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#||
general-theorems.lisp
~~~~~~~~~~~~~~~~~~~~~~
Author: Disha Puri
Last Updated: 12th April 2014
This file the general theorems about syntax and semantics
to be used in other files.
||#
(in-package "ACL2")
(include-book "semantics")
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Theorems when there is no phi-statement in the CCDFG
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm phi-restriction-next-block
(implies (phi-restriction-list l)
(phi-restriction-list (next-to-execute l a))))
(defthm phi-restriction-run-lists
(implies (phi-restriction-list l)
(equal (run-lists l i next pre1)
(run-lists l i next pre2)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(phi-restriction-list
run-lists
execute-statement
phi-restriction-next-block)))))
;; if no phi-step then previous scheduling step does not matter in execution
(defthm phi-restriction-run-block
(implies (phi-restriction-block b)
(equal (run-block b i next pre1)
(run-block b i next pre2)))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block
phi-restriction-block))
:induct t
:do-not-induct t)
("Subgoal *1/2''"
:use
((:instance phi-restriction-run-lists
(l (car b)))
(:instance (:definition run-block)
(init-state i)
(prev pre2))))))
(defthm phi-restriction-block-set
(implies (phi-restriction-ccdfg c)
(equal (run-block-set c i next pre1)
(run-block-set c i next pre2)))
:otf-flg t
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(phi-restriction-ccdfg
run-block-set))
:do-not '(eliminate-destructors fertilize generalize)
:induct (run-block-set c i next pre1)
:do-not-induct t)
("Subgoal *1/2''"
:use
((:instance (:definition run-block-set)
(init-state i)
(next-bb next)
(prev pre2))
(:instance phi-restriction-run-block
(b (cadar c)))))))
(in-theory (disable phi-restriction-block-set))
(defthm phi-restriction-block-iters
(implies (phi-restriction-ccdfg c)
(equal (run-blocks-iters c i iter pre1)
(run-blocks-iters c i iter pre2)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-blocks-iters)))
("Subgoal *1/2'"
:use
((:instance (:definition run-blocks-iters)
(init-state i)
(prev pre2))
(:instance phi-restriction-block-set
(next nil))))))
(in-theory (disable phi-restriction-block-iters))
(defthm phi-restriction-append-reverse
(implies (and (phi-restriction-ccdfg a)
(phi-restriction-ccdfg b))
(phi-restriction-ccdfg (append a b))))
(defthm phi-restriction-remove-n
(implies (phi-restriction-ccdfg a)
(phi-restriction-ccdfg (remove-n x a))))
(defthm phi-restriction-take-n
(implies (phi-restriction-ccdfg a)
(phi-restriction-ccdfg (take-n x a))))
(defthm phi-restriction-cdr
(implies (phi-restriction-ccdfg a)
(phi-restriction-ccdfg (cdr a))))
(defthm phi-restriction-nil
(phi-restriction-ccdfg nil))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Theorems when there are no branch statements in the CCDFG
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm branch-restriction-list-get-next-block
(implies (branch-restriction-list b)
(equal (get-next-lists b i) nil))
:hints
(("Goal"
:in-theory (disable branch-statement-p)
:induct t
:do-not-induct t)))
(defthm branch-restriction-block-get-next-block
(implies (branch-restriction-block b)
(equal (get-next-block b i) nil))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(branch-restriction-block
branch-restriction-list-get-next-block
get-next-block)))))
(defthm branch-restriction-ccdfg-cdr
(implies (branch-restriction-ccdfg c)
(branch-restriction-ccdfg (cdr c))))
(defthm branch-restriction-ccdfg-car
(implies (branch-restriction-ccdfg c)
(branch-restriction-ccdfg (list (car c)))))
(defthm branch-restriction-remove-n
(implies (branch-restriction-ccdfg a)
(branch-restriction-ccdfg (remove-n x a))))
(defthm branch-restriction-take-n
