src/HOL/Real/PReal.thy
author haftmann
Tue Mar 20 15:52:40 2007 +0100 (2007-03-20)
changeset 22483 86064f2f2188
parent 21404 eb85850d3eb7
child 22710 f44439cdce77
permissions -rw-r--r--
added instance for lattice
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(*  Title       : PReal.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Description : The positive reals as Dedekind sections of positive
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         rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
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                  provides some of the definitions.
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*)
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header {* Positive real numbers *}
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theory PReal
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imports Rational
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begin
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text{*Could be generalized and moved to @{text Ring_and_Field}*}
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lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"
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by (rule_tac x="b-a" in exI, simp)
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text{*As a special case, the sum of two positives is positive.  One of the
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premises could be weakened to the relation @{text "\<le>"}.*}
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lemma pos_add_strict: "[|0<a; b<c|] ==> b < a + (c::'a::ordered_semidom)"
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by (insert add_strict_mono [of 0 a b c], simp)
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lemma interval_empty_iff:
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     "({y::'a::ordered_field. x < y & y < z} = {}) = (~(x < z))"
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by (auto dest: dense)
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definition
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  cut :: "rat set => bool" where
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  "cut A = ({} \<subset> A &
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            A < {r. 0 < r} &
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            (\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"
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lemma cut_of_rat: 
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  assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}" (is "cut ?A")
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proof -
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  from q have pos: "?A < {r. 0 < r}" by force
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  have nonempty: "{} \<subset> ?A"
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  proof
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    show "{} \<subseteq> ?A" by simp
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    show "{} \<noteq> ?A"
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      by (force simp only: q eq_commute [of "{}"] interval_empty_iff)
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  qed
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  show ?thesis
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    by (simp add: cut_def pos nonempty,
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        blast dest: dense intro: order_less_trans)
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qed
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typedef preal = "{A. cut A}"
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  by (blast intro: cut_of_rat [OF zero_less_one])
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instance preal :: "{ord, plus, minus, times, inverse}" ..
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definition
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  preal_of_rat :: "rat => preal" where
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  "preal_of_rat q = Abs_preal {x::rat. 0 < x & x < q}"
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definition
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  psup :: "preal set => preal" where
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  "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"
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definition
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  add_set :: "[rat set,rat set] => rat set" where
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  "add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
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definition
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  diff_set :: "[rat set,rat set] => rat set" where
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  "diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"
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definition
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  mult_set :: "[rat set,rat set] => rat set" where
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  "mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
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definition
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  inverse_set :: "rat set => rat set" where
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  "inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"
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defs (overloaded)
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  preal_less_def:
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    "R < S == Rep_preal R < Rep_preal S"
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  preal_le_def:
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    "R \<le> S == Rep_preal R \<subseteq> Rep_preal S"
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  preal_add_def:
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    "R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"
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  preal_diff_def:
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    "R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"
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  preal_mult_def:
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    "R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"
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  preal_inverse_def:
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    "inverse R == Abs_preal (inverse_set (Rep_preal R))"
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text{*Reduces equality on abstractions to equality on representatives*}
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declare Abs_preal_inject [simp]
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declare Abs_preal_inverse [simp]
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lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
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by (simp add: preal_def cut_of_rat)
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lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"
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by (unfold preal_def cut_def, blast)
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lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"
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by (drule preal_nonempty, fast)
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lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"
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by (force simp add: preal_def cut_def)
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lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"
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by (drule preal_imp_psubset_positives, auto)
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lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
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by (unfold preal_def cut_def, blast)
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lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
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by (unfold preal_def cut_def, blast)
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text{*Relaxing the final premise*}
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lemma preal_downwards_closed':
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     "[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"
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apply (simp add: order_le_less)
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apply (blast intro: preal_downwards_closed)
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done
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text{*A positive fraction not in a positive real is an upper bound.
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 Gleason p. 122 - Remark (1)*}
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lemma not_in_preal_ub:
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  assumes A: "A \<in> preal"
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    and notx: "x \<notin> A"
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    and y: "y \<in> A"
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    and pos: "0 < x"
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  shows "y < x"
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proof (cases rule: linorder_cases)
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  assume "x<y"
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  with notx show ?thesis
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    by (simp add:  preal_downwards_closed [OF A y] pos)
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next
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  assume "x=y"
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  with notx and y show ?thesis by simp
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next
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  assume "y<x"
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  thus ?thesis .
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qed
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text {* preal lemmas instantiated to @{term "Rep_preal X"} *}
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lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
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by (rule preal_Ex_mem [OF Rep_preal])
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lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X"
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by (rule preal_exists_bound [OF Rep_preal])
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lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
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subsection{*@{term preal_of_prat}: the Injection from prat to preal*}
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lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal"
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by (simp add: preal_def cut_of_rat)
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lemma rat_subset_imp_le:
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     "[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y"
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apply (simp add: linorder_not_less [symmetric])
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apply (blast dest: dense intro: order_less_trans)
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done
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lemma rat_set_eq_imp_eq:
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     "[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y};
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        0 < x; 0 < y|] ==> x = y"
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by (blast intro: rat_subset_imp_le order_antisym)
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subsection{*Properties of Ordering*}
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lemma preal_le_refl: "w \<le> (w::preal)"
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by (simp add: preal_le_def)
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lemma preal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::preal)"
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by (force simp add: preal_le_def)
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lemma preal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::preal)"
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apply (simp add: preal_le_def)
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apply (rule Rep_preal_inject [THEN iffD1], blast)
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done
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(* Axiom 'order_less_le' of class 'order': *)
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lemma preal_less_le: "((w::preal) < z) = (w \<le> z & w \<noteq> z)"
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by (simp add: preal_le_def preal_less_def Rep_preal_inject psubset_def)
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instance preal :: order
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  by intro_classes
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    (assumption |
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      rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+
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lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"
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by (insert preal_imp_psubset_positives, blast)
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lemma preal_le_linear: "x <= y | y <= (x::preal)"
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apply (auto simp add: preal_le_def)
