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(in-package "ACL2")
;; This file contains the examples from our paper that we
;; were able to prove using Brute force through the ACL2
;; simplifier. Recursive calls are unrolled and ifs are
;; turned into case splits.
;; For a description of the example and a description of
;; how this relates to our SAT system see our workshop paper,
;; A SAT-Based Procedure for Verifying Finite State Machines in ACL2.
(defun n-bleq (n x y)
(if (zp n)
t
(and (iff (car x) (car y))
(n-bleq (1- n) (cdr x) (cdr y)))))
(defun xor3 (x y z)
(xor x (xor y z)))
(defun maj3 (x y z)
(if x (or y z)
(and y z)))
;; Returns a n+1 length sum of the first
;; n bits of a and b (plus the carray).
(defun v-adder (n c a b)
(if (zp n)
(list c)
(cons (xor3 c (car a) (car b))
(v-adder (1- n)
(maj3 c (car a) (car b))
(cdr a) (cdr b)))))
(defthm v-adder-rewrite
(equal
(v-adder n c a b)
(if (zp n)
(list c)
(cons (xor3 c (car a) (car b))
(v-adder (1- n)
(maj3 c (car a) (car b))
(cdr a) (cdr b))))))
(defthm n-bleq-rewrite
(equal
(n-bleq n x y)
(if (zp n)
t
(and (iff (car x) (car y))
(n-bleq (1- n) (cdr x) (cdr y))))))
;; ;; 4 Bit Adder Associativity
;; (defthm 4-adder-assoc
;; (n-bleq 4 (v-adder 4 nil (v-adder 4 nil a b) c)
;; (v-adder 4 nil a (v-adder 4 nil b c))))))
;; ;; 32 bit adder associativity
;; (defthm 32-adder-assoc
;; (n-bleq 32 (v-adder 32 nil (v-adder 32 nil a b) c)
;; (v-adder 32 nil a (v-adder 32 nil b c))))
;; ;; 200 Bit adder associativity
;; (defthm 200-adder-assoc
;; (n-bleq 200 (v-adder 200 nil (v-adder 200 nil a b) c)
;; (v-adder 200 nil a (v-adder 200 nil b c))))
(defun nth-cdr (n x)
(if (zp n)
x
(nth-cdr (1- n) (cdr x))))
(defun nth-sublist (n lst)
(if (zp n)
nil
(cons (car lst) (nth-sublist (1- n) (cdr lst)))))
(defun append-n (n x y)
(if (zp n)
y
(cons (car x) (append-n (1- n) (cdr x) y))))
(defun n-nills (n)
(if (zp n)
nil
(cons nil (n-nills (1- n)))))
(defun rev-n (n x ans)
(if (zp n)
ans
(rev-n (1- n) (cdr x) (cons (car x) ans))))
(defun mux-n-help (n in rsel)
(if (zp n)
(car in)
(if (car rsel)
(mux-n-help (1- n) (nth-cdr (expt 2 (1- n)) in) (cdr rsel))
(mux-n-help (1- n) in (cdr rsel)))))
(defun mux-n (n in sel)
(mux-n-help n in (rev-n n sel nil)))
(defun mux-n-w-help (n w in)
(if (zp n)
nil
(cons (car in) (mux-n-w-help (1- n) w (nth-cdr w in)))))
(defun mux-n-w1 (nw sn w in sel)
(if (zp nw)
nil
(cons (mux-n sn (mux-n-w-help (expt 2 sn) w in) sel)
(mux-n-w1 (1- nw) sn w (cdr in) sel))))
(defun mux-n-w (n w in sel)
(mux-n-w1 w n w in sel))
(defun shift-mux-help (n w reg)
(if (zp n)
nil
(append-n w reg (shift-mux-help (1- n) w (cons nil reg)))))
(defun shifter (sn rn rshift reg)
(if (zp sn)
reg
(mux-n-w sn rn (shift-mux-help (expt 2 sn) rn reg) rshift)))
(defthm append-n-rewrite
(equal
(append-n n x y)
(if (zp n)
y
(cons (car x) (append-n (1- n) (cdr x) y)))))
(defthm nth-cdr-rewrite
(equal
(nth-cdr n x)
(if (zp n)
x
(nth-cdr (1- n) (cdr x)))))
(defthm nth-sublist-rewrite
(equal
(nth-sublist n lst)
(if (zp n)
nil
(cons (car lst) (nth-sublist (1- n) (cdr lst))))))
(defthm n-nills-rewrite
