src/HOL/Real/PReal.thy
author paulson
Tue Jan 27 15:39:51 2004 +0100 (2004-01-27)
changeset 14365 3d4df8c166ae
parent 14335 9c0b5e081037
child 14369 c50188fe6366
permissions -rw-r--r--
replacing HOL/Real/PRat, PNat by the rational number development
of Markus Wenzel
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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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theory PReal = RatArith:
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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_semiring)"
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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 (blast dest: dense intro: order_less_trans)
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constdefs
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  cut :: "rat set => bool"
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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}"
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proof -
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  let ?A = "{r::rat. 0 < r & r < q}"
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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 ..
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instance preal :: plus ..
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instance preal :: minus ..
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instance preal :: times ..
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instance preal :: inverse ..
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constdefs
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  preal_of_rat :: "rat => preal"
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     "preal_of_rat q == Abs_preal({x::rat. 0 < x & x < q})"
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  psup       :: "preal set => preal"
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    "psup(P)   == Abs_preal(\<Union>X \<in> P. Rep_preal(X))"
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  add_set :: "[rat set,rat set] => rat set"
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    "add_set A B == {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
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  diff_set :: "[rat set,rat set] => rat set"
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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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  mult_set :: "[rat set,rat set] => rat set"
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    "mult_set A B == {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
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  inverse_set :: "rat set => rat set"
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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::preal) == Rep_preal R < Rep_preal S"
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  preal_le_def:
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    "R \<le> (S::preal) == 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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lemma inj_on_Abs_preal: "inj_on Abs_preal preal"
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apply (rule inj_on_inverseI)
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apply (erule Abs_preal_inverse)
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done
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declare inj_on_Abs_preal [THEN inj_on_iff, simp]
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lemma inj_Rep_preal: "inj(Rep_preal)"
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apply (rule inj_on_inverseI)
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apply (rule Rep_preal_inverse)
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done
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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_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 mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
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apply (insert Rep_preal [of X])
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apply (unfold preal_def cut_def, blast)
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done
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declare Abs_preal_inverse [simp]
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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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lemma Rep_preal_exists_bound: "\<exists>x. 0 < x & x \<notin> Rep_preal X"
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apply (cut_tac x = X in Rep_preal)
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apply (drule preal_imp_psubset_positives)
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apply (auto simp add: psubset_def)
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done
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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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apply (auto simp add: preal_def cut_def intro: order_less_trans)
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apply (force simp only: eq_commute [of "{}"] interval_empty_iff)
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apply (blast dest: dense intro: order_less_trans)
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done
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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{*Theorems for Ordering*}
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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 by assumption
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qed
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lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
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subsection{*The @{text "\<le>"} 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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proof qed
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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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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 (insert preal_nonempty [of A] preal_nonempty [of B]) 
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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 & 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 not_in_preal_ub, assumption+)+
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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
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    show "\<exists>y' \<in> B. z = x*?f + y'"
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    proof
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      show "z = x*?f + y*?f"
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	by (simp add: left_distrib [symmetric] divide_inverse_zero 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 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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    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 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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   321
qed
paulson@14365
   322
paulson@14365
   323
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   324
lemma add_set_lemma4:
paulson@14365
   325
     "[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
paulson@14365
   326
apply (auto simp add: add_set_def)
paulson@14365
   327
apply (frule preal_exists_greater [of A], auto) 
paulson@14365
   328
apply (rule_tac x="u + y" in exI)
paulson@14365
   329
apply (auto intro: add_strict_left_mono)
paulson@14335
   330
done
paulson@14335
   331
paulson@14365
   332
lemma mem_add_set:
paulson@14365
   333
     "[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
paulson@14365
   334
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   335
apply (blast intro!: add_set_not_empty add_set_not_rat_set
paulson@14365
   336
                     add_set_lemma3 add_set_lemma4)
paulson@14335
   337
done
paulson@14335
   338
paulson@14335
   339
lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
paulson@14365
   340
apply (simp add: preal_add_def mem_add_set Rep_preal)
paulson@14365
   341
apply (force simp add: add_set_def add_ac)
paulson@14335
   342
done
paulson@14335
   343
paulson@14335
   344
lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
paulson@14335
   345
  apply (rule mk_left_commute [of "op +"])
paulson@14335
   346
  apply (rule preal_add_assoc)
paulson@14335
   347
  apply (rule preal_add_commute)
paulson@14335
   348
  done
paulson@14335
   349
paulson@14365
   350
text{* Positive Real addition is an AC operator *}
paulson@14335
   351
lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
paulson@14335
   352
paulson@14335
   353
paulson@14335
   354
subsection{*Properties of Multiplication*}
paulson@14335
   355
paulson@14335
   356
text{*Proofs essentially same as for addition*}
paulson@14335
   357
paulson@14335
   358
lemma preal_mult_commute: "(x::preal) * y = y * x"
paulson@14365
   359
apply (unfold preal_mult_def mult_set_def)
paulson@14335
   360
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
   361
apply (force simp add: mult_commute)
paulson@14335
   362
done
paulson@14335
   363
paulson@14335
   364
text{*Multiplication of two positive reals gives a positive real.}
paulson@14335
   365
paulson@14335
   366
text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
paulson@14335
   367
paulson@14335
   368
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   369
lemma mult_set_not_empty:
paulson@14365
   370
     "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
paulson@14365
   371
apply (insert preal_nonempty [of A] preal_nonempty [of B]) 
paulson@14365
   372
apply (auto simp add: mult_set_def)
paulson@14335
   373
done
paulson@14335
   374
paulson@14335
   375
text{*Part 2 of Dedekind sections definition*}
paulson@14335
   376
lemma preal_not_mem_mult_set_Ex:
paulson@14365
   377
   assumes A: "A \<in> preal" 
paulson@14365
   378
       and B: "B \<in> preal"
paulson@14365
   379
     shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
paulson@14365
   380
proof -
paulson@14365
   381
  from preal_exists_bound [OF A]
paulson@14365
   382
  obtain x where [simp]: "0 < x" "x \<notin> A" by blast
paulson@14365
   383
  from preal_exists_bound [OF B]
paulson@14365
   384
  obtain y where [simp]: "0 < y" "y \<notin> B" by blast
paulson@14365
   385
  show ?thesis
paulson@14365
   386
  proof (intro exI conjI)
paulson@14365
   387
    show "0 < x*y" by (simp add: mult_pos)
paulson@14365
   388
    show "x * y \<notin> mult_set A B"
paulson@14365
   389
    proof (auto simp add: mult_set_def)
paulson@14365
   390
      fix u::rat and v::rat
paulson@14365
   391
      assume "u \<in> A" and "v \<in> B" and "x*y = u*v"
paulson@14365
   392
      moreover
paulson@14365
   393
      with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
paulson@14365
   394
      moreover
paulson@14365
   395
      with prems have "0\<le>v"
paulson@14365
   396
        by (blast intro: preal_imp_pos [OF B]  order_less_imp_le prems)
paulson@14365
   397
      moreover
paulson@14365
   398
      hence "u*v < x*y" by (blast intro: mult_strict_mono prems)
paulson@14365
   399
      ultimately show False by force
paulson@14365
   400
    qed
paulson@14365
   401
  qed
paulson@14365
   402
qed
paulson@14335
   403
paulson@14365
   404
lemma mult_set_not_rat_set:
paulson@14365
   405
   assumes A: "A \<in> preal" 
paulson@14365
   406
       and B: "B \<in> preal"
paulson@14365
   407
     shows "mult_set A B < {r. 0 < r}"
paulson@14365
   408
proof
paulson@14365
   409
  show "mult_set A B \<subseteq> {r. 0 < r}"
paulson@14365
   410
    by (force simp add: mult_set_def
paulson@14365
   411
              intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos)
paulson@14365
   412
next
paulson@14365
   413
  show "mult_set A B \<noteq> {r. 0 < r}"
paulson@14365
   414
    by (insert preal_not_mem_mult_set_Ex [OF A B], blast)
paulson@14365
   415
qed
paulson@14365
   416
paulson@14365
   417
paulson@14335
   418
paulson@14335
   419
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   420
lemma mult_set_lemma3:
paulson@14365
   421
     "[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|] 
paulson@14365
   422
      ==> z \<in> mult_set A B"
paulson@14365
   423
proof (unfold mult_set_def, clarify)
paulson@14365
   424
  fix x::rat and y::rat
paulson@14365
   425
  assume A: "A \<in> preal" 
paulson@14365
   426
     and B: "B \<in> preal"
paulson@14365
   427
     and [simp]: "0 < z"
paulson@14365
   428
     and zless: "z < x * y"
paulson@14365
   429
     and x:  "x \<in> A"
paulson@14365
   430
     and y:  "y \<in> B"
paulson@14365
   431
  have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
paulson@14365
   432
  show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
paulson@14365
   433
  proof
paulson@14365
   434
    show "\<exists>y'\<in>B. z = (z/y) * y'"
paulson@14365
   435
    proof
paulson@14365
   436
      show "z = (z/y)*y"
paulson@14365
   437
	by (simp add: divide_inverse_zero mult_commute [of y] mult_assoc
paulson@14365
   438
		      order_less_imp_not_eq2)
paulson@14365
   439
      show "y \<in> B" .