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apply (rule ccontr)
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apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
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             elim: order_less_asym)
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done
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instance preal :: linorder
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  by intro_classes (rule preal_le_linear)
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instance preal :: distrib_lattice
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  "inf \<equiv> min"
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  "sup \<equiv> max"
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  by intro_classes
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    (auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1)
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subsection{*Properties of Addition*}
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lemma preal_add_commute: "(x::preal) + y = y + x"
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apply (unfold preal_add_def add_set_def)
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apply (rule_tac f = Abs_preal in arg_cong)
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apply (force simp add: add_commute)
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done
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text{*Lemmas for proving that addition of two positive reals gives
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 a positive real*}
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lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A"
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by blast
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text{*Part 1 of Dedekind sections definition*}
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lemma add_set_not_empty:
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     "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"
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apply (drule preal_nonempty)+
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apply (auto simp add: add_set_def)
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done
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text{*Part 2 of Dedekind sections definition.  A structured version of
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this proof is @{text preal_not_mem_mult_set_Ex} below.*}
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lemma preal_not_mem_add_set_Ex:
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     "[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B"
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apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto) 
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apply (rule_tac x = "x+xa" in exI)
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apply (simp add: add_set_def, clarify)
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apply (drule (3) not_in_preal_ub)+
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apply (force dest: add_strict_mono)
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done
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lemma add_set_not_rat_set:
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   assumes A: "A \<in> preal" 
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       and B: "B \<in> preal"
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     shows "add_set A B < {r. 0 < r}"
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proof
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  from preal_imp_pos [OF A] preal_imp_pos [OF B]
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  show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def) 
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next
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  show "add_set A B \<noteq> {r. 0 < r}"
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    by (insert preal_not_mem_add_set_Ex [OF A B], blast) 
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qed
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text{*Part 3 of Dedekind sections definition*}
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lemma add_set_lemma3:
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     "[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|] 
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      ==> z \<in> add_set A B"
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proof (unfold add_set_def, clarify)
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  fix x::rat and y::rat
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  assume A: "A \<in> preal" 
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    and B: "B \<in> preal"
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    and [simp]: "0 < z"
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    and zless: "z < x + y"
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    and x:  "x \<in> A"
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    and y:  "y \<in> B"
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  have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])
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  have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])
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  have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)
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  let ?f = "z/(x+y)"
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  have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
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  show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"
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  proof (intro bexI)
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    show "z = x*?f + y*?f"
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      by (simp add: left_distrib [symmetric] divide_inverse mult_ac
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          order_less_imp_not_eq2)
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  next
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    show "y * ?f \<in> B"
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    proof (rule preal_downwards_closed [OF B y])
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      show "0 < y * ?f"
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        by (simp add: divide_inverse zero_less_mult_iff)
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    next
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      show "y * ?f < y"
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        by (insert mult_strict_left_mono [OF fless ypos], simp)
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    qed
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  next
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    show "x * ?f \<in> A"
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    proof (rule preal_downwards_closed [OF A x])
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      show "0 < x * ?f"
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	by (simp add: divide_inverse zero_less_mult_iff)
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    next
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      show "x * ?f < x"
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	by (insert mult_strict_left_mono [OF fless xpos], simp)
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    qed
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  qed
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qed
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text{*Part 4 of Dedekind sections definition*}
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lemma add_set_lemma4:
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     "[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
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apply (auto simp add: add_set_def)
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apply (frule preal_exists_greater [of A], auto) 
paulson@14365
   321
apply (rule_tac x="u + y" in exI)
paulson@14365
   322
apply (auto intro: add_strict_left_mono)
paulson@14335
   323
done
paulson@14335
   324
paulson@14365
   325
lemma mem_add_set:
paulson@14365
   326
     "[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
paulson@14365
   327
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   328
apply (blast intro!: add_set_not_empty add_set_not_rat_set
paulson@14365
   329
                     add_set_lemma3 add_set_lemma4)
paulson@14335
   330
done
paulson@14335
   331
paulson@14335
   332
lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
paulson@14365
   333
apply (simp add: preal_add_def mem_add_set Rep_preal)
paulson@14365
   334
apply (force simp add: add_set_def add_ac)
paulson@14335
   335
done
paulson@14335
   336
paulson@14335
   337
lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
paulson@14335
   338
  apply (rule mk_left_commute [of "op +"])
paulson@14335
   339
  apply (rule preal_add_assoc)
paulson@14335
   340
  apply (rule preal_add_commute)
paulson@14335
   341
  done
paulson@14335
   342
paulson@14365
   343
text{* Positive Real addition is an AC operator *}
paulson@14335
   344
lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
paulson@14335
   345
paulson@14335
   346
paulson@14335
   347
subsection{*Properties of Multiplication*}
paulson@14335
   348
paulson@14335
   349
text{*Proofs essentially same as for addition*}
paulson@14335
   350
paulson@14335
   351
lemma preal_mult_commute: "(x::preal) * y = y * x"
paulson@14365
   352
apply (unfold preal_mult_def mult_set_def)
paulson@14335
   353
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
   354
apply (force simp add: mult_commute)
paulson@14335
   355
done
paulson@14335
   356
nipkow@15055
   357
text{*Multiplication of two positive reals gives a positive real.*}
paulson@14335
   358
paulson@14335
   359
text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
paulson@14335
   360
paulson@14335
   361
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   362
lemma mult_set_not_empty:
paulson@14365
   363
     "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
paulson@14365
   364
apply (insert preal_nonempty [of A] preal_nonempty [of B]) 
paulson@14365
   365
apply (auto simp add: mult_set_def)
paulson@14335
   366
done
paulson@14335
   367
paulson@14335
   368
text{*Part 2 of Dedekind sections definition*}
paulson@14335
   369
lemma preal_not_mem_mult_set_Ex:
paulson@14365
   370
   assumes A: "A \<in> preal" 
paulson@14365
   371
       and B: "B \<in> preal"
paulson@14365
   372
     shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
paulson@14365
   373
proof -
paulson@14365
   374
  from preal_exists_bound [OF A]
paulson@14365
   375
  obtain x where [simp]: "0 < x" "x \<notin> A" by blast
paulson@14365
   376
  from preal_exists_bound [OF B]
paulson@14365
   377
  obtain y where [simp]: "0 < y" "y \<notin> B" by blast
paulson@14365
   378
  show ?thesis
paulson@14365
   379
  proof (intro exI conjI)
avigad@16775
   380
    show "0 < x*y" by (simp add: mult_pos_pos)
paulson@14365
   381
    show "x * y \<notin> mult_set A B"
paulson@14377
   382
    proof -
paulson@14377
   383
      { fix u::rat and v::rat
kleing@14550
   384
	      assume "u \<in> A" and "v \<in> B" and "x*y = u*v"
kleing@14550
   385
	      moreover
kleing@14550
   386
	      with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
kleing@14550
   387
	      moreover
kleing@14550
   388
	      with prems have "0\<le>v"
kleing@14550
   389
	        by (blast intro: preal_imp_pos [OF B]  order_less_imp_le prems)
kleing@14550
   390
	      moreover
kleing@14550
   391
        from calculation
kleing@14550
   392
	      have "u*v < x*y" by (blast intro: mult_strict_mono prems)
kleing@14550
   393
	      ultimately have False by force }
paulson@14377
   394
      thus ?thesis by (auto simp add: mult_set_def)
paulson@14365
   395
    qed
paulson@14365
   396
  qed
paulson@14365
   397
qed
paulson@14335
   398
paulson@14365
   399
lemma mult_set_not_rat_set:
wenzelm@19765
   400
  assumes A: "A \<in> preal" 
wenzelm@19765
   401
    and B: "B \<in> preal"
wenzelm@19765
   402
  shows "mult_set A B < {r. 0 < r}"
paulson@14365
   403
proof
paulson@14365
   404
  show "mult_set A B \<subseteq> {r. 0 < r}"
paulson@14365
   405
    by (force simp add: mult_set_def
wenzelm@19765
   406
      intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)
paulson@14365
   407
  show "mult_set A B \<noteq> {r. 0 < r}"
wenzelm@19765
   408
    using preal_not_mem_mult_set_Ex [OF A B] by blast
paulson@14365
   409
qed
paulson@14365
   410
paulson@14365
   411
paulson@14335
   412
paulson@14335
   413
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   414
lemma mult_set_lemma3:
paulson@14365
   415
     "[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|] 
paulson@14365
   416
      ==> z \<in> mult_set A B"
paulson@14365
   417
proof (unfold mult_set_def, clarify)
paulson@14365
   418
  fix x::rat and y::rat
paulson@14365
   419
  assume A: "A \<in> preal" 
wenzelm@19765
   420
    and B: "B \<in> preal"
wenzelm@19765
   421
    and [simp]: "0 < z"
wenzelm@19765
   422
    and zless: "z < x * y"
wenzelm@19765
   423
    and x:  "x \<in> A"
wenzelm@19765
   424
    and y:  "y \<in> B"
paulson@14365
   425
  have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
paulson@14365
   426
  show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
paulson@14365
   427
  proof
paulson@14365
   428
    show "\<exists>y'\<in>B. z = (z/y) * y'"
paulson@14365
   429
    proof
paulson@14365
   430
      show "z = (z/y)*y"
paulson@14430
   431
	by (simp add: divide_inverse mult_commute [of y] mult_assoc
paulson@14365
   432
		      order_less_imp_not_eq2)
paulson@14365
   433
      show "y \<in> B" .