(equal
(n-nills n)
(if (zp n)
nil
(cons nil (n-nills (1- n))))))
(defthm rev-n-rewrite
(equal
(rev-n n x ans)
(if (zp n)
ans
(rev-n (1- n) (cdr x) (cons (car x) ans)))))
(defthm mux-n-help-rewrite
(equal
(mux-n-help n in rsel)
(if (zp n)
(car in)
(if (car rsel)
(mux-n-help (1- n) (nth-cdr (expt 2 (1- n)) in) (cdr rsel))
(mux-n-help (1- n) in (cdr rsel)))))
:hints (("Goal" :in-theory (disable nth-cdr-rewrite))))
(defthm mux-n-w-help-rewrite
(equal
(mux-n-w-help n w in)
(if (zp n)
nil
(cons (car in) (mux-n-w-help (1- n) w (nth-cdr w in)))))
:hints (("Goal" :in-theory (disable nth-cdr-rewrite))))
(defthm mux-n-w1-rewrite
(equal
(mux-n-w1 nw sn w in sel)
(if (zp nw)
nil
(cons (mux-n sn (mux-n-w-help (expt 2 sn) w in) sel)
(mux-n-w1 (1- nw) sn w (cdr in) sel))))
:hints (("Goal" :in-theory (disable mux-n-w-help-rewrite mux-n-w-help mux-n mux-n-help-rewrite))))
(defthm shift-mux-help-rewrite
(equal
(shift-mux-help n w reg)
(if (zp n)
nil
(append-n w reg (shift-mux-help (1- n) w (cons nil reg)))))
:hints (("Goal" :in-theory (disable append-n-rewrite))))
;; ;; 32x6 Shift-0
;; (defthm 32x6-shift-0
;; (implies
;; (car (nth-cdr 5 shift0))
;; (n-bleq 32
;; (shifter 6 32 shift0 reg)
;; (n-nills 32))))
;; ;; 64x7 Shift-0
;; (defthm 64x7-shift-0
;; (implies
;; (car (nth-cdr 6 shift0))
;; (n-bleq 64 (shifter 7 64 shift0 reg)
;; (n-nills 64))))
;; ;; 32x4 Add-shift
;; (defthm 32x4-add-shift
;; (n-bleq 32
;; (shifter 4 32 shift0 (shifter 4 32 shift1 reg))
;; (shifter 5 32 (v-adder 4 nil shift0 shift1) reg)))
;; ;; 64x6 Add-shift
;; (defthm 64x6-add-shift
;; (n-bleq 64
;; (shifter 6 64 shift0 (shifter 6 64 shift1 reg))
;; (shifter 7 64 (v-adder 6 nil shift0 shift1) reg)))
(defun increment (n x)
(if (zp n)
nil
(if (car x)
(cons nil (increment (1- n) (cdr x)))
(cons t (cdr x)))))
(defthm increment-rewrite
(implies
(and (syntaxp (quotep n))
(not (zp n)))
(equal (increment n x)
(if (car x)
(cons nil (increment (1- n) (cdr x)))
(cons t (cdr x))))))
(defthm increment-rewrite0
(equal (increment 0 x)
nil))
(defun next_digit_counter_state (x)
(if (n-bleq 4 x '(t nil nil t))
(list '(nil nil nil nil) t)
(list (increment 4 x) nil)))
(defun next_counter_state (n x)
(let* ((curr_d_out (next_digit_counter_state (car x)))
(curr_d_val (car curr_d_out))
(curr_d_reset (cadr curr_d_out)))
(if (zp n)
nil
(if curr_d_reset
(cons curr_d_val (next_counter_state (1- n) (cdr x)))
(cons curr_d_val (cdr x))))))
(defun valid_digit (a)
(let ((a1 (cadr a))
(a2 (caddr a))
(a3 (cadddr a)))
(not (and a3 (or a2 a1)))))
(defun valid_digits (n x)
(if (zp n)
(not (consp x))
(and (valid_digit (car x))
(valid_digits (1- n) (cdr x)))))
(defthm valid_digits-rewrite
(implies
(and (syntaxp (quotep n))
(not (zp n)))
(equal
(valid_digits n x)
(and (valid_digit (car x))
(valid_digits (1- n) (cdr x))))))
(defthm valid_digits-rewrite0
(equal (valid_digits 0 x)
(not (consp x))))
;; 100 Digit Counter Invariant
;; (defthm 100-digit-counter-inv
;; (implies
;; (valid_digits 100 x)
;; (valid_digits 100 (next_counter_state 100 x))))
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