src/HOL/Real/RealDef.thy
 author paulson Wed, 28 Jan 2004 17:01:01 +0100 changeset 14369 c50188fe6366 parent 14365 3d4df8c166ae child 14378 69c4d5997669 permissions -rw-r--r--
tidying up arithmetic for the hyperreals
```
(*  Title       : Real/RealDef.thy
ID          : \$Id\$
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
Description : The reals
*)

theory RealDef = PReal:

constdefs
realrel   ::  "((preal * preal) * (preal * preal)) set"
"realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"

typedef (REAL)  real = "UNIV//realrel"

instance real :: ord ..
instance real :: zero ..
instance real :: one ..
instance real :: plus ..
instance real :: times ..
instance real :: minus ..
instance real :: inverse ..

consts
(*Overloaded constants denoting the Nat and Real subsets of enclosing
types such as hypreal and complex*)
Nats  :: "'a set"
Reals :: "'a set"

(*overloaded constant for injecting other types into "real"*)
real :: "'a => real"

real_zero_def:
"0 == Abs_REAL(realrel``{(preal_of_rat 1, preal_of_rat 1)})"

real_one_def:
"1 == Abs_REAL(realrel``
{(preal_of_rat 1 + preal_of_rat 1,
preal_of_rat 1)})"

real_minus_def:
"- R ==  Abs_REAL(UN (x,y):Rep_REAL(R). realrel``{(y,x)})"

real_diff_def:
"R - (S::real) == R + - S"

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

real_divide_def:
"R / (S::real) == R * inverse S"

constdefs

(** these don't use the overloaded "real" function: users don't see them **)

real_of_preal :: "preal => real"
"real_of_preal m     ==
Abs_REAL(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})"

"P+Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"

real_mult_def:
"P*Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})
p2) p1)"

real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)"

real_le_def:
"P \<le> (Q::real) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
(x1,y1) \<in> Rep_REAL(P) & (x2,y2) \<in> Rep_REAL(Q)"

real_abs_def:  "abs (r::real) == (if 0 \<le> r then r else -r)"

syntax (xsymbols)
Reals     :: "'a set"                   ("\<real>")
Nats      :: "'a set"                   ("\<nat>")

real_of_int_def:
"real z == Abs_REAL(\<Union>(i,j) \<in> Rep_Integ z. realrel ``
{(preal_of_rat(rat(int(Suc i))),
preal_of_rat(rat(int(Suc j))))})"

real_of_nat_def:   "real n == real (int n)"

subsection{*Proving that realrel is an equivalence relation*}

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: preal_add_ac)
also have "... = (x2 + y) + x1"  by (simp add: prems)
also have "... = x2 + (x1 + y)"  by (simp add: preal_add_ac)
also have "... = x2 + (x + y1)"  by (simp add: prems)
also have "... = (x2 + y1) + x"  by (simp add: preal_add_ac)
finally have "(x1 + y2) + x = (x2 + y1) + x" .