paulson@14365
   434
    qed
paulson@14365
   435
  next
paulson@14365
   436
    show "z/y \<in> A"
paulson@14365
   437
    proof (rule preal_downwards_closed [OF A x])
paulson@14365
   438
      show "0 < z/y"
paulson@14365
   439
	by (simp add: zero_less_divide_iff)
paulson@14365
   440
      show "z/y < x" by (simp add: pos_divide_less_eq zless)
paulson@14365
   441
    qed
paulson@14365
   442
  qed
paulson@14365
   443
qed
paulson@14365
   444
paulson@14365
   445
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   446
lemma mult_set_lemma4:
paulson@14365
   447
     "[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
paulson@14365
   448
apply (auto simp add: mult_set_def)
paulson@14365
   449
apply (frule preal_exists_greater [of A], auto) 
paulson@14365
   450
apply (rule_tac x="u * y" in exI)
paulson@14365
   451
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B] 
paulson@14365
   452
                   mult_strict_right_mono)
paulson@14335
   453
done
paulson@14335
   454
paulson@14335
   455
paulson@14365
   456
lemma mem_mult_set:
paulson@14365
   457
     "[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
paulson@14365
   458
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   459
apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
paulson@14365
   460
                     mult_set_lemma3 mult_set_lemma4)
paulson@14335
   461
done
paulson@14335
   462
paulson@14335
   463
lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
paulson@14365
   464
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
paulson@14365
   465
apply (force simp add: mult_set_def mult_ac)
paulson@14335
   466
done
paulson@14335
   467
paulson@14335
   468
lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
paulson@14335
   469
  apply (rule mk_left_commute [of "op *"])
paulson@14335
   470
  apply (rule preal_mult_assoc)
paulson@14335
   471
  apply (rule preal_mult_commute)
paulson@14335
   472
  done
paulson@14335
   473
paulson@14365
   474
paulson@14365
   475
text{* Positive Real multiplication is an AC operator *}
paulson@14335
   476
lemmas preal_mult_ac =
paulson@14335
   477
       preal_mult_assoc preal_mult_commute preal_mult_left_commute
paulson@14335
   478
paulson@14365
   479
paulson@14365
   480
text{* Positive real 1 is the multiplicative identity element *}
paulson@14365
   481
paulson@14365
   482
lemma preal_mult_1: "(preal_of_rat 1) * z = z"
paulson@14365
   483
proof (induct z)
paulson@14365
   484
  fix A :: "rat set"
paulson@14365
   485
  assume A: "A \<in> preal"
paulson@14365
   486
  have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
paulson@14365
   487
  proof
paulson@14365
   488
    show "?lhs \<subseteq> A"
paulson@14365
   489
    proof clarify
paulson@14365
   490
      fix x::rat and u::rat and v::rat
paulson@14365
   491
      assume upos: "0<u" and "u<1" and v: "v \<in> A"
paulson@14365
   492
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
paulson@14365
   493
      hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems)
paulson@14365
   494
      thus "u * v \<in> A"
avigad@16775
   495
        by (force intro: preal_downwards_closed [OF A v] mult_pos_pos 
avigad@16775
   496
          upos vpos)
paulson@14365
   497
    qed
paulson@14365
   498
  next
paulson@14365
   499
    show "A \<subseteq> ?lhs"
paulson@14365
   500
    proof clarify
paulson@14365
   501
      fix x::rat
paulson@14365
   502
      assume x: "x \<in> A"
paulson@14365
   503
      have xpos: "0<x" by (rule preal_imp_pos [OF A x])
paulson@14365
   504
      from preal_exists_greater [OF A x]
paulson@14365
   505
      obtain v where v: "v \<in> A" and xlessv: "x < v" ..
paulson@14365
   506
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
paulson@14365
   507
      show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
paulson@14365
   508
      proof (intro exI conjI)
paulson@14365
   509
        show "0 < x/v"
paulson@14365
   510
          by (simp add: zero_less_divide_iff xpos vpos)
paulson@14365
   511
	show "x / v < 1"
paulson@14365
   512
          by (simp add: pos_divide_less_eq vpos xlessv)
paulson@14365
   513
        show "\<exists>v'\<in>A. x = (x / v) * v'"
paulson@14365
   514
        proof
paulson@14365
   515
          show "x = (x/v)*v"
paulson@14430
   516
	    by (simp add: divide_inverse mult_assoc vpos
paulson@14365
   517
                          order_less_imp_not_eq2)
paulson@14365
   518
          show "v \<in> A" .
paulson@14365
   519
        qed
paulson@14365
   520
      qed
paulson@14365
   521
    qed
paulson@14365
   522
  qed
paulson@14365
   523
  thus "preal_of_rat 1 * Abs_preal A = Abs_preal A"
paulson@14365
   524
    by (simp add: preal_of_rat_def preal_mult_def mult_set_def 
paulson@14365
   525
                  rat_mem_preal A)
paulson@14365
   526
qed
paulson@14365
   527
paulson@14365
   528
paulson@14365
   529
lemma preal_mult_1_right: "z * (preal_of_rat 1) = z"
paulson@14335
   530
apply (rule preal_mult_commute [THEN subst])
paulson@14335
   531
apply (rule preal_mult_1)
paulson@14335
   532
done
paulson@14335
   533
paulson@14335
   534
paulson@14335
   535
subsection{*Distribution of Multiplication across Addition*}
paulson@14335
   536
paulson@14335
   537
lemma mem_Rep_preal_add_iff:
paulson@14365
   538
      "(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"
paulson@14365
   539
apply (simp add: preal_add_def mem_add_set Rep_preal)
paulson@14365
   540
apply (simp add: add_set_def) 
paulson@14335
   541
done
paulson@14335
   542
paulson@14335
   543
lemma mem_Rep_preal_mult_iff:
paulson@14365
   544
      "(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
paulson@14365
   545
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
paulson@14365
   546
apply (simp add: mult_set_def) 
paulson@14365
   547
done
paulson@14335
   548
paulson@14365
   549
lemma distrib_subset1:
paulson@14365
   550
     "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
paulson@14365
   551
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
paulson@14365
   552
apply (force simp add: right_distrib)
paulson@14335
   553
done
paulson@14335
   554
paulson@14365
   555
lemma preal_add_mult_distrib_mean:
paulson@14365
   556
  assumes a: "a \<in> Rep_preal w"
wenzelm@19765
   557
    and b: "b \<in> Rep_preal w"
wenzelm@19765
   558
    and d: "d \<in> Rep_preal x"
wenzelm@19765
   559
    and e: "e \<in> Rep_preal y"
wenzelm@19765
   560
  shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
paulson@14365
   561
proof
paulson@14365
   562
  let ?c = "(a*d + b*e)/(d+e)"
paulson@14365
   563
  have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
paulson@14365
   564
    by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
paulson@14365
   565
  have cpos: "0 < ?c"
paulson@14365
   566
    by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
paulson@14365
   567
  show "a * d + b * e = ?c * (d + e)"
paulson@14430
   568
    by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
paulson@14365
   569
  show "?c \<in> Rep_preal w"
huffman@20495
   570
  proof (cases rule: linorder_le_cases)
huffman@20495
   571
    assume "a \<le> b"
huffman@20495
   572
    hence "?c \<le> b"
huffman@20495
   573
      by (simp add: pos_divide_le_eq right_distrib mult_right_mono
huffman@20495
   574
                    order_less_imp_le)
huffman@20495
   575
    thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
huffman@20495
   576
  next
huffman@20495
   577
    assume "b \<le> a"
huffman@20495
   578
    hence "?c \<le> a"
huffman@20495
   579
      by (simp add: pos_divide_le_eq right_distrib mult_right_mono
huffman@20495
   580
                    order_less_imp_le)
huffman@20495
   581
    thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
paulson@14365
   582
  qed
huffman@20495
   583
qed
paulson@14365
   584
paulson@14365
   585
lemma distrib_subset2:
paulson@14365
   586
     "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
paulson@14365
   587
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
paulson@14365
   588
apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
paulson@14335
   589
done
paulson@14335
   590
paulson@14365
   591
lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
paulson@15413
   592
apply (rule Rep_preal_inject [THEN iffD1])
paulson@14365
   593
apply (rule equalityI [OF distrib_subset1 distrib_subset2])
paulson@14335
   594
done
paulson@14335
   595
paulson@14365
   596
lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
paulson@14365
   597
by (simp add: preal_mult_commute preal_add_mult_distrib2)
paulson@14365
   598
paulson@14335
   599
paulson@14335
   600
subsection{*Existence of Inverse, a Positive Real*}
paulson@14335
   601
paulson@14365
   602
lemma mem_inv_set_ex:
paulson@14365
   603
  assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
paulson@14365
   604
proof -
paulson@14365
   605
  from preal_exists_bound [OF A]
paulson@14365
   606
  obtain x where [simp]: "0<x" "x \<notin> A" by blast
paulson@14365
   607
  show ?thesis
paulson@14365
   608
  proof (intro exI conjI)
paulson@14365
   609
    show "0 < inverse (x+1)"
paulson@14365
   610
      by (simp add: order_less_trans [OF _ less_add_one]) 
paulson@14365
   611
    show "inverse(x+1) < inverse x"
paulson@14365
   612
      by (simp add: less_imp_inverse_less less_add_one)
paulson@14365
   613
    show "inverse (inverse x) \<notin> A"
paulson@14365
   614
      by (simp add: order_less_imp_not_eq2)
paulson@14365
   615
  qed
paulson@14365
   616
qed
paulson@14335
   617
paulson@14335
   618
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   619
lemma inverse_set_not_empty:
paulson@14365
   620
     "A \<in> preal ==> {} \<subset> inverse_set A"
paulson@14365
   621
apply (insert mem_inv_set_ex [of A])
paulson@14365
   622
apply (auto simp add: inverse_set_def)
paulson@14335
   623
done
paulson@14335
   624
paulson@14335
   625
text{*Part 2 of Dedekind sections definition*}
paulson@14335
   626
paulson@14365
   627
lemma preal_not_mem_inverse_set_Ex:
paulson@14365
   628
   assumes A: "A \<in> preal"  shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
paulson@14365
   629
proof -
paulson@14365
   630
  from preal_nonempty [OF A]
paulson@14365
   631
  obtain x where x: "x \<in> A" and  xpos [simp]: "0<x" ..