paulson@14365
   440
    qed
paulson@14365
   441
  next
paulson@14365
   442
    show "z/y \<in> A"
paulson@14365
   443
    proof (rule preal_downwards_closed [OF A x])
paulson@14365
   444
      show "0 < z/y"
paulson@14365
   445
	by (simp add: zero_less_divide_iff)
paulson@14365
   446
      show "z/y < x" by (simp add: pos_divide_less_eq zless)
paulson@14365
   447
    qed
paulson@14365
   448
  qed
paulson@14365
   449
qed
paulson@14365
   450
paulson@14365
   451
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   452
lemma mult_set_lemma4:
paulson@14365
   453
     "[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
paulson@14365
   454
apply (auto simp add: mult_set_def)
paulson@14365
   455
apply (frule preal_exists_greater [of A], auto) 
paulson@14365
   456
apply (rule_tac x="u * y" in exI)
paulson@14365
   457
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B] 
paulson@14365
   458
                   mult_strict_right_mono)
paulson@14335
   459
done
paulson@14335
   460
paulson@14335
   461
paulson@14365
   462
lemma mem_mult_set:
paulson@14365
   463
     "[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
paulson@14365
   464
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   465
apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
paulson@14365
   466
                     mult_set_lemma3 mult_set_lemma4)
paulson@14335
   467
done
paulson@14335
   468
paulson@14335
   469
lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
paulson@14365
   470
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
paulson@14365
   471
apply (force simp add: mult_set_def mult_ac)
paulson@14335
   472
done
paulson@14335
   473
paulson@14335
   474
lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
paulson@14335
   475
  apply (rule mk_left_commute [of "op *"])
paulson@14335
   476
  apply (rule preal_mult_assoc)
paulson@14335
   477
  apply (rule preal_mult_commute)
paulson@14335
   478
  done
paulson@14335
   479
paulson@14365
   480
paulson@14365
   481
text{* Positive Real multiplication is an AC operator *}
paulson@14335
   482
lemmas preal_mult_ac =
paulson@14335
   483
       preal_mult_assoc preal_mult_commute preal_mult_left_commute
paulson@14335
   484
paulson@14365
   485
paulson@14365
   486
text{* Positive real 1 is the multiplicative identity element *}
paulson@14365
   487
paulson@14365
   488
lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
paulson@14365
   489
by (simp add: preal_def cut_of_rat)
paulson@14335
   490
paulson@14365
   491
lemma preal_mult_1: "(preal_of_rat 1) * z = z"
paulson@14365
   492
proof (induct z)
paulson@14365
   493
  fix A :: "rat set"
paulson@14365
   494
  assume A: "A \<in> preal"
paulson@14365
   495
  have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
paulson@14365
   496
  proof
paulson@14365
   497
    show "?lhs \<subseteq> A"
paulson@14365
   498
    proof clarify
paulson@14365
   499
      fix x::rat and u::rat and v::rat
paulson@14365
   500
      assume upos: "0<u" and "u<1" and v: "v \<in> A"
paulson@14365
   501
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
paulson@14365
   502
      hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems)
paulson@14365
   503
      thus "u * v \<in> A"
paulson@14365
   504
        by (force intro: preal_downwards_closed [OF A v] mult_pos upos vpos)
paulson@14365
   505
    qed
paulson@14365
   506
  next
paulson@14365
   507
    show "A \<subseteq> ?lhs"
paulson@14365
   508
    proof clarify
paulson@14365
   509
      fix x::rat
paulson@14365
   510
      assume x: "x \<in> A"
paulson@14365
   511
      have xpos: "0<x" by (rule preal_imp_pos [OF A x])
paulson@14365
   512
      from preal_exists_greater [OF A x]
paulson@14365
   513
      obtain v where v: "v \<in> A" and xlessv: "x < v" ..
paulson@14365
   514
      have vpos: "0<v" by (rule preal_imp_pos [OF A v])
paulson@14365
   515
      show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
paulson@14365
   516
      proof (intro exI conjI)
paulson@14365
   517
        show "0 < x/v"
paulson@14365
   518
          by (simp add: zero_less_divide_iff xpos vpos)
paulson@14365
   519
	show "x / v < 1"
paulson@14365
   520
          by (simp add: pos_divide_less_eq vpos xlessv)
paulson@14365
   521
        show "\<exists>v'\<in>A. x = (x / v) * v'"
paulson@14365
   522
        proof
paulson@14365
   523
          show "x = (x/v)*v"
paulson@14365
   524
	    by (simp add: divide_inverse_zero mult_assoc vpos
paulson@14365
   525
                          order_less_imp_not_eq2)
paulson@14365
   526
          show "v \<in> A" .
