src/HOL/Real/RealDef.thy
 author paulson Tue, 02 Dec 2003 11:48:15 +0100 changeset 14270 342451d763f9 parent 14269 502a7c95de73 child 14329 ff3210fe968f permissions -rw-r--r--
More re-organising of numerical theorems
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(*  Title       : Real/RealDef.thy
ID          : \$Id\$
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
Description : The reals
*)

theory RealDef = PReal:

(*MOVE TO THEORY PREAL*)
instance preal :: order
proof qed
(assumption |
rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+

instance preal :: order
by (intro_classes,
(assumption |
rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+)

lemma preal_le_linear: "x <= y | y <= (x::preal)"
apply (insert preal_linear [of x y])
done

instance preal :: linorder
by (intro_classes, rule preal_le_linear)

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_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 1))})"

real_one_def:
"1 == Abs_REAL(realrel``
{(preal_of_prat(prat_of_pnat 1) + preal_of_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 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_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 1))})"

real_of_posnat :: "nat => real"
"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"

real_of_nat_def:   "real n == real_of_posnat n + (- 1)"

"P+Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"

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

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

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

(*** Proving that realrel is an equivalence relation ***)

lemma preal_trans_lemma:
"[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |]
==> x1 + y3 = x3 + y1"
apply (rule_tac C = y2 in preal_add_right_cancel)
apply (rotate_tac 1, drule sym)
apply (rule_tac x1 = x1 in preal_add_assoc [THEN subst])
done

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 (unfold equiv_def refl_def sym_def trans_def realrel_def)
apply (fast elim!: sym 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

(**** real_minus: additive inverse on real ****)

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_minus_minus: "- (- z) = (z::real)"
apply (rule_tac z = z in eq_Abs_REAL)
done

declare real_minus_minus [simp]

lemma inj_real_minus: "inj(%r::real. -r)"
apply (rule inj_onI)
apply (drule_tac f = uminus in arg_cong)
done

lemma real_minus_zero: "- 0 = (0::real)"
apply (unfold real_zero_def)
done

declare real_minus_zero [simp]

lemma real_minus_zero_iff: "(-x = 0) = (x = (0::real))"
apply (rule_tac z = x in eq_Abs_REAL)
done

declare real_minus_zero_iff [simp]

(*** Congruence property for addition ***)

"[|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

(*For AC rewriting*)
apply (rule mk_left_commute [of "op +"])
done

(* real addition is an AC operator *)

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

lemma real_add_minus_left: "(-z) + z = (0::real)"

(*** 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

(*For AC rewriting*)
lemma real_mult_left_commute: "(x::real)*(y*z)=y*(x*z)"
apply (rule mk_left_commute [of "op *"])
apply (rule real_mult_assoc)
apply (rule real_mult_commute)
done

(* real multiplication is an AC operator *)
lemmas real_mult_ac = real_mult_assoc real_mult_commute real_mult_left_commute

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

declare real_mult_1 [simp]

lemma real_mult_1_right: "z * (1::real) = z"

declare real_mult_1_right [simp]

lemma real_mult_0: "0 * z = (0::real)"
apply (unfold real_zero_def pnat_one_def)
apply (rule_tac z = z in eq_Abs_REAL)
done

lemma real_mult_0_right: "z * 0 = (0::real)"

declare real_mult_0_right [simp] real_mult_0 [simp]

lemma real_mult_minus_eq1: "(-x) * (y::real) = -(x * y)"
apply (rule_tac z = x in eq_Abs_REAL)
apply (rule_tac z = y in eq_Abs_REAL)
done
declare real_mult_minus_eq1 [simp]

lemmas real_minus_mult_eq1 = real_mult_minus_eq1 [symmetric, standard]

lemma real_mult_minus_eq2: "x * (- y :: real) = -(x * y)"
by (simp add: real_mult_commute [of x])
declare real_mult_minus_eq2 [simp]

lemmas real_minus_mult_eq2 = real_mult_minus_eq2 [symmetric, standard]

lemma real_mult_minus_1: "(- (1::real)) * z = -z"
by simp
declare real_mult_minus_1 [simp]

lemma real_mult_minus_1_right: "z * (- (1::real)) = -z"
by (subst real_mult_commute, simp)
declare real_mult_minus_1_right [simp]

lemma real_minus_mult_cancel: "(-x) * (-y) = x * (y::real)"
by simp

declare real_minus_mult_cancel [simp]

lemma real_minus_mult_commute: "(-x) * y = x * (- y :: real)"
by simp

(** Lemmas **)

lemma real_add_assoc_cong: "(z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"

