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|
#||
seq-pp-equivalence.lisp
~~~~~~~~~~~~~~~~~~~~~~~~
Author: Disha Puri
Last Updated: 12th April 2014
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Invariant
;; We are claiming that executing pre pipeline + k iterations of loop pipeline, where you do not exit
;; is same as executing seq-loop for pre + (k-1) iterations of seq-loop + m blocks of seq-loop + m-I blocks
;; of seq-loop and so on till 0 or 1
;; From this invariant, we want to claim that running pre-pp-loop + k iterations of pp-loop + post-pp-loop
;; is same as running seq-loop for pre + (k-1) iterations of seq-loop + x iterations of seq-loop where x =
;; number of iterations created from pre and post
||#
(in-package "ACL2")
(include-book "superstep-construction")
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Defining the invariant
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; x is state after k iterations of pipelined loop
;; run seq for k iterations completely + (run-block-seq blocks m)
;; seq-ccdfg == (list '(1 2 3) '(1 2 3) nil)
(defun get-m-blocks-seq (m c pp-interval)
(if (or (not (posp m))
(not (posp pp-interval))) nil
(if (<= m pp-interval) (take-n m c) ; why doesn't it terminate without this?
(append (take-n m c)
(get-m-blocks-seq (- m pp-interval) c pp-interval)))))
(defun pipeline-loop-invariant (pp-state k pre loop pp-interval init-state prev m)
(let* ((seq-loop-top (run-block-set pre init-state nil prev))
(seq-loop-k (run-blocks-iters loop seq-loop-top (- k 1) prev))
(seq-loop-x (run-block-set (get-m-blocks-seq m loop pp-interval) seq-loop-k nil prev)))
(equal pp-state
seq-loop-x)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Theorems
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm get-m-block-seq-equal-pre-superstep
(equal (pre-superstep-from-loop m loop pp-interval)
(get-m-blocks-seq m loop pp-interval))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(pre-superstep-from-loop
get-m-blocks-seq)))))
(defthm ceiling-m-pp-interval
(implies (and (posp m)
(posp pp-interval)
(<= m pp-interval))
(equal (ceiling m pp-interval) 1)))
(defthm ceiling-m-pp-interval-add
(implies (and (posp m)
(posp pp-interval)
(> m pp-interval))
(equal (ceiling m pp-interval)
(+ (ceiling (- m pp-interval) pp-interval) 1))))
(defthm ceiling-+
(implies (and (posp m)
(posp pp-interval)
(< pp-interval m))
(> (ceiling (- m pp-interval) pp-interval) 0)))
(defthm ceiling-same
(implies (posp m)
(equal (ceiling m m) 1)))
(defthm ceiling-natp
(implies (and (posp m)
(posp pp-interval))
(natp (ceiling m pp-interval))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Invariant base case
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun induction-hint-invariant-case (m loop init-state pp-interval prev)
(if (or (not (posp m))
(not (posp pp-interval))
(<= m pp-interval)) (list m loop init-state pp-interval prev)
(induction-hint-invariant-case (- m pp-interval) loop
(run-block-set (take-n (+ m pp-interval) loop) init-state nil prev)
pp-interval
prev)))
(defthm branch-restriction-get-m-seq
(implies (branch-restriction-ccdfg loop)
(branch-restriction-ccdfg (get-m-blocks-seq m loop pp-interval)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(branch-restriction-ccdfg
get-m-blocks-seq)))))
(defthm invariant-base-case-loop
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg loop)
(not (equal (loop-superstep-from-loop m loop pp-interval) "error")))
(equal (run-block-set
(loop-superstep-from-loop m loop pp-interval)
(run-block-set (pre-superstep-from-loop m loop pp-interval)
init-state nil prev)
nil prev)
(run-block-set (get-m-blocks-seq m loop pp-interval)
