--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/PReal.thy Wed Dec 03 15:58:44 2008 +0100
@@ -0,0 +1,1343 @@
+(* Title : PReal.thy
+ 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]
+ provides some of the definitions.
+*)
+
+header {* Positive real numbers *}
+
+theory PReal
+imports Rational "~~/src/HOL/Library/Dense_Linear_Order"
+begin
+
+text{*Could be generalized and moved to @{text Ring_and_Field}*}
+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 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
+ preal_of_rat :: "rat => preal" where
+ "preal_of_rat q = Abs_preal {x::rat. 0 < x & x < q}"
+
+definition
+ psup :: "preal set => preal" where
+ "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"
+
+definition
+ add_set :: "[rat set,rat set] => rat set" where
+ "add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
+
+definition
+ diff_set :: "[rat set,rat set] => rat set" where
+ [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 == preal_of_rat 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{*@{term preal_of_prat}: the Injection from prat to preal*}
+
+lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal"
+by (simp add: preal_def cut_of_rat)
+
+lemma rat_subset_imp_le:
+ "[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y"
+apply (simp add: linorder_not_less [symmetric])
+apply (blast dest: dense intro: order_less_trans)
+done
+
+lemma rat_set_eq_imp_eq:
+ "[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y};
+ 0 < x; 0 < y|] ==> x = y"
+by (blast intro: rat_subset_imp_le order_antisym)
+
+
+
+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
+
+lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
+by (rule add_left_commute)
+
+text{* Positive Real 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 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
+
+lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
+by (rule mult_left_commute)
+
+
+text{* Positive Real multiplication is an AC operator *}
+lemmas preal_mult_ac =
+ preal_mult_assoc preal_mult_commute preal_mult_left_commute
+
+
+text{* Positive real 1 is the multiplicative identity element *}
+
+lemma preal_mult_1: "(1::preal) * z = z"
+unfolding preal_one_def
+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 "preal_of_rat 1 * Abs_preal A = Abs_preal A"
+ by (simp add: preal_of_rat_def preal_mult_def mult_set_def
+ rat_mem_preal A)
+qed
+
+instance preal :: comm_monoid_mult
+by intro_classes (rule preal_mult_1)
+
+lemma preal_mult_1_right: "z * (1::preal) = z"
+by (rule mult_1_right)
+
+
+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: left_distrib add_ac 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: mult_rat le_rat 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: right_distrib divide_inverse mult_ac, simp)
+ also have "... = r*x" using ypos
+ by (simp add: times_divide_eq_left)
+ 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_of_rat:
+ "0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
+by (simp add: preal_of_rat_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(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
+apply (auto simp add: Bex_def Rep_preal_of_rat 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(preal_of_rat 1)"
+apply (auto simp add: Bex_def Rep_preal_of_rat 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)"
+unfolding preal_one_def
+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
+
+lemma preal_self_less_add_right: "(R::preal) < S + R"
+by (simp add: preal_add_commute preal_self_less_add_left)
+
+lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
+by (insert preal_self_less_add_left [of x y], auto)
+
+
+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_right: "((R::preal) + T < S + T) = (R < S)"
+by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
+
+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_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
+by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right)
+
+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_less_mono:
+ "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
+apply (auto dest!: 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_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)
+
+lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
+by (fast intro: preal_add_left_cancel)
+
+lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
+by (fast intro: preal_add_right_cancel)
+
+lemmas preal_cancels =
+ preal_add_less_cancel_right preal_add_less_cancel_left
+ preal_add_le_cancel_right preal_add_le_cancel_left
+ preal_add_left_cancel_iff preal_add_right_cancel_iff
+
+instance preal :: ordered_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
+
+
+subsection{*The Embedding from @{typ rat} into @{typ preal}*}
+
+lemma preal_of_rat_add_lemma1:
+ "[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
+apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
+apply (simp add: zero_less_mult_iff)
+apply (simp add: mult_ac)
+done
+
+lemma preal_of_rat_add_lemma2:
+ assumes "u < x + y"
+ and "0 < x"
+ and "0 < y"
+ and "0 < u"
+ shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
+proof (intro exI conjI)
+ show "u * x * inverse(x+y) < x" using prems
+ by (simp add: preal_of_rat_add_lemma1)
+ show "u * y * inverse(x+y) < y" using prems
+ by (simp add: preal_of_rat_add_lemma1 add_commute [of x])
+ show "0 < u * x * inverse (x + y)" using prems
+ by (simp add: zero_less_mult_iff)
+ show "0 < u * y * inverse (x + y)" using prems
+ by (simp add: zero_less_mult_iff)
+ show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
+ by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
+qed
+
+lemma preal_of_rat_add:
+ "[| 0 < x; 0 < y|]
+ ==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
+apply (unfold preal_of_rat_def preal_add_def)
+apply (simp add: rat_mem_preal)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (auto simp add: add_set_def)
+apply (blast dest: preal_of_rat_add_lemma2)
+done
+
+lemma preal_of_rat_mult_lemma1:
+ "[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
+apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
+apply (simp add: zero_less_mult_iff)
+apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
+apply (simp_all add: mult_ac)
+done
+
+lemma preal_of_rat_mult_lemma2:
+ assumes xless: "x < y * z"
+ and xpos: "0 < x"
+ and ypos: "0 < y"
+ shows "x * z * inverse y * inverse z < (z::rat)"
+proof -
+ have "0 < y * z" using prems by simp
+ hence zpos: "0 < z" using prems by (simp add: zero_less_mult_iff)
+ have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
+ by (simp add: mult_ac)
+ also have "... = x/y" using zpos
+ by (simp add: divide_inverse)
+ also from xless have "... < z"
+ by (simp add: pos_divide_less_eq [OF ypos] mult_commute)
+ finally show ?thesis .
+qed
+
+lemma preal_of_rat_mult_lemma3:
+ assumes uless: "u < x * y"
+ and "0 < x"
+ and "0 < y"
+ and "0 < u"
+ shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
+proof -
+ from dense [OF uless]
+ obtain r where "u < r" "r < x * y" by blast
+ thus ?thesis
+ proof (intro exI conjI)
+ show "u * x * inverse r < x" using prems
+ by (simp add: preal_of_rat_mult_lemma1)
+ show "r * y * inverse x * inverse y < y" using prems
+ by (simp add: preal_of_rat_mult_lemma2)
+ show "0 < u * x * inverse r" using prems
+ by (simp add: zero_less_mult_iff)
+ show "0 < r * y * inverse x * inverse y" using prems
+ by (simp add: zero_less_mult_iff)
+ have "u * x * inverse r * (r * y * inverse x * inverse y) =
+ u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
+ by (simp only: mult_ac)
+ thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
+ by simp
+ qed
+qed
+
+lemma preal_of_rat_mult:
+ "[| 0 < x; 0 < y|]
+ ==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
+apply (unfold preal_of_rat_def preal_mult_def)
+apply (simp add: rat_mem_preal)
+apply (rule_tac f = Abs_preal in arg_cong)
+apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def)
+apply (blast dest: preal_of_rat_mult_lemma3)
+done
+
+lemma preal_of_rat_less_iff:
+ "[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
+by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal)
+
+lemma preal_of_rat_le_iff:
+ "[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
+by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric])
+
+lemma preal_of_rat_eq_iff:
+ "[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
+by (simp add: preal_of_rat_le_iff order_eq_iff)
+
+end