paulson@14365
   527
        qed
paulson@14365
   528
      qed
paulson@14365
   529
    qed
paulson@14365
   530
  qed
paulson@14365
   531
  thus "preal_of_rat 1 * Abs_preal A = Abs_preal A"
paulson@14365
   532
    by (simp add: preal_of_rat_def preal_mult_def mult_set_def 
paulson@14365
   533
                  rat_mem_preal A)
paulson@14365
   534
qed
paulson@14365
   535
paulson@14365
   536
paulson@14365
   537
lemma preal_mult_1_right: "z * (preal_of_rat 1) = z"
paulson@14335
   538
apply (rule preal_mult_commute [THEN subst])
paulson@14335
   539
apply (rule preal_mult_1)
paulson@14335
   540
done
paulson@14335
   541
paulson@14335
   542
paulson@14335
   543
subsection{*Distribution of Multiplication across Addition*}
paulson@14335
   544
paulson@14335
   545
lemma mem_Rep_preal_add_iff:
paulson@14365
   546
      "(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"
paulson@14365
   547
apply (simp add: preal_add_def mem_add_set Rep_preal)
paulson@14365
   548
apply (simp add: add_set_def) 
paulson@14335
   549
done
paulson@14335
   550
paulson@14335
   551
lemma mem_Rep_preal_mult_iff:
paulson@14365
   552
      "(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
paulson@14365
   553
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
paulson@14365
   554
apply (simp add: mult_set_def) 
paulson@14365
   555
done
paulson@14335
   556
paulson@14365
   557
lemma distrib_subset1:
paulson@14365
   558
     "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
paulson@14365
   559
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
paulson@14365
   560
apply (force simp add: right_distrib)
paulson@14335
   561
done
paulson@14335
   562
paulson@14365
   563
lemma linorder_le_cases [case_names le ge]:
paulson@14365
   564
    "((x::'a::linorder) <= y ==> P) ==> (y <= x ==> P) ==> P"
paulson@14365
   565
  apply (insert linorder_linear, blast)
paulson@14365
   566
  done
paulson@14335
   567
paulson@14365
   568
lemma preal_add_mult_distrib_mean:
paulson@14365
   569
  assumes a: "a \<in> Rep_preal w"
paulson@14365
   570
      and b: "b \<in> Rep_preal w"
paulson@14365
   571
      and d: "d \<in> Rep_preal x"
paulson@14365
   572
      and e: "e \<in> Rep_preal y"
paulson@14365
   573
     shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
paulson@14365
   574
proof
paulson@14365
   575
  let ?c = "(a*d + b*e)/(d+e)"
paulson@14365
   576
  have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
paulson@14365
   577
    by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
paulson@14365
   578
  have cpos: "0 < ?c"
paulson@14365
   579
    by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
paulson@14365
   580
  show "a * d + b * e = ?c * (d + e)"
paulson@14365
   581
    by (simp add: divide_inverse_zero mult_assoc order_less_imp_not_eq2)
paulson@14365
   582
  show "?c \<in> Rep_preal w"
paulson@14365
   583
    proof (cases rule: linorder_le_cases)
paulson@14365
   584
      assume "a \<le> b"
paulson@14365
   585
      hence "?c \<le> b"
paulson@14365
   586
	by (simp add: pos_divide_le_eq right_distrib mult_right_mono
paulson@14365
   587
                      order_less_imp_le)
paulson@14365
   588
      thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
paulson@14365
   589
    next
paulson@14365
   590
      assume "b \<le> a"
paulson@14365
   591
      hence "?c \<le> a"
paulson@14365
   592
	by (simp add: pos_divide_le_eq right_distrib mult_right_mono
paulson@14365
   593
                      order_less_imp_le)
paulson@14365
   594
      thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
paulson@14365
   595
    qed
paulson@14365
   596
  qed
paulson@14365
   597
paulson@14365
   598
lemma distrib_subset2:
paulson@14365
   599
     "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
paulson@14365
   600
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
paulson@14365
   601
apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
paulson@14335
   602
done
paulson@14335
   603
paulson@14365
   604
lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
paulson@14365
   605
apply (rule inj_Rep_preal [THEN injD])
paulson@14365
   606
apply (rule equalityI [OF distrib_subset1 distrib_subset2])
paulson@14335
   607
done
paulson@14335
   608
paulson@14365
   609
lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
paulson@14365
   610
by (simp add: preal_mult_commute preal_add_mult_distrib2)
paulson@14365
   611
paulson@14335
   612
paulson@14335
   613
subsection{*Existence of Inverse, a Positive Real*}
paulson@14335
   614
paulson@14365
   615
lemma mem_inv_set_ex:
paulson@14365
   616
  assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
paulson@14365
   617
proof -
paulson@14365
   618
  from preal_exists_bound [OF A]
paulson@14365
   619
  obtain x where [simp]: "0<x" "x \<notin> A" by blast
paulson@14365
   620
  show ?thesis
paulson@14365
   621
  proof (intro exI conjI)
paulson@14365
   622
    show "0 < inverse (x+1)"
paulson@14365
   623
      by (simp add: order_less_trans [OF _ less_add_one]) 
paulson@14365
   624
    show "inverse(x+1) < inverse x"
paulson@14365
   625
      by (simp add: less_imp_inverse_less less_add_one)
paulson@14365
   626
    show "inverse (inverse x) \<notin> A"
paulson@14365
   627
      by (simp add: order_less_imp_not_eq2)
paulson@14365
   628
  qed
paulson@14365
   629
qed
paulson@14335
   630
paulson@14335
   631
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   632
lemma inverse_set_not_empty:
paulson@14365
   633
     "A \<in> preal ==> {} \<subset> inverse_set A"
paulson@14365
   634
apply (insert mem_inv_set_ex [of A])
paulson@14365
   635
apply (auto simp add: inverse_set_def)
paulson@14335
   636
done
paulson@14335
   637
paulson@14335
   638
text{*Part 2 of Dedekind sections definition*}
paulson@14335
   639
paulson@14365
   640
lemma preal_not_mem_inverse_set_Ex:
paulson@14365
   641
   assumes A: "A \<in> preal"  shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
paulson@14365
   642
proof -
paulson@14365
   643
  from preal_nonempty [OF A]
paulson@14365
   644
  obtain x where x: "x \<in> A" and  xpos [simp]: "0<x" ..
paulson@14365
   645
  show ?thesis
paulson@14365
   646
  proof (intro exI conjI)
paulson@14365
   647
    show "0 < inverse x" by simp
paulson@14365
   648
    show "inverse x \<notin> inverse_set A"
paulson@14365
   649
    proof (auto simp add: inverse_set_def)
paulson@14365
   650
      fix y::rat 
paulson@14365
   651
      assume ygt: "inverse x < y"
paulson@14365
   652
      have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
paulson@14365
   653
      have iyless: "inverse y < x" 
paulson@14365
   654
        by (simp add: inverse_less_imp_less [of x] ygt)
paulson@14365
   655
      show "inverse y \<in> A"
paulson@14365
   656
        by (simp add: preal_downwards_closed [OF A x] iyless) 
paulson@14365
   657
    qed
paulson@14365
   658
  qed
paulson@14365
   659
qed
paulson@14335
   660
paulson@14365
   661
lemma inverse_set_not_rat_set:
paulson@14365
   662
   assumes A: "A \<in> preal"  shows "inverse_set A < {r. 0 < r}"
paulson@14365
   663
proof
paulson@14365
   664
  show "inverse_set A \<subseteq> {r. 0 < r}"  by (force simp add: inverse_set_def)
paulson@14365
   665
next
paulson@14365
   666
  show "inverse_set A \<noteq> {r. 0 < r}"
paulson@14365
   667
    by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
paulson@14365
   668
qed
paulson@14335
   669
paulson@14335
   670
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   671
lemma inverse_set_lemma3:
paulson@14365
   672
     "[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|] 
paulson@14365
   673
      ==> z \<in> inverse_set A"
paulson@14365
   674
apply (auto simp add: inverse_set_def)
paulson@14365
   675
apply (auto intro: order_less_trans)
paulson@14335
   676
done
paulson@14335
   677
paulson@14365
   678
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   679
lemma inverse_set_lemma4:
paulson@14365
   680
     "[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
paulson@14365
   681
apply (auto simp add: inverse_set_def)
paulson@14365
   682
apply (drule dense [of y]) 
paulson@14365
   683
apply (blast intro: order_less_trans)
paulson@14335
   684
done
paulson@14335
   685
paulson@14365
   686
paulson@14365
   687
lemma mem_inverse_set:
paulson@14365
   688
     "A \<in> preal ==> inverse_set A \<in> preal"
paulson@14365
   689
apply (simp (no_asm_simp) add: preal_def cut_def)
paulson@14365
   690
apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
paulson@14365
   691
                     inverse_set_lemma3 inverse_set_lemma4)
paulson@14335
   692
done
paulson@14335
   693
paulson@14365
   694
paulson@14335
   695
subsection{*Gleason's Lemma 9-3.4, page 122*}
paulson@14335
   696
paulson@14365
   697
(*????Why can't I get case_names like nonneg to work?*)
paulson@14365
   698
lemma Gleason9_34_exists:
paulson@14365
   699
  assumes A: "A \<in> preal"
paulson@14365
   700
      and closed: "\<forall>x\<in>A. x + u \<in> A"
paulson@14365
   701
      and nonneg: "0 \<le> z"
paulson@14365
   702
     shows "\<exists>b\<in>A. b + (rat z) * u \<in> A"
paulson@14365
   703
proof (cases z)
paulson@14365
   704
  case (1 n)
paulson@14365
   705
  show ?thesis
paulson@14365
   706
  proof (simp add: prems, induct n)
paulson@14365
   707
    case 0
paulson@14365
   708
      from preal_nonempty [OF A]
paulson@14365
   709
      show ?case  by force 
paulson@14365
   710
    case (Suc k)
paulson@14365
   711
      from this obtain b where "b \<in> A" "b + rat (int k) * u \<in> A" ..