paulson@14365
   632
  show ?thesis
paulson@14365
   633
  proof (intro exI conjI)
paulson@14365
   634
    show "0 < inverse x" by simp
paulson@14365
   635
    show "inverse x \<notin> inverse_set A"
paulson@14377
   636
    proof -
paulson@14377
   637
      { fix y::rat 
paulson@14377
   638
	assume ygt: "inverse x < y"
paulson@14377
   639
	have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
paulson@14377
   640
	have iyless: "inverse y < x" 
paulson@14377
   641
	  by (simp add: inverse_less_imp_less [of x] ygt)
paulson@14377
   642
	have "inverse y \<in> A"
paulson@14377
   643
	  by (simp add: preal_downwards_closed [OF A x] iyless)}
paulson@14377
   644
     thus ?thesis by (auto simp add: inverse_set_def)
paulson@14365
   645
    qed
paulson@14365
   646
  qed
paulson@14365
   647
qed
paulson@14335
   648
paulson@14365
   649
lemma inverse_set_not_rat_set:
paulson@14365
   650
   assumes A: "A \<in> preal"  shows "inverse_set A < {r. 0 < r}"
paulson@14365
   651
proof
paulson@14365
   652
  show "inverse_set A \<subseteq> {r. 0 < r}"  by (force simp add: inverse_set_def)
paulson@14365
   653
next
paulson@14365
   654
  show "inverse_set A \<noteq> {r. 0 < r}"
paulson@14365
   655
    by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
paulson@14365
   656
qed
paulson@14335
   657
paulson@14335
   658
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   659
lemma inverse_set_lemma3:
paulson@14365
   660
     "[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|] 
paulson@14365
   661
      ==> z \<in> inverse_set A"
paulson@14365
   662
apply (auto simp add: inverse_set_def)
paulson@14365
   663
apply (auto intro: order_less_trans)
paulson@14335
   664
done
paulson@14335
   665
paulson@14365
   666
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   667
lemma inverse_set_lemma4:
paulson@14365
   668
     "[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
paulson@14365
   669
apply (auto simp add: inverse_set_def)
paulson@14365
   670
apply (drule dense [of y]) 
paulson@14365
   671
apply (blast intro: order_less_trans)
paulson@14335
   672
done
paulson@14335
   673
paulson@14365
   674
paulson@14365
   675
lemma mem_inverse_set:
paulson@14365
   676
     "A \<in> preal ==> inverse_set A \<in> preal"
paulson@14365
   677
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   678
apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
paulson@14365
   679
                     inverse_set_lemma3 inverse_set_lemma4)
paulson@14335
   680
done
paulson@14335
   681
paulson@14365
   682
paulson@14335
   683
subsection{*Gleason's Lemma 9-3.4, page 122*}
paulson@14335
   684
paulson@14365
   685
lemma Gleason9_34_exists:
paulson@14365
   686
  assumes A: "A \<in> preal"
wenzelm@19765
   687
    and "\<forall>x\<in>A. x + u \<in> A"
wenzelm@19765
   688
    and "0 \<le> z"
wenzelm@19765
   689
  shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
paulson@14369
   690
proof (cases z rule: int_cases)
paulson@14369
   691
  case (nonneg n)
paulson@14365
   692
  show ?thesis
paulson@14365
   693
  proof (simp add: prems, induct n)
paulson@14365
   694
    case 0
paulson@14365
   695
      from preal_nonempty [OF A]
paulson@14365
   696
      show ?case  by force 
paulson@14365
   697
    case (Suc k)
paulson@15013
   698
      from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
paulson@14378
   699
      hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
paulson@14365
   700
      thus ?case by (force simp add: left_distrib add_ac prems) 
paulson@14365
   701
  qed
paulson@14365
   702
next
paulson@14369
   703
  case (neg n)
paulson@14369
   704
  with prems show ?thesis by simp
paulson@14365
   705
qed
paulson@14365
   706
paulson@14365
   707
lemma Gleason9_34_contra:
paulson@14365
   708
  assumes A: "A \<in> preal"
paulson@14365
   709
    shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
paulson@14365
   710
proof (induct u, induct y)
paulson@14365
   711
  fix a::int and b::int
paulson@14365
   712
  fix c::int and d::int
paulson@14365
   713
  assume bpos [simp]: "0 < b"
wenzelm@19765
   714
    and dpos [simp]: "0 < d"
wenzelm@19765
   715
    and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
wenzelm@19765
   716
    and upos: "0 < Fract c d"
wenzelm@19765
   717
    and ypos: "0 < Fract a b"
wenzelm@19765
   718
    and notin: "Fract a b \<notin> A"
paulson@14365
   719
  have cpos [simp]: "0 < c" 
paulson@14365
   720
    by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos) 
paulson@14365
   721
  have apos [simp]: "0 < a" 
paulson@14365
   722
    by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) 
paulson@14365
   723
  let ?k = "a*d"
paulson@14378
   724
  have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)" 
paulson@14365
   725
  proof -
paulson@14365
   726
    have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
paulson@14378
   727
      by (simp add: mult_rat le_rat order_less_imp_not_eq2 mult_ac) 
paulson@14365
   728
    moreover
paulson@14365
   729
    have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
paulson@14365
   730
      by (rule mult_mono, 
paulson@14365
   731
          simp_all add: int_one_le_iff_zero_less zero_less_mult_iff 
paulson@14365
   732
                        order_less_imp_le)
paulson@14365
   733
    ultimately
paulson@14365
   734
    show ?thesis by simp
paulson@14365
   735
  qed
paulson@14365
   736
  have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)  
paulson@14365
   737
  from Gleason9_34_exists [OF A closed k]
paulson@14365
   738
  obtain z where z: "z \<in> A" 
paulson@14378
   739
             and mem: "z + of_int ?k * Fract c d \<in> A" ..
