(* Title : PReal.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : The positive reals as Dedekind sections of positive
rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
provides some of the definitions.
*)
header {* Positive real numbers *}
theory PReal
imports Rational
begin
text{*Could be generalized and moved to @{text Ring_and_Field}*}
lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"
by (rule_tac x="b-a" in exI, simp)
definition
cut :: "rat set => bool" where
"cut A = ({} \<subset> A &
A < {r. 0 < r} &
(\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"
lemma cut_of_rat:
assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}" (is "cut ?A")
proof -
from q have pos: "?A < {r. 0 < r}" by force
have nonempty: "{} \<subset> ?A"
proof
show "{} \<subseteq> ?A" by simp
show "{} \<noteq> ?A"
by (force simp only: q eq_commute [of "{}"] interval_empty_iff)
qed
show ?thesis
by (simp add: cut_def pos nonempty,
blast dest: dense intro: order_less_trans)
qed
typedef preal = "{A. cut A}"
by (blast intro: cut_of_rat [OF zero_less_one])
definition
preal_of_rat :: "rat => preal" where
"preal_of_rat q = Abs_preal {x::rat. 0 < x & x < q}"
definition
psup :: "preal set => preal" where
"psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"
definition
add_set :: "[rat set,rat set] => rat set" where
"add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
definition
diff_set :: "[rat set,rat set] => rat set" where
"diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"
definition
mult_set :: "[rat set,rat set] => rat set" where
"mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
definition
inverse_set :: "rat set => rat set" where
"inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"
instantiation preal :: "{ord, plus, minus, times, inverse, one}"
begin
definition
preal_less_def:
"R < S == Rep_preal R < Rep_preal S"
definition
preal_le_def:
"R \<le> S == Rep_preal R \<subseteq> Rep_preal S"
definition
preal_add_def:
"R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"
definition
preal_diff_def:
"R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"
definition
preal_mult_def:
"R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"
definition
preal_inverse_def:
"inverse R == Abs_preal (inverse_set (Rep_preal R))"
definition
preal_one_def:
"1 == preal_of_rat 1"
instance ..
end
text{*Reduces equality on abstractions to equality on representatives*}
declare Abs_preal_inject [simp]
declare Abs_preal_inverse [simp]
lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
by (simp add: preal_def cut_of_rat)
lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"
by (unfold preal_def cut_def, blast)
lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"
by (drule preal_nonempty, fast)
lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"
by (force simp add: preal_def cut_def)
lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"
by (drule preal_imp_psubset_positives, auto)
lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
by (unfold preal_def cut_def, blast)
lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
by (unfold preal_def cut_def, blast)
text{*Relaxing the final premise*}
lemma preal_downwards_closed':
"[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"
apply (simp add: order_le_less)
apply (blast intro: preal_downwards_closed)
done
text{*A positive fraction not in a positive real is an upper bound.
Gleason p. 122 - Remark (1)*}
lemma not_in_preal_ub:
assumes A: "A \<in> preal"
and notx: "x \<notin> A"
and y: "y \<in> A"
and pos: "0 < x"
shows "y < x"
proof (cases rule: linorder_cases)
assume "x<y"
with notx show ?thesis
by (simp add: preal_downwards_closed [OF A y] pos)
next
assume "x=y"
with notx and y show ?thesis by simp
next
assume "y<x"
thus ?thesis .