paulson@14365
   712
      hence "b + rat (int k)*u + u \<in> A" by (simp add: closed)
paulson@14365
   713
      thus ?case by (force simp add: left_distrib add_ac prems) 
paulson@14365
   714
  qed
paulson@14365
   715
next
paulson@14365
   716
  case (2 n)
paulson@14365
   717
  with nonneg show ?thesis by simp
paulson@14365
   718
qed
paulson@14365
   719
paulson@14335
   720
paulson@14365
   721
lemma Gleason9_34_contra:
paulson@14365
   722
  assumes A: "A \<in> preal"
paulson@14365
   723
    shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
paulson@14365
   724
proof (induct u, induct y)
paulson@14365
   725
  fix a::int and b::int
paulson@14365
   726
  fix c::int and d::int
paulson@14365
   727
  assume bpos [simp]: "0 < b"
paulson@14365
   728
     and dpos [simp]: "0 < d"
paulson@14365
   729
     and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
paulson@14365
   730
     and upos: "0 < Fract c d"
paulson@14365
   731
     and ypos: "0 < Fract a b"
paulson@14365
   732
     and notin: "Fract a b \<notin> A"
paulson@14365
   733
  have cpos [simp]: "0 < c" 
paulson@14365
   734
    by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos) 
paulson@14365
   735
  have apos [simp]: "0 < a" 
paulson@14365
   736
    by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) 
paulson@14365
   737
  let ?k = "a*d"
paulson@14365
   738
  have frle: "Fract a b \<le> rat ?k * (Fract c d)" 
paulson@14365
   739
  proof -
paulson@14365
   740
    have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
paulson@14365
   741
      by (simp add: rat_def mult_rat le_rat order_less_imp_not_eq2 mult_ac) 
paulson@14365
   742
    moreover
paulson@14365
   743
    have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
paulson@14365
   744
      by (rule mult_mono, 
paulson@14365
   745
          simp_all add: int_one_le_iff_zero_less zero_less_mult_iff 
paulson@14365
   746
                        order_less_imp_le)
paulson@14365
   747
    ultimately
paulson@14365
   748
    show ?thesis by simp
paulson@14365
   749
  qed
paulson@14365
   750
  have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)  
paulson@14365
   751
  from Gleason9_34_exists [OF A closed k]
paulson@14365
   752
  obtain z where z: "z \<in> A" 
paulson@14365
   753
             and mem: "z + rat ?k * Fract c d \<in> A" ..
paulson@14365
   754
  have less: "z + rat ?k * Fract c d < Fract a b"
paulson@14365
   755
    by (rule not_in_preal_ub [OF A notin mem ypos])
paulson@14365
   756
  have "0<z" by (rule preal_imp_pos [OF A z])
paulson@14365
   757
  with frle and less show False by arith 
paulson@14365
   758
qed
paulson@14335
   759
paulson@14335
   760
paulson@14365
   761
lemma Gleason9_34:
paulson@14365
   762
  assumes A: "A \<in> preal"
paulson@14365
   763
      and upos: "0 < u"
paulson@14365
   764
    shows "\<exists>r \<in> A. r + u \<notin> A"
paulson@14365
   765
proof (rule ccontr, simp)
paulson@14365
   766
  assume closed: "\<forall>r\<in>A. r + u \<in> A"
paulson@14365
   767
  from preal_exists_bound [OF A]
paulson@14365
   768
  obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
paulson@14365
   769
  show False
paulson@14365
   770
    by (rule Gleason9_34_contra [OF A closed upos ypos y])
paulson@14365
   771
qed
paulson@14365
   772
paulson@14335
   773
paulson@14335
   774
paulson@14335
   775
subsection{*Gleason's Lemma 9-3.6*}
paulson@14335
   776
paulson@14365
   777
lemma lemma_gleason9_36:
paulson@14365
   778
  assumes A: "A \<in> preal"
paulson@14365
   779
      and x: "1 < x"
paulson@14365
   780
    shows "\<exists>r \<in> A. r*x \<notin> A"
paulson@14365
   781
proof -
paulson@14365
   782
  from preal_nonempty [OF A]
paulson@14365
   783
  obtain y where y: "y \<in> A" and  ypos: "0<y" ..
paulson@14365
   784
  show ?thesis 
paulson@14365
   785
  proof (rule classical)
paulson@14365
   786
    assume "~(\<exists>r\<in>A. r * x \<notin> A)"
paulson@14365
   787
    with y have ymem: "y * x \<in> A" by blast 
paulson@14365
   788
    from ypos mult_strict_left_mono [OF x]
paulson@14365
   789
    have yless: "y < y*x" by simp 
paulson@14365
   790
    let ?d = "y*x - y"
paulson@14365
   791
    from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
paulson@14365
   792
    from Gleason9_34 [OF A dpos]
paulson@14365
   793
    obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
paulson@14365
   794
    have rpos: "0<r" by (rule preal_imp_pos [OF A r])
paulson@14365
   795
    with dpos have rdpos: "0 < r + ?d" by arith
paulson@14365
   796
    have "~ (r + ?d \<le> y + ?d)"
paulson@14365
   797
    proof
paulson@14365
   798
      assume le: "r + ?d \<le> y + ?d" 
paulson@14365
   799
      from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
paulson@14365
   800
      have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
paulson@14365
   801
      with notin show False by simp
paulson@14365
   802
    qed
paulson@14365
   803
    hence "y < r" by simp
paulson@14365
   804
    with ypos have  dless: "?d < (r * ?d)/y"
paulson@14365
   805
      by (simp add: pos_less_divide_eq mult_commute [of ?d]
paulson@14365
   806
                    mult_strict_right_mono dpos)
paulson@14365
   807
    have "r + ?d < r*x"
paulson@14365
   808
    proof -
paulson@14365
   809
      have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
paulson@14365
   810
      also with ypos have "... = (r/y) * (y + ?d)"
paulson@14365
   811
	by (simp only: right_distrib divide_inverse_zero mult_ac, simp)
paulson@14365
   812
      also have "... = r*x" using ypos
paulson@14365
   813
	by simp
paulson@14365
   814
      finally show "r + ?d < r*x" .
