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;; Copyright (C) 2019, 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>
;; May 2019
(in-package "ADE")
(include-book "interl2")
(include-book "../gcd/gcd1")
(local (include-book "arithmetic-3/top" :dir :system))
(local (include-book "std/lists/sets" :dir :system))
(local (in-theory (disable nth)))
;; ======================================================================
;;; Table of Contents:
;;;
;;; 1. DE Module Generator of IGCD
;;; 2. Multi-Step State Lemma
;;; 3. Single-Step-Update Property
;;; 4. Relationship Between the Input and Output Sequences
;; ======================================================================
;; 1. DE Module Generator of IGCD
;;
;; Construct a DE module generator for circuits calculating the Greatest Common
;; Divisor (GCD) of two natural numbers. There are two mutually exclusive
;; input streams to the GCD submodule that are served on a
;; first-come-first-served basis via an arbitrated merge joint.
(defconst *igcd$select-num* *interl$select-num*)
(defconst *igcd$go-num* (+ *interl$go-num*
*gcd1$go-num*))
(defun igcd$data-ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ 3 (* 4 (mbe :logic (nfix data-size)
:exec data-size))))
(defun igcd$ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ (igcd$data-ins-len data-size)
*igcd$select-num*
*igcd$go-num*))
;; DE module generator of IGCD
(module-generator
igcd* (data-size)
(si 'igcd data-size)
(list* 'full-in0 'full-in1 'empty-out-
(append (sis 'data0-in 0 (* 2 data-size))
(sis 'data1-in 0 (* 2 data-size))
(cons 'select (sis 'go 0 *igcd$go-num*))))
(list* 'in0-act 'in1-act 'out-act
(sis 'data-out 0 data-size))
'(l interl gcd1)
(list
;; LINK
;; L
(list 'l
(list* 'l-status (sis 'd-out 0 (* 2 data-size)))
(si 'link (* 2 data-size))
(list* 'interl-out-act 'gcd1-in-act (sis 'd-in 0 (* 2 data-size))))
;; JOINTS
;; INTERL
(list 'interl
(list* 'in0-act 'in1-act 'interl-out-act
(sis 'd-in 0 (* 2 data-size)))
(si 'interl (* 2 data-size))
(list* 'full-in0 'full-in1 'l-status
(append (sis 'data0-in 0 (* 2 data-size))
(sis 'data1-in 0 (* 2 data-size))
(cons 'select (sis 'go 0 *interl$go-num*)))))
;; GCD1
(list 'gcd1
(list* 'gcd1-in-act 'out-act
(sis 'data-out 0 data-size))
(si 'gcd1 data-size)
(list* 'l-status 'empty-out-
(append (sis 'd-out 0 (* 2 data-size))
(sis 'go
*interl$go-num*
*gcd1$go-num*)))))
(declare (xargs :guard (natp data-size))))
(make-event
`(progn
,@(state-accessors-gen 'igcd '(l interl gcd1) 0)))
;; DE netlist generator. A generated netlist will contain an instance of
;; IGCD.
