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(in-package "ACL2")
(include-book "nonstd/integrals/integrable-functions" :dir :system)
(local (include-book "arithmetic-5/top" :dir :system))
(local (include-book "nonstd/nsa/nsa" :dir :system))
(defun make-partition-rec (a delta n)
(if (zp n)
(list a)
(cons a
(make-partition-rec (+ a delta) delta (1- n)))))
(defun make-partition (a b n)
(make-partition-rec a (/ (- b a) n) n))
(defun make-small-partition (a b)
(make-partition a b (i-large-integer)))
(defthm partitionp-append
(implies (and (partitionp p1)
(partitionp p2)
(equal (car (last p1)) (car p2)))
(partitionp (append p1 (cdr p2)))))
(local
(defthm deltas-append
(implies (and (partitionp p1)
(partitionp p2)
(equal (car (last p1)) (car p2)))
(equal (deltas (append p1 (cdr p2)))
(append (deltas p1)
(deltas p2))))))
(local
(defthm maxlist-non-negative
(implies (real-listp x)
(and (realp (maxlist x))
(<= 0 (maxlist x))))))
(local
(defthm maxlist-append
(implies (and (real-listp x1)
(real-listp x2)
)
(equal (maxlist (append x1 x2))
(max (maxlist x1)
(maxlist x2))))))
(local
(defthm real-listp-deltas
(implies (partitionp p)
(real-listp (deltas p)))))
(local
(defthm maxlist-deltas-append
(implies (and (partitionp p1)
(partitionp p2)
(equal (car (last p1)) (car p2)))
(equal (maxlist (deltas (append p1 (cdr p2))))
(max (maxlist (deltas p1))
(maxlist (deltas p2)))))
:hints (("Goal" :do-not-induct t))
))
(defthm mesh-append
(implies (and (partitionp p1)
(partitionp p2)
(equal (car (last p1)) (car p2)))
(equal (mesh (append p1 (cdr p2)))
(max (mesh p1)
(mesh p2)))))
(local
(defthm mesh-append
(implies (and (partitionp p1)
(partitionp p2)
(equal (car (last p1)) (car p2)))
(equal (mesh (append p1 (cdr p2)))
(max (mesh p1)
(mesh p2))))))
(local
(defthmd i-close-plus
(implies (and (i-close x1 x2)
(i-close y1 y2))
(i-close (+ x1 y1)
(+ x2 y2)))
:hints (("Goal"
:in-theory (enable i-close i-small)))))
(local
(defthm car-make-partition-rec
(equal (car (make-partition-rec a delta n))
a)))
(local
(defthm car-make-partition
(equal (car (make-partition a b n))
a)))
(local
(defthm car-last-make-partition-rec
(implies (and (acl2-numberp a)
(acl2-numberp delta)
(natp n))
(equal (car (last (make-partition-rec a delta n)))
(+ a (* n delta))))))
(local
(defthm car-last-make-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(posp n))
(equal (car (last (make-partition a b n)))
b))))
(local
(defthm partitionp-make-partition-rec
(implies (and (realp a)
(realp delta)
(< 0 delta)
(natp n)
)
(partitionp (make-partition-rec a delta n)))))
(local
(defthm partitionp-make-partition
(implies (and (realp a)
(realp b)
(< a b)
(natp n)
)
(partitionp (make-partition a b n)))))
(local
(defthm maxlist-deltas-make-partition-rec
(implies (and (acl2-numberp a)
(acl2-numberp delta)
(<= 0 delta))
(equal (maxlist (deltas (make-partition-rec a delta n)))
(if (zp n)
0
delta)))))
(local
(defthm maxlist-deltas-make-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(< a b)
(posp n))
(equal (maxlist (deltas (make-partition a b n)))
(/ (- b a) n)))))
(local
(defthm consp-cdr-make-partition-rec-2
(consp (cdr (make-partition-rec a delta 2)))
:hints (("Goal"
:expand ((make-partition-rec a delta 2))))
))
(local
(defthm consp-cdr-make-partition-rec-1
(consp (cdr (make-partition-rec a delta 1)))
:hints (("Goal"
:expand ((make-partition-rec a delta 1))))
))
(local
(defthm consp-cdr-make-partition-rec
(implies (posp n)
(consp (cdr (make-partition-rec a delta n))))))
(local
(defthm consp-cdr-make-partition
(implies (posp n)
(consp (cdr (make-partition a b n))))))
(defthm partitionp-make-small-partition
(implies (and (realp a)
(realp b)
(< a b))
(partitionp (make-small-partition a b))))
(defthm car-make-small-partition
(equal (car (make-small-partition a b))
a))
(defthm car-last-make-small-partition
(implies (and (acl2-numberp a)
(acl2-numberp b))
(equal (car (last (make-small-partition a b)))
b)))
(defthm car-last-cdr-make-small-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(< a b))
(equal (car (last (cdr (make-small-partition a b))))
b))
:hints (("Goal"
:use ((:instance car-last-make-small-partition))
:in-theory (disable car-last-make-small-partition
make-small-partition))))
(local
(defthm maxlist-deltas-make-small-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(< a b))
(equal (maxlist (deltas (make-small-partition a b)))
(/ (- b a) (i-large-integer))))))
(local
(defthm mesh-make-small-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(< a b))
(equal (mesh (make-small-partition a b))
(/ (- b a) (i-large-integer))))))
(defthm small-mesh-make-small-partition
(implies (and (acl2-numberp a) (standardp a)
(acl2-numberp b) (standardp b)
(< a b))
(i-small (mesh (make-small-partition a b)))))
(defthm consp-cdr-make-small-partition
(implies (and (acl2-numberp a)
(acl2-numberp b)
(< a b))
(consp (cdr (make-small-partition a b)))))
(in-theory (disable make-small-partition))
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