(run-block-set (take-n (+ m pp-interval) loop)
init-state
nil prev)
nil prev)))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(induction-hint-invariant-case
branch-restriction-loop-from-loop
branch-restriction-pre-from-loop
branch-restriction-get-m-seq
loop-superstep-from-loop
pre-superstep-from-loop
get-m-blocks-seq
take-return-take
run-block-set-append
run-block-set-nil
branch-restriction-take-n
branch-restriction-remove-n))
:induct (induction-hint-invariant-case m loop init-state pp-interval prev)
:do-not-induct t
:do-not '(eliminate-destructors fertilize generalize))
("Subgoal *1/2.1"
:use
((:instance run-block-set-no-conflict
(a (take-n pp-interval (remove-n m loop)))
(b (pre-superstep-from-loop (- m pp-interval) loop pp-interval))
(init-state (run-block-set (take-n m loop) init-state nil prev)))))))
(defthm invariant-base-case-subgoal-1
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error")))
(equal (run-block-set (loop-superstep-in-order m pre loop pp-interval)
(run-block-set (pre-superstep-in-order m pre loop pp-interval)
init-state nil prev)
nil prev)
(run-block-set (get-m-blocks-seq m loop pp-interval)
(run-block-set pre init-state nil prev)
nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(branch-restriction-ccdfg-cdr
branch-restriction-remove-n
branch-restriction-take-n
branch-restriction-loop-from-loop
branch-restriction-pre-from-loop
branch-restriction-get-m-seq
pre-superstep-in-order
loop-superstep-in-order
get-m-blocks-seq
run-block-set-append
run-block-set-cons
run-block-set-nil
run-block-set-remove-take-n
error-loop
loop-superstep-error))
:do-not-induct t
:use
((:instance run-block-set-no-conflict
(a (remove-n m pre))
(b (pre-superstep-from-loop (- m pp-interval) loop pp-interval))
(init-state (run-block-set (take-n m pre) init-state nil prev)))
(:instance invariant-base-case-loop
(init-state (RUN-BLOCK-SET PRE INIT-STATE NIL PREV))
(m (- m pp-interval)))))))
;; pp-ccdfg is the pipelined ccdfg created by superstep-construction
;; pp-state is state after running pp-ccdfg for pre + k iterations
;; then, invariant is true, which means pp-state is equal to executing seq-ccdfg's pre + (- k 1) iterations + sequence of m
;; (let* ((m (+ (* (floor (- (len (car seq-ccdfg)) (+ pp-interval 1)) pp-interval) pp-interval) 1))
(defthm invariant-base-case
(implies (and (phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop)
(equal k 1)
(posp m)
(posp pp-interval)
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(true-listp pre)
(true-listp loop)
(not (equal (pre-supersteps-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error"))
(equal pp-ccdfg (superstep-construction pre loop pp-interval m))
(equal pp-state (run-ccdfg-k (car pp-ccdfg) (cadr pp-ccdfg)
1 init-state prev)))
(pipeline-loop-invariant pp-state 1 pre loop pp-interval init-state prev m))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(pipeline-loop-invariant
superstep-construction
branch-restriction-ccdfg-cdr
branch-restriction-remove-n
branch-restriction-take-n
branch-restriction-loop-from-loop
branch-restriction-pre-from-loop
branch-restriction-get-m-seq
pre-parallel-order-same
loop-parallel-order-same
run-blocks-iters
run-ccdfg-k
run-block-set-from-iters
invariant-base-case-subgoal-1
phi-restriction-loop-superstep-in-order))
:use
((:instance phi-restriction-block-set
(c (loop-superstep-in-order m pre loop pp-interval))
(i (run-block-set (pre-superstep-in-order m pre loop pp-interval)
init-state nil prev))
(next nil)
(pre1 (prefix (car (last (cadar (last (loop-superstep-in-parallel m pre loop pp-interval)))))))
(pre2 prev)))
:do-not-induct t)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Prove for k > 1
