# Theory Dedekind_Real

theory Dedekind_Real
imports Complex_Main
```(*  Title:      HOL/ex/Dedekind_Real.thy
Author:     Jacques D. Fleuriot, University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

The positive reals as Dedekind sections of positive
rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
provides some of the definitions.
*)

theory Dedekind_Real
imports Complex_Main
begin

section ‹Positive real numbers›

text‹Could be generalized and moved to ‹Groups››
lemma add_eq_exists: "∃x. a+x = (b::rat)"
by (rule_tac x="b-a" in exI, simp)

definition
cut :: "rat set => bool" where
"cut A = ({} ⊂ A &
A < {r. 0 < r} &
(∀y ∈ A. ((∀z. 0<z & z < y --> z ∈ A) & (∃u ∈ A. y < u))))"

lemma interval_empty_iff:
"{y. (x::'a::unbounded_dense_linorder) < y ∧ y < z} = {} ⟷ ¬ x < z"
by (auto dest: dense)

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: "{} ⊂ ?A"
proof
show "{} ⊆ ?A" by simp
show "{} ≠ ?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 = "Collect cut"
by (blast intro: cut_of_rat [OF zero_less_one])

lemma Abs_preal_induct [induct type: preal]:
"(⋀x. cut x ⟹ P (Abs_preal x)) ⟹ P x"
using Abs_preal_induct [of P x] by simp

lemma Rep_preal:
"cut (Rep_preal x)"
using Rep_preal [of x] by simp

definition
psup :: "preal set => preal" where
"psup P = Abs_preal (⋃X ∈ P. Rep_preal X)"

definition
add_set :: "[rat set,rat set] => rat set" where
"add_set A B = {w. ∃x ∈ A. ∃y ∈ B. w = x + y}"

definition
diff_set :: "[rat set,rat set] => rat set" where
"diff_set A B = {w. ∃x. 0 < w & 0 < x & x ∉ B & x + w ∈ A}"

definition
mult_set :: "[rat set,rat set] => rat set" where
"mult_set A B = {w. ∃x ∈ A. ∃y ∈ B. w = x * y}"

definition
inverse_set :: "rat set => rat set" where
"inverse_set A = {x. ∃y. 0 < x & x < y & inverse y ∉ 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 ≤ S == Rep_preal R ⊆ Rep_preal S"

definition
"R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"

definition
preal_diff_def:
"R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"

definition
preal_mult_def:
"R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"

definition
preal_inverse_def:
"inverse R == Abs_preal (inverse_set (Rep_preal R))"

definition "R div S = R * inverse (S::preal)"

definition
preal_one_def:
"1 == Abs_preal {x. 0 < x & x < 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 ==> cut {r::rat. 0 < r & r < q}"

lemma preal_nonempty: "cut A ==> ∃x∈A. 0 < x"
unfolding cut_def [abs_def] by blast

lemma preal_Ex_mem: "cut A ⟹ ∃x. x ∈ A"
apply (drule preal_nonempty)
apply fast
done

lemma preal_imp_psubset_positives: "cut A ==> A < {r. 0 < r}"

lemma preal_exists_bound: "cut A ==> ∃x. 0 < x & x ∉ A"
apply (drule preal_imp_psubset_positives)
apply auto
done

lemma preal_exists_greater: "[| cut A; y ∈ A |] ==> ∃u ∈ A. y < u"
unfolding cut_def [abs_def] by blast

lemma preal_downwards_closed: "[| cut A; y ∈ A; 0 < z; z < y |] ==> z ∈ A"
unfolding cut_def [abs_def] by blast

text‹Relaxing the final premise›
lemma preal_downwards_closed':
"[| cut A; y ∈ A; 0 < z; z ≤ y |] ==> z ∈ A"
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: "cut A"
and notx: "x ∉ A"
and y: "y ∈ 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: "∃x. x ∈ Rep_preal X"
thm preal_Ex_mem
by (rule preal_Ex_mem [OF Rep_preal])

lemma Rep_preal_exists_bound: "∃x>0. x ∉ 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‹Properties of Ordering›

instance preal :: order
proof
fix w :: preal
show "w ≤ w" by (simp add: preal_le_def)
next
fix i j k :: preal
assume "i ≤ j" and "j ≤ k"
then show "i ≤ k" by (simp add: preal_le_def)
next
fix z w :: preal
assume "z ≤ w" and "w ≤ z"
then show "z = w" by (simp add: preal_le_def Rep_preal_inject)
next
fix z w :: preal
show "z < w ⟷ z ≤ w ∧ ¬ w ≤ z"
by (auto simp add: preal_le_def preal_less_def Rep_preal_inject)
qed

lemma preal_imp_pos: "[|cut A; r ∈ A|] ==> 0 < r"
by (insert preal_imp_psubset_positives, blast)

instance preal :: linorder
proof
fix x y :: preal
show "x <= y | y <= x"
apply (rule ccontr)
apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
elim: order_less_asym)
done
qed

instantiation preal :: distrib_lattice
begin

definition
"(inf :: preal ⇒ preal ⇒ preal) = min"

definition
"(sup :: preal ⇒ preal ⇒ preal) = max"

instance
by intro_classes
(auto simp add: inf_preal_def sup_preal_def max_min_distrib2)

end

lemma preal_add_commute: "(x::preal) + y = y + x"
apply (rule_tac f = Abs_preal in arg_cong)
done

text‹Lemmas for proving that addition of two positive reals gives
a positive real›

text‹Part 1 of Dedekind sections definition›