(defund igcd$netlist (data-size)
(declare (xargs :guard (and (natp data-size)
(<= 2 data-size))))
(cons (igcd* data-size)
(union$ (interl$netlist (* 2 data-size))
(gcd1$netlist data-size)
:test 'equal)))
;; Recognizer for IGCD
(defund igcd& (netlist data-size)
(declare (xargs :guard (and (alistp netlist)
(natp data-size)
(<= 2 data-size))))
(b* ((subnetlist (delete-to-eq (si 'igcd data-size) netlist)))
(and (equal (assoc (si 'igcd data-size) netlist)
(igcd* data-size))
(link& subnetlist (* 2 data-size))
(interl& subnetlist (* 2 data-size))
(gcd1& subnetlist data-size))))
;; Sanity check
(local
(defthmd check-igcd$netlist-64
(and (net-syntax-okp (igcd$netlist 64))
(net-arity-okp (igcd$netlist 64))
(igcd& (igcd$netlist 64) 64))))
;; Constraints on the state of IGCD
(defund igcd$st-format (st data-size)
(b* ((l (nth *igcd$l* st))
(interl (nth *igcd$interl* st))
(gcd1 (nth *igcd$gcd1* st)))
(and (link$st-format l (* 2 data-size))
(interl$st-format interl (* 2 data-size))
(gcd1$st-format gcd1 data-size))))
(defthm igcd$st-format=>constraint
(implies (igcd$st-format st data-size)
(and (natp data-size)
(<= 3 data-size)))
:hints (("Goal" :in-theory (enable igcd$st-format)))
:rule-classes :forward-chaining)
(defund igcd$valid-st (st data-size)
(b* ((l (nth *igcd$l* st))
(interl (nth *igcd$interl* st))
(gcd1 (nth *igcd$gcd1* st)))
(and (link$valid-st l (* 2 data-size))
(interl$valid-st interl (* 2 data-size))
(gcd1$valid-st gcd1 data-size))))
(defthmd igcd$valid-st=>constraint
(implies (igcd$valid-st st data-size)
(and (natp data-size)
(<= 3 data-size)))
:hints (("Goal" :in-theory (enable gcd1$valid-st=>constraint
igcd$valid-st)))
:rule-classes :forward-chaining)
(defthmd igcd$valid-st=>st-format
(implies (igcd$valid-st st data-size)
(igcd$st-format st data-size))
:hints (("Goal" :in-theory (e/d (interl$valid-st=>st-format
gcd1$valid-st=>st-format
igcd$st-format
igcd$valid-st)
(link$st-format)))))
;; Extract the input and output signals for IGCD
(progn
;; Extract the 1st input data item
(defun igcd$data0-in (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(take (* 2 (mbe :logic (nfix data-size)
:exec data-size))
(nthcdr 3 inputs)))
(defthm len-igcd$data0-in
(equal (len (igcd$data0-in inputs data-size))
(* 2 (nfix data-size))))
(in-theory (disable igcd$data0-in))
;; Extract the 2nd input data item
(defun igcd$data1-in (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(b* ((size (* 2 (mbe :logic (nfix data-size)
:exec data-size))))
(take size
(nthcdr (+ 3 size) inputs))))
(defthm len-igcd$data1-in
(equal (len (igcd$data1-in inputs data-size))
(* 2 (nfix data-size))))
(in-theory (disable igcd$data1-in))
;; Extract the inputs for joint INTERL
(defund igcd$interl-inputs (inputs st data-size)
(b* ((full-in0 (nth 0 inputs))
(full-in1 (nth 1 inputs))
(data0-in (igcd$data0-in inputs data-size))
(data1-in (igcd$data1-in inputs data-size))
(select (nth (igcd$data-ins-len data-size) inputs))
(go-signals (nthcdr (+ (igcd$data-ins-len data-size)
*igcd$select-num*)
inputs))
(interl-go-signals (take *interl$go-num* go-signals))
(l (nth *igcd$l* st))
(l.s (nth *link$s* l)))
(list* full-in0 full-in1 (f-buf (car l.s))
(append data0-in data1-in
(cons select interl-go-signals)))))
;; Extract the "out-act0" signal for joint INTERL
(defund igcd$interl-out-act0 (inputs st data-size)
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size))
(interl (nth *igcd$interl* st)))
(interl$out-act0 interl-inputs interl (* 2 data-size))))
;; Extract the "out-act1" signal for joint INTERL
(defund igcd$interl-out-act1 (inputs st data-size)
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size))
(interl (nth *igcd$interl* st)))
(interl$out-act1 interl-inputs interl (* 2 data-size))))