(implies (branch-restriction-ccdfg a)
(branch-restriction-ccdfg (take-n x a))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Trivial theorems
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm take-n-1
(implies (consp pre)
(equal (take-n 1 pre)
(list (car pre)))))
(defthm remove-n-1
(implies (consp pre)
(equal (remove-n 1 pre)
(cdr pre))))
(defthm m
(implies (acl2-numberp m)
(equal (+ -1 1 m) m)))
(defthm add-3
(equal (+ a b c) (+ b c a)))
(defthm not-posp
(implies (<= m pp-interval)
(not (posp (- m pp-interval)))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Theorems related to run-block-set
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; not proving this theorem in this script
;; The theorem says that if there are no read-write hazards and no conditional
;; branches, then executing a followed by b is same as executing b followed by a
;; not part of invariant, part of algorithm
;; we have not worked on this proof yet, as we were more focussed on proving
;; that if we can create a pipelined CCDFG with this acceptable assumption, then
;; that CCDFG follows our invariant
(defthms run-block-set-no-conflict
(implies (and (no-conflict a b)
(branch-restriction-ccdfg a)
(branch-restriction-ccdfg b))
(equal (run-block-set a (run-block-set b init-state nil prev) nil prev)
(run-block-set b (run-block-set a init-state nil prev) nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set))
:induct t
:do-not-induct t)))
;; will prove theorem related to no-conflict later
(defthms no-conflict-extended
(implies (no-conflict b c)
(no-conflict c (list (list name (cadar b))))))
(defthm run-block-set-append
(implies (and (branch-restriction-ccdfg a)
(branch-restriction-ccdfg b))
(equal (run-block-set (append a b) init-state nil prev)
(run-block-set b
(run-block-set a init-state nil prev)
nil
prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set
branch-restriction-ccdfg
branch-restriction-block-get-next-block))
:induct t
:do-not-induct t)))
(defthm run-block-set-from-iters
(equal (run-blocks-iters c init-state 1 prev)
(run-block-set c init-state nil prev))
:hints
(("Goal"
:in-theory (theory 'ground-zero)
:use
((:instance run-blocks-iters
(iter 1))
(:instance run-blocks-iters
(iter 0)
(init-state (run-block-set c init-state nil prev)))))))
(defthm not-consp-run-block-set
(implies (not (consp c))
(equal (run-block-set c init-state nil prev) init-state)))
(defthm run-block-set-reverse-append
(implies (and (branch-restriction-ccdfg a)
(branch-restriction-ccdfg b))
(equal (run-block-set a (run-block-set b init-state nil prev)
nil prev)
(run-block-set (append b a) init-state nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set-append)))))
(defthm run-block->run-set
(equal (run-block (cadr b) init-state nil prev)
(run-block-set (list b) init-state nil prev))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set)))))
(defthm run-block-set-cons
(implies (and (branch-restriction-ccdfg (list a))
(branch-restriction-ccdfg b))
(equal (run-block-set (cons a b) init-state nil prev)
(run-block-set b
(run-block-set (list a) init-state nil prev)
nil prev))))
(defthm run-block-set-nil
(equal (run-block-set nil init-state next prev)
init-state))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Theorems related to execution of pre and loop
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun induction-hint-run-block-set-remove-take-n (m init-state pre prev)
(if (or (endp pre)
(zp m)) (list m init-state pre prev)
(induction-hint-run-block-set-remove-take-n (- m 1)
(run-block-set (list (car pre))
init-state nil prev)
(cdr pre)
prev)))
(defthm run-block-set-remove-take-n
(implies (and (posp m)
(branch-restriction-ccdfg pre))
(equal (run-block-set (remove-n m pre)
(run-block-set (take-n m pre)
init-state nil prev)