qed

lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"
by (unfold realrel_def, blast)

lemma realrel_refl: "(x,x): realrel"
apply (case_tac "x")
done

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

(* (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 (unfold REAL_def realrel_def quotient_def, blast)

lemma inj_on_Abs_REAL: "inj_on Abs_REAL REAL"
apply (rule inj_on_inverseI)
apply (erule Abs_REAL_inverse)
done

declare inj_on_Abs_REAL [THEN inj_on_iff, simp]
declare Abs_REAL_inverse [simp]

lemmas eq_realrelD = equiv_realrel [THEN [2] eq_equiv_class]

lemma inj_Rep_REAL: "inj Rep_REAL"
apply (rule inj_on_inverseI)
apply (rule Rep_REAL_inverse)
done

(** real_of_preal: the injection from preal to real **)
lemma inj_real_of_preal: "inj(real_of_preal)"
apply (rule inj_onI)
apply (unfold real_of_preal_def)
apply (drule inj_on_Abs_REAL [THEN inj_onD])
apply (rule realrel_in_real)+
apply (drule eq_equiv_class)
apply (rule equiv_realrel, blast)
done

lemma eq_Abs_REAL:
"(!!x y. z = Abs_REAL(realrel``{(x,y)}) ==> P) ==> P"
apply (rule_tac x1 = z in Rep_REAL [unfolded REAL_def, THEN quotientE])
apply (drule_tac f = Abs_REAL in arg_cong)
apply (case_tac "x")
done

lemma real_eq_iff:
"[|(x1,y1) \<in> Rep_REAL w; (x2,y2) \<in> Rep_REAL z|]
==> (z = w) = (x1+y2 = x2+y1)"
apply (insert quotient_eq_iff
[OF equiv_realrel,
of "Rep_REAL w" "Rep_REAL z" "(x1,y1)" "(x2,y2)"])
apply (simp add: Rep_REAL [unfolded REAL_def] Rep_REAL_inject eq_commute)
done

lemma mem_REAL_imp_eq:
"[|R \<in> REAL; (x1,y1) \<in> R; (x2,y2) \<in> R|] ==> x1+y2 = x2+y1"
apply (auto simp add: REAL_def realrel_def quotient_def)
apply (blast dest: preal_trans_lemma)
done

lemma Rep_REAL_cancel_right:
"((x + z, y + z) \<in> Rep_REAL R) = ((x, y) \<in> Rep_REAL R)"
apply (rule_tac z = R in eq_Abs_REAL, simp)
apply (rename_tac u v)
apply (subgoal_tac "(u + (y + z) = x + z + v) = ((u + y) + z = (x + v) + z)")
done

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

"Abs_REAL(realrel``{(x1,y1)}) + Abs_REAL(realrel``{(x2,y2)}) =
Abs_REAL(realrel``{(x1+x2, y1+y2)})"
apply (subst equiv_realrel [THEN UN_equiv_class2])
done

lemma real_add_commute: "(z::real) + w = w + z"
apply (rule_tac z = z in eq_Abs_REAL)
apply (rule_tac z = w in eq_Abs_REAL)
done

lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
apply (rule_tac z = z1 in eq_Abs_REAL)
apply (rule_tac z = z2 in eq_Abs_REAL)
apply (rule_tac z = z3 in eq_Abs_REAL)
done

lemma real_add_zero_left: "(0::real) + z = z"
apply (unfold real_of_preal_def real_zero_def)
apply (rule_tac z = z in eq_Abs_REAL)
done

lemma real_add_zero_right: "z + (0::real) = z"

instance real :: plus_ac0
by (intro_classes,
(assumption |

lemma real_minus_congruent:
"congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)"
apply (unfold congruent_def, clarify)
done

lemma real_minus:
"- (Abs_REAL(realrel``{(x,y)})) = Abs_REAL(realrel `` {(y,x)})"
apply (unfold real_minus_def)
apply (rule_tac f = Abs_REAL in arg_cong)
apply (simp add: realrel_in_real [THEN Abs_REAL_inverse]
UN_equiv_class [OF equiv_realrel real_minus_congruent])
done

lemma real_add_minus_left: "(-z) + z = (0::real)"
apply (unfold real_zero_def)
apply (rule_tac z = z in eq_Abs_REAL)
done

subsection{*Congruence property for multiplication*}

lemma real_mult_congruent2_lemma:
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
x * x1 + y * y1 + (x * y2 + x2 * y) =
x * x2 + y * y2 + (x * y1 + x1 * y)"
apply (rule preal_mult_commute [THEN subst])
apply (rule_tac y1 = x2 in preal_mult_commute [THEN subst])
done

lemma real_mult_congruent2:
"congruent2 realrel (%p1 p2.