"[|cut A; cut B|] ==> {} ⊂ add_set A B"
apply (drule preal_nonempty)+
done

text‹Part 2 of Dedekind sections definition.  A structured version of
this proof is ‹preal_not_mem_mult_set_Ex› below.›
"[|cut A; cut B|] ==> ∃q>0. q ∉ 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 (drule (3) not_in_preal_ub)+
done

assumes A: "cut A"
and B: "cut B"
shows "add_set A B < {r. 0 < r}"
proof
from preal_imp_pos [OF A] preal_imp_pos [OF B]
next
show "add_set A B ≠ {r. 0 < r}"
by (insert preal_not_mem_add_set_Ex [OF A B], blast)
qed

text‹Part 3 of Dedekind sections definition›
"[|cut A; cut B; u ∈ add_set A B; 0 < z; z < u|]
==> z ∈ add_set A B"
fix x::rat and y::rat
assume A: "cut A"
and B: "cut B"
and [simp]: "0 < z"
and zless: "z < x + y"
and x:  "x ∈ A"
and y:  "y ∈ 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])
let ?f = "z/(x+y)"
have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
show "∃x' ∈ A. ∃y'∈B. z = x' + y'"
proof (intro bexI)
show "z = x*?f + y*?f"
by (simp add: distrib_right [symmetric] divide_inverse ac_simps
order_less_imp_not_eq2)
next
show "y * ?f ∈ B"
proof (rule preal_downwards_closed [OF B y])
show "0 < y * ?f"
next
show "y * ?f < y"
by (insert mult_strict_left_mono [OF fless ypos], simp)
qed
next
show "x * ?f ∈ A"
proof (rule preal_downwards_closed [OF A x])
show "0 < x * ?f"
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›
"[|cut A; cut B; y ∈ add_set A B|] ==> ∃u ∈ add_set A B. y < u"
apply (frule preal_exists_greater [of A], auto)
apply (rule_tac x="u + ya" in exI)
done

"[|cut A; cut B|] ==> cut (add_set A B)"
done

lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
done

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

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)
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:
"[|cut A; cut B|] ==> {} ⊂ mult_set A B"
apply (insert preal_nonempty [of A] preal_nonempty [of B])
done

text‹Part 2 of Dedekind sections definition›
lemma preal_not_mem_mult_set_Ex:
assumes A: "cut A"
and B: "cut B"
shows "∃q. 0 < q & q ∉ mult_set A B"
proof -
from preal_exists_bound [OF A] obtain x where 1 [simp]: "0 < x" "x ∉ A" by blast
from preal_exists_bound [OF B] obtain y where 2 [simp]: "0 < y" "y ∉ B" by blast
show ?thesis
proof (intro exI conjI)
show "0 < x*y" by simp
show "x * y ∉ mult_set A B"
proof -
{
fix u::rat and v::rat
assume u: "u ∈ A" and v: "v ∈ B" and xy: "x*y = u*v"
moreover from A B 1 2 u v have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
moreover
from A B 1 2 u v have "0≤v"
by (blast intro: preal_imp_pos [OF B] order_less_imp_le)
moreover
from A B 1 ‹u < x› ‹v < y› ‹0 ≤ v›
have "u*v < x*y" by (blast intro: mult_strict_mono)
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: "cut A"
and B: "cut B"
shows "mult_set A B < {r. 0 < r}"
proof
show "mult_set A B ⊆ {r. 0 < r}"
intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)
show "mult_set A B ≠ {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:
"[|cut A; cut B; u ∈ mult_set A B; 0 < z; z < u|]
==> z ∈ mult_set A B"
proof (unfold mult_set_def, clarify)
fix x::rat and y::rat
assume A: "cut A"
and B: "cut B"
and [simp]: "0 < z"
and zless: "z < x * y"
and x:  "x ∈ A"
and y:  "y ∈ B"
have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
show "∃x' ∈ A. ∃y' ∈ B. z = x' * y'"
proof
show "∃y'∈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 ∈ B" by fact
qed
next
show "z/y ∈ A"
proof (rule preal_downwards_closed [OF A x])
show "0 < z/y"
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:
"[|cut A; cut B; y ∈ mult_set A B|] ==> ∃u ∈ mult_set A B. y < u"
apply (frule preal_exists_greater [of A], auto)
apply (rule_tac x="u * ya" in exI)
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B]
mult_strict_right_mono)
done

lemma mem_mult_set:
"[|cut A; cut B|] ==> cut (mult_set A B)"
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 ac_simps)
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

text‹Positive real 1 is the multiplicative identity element›

lemma preal_mult_1: "(1::preal) * z = z"
proof (induct z)
fix A :: "rat set"
assume A: "cut A"
have "{w. ∃u. 0 < u ∧ u < 1 & (∃v ∈ A. w = u * v)} = A" (is "?lhs = A")
proof
show "?lhs ⊆ A"
proof clarify
fix x::rat and u::rat and v::rat
assume upos: "0<u" and "u<1" and v: "v ∈ 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 upos ‹u < 1› v)
thus "u * v ∈ A"
by (force intro: preal_downwards_closed [OF A v] mult_pos_pos
upos vpos)
qed
next
show "A ⊆ ?lhs"
proof clarify
fix x::rat
assume x: "x ∈ 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 ∈ A" and xlessv: "x < v" ..