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm invariant-greater-1-subgoal-2
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg loop)
(not (equal (loop-superstep-from-loop m loop pp-interval) "error")))
(equal (run-block-set (loop-superstep-from-loop m
loop pp-interval)
(run-block-set (get-m-blocks-seq
m
loop pp-interval)
init-state nil prev)
nil prev)
(run-block-set (get-m-blocks-seq m loop pp-interval)
(run-block-set (take-n (+ m pp-interval) loop) init-state nil prev)
nil prev)))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(induction-hint-invariant-case
branch-restriction-ccdfg-cdr
branch-restriction-remove-n
branch-restriction-take-n
branch-restriction-loop-from-loop
branch-restriction-pre-from-loop
branch-restriction-get-m-seq
loop-superstep-error
loop-superstep-from-loop
get-m-blocks-seq
run-block-set-append
take-n-plus-interval
get-m-block-seq-equal-pre-superstep))
:induct (induction-hint-invariant-case m loop init-state pp-interval prev)
:do-not-induct t
:do-not '(eliminate-destructors fertilize generalize))
("Subgoal *1/2.1"
:use
((:instance run-block-set-no-conflict
(a (take-n pp-interval (remove-n m loop)))
(b (get-m-blocks-seq (- m pp-interval) loop pp-interval))
(init-state (run-block-set (take-n m loop) init-state nil prev)))))))
(defthm invariant-greater-1-subgoal
(implies (and (posp m)
(posp pp-interval)
(> m pp-interval)
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error")))
(equal (run-block-set (loop-superstep-from-loop (- m pp-interval)
loop pp-interval)
(run-block-set (get-m-blocks-seq
(- m pp-interval)
loop pp-interval)
init-state nil prev)
nil prev)
(run-block-set (get-m-blocks-seq (- m pp-interval) loop pp-interval)
(run-block-set (take-n m loop) init-state nil prev)
nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(loop-superstep-error
error-loop
get-m-block-seq-equal-pre-superstep
branch-restriction-ccdfg-cdr
branch-restriction-remove-n
branch-restriction-take-n
branch-restriction-loop-from-loop
branch-restriction-pre-from-loop
branch-restriction-get-m-seq))
:use
((:instance invariant-greater-1-subgoal-2
(m (- m pp-interval)))))))
(defthm phi-restriction-ccdfg-append
(implies (and (phi-restriction-ccdfg a)
(phi-restriction-ccdfg b))
(phi-restriction-ccdfg (append a b))))
(defthm phi-restriction-loop-from-loop
(implies (phi-restriction-ccdfg loop)
(phi-restriction-ccdfg (loop-superstep-from-loop x loop pp-interval)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(phi-restriction-ccdfg
loop-superstep-from-loop
phi-restriction-cdr
phi-restriction-ccdfg-append
phi-restriction-remove-n
phi-restriction-take-n)))))
(defthm phi-restriction-loop-in-order
(implies (and (phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop))
(phi-restriction-ccdfg (loop-superstep-in-order m pre loop pp-interval)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(phi-restriction-ccdfg
loop-superstep-in-order
phi-restriction-cdr
phi-restriction-ccdfg-append
phi-restriction-remove-n
phi-restriction-take-n
phi-restriction-loop-from-loop)))))
(DEFTHM
phi-RESTRICTION-GET-M-SEQ
(IMPLIES (phi-RESTRICTION-CCDFG LOOP)
(phi-RESTRICTION-CCDFG
(GET-M-BLOCKS-SEQ M LOOP PP-INTERVAL)))
:HINTS
(("Goal" :IN-THEORY
(UNION-THEORIES
(THEORY 'GROUND-ZERO)
'(phi-RESTRICTION-CCDFG GET-M-BLOCKS-SEQ)))))
(defthm invariant-holds-k-greater-1
(implies (and (posp k)
(posp m)
(posp pp-interval)
(<= m (len pre))
(consp pre)
(consp loop)
(seq-ccdfg-p (list pre loop))
(phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop)
(branch-restriction-ccdfg loop)
(branch-restriction-ccdfg pre)
(true-listp pre)
(true-listp loop)