qed
text {* preal lemmas instantiated to @{term "Rep_preal X"} *}
lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
by (rule preal_Ex_mem [OF Rep_preal])
lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X"
by (rule preal_exists_bound [OF Rep_preal])
lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
subsection{*@{term preal_of_prat}: the Injection from prat to preal*}
lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal"
by (simp add: preal_def cut_of_rat)
lemma rat_subset_imp_le:
"[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y"
apply (simp add: linorder_not_less [symmetric])
apply (blast dest: dense intro: order_less_trans)
done
lemma rat_set_eq_imp_eq:
"[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y};
0 < x; 0 < y|] ==> x = y"
by (blast intro: rat_subset_imp_le order_antisym)
subsection{*Properties of Ordering*}
lemma preal_le_refl: "w \<le> (w::preal)"
by (simp add: preal_le_def)
lemma preal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::preal)"
by (force simp add: preal_le_def)
lemma preal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::preal)"
apply (simp add: preal_le_def)
apply (rule Rep_preal_inject [THEN iffD1], blast)
done
(* Axiom 'order_less_le' of class 'order': *)
lemma preal_less_le: "((w::preal) < z) = (w \<le> z & w \<noteq> z)"
by (simp add: preal_le_def preal_less_def Rep_preal_inject psubset_def)
instance preal :: order
by intro_classes
(assumption |
rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+
lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"
by (insert preal_imp_psubset_positives, blast)
lemma preal_le_linear: "x <= y | y <= (x::preal)"
apply (auto simp add: preal_le_def)
apply (rule ccontr)
apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
elim: order_less_asym)
done
instance preal :: linorder
by intro_classes (rule preal_le_linear)
instantiation preal :: distrib_lattice
begin
definition
"(inf \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = min"
definition
"(sup \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = max"
instance
by intro_classes
(auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1)
end
subsection{*Properties of Addition*}
lemma preal_add_commute: "(x::preal) + y = y + x"
apply (unfold preal_add_def add_set_def)
apply (rule_tac f = Abs_preal in arg_cong)
apply (force simp add: add_commute)
done
text{*Lemmas for proving that addition of two positive reals gives
a positive real*}
lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A"
by blast
text{*Part 1 of Dedekind sections definition*}
lemma add_set_not_empty:
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"
apply (drule preal_nonempty)+
apply (auto simp add: add_set_def)
done
text{*Part 2 of Dedekind sections definition. A structured version of
this proof is @{text preal_not_mem_mult_set_Ex} below.*}
lemma preal_not_mem_add_set_Ex:
"[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B"
apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto)
apply (rule_tac x = "x+xa" in exI)
apply (simp add: add_set_def, clarify)
apply (drule (3) not_in_preal_ub)+
apply (force dest: add_strict_mono)
done
lemma add_set_not_rat_set:
assumes A: "A \<in> preal"
and B: "B \<in> preal"
shows "add_set A B < {r. 0 < r}"
proof
from preal_imp_pos [OF A] preal_imp_pos [OF B]
show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def)
next
show "add_set A B \<noteq> {r. 0 < r}"
by (insert preal_not_mem_add_set_Ex [OF A B], blast)
qed
text{*Part 3 of Dedekind sections definition*}
lemma add_set_lemma3:
"[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|]
==> z \<in> add_set A B"
proof (unfold add_set_def, clarify)
fix x::rat and y::rat
assume A: "A \<in> preal"
and B: "B \<in> preal"
and [simp]: "0 < z"
and zless: "z < x + y"
and x: "x \<in> A"
and y: "y \<in> B"
have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])
have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])
have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)
let ?f = "z/(x+y)"
have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"
proof (intro bexI)
show "z = x*?f + y*?f"
by (simp add: left_distrib [symmetric] divide_inverse mult_ac
order_less_imp_not_eq2)
next
show "y * ?f \<in> B"
proof (rule preal_downwards_closed [OF B y])
show "0 < y * ?f"
by (simp add: divide_inverse zero_less_mult_iff)
next
show "y * ?f < y"
by (insert mult_strict_left_mono [OF fless ypos], simp)
qed
next
show "x * ?f \<in> A"
proof (rule preal_downwards_closed [OF A x])
show "0 < x * ?f"
by (simp add: divide_inverse zero_less_mult_iff)
next
show "x * ?f < x"
by (insert mult_strict_left_mono [OF fless xpos], simp)
qed
qed
qed
text{*Part 4 of Dedekind sections definition*}
lemma add_set_lemma4:
"[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
apply (auto simp add: add_set_def)
apply (frule preal_exists_greater [of A], auto)
apply (rule_tac x="u + y" in exI)
apply (auto intro: add_strict_left_mono)
done
lemma mem_add_set:
"[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
apply (simp (no_asm_simp) add: preal_def cut_def)
apply (blast intro!: add_set_not_empty add_set_not_rat_set