paulson@14365
   815
    qed
paulson@14365
   816
    with r notin rdpos
paulson@14365
   817
    show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest:  preal_downwards_closed [OF A])
paulson@14365
   818
  qed  
paulson@14365
   819
qed
paulson@14335
   820
paulson@14365
   821
subsection{*Existence of Inverse: Part 2*}
paulson@14365
   822
paulson@14365
   823
lemma mem_Rep_preal_inverse_iff:
paulson@14365
   824
      "(z \<in> Rep_preal(inverse R)) = 
paulson@14365
   825
       (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
paulson@14365
   826
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
paulson@14365
   827
apply (simp add: inverse_set_def) 
paulson@14335
   828
done
paulson@14335
   829
paulson@14365
   830
lemma Rep_preal_of_rat:
paulson@14365
   831
     "0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
paulson@14365
   832
by (simp add: preal_of_rat_def rat_mem_preal) 
paulson@14365
   833
paulson@14365
   834
lemma subset_inverse_mult_lemma:
paulson@14365
   835
      assumes xpos: "0 < x" and xless: "x < 1"
paulson@14365
   836
         shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R & 
paulson@14365
   837
                        u \<in> Rep_preal R & x = r * u"
paulson@14365
   838
proof -
paulson@14365
   839
  from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
paulson@14365
   840
  from lemma_gleason9_36 [OF Rep_preal this]
paulson@14365
   841
  obtain r where r: "r \<in> Rep_preal R" 
paulson@14365
   842
             and notin: "r * (inverse x) \<notin> Rep_preal R" ..
paulson@14365
   843
  have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
paulson@14365
   844
  from preal_exists_greater [OF Rep_preal r]
paulson@14365
   845
  obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
paulson@14365
   846
  have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
paulson@14365
   847
  show ?thesis
paulson@14365
   848
  proof (intro exI conjI)
paulson@14365
   849
    show "0 < x/u" using xpos upos
paulson@14365
   850
      by (simp add: zero_less_divide_iff)  
paulson@14365
   851
    show "x/u < x/r" using xpos upos rpos
paulson@14365
   852
      by (simp add: divide_inverse_zero mult_less_cancel_left rless) 
paulson@14365
   853
    show "inverse (x / r) \<notin> Rep_preal R" using notin
paulson@14365
   854
      by (simp add: divide_inverse_zero mult_commute) 
paulson@14365
   855
    show "u \<in> Rep_preal R" by (rule u) 
paulson@14365
   856
    show "x = x / u * u" using upos 
paulson@14365
   857
      by (simp add: divide_inverse_zero mult_commute) 
paulson@14365
   858
  qed
paulson@14365
   859
qed
paulson@14365
   860
paulson@14365
   861
lemma subset_inverse_mult: 
paulson@14365
   862
     "Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
paulson@14365
   863
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff 
paulson@14365
   864
                      mem_Rep_preal_mult_iff)
paulson@14365
   865
apply (blast dest: subset_inverse_mult_lemma) 
paulson@14335
   866
done
paulson@14335
   867
paulson@14365
   868
lemma inverse_mult_subset_lemma:
paulson@14365
   869
     assumes rpos: "0 < r" 
paulson@14365
   870
         and rless: "r < y"
paulson@14365
   871
         and notin: "inverse y \<notin> Rep_preal R"
paulson@14365
   872
         and q: "q \<in> Rep_preal R"
paulson@14365
   873
     shows "r*q < 1"
paulson@14365
   874
proof -
paulson@14365
   875
  have "q < inverse y" using rpos rless
paulson@14365
   876
    by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
paulson@14365
   877
  hence "r * q < r/y" using rpos
paulson@14365
   878
    by (simp add: divide_inverse_zero mult_less_cancel_left)
paulson@14365
   879
  also have "... \<le> 1" using rpos rless
paulson@14365
   880
    by (simp add: pos_divide_le_eq)
paulson@14365
   881
  finally show ?thesis .
paulson@14365
   882
qed
paulson@14365
   883
paulson@14365
   884
lemma inverse_mult_subset:
paulson@14365
   885
     "Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)"
paulson@14365
   886
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff 
paulson@14365
   887
                      mem_Rep_preal_mult_iff)
paulson@14365
   888
apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal]) 
paulson@14365
   889
apply (blast intro: inverse_mult_subset_lemma) 
paulson@14365
   890
done
paulson@14365
   891
paulson@14365
   892
lemma preal_mult_inverse:
paulson@14365
   893
     "inverse R * R = (preal_of_rat 1)"
paulson@14365
   894
apply (rule inj_Rep_preal [THEN injD])
paulson@14365
   895
apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult]) 
paulson@14365
   896
done
paulson@14365
   897
paulson@14365
   898
lemma preal_mult_inverse_right:
paulson@14365
   899
     "R * inverse R = (preal_of_rat 1)"
paulson@14365
   900
apply (rule preal_mult_commute [THEN subst])
paulson@14365
   901
apply (rule preal_mult_inverse)
paulson@14335
   902
done
paulson@14335
   903
paulson@14335
   904
paulson@14365
   905
text{*Theorems needing @{text Gleason9_34}*}
paulson@14335
   906
paulson@14365
   907
lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
paulson@14365
   908
proof 
paulson@14365
   909
  fix r
paulson@14365
   910
  assume r: "r \<in> Rep_preal R"
paulson@14365
   911
  have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
paulson@14365
   912
  from mem_Rep_preal_Ex 
paulson@14365
   913
  obtain y where y: "y \<in> Rep_preal S" ..
paulson@14365
   914
  have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
paulson@14365
   915
  have ry: "r+y \<in> Rep_preal(R + S)" using r y
paulson@14365
   916
    by (auto simp add: mem_Rep_preal_add_iff)
paulson@14365
   917
  show "r \<in> Rep_preal(R + S)" using r ypos rpos 
paulson@14365
   918
    by (simp add:  preal_downwards_closed [OF Rep_preal ry]) 
paulson@14365
   919
qed
paulson@14335
   920
paulson@14365
   921
lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
paulson@14365
   922
proof -
paulson@14365
   923
  from mem_Rep_preal_Ex 
paulson@14365
   924
  obtain y where y: "y \<in> Rep_preal S" ..
