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; Copyright (C) 2019, ForrestHunt, Inc.
; Written by Matt Kaufmann and J Moore
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "MODAPP")
(include-book "user-defs")
; -----------------------------------------------------------------
; Some Sample Theorems about Mapping Functions
; Theorems just showing some evaluations:
(defthm collect-example
(equal (collect '(1 2 3) '(LAMBDA (X) (CONS 'A X)))
'((A . 1) (A . 2) (A . 3)))
:rule-classes nil)
(defthm collect-collect-example
(implies (warrant collect)
(equal (collect '((1 2 3) (4 5 6))
'(LAMBDA (X)
(COLLECT X '(LAMBDA (X) (CONS 'A X)))))
'(((A . 1) (A . 2) (A . 3))
((A . 4) (A . 5) (A . 6)))))
:rule-classes nil
:hints (("Goal" :expand ((:free (x) (hide x))))))
(defthm recur-by-collect-example
(equal (recur-by-collect '(1 1 1)
'(LAMBDA (X) (BINARY-+ '1 X)))
'(1 2 3))
:rule-classes nil)
; A theorem showing that functions can be data, i.e., we can apply mapping functions
; to mapping functions, as long as the data functions are tame.
(defthm collect*-collect-example
(implies (warrant square collect)
(equal (collect* '(((1 2 3) (LAMBDA (X) (CONS 'A X)))
((4 5 6 7) SQUARE)
(((20 21 22)
(30 31 32)
(40 41 42))
(LAMBDA (Y)
(COLLECT Y '(LAMBDA (Z) (CONS 'C Z)))))
)
'COLLECT)
'(((A . 1)(A . 2)(A . 3))
(16 25 36 49)
(((C . 20) (C . 21) (C . 22))
((C . 30) (C . 31) (C . 32))
((C . 40) (C . 41) (C . 42)))
)))
:rule-classes nil
:hints
(("Goal"
:in-theory
(disable (:executable-counterpart collect)
(:executable-counterpart collect*)))))
; Theorems about general relationships
(defthm collect-my-append1
(equal (collect (my-append1 a b) fn)
(my-append1 (collect a fn)
(collect b fn))))
(defthm sumlist-my-append1
(equal (sumlist (my-append1 a b) u)
(+ (sumlist a u)
(sumlist b u))))
(defthm all-my-append1
(equal (all (my-append1 a b) fn)
(and (all a fn) (all b fn))))
(defthm xists-my-append1
(equal (xists (my-append1 a b) fn)
(if (xists a fn)
t
(xists b fn))))
(defthm len-filter
(<= (len (filter lst fn)) (len lst))
:rule-classes :linear)
(defthm filter-v-all
(implies (true-listp lst)
(iff (equal (filter lst fn) lst)
(all lst fn)))
:hints (("Subgoal *1/3.2'" :in-theory (disable len-filter)
:use ((:instance len-filter (lst (cdr lst)) (fn fn))))))
(defthm filter-v-xists
(iff (filter lst fn)
(xists lst fn))
:rule-classes nil)
; We show some theorems about FOLDR further below.
; Theorems about concrete uses of mapping functions.
(encapsulate nil
(local (include-book "arithmetic-5/top" :dir :system))
(defthm sumlist-nats
(implies (and (warrant square)
(natp n))
(equal (sumlist (nats n) 'SQUARE)
(/ (* n (+ n 1) (+ (* 2 n) 1))
6)))
:hints (("Goal" :expand ((:free (x) (HIDE x)))))))
; If lst is a true-list and every element of lst is also, then reversing each element twice
; leaves lst unchanged:
(defthm rev-rev-version-1
(implies (and (warrant my-rev)
(true-listp lst)
(all lst 'true-listp))
(equal (collect (collect lst 'my-rev) 'my-rev) lst)))
; We can actually accomplish an append and a reverse via FOLDRs. So after we
; establish that, we repeat the above theorem but this time without referring
; to my-rev.
(defthm foldr-as-my-append1
(equal (foldr x 'CONS y)
(my-append1 x y)))
(defthm foldr-as-my-rev
(implies (warrant foldr)
(equal (foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)
(my-rev x))))
(defthm rev-rev-version-2
(implies (and (warrant foldr)
(true-listp lst)
(all lst 'true-listp))
(equal (collect (collect lst
'(lambda (x)
(foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)))
'(lambda (x)
(foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)))
lst)))
; And ALL and COLLECT are also FOLDRs, so we can make this an all FOLDR show:
(defthm rev-rev-version-3
(implies (and (warrant foldr)
(true-listp lst)
(foldr lst
'(LAMBDA (X Y)
(IF (TRUE-LISTP X) Y 'NIL))
t))
(equal (foldr
(foldr lst
'(LAMBDA (X Y)
(CONS (foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)
Y))
nil)
'(LAMBDA (X Y)
(CONS (foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)
Y))
nil)
lst)))
; A few theorems manipulating mapping functions and the functions they
; apply.
