_empty = thm"preal_less_set_not_empty";
+val preal_less_set_not_prat_set = thm"preal_less_set_not_prat_set";
+val preal_mem_less_set = thm"preal_mem_less_set";
+val preal_less_add_left_subsetI = thm"preal_less_add_left_subsetI";
+val preal_less_add_left_subsetI2 = thm"preal_less_add_left_subsetI2";
+val preal_less_add_left = thm"preal_less_add_left";
+val preal_less_add_left_Ex = thm"preal_less_add_left_Ex";
+val preal_add_less2_mono1 = thm"preal_add_less2_mono1";
+val preal_add_less2_mono2 = thm"preal_add_less2_mono2";
+val preal_mult_less_mono1 = thm"preal_mult_less_mono1";
+val preal_mult_left_less_mono1 = thm"preal_mult_left_less_mono1";
+val preal_mult_left_le_mono1 = thm"preal_mult_left_le_mono1";
+val preal_mult_le_mono1 = thm"preal_mult_le_mono1";
+val preal_add_left_le_mono1 = thm"preal_add_left_le_mono1";
+val preal_add_le_mono1 = thm"preal_add_le_mono1";
+val preal_add_right_less_cancel = thm"preal_add_right_less_cancel";
+val preal_add_left_less_cancel = thm"preal_add_left_less_cancel";
+val preal_add_less_iff1 = thm"preal_add_less_iff1";
+val preal_add_less_iff2 = thm"preal_add_less_iff2";
+val preal_add_less_mono = thm"preal_add_less_mono";
+val preal_mult_less_mono = thm"preal_mult_less_mono";
+val preal_add_right_cancel = thm"preal_add_right_cancel";
+val preal_add_left_cancel = thm"preal_add_left_cancel";
+val preal_add_left_cancel_iff = thm"preal_add_left_cancel_iff";
+val preal_add_right_cancel_iff = thm"preal_add_right_cancel_iff";
+val preal_sup_mem_Ex = thm"preal_sup_mem_Ex";
+val preal_sup_set_not_empty = thm"preal_sup_set_not_empty";
+val preal_sup_not_mem_Ex = thm"preal_sup_not_mem_Ex";
+val preal_sup_not_mem_Ex1 = thm"preal_sup_not_mem_Ex1";
+val preal_sup_set_not_prat_set = thm"preal_sup_set_not_prat_set";
+val preal_sup_set_not_prat_set1 = thm"preal_sup_set_not_prat_set1";
+val preal_sup = thm"preal_sup";
+val preal_sup1 = thm"preal_sup1";
+val preal_psup_leI = thm"preal_psup_leI";
+val preal_psup_leI2 = thm"preal_psup_leI2";
+val preal_psup_leI2b = thm"preal_psup_leI2b";
+val preal_psup_leI2a = thm"preal_psup_leI2a";
+val psup_le_ub = thm"psup_le_ub";
+val psup_le_ub1 = thm"psup_le_ub1";
+val preal_complete = thm"preal_complete";
+val preal_of_prat_add = thm"preal_of_prat_add";
+val preal_of_prat_mult = thm"preal_of_prat_mult";
+
+val preal_add_ac = thms"preal_add_ac";
+val preal_mult_ac = thms"preal_mult_ac";
+*}
+
 end
-