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

lemma real_add_mult_distrib2: "(w::real) * (z1 + z2) = (w * z1) + (w * z2)"

lemma real_diff_mult_distrib: "((z1::real) - z2) * w = (z1 * w) - (z2 * w)"
apply (unfold real_diff_def)
done

lemma real_diff_mult_distrib2: "(w::real) * (z1 - z2) = (w * z1) - (w * z2)"
by (simp add: real_mult_commute [of w] real_diff_mult_distrib)

(*** one and zero are distinct ***)
lemma real_zero_not_eq_one: "0 ~= (1::real)"
apply (unfold real_zero_def real_one_def)
done

(*** existence of inverse ***)
(** lemma -- alternative definition of 0 **)
lemma real_zero_iff: "0 = Abs_REAL (realrel `` {(x, x)})"
apply (unfold real_zero_def)
done

lemma real_mult_inv_right_ex:
"!!(x::real). x ~= 0 ==> \<exists>y. x*y = (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_prat (prat_of_pnat 1), pinv (D) + preal_of_prat (prat_of_pnat 1))}) " in exI)
apply (rule_tac [2] x = "Abs_REAL (realrel `` { (pinv (D) + preal_of_prat (prat_of_pnat 1), preal_of_prat (prat_of_pnat 1))}) " in exI)
done

lemma real_mult_inv_left_ex: "x ~= 0 ==> \<exists>y. y*x = (1::real)"
apply (drule real_mult_inv_right_ex)
done

lemma real_mult_inv_left: "x ~= 0 ==> inverse(x)*x = (1::real)"
apply (unfold real_inverse_def)
apply (frule real_mult_inv_left_ex, safe)
apply (rule someI2, auto)
done
declare real_mult_inv_left [simp]

lemma real_mult_inv_right: "x ~= 0 ==> x*inverse(x) = (1::real)"
apply (subst real_mult_commute)
done
declare real_mult_inv_right [simp]

(*---------------------------------------------------------
Theorems for ordering
--------------------------------------------------------*)
(* prove introduction and elimination rules for real_less *)

(* real_less is a strong order i.e. nonreflexive and transitive *)

(*** lemmas ***)
lemma preal_lemma_eq_rev_sum: "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"

lemma preal_add_left_commute_cancel: "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"

lemma preal_lemma_for_not_refl: "!!(x::preal). [| x + y2a = x2a + y;
x + y2b = x2b + y |]
==> x2a + y2b = x2b + y2a"
apply (drule preal_lemma_eq_rev_sum, assumption)
apply (erule_tac V = "x + y2b = x2b + y" in thin_rl)
done

lemma real_less_not_refl: "~ (R::real) < R"
apply (rule_tac z = R in eq_Abs_REAL)
apply (drule preal_lemma_for_not_refl, assumption, auto)
done

(*** y < y ==> P ***)
lemmas real_less_irrefl = real_less_not_refl [THEN notE, standard]
declare real_less_irrefl [elim!]

lemma real_not_refl2: "!!(x::real). x < y ==> x ~= y"

(* lemma re-arranging and eliminating terms *)
lemma preal_lemma_trans: "!! (a::preal). [| a + b = c + d;
x2b + d + (c + y2e) < a + y2b + (x2e + b) |]
==> x2b + y2e < x2e + y2b"
apply (rule_tac C = "c+d" in preal_add_left_less_cancel)
done

(** A MESS!  heavy re-writing involved*)
lemma real_less_trans: "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3"
apply (rule_tac z = R1 in eq_Abs_REAL)
apply (rule_tac z = R2 in eq_Abs_REAL)
apply (rule_tac z = R3 in eq_Abs_REAL)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
prefer 2 apply blast
prefer 2 apply blast
apply (drule preal_lemma_for_not_refl, assumption)
apply (blast dest: preal_add_less_mono intro: preal_lemma_trans)
done

lemma real_less_not_sym: "!! (R1::real). R1 < R2 ==> ~ (R2 < R1)"
apply (rule notI)
apply (drule real_less_trans, assumption)
done

(* [| x < y;  ~P ==> y < x |] ==> P *)
lemmas real_less_asym = real_less_not_sym [THEN contrapos_np, standard]