have vpos: "0<v" by (rule preal_imp_pos [OF A v])
show "∃u. 0 < u ∧ u < 1 ∧ (∃v∈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 "∃v'∈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 ∈ A" by fact
qed
qed
qed
qed
thus "1 * Abs_preal A = Abs_preal A"
by (simp add: preal_one_def preal_mult_def mult_set_def
rat_mem_preal A)
qed

instance preal :: comm_monoid_mult
by intro_classes (rule preal_mult_1)

"(z ∈ Rep_preal(R+S)) = (∃x ∈ Rep_preal R. ∃y ∈ Rep_preal S. z = x + y)"
done

lemma mem_Rep_preal_mult_iff:
"(z ∈ Rep_preal(R*S)) = (∃x ∈ Rep_preal R. ∃y ∈ Rep_preal S. z = x * y)"
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
done

lemma distrib_subset1:
"Rep_preal (w * (x + y)) ⊆ Rep_preal (w * x + w * y)"
done

assumes a: "a ∈ Rep_preal w"
and b: "b ∈ Rep_preal w"
and d: "d ∈ Rep_preal x"
and e: "e ∈ Rep_preal y"
shows "∃c ∈ 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"
show "a * d + b * e = ?c * (d + e)"
by (simp add: divide_inverse mult.assoc order_less_imp_not_eq2)
show "?c ∈ Rep_preal w"
proof (cases rule: linorder_le_cases)
assume "a ≤ b"
hence "?c ≤ b"
by (simp add: pos_divide_le_eq distrib_left mult_right_mono
order_less_imp_le)
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
next
assume "b ≤ a"
hence "?c ≤ a"
by (simp add: pos_divide_le_eq distrib_left 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) ⊆ Rep_preal (w * (x + y))"
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)"

instance preal :: comm_semiring

subsection‹Existence of Inverse, a Positive Real›

lemma mem_inv_set_ex:
assumes A: "cut A" shows "∃x y. 0 < x & x < y & inverse y ∉ A"
proof -
from preal_exists_bound [OF A]
obtain x where [simp]: "0<x" "x ∉ A" by blast
show ?thesis
proof (intro exI conjI)
show "0 < inverse (x+1)"
show "inverse(x+1) < inverse x"
show "inverse (inverse x) ∉ A"
qed
qed

text‹Part 1 of Dedekind sections definition›
lemma inverse_set_not_empty:
"cut A ==> {} ⊂ inverse_set A"
apply (insert mem_inv_set_ex [of A])
done

text‹Part 2 of Dedekind sections definition›

lemma preal_not_mem_inverse_set_Ex:
assumes A: "cut A"  shows "∃q. 0 < q & q ∉ inverse_set A"
proof -
from preal_nonempty [OF A]
obtain x where x: "x ∈ A" and  xpos [simp]: "0<x" ..
show ?thesis
proof (intro exI conjI)
show "0 < inverse x" by simp
show "inverse x ∉ 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 ∈ 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: "cut A"  shows "inverse_set A < {r. 0 < r}"
proof
show "inverse_set A ⊆ {r. 0 < r}"  by (force simp add: inverse_set_def)
next
show "inverse_set A ≠ {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:
"[|cut A; u ∈ inverse_set A; 0 < z; z < u|]
==> z ∈ inverse_set A"
apply (auto intro: order_less_trans)
done

text‹Part 4 of Dedekind sections definition›
lemma inverse_set_lemma4:
"[|cut A; y ∈ inverse_set A|] ==> ∃u ∈ inverse_set A. y < u"
apply (drule dense [of y])
apply (blast intro: order_less_trans)
done

lemma mem_inverse_set:
"cut A ==> cut (inverse_set A)"
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: "cut A"
and "∀x∈A. x + u ∈ A"
and "0 ≤ z"
shows "∃b∈A. b + (of_int z) * u ∈ A"
proof (cases z rule: int_cases)
case (nonneg n)
show ?thesis
proof (simp add: nonneg, induct n)
case 0
from preal_nonempty [OF A]
show ?case  by force
next
case (Suc k)
then obtain b where b: "b ∈ A" "b + of_nat k * u ∈ A" ..
hence "b + of_int (int k)*u + u ∈ A" by (simp add: assms)
thus ?case by (force simp add: algebra_simps b)
qed
next
case (neg n)
with assms show ?thesis by simp
qed

lemma Gleason9_34_contra:
assumes A: "cut A"
shows "[|∀x∈A. x + u ∈ A; 0 < u; 0 < y; y ∉ 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: "∀x∈A. x + (Fract c d) ∈ A"
and upos: "0 < Fract c d"
and ypos: "0 < Fract a b"
and notin: "Fract a b ∉ 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 ≤ Fract ?k 1 * (Fract c d)"
proof -
have "?thesis = ((a * d * b * d) ≤ c * b * (a * d * b * d))"
moreover
have "(1 * (a * d * b * d)) ≤ c * b * (a * d * b * d)"
by (rule mult_mono,
order_less_imp_le)
ultimately
show ?thesis by simp
qed
have k: "0 ≤ ?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 ∈ A"
and mem: "z + of_int ?k * Fract c d ∈ 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: "cut A"
and upos: "0 < u"
shows "∃r ∈ A. r + u ∉ A"
proof (rule ccontr, simp)
assume closed: "∀r∈A. r + u ∈ A"
from preal_exists_bound [OF A]
obtain y where y: "y ∉ 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: "cut A"
and x: "1 < x"
shows "∃r ∈ A. r*x ∉ A"
proof -
from preal_nonempty [OF A]
obtain y where y: "y ∈ A" and  ypos: "0<y" ..
show ?thesis
proof (rule classical)
assume "~(∃r∈A. r * x ∉ A)"
with y have ymem: "y * x ∈ 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∈A" and notin: "r + ?d ∉ 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 ≤ y + ?d)"
proof
assume le: "r + ?d ≤ y + ?d"
from ymem have yd: "y + ?d ∈ A" by (simp add: eq)
have "r + ?d ∈ 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"
using dpos less_divide_eq_1 by fastforce
have "r + ?d < r*x"
proof -
have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
also from ypos have "... = (r/y) * (y + ?d)"
by (simp only: algebra_simps divide_inverse, simp)
also have "... = r*x" using ypos
by simp
finally show "r + ?d < r*x" .
qed
with r notin rdpos
show "∃r∈A. r * x ∉ A" by (blast dest:  preal_downwards_closed [OF A])
qed
qed

subsection‹Existence of Inverse: Part 2›

lemma mem_Rep_preal_inverse_iff:
"(z ∈ Rep_preal(inverse R)) =
(0 < z ∧ (∃y. z < y ∧ inverse y ∉ Rep_preal R))"
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
done

lemma Rep_preal_one:
"Rep_preal 1 = {x. 0 < x ∧ x < 1}"

lemma subset_inverse_mult_lemma:
assumes xpos: "0 < x" and xless: "x < 1"
shows "∃r u y. 0 < r & r < y & inverse y ∉ Rep_preal R &
u ∈ 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 ∈ Rep_preal R"
and notin: "r * (inverse x) ∉ 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 ∈ 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
show "x/u < x/r" using xpos upos rpos
by (simp add: divide_inverse mult_less_cancel_left rless)
show "inverse (x / r) ∉ Rep_preal R" using notin
show "u ∈ Rep_preal R" by (rule u)
show "x = x / u * u" using upos
qed
qed

lemma subset_inverse_mult:
"Rep_preal 1 ⊆ Rep_preal(inverse R * R)"
apply (auto simp add: Bex_def Rep_preal_one 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 ∉ Rep_preal R"
and q: "q ∈ 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
also have "... ≤ 1" using rpos rless
finally show ?thesis .
qed

lemma inverse_mult_subset:
"Rep_preal(inverse R * R) ⊆ Rep_preal 1"
apply (auto simp add: Bex_def Rep_preal_one 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)"
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 ‹Gleason9_34››

lemma Rep_preal_self_subset: "Rep_preal (R) ⊆ Rep_preal(R + S)"
proof
fix r
assume r: "r ∈ 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 ∈ Rep_preal S" ..
