#|
Copyright (C) 1994 by Alex Bronstein and Carolyn Talcott.  All Rights
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|#

Event: Start with the library "mlp" using the compiled version.

; countstut.bm: a simple clock counter, and a STUTTERING version of it,
;     to test Stutter formalizaton.
;
; . A2PC blows up on stuttering version, but that's been reduced to a
; a very trivial version (3 lines) which also blows up and on which one
; can see that BM loops without any rule: one of the equality hyp. is used
; in the opposite direction as the definition expansion.
; We use ADD-AXIOM on A2-PCs which blow.
;
;;; DEFINITION OF CIRCUITs:

#|
(setq A '(sy-A (x)
(Yout R 0 Y1)
(Y1 S Inc Yout)
))
(setq B '(sy-B (x)
(Yout R 0 Y1m)
(Y1m S Mux Yst Yout Y1)
(Y1 S Inc Yout)
(Yst R T Yst1)
(Yst1 S Not Yst)
))
(setq C '(sy-C (xst)
(Yout R 0 Y1m)
(Y1m S Mux xst Yout Y1)
(Y1 S Inc Yout)
))
(setq countstut '( |#
; BM DEFINITIONS and A2 LEMMAS, generated by BMSYSD:
; comb_inc.bm: INCrement combinational element
; U7-DONE

Definition: inc (u) = (1 + u)

; Everything below generated by: (bmcomb 'inc '() '(x))

Definition: s-inc (x) = if empty (x) then e

else a (s-inc (p (x)), inc (l (x))) endif

;; A2-Begin-S-INC

Theorem: a2-empty-s-inc empty (s-inc (x)) = empty (x)

Theorem: a2-e-s-inc (s-inc (x) = e) = empty (x)

Theorem: a2-lp-s-inc len (s-inc (x)) = len (x)

Theorem: a2-lpe-s-inc eqlen (s-inc (x), x)

Theorem: a2-ic-s-inc s-inc (i (c x, x)) = i (inc (c x), s-inc (x))

Theorem: a2-lc-s-inc (¬ empty (x)) → (l (s-inc (x)) = inc (l (x)))

Theorem: a2-pc-s-inc p (s-inc (x)) = s-inc (p (x))

Theorem: a2-hc-s-inc (¬ empty (x)) → (h (s-inc (x)) = inc (h (x)))

Theorem: a2-bc-s-inc b (s-inc (x)) = s-inc (b (x))

Theorem: a2-bnc-s-inc bn (n, s-inc (x)) = s-inc (bn (n, x))

;; A2-End-S-INC
; eof:comb_inc.bm

Definition: topor-sy-a (ln) = if ln = 'yout then 0

elseif ln = 'y1 then 1

else 0 endif

Definition: sy-a (ln, x) = if ln = 'yout

then if empty (x) then e

else i (0, sy-a ('y1, p (x))) endif elseif ln = 'y1 then s-inc (sy-a ('yout, x)) else sfix (x) endif

;; A2-Begin-SY-A

Theorem: a2-empty-sy-a empty (sy-a (ln, x)) = empty (x)

Theorem: a2-e-sy-a (sy-a (ln, x) = e) = empty (x)

Theorem: a2-lp-sy-a len (sy-a (ln, x)) = len (x)

Theorem: a2-lpe-sy-a eqlen (sy-a (ln, x), x)

Theorem: a2-pc-sy-a p (sy-a (ln, x)) = sy-a (ln, p (x))

;; A2-End-SY-A
; BM DEFINITIONS and A2 LEMMAS, generated by BMSYSD:
; comb_mux.bm: Mux combinational element, i.e. "if".
; U7-DONE

Definition: mux (u1 , u2 , u3 ) = if u1 then u2

else u3 endif

; everything below generated by: (bmcomb 'mux '() '(x1 x2 x3))
; with the EXCEPTIONS/HAND-MODIFICATIONS given below.

Definition: s-mux (x1 , x2 , x3 ) = if empty (x1 ) then e

else a (s-mux (p (x1 ), p (x2 ), p (x3 )), mux (l (x1 ), l (x2 ), l (x3 ))) endif

; SMUX-is-SIF can make things much simpler on occasions:

Theorem: smux-is-sif s-mux (x1 , x2 , x3 ) = s-if (x1 , x2 , x3 )

Event: Disable smux-is-sif.

; We take advantage of SMUX-is-SIF for all inductive proofs. To do so we
; HAND-MODIFY the code generated by Sugar to replace all the hints by
;   - A2-EMPTY, A2-PC replace hint with: ((enable smux-is-sif))

;   - A2-LP, A2-IC, A2-HC, A2-BC: ((enable smux-is-sif) (disable len))
;   - A2-BNC: ((enable smux-is-sif) (disable bn len))
;; A2-Begin-S-MUX

Theorem: a2-empty-s-mux empty (s-mux (x1 , x2 , x3 )) = empty (x1 )

Theorem: a2-e-s-mux (s-mux (x1 , x2 , x3 ) = e) = empty (x1 )

Theorem: a2-lp-s-mux len (s-mux (x1 , x2 , x3 )) = len (x1 )

Theorem: a2-lpe-s-mux eqlen (s-mux (x1 , x2 , x3 ), x1 )