(%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"
apply (rule equiv_realrel [THEN congruent2_commuteI], clarify)
apply (unfold split_def)
done

lemma real_mult:
"Abs_REAL((realrel``{(x1,y1)})) * Abs_REAL((realrel``{(x2,y2)})) =
Abs_REAL(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})"
apply (unfold real_mult_def)
apply (simp add: equiv_realrel [THEN UN_equiv_class2] real_mult_congruent2)
done

lemma real_mult_commute: "(z::real) * w = w * z"
apply (rule_tac z = z in eq_Abs_REAL)
apply (rule_tac z = w in eq_Abs_REAL)
done

lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
apply (rule_tac z = z1 in eq_Abs_REAL)
apply (rule_tac z = z2 in eq_Abs_REAL)
apply (rule_tac z = z3 in eq_Abs_REAL)
done

lemma real_mult_1: "(1::real) * z = z"
apply (unfold real_one_def)
apply (rule_tac z = z in eq_Abs_REAL)
done

lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
apply (rule_tac z = z1 in eq_Abs_REAL)
apply (rule_tac z = z2 in eq_Abs_REAL)
apply (rule_tac z = w in eq_Abs_REAL)
done

text{*one and zero are distinct*}
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
apply (subgoal_tac "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1")
apply (unfold real_zero_def real_one_def)
done

subsection{*existence of inverse*}

lemma real_zero_iff: "Abs_REAL (realrel `` {(x, x)}) = 0"
apply (unfold real_zero_def)
done

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 \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
apply (unfold real_zero_def real_one_def)
apply (rule_tac z = x in eq_Abs_REAL)
apply (cut_tac x = xa and y = y in linorder_less_linear)
apply (rule_tac
x = "Abs_REAL (realrel `` { (preal_of_rat 1,
inverse (D) + preal_of_rat 1)}) "
in exI)
apply (rule_tac [2]
x = "Abs_REAL (realrel `` { (inverse (D) + preal_of_rat 1,
preal_of_rat 1)})"
in exI)
apply (auto simp add: real_mult preal_mult_1_right
done

lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
apply (unfold real_inverse_def)
apply (frule real_mult_inverse_left_ex, safe)
apply (rule someI2, auto)
done

subsection{*The Real Numbers form a Field*}

instance real :: field
proof
fix x y z :: real
show "(x + y) + z = x + (y + z)" by (rule real_add_assoc)
show "x + y = y + x" by (rule real_add_commute)
show "0 + x = x" by simp
show "- x + x = 0" by (rule real_add_minus_left)
show "x - y = x + (-y)" by (simp add: real_diff_def)
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 (simp add: real_add_mult_distrib)
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def)
assume eq: "z+x = z+y"
hence "(-z + z) + x = (-z + z) + y" by (simp only: eq real_add_assoc)
qed

text{*Inverse of zero!  Useful to simplify certain equations*}

lemma INVERSE_ZERO: "inverse 0 = (0::real)"
apply (unfold real_inverse_def)
apply (rule someI2)
done

lemma DIVISION_BY_ZERO: "a / (0::real) = 0"

instance real :: division_by_zero
proof
fix x :: real
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
show "x/0 = 0" by (rule DIVISION_BY_ZERO)
qed

(*Pull negations out*)
declare minus_mult_right [symmetric, simp]
minus_mult_left [symmetric, simp]

lemma real_mult_1_right: "z * (1::real) = z"
by (rule Ring_and_Field.mult_1_right)

subsection{*The @{text "\<le>"} Ordering*}

lemma real_le_refl: "w \<le> (w::real)"
apply (rule_tac z = w in eq_Abs_REAL)
done

lemma real_le_trans_lemma:
assumes le1: "x1 + y2 \<le> x2 + y1"
and le2: "u1 + v2 \<le> u2 + v1"
and eq: "x2 + v1 = u1 + y2"
shows "x1 + v2 + u1 + y2 \<le> u2 + u1 + y2 + (y1::preal)"
proof -
have "x1 + v2 + u1 + y2 = (x1 + y2) + (u1 + v2)" by (simp add: preal_add_ac)
also have "... \<le> (x2 + y1) + (u1 + v2)"
also have "... \<le> (x2 + y1) + (u2 + v1)"
also have "... = (x2 + v1) + (u2 + y1)" by (simp add: preal_add_ac)
also have "... = (u1 + y2) + (u2 + y1)" by (simp add: prems)
also have "... = u2 + u1 + y2 + y1" by (simp add: preal_add_ac)
finally show ?thesis .