paulson@14365
   925
  have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
paulson@14365
   926
  from  Gleason9_34 [OF Rep_preal ypos]
paulson@14365
   927
  obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
paulson@14365
   928
  have "r + y \<in> Rep_preal (R + S)" using r y
paulson@14365
   929
    by (auto simp add: mem_Rep_preal_add_iff)
paulson@14365
   930
  thus ?thesis using notin by blast
paulson@14365
   931
qed
paulson@14335
   932
paulson@14365
   933
lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
paulson@14365
   934
by (insert Rep_preal_sum_not_subset, blast)
paulson@14335
   935
paulson@14335
   936
text{*at last, Gleason prop. 9-3.5(iii) page 123*}
paulson@14365
   937
lemma preal_self_less_add_left: "(R::preal) < R + S"
paulson@14335
   938
apply (unfold preal_less_def psubset_def)
paulson@14335
   939
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
paulson@14335
   940
done
paulson@14335
   941
paulson@14365
   942
lemma preal_self_less_add_right: "(R::preal) < S + R"
paulson@14365
   943
by (simp add: preal_add_commute preal_self_less_add_left)
paulson@14365
   944
paulson@14365
   945
lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
paulson@14365
   946
by (insert preal_self_less_add_left [of x y], auto)
paulson@14335
   947
paulson@14335
   948
paulson@14365
   949
subsection{*Subtraction for Positive Reals*}
paulson@14335
   950
paulson@14365
   951
text{*Gleason prop. 9-3.5(iv), page 123: proving @{term "A < B ==> \<exists>D. A + D =
paulson@14365
   952
B"}. We define the claimed @{term D} and show that it is a positive real*}
paulson@14335
   953
paulson@14335
   954
text{*Part 1 of Dedekind sections definition*}
paulson@14365
   955
lemma diff_set_not_empty:
paulson@14365
   956
     "R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   957
apply (auto simp add: preal_less_def diff_set_def elim!: equalityE) 
paulson@14365
   958
apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
paulson@14365
   959
apply (drule preal_imp_pos [OF Rep_preal], clarify)
paulson@14365
   960
apply (cut_tac a=x and b=u in add_eq_exists, force) 
paulson@14335
   961
done
paulson@14335
   962
paulson@14335
   963
text{*Part 2 of Dedekind sections definition*}
paulson@14365
   964
lemma diff_set_nonempty:
paulson@14365
   965
     "\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   966
apply (cut_tac X = S in Rep_preal_exists_bound)
paulson@14335
   967
apply (erule exE)
paulson@14335
   968
apply (rule_tac x = x in exI, auto)
paulson@14365
   969
apply (simp add: diff_set_def) 
paulson@14365
   970
apply (auto dest: Rep_preal [THEN preal_downwards_closed])
paulson@14335
   971
done
paulson@14335
   972
paulson@14365
   973
lemma diff_set_not_rat_set:
paulson@14365
   974
     "diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
paulson@14365
   975
proof
paulson@14365
   976
  show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def) 
paulson@14365
   977
  show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
paulson@14365
   978
qed
paulson@14335
   979
paulson@14335
   980
text{*Part 3 of Dedekind sections definition*}
paulson@14365
   981
lemma diff_set_lemma3:
paulson@14365
   982
     "[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|] 
paulson@14365
   983
      ==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
paulson@14365
   984
apply (auto simp add: diff_set_def) 
paulson@14365
   985
apply (rule_tac x=x in exI) 
paulson@14365
   986
apply (drule Rep_preal [THEN preal_downwards_closed], auto)
paulson@14335
   987
done
paulson@14335
   988
paulson@14365
   989
text{*Part 4 of Dedekind sections definition*}
paulson@14365
   990
lemma diff_set_lemma4:
paulson@14365
   991
     "[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|] 
paulson@14365
   992
      ==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
paulson@14365
   993
apply (auto simp add: diff_set_def) 
paulson@14365
   994
apply (drule Rep_preal [THEN preal_exists_greater], clarify) 
paulson@14365
   995
apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)  
paulson@14365
   996
apply (rule_tac x="y+xa" in exI) 
paulson@14365
   997
apply (auto simp add: add_ac)
paulson@14335
   998
done
paulson@14335
   999
paulson@14365
  1000
lemma mem_diff_set:
paulson@14365
  1001
     "R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
paulson@14365
  1002
apply (unfold preal_def cut_def)
paulson@14365
  1003
apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
paulson@14365
  1004
                     diff_set_lemma3 diff_set_lemma4)
paulson@14365
  1005
done
paulson@14365
  1006
paulson@14365
  1007
lemma mem_Rep_preal_diff_iff:
paulson@14365
  1008
      "R < S ==>
paulson@14365
  1009
       (z \<in> Rep_preal(S-R)) = 
paulson@14365
  1010
       (\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
paulson@14365
  1011
apply (simp add: preal_diff_def mem_diff_set Rep_preal)
paulson@14365
  1012
apply (force simp add: diff_set_def) 
paulson@14335
  1013
done
paulson@14335
  1014
paulson@14365
  1015
paulson@14365
  1016
text{*proving that @{term "R + D \<le> S"}*}
paulson@14365
  1017
paulson@14365
  1018
lemma less_add_left_lemma:
paulson@14365
  1019
  assumes Rless: "R < S"
paulson@14365
  1020
      and a: "a \<in> Rep_preal R"
paulson@14365
  1021
      and cb: "c + b \<in> Rep_preal S"
paulson@14365
  1022
      and "c \<notin> Rep_preal R"
paulson@14365
  1023
      and "0 < b"
paulson@14365
  1024
      and "0 < c"
paulson@14365
  1025
  shows "a + b \<in> Rep_preal S"
paulson@14365
  1026
proof -
paulson@14365
  1027
  have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
paulson@14365
  1028
  moreover
paulson@14365
  1029
  have "a < c" using prems
paulson@14365
  1030
    by (blast intro: not_in_Rep_preal_ub ) 
paulson@14365
  1031
  ultimately show ?thesis using prems
paulson@14365
  1032
    by (simp add: preal_downwards_closed [OF Rep_preal cb]) 
paulson@14365
  1033
qed
paulson@14365
  1034
paulson@14365
  1035
lemma less_add_left_le1:
paulson@14365
  1036
       "R < (S::preal) ==> R + (S-R) \<le> S"
paulson@14365
  1037
apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff 
paulson@14365
  1038
                      mem_Rep_preal_diff_iff)
paulson@14365
  1039
apply (blast intro: less_add_left_lemma) 
paulson@14335
  1040
done
paulson@14335
  1041
paulson@14365
  1042
subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
paulson@14335
  1043
paulson@14335
  1044
lemma lemma_sum_mem_Rep_preal_ex:
paulson@14365
  1045
     "x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
paulson@14365
  1046
apply (drule Rep_preal [THEN preal_exists_greater], clarify) 
paulson@14365
  1047
apply (cut_tac a=x and b=u in add_eq_exists, auto) 
paulson@14335
  1048
done
paulson@14335
  1049
paulson@14365
  1050
lemma less_add_left_lemma2:
paulson@14365
  1051
  assumes Rless: "R < S"
paulson@14365
  1052
      and x:     "x \<in> Rep_preal S"
paulson@14365
  1053
      and xnot: "x \<notin>  Rep_preal R"
paulson@14365
  1054
  shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R & 
paulson@14365
  1055
                     z + v \<in> Rep_preal S & x = u + v"
paulson@14365
  1056
proof -
paulson@14365
  1057
  have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
paulson@14365
  1058
  from lemma_sum_mem_Rep_preal_ex [OF x]
paulson@14365
  1059
  obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
paulson@14365
  1060
  from  Gleason9_34 [OF Rep_preal epos]
paulson@14365
  1061
  obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
paulson@14365
  1062
  with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
paulson@14365
  1063
  from add_eq_exists [of r x]
paulson@14365
  1064
  obtain y where eq: "x = r+y" by auto
paulson@14365
  1065
  show ?thesis 
paulson@14365
  1066
  proof (intro exI conjI)
paulson@14365
  1067
    show "r \<in> Rep_preal R" by (rule r)
paulson@14365
  1068
    show "r + e \<notin> Rep_preal R" by (rule notin)
paulson@14365
  1069
    show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
paulson@14365
  1070
    show "x = r + y" by (simp add: eq)
paulson@14365
  1071
    show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
paulson@14365
  1072
      by simp
paulson@14365
  1073
    show "0 < y" using rless eq by arith
paulson@14365
  1074
  qed
paulson@14365
  1075
qed
paulson@14365
  1076
paulson@14365
  1077
lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
paulson@14365
  1078
apply (auto simp add: preal_le_def)
paulson@14365
  1079
apply (case_tac "x \<in> Rep_preal R")
paulson@14365
  1080
apply (cut_tac Rep_preal_self_subset [of R], force)
paulson@14365
  1081
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
paulson@14365
  1082
apply (blast dest: less_add_left_lemma2)
paulson@14335
  1083
done
paulson@14335
  1084
paulson@14365
  1085
lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
paulson@14365
  1086
by (blast intro: preal_le_anti_sym [OF less_add_left_le1 less_add_left_le2])
paulson@14335
  1087
paulson@14365
  1088
lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
paulson@14365
  1089
by (fast dest: less_add_left)
paulson@14335
  1090
paulson@14365
  1091
lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
paulson@14365
  1092
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
paulson@14335
  1093
apply (rule_tac y1 = D in preal_add_commute [THEN subst])
paulson@14335
  1094
apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
paulson@14335
  1095
done
paulson@14335
  1096
paulson@14365
  1097
lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
paulson@14365
  1098
by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
paulson@14335
