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; Copyright (C) 2007 by Shant Harutunian
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
; Definition of the measure structure less than operator
(defun o<-1 (x y)
(cond ((o-finp x)
(or (not (o-finp y)) (<= (+ X 1) Y)))
((o-finp y) nil)
((equal (o-first-expt x)
(o-first-expt y))
(if (equal (o-first-coeff x)
(o-first-coeff y))
(o<-1 (o-rst x) (o-rst y))
(<= (+ (o-first-coeff x) 1)
(o-first-coeff y))))
(t (o<-1 (o-first-expt x)
(o-first-expt y)))))
; The following defines a predicate which
; is true if x is a measure structure
(defun o-real-p (x)
(cond ((o-finp x)
(and
(realp X)
(<= 0 x)))
((consp (car x))
(and (o-real-p (o-first-expt x))
(if (acl2-numberp (o-first-expt x))
(<= 1 (o-first-expt x))
t)
(realp (o-first-coeff x))
(<= 1 (o-first-coeff x))
(o-real-p (o-rst x))
(o<-1 (o-first-expt (o-rst x))
(o-first-expt x))))
(t nil)))
; Recursively traverse
; a measure structure applying floor
; to the real values in the structure
; The intent is to convert a measure structure to
; an ordinal.
(defun o-floor1 (x)
(cond ((o-finp x) (floor1 x))
((consp (car x))
(make-ord (o-floor1 (o-first-expt x))
(floor1 (o-first-coeff x))
(o-floor1 (o-rst x))))
(t nil)))
(defthm o<-1-floor1-neq-thm
(implies
(and
(o-real-p x)
(o-real-p y)
(o<-1 x y))
(not (equal (o-floor1 x) (o-floor1 y)))))
; The following theorem states that
; showing a measure structure x is less than y,
; using the o<-1 operator
; implies that the ordinals attained by
; applying o-floor1 to x and y, respectively,
; are less than each other.
; Hence, in our proof obligation, we need only
; show that (o<-1 x y), this theorem may then
; be applied to show the respective ordinals
; attained by applying o-floor1 are less
; than each other.
(defthm o<-1-floor1-o<-thm
(implies
(and
(o-real-p x)
(o-real-p y)
(o<-1 x y))
(o< (o-floor1 x) (o-floor1 y))))
(defthm floor1-posp
(implies
(and
(realp x)
(<= 1 x))
(posp (floor1 x))))
(defthm o-floor1-non-zero
(implies
(and
(o-real-p x)
(o-real-p y)
(o<-1 y x))
(not (equal 0 (o-floor1 x)))))
(defthm o-real-p-caadr
(implies
(and
(o-real-p x)
(consp x)
(consp (cdr x)))
(o-real-p (caadr x))))
(defthm o-real-p-caadr-2
(implies
(and
(o-real-p x)
(consp x))
(equal (caar (o-floor1 x)) (o-floor1 (caar x))))
:hints (("Goal" :do-not-induct t)))
(defthm o-floor1-consp
(implies
(and
(consp x)
(o-real-p x))
(consp (o-floor1 x))))
(defthm consp-not-zero
(implies
(consp x)
(not (equal 0 x))))
; The following states that if x is a measure
; structure, then (o-floor1 x) is an ordinal.
(defthm o-floor1-thm
(implies
(o-real-p x)
(o-p (o-floor1 x)))
:hints (("Goal" :do-not '(generalize)
:induct (o-real-p x)
:do-not-induct t)))
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