qed

lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
apply (rename_tac x1 u1 y1 v1 x2 u2 y2 v2)
apply (drule mem_REAL_imp_eq [OF Rep_REAL], assumption)
apply (rule_tac x=x1 in exI)
apply (rule_tac x=y1 in exI)
apply (rule_tac x="u2 + (x2 + v1)" in exI)
apply (rule_tac x="v2 + (u1 + y2)" in exI)
done

lemma real_le_anti_sym_lemma:
assumes le1: "x1 + y2 \<le> x2 + y1"
and le2: "u1 + v2 \<le> u2 + v1"
and eq1: "x1 + v2 = u2 + y1"
and eq2: "x2 + v1 = u1 + y2"
shows "x2 + y1 = x1 + (y2::preal)"
proof (rule order_antisym)
show "x1 + y2 \<le> x2 + y1" .
have "(x2 + y1) + (u1+u2) = x2 + u1 + (u2 + y1)" by (simp add: preal_add_ac)
also have "... = x2 + u1 + (x1 + v2)" by (simp add: prems)
also have "... = (x2 + x1) + (u1 + v2)" by (simp add: preal_add_ac)
also have "... \<le> (x2 + x1) + (u2 + v1)"
also have "... = (x1 + u2) + (x2 + v1)" by (simp add: preal_add_ac)
also have "... = (x1 + u2) + (u1 + y2)" by (simp add: prems)
also have "... = (x1 + y2) + (u1 + u2)" by (simp add: preal_add_ac)
finally show "x2 + y1 \<le> x1 + y2" by (simp add: preal_add_le_cancel_right)
qed

lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
apply (rule real_eq_iff [THEN iffD2], assumption+)
apply (drule mem_REAL_imp_eq [OF Rep_REAL], assumption)+
apply (blast intro: real_le_anti_sym_lemma)
done

(* Axiom 'order_less_le' of class 'order': *)
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"

instance real :: order
proof qed
(assumption |
rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+

text{*Simplifies a strange formula that occurs quantified.*}
lemma preal_strange_le_eq: "(x1 + x2 \<le> x2 + y1) = (x1 \<le> (y1::preal))"

text{*This is the nicest way to prove linearity*}
lemma real_le_linear_0: "(z::real) \<le> 0 | 0 \<le> z"
apply (rule_tac z = z in eq_Abs_REAL)
preal_cancels preal_strange_le_eq)
apply (cut_tac x=x and y=y in linorder_linear, auto)
done

lemma real_minus_zero_le_iff: "(0 \<le> -R) = (R \<le> (0::real))"
apply (rule_tac z = R in eq_Abs_REAL)
preal_cancels preal_strange_le_eq)
done

lemma real_le_imp_diff_le_0: "x \<le> y ==> x-y \<le> (0::real)"
apply (rule_tac z = x in eq_Abs_REAL)
apply (rule_tac z = y in eq_Abs_REAL)
apply (auto simp add: real_le_def real_zero_def real_diff_def real_minus
apply (rule exI)+
apply (rule conjI, assumption)
apply (subgoal_tac " x + (x2 + y1 + ya) = (x + y1) + (x2 + ya)")
prefer 2 apply (simp (no_asm) only: preal_add_ac)
apply (subgoal_tac "x1 + y2 + (xa + y) = (x1 + y) + (xa + y2)")
prefer 2 apply (simp (no_asm) only: preal_add_ac)
apply simp
done

lemma real_diff_le_0_imp_le: "x-y \<le> (0::real) ==> x \<le> y"
apply (rule_tac z = x in eq_Abs_REAL)
apply (rule_tac z = y in eq_Abs_REAL)
apply (auto simp add: real_le_def real_zero_def real_diff_def real_minus
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
apply (subgoal_tac "(x + ya) + (x1 + y1) \<le> (xa + y) + (x1 + y1)")
apply (subgoal_tac "x1 + (x + (y1 + ya)) \<le> y1 + (x1 + (xa + y))")
done

lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
by (blast intro!: real_diff_le_0_imp_le real_le_imp_diff_le_0)

(* Axiom 'linorder_linear' of class 'linorder': *)
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
apply (insert real_le_linear_0 [of "z-w"])
apply (auto simp add: real_le_eq_diff [of w] real_le_eq_diff [of z]
real_minus_zero_le_iff [symmetric])
done

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

lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)"
apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"])
apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)")
done

lemma real_minus_zero_le_iff2: "(-R \<le> 0) = (0 \<le> (R::real))"
apply (rule_tac z = R in eq_Abs_REAL)
preal_cancels preal_strange_le_eq)
done