(defthm igcd$interl-out-act-mutually-exclusive
(implies (and (igcd$valid-st st data-size)
(igcd$interl-out-act0 inputs st data-size))
(not (igcd$interl-out-act1 inputs st data-size)))
:hints (("Goal" :in-theory (enable igcd$valid-st
igcd$interl-out-act0
igcd$interl-out-act1))))
;; Extract the "out-act" signal for joint INTERL
(defund igcd$interl-out-act (inputs st data-size)
(f-or (igcd$interl-out-act0 inputs st data-size)
(igcd$interl-out-act1 inputs st data-size)))
;; Extract the output data from joint INTERL
(defund igcd$interl-data-out (inputs st data-size)
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size))
(interl (nth *igcd$interl* st)))
(interl$data-out interl-inputs interl (* 2 data-size))))
;; Extract the inputs for joint GCD1
(defund igcd$gcd1-inputs (inputs st data-size)
(b* ((empty-out- (nth 2 inputs))
(go-signals (nthcdr (+ (igcd$data-ins-len data-size)
*igcd$select-num*)
inputs))
(gcd1-go-signals (take *gcd1$go-num*
(nthcdr *interl$go-num* go-signals)))
(l (nth *igcd$l* st))
(l.s (nth *link$s* l))
(l.d (nth *link$d* l)))
(list* (f-buf (car l.s)) empty-out-
(append (v-threefix (strip-cars l.d))
gcd1-go-signals))))
;; Extract the "in0-act" signal
(defund igcd$in0-act (inputs st data-size)
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size))
(interl (nth *igcd$interl* st)))
(interl$in0-act interl-inputs interl (* 2 data-size))))
;; Extract the "in1-act" signal
(defund igcd$in1-act (inputs st data-size)
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size))
(interl (nth *igcd$interl* st)))
(interl$in1-act interl-inputs interl (* 2 data-size))))
;; Extract the "out-act" signal
(defund igcd$out-act (inputs st data-size)
(gcd1$out-act (igcd$gcd1-inputs inputs st data-size)
(nth *igcd$gcd1* st)
data-size))
;; Extract the output data
(defund igcd$data-out (inputs st data-size)
(gcd1$data-out (igcd$gcd1-inputs inputs st data-size)
(nth *igcd$gcd1* st)
data-size))
(defthm len-igcd$data-out-1
(implies (igcd$st-format st data-size)
(equal (len (igcd$data-out inputs st data-size))
data-size))
:hints (("Goal" :in-theory (enable igcd$st-format
igcd$data-out))))
(defthm len-igcd$data-out-2
(implies (igcd$valid-st st data-size)
(equal (len (igcd$data-out inputs st data-size))
data-size))
:hints (("Goal" :in-theory (enable igcd$valid-st
igcd$data-out))))
(defthm bvp-igcd$data-out
(implies (and (igcd$valid-st st data-size)
(igcd$out-act inputs st data-size))
(bvp (igcd$data-out inputs st data-size)))
:hints (("Goal" :in-theory (enable igcd$valid-st
igcd$out-act
igcd$data-out))))
(defun igcd$outputs (inputs st data-size)
(list* (igcd$in0-act inputs st data-size)
(igcd$in1-act inputs st data-size)
(igcd$out-act inputs st data-size)
(igcd$data-out inputs st data-size)))
)
;; The value lemma for IGCD
(encapsulate
()
(local
(defthm arith-lemma
(implies (equal m (* 2 n))
(equal (* 2 m) (* 4 n)))))
(defthm igcd$value
(b* ((inputs (list* full-in0 full-in1 empty-out-
(append data0-in data1-in
(cons select go-signals)))))
(implies (and (igcd& netlist data-size)
(true-listp data0-in)
(equal (len data0-in) (* 2 data-size))
(true-listp data1-in)
(equal (len data1-in) (* 2 data-size))
(true-listp go-signals)
(equal (len go-signals) *igcd$go-num*)
(igcd$st-format st data-size))
(equal (se (si 'igcd data-size) inputs st netlist)
(igcd$outputs inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(se (si 'igcd data-size) inputs st netlist))
:in-theory (e/d (de-rules
igcd&
igcd*$destructure
igcd$st-format
igcd$data0-in
igcd$data1-in
igcd$interl-inputs
igcd$gcd1-inputs
igcd$in0-act
igcd$in1-act
igcd$out-act
igcd$data-out)
(de-module-disabled-rules)))))
;; This function specifies the next state of IGCD.
