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;; Copyright (C) 2017, Regents of the University of Texas
;; Written by Cuong Chau
;; License: A 3-clause BSD license. See the LICENSE file distributed with
;; ACL2.
;; Cuong Chau <ckcuong@cs.utexas.edu>
;; November 2018
;; An n-bit, big-endian "less than" comparator
(in-package "ADE")
(include-book "../unbound")
;; ======================================================================
(module-generator
1-bit-<* ()
'1-bit-<
'(ind-in flag-in a b)
'(ind-out flag-out)
()
'((xnor (x) b-xnor (a b))
(f (flag-out) b-and (x flag-in))
(neg (a~) b-not (a))
(g (c) b-and3 (flag-in a~ b))
(h (ind-out) b-or (c ind-in)))
(declare (xargs :guard t)))
(defund 1-bit-<$netlist ()
(declare (xargs :guard t))
(list (1-bit-<*)))
(defund 1-bit-<& (netlist)
(declare (xargs :guard (alistp netlist)))
(equal (assoc '1-bit-< netlist)
(1-bit-<*)))
(local
(defthmd check-1-bit-<$netlist
(and (net-syntax-okp (1-bit-<$netlist))
(net-arity-okp (1-bit-<$netlist))
(1-bit-<& (1-bit-<$netlist)))))
(defthm 1-bit-<$value
(implies (1-bit-<& netlist)
(equal (se '1-bit-< (list ind-in flag-in a b) st netlist)
(list (f-or (f-and3 flag-in (f-not a) b)
ind-in)
(f-and (f-xnor a b) flag-in))))
:hints (("Goal"
:expand (:free (inputs)
(se '1-bit-< inputs st netlist))
:in-theory (enable de-rules 1-bit-<&))))
;; ======================================================================
(defun fv-< (ind flag a b)
(declare (xargs :guard (true-listp b)))
(if (atom a)
(3v-fix ind)
(fv-< (f-or (f-and3 flag (f-not (car a)) (car b))
ind)
(f-and (f-xnor (car a) (car b))
flag)
(cdr a)
(cdr b))))
(defthm 3v-fix-fv-<
(equal (3v-fix (fv-< ind flag a b))
(fv-< ind flag a b)))
(defthm fv-<-of-f-buf-canceled
(and (equal (fv-< (f-buf ind) flag a b)
(fv-< ind flag a b))
(equal (fv-< ind (f-buf flag) a b)
(fv-< ind flag a b))))
(defthm fv-<-of-v-threefix-canceled-1
(equal (fv-< ind flag (v-threefix a) b)
(fv-< ind flag a b))
:hints (("Goal" :in-theory (enable 3vp f-gates))))
(defthm fv-<-of-v-threefix-canceled-2
(equal (fv-< ind flag a (v-threefix b))
(fv-< ind flag a b))
:hints (("Goal" :in-theory (enable f-gates))))
(in-theory (disable fv-<))
(defun v-< (ind flag a b)
(declare (xargs :guard (true-listp b)))
(if (atom a)
(bool-fix ind)
(v-< (b-or (b-and3 flag (b-not (car a)) (car b))
ind)
(b-and (b-xnor (car a) (car b))
flag)
(cdr a)
(cdr b))))
(defthm booleanp-v-<
(booleanp (v-< ind flag a b))
:rule-classes :type-prescription)
(defthm fv-<=v-<
(implies (and (booleanp ind)
(booleanp flag)
(bvp a)
(bvp b))
(equal (fv-< ind flag a b)
(v-< ind flag a b)))
:hints (("Goal" :in-theory (enable bvp fv-<))))
(local
(defthm v-<-lemma
(and (implies ind
(equal (v-< ind flag a b) t))
(implies (not flag)
(equal (v-< ind flag a b)
(bool-fix ind))))))
(local
(defthm v-<-append
(implies (bv2p x1 x2)
(equal (v-< ind flag (append x1 y1) (append x2 y2))
(or (v-< ind flag x1 x2)
(and (equal x1 x2)
(v-< ind flag y1 y2)))))
:hints (("Goal" :in-theory (enable bool-fix bvp)))))
;; (defthmd v-<-works
;; (implies (and (bv2p a b)
;; (v-< ind flag (rev a) (rev b)))
;; (or ind
;; (< (v-to-nat a) (v-to-nat b))))
;; :hints (("Goal"
;; :induct (v-<-works-induct a b)
;; :in-theory (enable bvp v-to-nat))))
;; (defthm v-<-correct
;; (implies (and (bv2p a b)
;; (v-< nil t (rev a) (rev b)))
;; (< (v-to-nat a) (v-to-nat b)))
;; :hints (("Goal" :use (:instance v-<-works
;; (ind nil)
;; (flag t))))
;; :rule-classes :linear)
(defun v-<-correct-induct (a b)
(if (atom a)
b
(v-<-correct-induct (cdr a) (cdr b))))
(defthm v-<-correct-1
(implies (and (bv2p a b)
(v-< nil t (rev a) (rev b)))
(< (v-to-nat a) (v-to-nat b)))
:hints (("Goal"
:induct (v-<-correct-induct a b)
:in-theory (enable bvp v-to-nat)))
:rule-classes (:rewrite :linear))
(defthm v-<-correct-2
(implies (and (bv2p a b)
(not (v-< nil t (rev a) (rev b))))
(>= (v-to-nat a) (v-to-nat b)))
:hints (("Goal"
:induct (v-<-correct-induct a b)
:in-theory (enable bvp v-to-nat)))
:rule-classes (:rewrite :linear))
(in-theory (disable v-<))
(defun v-<-body (m n)
(declare (xargs :guard (and (natp m)
(natp n))))
;; m is the current index and n is the number of occurrences.