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
have ry: "r+y ∈ Rep_preal(R + S)" using r y
show "r ∈ 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) ⊆ Rep_preal(R)"
proof -
from mem_Rep_preal_Ex
obtain y where y: "y ∈ 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 ∈ Rep_preal R" and notin: "r + y ∉ Rep_preal R" ..
have "r + y ∈ Rep_preal (R + S)" using r y
thus ?thesis using notin by blast
qed

lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) ≠ Rep_preal(R)"
by (insert Rep_preal_sum_not_subset, blast)

text‹at last, Gleason prop. 9-3.5(iii) page 123›
lemma preal_self_less_add_left: "(R::preal) < R + S"
apply (unfold preal_less_def less_le)
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
done

subsection‹Subtraction for Positive Reals›

text‹Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> ∃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 ==> {} ⊂ 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:
"∃q. 0 < q & q ∉ 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 (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 ⊆ ?rhs" by (auto simp add: diff_set_def)
show "?lhs ≠ ?rhs" using diff_set_nonempty by blast
qed

text‹Part 3 of Dedekind sections definition›
lemma diff_set_lemma3:
"[|R < S; u ∈ diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|]
==> z ∈ diff_set (Rep_preal S) (Rep_preal R)"
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 ∈ diff_set (Rep_preal S) (Rep_preal R)|]
==> ∃u ∈ diff_set (Rep_preal S) (Rep_preal R). y < u"
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)
done

lemma mem_diff_set:
"R < S ==> cut (diff_set (Rep_preal S) (Rep_preal R))"
apply (unfold 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 ∈ Rep_preal(S-R)) =
(∃x. 0 < x & 0 < z & x ∉ Rep_preal R & x + z ∈ Rep_preal S)"
apply (simp add: preal_diff_def mem_diff_set Rep_preal)
done

text‹proving that @{term "R + D ≤ S"}›

assumes Rless: "R < S"
and a: "a ∈ Rep_preal R"
and cb: "c + b ∈ Rep_preal S"
and "c ∉ Rep_preal R"
and "0 < b"
and "0 < c"
shows "a + b ∈ Rep_preal S"
proof -
have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
moreover
have "a < c" using assms by (blast intro: not_in_Rep_preal_ub )
ultimately show ?thesis
using assms by (simp add: preal_downwards_closed [OF Rep_preal cb])
qed

"R < (S::preal) ==> R + (S-R) ≤ S"
mem_Rep_preal_diff_iff)
done

subsection‹proving that @{term "S ≤ R + D"} --- trickier›

lemma lemma_sum_mem_Rep_preal_ex:
"x ∈ Rep_preal S ==> ∃e. 0 < e & x + e ∈ 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

assumes Rless: "R < S"
and x:     "x ∈ Rep_preal S"
and xnot: "x ∉  Rep_preal R"
shows "∃u v z. 0 < v & 0 < z & u ∈ Rep_preal R & z ∉ Rep_preal R &
z + v ∈ 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 ∈ Rep_preal S" by blast
from  Gleason9_34 [OF Rep_preal epos]
obtain r where r: "r ∈ Rep_preal R" and notin: "r + e ∉ Rep_preal R" ..
with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
obtain y where eq: "x = r+y" by auto
show ?thesis
proof (intro exI conjI)
show "r ∈ Rep_preal R" by (rule r)
show "r + e ∉ Rep_preal R" by (rule notin)
show "r + e + y ∈ Rep_preal S" using xe eq by (simp add: ac_simps)
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 ≤ R + (S-R)"
apply (case_tac "x ∈ Rep_preal R")
apply (cut_tac Rep_preal_self_subset [of R], force)
done

lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"

lemma less_add_left_Ex: "R < (S::preal) ==> ∃D. R + D = S"

lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
apply (rule_tac y1 = D in preal_add_commute [THEN subst])
done

lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"

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)"

lemma preal_add_less_cancel_left [simp]: "(T + (R::preal) < T + S) = (R < S)"

lemma preal_add_less_cancel_right [simp]: "((R::preal) + T < S + T) = (R < S)"

lemma preal_add_le_cancel_left [simp]: "(T + (R::preal) ≤ T + S) = (R ≤ S)"

lemma preal_add_le_cancel_right [simp]: "((R::preal) + T ≤ S + T) = (R ≤ S)"

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)"

proof
fix a b c :: preal
show "a ≤ b ⟹ c + a ≤ 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 ≠ {} ==> {} ⊂ (⋃X ∈ 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:
"∀X ∈ P. X ≤ Y ==> ∃q. 0 < q & q ∉ (⋃X ∈ P. Rep_preal(X))"
apply (cut_tac X = Y in Rep_preal_exists_bound)
done

lemma preal_sup_set_not_rat_set:
"∀X ∈ P. X ≤ Y ==> (⋃X ∈ 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 ≠ {}; ∀X ∈ P. X ≤ Y; u ∈ (⋃X ∈ P. Rep_preal(X)); 0 < z; z < u|]
==> z ∈ (⋃X ∈ 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 ≠ {}; ∀X ∈ P. X ≤ Y; y ∈ (⋃X ∈ P. Rep_preal(X)) |]
==> ∃u ∈ (⋃X ∈ P. Rep_preal(X)). y < u"
by (blast dest: Rep_preal [THEN preal_exists_greater])

lemma preal_sup:
"[|P ≠ {}; ∀X ∈ P. X ≤ Y|] ==> cut (⋃X ∈ P. Rep_preal(X))"
apply (unfold 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:
"[| ∀X ∈ P. X ≤ Y;  x ∈ P |] ==> x ≤ psup P"
apply (subgoal_tac "P ≠ {}")
apply (auto simp add: psup_def preal_sup)
done

lemma psup_le_ub: "[| P ≠ {}; ∀X ∈ P. X ≤ Y |] ==> psup P ≤ Y"
done

text‹Supremum property›
lemma preal_complete:
"[| P ≠ {}; ∀X ∈ P. X ≤ Y |] ==> (∃X ∈ P. Z < X) = (Z < psup P)"
apply (simp add: preal_less_def psup_def preal_sup)
apply (rename_tac U)
apply (cut_tac x = U and y = Z in linorder_less_linear)
done

section ‹Defining the Reals from the Positive Reals›

definition
realrel   ::  "((preal * preal) * (preal * preal)) set" where
"realrel = {p. ∃x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"

definition "Real = UNIV//realrel"

typedef real = Real
morphisms Rep_Real Abs_Real
unfolding Real_def by (auto simp add: quotient_def)

definition
(** these don't use the overloaded "real" function: users don't see them **)
real_of_preal :: "preal => real" where
"real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"

instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
begin

definition
real_zero_def: "0 = Abs_Real(realrel``{(1, 1)})"

definition
real_one_def: "1 = Abs_Real(realrel``{(1 + 1, 1)})"

definition
the_elem (⋃(x,y) ∈ Rep_Real(z). ⋃(u,v) ∈ Rep_Real(w).