(defun igcd$step (inputs st data-size)
(b* ((l (nth *igcd$l* st))
(interl (nth *igcd$interl* st))
(gcd1 (nth *igcd$gcd1* st))
(interl-inputs (igcd$interl-inputs inputs st data-size))
(gcd1-inputs (igcd$gcd1-inputs inputs st data-size))
(interl-out-act (interl$out-act interl-inputs interl (* 2 data-size)))
(gcd1-in-act (gcd1$in-act gcd1-inputs gcd1 data-size))
(d-in (interl$data-out interl-inputs interl (* 2 data-size)))
(l-inputs (list* interl-out-act gcd1-in-act d-in)))
(list
;; L
(link$step l-inputs l (* 2 data-size))
;; Joint INTERL
(interl$step interl-inputs interl (* 2 data-size))
;; Joint GCD1
(gcd1$step gcd1-inputs gcd1 data-size))))
;; The state lemma for IGCD
(defthm igcd$state
(b* ((inputs (list* full-in0 full-in1 empty-out-
(append data0-in data1-in
(cons select go-signals)))))
(implies (and (igcd& netlist data-size)
(true-listp data0-in)
(equal (len data0-in) (* 2 data-size))
(true-listp data1-in)
(equal (len data1-in) (* 2 data-size))
(true-listp go-signals)
(equal (len go-signals) *igcd$go-num*)
(igcd$st-format st data-size))
(equal (de (si 'igcd data-size) inputs st netlist)
(igcd$step inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(de (si 'igcd data-size) inputs st netlist))
:in-theory (e/d (de-rules
igcd&
igcd*$destructure
igcd$st-format
igcd$data0-in
igcd$data1-in
igcd$interl-inputs
igcd$gcd1-inputs
igcd$in0-act
igcd$in1-act
igcd$out-act)
(de-module-disabled-rules)))))
(in-theory (disable igcd$step))
)
;; ======================================================================
;; 2. Multi-Step State Lemma
;; Conditions on the inputs
(defund igcd$input-format (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(b* ((full-in0 (nth 0 inputs))
(full-in1 (nth 1 inputs))
(empty-out- (nth 2 inputs))
(data0-in (igcd$data0-in inputs data-size))
(data1-in (igcd$data1-in inputs data-size))
(select (nth (igcd$data-ins-len data-size) inputs))
(go-signals (nthcdr (+ (igcd$data-ins-len data-size)
*igcd$select-num*)
inputs)))
(and
(booleanp full-in0)
(booleanp full-in1)
(booleanp empty-out-)
(or (not full-in0) (bvp data0-in))
(or (not full-in1) (bvp data1-in))
(true-listp go-signals)
(= (len go-signals) *igcd$go-num*)
(equal inputs
(list* full-in0 full-in1 empty-out-
(append data0-in data1-in (cons select go-signals)))))))
(local
(defthm igcd$input-format=>interl$input-format
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(interl$input-format
(igcd$interl-inputs inputs st data-size)
(* 2 data-size)))
:hints (("Goal"
:in-theory (e/d (open-nth
gcd1$valid-st=>constraint
interl$input-format
interl$data0-in
interl$data1-in
igcd$input-format
igcd$valid-st
igcd$interl-inputs)
())))))
(local
(defthm igcd$input-format=>gcd1$input-format
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(gcd1$input-format
(igcd$gcd1-inputs inputs st data-size)
data-size))
:hints (("Goal"
:in-theory (e/d (gcd1$valid-st=>constraint
gcd1$input-format
gcd1$data-in
igcd$input-format
igcd$valid-st
igcd$gcd1-inputs)
())))))
(defthm booleanp-igcd$in0-act
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(booleanp (igcd$in0-act inputs st data-size)))
:hints (("Goal"
:in-theory (enable igcd$valid-st
igcd$in0-act)))
:rule-classes (:rewrite :type-prescription))
(defthm booleanp-igcd$in1-act
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(booleanp (igcd$in1-act inputs st data-size)))