lemma real_minus_zero_less_iff: "(0 < -R) = (R < (0::real))"
by (simp add: linorder_not_le [symmetric] real_minus_zero_le_iff2)

lemma real_minus_zero_less_iff2: "(-R < 0) = ((0::real) < R)"
by (simp add: linorder_not_le [symmetric] real_minus_zero_le_iff)

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] diff_minus)

text{*Used a few times in Lim and Transcendental*}
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] diff_minus)

text{*Handles other strange cases that arise in these proofs.*}
lemma forall_imp_less: "\<forall>u v. u \<le> v \<longrightarrow> x + v \<noteq> u + (y::preal) ==> y < x";
apply (drule_tac x=x in spec)
apply (drule_tac x=y in spec)
done

text{*The arithmetic decision procedure is not set up for type preal.*}
lemma preal_eq_le_imp_le:
assumes eq: "a+b = c+d" and le: "c \<le> a"
shows "b \<le> (d::preal)"
proof -
have "c+d \<le> a+d" by (simp add: prems preal_cancels)
hence "a+b \<le> a+d" by (simp add: prems)
thus "b \<le> d" by (simp add: preal_cancels)
qed

lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
--{*Reduce to the (simpler) @{text "\<le>"} relation *}
apply (rule_tac z = x in eq_Abs_REAL)
apply (rule_tac z = y in eq_Abs_REAL)
preal_cancels preal_strange_le_eq)
apply (drule preal_eq_le_imp_le, assumption)
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

text{*lemma for proving @{term "0<(1::real)"}*}
lemma real_zero_le_one: "0 \<le> (1::real)"
preal_cancels)
apply (rule_tac x="preal_of_rat 1 + preal_of_rat 1" in exI)
apply (rule_tac x="preal_of_rat 1" in exI)
done

subsection{*The Reals Form an Ordered Field*}

instance real :: ordered_field
proof
fix x y z :: real
show "0 < (1::real)"
by (simp add: real_less_def real_zero_le_one real_zero_not_eq_one)
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
show "\<bar>x\<bar> = (if x < 0 then -x else x)"
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
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"

text{*Gleason prop 9-4.4 p 127*}
lemma real_of_preal_trichotomy:
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
apply (unfold real_of_preal_def real_zero_def)
apply (rule_tac z = x in eq_Abs_REAL)
apply (cut_tac x = x and y = y in linorder_less_linear)
done

lemma real_of_preal_leD:
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
apply (unfold real_of_preal_def)
done

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"
apply (drule real_of_preal_leD)
done

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 \<le> real_of_preal m2) = (m1 \<le> 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
apply (insert preal_not_eq_self [of "preal_of_rat 1" m])
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"
apply (cut_tac real_of_preal_minus_less_zero)
apply (fast dest: order_less_trans)
done

text{*obsolete but used a lot*}

lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
by blast

lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"

lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>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 ==> \<exists>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 \<le> x ==> \<exists>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 \<le> 0 ==> \<forall>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 ==> \<forall>x. y < real_of_preal x"
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])

lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"

"!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"

lemma real_zero_less_one: "0 < (1::real)"
by (rule Ring_and_Field.zero_less_one)

lemma real_le_square [simp]: "(0::real) \<le> x*x"
by (rule Ring_and_Field.zero_le_square)

subsection{*More Lemmas*}

lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
by auto

lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
by auto

text{*The precondition could be weakened to @{term "0\<le>x"}*}
lemma real_mult_less_mono:
"[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"

lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
by (force elim: order_less_asym

lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
apply (blast intro: elim: order_less_asym)
done

lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
by (force elim: order_less_asym

text{*Only two uses?*}
lemma real_mult_less_mono':
"[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
by (rule Ring_and_Field.mult_strict_mono')

text{*FIXME: delete or at least combine the next two lemmas*}
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
apply (drule Ring_and_Field.equals_zero_I [THEN sym])
apply (cut_tac x = y in real_le_square)
apply (auto, drule real_le_anti_sym, auto)
done

lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
apply (rule_tac y = x in real_sum_squares_cancel)
done

lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
by (drule add_strict_mono [of concl: 0 0], assumption, simp)

lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
apply (drule order_le_imp_less_or_eq)+
done

lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
apply (case_tac "x \<noteq> 0")
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
done

lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
by (auto dest: less_imp_inverse_less)

lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
proof -
have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
thus ?thesis by simp
qed

subsection{*Embedding the Integers into the Reals*}

lemma real_of_int_congruent:
"congruent intrel (%p. (%(i,j). realrel ``
{(preal_of_rat (rat (int(Suc i))), preal_of_rat (rat (int(Suc j))))}) p)"
(*OPTION raised if only is changed to add?????????*)
done

lemma real_of_int:
"real (Abs_Integ (intrel `` {(i, j)})) =
Abs_REAL(realrel ``
{(preal_of_rat (rat (int(Suc i))),
preal_of_rat (rat (int(Suc j))))})"
apply (unfold real_of_int_def)
apply (rule_tac f = Abs_REAL in arg_cong)
apply (simp del: int_Suc
UN_equiv_class [OF equiv_intrel real_of_int_congruent])
done

lemma inj_real_of_int: "inj(real :: int => real)"
apply (rule inj_onI)
apply (rule_tac z = x in eq_Abs_Integ)
apply (rule_tac z = y in eq_Abs_Integ, clarify)
apply (simp del: int_Suc
done

lemma real_of_int_int_zero: "real (int 0) = 0"

lemma real_of_int_zero [simp]: "real (0::int) = 0"
by (insert real_of_int_int_zero, simp)

lemma real_of_one [simp]: "real (1::int) = (1::real)"
apply (subst int_1 [symmetric])
done

lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
apply (rule_tac z = x in eq_Abs_Integ)
apply (rule_tac z = y in eq_Abs_Integ, clarify)
apply (simp del: int_Suc
done

lemma real_of_int_minus: "-real (x::int) = real (-x)"
apply (rule_tac z = x in eq_Abs_Integ)
apply (auto simp add: real_of_int real_minus zminus)
done
declare real_of_int_minus [symmetric, simp]

lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
by (simp only: zdiff_def real_diff_def real_of_int_add real_of_int_minus)
declare real_of_int_diff [symmetric, simp]

lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
apply (rule_tac z = x in eq_Abs_Integ)
apply (rule_tac z = y in eq_Abs_Integ)
apply (rename_tac a b c d)
apply (simp del: int_Suc
apply (rule_tac f=preal_of_rat in arg_cong)
rat_inject)
done
declare real_of_int_mult [symmetric, simp]

lemma real_of_int_Suc: "real (int (Suc n)) = real (int n) + (1::real)"

lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
by (auto intro: inj_real_of_int [THEN injD])

lemma zero_le_real_of_int: "0 \<le> real y ==> 0 \<le> (y::int)"
apply (rule_tac z = y in eq_Abs_Integ)
apply (rename_tac a b c d)