"real_of_preal ((z1::preal) + z2) =
real_of_preal z1 + real_of_preal z2"
apply (unfold real_of_preal_def)
done

lemma real_of_preal_mult:
"real_of_preal ((z1::preal) * z2) =
real_of_preal z1* real_of_preal z2"
apply (unfold real_of_preal_def)
done

lemma real_of_preal_ExI:
"!!(x::preal). y < x ==>
\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m"
apply (unfold real_of_preal_def)
done

lemma real_of_preal_ExD:
"!!(x::preal). \<exists>m. Abs_REAL (realrel `` {(x,y)}) =
real_of_preal m ==> y < x"
apply (unfold real_of_preal_def)
done

lemma real_of_preal_iff: "(\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m) = (y < x)"
by (blast intro!: real_of_preal_ExI real_of_preal_ExD)

(*** 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_trichotomyE: "!!P. [| !!m. x = real_of_preal m ==> P;
x = 0 ==> P;
!!m. x = -(real_of_preal m) ==> P |] ==> P"
apply (cut_tac x = x in real_of_preal_trichotomy, auto)
done

lemma real_of_preal_lessD:
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
apply (unfold real_of_preal_def)
done

lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_less_iff1: "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
by (blast intro: real_of_preal_lessI real_of_preal_lessD)

declare real_of_preal_less_iff1 [simp]

lemma real_of_preal_minus_less_self: "- real_of_preal m < real_of_preal m"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
apply (unfold real_zero_def)
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

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: real_less_trans elim: real_less_irrefl)
done

lemma real_of_preal_zero_less: "0 < real_of_preal m"
apply (unfold real_zero_def)
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_not_less_zero: "~ real_of_preal m < 0"
apply (cut_tac real_of_preal_zero_less)
apply (blast dest: real_less_trans elim: real_less_irrefl)
done

lemma real_minus_minus_zero_less: "0 < - (- real_of_preal m)"

(* another lemma *)
lemma real_of_preal_sum_zero_less:
"0 < real_of_preal m + real_of_preal m1"
apply (unfold real_zero_def)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_minus_less_all: "- real_of_preal m < real_of_preal m1"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_not_minus_gt_all: "~ real_of_preal m < - real_of_preal m1"
apply (cut_tac real_of_preal_minus_less_all)
apply (blast dest: real_less_trans elim: real_less_irrefl)
done

lemma real_of_preal_minus_less_rev1: "- real_of_preal m1 < - real_of_preal m2
==> real_of_preal m2 < real_of_preal m1"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_minus_less_rev2: "real_of_preal m1 < real_of_preal m2
==> - real_of_preal m2 < - real_of_preal m1"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
apply (rule conjI, rule_tac [2] conjI)
apply (rule_tac [2] refl)+
done

lemma real_of_preal_minus_less_rev_iff: "(- real_of_preal m1 < - real_of_preal m2) =
(real_of_preal m2 < real_of_preal m1)"
apply (blast intro!: real_of_preal_minus_less_rev1 real_of_preal_minus_less_rev2)
done

declare real_of_preal_minus_less_rev_iff [simp]

subsection{*Linearity of the Ordering*}

lemma real_linear: "(x::real) < y | x = y | y < x"
apply (rule_tac x = x in real_of_preal_trichotomyE)
apply (rule_tac [!] x = y in real_of_preal_trichotomyE)
apply (auto dest!: preal_le_anti_sym
real_of_preal_zero_less real_of_preal_minus_less_all)
done

lemma real_neq_iff: "!!w::real. (w ~= z) = (w<z | z<w)"
by (cut_tac real_linear, blast)

lemma real_linear_less2: "!!(R1::real). [| R1 < R2 ==> P;  R1 = R2 ==> P;
R2 < R1 ==> P |] ==> P"
apply (cut_tac x = R1 and y = R2 in real_linear, auto)
done

lemma real_minus_zero_less_iff: "(0 < -R) = (R < (0::real))"
apply (rule_tac x = R in real_of_preal_trichotomyE)
apply (auto simp add: real_of_preal_not_minus_gt_zero real_of_preal_not_less_zero real_of_preal_zero_less real_of_preal_minus_less_zero)
done
declare real_minus_zero_less_iff [simp]

lemma real_minus_zero_less_iff2: "(-R < 0) = ((0::real) < R)"
apply (rule_tac x = R in real_of_preal_trichotomyE)
apply (auto simp add: real_of_preal_not_minus_gt_zero real_of_preal_not_less_zero real_of_preal_zero_less real_of_preal_minus_less_zero)
done
declare real_minus_zero_less_iff2 [simp]

ML
{*
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 real_of_nat_def = thm "real_of_nat_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_minus_minus = thm"real_minus_minus";
val inj_real_minus = thm"inj_real_minus";
val real_minus_zero = thm"real_minus_zero";
val real_minus_zero_iff = thm"real_minus_zero_iff";