(not (equal (pre-supersteps-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error"))
(equal pp-ccdfg (superstep-construction pre loop pp-interval m))
(equal pp-state (run-ccdfg-k (car pp-ccdfg) (second pp-ccdfg) k init-state prev))
(pipeline-loop-invariant pp-state k pre loop pp-interval init-state prev m))
(pipeline-loop-invariant (run-ccdfg-k (car pp-ccdfg) (second pp-ccdfg) (+ k 1) init-state prev)
(+ k 1) pre loop pp-interval init-state prev m))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set-append
run-ccdfg-k
pre-parallel-order-same
loop-parallel-order-same
phi-restriction-get-m-seq
superstep-construction
pipeline-loop-invariant
run-block-iters-append
branch-restriction-take-n
branch-restriction-remove-n
branch-restriction-pre-from-loop
branch-restriction-loop-from-loop
branch-restriction-get-m-seq
phi-restriction-take-n
phi-restriction-remove-n
phi-restriction-ccdfg-append
phi-restriction-loop-from-loop
loop-superstep-in-order
get-m-blocks-seq
pre-loop-cdr-same
run-block-set-append
run-block-set-remove-take-n
error-loop
invariant-greater-1-subgoal
loop-superstep-error
get-m-block-seq-equal-pre-superstep))
:do-not-induct t
:use
((:instance phi-restriction-block-iters
(c (loop-superstep-in-order m pre loop pp-interval))
(i (run-block-set (pre-superstep-in-order m pre loop pp-interval) init-state nil prev))
(iter k)
(pre1 (prefix (last (cadar (loop-superstep-in-parallel m pre loop pp-interval)))))
(pre2 prev))
(:instance phi-restriction-block-iters
(c (loop-superstep-in-order m pre loop pp-interval))
(i (run-block-set (pre-superstep-in-order m pre loop pp-interval) init-state nil prev))
(iter (+ k 1))
(pre1 (prefix (last (cadar (loop-superstep-in-parallel m pre loop pp-interval)))))
(pre2 prev))
(:instance phi-restriction-block-set
(c (loop-superstep-in-order m pre loop pp-interval))
(i (run-block-set (get-m-blocks-seq m loop pp-interval)
(run-blocks-iters loop (run-block-set pre init-state nil prev)
(- k 1)
prev) nil prev))
(next nil)
(pre1 (prefix (car (last (cadar (last (loop-superstep-in-parallel m pre loop pp-interval)))))))
(pre2 prev))
(:instance run-block-iters-reverse-append
(c loop)
(i (run-block-set pre init-state nil prev)))
(:instance run-block-set-remove-take-n
(pre loop)
(init-state (run-block-set (pre-superstep-from-loop m loop pp-interval)
(run-blocks-iters loop
(run-block-set pre init-state nil prev)
(+ -1 k)
prev)
nil prev)))
(:instance run-block-set-remove-take-n
(pre loop)
(init-state (run-blocks-iters loop (run-block-set pre init-state nil prev)
(+ -1 k) prev))
(prev (prefix (car (last (cadar (last (loop-superstep-in-parallel m pre loop pp-interval))))))))
(:instance run-block-set-no-conflict
(a (remove-n m loop))
(b (get-m-blocks-seq (- m pp-interval)
loop pp-interval))
(init-state (run-block-set (take-n m loop)
(run-blocks-iters loop
(run-block-set pre init-state nil prev)
(+ -1 k)
prev)
nil prev)))))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Invariant holds
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun induction-hint-invariant-function (k m pre loop pp-interval prev init-state)
(if (not (posp k)) (list m pre loop pp-interval prev init-state)
(induction-hint-invariant-function (- k 1) m pre loop pp-interval prev init-state)))
(defthm invariant-holds
(implies (and (posp k)
(posp m)
(posp pp-interval)
(true-listp pre)
(true-listp loop)
(consp pre)
(consp loop)
(<= m (len pre))
(phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop)
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(seq-ccdfg-p (list pre loop))
(not (equal (pre-supersteps-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-parallel m pre loop pp-interval) "error"))