:hints (("Goal"
:in-theory (enable igcd$valid-st
igcd$in1-act)))
:rule-classes (:rewrite :type-prescription))
(defthm booleanp-igcd$out-act
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(booleanp (igcd$out-act inputs st data-size)))
:hints (("Goal"
:in-theory (enable igcd$valid-st
igcd$out-act)))
:rule-classes (:rewrite :type-prescription))
(simulate-lemma igcd)
;; ======================================================================
;; 3. Single-Step-Update Property
;; The operation of IGCD over a data sequence
(defun igcd$op-map (x)
(gcd$op-map x))
;; The extraction functions for IGCD
(defund igcd$extract0 (st)
(b* ((interl (nth *igcd$interl* st)))
(igcd$op-map (interl$extract0 interl))))
(defund igcd$extract1 (st)
(b* ((interl (nth *igcd$interl* st)))
(igcd$op-map (interl$extract1 interl))))
(defund igcd$extract2 (st)
(b* ((l (nth *igcd$l* st))
(gcd1 (nth *igcd$gcd1* st)))
(append (igcd$op-map (extract-valid-data (list l)))
(gcd1$extract gcd1))))
(defthm igcd$extract0-not-empty
(implies (and (igcd$interl-out-act0 inputs st data-size)
(igcd$valid-st st data-size))
(< 0 (len (igcd$extract0 st))))
:hints (("Goal"
:in-theory (e/d (igcd$interl-out-act0
igcd$valid-st
igcd$extract0)
())))
:rule-classes :linear)
(defthm igcd$extract1-not-empty
(implies (and (igcd$interl-out-act1 inputs st data-size)
(igcd$valid-st st data-size))
(< 0 (len (igcd$extract1 st))))
:hints (("Goal"
:in-theory (e/d (igcd$interl-out-act1
igcd$valid-st
igcd$extract1)
())))
:rule-classes :linear)
(defthm igcd$extract2-not-empty
(implies (and (igcd$out-act inputs st data-size)
(igcd$valid-st st data-size))
(< 0 (len (igcd$extract2 st))))
:hints (("Goal"
:in-theory (e/d (igcd$out-act
igcd$valid-st
igcd$extract2)
())))
:rule-classes :linear)
;; Specify and prove a state invariant
(progn
(defund igcd$inv (st)
(b* ((gcd1 (nth *igcd$gcd1* st)))
(gcd1$inv gcd1)))
(defthm igcd$inv-preserved
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size)
(igcd$inv st))
(igcd$inv (igcd$step inputs st data-size)))
:hints (("Goal"
:in-theory (e/d (igcd$valid-st
igcd$inv
igcd$step)
()))))
)
;; The extracted next-state functions for IGCD. Note that these functions
;; avoid exploring the internal computation of IGCD.
(defund igcd$extracted0-step (inputs st data-size)
(b* ((data (gcd$op (igcd$data0-in inputs data-size)))
(extracted-st (igcd$extract0 st))
(n (1- (len extracted-st))))
(cond
((equal (igcd$interl-out-act0 inputs st data-size) t)
(cond
((equal (igcd$in0-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (igcd$in0-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
(defund igcd$extracted1-step (inputs st data-size)
(b* ((data (gcd$op (igcd$data1-in inputs data-size)))
(extracted-st (igcd$extract1 st))
(n (1- (len extracted-st))))
(cond
((equal (igcd$interl-out-act1 inputs st data-size) t)
(cond
((equal (igcd$in1-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (igcd$in1-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
(defund igcd$extracted2-step (inputs st data-size)
(b* ((data (gcd$op (igcd$interl-data-out inputs st data-size)))
(extracted-st (igcd$extract2 st))
(n (1- (len extracted-st))))
(cond
((equal (igcd$out-act inputs st data-size) t)
(cond
((equal (igcd$interl-out-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (igcd$interl-out-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