add_set_lemma3 add_set_lemma4)
done
lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
apply (simp add: preal_add_def mem_add_set Rep_preal)
apply (force simp add: add_set_def add_ac)
done
instance preal :: ab_semigroup_add
proof
fix a b c :: preal
show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc)
show "a + b = b + a" by (rule preal_add_commute)
qed
lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
by (rule add_left_commute)
text{* Positive Real addition is an AC operator *}
lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
subsection{*Properties of Multiplication*}
text{*Proofs essentially same as for addition*}
lemma preal_mult_commute: "(x::preal) * y = y * x"
apply (unfold preal_mult_def mult_set_def)
apply (rule_tac f = Abs_preal in arg_cong)
apply (force simp add: mult_commute)
done
text{*Multiplication of two positive reals gives a positive real.*}
text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
text{*Part 1 of Dedekind sections definition*}
lemma mult_set_not_empty:
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
apply (insert preal_nonempty [of A] preal_nonempty [of B])
apply (auto simp add: mult_set_def)
done
text{*Part 2 of Dedekind sections definition*}
lemma preal_not_mem_mult_set_Ex:
assumes A: "A \<in> preal"
and B: "B \<in> preal"
shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
proof -
from preal_exists_bound [OF A]
obtain x where [simp]: "0 < x" "x \<notin> A" by blast
from preal_exists_bound [OF B]
obtain y where [simp]: "0 < y" "y \<notin> B" by blast
show ?thesis
proof (intro exI conjI)
show "0 < x*y" by (simp add: mult_pos_pos)
show "x * y \<notin> mult_set A B"
proof -
{ fix u::rat and v::rat
assume "u \<in> A" and "v \<in> B" and "x*y = u*v"
moreover
with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
moreover
with prems have "0\<le>v"
by (blast intro: preal_imp_pos [OF B] order_less_imp_le prems)
moreover
from calculation
have "u*v < x*y" by (blast intro: mult_strict_mono prems)
ultimately have False by force }
thus ?thesis by (auto simp add: mult_set_def)
qed
qed
qed
lemma mult_set_not_rat_set:
assumes A: "A \<in> preal"
and B: "B \<in> preal"
shows "mult_set A B < {r. 0 < r}"
proof
show "mult_set A B \<subseteq> {r. 0 < r}"
by (force simp add: mult_set_def
intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)
show "mult_set A B \<noteq> {r. 0 < r}"
using preal_not_mem_mult_set_Ex [OF A B] by blast
qed
text{*Part 3 of Dedekind sections definition*}
lemma mult_set_lemma3:
"[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|]
==> z \<in> mult_set A B"
proof (unfold mult_set_def, clarify)
fix x::rat and y::rat
assume A: "A \<in> preal"
and B: "B \<in> preal"
and [simp]: "0 < z"
and zless: "z < x * y"
and x: "x \<in> A"
and y: "y \<in> B"
have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
proof
show "\<exists>y'\<in>B. z = (z/y) * y'"
proof
show "z = (z/y)*y"
by (simp add: divide_inverse mult_commute [of y] mult_assoc
order_less_imp_not_eq2)
show "y \<in> B" by fact
qed
next
show "z/y \<in> A"
proof (rule preal_downwards_closed [OF A x])
show "0 < z/y"
by (simp add: zero_less_divide_iff)
show "z/y < x" by (simp add: pos_divide_less_eq zless)
qed
qed
qed
text{*Part 4 of Dedekind sections definition*}
lemma mult_set_lemma4:
"[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
apply (auto simp add: mult_set_def)
apply (frule preal_exists_greater [of A], auto)
apply (rule_tac x="u * y" in exI)
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B]
mult_strict_right_mono)
done
lemma mem_mult_set:
"[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
apply (simp (no_asm_simp) add: preal_def cut_def)
apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
mult_set_lemma3 mult_set_lemma4)
done
lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
apply (force simp add: mult_set_def mult_ac)
done
instance preal :: ab_semigroup_mult
proof
fix a b c :: preal
show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc)
show "a * b = b * a" by (rule preal_mult_commute)
qed
lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
by (rule mult_left_commute)
text{* Positive Real multiplication is an AC operator *}
lemmas preal_mult_ac =
preal_mult_assoc preal_mult_commute preal_mult_left_commute
text{* Positive real 1 is the multiplicative identity element *}
lemma preal_mult_1: "(1::preal) * z = z"
unfolding preal_one_def
proof (induct z)
fix A :: "rat set"
assume A: "A \<in> preal"
have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
proof
show "?lhs \<subseteq> A"
proof clarify
fix x::rat and u::rat and v::rat
assume upos: "0<u" and "u<1" and v: "v \<in> A"
have vpos: "0<v" by (rule preal_imp_pos [OF A v])
hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems)
thus "u * v \<in> A"
by (force intro: preal_downwards_closed [OF A v] mult_pos_pos
upos vpos)
qed
next
show "A \<subseteq> ?lhs"
proof clarify
fix x::rat
assume x: "x \<in> A"
have xpos: "0<x" by (rule preal_imp_pos [OF A x])
from preal_exists_greater [OF A x]
obtain v where v: "v \<in> A" and xlessv: "x < v" ..