paulson@14378
   740
  have less: "z + of_int ?k * Fract c d < Fract a b"
paulson@14365
   741
    by (rule not_in_preal_ub [OF A notin mem ypos])
paulson@14365
   742
  have "0<z" by (rule preal_imp_pos [OF A z])
paulson@14378
   743
  with frle and less show False by (simp add: Fract_of_int_eq) 
paulson@14365
   744
qed
paulson@14335
   745
paulson@14335
   746
paulson@14365
   747
lemma Gleason9_34:
paulson@14365
   748
  assumes A: "A \<in> preal"
wenzelm@19765
   749
    and upos: "0 < u"
wenzelm@19765
   750
  shows "\<exists>r \<in> A. r + u \<notin> A"
paulson@14365
   751
proof (rule ccontr, simp)
paulson@14365
   752
  assume closed: "\<forall>r\<in>A. r + u \<in> A"
paulson@14365
   753
  from preal_exists_bound [OF A]
paulson@14365
   754
  obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
paulson@14365
   755
  show False
paulson@14365
   756
    by (rule Gleason9_34_contra [OF A closed upos ypos y])
paulson@14365
   757
qed
paulson@14365
   758
paulson@14335
   759
paulson@14335
   760
paulson@14335
   761
subsection{*Gleason's Lemma 9-3.6*}
paulson@14335
   762
paulson@14365
   763
lemma lemma_gleason9_36:
paulson@14365
   764
  assumes A: "A \<in> preal"
wenzelm@19765
   765
    and x: "1 < x"
wenzelm@19765
   766
  shows "\<exists>r \<in> A. r*x \<notin> A"
paulson@14365
   767
proof -
paulson@14365
   768
  from preal_nonempty [OF A]
paulson@14365
   769
  obtain y where y: "y \<in> A" and  ypos: "0<y" ..
paulson@14365
   770
  show ?thesis 
paulson@14365
   771
  proof (rule classical)
paulson@14365
   772
    assume "~(\<exists>r\<in>A. r * x \<notin> A)"
paulson@14365
   773
    with y have ymem: "y * x \<in> A" by blast 
paulson@14365
   774
    from ypos mult_strict_left_mono [OF x]
paulson@14365
   775
    have yless: "y < y*x" by simp 
paulson@14365
   776
    let ?d = "y*x - y"
paulson@14365
   777
    from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
paulson@14365
   778
    from Gleason9_34 [OF A dpos]
paulson@14365
   779
    obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
paulson@14365
   780
    have rpos: "0<r" by (rule preal_imp_pos [OF A r])
paulson@14365
   781
    with dpos have rdpos: "0 < r + ?d" by arith
paulson@14365
   782
    have "~ (r + ?d \<le> y + ?d)"
paulson@14365
   783
    proof
paulson@14365
   784
      assume le: "r + ?d \<le> y + ?d" 
paulson@14365
   785
      from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
paulson@14365
   786
      have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
paulson@14365
   787
      with notin show False by simp
paulson@14365
   788
    qed
paulson@14365
   789
    hence "y < r" by simp
paulson@14365
   790
    with ypos have  dless: "?d < (r * ?d)/y"
paulson@14365
   791
      by (simp add: pos_less_divide_eq mult_commute [of ?d]
paulson@14365
   792
                    mult_strict_right_mono dpos)
paulson@14365
   793
    have "r + ?d < r*x"
paulson@14365
   794
    proof -
paulson@14365
   795
      have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
paulson@14365
   796
      also with ypos have "... = (r/y) * (y + ?d)"
paulson@14430
   797
	by (simp only: right_distrib divide_inverse mult_ac, simp)
paulson@14365
   798
      also have "... = r*x" using ypos
paulson@15234
   799
	by (simp add: times_divide_eq_left) 
paulson@14365
   800
      finally show "r + ?d < r*x" .
paulson@14365
   801
    qed
paulson@14365
   802
    with r notin rdpos
paulson@14365
   803
    show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest:  preal_downwards_closed [OF A])
paulson@14365
   804
  qed  
paulson@14365
   805
qed
paulson@14335
   806
paulson@14365
   807
subsection{*Existence of Inverse: Part 2*}
paulson@14365
   808
paulson@14365
   809
lemma mem_Rep_preal_inverse_iff:
paulson@14365
   810
      "(z \<in> Rep_preal(inverse R)) = 
paulson@14365
   811
       (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
paulson@14365
   812
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
paulson@14365
   813
apply (simp add: inverse_set_def) 
paulson@14335
   814
done
paulson@14335
   815
paulson@14365
   816
lemma Rep_preal_of_rat:
paulson@14365
   817
     "0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
paulson@14365
   818
by (simp add: preal_of_rat_def rat_mem_preal) 
paulson@14365
   819
paulson@14365
   820
lemma subset_inverse_mult_lemma:
wenzelm@19765
   821
  assumes xpos: "0 < x" and xless: "x < 1"
wenzelm@19765
   822
  shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R & 
wenzelm@19765
   823
    u \<in> Rep_preal R & x = r * u"
paulson@14365
   824
proof -
paulson@14365
   825
  from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
paulson@14365
   826
  from lemma_gleason9_36 [OF Rep_preal this]
paulson@14365
   827
  obtain r where r: "r \<in> Rep_preal R" 
paulson@14365
   828
             and notin: "r * (inverse x) \<notin> Rep_preal R" ..
paulson@14365
   829
  have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
paulson@14365
   830
  from preal_exists_greater [OF Rep_preal r]
paulson@14365
   831
  obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
paulson@14365
   832
  have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
paulson@14365
   833
  show ?thesis
paulson@14365
   834
  proof (intro exI conjI)
paulson@14365
   835
    show "0 < x/u" using xpos upos
paulson@14365
   836
      by (simp add: zero_less_divide_iff)  
paulson@14365
   837
    show "x/u < x/r" using xpos upos rpos
paulson@14430
   838
      by (simp add: divide_inverse mult_less_cancel_left rless) 
paulson@14365
   839
    show "inverse (x / r) \<notin> Rep_preal R" using notin
paulson@14430
   840
      by (simp add: divide_inverse mult_commute) 
paulson@14365
   841
    show "u \<in> Rep_preal R" by (rule u) 
paulson@14365
   842
    show "x = x / u * u" using upos 
paulson@14430
   843
      by (simp add: divide_inverse mult_commute) 
paulson@14365
   844
  qed
paulson@14365
   845
qed
paulson@14365
   846
paulson@14365
   847
lemma subset_inverse_mult: 
paulson@14365
   848
     "Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
paulson@14365
   849
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff 
paulson@14365
   850
                      mem_Rep_preal_mult_iff)
paulson@14365
   851
apply (blast dest: subset_inverse_mult_lemma) 
paulson@14335
   852
done
paulson@14335
   853
paulson@14365
   854
lemma inverse_mult_subset_lemma:
wenzelm@19765
   855
  assumes rpos: "0 < r" 
wenzelm@19765
   856
    and rless: "r < y"
wenzelm@19765
   857
    and notin: "inverse y \<notin> Rep_preal R"
wenzelm@19765
   858
    and q: "q \<in> Rep_preal R"
wenzelm@19765
   859
  shows "r*q < 1"
paulson@14365
   860
proof -
paulson@14365
   861
  have "q < inverse y" using rpos rless
paulson@14365
   862
    by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
paulson@14365
   863
  hence "r * q < r/y" using rpos
paulson@14430
   864
    by (simp add: divide_inverse mult_less_cancel_left)
paulson@14365
   865
  also have "... \<le> 1" using rpos rless
paulson@14365
   866
    by (simp add: pos_divide_le_eq)
paulson@14365
   867
  finally show ?thesis .