; Here is a way to move a constant factor out of a sumlist regardless of the
; names of the variables.
(defthm sumlist-binary-*-constant
(implies (tamep body)
(equal (sumlist lst (lamb (list v) (list 'binary-* (list 'quote const) body)))
(* const (sumlist lst (lamb (list v) body))))))
(defthm lamb-x-x-is-identity
(implies (symbolp x)
(fn-equal (lamb (list x) x) 'identity))
:hints (("Goal" :in-theory (enable fn-equal))))
; The hint below is only provided to show that the theorem is proved by
; rewriting, not induction.
(defthm example-of-loop$-rewriting
(equal (sumlist (my-append1 aaa bbb) (lamb '(x) '(binary-* '2 x)))
(+ (* 2 (sumlist aaa 'identity))
(* 2 (sumlist bbb 'identity))))
:hints (("Goal" :do-not-induct t))
:rule-classes nil)
; A couple of nice foldr theorems. The first two are redundant; they were
; critical as rewrite rules in our rev-rev versions above. We repeat them here
; just to summarize the relations with foldr. The rest These are really just
; designed to illustrate the relationships, not to be used as rewrites.
(defthm foldr-as-my-append1
(equal (foldr x 'CONS y)
(my-append1 x y)))
(defthm foldr-as-my-rev
(implies (warrant foldr)
(equal (foldr x
'(LAMBDA (X Y)
(FOLDR Y 'CONS (CONS X 'NIL)))
nil)
(my-rev x))))
; The theorem below, which is not being stored as a rule, illustrates the most
; general fact we can think of relating COLLECT and FOLDR:
(defthm general-collect-is-a-foldr
(implies (and (not (equal fn 'QUOTE))
(not (equal fn 'IF))
(symbolp x)
(symbolp y)
(not (eq x y))
(tamep `(,fn ,x)))
(equal (collect lst fn)
(foldr lst
`(lambda (,x ,y)
(cons (,fn ,x) ,y))
nil)))
:rule-classes nil)
; But this rule is not useful as a rewrite rule because x and y are free
; variables. Furthermore, if the goal is to rewrite COLLECTs into FOLDRs, it's
; not necessary to be so general: we could just choose the x and y we want to
; use. But we're not interested in rewriting in either direction here so we
; make this (and subsequent theorems) :rule-classes nil.
; We use a convenient abbreviation for the hypotheses we'll see over and over
; here.
(defthm foldr-as-collect
(implies (force (ok-fnp fn))
(equal (foldr lst
`(LAMBDA (X Y)
(CONS (,fn X) Y))
nil)
(collect lst fn)))
:rule-classes nil)
(defthm foldr-as-sumlist
(implies (ok-fnp fn)
(equal (foldr lst
`(LAMBDA (X Y)
(BINARY-+ (,fn X) Y))
0)
(sumlist lst fn)))
:rule-classes nil)
(defthm foldr-as-filter
(implies (ok-fnp fn)
(equal (foldr lst
`(LAMBDA (X Y)
(IF (,fn X) (CONS X Y) Y))
nil)
(filter lst fn)))
:rule-classes nil)
(defthm foldr-as-all
(implies (ok-fnp fn)
(equal (foldr lst
`(LAMBDA (X Y)
(IF (,fn X) Y 'NIL))
t)
(all lst fn)))
:rule-classes nil)
(defthm foldr-as-xists
(implies (ok-fnp fn)
(equal (foldr lst
`(LAMBDA (X Y)
(IF (,fn X) 'T Y))
nil)
(xists lst fn)))
:rule-classes nil)
(defthm maxlist-non-nil
(implies (and (ok-fnp fn)
(all (collect lst fn) 'ACL2-NUMBERP))
(iff (maxlist lst fn)
(consp lst)))
:rule-classes nil)
(defthm foldr-as-maxlist
(implies (and (ok-fnp fn)
(all (collect lst fn) 'ACL2-NUMBERP))
(equal (foldr lst `(LAMBDA (X Y)
(IF (EQUAL Y 'NIL)
(,fn X)
(MAX (,fn X) Y)))
nil)
(maxlist lst fn)))
:rule-classes nil)
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