(equal pp-ccdfg (superstep-construction pre loop pp-interval m)))
(pipeline-loop-invariant
(run-ccdfg-k (car pp-ccdfg) (second pp-ccdfg) k init-state prev)
k pre loop pp-interval init-state prev m))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(induction-hint-invariant-function))
:induct (induction-hint-invariant-function k m pre loop pp-interval prev init-state)
:do-not-induct t)
("Subgoal *1/2'"
:in-theory (union-theories (theory 'ground-zero)
'(invariant-base-case)))
("Subgoal *1/1"
:in-theory (union-theories (theory 'ground-zero)
'(invariant-base-case))
:use
((:instance invariant-holds-k-greater-1
(k (- k 1))
(pp-ccdfg (superstep-construction pre loop pp-interval m))
(pp-state (run-ccdfg-k (car (superstep-construction pre loop pp-interval m))
(cadr (superstep-construction pre loop pp-interval m))
(+ -1 k)
init-state prev)))
(:instance m
(m k))))))
(defthm usable-invariant-holds
(implies (and (posp k)
(posp m)
(posp pp-interval)
(true-listp pre)
(true-listp loop)
(consp pre)
(consp loop)
(<= m (len pre))
(phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop)
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(seq-ccdfg-p (list pre loop))
(not (equal (loop-superstep-in-parallel m pre loop pp-interval) "error"))
(not (equal (pre-supersteps-in-parallel m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error")))
(pipeline-loop-invariant
(run-ccdfg-k
(pre-supersteps-in-parallel m pre loop pp-interval)
(loop-superstep-in-parallel m pre loop pp-interval)
k init-state prev)
k pre loop pp-interval init-state prev m))
:otf-flg t
:hints
(("Goal"
:do-not-induct t
:in-theory (union-theories (theory 'ground-zero)
'(superstep-construction
branch-restriction-take-n
branch-restriction-remove-n))
:use
((:instance invariant-holds
(pp-ccdfg (superstep-construction pre loop pp-interval m)))))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Main correctness theorem
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; run-ccdfg in terms of run-ccdfg-k in pipelined ccdfg
(defthm run-ccdfg->run-k
(implies (phi-restriction-ccdfg post)
(equal (run-ccdfg pre loop post k init-state prev)
(run-block-set post
(run-ccdfg-k pre loop k init-state prev)
nil
prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-ccdfg
run-ccdfg-k))
:use
((:instance phi-restriction-block-set
(c post)
(i (run-blocks-iters loop (run-block-set pre
init-state nil prev)
k
(prefix (car (last (cadar (last loop)))))))
(next nil)
(pre1 (prefix (car (last (cadar (last post))))))
(pre2 prev))))))
(defun induction-hint-post (m pp-interval init-state loop prev)
(if (or (not (posp m))
(not (posp pp-interval))
(< m pp-interval)) (list m pp-interval init-state loop prev)
(induction-hint-post (- m pp-interval) pp-interval
(run-block-set loop init-state nil prev)
loop prev)))
(defthm branch-restriction-post-from-loop
(implies (branch-restriction-ccdfg loop)
(branch-restriction-ccdfg (post-superstep-in-order x loop pp-interval))))
(defthm run-post-subgoal
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg loop)
(not (equal (post-superstep-in-order m loop pp-interval) "error")))
(equal (run-block-set (post-superstep-in-order m loop pp-interval)
(run-block-set (get-m-blocks-seq m loop pp-interval)
init-state
nil
prev)
nil prev)
(run-blocks-iters loop init-state (ceiling m pp-interval) prev)))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories
(set-difference-theories (theory 'ground-zero)
(list 'ceiling))
(list 'induction-hint-post
'post-superstep-in-order
'get-m-blocks-seq
'run-block-set-from-iters
'run-block-set-nil
'run-block-set-remove-take-n
'ceiling-same
'get-m-block-seq-equal-pre-superstep
'run-block-set-append