{ Abs_Real(realrel``{(x+u, y+v)}) })"

definition
real_minus_def: "- r =  the_elem (⋃(x,y) ∈ Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"

definition
real_diff_def: "r - (s::real) = r + - s"

definition
real_mult_def:
"z * w =
the_elem (⋃(x,y) ∈ Rep_Real(z). ⋃(u,v) ∈ Rep_Real(w).
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"

definition
real_inverse_def: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"

definition
real_divide_def: "R div (S::real) = R * inverse S"

definition
real_le_def: "z ≤ (w::real) ⟷
(∃x y u v. x+v ≤ u+y & (x,y) ∈ Rep_Real z & (u,v) ∈ Rep_Real w)"

definition
real_less_def: "x < (y::real) ⟷ x ≤ y ∧ x ≠ y"

definition
real_abs_def: "¦r::real¦ = (if r < 0 then - r else r)"

definition
real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"

instance ..

end

subsection ‹Equivalence relation over positive reals›

lemma preal_trans_lemma:
assumes "x + y1 = x1 + y"
and "x + y2 = x2 + y"
shows "x1 + y2 = x2 + (y1::preal)"
proof -
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: ac_simps)
also have "... = (x2 + y) + x1"  by (simp add: assms)
also have "... = x2 + (x1 + y)"  by (simp add: ac_simps)
also have "... = x2 + (x + y1)"  by (simp add: assms)
also have "... = (x2 + y1) + x"  by (simp add: ac_simps)
finally have "(x1 + y2) + x = (x2 + y1) + x" .
qed

lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) ∈ realrel) = (x1 + y2 = x2 + y1)"

lemma equiv_realrel: "equiv UNIV realrel"
apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def)
apply (blast dest: preal_trans_lemma)
done

text‹Reduces equality of equivalence classes to the @{term realrel} relation:
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) ∈ realrel)"}›
lemmas equiv_realrel_iff =
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]

declare equiv_realrel_iff [simp]

lemma realrel_in_real [simp]: "realrel``{(x,y)} ∈ Real"
by (simp add: Real_def realrel_def quotient_def, blast)

declare Abs_Real_inject [simp]
declare Abs_Real_inverse [simp]

text‹Case analysis on the representation of a real number as an equivalence
class of pairs of positive reals.›
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
"(⋀x y. z = Abs_Real(realrel``{(x,y)}) ⟹ P) ⟹ P"
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
apply (drule arg_cong [where f=Abs_Real])
done

"[|a + ba = aa + b; ab + bc = ac + bb|]
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
apply (rule add.left_commute [of ab, THEN ssubst])
done

"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
Abs_Real (realrel``{(x+u, y+v)})"
proof -
have "(λz w. (λ(x,y). (λ(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
respects2 realrel"
thus ?thesis
UN_equiv_class2 [OF equiv_realrel equiv_realrel])
qed

lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
proof -
have "(λ(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
thus ?thesis
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
qed

proof
fix x y z :: real
show "(x + y) + z = x + (y + z)"
show "x + y = y + x"
show "0 + x = x"
show "- x + x = 0"
show "x - y = x + - y"
qed

subsection ‹Multiplication›

lemma real_mult_congruent2_lemma:
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
x * x1 + y * y1 + (x * y2 + y * x2) =
x * x2 + y * y2 + (x * y1 + y * x1)"
done

lemma real_mult_congruent2:
"(%p1 p2.
(%(x1,y1). (%(x2,y2).