apply (simp del: int_Suc zdiff_def [symmetric]
apply (drule sym, drule preal_eq_le_imp_le, assumption)
apply (simp del: int_Suc add: preal_of_rat_le_iff)
done

lemma real_of_int_le_cancel:
assumes le: "real (x::int) \<le> real y"
shows "x \<le> y"
proof -
have "real x - real x \<le> real y - real x" using le
hence "0 \<le> real y - real x" by simp
hence "0 \<le> y - x" by (simp only: real_of_int_diff zero_le_real_of_int)
hence "0 + x \<le> (y - x) + x" by (simp only: add_le_cancel_right)
thus  "x \<le> y" by simp
qed

lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
by (blast dest!: inj_real_of_int [THEN injD])

lemma real_of_int_less_cancel: "real (x::int) < real y ==> x < y"
by (auto simp add: order_less_le real_of_int_le_cancel)

lemma real_of_int_less_mono: "x < y ==> (real (x::int) < real y)"
apply (auto dest!: real_of_int_less_cancel simp add: order_le_less)
done

lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
by (blast dest: real_of_int_less_cancel intro: real_of_int_less_mono)

lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"

subsection{*Embedding the Naturals into the Reals*}

lemma real_of_nat_zero [simp]: "real (0::nat) = 0"

lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"

lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"

(*Not for addsimps: often the LHS is used to represent a positive natural*)
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"

lemma real_of_nat_less_iff [iff]:
"(real (n::nat) < real m) = (n < m)"

lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"

lemma inj_real_of_nat: "inj (real :: nat => real)"
apply (rule inj_onI)
done

lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
apply (insert real_of_int_le_iff [of 0 "int n"])
done

lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
by (insert real_of_nat_less_iff [of 0 "Suc n"], simp)

lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
by (simp add: real_of_nat_def zmult_int [symmetric])

lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
by (auto dest: inj_real_of_nat [THEN injD])

lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)"
proof
assume "real n = 0"
have "real n = real (0::nat)" by simp
then show "n = 0" by (simp only: real_of_nat_inject)
next
show "n = 0 \<Longrightarrow> real n = 0" by simp
qed

lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
by (simp add: real_of_nat_def zdiff_int [symmetric])

lemma real_of_nat_neg_int [simp]: "neg z ==> real (nat z) = 0"

lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
by (rule real_of_nat_less_iff [THEN subst], auto)

lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
apply (rule real_of_nat_zero [THEN subst])
apply (simp only: real_of_nat_le_iff, simp)
done

lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"

lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"

lemma real_of_int_real_of_nat: "real (int n) = real n"

lemma real_of_nat_real_of_int: "~neg z ==> real (nat z) = real z"

ML
{*
val real_abs_def = thm "real_abs_def";

val real_le_def = thm "real_le_def";
val real_diff_def = thm "real_diff_def";
val real_divide_def = thm "real_divide_def";

val preal_trans_lemma = thm"preal_trans_lemma";
val realrel_iff = thm"realrel_iff";
val realrel_refl = thm"realrel_refl";
val equiv_realrel = thm"equiv_realrel";
val equiv_realrel_iff = thm"equiv_realrel_iff";
val realrel_in_real = thm"realrel_in_real";
val inj_on_Abs_REAL = thm"inj_on_Abs_REAL";
val eq_realrelD = thm"eq_realrelD";
val inj_Rep_REAL = thm"inj_Rep_REAL";
val inj_real_of_preal = thm"inj_real_of_preal";
val eq_Abs_REAL = thm"eq_Abs_REAL";
val real_minus_congruent = thm"real_minus_congruent";
val real_minus = thm"real_minus";

val real_mult = thm"real_mult";