'branch-restriction-remove-n
'branch-restriction-take-n
'branch-restriction-combine-iters
'branch-restriction-get-m-seq
'branch-restriction-post-from-loop))
:induct (induction-hint-post m pp-interval init-state loop prev)
:do-not-induct t
:do-not '(eliminate-destructors fertilize generalize))
("Subgoal *1/2.2"
:use
((:instance run-block-set-no-conflict
(a (remove-n m loop))
(b (get-m-blocks-seq (- m pp-interval) loop pp-interval))
(init-state (run-block-set (take-n m loop) init-state nil prev)))
(:instance run-block-iters-from-set
(c loop))
(:instance run-block-iters-add
(c loop)
(m 1)
(k (ceiling (- m pp-interval) pp-interval))
(i init-state))
(:instance ceiling-+)
(:instance ceiling-m-pp-interval-add)))
("Subgoal *1/1''"
:use
((:instance ceiling-m-pp-interval)
(:instance (:definition post-superstep-in-order))
(:instance (:definition get-m-blocks-seq))))))
;; pp-ccdfg is the pipelined ccdfg created by superstep-construction
;; pp-state is state after running pp-ccdfg for pre + k iterations
;; invariant is true which means pp-state is equal to executing seq-ccdfg's pre + (- k 1) iterations + sequence of m
(defthm invariant-implies-correctness
(implies (and (posp k)
(seq-ccdfg-p (list pre loop)) ;; needed to say cdr of pre = cdr of loop
(posp pp-interval)
(posp m)
(true-listp pre)
(true-listp loop)
(consp pre)
(consp loop)
(<= m (len pre))
(branch-restriction-ccdfg pre)
(branch-restriction-ccdfg loop)
(phi-restriction-ccdfg pre)
(phi-restriction-ccdfg loop)
(not (equal (pre-supersteps-in-parallel m pre loop pp-interval) "error"))
(not (equal (post-superstep-in-order m loop pp-interval) "error"))
(not (equal (loop-superstep-in-order m pre loop pp-interval) "error"))
(not (equal (loop-superstep-in-parallel m pre loop pp-interval) "error"))
(not (equal (post-superstep-in-parallel m loop pp-interval) "error"))
(equal pp-ccdfg (superstep-construction pre loop pp-interval m)))
(equal (run-ccdfg (car pp-ccdfg) (second pp-ccdfg) (third pp-ccdfg) k init-state prev)
(run-ccdfg pre loop nil (+ (+ -1 k) (ceiling m pp-interval)) init-state prev)))
:hints
(("Goal"
:do-not '(preprocess)
:in-theory (union-theories
(set-difference-theories (theory 'ground-zero)
(list 'ceiling))
(list 'superstep-construction
'run-ccdfg
'run-post-subgoal
'ceiling-m-pp-interval
'run-block-set-nil
'run-block-set-remove-take-n
'run-block-iters-reverse-append
'get-m-block-seq-equal-pre-superstep
'run-block-set-append
'run-block-iters-add
'pre-parallel-order-same
'post-parallel-order-same
'loop-parallel-order-same
'ceiling-natp
'phi-restriction-loop-superstep-in-order
'phi-restriction-loop-superstep-from-loop
'phi-restriction-loop-superstep-in-parallel
'phi-restriction-post-superstep-in-parallel
'branch-restriction-remove-n
'branch-restriction-take-n
'branch-restriction-combine-iters
'branch-restriction-get-m-seq
'branch-restriction-post-from-loop
'branch-restriction-pre-from-loop
'branch-restriction-loop-from-loop))
:do-not-induct t
:use
((:instance usable-invariant-holds)
(:instance run-ccdfg->run-k
(pre (pre-supersteps-in-parallel m pre loop pp-interval))
(loop (loop-superstep-in-parallel m pre loop pp-interval))
(post (post-superstep-in-parallel m loop pp-interval)))
(:instance pipeline-loop-invariant
(pp-state (run-ccdfg-k (pre-supersteps-in-parallel m pre loop pp-interval)
(loop-superstep-in-parallel m pre loop pp-interval)
k init-state prev)))
(:instance phi-restriction-block-iters
(c loop)
(i (run-block-set pre init-state nil prev))
(iter (+ -1 k (ceiling m pp-interval)))
(pre1 (prefix (car (last (cadar (last loop))))))
(pre2 prev))))))#|ACL2s-ToDo-Line|#
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