nil prev)
(run-block-set pre init-state nil prev)))
:hints
(("Goal"
:do-not '(preprocess generalize fertilize)
:induct (induction-hint-run-block-set-remove-take-n m init-state pre prev)
:do-not-induct t
:in-theory (union-theories (theory 'ground-zero)
'(induction-hint-run-block-set-remove-take-n
remove-n-1
take-n-1
branch-restriction-ccdfg-cdr
branch-restriction-ccdfg-car
branch-restriction-take-n
branch-restriction-remove-n
run-block-set-reverse-append
run-block-set-nil
take-n
remove-n
append)))))
(defthm take-return-take
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg loop))
(equal (run-block-set (take-n pp-interval (remove-n m loop))
(run-block-set (take-n m loop) init-state nil prev)
nil prev)
(run-block-set (take-n (+ m pp-interval) loop)
init-state nil prev)))
:otf-flg t
:hints
(("Goal"
:do-not '(preprocess generalize fertilize)
:induct (induction-hint-run-block-set-remove-take-n m init-state loop prev)
:do-not-induct t
:in-theory (union-theories (theory 'ground-zero)
'(induction-hint-run-block-set-remove-take-n
remove-n-1
take-n-1
branch-restriction-take-n
branch-restriction-remove-n
branch-restriction-ccdfg-cdr
branch-restriction-ccdfg-car
run-block-set-reverse-append
take-n
append
run-block-set-nil
remove-n)))
("Subgoal *1/2.3''"
:use
((:instance (:definition take-n)
(n (+ 1 pp-interval))
(l loop))
(:instance m
(m pp-interval))))
("Subgoal *1/2.1'"
:use
((:instance (:definition take-n)
(n (+ m pp-interval))
(l loop))
(:instance add-3
(a pp-interval)
(b -1)
(c m))))))
(defthm append-three
(equal (append (append a b) c) (append a (append b c))))
(defthm run-block-iters-from-set
(equal (run-block-set c init-state nil prev)
(run-blocks-iters c init-state 1 prev))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-set-from-iters)))))
(defthm k
(equal (+ 1 -1 k) (+ -1 1 k)))
(defthm run-block-iters-append-subgoal
(implies (and (natp k)
(natp m))
(equal (run-blocks-iters c i (+ m k) prev)
(run-blocks-iters c (run-blocks-iters c i k prev)
m prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-blocks-iters
run-block-iters-from-set)))
("Subgoal *1/3'"
:use
((:instance (:definition run-blocks-iters)
(init-state i)
(iter (+ 1 m)))
(:instance m)))))
(defthm run-block-iters-append
(implies (natp k)
(equal (run-blocks-iters c i (+ 1 k) prev)
(run-block-set c (run-blocks-iters c i k prev)
nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-iters-append-subgoal
run-block-set-from-iters)))))
(defthm run-block-iters-add
(implies (and (natp m)
(natp k))
(equal (run-blocks-iters c (run-blocks-iters c i m prev) k prev)
(run-blocks-iters c i (+ k m) prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(run-block-iters-append-subgoal)))))
;; you need this notion that apart from first block, pre and loop are same
;; true by assumption
(defthm pre-loop-cdr-same
(implies (and (posp m)
(consp pre)
(consp loop)
(seq-ccdfg-p (list pre loop)))
(equal (remove-n m pre)
(remove-n m loop)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(seq-ccdfg-p
remove-n)))))
(defthm run-block-iters-reverse-append
(implies (posp k)
(equal (run-block-set c (run-blocks-iters c i (+ -1 k) prev)
nil prev)
(run-blocks-iters c i (+ -1 1 k) prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(k))
:use
((:instance run-block-iters-append
(k (- k 1)))))))
(defthm take-n-plus-interval
(implies (and (posp m)
(posp pp-interval)
(branch-restriction-ccdfg loop))
(equal (run-block-set (take-n (+ m pp-interval) loop) init-state nil prev)
(run-block-set (take-n pp-interval (remove-n m loop))
(run-block-set (take-n m loop) init-state nil prev)
nil prev)))
:hints
(("Goal"
:in-theory (union-theories (theory 'ground-zero)
'(take-return-take
branch-restriction-ccdfg-cdr
branch-restriction-take-n
branch-restriction-remove-n)))))
|