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
respects2 realrel"
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
done

lemma real_mult:
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])

lemma real_mult_commute: "(z::real) * w = w * z"
by (cases z, cases w, simp add: real_mult ac_simps)

lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
apply (cases z1, cases z2, cases z3)
done

lemma real_mult_1: "(1::real) * z = z"
apply (cases z)
apply (simp add: real_mult real_one_def algebra_simps)
done

lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
apply (cases z1, cases z2, cases w)
done

text‹one and zero are distinct›
lemma real_zero_not_eq_one: "0 ≠ (1::real)"
proof -
have "(1::preal) < 1 + 1"
then show ?thesis
by (simp add: real_zero_def real_one_def neq_iff)
qed

instance real :: comm_ring_1
proof
fix x y z :: real
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
show "x * y = y * x" by (rule real_mult_commute)
show "1 * x = x" by (rule real_mult_1)
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
show "0 ≠ (1::real)" by (rule real_zero_not_eq_one)
qed

subsection ‹Inverse and Division›

lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"

text‹Instead of using an existential quantifier and constructing the inverse
within the proof, we could define the inverse explicitly.›

lemma real_mult_inverse_left_ex: "x ≠ 0 ==> ∃y. y*x = (1::real)"
apply (simp add: real_zero_def real_one_def, cases x)
apply (cut_tac x = xa and y = y in linorder_less_linear)
apply (rule_tac
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
in exI)
apply (rule_tac [2]
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
in exI)
apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
done

lemma real_mult_inverse_left: "x ≠ 0 ==> inverse(x)*x = (1::real)"
apply (drule real_mult_inverse_left_ex, safe)
apply (rule theI, assumption, rename_tac z)
apply (subgoal_tac "(z * x) * y = z * (x * y)")
apply (rule mult.assoc)
done

subsection‹The Real Numbers form a Field›

instance real :: field
proof
fix x y z :: real
show "x ≠ 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
show "x / y = x * inverse y" by (simp add: real_divide_def)
show "inverse 0 = (0::real)" by (simp add: real_inverse_def)
qed

subsection‹The ‹≤› Ordering›

lemma real_le_refl: "w ≤ (w::real)"
by (cases w, force simp add: real_le_def)

text‹The arithmetic decision procedure is not set up for type preal.
This lemma is currently unused, but it could simplify the proofs of the
following two lemmas.›
lemma preal_eq_le_imp_le:
assumes eq: "a+b = c+d" and le: "c ≤ a"
shows "b ≤ (d::preal)"
proof -
from le have "c+d ≤ a+d" by simp
hence "a+b ≤ a+d" by (simp add: eq)
thus "b ≤ d" by simp
qed

lemma real_le_lemma:
assumes l: "u1 + v2 ≤ u2 + v1"
and "x1 + v1 = u1 + y1"
and "x2 + v2 = u2 + y2"
shows "x1 + y2 ≤ x2 + (y1::preal)"
proof -
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: assms)
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: ac_simps)
also have "... ≤ (x2+y1) + (u2+v1)" by (simp add: assms)
finally show ?thesis by simp
qed

lemma real_le:
"(Abs_Real(realrel``{(x1,y1)}) ≤ Abs_Real(realrel``{(x2,y2)})) =
(x1 + y2 ≤ x2 + y1)"
apply (auto intro: real_le_lemma)
done

lemma real_le_antisym: "[| z ≤ w; w ≤ z |] ==> z = (w::real)"
by (cases z, cases w, simp add: real_le)

lemma real_trans_lemma:
assumes "x + v ≤ u + y"
and "u + v' ≤ u' + v"
and "x2 + v2 = u2 + y2"
shows "x + v' ≤ u' + (y::preal)"
proof -
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: ac_simps)
also have "... ≤ (u+y) + (u+v')" by (simp add: assms)
also have "... ≤ (u+y) + (u'+v)" by (simp add: assms)
also have "... = (u'+y) + (u+v)"  by (simp add: ac_simps)
finally show ?thesis by simp
qed

lemma real_le_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::real)"
apply (cases i, cases j, cases k)
apply (blast intro: real_trans_lemma)
done

instance real :: order
proof
fix u v :: real
show "u < v ⟷ u ≤ v ∧ ¬ v ≤ u"
by (auto simp add: real_less_def intro: real_le_antisym)
qed (assumption | rule real_le_refl real_le_trans real_le_antisym)+

(* Axiom 'linorder_linear' of class 'linorder': *)
lemma real_le_linear: "(z::real) ≤ w | w ≤ z"
apply (cases z, cases w)
apply (auto simp add: real_le real_zero_def ac_simps)
done

instance real :: linorder
by (intro_classes, rule real_le_linear)

lemma real_le_eq_diff: "(x ≤ y) = (x-y ≤ (0::real))"
apply (cases x, cases y)
ac_simps)
done

assumes le: "x ≤ y" shows "z + x ≤ z + (y::real)"
proof -
have "z + x - (z + y) = (z + -z) + (x - y)"
with le show ?thesis
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"])
qed

lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S])

lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S])

lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
apply (cases x, cases y)
apply (simp add: linorder_not_le [where 'a = real, symmetric]
linorder_not_le [where 'a = preal]
real_zero_def real_le real_mult)
― ‹Reduce to the (simpler) ‹≤› relation›
done

lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
apply (rule real_sum_gt_zero_less)
apply (drule real_less_sum_gt_zero [of x y])
apply (drule real_mult_order, assumption)
done

instantiation real :: distrib_lattice
begin

definition
"(inf :: real ⇒ real ⇒ real) = min"

definition
"(sup :: real ⇒ real ⇒ real) = max"

instance
by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)

end

subsection‹The Reals Form an Ordered Field›

instance real :: linordered_field
proof
fix x y z :: real
show "x ≤ y ==> z + x ≤ z + y" by (rule real_add_left_mono)
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
show "¦x¦ = (if x < 0 then -x else x)" by (simp only: real_abs_def)
show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
by (simp only: real_sgn_def)
qed

text‹The function @{term real_of_preal} requires many proofs, but it seems
to be essential for proving completeness of the reals from that of the
positive reals.›

"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"

lemma real_of_preal_mult:
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
by (simp add: real_of_preal_def real_mult algebra_simps)

text‹Gleason prop 9-4.4 p 127›
lemma real_of_preal_trichotomy:
"∃m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
apply (simp add: real_of_preal_def real_zero_def, cases x)
apply (auto simp add: real_minus ac_simps)
apply (cut_tac x = xa and y = y in linorder_less_linear)
done

lemma real_of_preal_leD:
"real_of_preal m1 ≤ real_of_preal m2 ==> m1 ≤ m2"

lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])

lemma real_of_preal_lessD:
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])

lemma real_of_preal_less_iff [simp]:
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
by (blast intro: real_of_preal_lessI real_of_preal_lessD)

lemma real_of_preal_le_iff:
"(real_of_preal m1 ≤ real_of_preal m2) = (m1 ≤ m2)"

lemma real_of_preal_zero_less: "0 < real_of_preal m"
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def ac_simps neq_iff)
done

lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"

lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
proof -
from real_of_preal_minus_less_zero
show ?thesis by (blast dest: order_less_trans)
qed

lemma real_gt_zero_preal_Ex: "(0 < x) = (∃y. x = real_of_preal y)"
apply (cut_tac x = x in real_of_preal_trichotomy)
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
done

lemma real_gt_preal_preal_Ex:
"real_of_preal z < x ==> ∃y. x = real_of_preal y"
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
intro: real_gt_zero_preal_Ex [THEN iffD1])

lemma real_ge_preal_preal_Ex:
"real_of_preal z ≤ x ==> ∃y. x = real_of_preal y"
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)

lemma real_less_all_preal: "y ≤ 0 ==> ∀x. y < real_of_preal x"
by (auto elim: order_le_imp_less_or_eq [THEN disjE]
intro: real_of_preal_zero_less [THEN [2] order_less_trans]

lemma real_less_all_real2: "~ 0 < y ==> ∀x. y < real_of_preal x"
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])

subsection ‹Completeness of Positive Reals›

text ‹
Supremum property for the set of positive reals

Let ‹P› be a non-empty set of positive reals, with an upper
bound ‹y›.  Then ‹P› has a least upper bound
(written ‹S›).