  1099
paulson@14365
  1100
lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
paulson@14365
  1101
apply (insert linorder_less_linear [of R S], auto)
paulson@14365
  1102
apply (drule_tac R = S and T = T in preal_add_less2_mono1)
paulson@14365
  1103
apply (blast dest: order_less_trans) 
paulson@14335
  1104
done
paulson@14335
  1105
paulson@14365
  1106
lemma preal_add_left_less_cancel: "T + R < T + S ==> R <  (S::preal)"
paulson@14365
  1107
by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
paulson@14335
  1108
paulson@14365
  1109
lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)"
paulson@14335
  1110
by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
paulson@14335
  1111
paulson@14365
  1112
lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
paulson@14335
  1113
by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
paulson@14335
  1114
paulson@14365
  1115
lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
paulson@14365
  1116
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right) 
paulson@14365
  1117
paulson@14365
  1118
lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
paulson@14365
  1119
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left) 
paulson@14365
  1120
paulson@14335
  1121
lemma preal_add_less_mono:
paulson@14335
  1122
     "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
paulson@14365
  1123
apply (auto dest!: less_add_left_Ex simp add: preal_add_ac)
paulson@14335
  1124
apply (rule preal_add_assoc [THEN subst])
paulson@14335
  1125
apply (rule preal_self_less_add_right)
paulson@14335
  1126
done
paulson@14335
  1127
paulson@14365
  1128
lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
paulson@14365
  1129
apply (insert linorder_less_linear [of R S], safe)
paulson@14365
  1130
apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
paulson@14335
  1131
done
paulson@14335
  1132
paulson@14365
  1133
lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
paulson@14335
  1134
by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
paulson@14335
  1135
paulson@14365
  1136
lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
paulson@14335
  1137
by (fast intro: preal_add_left_cancel)
paulson@14335
  1138
paulson@14365
  1139
lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
paulson@14335
  1140
by (fast intro: preal_add_right_cancel)
paulson@14335
  1141
paulson@14365
  1142
lemmas preal_cancels =
paulson@14365
  1143
    preal_add_less_cancel_right preal_add_less_cancel_left
paulson@14365
  1144
    preal_add_le_cancel_right preal_add_le_cancel_left
paulson@14365
  1145
    preal_add_left_cancel_iff preal_add_right_cancel_iff
paulson@14335
  1146
paulson@14335
  1147
paulson@14335
  1148
subsection{*Completeness of type @{typ preal}*}
paulson@14335
  1149
paulson@14335
  1150
text{*Prove that supremum is a cut*}
paulson@14335
  1151
paulson@14365
  1152
text{*Part 1 of Dedekind sections definition*}
paulson@14365
  1153
paulson@14365
  1154
lemma preal_sup_set_not_empty:
paulson@14365
  1155
     "P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1156
apply auto
paulson@14365
  1157
apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
paulson@14335
  1158
done
paulson@14335
  1159
paulson@14335
  1160
paulson@14335
  1161
text{*Part 2 of Dedekind sections definition*}
paulson@14365
  1162
paulson@14365
  1163
lemma preal_sup_not_exists:
paulson@14365
  1164
     "\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1165
apply (cut_tac X = Y in Rep_preal_exists_bound)
paulson@14365
  1166
apply (auto simp add: preal_le_def)
paulson@14335
  1167
done
paulson@14335
  1168
paulson@14365
  1169
lemma preal_sup_set_not_rat_set:
paulson@14365
  1170
     "\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
paulson@14365
  1171
apply (drule preal_sup_not_exists)
paulson@14365
  1172
apply (blast intro: preal_imp_pos [OF Rep_preal])  
paulson@14335
  1173
done
paulson@14335
  1174
paulson@14335
  1175
text{*Part 3 of Dedekind sections definition*}
paulson@14335
  1176
lemma preal_sup_set_lemma3:
paulson@14365
  1177
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
paulson@14365
  1178
      ==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
paulson@14365
  1179
by (auto elim: Rep_preal [THEN preal_downwards_closed])
paulson@14335
  1180
paulson@14365
  1181
text{*Part 4 of Dedekind sections definition*}
paulson@14335
  1182
lemma preal_sup_set_lemma4:
paulson@14365
  1183
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
paulson@14365
  1184
          ==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
paulson@14365
  1185
by (blast dest: Rep_preal [THEN preal_exists_greater])
paulson@14335
  1186
paulson@14335
  1187
lemma preal_sup:
paulson@14365
  1188
     "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
paulson@14365
  1189
apply (unfold preal_def cut_def)
paulson@14365
  1190
apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
paulson@14365
  1191
                     preal_sup_set_lemma3 preal_sup_set_lemma4)
paulson@14335
  1192
done
paulson@14335
  1193
paulson@14365
  1194
lemma preal_psup_le:
paulson@14365
  1195
     "[| \<forall>X \<in> P. X \<le> Y;  x \<in> P |] ==> x \<le> psup P"
paulson@14365
  1196
apply (simp (no_asm_simp) add: preal_le_def) 
paulson@14365
  1197
apply (subgoal_tac "P \<noteq> {}") 
paulson@14365
  1198
apply (auto simp add: psup_def preal_sup) 
paulson@14335
  1199
done
paulson@14335
  1200
paulson@14365
  1201
lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
paulson@14365
  1202
apply (simp (no_asm_simp) add: preal_le_def)
paulson@14365
  1203
apply (simp add: psup_def preal_sup) 
paulson@14335
  1204
apply (auto simp add: preal_le_def)
paulson@14335
  1205
done
paulson@14335
  1206
paulson@14335
  1207
text{*Supremum property*}
paulson@14335
  1208
lemma preal_complete:
paulson@14365
  1209
     "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
paulson@14365
  1210
apply (simp add: preal_less_def psup_def preal_sup)
paulson@14365
  1211
apply (auto simp add: preal_le_def)
paulson@14365
  1212
apply (rename_tac U) 
paulson@14365
  1213
apply (cut_tac x = U and y = Z in linorder_less_linear)
paulson@14365
  1214
apply (auto simp add: preal_less_def)
paulson@14335
  1215
done
paulson@14335
  1216
paulson@14335
  1217
paulson@14365
  1218
subsection{*The Embadding from @{typ rat} into @{typ preal}*}
paulson@14335
  1219
paulson@14365
  1220
lemma preal_of_rat_add_lemma1:
paulson@14365
  1221
     "[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
paulson@14365
  1222
apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
paulson@14365
  1223
apply (simp add: zero_less_mult_iff) 
paulson@14365
  1224
apply (simp add: mult_ac)
paulson@14335
  1225
done
paulson@14335
  1226
paulson@14365
  1227
lemma preal_of_rat_add_lemma2:
paulson@14365
  1228
  assumes "u < x + y"
paulson@14365
  1229
      and "0 < x"
paulson@14365
  1230
      and "0 < y"
paulson@14365
  1231
      and "0 < u"
paulson@14365
  1232
  shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
paulson@14365
  1233
proof (intro exI conjI)
paulson@14365
  1234
  show "u * x * inverse(x+y) < x" using prems 
paulson@14365
  1235
    by (simp add: preal_of_rat_add_lemma1) 
paulson@14365
  1236
  show "u * y * inverse(x+y) < y" using prems 
paulson@14365
  1237
    by (simp add: preal_of_rat_add_lemma1 add_commute [of x]) 
paulson@14365
  1238
  show "0 < u * x * inverse (x + y)" using prems
paulson@14365
  1239
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1240
  show "0 < u * y * inverse (x + y)" using prems
paulson@14365
  1241
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1242
  show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
paulson@14365
  1243
    by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
paulson@14365
  1244
qed
paulson@14365
  1245
paulson@14365
  1246
lemma preal_of_rat_add:
paulson@14365
  1247
     "[| 0 < x; 0 < y|] 
paulson@14365
  1248
      ==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
paulson@14365
  1249
apply (unfold preal_of_rat_def preal_add_def)
paulson@14365
  1250
apply (simp add: rat_mem_preal) 
paulson@14335
  1251
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
  1252
apply (auto simp add: add_set_def) 
paulson@14365
  1253
apply (blast dest: preal_of_rat_add_lemma2) 
paulson@14365
  1254
done
paulson@14365
  1255
paulson@14365
  1256
lemma preal_of_rat_mult_lemma1:
paulson@14365
  1257
     "[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
paulson@14365
  1258
apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
paulson@14365
  1259
apply (simp add: zero_less_mult_iff)
paulson@14365
  1260
apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
paulson@14365
  1261
apply (simp_all add: mult_ac)
paulson@14335
  1262
done
paulson@14335
  1263
paulson@14365
  1264
lemma preal_of_rat_mult_lemma2: 
paulson@14365
  1265
  assumes xless: "x < y * z"
paulson@14365
  1266
      and xpos: "0 < x"
paulson@14365
  1267
      and ypos: "0 < y"
paulson@14365
  1268
  shows "x * z * inverse y * inverse z < (z::rat)"
paulson@14365
  1269
proof -
paulson@14365
  1270
  have "0 < y * z" using prems by simp
paulson@14365
  1271
  hence zpos:  "0 < z" using prems by (simp add: zero_less_mult_iff)
paulson@14365
  1272
  have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
paulson@14365
  1273
    by (simp add: mult_ac)
paulson@14365
  1274
  also have "... = x/y" using zpos
paulson@14365
  1275
    by (simp add: divide_inverse_zero)
paulson@14365
  1276
  also have "... < z"
paulson@14365
  1277
    by (simp add: pos_divide_less_eq [OF ypos] mult_commute) 
paulson@14365
  1278
  finally show ?thesis .