paulson@14365
   868
qed
paulson@14365
   869
paulson@14365
   870
lemma inverse_mult_subset:
paulson@14365
   871
     "Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)"
paulson@14365
   872
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff 
paulson@14365
   873
                      mem_Rep_preal_mult_iff)
paulson@14365
   874
apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal]) 
paulson@14365
   875
apply (blast intro: inverse_mult_subset_lemma) 
paulson@14365
   876
done
paulson@14365
   877
paulson@15413
   878
lemma preal_mult_inverse: "inverse R * R = (preal_of_rat 1)"
paulson@15413
   879
apply (rule Rep_preal_inject [THEN iffD1])
paulson@14365
   880
apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult]) 
paulson@14365
   881
done
paulson@14365
   882
paulson@15413
   883
lemma preal_mult_inverse_right: "R * inverse R = (preal_of_rat 1)"
paulson@14365
   884
apply (rule preal_mult_commute [THEN subst])
paulson@14365
   885
apply (rule preal_mult_inverse)
paulson@14335
   886
done
paulson@14335
   887
paulson@14335
   888
paulson@14365
   889
text{*Theorems needing @{text Gleason9_34}*}
paulson@14335
   890
paulson@14365
   891
lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
paulson@14365
   892
proof 
paulson@14365
   893
  fix r
paulson@14365
   894
  assume r: "r \<in> Rep_preal R"
paulson@14365
   895
  have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
paulson@14365
   896
  from mem_Rep_preal_Ex 
paulson@14365
   897
  obtain y where y: "y \<in> Rep_preal S" ..
paulson@14365
   898
  have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
paulson@14365
   899
  have ry: "r+y \<in> Rep_preal(R + S)" using r y
paulson@14365
   900
    by (auto simp add: mem_Rep_preal_add_iff)
paulson@14365
   901
  show "r \<in> Rep_preal(R + S)" using r ypos rpos 
paulson@14365
   902
    by (simp add:  preal_downwards_closed [OF Rep_preal ry]) 
paulson@14365
   903
qed
paulson@14335
   904
paulson@14365
   905
lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
paulson@14365
   906
proof -
paulson@14365
   907
  from mem_Rep_preal_Ex 
paulson@14365
   908
  obtain y where y: "y \<in> Rep_preal S" ..
paulson@14365
   909
  have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
paulson@14365
   910
  from  Gleason9_34 [OF Rep_preal ypos]
paulson@14365
   911
  obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
paulson@14365
   912
  have "r + y \<in> Rep_preal (R + S)" using r y
paulson@14365
   913
    by (auto simp add: mem_Rep_preal_add_iff)
paulson@14365
   914
  thus ?thesis using notin by blast
paulson@14365
   915
qed
paulson@14335
   916
paulson@14365
   917
lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
paulson@14365
   918
by (insert Rep_preal_sum_not_subset, blast)
paulson@14335
   919
paulson@14335
   920
text{*at last, Gleason prop. 9-3.5(iii) page 123*}
paulson@14365
   921
lemma preal_self_less_add_left: "(R::preal) < R + S"
paulson@14335
   922
apply (unfold preal_less_def psubset_def)
paulson@14335
   923
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
paulson@14335
   924
done
paulson@14335
   925
paulson@14365
   926
lemma preal_self_less_add_right: "(R::preal) < S + R"
paulson@14365
   927
by (simp add: preal_add_commute preal_self_less_add_left)
paulson@14365
   928
paulson@14365
   929
lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
paulson@14365
   930
by (insert preal_self_less_add_left [of x y], auto)
paulson@14335
   931
paulson@14335
   932
paulson@14365
   933
subsection{*Subtraction for Positive Reals*}
paulson@14335
   934
paulson@14365
   935
text{*Gleason prop. 9-3.5(iv), page 123: proving @{term "A < B ==> \<exists>D. A + D =
paulson@14365
   936
B"}. We define the claimed @{term D} and show that it is a positive real*}
paulson@14335
   937
paulson@14335
   938
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   939
lemma diff_set_not_empty:
paulson@14365
   940
     "R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   941
apply (auto simp add: preal_less_def diff_set_def elim!: equalityE) 
paulson@14365
   942
apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
paulson@14365
   943
apply (drule preal_imp_pos [OF Rep_preal], clarify)
paulson@14365
   944
apply (cut_tac a=x and b=u in add_eq_exists, force) 
paulson@14335
   945
done
paulson@14335
   946
paulson@14335
   947
text{*Part 2 of Dedekind sections definition*}
paulson@14365
   948
lemma diff_set_nonempty:
paulson@14365
   949
     "\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   950
apply (cut_tac X = S in Rep_preal_exists_bound)
paulson@14335
   951
apply (erule exE)
paulson@14335
   952
apply (rule_tac x = x in exI, auto)
paulson@14365
   953
apply (simp add: diff_set_def) 
paulson@14365
   954
apply (auto dest: Rep_preal [THEN preal_downwards_closed])
paulson@14335
   955
done
paulson@14335
   956
paulson@14365
   957
lemma diff_set_not_rat_set:
wenzelm@19765
   958
  "diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
paulson@14365
   959
proof
paulson@14365
   960
  show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def) 
paulson@14365
   961
  show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
paulson@14365
   962
qed
paulson@14335
   963
paulson@14335
   964
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   965
lemma diff_set_lemma3:
paulson@14365
   966
     "[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|] 
paulson@14365
   967
      ==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   968
apply (auto simp add: diff_set_def) 
paulson@14365
   969
apply (rule_tac x=x in exI) 
paulson@14365
   970
apply (drule Rep_preal [THEN preal_downwards_closed], auto)
paulson@14335
   971
done
paulson@14335
   972
paulson@14365
   973
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   974
lemma diff_set_lemma4:
paulson@14365
   975
     "[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|] 
paulson@14365
   976
      ==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
paulson@14365
   977
apply (auto simp add: diff_set_def) 
paulson@14365
   978
apply (drule Rep_preal [THEN preal_exists_greater], clarify) 
paulson@14365
   979
apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)  
paulson@14365
   980
apply (rule_tac x="y+xa" in exI) 
paulson@14365
   981
apply (auto simp add: add_ac)
paulson@14335
   982
done
paulson@14335
   983
paulson@14365
   984
lemma mem_diff_set:
paulson@14365
   985
     "R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
paulson@14365
   986
apply (unfold preal_def cut_def)
paulson@14365
   987
apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
paulson@14365
   988
                     diff_set_lemma3 diff_set_lemma4)
paulson@14365
   989
done
paulson@14365
   990
paulson@14365
   991
lemma mem_Rep_preal_diff_iff:
paulson@14365
   992
      "R < S ==>
paulson@14365
   993
       (z \<in> Rep_preal(S-R)) = 
paulson@14365
   994
       (\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
paulson@14365
   995
apply (simp add: preal_diff_def mem_diff_set Rep_preal)
paulson@14365
   996
apply (force simp add: diff_set_def) 
paulson@14335
   997
done
paulson@14335
   998
paulson@14365
   999
paulson@14365
  1000
text{*proving that @{term "R + D \<le> S"}*}
paulson@14365
  1001
paulson@14365
  1002
lemma less_add_left_lemma:
paulson@14365
  1003
  assumes Rless: "R < S"
wenzelm@19765
  1004
    and a: "a \<in> Rep_preal R"
wenzelm@19765
  1005
    and cb: "c + b \<in> Rep_preal S"
wenzelm@19765
  1006
    and "c \<notin> Rep_preal R"
wenzelm@19765
  1007
    and "0 < b"
wenzelm@19765
  1008
    and "0 < c"
paulson@14365
  1009
  shows "a + b \<in> Rep_preal S"
paulson@14365
  1010
proof -
paulson@14365
  1011
  have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
paulson@14365
  1012
  moreover
paulson@14365
  1013
  have "a < c" using prems
paulson@14365
  1014
    by (blast intro: not_in_Rep_preal_ub ) 
paulson@14365
  1015
  ultimately show ?thesis using prems
paulson@14365
  1016
    by (simp add: preal_downwards_closed [OF Rep_preal cb]) 