;; The single-step-update property
(progn
(local
(defthm take-of-gcd$op-map
(equal (take n (gcd$op-map l))
(gcd$op-map (take n l)))
:hints (("Goal" :in-theory (enable repeat)))))
(local
(defthm igcd-aux-1
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size)))
(implies (natp data-size)
(equal (interl$data0-in interl-inputs (* 2 data-size))
(take (* 2 data-size)
(nthcdr 3 inputs)))))
:hints (("Goal" :in-theory (enable igcd$interl-inputs
igcd$data0-in
interl$data0-in)))))
(local
(defthm igcd-aux-2
(b* ((interl-inputs (igcd$interl-inputs inputs st data-size)))
(implies (natp data-size)
(equal (interl$data1-in interl-inputs (* 2 data-size))
(take (* 2 data-size)
(nthcdr (+ 3 (* 2 data-size)) inputs)))))
:hints (("Goal" :in-theory (enable igcd$interl-inputs
igcd$data1-in
interl$data1-in)))))
(defthm igcd$extracted0+1-step-correct
(b* ((next-st (igcd$step inputs st data-size)))
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(and (equal (igcd$extract0 next-st)
(igcd$extracted0-step inputs st data-size))
(equal (igcd$extract1 next-st)
(igcd$extracted1-step inputs st data-size)))))
:hints (("Goal"
:in-theory (e/d (f-sr
gcd1$valid-st=>constraint
interl$extracted0-step
interl$extracted1-step
igcd$extracted0-step
igcd$extracted1-step
igcd$valid-st
igcd$step
igcd$data0-in
igcd$data1-in
igcd$interl-out-act0
igcd$interl-out-act1
igcd$interl-out-act
igcd$in0-act
igcd$in1-act
igcd$extract0
igcd$extract1)
(link$valid-st
link$step)))))
(local
(defthm igcd$interl-out-act-inactive
(implies (equal (nth *link$s*
(nth *igcd$l* st))
'(t))
(and (not (interl$out-act0
(igcd$interl-inputs inputs st data-size)
(nth *igcd$interl* st)
(* 2 data-size)))
(not (interl$out-act1
(igcd$interl-inputs inputs st data-size)
(nth *igcd$interl* st)
(* 2 data-size)))))
:hints (("Goal"
:in-theory (e/d (igcd$interl-inputs)
(nfix))))))
(local
(defthm igcd$gcd1-in-act-inactive
(implies (equal (nth *link$s*
(nth *igcd$l* st))
'(nil))
(not (gcd1$in-act (igcd$gcd1-inputs inputs st data-size)
(nth *igcd$gcd1* st)
data-size)))
:hints (("Goal"
:in-theory (e/d (igcd$gcd1-inputs)
(nfix))))))
(local
(defthm igcd-aux-3
(b* ((gcd1-inputs (igcd$gcd1-inputs inputs st data-size))
(l (nth *igcd$l* st))
(l.d (nth *link$d* l)))
(implies (and (natp data-size)
(equal (len l.d) (* 2 data-size))
(bvp (strip-cars l.d)))
(equal (gcd1$data-in gcd1-inputs data-size)
(strip-cars l.d))))
:hints (("Goal" :in-theory (enable igcd$gcd1-inputs
gcd1$data-in)))))
(defthm igcd$extracted2-step-correct
(b* ((next-st (igcd$step inputs st data-size)))
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size)
(igcd$inv st))
(equal (igcd$extract2 next-st)
(igcd$extracted2-step inputs st data-size))))
:hints (("Goal"
:use igcd$input-format=>gcd1$input-format
:in-theory (e/d (f-sr
interl$out-act
gcd1$valid-st=>constraint
gcd1$extracted-step
igcd$extracted2-step
igcd$valid-st
igcd$inv
igcd$step
igcd$interl-out-act0
igcd$interl-out-act1
igcd$interl-out-act
igcd$interl-data-out
igcd$out-act
igcd$extract2)
(igcd$input-format=>gcd1$input-format
interl$extract0-lemma
interl$extract1-lemma)))))
)
;; ======================================================================
;; 4. Relationship Between the Input and Output Sequences
;; Prove that igcd$valid-st is an invariant.