have vpos: "0<v" by (rule preal_imp_pos [OF A v])
show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
proof (intro exI conjI)
show "0 < x/v"
by (simp add: zero_less_divide_iff xpos vpos)
show "x / v < 1"
by (simp add: pos_divide_less_eq vpos xlessv)
show "\<exists>v'\<in>A. x = (x / v) * v'"
proof
show "x = (x/v)*v"
by (simp add: divide_inverse mult_assoc vpos
order_less_imp_not_eq2)
show "v \<in> A" by fact
qed
qed
qed
qed
thus "preal_of_rat 1 * Abs_preal A = Abs_preal A"
by (simp add: preal_of_rat_def preal_mult_def mult_set_def
rat_mem_preal A)
qed
instance preal :: comm_monoid_mult
by intro_classes (rule preal_mult_1)
lemma preal_mult_1_right: "z * (1::preal) = z"
by (rule mult_1_right)
subsection{*Distribution of Multiplication across Addition*}
lemma mem_Rep_preal_add_iff:
"(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"
apply (simp add: preal_add_def mem_add_set Rep_preal)
apply (simp add: add_set_def)
done
lemma mem_Rep_preal_mult_iff:
"(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
apply (simp add: mult_set_def)
done
lemma distrib_subset1:
"Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
apply (force simp add: right_distrib)
done
lemma preal_add_mult_distrib_mean:
assumes a: "a \<in> Rep_preal w"
and b: "b \<in> Rep_preal w"
and d: "d \<in> Rep_preal x"
and e: "e \<in> Rep_preal y"
shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
proof
let ?c = "(a*d + b*e)/(d+e)"
have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
have cpos: "0 < ?c"
by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
show "a * d + b * e = ?c * (d + e)"
by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
show "?c \<in> Rep_preal w"
proof (cases rule: linorder_le_cases)
assume "a \<le> b"
hence "?c \<le> b"
by (simp add: pos_divide_le_eq right_distrib mult_right_mono
order_less_imp_le)
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
next
assume "b \<le> a"
hence "?c \<le> a"
by (simp add: pos_divide_le_eq right_distrib mult_right_mono
order_less_imp_le)
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
qed
qed
lemma distrib_subset2:
"Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
done
lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
apply (rule Rep_preal_inject [THEN iffD1])
apply (rule equalityI [OF distrib_subset1 distrib_subset2])
done
lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
by (simp add: preal_mult_commute preal_add_mult_distrib2)
instance preal :: comm_semiring
by intro_classes (rule preal_add_mult_distrib)
subsection{*Existence of Inverse, a Positive Real*}
lemma mem_inv_set_ex:
assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
proof -
from preal_exists_bound [OF A]
obtain x where [simp]: "0<x" "x \<notin> A" by blast
show ?thesis
proof (intro exI conjI)
show "0 < inverse (x+1)"
by (simp add: order_less_trans [OF _ less_add_one])
show "inverse(x+1) < inverse x"
by (simp add: less_imp_inverse_less less_add_one)
show "inverse (inverse x) \<notin> A"
by (simp add: order_less_imp_not_eq2)
qed
qed
text{*Part 1 of Dedekind sections definition*}
lemma inverse_set_not_empty:
"A \<in> preal ==> {} \<subset> inverse_set A"
apply (insert mem_inv_set_ex [of A])
apply (auto simp add: inverse_set_def)
done
text{*Part 2 of Dedekind sections definition*}
lemma preal_not_mem_inverse_set_Ex:
assumes A: "A \<in> preal" shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
proof -
from preal_nonempty [OF A]
obtain x where x: "x \<in> A" and xpos [simp]: "0<x" ..