(if (zp n)
nil
(list*
(list
;; occurrence name
(si 'g m)
;; outputs
(list (si 'ind (1+ m))
(si 'flag (1+ m)))
;; inferior module reference
'1-bit-<
;; inputs
(list (si 'ind m)
(si 'flag m)
(si 'a m)
(si 'b m)))
(v-<-body (1+ m) (1- n)))))
(module-generator
v-<* (n)
(si 'v-< n)
(append (sis 'a 0 n)
(sis 'b 0 n))
(list (si 'ind n))
()
(list*
(list 'ind-in (list (si 'ind 0)) 'vss ())
(list 'flag-in (list (si 'flag 0)) 'vdd ())
(v-<-body 0 n))
(declare (xargs :guard (natp n))))
(defund v-<$netlist (n)
(declare (xargs :guard (natp n)))
(cons (v-<* n)
(1-bit-<$netlist)))
(defund v-<& (netlist n)
(declare (xargs :guard (and (alistp netlist)
(natp n))))
(and (equal (assoc (si 'v-< n) netlist)
(v-<* n))
(1-bit-<& (delete-to-eq (si 'v-< n)
netlist))))
(local
(defthmd check-v-<$netlist-64
(and (net-syntax-okp (v-<$netlist 64))
(net-arity-okp (v-<$netlist 64))
(v-<& (v-<$netlist 64) 64))))
(defun v-<-body-induct (m n wire-alist st-alist netlist)
(if (zp n)
wire-alist
(v-<-body-induct
(1+ m)
(1- n)
(se-occ-bindings 1
(v-<-body m n)
wire-alist
st-alist
netlist)
st-alist
netlist)))
(local
(defthm v-<-body$value
(implies (and (1-bit-<& netlist)
(natp m)
(natp n)
(equal m+n (+ m n))
;; We need the following hypothesis for the case of (zp n)
(3vp (assoc-eq-value (si 'ind m) wire-alist)))
(equal (assoc-eq-value (si 'ind m+n)
(se-occ (v-<-body m n)
wire-alist
st-alist
netlist))
(fv-<
(assoc-eq-value (si 'ind m) wire-alist)
(assoc-eq-value (si 'flag m) wire-alist)
(assoc-eq-values (sis 'a m n) wire-alist)
(assoc-eq-values (sis 'b m n) wire-alist))))
:hints (("Goal"
:in-theory (enable de-rules
fv-<
sis)
:induct (v-<-body-induct m n
wire-alist
st-alist
netlist)))))
(local
(defthm v-<-body$value-m=0
(implies (and (1-bit-<& netlist)
(natp n)
(3vp (assoc-eq-value (si 'ind 0) wire-alist)))
(equal (assoc-eq-value (si 'ind n)
(se-occ (v-<-body 0 n)
wire-alist
st-alist
netlist))
(fv-<
(assoc-eq-value (si 'ind 0) wire-alist)
(assoc-eq-value (si 'flag 0) wire-alist)
(assoc-eq-values (sis 'a 0 n) wire-alist)
(assoc-eq-values (sis 'b 0 n) wire-alist))))))
(defthm v-<$value
(implies (and (v-<& netlist n)
(natp n)
(true-listp a)
(true-listp b)
(equal (len a) n)
(equal (len b) n))
(equal (se (si 'v-< n)
(append a b)
st
netlist)
(list (fv-< nil t a b))))
:hints (("Goal"
:expand (:free (inputs n)
(se (si 'v-< n) inputs st netlist))
:in-theory (e/d* (de-rules
v-<&
v-<*$destructure)
(de-module-disabled-rules)))))
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