(encapsulate
()
(local
(defthm igcd$valid-st-preserved-aux-1
(implies (and (equal (nth 2 inputs2) (nth 0 inputs1))
(booleanp (nth 2 inputs2)))
(not (and (interl$out-act inputs2 st2 data-size2)
(gcd1$in-act inputs1 st1 data-size1))))
:hints (("Goal" :cases ((nth 2 inputs2))))))
(local
(defthm igcd$valid-st-preserved-aux-2
(implies (link$valid-st st data-size)
(booleanp (car (nth *link$s* st))))
:rule-classes (:rewrite :type-prescription)))
(defthm igcd$valid-st-preserved
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size))
(igcd$valid-st (igcd$step inputs st data-size)
data-size))
:hints (("Goal"
:use (igcd$input-format=>interl$input-format
igcd$input-format=>gcd1$input-format)
:in-theory (e/d (igcd$valid-st
igcd$step
igcd$interl-inputs
igcd$gcd1-inputs)
(igcd$input-format=>interl$input-format
igcd$input-format=>gcd1$input-format
link$valid-st
link$step
nfix)))))
)
(encapsulate
()
(local
(defthm nthcdr-gcd$op-map
(equal (nthcdr n (gcd$op-map l))
(gcd$op-map (nthcdr n l)))))
(local
(defthm interl$extract0-lemma-alt
(implies (and (interl$input-format inputs data-size)
(interl$valid-st st data-size)
(equal n (1- (len (interl$extract0 st))))
(interl$out-act0 inputs st data-size))
(equal (nthcdr n (interl$extract0 st))
(list (interl$data-out inputs st data-size))))))
(defthm igcd$extract0-lemma
(implies
(and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size)
(igcd$interl-out-act0 inputs st data-size))
(equal (list (gcd$op (igcd$interl-data-out inputs st data-size)))
(nthcdr (1- (len (igcd$extract0 st)))
(igcd$extract0 st))))
:hints (("Goal"
:use igcd$input-format=>interl$input-format
:in-theory (e/d (igcd$valid-st
igcd$interl-inputs
igcd$extract0
igcd$interl-out-act0
igcd$interl-data-out)
(igcd$input-format=>interl$input-format
interl$extract0-lemma)))))
(local
(defthm interl$extract1-lemma-alt
(implies (and (interl$input-format inputs data-size)
(interl$valid-st st data-size)
(equal n (1- (len (interl$extract1 st))))
(interl$out-act1 inputs st data-size))
(equal (nthcdr n (interl$extract1 st))
(list (interl$data-out inputs st data-size))))))
(defthm igcd$extract1-lemma
(implies
(and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size)
(igcd$interl-out-act1 inputs st data-size))
(equal (list (gcd$op (igcd$interl-data-out inputs st data-size)))
(nthcdr (1- (len (igcd$extract1 st)))
(igcd$extract1 st))))
:hints (("Goal"
:use igcd$input-format=>interl$input-format
:in-theory (e/d (igcd$valid-st
igcd$interl-inputs
igcd$extract1
igcd$interl-out-act1
igcd$interl-data-out)
(igcd$input-format=>interl$input-format
interl$extract1-lemma)))))
)
(defthm igcd$extract2-lemma
(implies (and (igcd$input-format inputs data-size)
(igcd$valid-st st data-size)
(igcd$out-act inputs st data-size))
(equal (list (igcd$data-out inputs st data-size))
(nthcdr (1- (len (igcd$extract2 st)))
(igcd$extract2 st))))
:hints (("Goal"
:in-theory (enable igcd$valid-st
igcd$extract2
igcd$out-act
igcd$data-out))))
;; Extract the accepted input sequences
(seq-gen igcd in0 in0-act 0
(igcd$data0-in inputs data-size))
(seq-gen igcd in1 in1-act 1
(igcd$data1-in inputs data-size))
;; Extract the valid output sequence
(seq-gen igcd out out-act 2
(igcd$data-out inputs st data-size)
:netlist-data (nthcdr 3 outputs))