show ?thesis
proof (intro exI conjI)
show "0 < inverse x" by simp
show "inverse x \<notin> inverse_set A"
proof -
{ fix y::rat
assume ygt: "inverse x < y"
have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
have iyless: "inverse y < x"
by (simp add: inverse_less_imp_less [of x] ygt)
have "inverse y \<in> A"
by (simp add: preal_downwards_closed [OF A x] iyless)}
thus ?thesis by (auto simp add: inverse_set_def)
qed
qed
qed
lemma inverse_set_not_rat_set:
assumes A: "A \<in> preal" shows "inverse_set A < {r. 0 < r}"
proof
show "inverse_set A \<subseteq> {r. 0 < r}" by (force simp add: inverse_set_def)
next
show "inverse_set A \<noteq> {r. 0 < r}"
by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
qed
text{*Part 3 of Dedekind sections definition*}
lemma inverse_set_lemma3:
"[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|]
==> z \<in> inverse_set A"
apply (auto simp add: inverse_set_def)
apply (auto intro: order_less_trans)
done
text{*Part 4 of Dedekind sections definition*}
lemma inverse_set_lemma4:
"[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
apply (auto simp add: inverse_set_def)
apply (drule dense [of y])
apply (blast intro: order_less_trans)
done
lemma mem_inverse_set:
"A \<in> preal ==> inverse_set A \<in> preal"
apply (simp (no_asm_simp) add: preal_def cut_def)
apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
inverse_set_lemma3 inverse_set_lemma4)
done
subsection{*Gleason's Lemma 9-3.4, page 122*}
lemma Gleason9_34_exists:
assumes A: "A \<in> preal"
and "\<forall>x\<in>A. x + u \<in> A"
and "0 \<le> z"
shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
proof (cases z rule: int_cases)
case (nonneg n)
show ?thesis
proof (simp add: prems, induct n)
case 0
from preal_nonempty [OF A]
show ?case by force
case (Suc k)
from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
thus ?case by (force simp add: left_distrib add_ac prems)
qed
next
case (neg n)
with prems show ?thesis by simp
qed
lemma Gleason9_34_contra:
assumes A: "A \<in> preal"
shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
proof (induct u, induct y)
fix a::int and b::int
fix c::int and d::int
assume bpos [simp]: "0 < b"
and dpos [simp]: "0 < d"
and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
and upos: "0 < Fract c d"
and ypos: "0 < Fract a b"
and notin: "Fract a b \<notin> A"
have cpos [simp]: "0 < c"
by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos)
have apos [simp]: "0 < a"
by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos)
let ?k = "a*d"
have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)"
proof -
have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
by (simp add: mult_rat le_rat order_less_imp_not_eq2 mult_ac)
moreover
have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
by (rule mult_mono,
simp_all add: int_one_le_iff_zero_less zero_less_mult_iff
order_less_imp_le)
ultimately
show ?thesis by simp
qed
have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)
from Gleason9_34_exists [OF A closed k]
obtain z where z: "z \<in> A"
and mem: "z + of_int ?k * Fract c d \<in> A" ..
have less: "z + of_int ?k * Fract c d < Fract a b"
by (rule not_in_preal_ub [OF A notin mem ypos])
have "0<z" by (rule preal_imp_pos [OF A z])
with frle and less show False by (simp add: Fract_of_int_eq)
qed
lemma Gleason9_34:
assumes A: "A \<in> preal"
and upos: "0 < u"
shows "\<exists>r \<in> A. r + u \<notin> A"
proof (rule ccontr, simp)
assume closed: "\<forall>r\<in>A. r + u \<in> A"
from preal_exists_bound [OF A]
obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
show False
by (rule Gleason9_34_contra [OF A closed upos ypos y])
qed
subsection{*Gleason's Lemma 9-3.6*}
lemma lemma_gleason9_36:
assumes A: "A \<in> preal"
and x: "1 < x"
shows "\<exists>r \<in> A. r*x \<notin> A"
proof -
from preal_nonempty [OF A]
obtain y where y: "y \<in> A" and ypos: "0<y" ..
show ?thesis
proof (rule classical)
assume "~(\<exists>r\<in>A. r * x \<notin> A)"
with y have ymem: "y * x \<in> A" by blast
from ypos mult_strict_left_mono [OF x]
have yless: "y < y*x" by simp
let ?d = "y*x - y"
from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
from Gleason9_34 [OF A dpos]
obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
have rpos: "0<r" by (rule preal_imp_pos [OF A r])
with dpos have rdpos: "0 < r + ?d" by arith
have "~ (r + ?d \<le> y + ?d)"
proof
assume le: "r + ?d \<le> y + ?d"
from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
with notin show False by simp
qed
hence "y < r" by simp
with ypos have dless: "?d < (r * ?d)/y"
by (simp add: pos_less_divide_eq mult_commute [of ?d]
mult_strict_right_mono dpos)
have "r + ?d < r*x"
proof -
have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
also with ypos have "... = (r/y) * (y + ?d)"
by (simp only: right_distrib divide_inverse mult_ac, simp)
also have "... = r*x" using ypos
by (simp add: times_divide_eq_left)
finally show "r + ?d < r*x" .
qed
with r notin rdpos
show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest: preal_downwards_closed [OF A])
qed
qed
subsection{*Existence of Inverse: Part 2*}
lemma mem_Rep_preal_inverse_iff:
"(z \<in> Rep_preal(inverse R)) =
(0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
apply (simp add: inverse_set_def)
done
lemma Rep_preal_of_rat:
"0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
by (simp add: preal_of_rat_def rat_mem_preal)
lemma subset_inverse_mult_lemma:
assumes xpos: "0 < x" and xless: "x < 1"
shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R &
u \<in> Rep_preal R & x = r * u"
proof -
from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
from lemma_gleason9_36 [OF Rep_preal this]
obtain r where r: "r \<in> Rep_preal R"
and notin: "r * (inverse x) \<notin> Rep_preal R" ..