FIXME: Can the premise be weakened to ‹∀x ∈ P. x≤ y›?
›

lemma posreal_complete:
assumes positive_P: "∀x ∈ P. (0::real) < x"
and not_empty_P: "∃x. x ∈ P"
and upper_bound_Ex: "∃y. ∀x ∈ P. x<y"
shows "∃S. ∀y. (∃x ∈ P. y < x) = (y < S)"
proof (rule exI, rule allI)
fix y
let ?pP = "{w. real_of_preal w ∈ P}"

show "(∃x∈P. y < x) = (y < real_of_preal (psup ?pP))"
proof (cases "0 < y")
assume neg_y: "¬ 0 < y"
show ?thesis
proof
assume "∃x∈P. y < x"
have "∀x. y < real_of_preal x"
using neg_y by (rule real_less_all_real2)
thus "y < real_of_preal (psup ?pP)" ..
next
assume "y < real_of_preal (psup ?pP)"
obtain "x" where x_in_P: "x ∈ P" using not_empty_P ..
hence "0 < x" using positive_P by simp
hence "y < x" using neg_y by simp
thus "∃x ∈ P. y < x" using x_in_P ..
qed
next
assume pos_y: "0 < y"

then obtain py where y_is_py: "y = real_of_preal py"

obtain a where "a ∈ P" using not_empty_P ..
with positive_P have a_pos: "0 < a" ..
then obtain pa where "a = real_of_preal pa"
hence "pa ∈ ?pP" using ‹a ∈ P› by auto
hence pP_not_empty: "?pP ≠ {}" by auto

obtain sup where sup: "∀x ∈ P. x < sup"
using upper_bound_Ex ..
from this and ‹a ∈ P› have "a < sup" ..
hence "0 < sup" using a_pos by arith
then obtain possup where "sup = real_of_preal possup"
hence "∀X ∈ ?pP. X ≤ possup"
using sup by (auto simp add: real_of_preal_lessI)
with pP_not_empty have psup: "⋀Z. (∃X ∈ ?pP. Z < X) = (Z < psup ?pP)"
by (rule preal_complete)

show ?thesis
proof
assume "∃x ∈ P. y < x"
then obtain x where x_in_P: "x ∈ P" and y_less_x: "y < x" ..
hence "0 < x" using pos_y by arith
then obtain px where x_is_px: "x = real_of_preal px"

have py_less_X: "∃X ∈ ?pP. py < X"
proof
show "py < px" using y_is_py and x_is_px and y_less_x
show "px ∈ ?pP" using x_in_P and x_is_px by simp
qed

have "(∃X ∈ ?pP. py < X) ==> (py < psup ?pP)"
using psup by simp
hence "py < psup ?pP" using py_less_X by simp
thus "y < real_of_preal (psup {w. real_of_preal w ∈ P})"
using y_is_py and pos_y by (simp add: real_of_preal_lessI)
next
assume y_less_psup: "y < real_of_preal (psup ?pP)"

hence "py < psup ?pP" using y_is_py
then obtain "X" where py_less_X: "py < X" and X_in_pP: "X ∈ ?pP"
using psup by auto
then obtain x where x_is_X: "x = real_of_preal X"
hence "y < x" using py_less_X and y_is_py

moreover have "x ∈ P" using x_is_X and X_in_pP by simp

ultimately show "∃ x ∈ P. y < x" ..
qed
qed
qed

text ‹
\medskip Completeness
›

lemma reals_complete:
fixes S :: "real set"
assumes notempty_S: "∃X. X ∈ S"
and exists_Ub: "bdd_above S"
shows "∃x. (∀s∈S. s ≤ x) ∧ (∀y. (∀s∈S. s ≤ y) ⟶ x ≤ y)"
proof -
obtain X where X_in_S: "X ∈ S" using notempty_S ..
obtain Y where Y_isUb: "∀s∈S. s ≤ Y"
using exists_Ub by (auto simp: bdd_above_def)
let ?SHIFT = "{z. ∃x ∈S. z = x + (-X) + 1} ∩ {x. 0 < x}"

{
fix x
assume S_le_x: "∀s∈S. s ≤ x"
{
fix s
assume "s ∈ {z. ∃x∈S. z = x + - X + 1}"
hence "∃ x ∈ S. s = x + -X + 1" ..
then obtain x1 where x1: "x1 ∈ S" "s = x1 + (-X) + 1" ..
then have "x1 ≤ x" using S_le_x by simp
with x1 have "s ≤ x + - X + 1" by arith
}
then have "∀s∈?SHIFT. s ≤ x + (-X) + 1"
by auto
} note S_Ub_is_SHIFT_Ub = this

have *: "∀s∈?SHIFT. s ≤ Y + (-X) + 1" using Y_isUb by (rule S_Ub_is_SHIFT_Ub)
have "∀s∈?SHIFT. s < Y + (-X) + 2"
proof
fix s assume "s∈?SHIFT"
with * have "s ≤ Y + (-X) + 1" by simp
also have "… < Y + (-X) + 2" by simp
finally show "s < Y + (-X) + 2" .