paulson@14365
  1279
qed
paulson@14335
  1280
paulson@14365
  1281
lemma preal_of_rat_mult_lemma3:
paulson@14365
  1282
  assumes uless: "u < x * y"
paulson@14365
  1283
      and "0 < x"
paulson@14365
  1284
      and "0 < y"
paulson@14365
  1285
      and "0 < u"
paulson@14365
  1286
  shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
paulson@14365
  1287
proof -
paulson@14365
  1288
  from dense [OF uless] 
paulson@14365
  1289
  obtain r where "u < r" "r < x * y" by blast
paulson@14365
  1290
  thus ?thesis
paulson@14365
  1291
  proof (intro exI conjI)
paulson@14365
  1292
  show "u * x * inverse r < x" using prems 
paulson@14365
  1293
    by (simp add: preal_of_rat_mult_lemma1) 
paulson@14365
  1294
  show "r * y * inverse x * inverse y < y" using prems
paulson@14365
  1295
    by (simp add: preal_of_rat_mult_lemma2)
paulson@14365
  1296
  show "0 < u * x * inverse r" using prems
paulson@14365
  1297
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1298
  show "0 < r * y * inverse x * inverse y" using prems
paulson@14365
  1299
    by (simp add: zero_less_mult_iff) 
paulson@14365
  1300
  have "u * x * inverse r * (r * y * inverse x * inverse y) =
paulson@14365
  1301
        u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
paulson@14365
  1302
    by (simp only: mult_ac)
paulson@14365
  1303
  thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
paulson@14365
  1304
    by simp
paulson@14365
  1305
  qed
paulson@14365
  1306
qed
paulson@14365
  1307
paulson@14365
  1308
lemma preal_of_rat_mult:
paulson@14365
  1309
     "[| 0 < x; 0 < y|] 
paulson@14365
  1310
      ==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
paulson@14365
  1311
apply (unfold preal_of_rat_def preal_mult_def)
paulson@14365
  1312
apply (simp add: rat_mem_preal) 
paulson@14365
  1313
apply (rule_tac f = Abs_preal in arg_cong)
paulson@14365
  1314
apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def) 
paulson@14365
  1315
apply (blast dest: preal_of_rat_mult_lemma3) 
paulson@14335
  1316
done
paulson@14335
  1317
paulson@14365
  1318
lemma preal_of_rat_less_iff:
paulson@14365
  1319
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
paulson@14365
  1320
by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal) 
paulson@14335
  1321
paulson@14365
  1322
lemma preal_of_rat_le_iff:
paulson@14365
  1323
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
paulson@14365
  1324
by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric]) 
paulson@14365
  1325
paulson@14365
  1326
lemma preal_of_rat_eq_iff:
paulson@14365
  1327
      "[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
paulson@14365
  1328
by (simp add: preal_of_rat_le_iff order_eq_iff) 
paulson@14335
  1329
paulson@14335
  1330
paulson@14335
  1331
ML
paulson@14335
  1332
{*
paulson@14335
  1333
val inj_on_Abs_preal = thm"inj_on_Abs_preal";
paulson@14335
  1334
val inj_Rep_preal = thm"inj_Rep_preal";
paulson@14335
  1335
val mem_Rep_preal_Ex = thm"mem_Rep_preal_Ex";
paulson@14335
  1336
val preal_add_commute = thm"preal_add_commute";
paulson@14335
  1337
val preal_add_assoc = thm"preal_add_assoc";
paulson@14335
  1338
val preal_add_left_commute = thm"preal_add_left_commute";
paulson@14335
  1339
val preal_mult_commute = thm"preal_mult_commute";
paulson@14335
  1340
val preal_mult_assoc = thm"preal_mult_assoc";
paulson@14335
  1341
val preal_mult_left_commute = thm"preal_mult_left_commute";
paulson@14335
  1342
val preal_add_mult_distrib2 = thm"preal_add_mult_distrib2";
paulson@14335
  1343
val preal_add_mult_distrib = thm"preal_add_mult_distrib";
paulson@14335
  1344
val preal_self_less_add_left = thm"preal_self_less_add_left";
paulson@14335
  1345
val preal_self_less_add_right = thm"preal_self_less_add_right";
paulson@14365
  1346
val less_add_left = thm"less_add_left";
paulson@14335
  1347
val preal_add_less2_mono1 = thm"preal_add_less2_mono1";
paulson@14335
  1348
val preal_add_less2_mono2 = thm"preal_add_less2_mono2";
paulson@14335
  1349
val preal_add_right_less_cancel = thm"preal_add_right_less_cancel";
paulson@14335
  1350
val preal_add_left_less_cancel = thm"preal_add_left_less_cancel";
paulson@14335
  1351
val preal_add_right_cancel = thm"preal_add_right_cancel";
paulson@14335
  1352
val preal_add_left_cancel = thm"preal_add_left_cancel";
paulson@14335
  1353
val preal_add_left_cancel_iff = thm"preal_add_left_cancel_iff";
paulson@14335
  1354
val preal_add_right_cancel_iff = thm"preal_add_right_cancel_iff";
paulson@14365
  1355
val preal_psup_le = thm"preal_psup_le";
paulson@14335
  1356
val psup_le_ub = thm"psup_le_ub";
paulson@14335
  1357
val preal_complete = thm"preal_complete";
paulson@14365
  1358
val preal_of_rat_add = thm"preal_of_rat_add";
paulson@14365
  1359
val preal_of_rat_mult = thm"preal_of_rat_mult";
paulson@14335
  1360
paulson@14335
  1361
val preal_add_ac = thms"preal_add_ac";
paulson@14335
  1362
val preal_mult_ac = thms"preal_mult_ac";
paulson@14335
  1363
*}
paulson@14335
  1364
paulson@5078
  1365
end