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
from preal_exists_greater [OF Rep_preal r]
obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
show ?thesis
proof (intro exI conjI)
show "0 < x/u" using xpos upos
by (simp add: zero_less_divide_iff)
show "x/u < x/r" using xpos upos rpos
by (simp add: divide_inverse mult_less_cancel_left rless)
show "inverse (x / r) \<notin> Rep_preal R" using notin
by (simp add: divide_inverse mult_commute)
show "u \<in> Rep_preal R" by (rule u)
show "x = x / u * u" using upos
by (simp add: divide_inverse mult_commute)
qed
qed
lemma subset_inverse_mult:
"Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
mem_Rep_preal_mult_iff)
apply (blast dest: subset_inverse_mult_lemma)
done
lemma inverse_mult_subset_lemma:
assumes rpos: "0 < r"
and rless: "r < y"
and notin: "inverse y \<notin> Rep_preal R"
and q: "q \<in> Rep_preal R"
shows "r*q < 1"
proof -
have "q < inverse y" using rpos rless
by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
hence "r * q < r/y" using rpos
by (simp add: divide_inverse mult_less_cancel_left)
also have "... \<le> 1" using rpos rless
by (simp add: pos_divide_le_eq)
finally show ?thesis .
qed
lemma inverse_mult_subset:
"Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)"
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
mem_Rep_preal_mult_iff)
apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal])
apply (blast intro: inverse_mult_subset_lemma)
done
lemma preal_mult_inverse: "inverse R * R = (1::preal)"
unfolding preal_one_def
apply (rule Rep_preal_inject [THEN iffD1])
apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult])
done
lemma preal_mult_inverse_right: "R * inverse R = (1::preal)"
apply (rule preal_mult_commute [THEN subst])
apply (rule preal_mult_inverse)
done
text{*Theorems needing @{text Gleason9_34}*}
lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
proof
fix r
assume r: "r \<in> Rep_preal R"
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
from mem_Rep_preal_Ex
obtain y where y: "y \<in> Rep_preal S" ..
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
have ry: "r+y \<in> Rep_preal(R + S)" using r y
by (auto simp add: mem_Rep_preal_add_iff)
show "r \<in> Rep_preal(R + S)" using r ypos rpos
by (simp add: preal_downwards_closed [OF Rep_preal ry])
qed
lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
proof -
from mem_Rep_preal_Ex
obtain y where y: "y \<in> Rep_preal S" ..
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
from Gleason9_34 [OF Rep_preal ypos]
obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
have "r + y \<in> Rep_preal (R + S)" using r y
by (auto simp add: mem_Rep_preal_add_iff)
thus ?thesis using notin by blast
qed
lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
by (insert Rep_preal_sum_not_subset, blast)
text{*at last, Gleason prop. 9-3.5(iii) page 123*}
lemma preal_self_less_add_left: "(R::preal) < R + S"
apply (unfold preal_less_def psubset_def)
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
done
lemma preal_self_less_add_right: "(R::preal) < S + R"
by (simp add: preal_add_commute preal_self_less_add_left)
lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
by (insert preal_self_less_add_left [of x y], auto)
subsection{*Subtraction for Positive Reals*}
text{*Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> \<exists>D. A + D =
B"}. We define the claimed @{term D} and show that it is a positive real*}
text{*Part 1 of Dedekind sections definition*}
lemma diff_set_not_empty:
"R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
apply (auto simp add: preal_less_def diff_set_def elim!: equalityE)
apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
apply (drule preal_imp_pos [OF Rep_preal], clarify)
apply (cut_tac a=x and b=u in add_eq_exists, force)
done
text{*Part 2 of Dedekind sections definition*}
lemma diff_set_nonempty:
"\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
apply (cut_tac X = S in Rep_preal_exists_bound)
apply (erule exE)
apply (rule_tac x = x in exI, auto)
apply (simp add: diff_set_def)
apply (auto dest: Rep_preal [THEN preal_downwards_closed])
done
lemma diff_set_not_rat_set:
"diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
proof
show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def)
show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
qed
text{*Part 3 of Dedekind sections definition*}
lemma diff_set_lemma3:
"[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|]
==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
apply (auto simp add: diff_set_def)
apply (rule_tac x=x in exI)
apply (drule Rep_preal [THEN preal_downwards_closed], auto)
done
text{*Part 4 of Dedekind sections definition*}
lemma diff_set_lemma4:
"[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|]
==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
apply (auto simp add: diff_set_def)