;; The multi-step input-output relationship
(encapsulate
()
(local
(defthm member-append-prepend-rec-instance-1
(implies (and (member (append a b c) (prepend-rec x y))
(equal y++x1 (append y x1)))
(member (append a b c x1)
(prepend-rec x y++x1)))
:hints (("Goal" :use (:instance member-append-prepend-rec
(e (append a b c))
(x (prepend-rec x y))
(e1 x1))))))
(local
(defthm member-append-prepend-rec-instance-2
(implies (and (member (append a b c)
(prepend-rec (interleave x y) (cons e z)))
(equal z++x1 (append z x1))
(equal xe (append x (list e)))
(true-listp y))
(member (append a b c x1)
(prepend-rec (interleave xe y) z++x1)))
:hints (("Goal"
:in-theory (disable member-append-prepend-rec-instance-1)
:use (:instance member-append-prepend-rec-instance-1
(x (interleave xe y))
(y z)
(y++x1 z++x1))))))
(local
(defthm member-append-prepend-rec-instance-3
(implies (and (member (append a b c)
(prepend-rec (interleave x y) (cons e z)))
(equal z++x1 (append z x1))
(equal ye (append y (list e)))
(true-listp x))
(member (append a b c x1)
(prepend-rec (interleave x ye) z++x1)))
:hints (("Goal"
:in-theory (disable member-append-prepend-rec-instance-1)
:use (:instance member-append-prepend-rec-instance-1
(x (interleave x ye))
(y z)
(y++x1 z++x1))))))
(defthmd igcd$dataflow-correct
(b* ((extracted0-st (igcd$extract0 st))
(extracted1-st (igcd$extract1 st))
(extracted2-st (igcd$extract2 st))
(final-st (igcd$run inputs-seq st data-size n))
(final-extracted0-st (igcd$extract0 final-st))
(final-extracted1-st (igcd$extract1 final-st))
(final-extracted2-st (igcd$extract2 final-st)))
(implies
(and (igcd$input-format-n inputs-seq data-size n)
(igcd$valid-st st data-size)
(igcd$inv st)
(member x (interleave final-extracted0-st final-extracted1-st)))
(member
(append x
final-extracted2-st
(igcd$out-seq inputs-seq st data-size n))
(prepend-rec
(interleave (append (igcd$op-map
(igcd$in0-seq inputs-seq st data-size n))
extracted0-st)
(append (igcd$op-map
(igcd$in1-seq inputs-seq st data-size n))
extracted1-st))
extracted2-st))))
:hints (("Goal" :in-theory (enable f-or
member-of-true-list-list-is-true-list
igcd$interl-out-act
igcd$extracted0-step
igcd$extracted1-step
igcd$extracted2-step))))
(defthmd igcd$functionally-correct
(b* ((extracted0-st (igcd$extract0 st))
(extracted1-st (igcd$extract1 st))
(extracted2-st (igcd$extract2 st))
(final-st (de-n (si 'igcd data-size) inputs-seq st netlist n))
(final-extracted0-st (igcd$extract0 final-st))
(final-extracted1-st (igcd$extract1 final-st))
(final-extracted2-st (igcd$extract2 final-st)))
(implies
(and (igcd& netlist data-size)
(igcd$input-format-n inputs-seq data-size n)
(igcd$valid-st st data-size)
(igcd$inv st)
(member x (interleave final-extracted0-st final-extracted1-st)))
(member
(append x
final-extracted2-st
(igcd$out-seq-netlist
inputs-seq st netlist data-size n))
(prepend-rec
(interleave (append (igcd$op-map
(igcd$in0-seq-netlist
inputs-seq st netlist data-size n))
extracted0-st)
(append (igcd$op-map
(igcd$in1-seq-netlist
inputs-seq st netlist data-size n))
extracted1-st))
extracted2-st))))
:hints (("Goal"
:use igcd$dataflow-correct
:in-theory (enable igcd$valid-st=>st-format
igcd$de-n))))
)
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