paulson@14365
  1017
qed
paulson@14365
  1018
paulson@14365
  1019
lemma less_add_left_le1:
paulson@14365
  1020
       "R < (S::preal) ==> R + (S-R) \<le> S"
paulson@14365
  1021
apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff 
paulson@14365
  1022
                      mem_Rep_preal_diff_iff)
paulson@14365
  1023
apply (blast intro: less_add_left_lemma) 
paulson@14335
  1024
done
paulson@14335
  1025
paulson@14365
  1026
subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
paulson@14335
  1027
paulson@14335
  1028
lemma lemma_sum_mem_Rep_preal_ex:
paulson@14365
  1029
     "x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
paulson@14365
  1030
apply (drule Rep_preal [THEN preal_exists_greater], clarify) 
paulson@14365
  1031
apply (cut_tac a=x and b=u in add_eq_exists, auto) 
paulson@14335
  1032
done
paulson@14335
  1033
paulson@14365
  1034
lemma less_add_left_lemma2:
paulson@14365
  1035
  assumes Rless: "R < S"
wenzelm@19765
  1036
    and x:     "x \<in> Rep_preal S"
wenzelm@19765
  1037
    and xnot: "x \<notin>  Rep_preal R"
paulson@14365
  1038
  shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R & 
paulson@14365
  1039
                     z + v \<in> Rep_preal S & x = u + v"
paulson@14365
  1040
proof -
paulson@14365
  1041
  have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
paulson@14365
  1042
  from lemma_sum_mem_Rep_preal_ex [OF x]
paulson@14365
  1043
  obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
paulson@14365
  1044
  from  Gleason9_34 [OF Rep_preal epos]
paulson@14365
  1045
  obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
paulson@14365
  1046
  with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
paulson@14365
  1047
  from add_eq_exists [of r x]
paulson@14365
  1048
  obtain y where eq: "x = r+y" by auto
paulson@14365
  1049
  show ?thesis 
paulson@14365
  1050
  proof (intro exI conjI)
paulson@14365
  1051
    show "r \<in> Rep_preal R" by (rule r)
paulson@14365
  1052
    show "r + e \<notin> Rep_preal R" by (rule notin)
paulson@14365
  1053
    show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
paulson@14365
  1054
    show "x = r + y" by (simp add: eq)
paulson@14365
  1055
    show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
paulson@14365
  1056
      by simp
paulson@14365
  1057
    show "0 < y" using rless eq by arith
paulson@14365
  1058
  qed
paulson@14365
  1059
qed
paulson@14365
  1060
paulson@14365
  1061
lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
paulson@14365
  1062
apply (auto simp add: preal_le_def)
paulson@14365
  1063
apply (case_tac "x \<in> Rep_preal R")
paulson@14365
  1064
apply (cut_tac Rep_preal_self_subset [of R], force)
paulson@14365
  1065
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
paulson@14365
  1066
apply (blast dest: less_add_left_lemma2)
paulson@14335
  1067
done
paulson@14335
  1068
paulson@14365
  1069
lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
paulson@14365
  1070
by (blast intro: preal_le_anti_sym [OF less_add_left_le1 less_add_left_le2])
paulson@14335
  1071
paulson@14365
  1072
lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
paulson@14365
  1073
by (fast dest: less_add_left)
paulson@14335
  1074
paulson@14365
  1075
lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
paulson@14365
  1076
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
paulson@14335
  1077
apply (rule_tac y1 = D in preal_add_commute [THEN subst])
paulson@14335
  1078
apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
paulson@14335
  1079
done
paulson@14335
  1080
paulson@14365
  1081
lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
paulson@14365
  1082
by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
paulson@14335
  1083
paulson@14365
  1084
lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
paulson@14365
  1085
apply (insert linorder_less_linear [of R S], auto)
paulson@14365
  1086
apply (drule_tac R = S and T = T in preal_add_less2_mono1)
paulson@14365
  1087
apply (blast dest: order_less_trans) 
paulson@14335
  1088
done
paulson@14335
  1089
paulson@14365
  1090
lemma preal_add_left_less_cancel: "T + R < T + S ==> R <  (S::preal)"
paulson@14365
  1091
by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
paulson@14335
  1092
paulson@14365
  1093
lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)"
paulson@14335
  1094
by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
paulson@14335
  1095
paulson@14365
  1096
lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
paulson@14335
  1097
by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
paulson@14335
  1098
paulson@14365
  1099
lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
paulson@14365
  1100
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right) 
paulson@14365
  1101
paulson@14365
  1102
lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
paulson@14365
  1103
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left) 
paulson@14365
  1104
paulson@14335
  1105
lemma preal_add_less_mono:
paulson@14335
  1106
     "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
paulson@14365
  1107
apply (auto dest!: less_add_left_Ex simp add: preal_add_ac)
paulson@14335
  1108
apply (rule preal_add_assoc [THEN subst])
paulson@14335
  1109
apply (rule preal_self_less_add_right)
paulson@14335
  1110
done
paulson@14335
  1111
paulson@14365
  1112
lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
paulson@14365
  1113
apply (insert linorder_less_linear [of R S], safe)
paulson@14365
  1114
apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
paulson@14335
  1115
done
paulson@14335
  1116
paulson@14365
  1117
lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
paulson@14335
  1118
by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
paulson@14335
  1119
paulson@14365
  1120
lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
paulson@14335
  1121
by (fast intro: preal_add_left_cancel)
paulson@14335
  1122
paulson@14365
  1123
lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
paulson@14335
  1124
by (fast intro: preal_add_right_cancel)
paulson@14335
  1125
paulson@14365
  1126
lemmas preal_cancels =
paulson@14365
  1127
    preal_add_less_cancel_right preal_add_less_cancel_left
paulson@14365
  1128
    preal_add_le_cancel_right preal_add_le_cancel_left
paulson@14365
  1129
    preal_add_left_cancel_iff preal_add_right_cancel_iff
paulson@14335
  1130
paulson@14335
  1131
paulson@14335
  1132
subsection{*Completeness of type @{typ preal}*}
paulson@14335
  1133
paulson@14335
  1134
text{*Prove that supremum is a cut*}
paulson@14335
  1135
paulson@14365
  1136
text{*Part 1 of Dedekind sections definition*}
paulson@14365
  1137
paulson@14365
  1138
lemma preal_sup_set_not_empty:
paulson@14365
  1139
     "P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1140
apply auto
paulson@14365
  1141
apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
paulson@14335
  1142
done
paulson@14335
  1143
paulson@14335
  1144
paulson@14335
  1145
text{*Part 2 of Dedekind sections definition*}
paulson@14365
  1146
paulson@14365
  1147
lemma preal_sup_not_exists:
paulson@14365
  1148
     "\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1149
apply (cut_tac X = Y in Rep_preal_exists_bound)
paulson@14365
  1150
apply (auto simp add: preal_le_def)
paulson@14335
  1151
done
paulson@14335
  1152
paulson@14365
  1153
lemma preal_sup_set_not_rat_set:
paulson@14365
  1154
     "\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
paulson@14365
  1155
apply (drule preal_sup_not_exists)
paulson@14365
  1156
apply (blast intro: preal_imp_pos [OF Rep_preal])  
paulson@14335
  1157
done
paulson@14335
  1158
paulson@14335
  1159
text{*Part 3 of Dedekind sections definition*}
paulson@14335
  1160
lemma preal_sup_set_lemma3:
paulson@14365
  1161
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
paulson@14365
  1162
      ==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1163
by (auto elim: Rep_preal [THEN preal_downwards_closed])
paulson@14335
  1164
paulson@14365
  1165