apply (drule Rep_preal [THEN preal_exists_greater], clarify)
apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)
apply (rule_tac x="y+xa" in exI)
apply (auto simp add: add_ac)
done
lemma mem_diff_set:
"R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
apply (unfold preal_def cut_def)
apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
diff_set_lemma3 diff_set_lemma4)
done
lemma mem_Rep_preal_diff_iff:
"R < S ==>
(z \<in> Rep_preal(S-R)) =
(\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
apply (simp add: preal_diff_def mem_diff_set Rep_preal)
apply (force simp add: diff_set_def)
done
text{*proving that @{term "R + D \<le> S"}*}
lemma less_add_left_lemma:
assumes Rless: "R < S"
and a: "a \<in> Rep_preal R"
and cb: "c + b \<in> Rep_preal S"
and "c \<notin> Rep_preal R"
and "0 < b"
and "0 < c"
shows "a + b \<in> Rep_preal S"
proof -
have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
moreover
have "a < c" using prems
by (blast intro: not_in_Rep_preal_ub )
ultimately show ?thesis using prems
by (simp add: preal_downwards_closed [OF Rep_preal cb])
qed
lemma less_add_left_le1:
"R < (S::preal) ==> R + (S-R) \<le> S"
apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff
mem_Rep_preal_diff_iff)
apply (blast intro: less_add_left_lemma)
done
subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
lemma lemma_sum_mem_Rep_preal_ex:
"x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
apply (drule Rep_preal [THEN preal_exists_greater], clarify)
apply (cut_tac a=x and b=u in add_eq_exists, auto)
done
lemma less_add_left_lemma2:
assumes Rless: "R < S"
and x: "x \<in> Rep_preal S"
and xnot: "x \<notin> Rep_preal R"
shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R &
z + v \<in> Rep_preal S & x = u + v"
proof -
have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
from lemma_sum_mem_Rep_preal_ex [OF x]
obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
from Gleason9_34 [OF Rep_preal epos]
obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
from add_eq_exists [of r x]
obtain y where eq: "x = r+y" by auto
show ?thesis
proof (intro exI conjI)
show "r \<in> Rep_preal R" by (rule r)
show "r + e \<notin> Rep_preal R" by (rule notin)
show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
show "x = r + y" by (simp add: eq)
show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
by simp
show "0 < y" using rless eq by arith
qed
qed
lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
apply (auto simp add: preal_le_def)
apply (case_tac "x \<in> Rep_preal R")
apply (cut_tac Rep_preal_self_subset [of R], force)
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
apply (blast dest: less_add_left_lemma2)
done
lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
by (blast intro: preal_le_anti_sym [OF less_add_left_le1 less_add_left_le2])
lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
by (fast dest: less_add_left)
lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
apply (rule_tac y1 = D in preal_add_commute [THEN subst])
apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
done
lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
apply (insert linorder_less_linear [of R S], auto)
apply (drule_tac R = S and T = T in preal_add_less2_mono1)
apply (blast dest: order_less_trans)
done
lemma preal_add_left_less_cancel: "T + R < T + S ==> R < (S::preal)"
by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)"
by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right)
lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left)
lemma preal_add_less_mono:
"[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
apply (auto dest!: less_add_left_Ex simp add: preal_add_ac)
apply (rule preal_add_assoc [THEN subst])
apply (rule preal_self_less_add_right)
done
lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
apply (insert linorder_less_linear [of R S], safe)
apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
done
lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
by (fast intro: preal_add_left_cancel)
lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
by (fast intro: preal_add_right_cancel)
lemmas preal_cancels =
preal_add_less_cancel_right preal_add_less_cancel_left
preal_add_le_cancel_right preal_add_le_cancel_left
preal_add_left_cancel_iff preal_add_right_cancel_iff
instance preal :: ordered_cancel_ab_semigroup_add
proof
fix a b c :: preal
show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel)
show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left)
qed
subsection{*Completeness of type @{typ preal}*}
text{*Prove that supremum is a cut*}
text{*Part 1 of Dedekind sections definition*}
lemma preal_sup_set_not_empty:
"P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
apply auto
apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
done
text{*Part 2 of Dedekind sections definition*}
lemma preal_sup_not_exists:
"\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
apply (cut_tac X = Y in Rep_preal_exists_bound)
apply (auto simp add: preal_le_def)
done
lemma preal_sup_set_not_rat_set:
"\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
apply (drule preal_sup_not_exists)
apply (blast intro: preal_imp_pos [OF Rep_preal])
done
text{*Part 3 of Dedekind sections definition*}
lemma preal_sup_set_lemma3:
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
by (auto elim: Rep_preal [THEN preal_downwards_closed])
text{*Part 4 of Dedekind sections definition*}
lemma preal_sup_set_lemma4:
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
by (blast dest: Rep_preal [THEN preal_exists_greater])
lemma preal_sup:
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
apply (unfold preal_def cut_def)
apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
preal_sup_set_lemma3 preal_sup_set_lemma4)
done
lemma preal_psup_le:
"[| \<forall>X \<in> P. X \<le> Y; x \<in> P |] ==> x \<le> psup P"
apply (simp (no_asm_simp) add: preal_le_def)
apply (subgoal_tac "P \<noteq> {}")
apply (auto simp add: psup_def preal_sup)
done
lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
apply (simp (no_asm_simp) add: preal_le_def)
apply (simp add: psup_def preal_sup)
apply (auto simp add: preal_le_def)
done
text{*Supremum property*}
lemma preal_complete:
"[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
apply (simp add: preal_less_def psup_def preal_sup)
apply (auto simp add: preal_le_def)
apply (rename_tac U)
apply (cut_tac x = U and y = Z in linorder_less_linear)
apply (auto simp add: preal_less_def)
done
subsection{*The Embedding from @{typ rat} into @{typ preal}*}
lemma preal_of_rat_add_lemma1:
"[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
apply (simp add: zero_less_mult_iff)
apply (simp add: mult_ac)
done
lemma preal_of_rat_add_lemma2:
assumes "u < x + y"
and "0 < x"
and "0 < y"
and "0 < u"
shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
proof (intro exI conjI)
show "u * x * inverse(x+y) < x" using prems
by (simp add: preal_of_rat_add_lemma1)
show "u * y * inverse(x+y) < y" using prems
by (simp add: preal_of_rat_add_lemma1 add_commute [of x])
show "0 < u * x * inverse (x + y)" using prems
by (simp add: zero_less_mult_iff)
show "0 < u * y * inverse (x + y)" using prems
by (simp add: zero_less_mult_iff)
show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
qed
lemma preal_of_rat_add:
"[| 0 < x; 0 < y|]
==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
apply (unfold preal_of_rat_def preal_add_def)
apply (simp add: rat_mem_preal)
apply (rule_tac f = Abs_preal in arg_cong)
apply (auto simp add: add_set_def)
apply (blast dest: preal_of_rat_add_lemma2)
done
lemma preal_of_rat_mult_lemma1:
"[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
apply (simp add: zero_less_mult_iff)
apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
apply (simp_all add: mult_ac)
done
lemma preal_of_rat_mult_lemma2:
assumes xless: "x < y * z"
and xpos: "0 < x"
and ypos: "0 < y"
shows "x * z * inverse y * inverse z < (z::rat)"
proof -
have "0 < y * z" using prems by simp
hence zpos: "0 < z" using prems by (simp add: zero_less_mult_iff)
have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
by (simp add: mult_ac)
also have "... = x/y" using zpos
by (simp add: divide_inverse)
also from xless have "... < z"
by (simp add: pos_divide_less_eq [OF ypos] mult_commute)
finally show ?thesis .
qed
lemma preal_of_rat_mult_lemma3:
assumes uless: "u < x * y"
and "0 < x"
and "0 < y"
and "0 < u"
shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
proof -
from dense [OF uless]
obtain r where "u < r" "r < x * y" by blast
thus ?thesis
proof (intro exI conjI)
show "u * x * inverse r < x" using prems
by (simp add: preal_of_rat_mult_lemma1)
show "r * y * inverse x * inverse y < y" using prems
by (simp add: preal_of_rat_mult_lemma2)
show "0 < u * x * inverse r" using prems
by (simp add: zero_less_mult_iff)
show "0 < r * y * inverse x * inverse y" using prems
by (simp add: zero_less_mult_iff)
have "u * x * inverse r * (r * y * inverse x * inverse y) =
u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
by (simp only: mult_ac)
thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
by simp
qed
qed
lemma preal_of_rat_mult:
"[| 0 < x; 0 < y|]
==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
apply (unfold preal_of_rat_def preal_mult_def)
apply (simp add: rat_mem_preal)
apply (rule_tac f = Abs_preal in arg_cong)
apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def)
apply (blast dest: preal_of_rat_mult_lemma3)
done
lemma preal_of_rat_less_iff:
"[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal)
lemma preal_of_rat_le_iff:
"[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric])
lemma preal_of_rat_eq_iff:
"[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
by (simp add: preal_of_rat_le_iff order_eq_iff)
end