qed
moreover have "∀y ∈ ?SHIFT. 0 < y" by auto
moreover have shifted_not_empty: "∃u. u ∈ ?SHIFT"
using X_in_S and Y_isUb by auto
ultimately obtain t where t_is_Lub: "∀y. (∃x∈?SHIFT. y < x) = (y < t)"
using posreal_complete [of ?SHIFT] unfolding bdd_above_def by blast

show ?thesis
proof
show "(∀s∈S. s ≤ (t + X + (-1))) ∧ (∀y. (∀s∈S. s ≤ y) ⟶ (t + X + (-1)) ≤ y)"
proof safe
fix x
assume "∀s∈S. s ≤ x"
hence "∀s∈?SHIFT. s ≤ x + (-X) + 1"
using S_Ub_is_SHIFT_Ub by simp
then have "¬ x + (-X) + 1 < t"
by (subst t_is_Lub[rule_format, symmetric]) (simp add: not_less)
thus "t + X + -1 ≤ x" by arith
next
fix y
assume y_in_S: "y ∈ S"
obtain "u" where u_in_shift: "u ∈ ?SHIFT" using shifted_not_empty ..
hence "∃ x ∈ S. u = x + - X + 1" by simp
then obtain "x" where x_and_u: "u = x + - X + 1" ..
have u_le_t: "u ≤ t"
proof (rule dense_le)
fix x assume "x < u" then have "x < t"
using u_in_shift t_is_Lub by auto
then show "x ≤ t"  by simp
qed

show "y ≤ t + X + -1"
proof cases
assume "y ≤ x"
moreover have "x = u + X + - 1" using x_and_u by arith
moreover have "u + X + - 1  ≤ t + X + -1" using u_le_t by arith
ultimately show "y  ≤ t + X + -1" by arith
next
assume "~(y ≤ x)"
hence x_less_y: "x < y" by arith

have "x + (-X) + 1 ∈ ?SHIFT" using x_and_u and u_in_shift by simp
hence "0 < x + (-X) + 1" by simp
hence "0 < y + (-X) + 1" using x_less_y by arith
hence *: "y + (-X) + 1 ∈ ?SHIFT" using y_in_S by simp
have "y + (-X) + 1 ≤ t"
proof (rule dense_le)
fix x assume "x < y + (-X) + 1" then have "x < t"
using * t_is_Lub by auto
then show "x ≤ t"  by simp
qed
thus ?thesis by simp
qed
qed
qed
qed

subsection ‹The Archimedean Property of the Reals›

theorem reals_Archimedean:
fixes x :: real
assumes x_pos: "0 < x"
shows "∃n. inverse (of_nat (Suc n)) < x"
proof (rule ccontr)
assume contr: "¬ ?thesis"
have "∀n. x * of_nat (Suc n) <= 1"
proof
fix n
from contr have "x ≤ inverse (of_nat (Suc n))"
hence "x ≤ (1 / (of_nat (Suc n)))"
moreover have "(0::real) ≤ of_nat (Suc n)"
by (rule of_nat_0_le_iff)
ultimately have "x * of_nat (Suc n) ≤ (1 / of_nat (Suc n)) * of_nat (Suc n)"
by (rule mult_right_mono)
thus "x * of_nat (Suc n) ≤ 1" by (simp del: of_nat_Suc)
qed
hence 2: "bdd_above {z. ∃n. z = x * (of_nat (Suc n))}"
by (auto intro!: bdd_aboveI[of _ 1])
have 1: "∃X. X ∈ {z. ∃n. z = x* (of_nat (Suc n))}" by auto
obtain t where
upper: "⋀z. z ∈ {z. ∃n. z = x * of_nat (Suc n)} ⟹ z ≤ t" and
least: "⋀y. (⋀a. a ∈ {z. ∃n. z = x * of_nat (Suc n)} ⟹ a ≤ y) ⟹ t ≤ y"
using reals_complete[OF 1 2] by auto

have "t ≤ t + - x"
proof (rule least)
fix a assume a: "a ∈ {z. ∃n. z = x * (of_nat (Suc n))}"
have "∀n::nat. x * of_nat n ≤ t + - x"
proof
fix n
have "x * of_nat (Suc n) ≤ t"
hence  "x * (of_nat n) + x ≤ t"
thus  "x * (of_nat n) ≤ t + - x" by arith
qed    hence "∀m. x * of_nat (Suc m) ≤ t + - x" by (simp del: of_nat_Suc)
with a show "a ≤ t + - x"
by auto
qed
thus False using x_pos by arith
qed

text ‹
There must be other proofs, e.g. ‹Suc› of the largest
integer in the cut representing ‹x›.
›

lemma reals_Archimedean2: "∃n. (x::real) < of_nat (n::nat)"
proof cases
assume "x ≤ 0"
hence "x < of_nat (1::nat)" by simp
thus ?thesis ..
next
assume "¬ x ≤ 0"
hence x_greater_zero: "0 < x" by simp
hence "0 < inverse x" by simp
then obtain n where "inverse (of_nat (Suc n)) < inverse x"
using reals_Archimedean by blast
hence "inverse (of_nat (Suc n)) * x < inverse x * x"
using x_greater_zero by (rule mult_strict_right_mono)
hence "inverse (of_nat (Suc n)) * x < 1"
using x_greater_zero by simp
hence "of_nat (Suc n) * (inverse (of_nat (Suc n)) * x) < of_nat (Suc n) * 1"
by (rule mult_strict_left_mono) (simp del: of_nat_Suc)
hence "x < of_nat (Suc n)"
by (simp add: algebra_simps del: of_nat_Suc)
thus "∃(n::nat). x < of_nat n" ..
qed

instance real :: archimedean_field
proof
fix r :: real
obtain n :: nat where "r < of_nat n"
using reals_Archimedean2 ..
then have "r ≤ of_int (int n)"
by simp
then show "∃z. r ≤ of_int z" ..
qed

end
```