val real_mult_commute = thm"real_mult_commute";
val real_mult_assoc = thm"real_mult_assoc";
val real_mult_1 = thm"real_mult_1";
val real_mult_1_right = thm"real_mult_1_right";
val preal_le_linear = thm"preal_le_linear";
val real_mult_inverse_left = thm"real_mult_inverse_left";
val real_not_refl2 = thm"real_not_refl2";
val real_of_preal_mult = thm"real_of_preal_mult";
val real_of_preal_trichotomy = thm"real_of_preal_trichotomy";
val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero";
val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero";
val real_of_preal_zero_less = thm"real_of_preal_zero_less";
val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq";
val real_le_refl = thm"real_le_refl";
val real_le_linear = thm"real_le_linear";
val real_le_trans = thm"real_le_trans";
val real_le_anti_sym = thm"real_le_anti_sym";
val real_less_le = thm"real_less_le";
val real_less_sum_gt_zero = thm"real_less_sum_gt_zero";
val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex";
val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex";
val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex";
val real_less_all_preal = thm "real_less_all_preal";
val real_less_all_real2 = thm "real_less_all_real2";
val real_of_preal_le_iff = thm "real_of_preal_le_iff";
val real_mult_order = thm "real_mult_order";
val real_zero_less_one = thm "real_zero_less_one";
val real_le_square = thm "real_le_square";
val real_mult_less_mono2 = thm "real_mult_less_mono2";

val real_mult_less_iff1 = thm "real_mult_less_iff1";
val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1";
val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2";
val real_mult_less_mono = thm "real_mult_less_mono";
val real_mult_less_mono' = thm "real_mult_less_mono'";
val real_sum_squares_cancel = thm "real_sum_squares_cancel";
val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2";

val real_mult_left_cancel = thm"real_mult_left_cancel";
val real_mult_right_cancel = thm"real_mult_right_cancel";
val real_inverse_unique = thm "real_inverse_unique";
val real_inverse_gt_one = thm "real_inverse_gt_one";

val real_of_int = thm"real_of_int";
val inj_real_of_int = thm"inj_real_of_int";
val real_of_int_zero = thm"real_of_int_zero";
val real_of_one = thm"real_of_one";
val real_of_int_minus = thm"real_of_int_minus";
val real_of_int_diff = thm"real_of_int_diff";
val real_of_int_mult = thm"real_of_int_mult";
val real_of_int_Suc = thm"real_of_int_Suc";
val real_of_int_real_of_nat = thm"real_of_int_real_of_nat";
val real_of_nat_real_of_int = thm"real_of_nat_real_of_int";
val real_of_int_less_cancel = thm"real_of_int_less_cancel";
val real_of_int_inject = thm"real_of_int_inject";
val real_of_int_less_mono = thm"real_of_int_less_mono";
val real_of_int_less_iff = thm"real_of_int_less_iff";
val real_of_int_le_iff = thm"real_of_int_le_iff";
val real_of_nat_zero = thm "real_of_nat_zero";
val real_of_nat_one = thm "real_of_nat_one";
val real_of_nat_Suc = thm "real_of_nat_Suc";
val real_of_nat_less_iff = thm "real_of_nat_less_iff";
val real_of_nat_le_iff = thm "real_of_nat_le_iff";
val inj_real_of_nat = thm "inj_real_of_nat";
val real_of_nat_ge_zero = thm "real_of_nat_ge_zero";
val real_of_nat_Suc_gt_zero = thm "real_of_nat_Suc_gt_zero";
val real_of_nat_mult = thm "real_of_nat_mult";
val real_of_nat_inject = thm "real_of_nat_inject";
val real_of_nat_diff = thm "real_of_nat_diff";
val real_of_nat_zero_iff = thm "real_of_nat_zero_iff";
val real_of_nat_neg_int = thm "real_of_nat_neg_int";
val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff";
val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff";
val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero";
val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff";
*}

end
```