text{*Part 4 of Dedekind sections definition*}
paulson@14335
  1166
lemma preal_sup_set_lemma4:
paulson@14365
  1167
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
paulson@14365
  1168
          ==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
paulson@14365
  1169
by (blast dest: Rep_preal [THEN preal_exists_greater])
paulson@14335
  1170
paulson@14335
  1171
lemma preal_sup:
paulson@14365
  1172
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
paulson@14365
  1173
apply (unfold preal_def cut_def)
paulson@14365
  1174
apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
paulson@14365
  1175
                     preal_sup_set_lemma3 preal_sup_set_lemma4)
paulson@14335
  1176
done
paulson@14335
  1177
paulson@14365
  1178
lemma preal_psup_le:
paulson@14365
  1179
     "[| \<forall>X \<in> P. X \<le> Y;  x \<in> P |] ==> x \<le> psup P"
paulson@14365
  1180
apply (simp (no_asm_simp) add: preal_le_def) 
paulson@14365
  1181
apply (subgoal_tac "P \<noteq> {}") 
paulson@14365
  1182
apply (auto simp add: psup_def preal_sup) 
paulson@14335
  1183
done
paulson@14335
  1184
paulson@14365
  1185
lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
paulson@14365
  1186
apply (simp (no_asm_simp) add: preal_le_def)
paulson@14365
  1187
apply (simp add: psup_def preal_sup) 
paulson@14335
  1188
apply (auto simp add: preal_le_def)
paulson@14335
  1189
done
paulson@14335
  1190
paulson@14335
  1191
text{*Supremum property*}
paulson@14335
  1192
lemma preal_complete:
paulson@14365
  1193
     "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
paulson@14365
  1194
apply (simp add: preal_less_def psup_def preal_sup)
paulson@14365
  1195
apply (auto simp add: preal_le_def)
paulson@14365
  1196
apply (rename_tac U) 
paulson@14365
  1197
apply (cut_tac x = U and y = Z in linorder_less_linear)
paulson@14365
  1198
apply (auto simp add: preal_less_def)
paulson@14335
  1199
done
paulson@14335
  1200
paulson@14335
  1201
huffman@20495
  1202
subsection{*The Embedding from @{typ rat} into @{typ preal}*}
paulson@14335
  1203
paulson@14365
  1204
lemma preal_of_rat_add_lemma1:
paulson@14365
  1205
     "[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
paulson@14365
  1206
apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
paulson@14365
  1207
apply (simp add: zero_less_mult_iff) 
paulson@14365
  1208
apply (simp add: mult_ac)
paulson@14335
  1209
done
paulson@14335
  1210
paulson@14365
  1211
lemma preal_of_rat_add_lemma2:
paulson@14365
  1212
  assumes "u < x + y"
wenzelm@19765
  1213
    and "0 < x"
wenzelm@19765
  1214
    and "0 < y"
wenzelm@19765
  1215
    and "0 < u"
paulson@14365
  1216
  shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
paulson@14365
  1217
proof (intro exI conjI)
paulson@14365
  1218
  show "u * x * inverse(x+y) < x" using prems 
paulson@14365
  1219
    by (simp add: preal_of_rat_add_lemma1) 
paulson@14365
  1220
  show "u * y * inverse(x+y) < y" using prems 
paulson@14365
  1221
    by (simp add: preal_of_rat_add_lemma1 add_commute [of x]) 
paulson@14365
  1222
  show "0 < u * x * inverse (x + y)" using prems
paulson@14365
  1223
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1224
  show "0 < u * y * inverse (x + y)" using prems
paulson@14365
  1225
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1226
  show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
paulson@14365
  1227
    by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
paulson@14365
  1228
qed
paulson@14365
  1229
paulson@14365
  1230
lemma preal_of_rat_add:
paulson@14365
  1231
     "[| 0 < x; 0 < y|] 
paulson@14365
  1232
      ==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
paulson@14365
  1233
apply (unfold preal_of_rat_def preal_add_def)
paulson@14365
  1234
apply (simp add: rat_mem_preal) 
paulson@14335
  1235
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
  1236
apply (auto simp add: add_set_def) 
paulson@14365
  1237
apply (blast dest: preal_of_rat_add_lemma2) 
paulson@14365
  1238
done
paulson@14365
  1239
paulson@14365
  1240
lemma preal_of_rat_mult_lemma1:
paulson@14365
  1241
     "[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
paulson@14365
  1242
apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
paulson@14365
  1243
apply (simp add: zero_less_mult_iff)
paulson@14365
  1244
apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
paulson@14365
  1245
apply (simp_all add: mult_ac)
paulson@14335
  1246
done
paulson@14335
  1247
paulson@14365
  1248
lemma preal_of_rat_mult_lemma2: 
paulson@14365
  1249
  assumes xless: "x < y * z"
wenzelm@19765
  1250
    and xpos: "0 < x"
wenzelm@19765
  1251
    and ypos: "0 < y"
paulson@14365
  1252
  shows "x * z * inverse y * inverse z < (z::rat)"
paulson@14365
  1253
proof -
paulson@14365
  1254
  have "0 < y * z" using prems by simp
paulson@14365
  1255
  hence zpos:  "0 < z" using prems by (simp add: zero_less_mult_iff)
paulson@14365
  1256
  have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
paulson@14365
  1257
    by (simp add: mult_ac)
paulson@14365
  1258
  also have "... = x/y" using zpos
paulson@14430
  1259
    by (simp add: divide_inverse)
paulson@14365
  1260
  also have "... < z"
paulson@14365
  1261
    by (simp add: pos_divide_less_eq [OF ypos] mult_commute) 
paulson@14365
  1262
  finally show ?thesis .
paulson@14365
  1263
qed
paulson@14335
  1264
paulson@14365
  1265
lemma preal_of_rat_mult_lemma3:
paulson@14365
  1266
  assumes uless: "u < x * y"
wenzelm@19765
  1267
    and "0 < x"
wenzelm@19765
  1268
    and "0 < y"
wenzelm@19765
  1269
    and "0 < u"
paulson@14365
  1270
  shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
paulson@14365
  1271
proof -
paulson@14365
  1272
  from dense [OF uless] 
paulson@14365
  1273
  obtain r where "u < r" "r < x * y" by blast
paulson@14365
  1274
  thus ?thesis
paulson@14365
  1275
  proof (intro exI conjI)
paulson@14365
  1276
  show "u * x * inverse r < x" using prems 
paulson@14365
  1277
    by (simp add: preal_of_rat_mult_lemma1) 
paulson@14365
  1278
  show "r * y * inverse x * inverse y < y" using prems
paulson@14365
  1279
    by (simp add: preal_of_rat_mult_lemma2)
paulson@14365
  1280
  show "0 < u * x * inverse r" using prems
paulson@14365
  1281
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1282
  show "0 < r * y * inverse x * inverse y" using prems
paulson@14365
  1283
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1284
  have "u * x * inverse r * (r * y * inverse x * inverse y) =
paulson@14365
  1285
        u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
paulson@14365
  1286
    by (simp only: mult_ac)
paulson@14365
  1287
  thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
paulson@14365
  1288
    by simp
paulson@14365
  1289
  qed
paulson@14365
  1290
qed
paulson@14365
  1291
paulson@14365
  1292
lemma preal_of_rat_mult:
paulson@14365
  1293
     "[| 0 < x; 0 < y|] 
paulson@14365
  1294
      ==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
paulson@14365
  1295
apply (unfold preal_of_rat_def preal_mult_def)
paulson@14365
  1296
apply (simp add: rat_mem_preal) 
paulson@14365
  1297
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
  1298
apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def) 
paulson@14365
  1299
apply (blast dest: preal_of_rat_mult_lemma3) 
paulson@14335
  1300
done
paulson@14335
  1301
paulson@14365
  1302
lemma preal_of_rat_less_iff:
paulson@14365
  1303
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
paulson@14365
  1304
by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal) 
paulson@14335
  1305
paulson@14365
  1306
lemma preal_of_rat_le_iff:
paulson@14365
  1307
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
paulson@14365
  1308
by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric]) 
paulson@14365
  1309
paulson@14365
  1310
lemma preal_of_rat_eq_iff:
paulson@14365
  1311
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
paulson@14365
  1312
by (simp add: preal_of_rat_le_iff order_eq_iff) 
paulson@14335
  1313
paulson@5078
  1314
end