Theorem: a2-ic-s-mux ((len (x1 ) = len (x2 )) ∧ (len (x2 ) = len (x3 ))) → (s-mux (i (c x1 , x1 ), i (c x2 , x2 ), i (c x3 , x3 ))

= i (mux (c x1 , c x2 , c x3 ), s-mux (x1 , x2 , x3 )))

Theorem: a2-lc-s-mux (¬ empty (x1 )) → (l (s-mux (x1 , x2 , x3 )) = mux (l (x1 ), l (x2 ), l (x3 )))

Theorem: a2-pc-s-mux p (s-mux (x1 , x2 , x3 )) = s-mux (p (x1 ), p (x2 ), p (x3 ))

Theorem: a2-hc-s-mux ((¬ empty (x1 )) ∧ ((len (x1 ) = len (x2 )) ∧ (len (x2 ) = len (x3 )))) → (h (s-mux (x1 , x2 , x3 )) = mux (h (x1 ), h (x2 ), h (x3 )))

;old:    ((DISABLE MUX S-MUX) (ENABLE H LEN) (INDUCT (S-MUX X1 X2 X3)))

Theorem: a2-bc-s-mux ((len (x1 ) = len (x2 )) ∧ (len (x2 ) = len (x3 ))) → (b (s-mux (x1 , x2 , x3 )) = s-mux (b (x1 ), b (x2 ), b (x3 )))

;old:    ((DISABLE MUX) (ENABLE B LEN) (INDUCT (S-MUX X1 X2 X3)))

Theorem: a2-bnc-s-mux ((len (x1 ) = len (x2 )) ∧ (len (x2 ) = len (x3 ))) → (bn (n, s-mux (x1 , x2 , x3 )) = s-mux (bn (n, x1 ), bn (n, x2 ), bn (n, x3 )))

;old: ((DISABLE MUX S-MUX))
;; A2-End-S-MUX
; eof:comb_mux.bm
;; already loaded in A: (LOAD "Comb/comb_inc.bm")

Definition: topor-sy-b (ln) = if ln = 'yout then 0

elseif ln = 'y1m then 2 elseif ln = 'y1 then 1 elseif ln = 'yst then 0 elseif ln = 'yst1 then 1 else 0 endif

Definition: sy-b (ln, x) = if ln = 'yout

then if empty (x) then e

else i (0, sy-b ('y1m, p (x))) endif elseif ln = 'y1m then s-mux (sy-b ('yst, x), sy-b ('yout, x), sy-b ('y1, x)) elseif ln = 'y1 then s-inc (sy-b ('yout, x)) elseif ln = 'yst then if empty (x) then e

else i (t, sy-b ('yst1, p (x))) endif elseif ln = 'yst1 then s-not (sy-b ('yst, x)) else sfix (x) endif

;; A2-Begin-SY-B

Theorem: a2-empty-sy-b empty (sy-b (ln, x)) = empty (x)

Theorem: a2-e-sy-b (sy-b (ln, x) = e) = empty (x)

Theorem: a2-lp-sy-b len (sy-b (ln, x)) = len (x)

Theorem: a2-lpe-sy-b eqlen (sy-b (ln, x), x)

; See note at top of file.
;(PROVE-LEMMA A2-PC-SY-B (REWRITE)
;    (EQUAL (P (SY-B LN X)) (SY-B LN (P X)))
;    ((DISABLE S-MUX S-INC S-NOT A2-IC-S-MUX A2-IC-S-INC A2-IC-S-NOT)))

Axiom: a2-pc-sy-b p (sy-b (ln, xst)) = sy-b (ln, p (xst))

;; A2-End-SY-B
; BM DEFINITIONS and A2 LEMMAS, generated by BMSYSD:
;; already loaded: (LOAD "Comb/comb_mux.bm")
;; already loaded: (LOAD "Comb/comb_inc.bm")

Definition: topor-sy-c (ln) = if ln = 'yout then 0

elseif ln = 'y1m then 2 elseif ln = 'y1 then 1 else 0 endif

Definition: sy-c (ln, xst) = if ln = 'yout

then if empty (xst) then e

else i (0, sy-c ('y1m, p (xst))) endif elseif ln = 'y1m then s-mux (xst, sy-c ('yout, xst), sy-c ('y1, xst)) elseif ln = 'y1 then s-inc (sy-c ('yout, xst)) else sfix (xst) endif

;; A2-Begin-SY-C

Theorem: a2-empty-sy-c empty (sy-c (ln, xst)) = empty (xst)

Theorem: a2-e-sy-c (sy-c (ln, xst) = e) = empty (xst)

Theorem: a2-lp-sy-c len (sy-c (ln, xst)) = len (xst)

Theorem: a2-lpe-sy-c eqlen (sy-c (ln, xst), xst)

; blows..
;(PROVE-LEMMA A2-PC-SY-C (REWRITE)
;    (EQUAL (P (SY-C LN XST)) (SY-C LN (P XST)))
;    ((DISABLE S-MUX S-INC A2-IC-S-MUX A2-IC-S-INC)))
; so TEMPORARILY:

Axiom: a2-pc-sy-c p (sy-c (ln, xst)) = sy-c (ln, p (xst))

;; A2-End-SY-C
;;; BEGIN: Circuit CORRECTNESS modulo Stuttering.
;; BEGIN: new 2nd order properties for combinationals.
;; END: new 2nd order properties for combinationals.
;;; Get STUTTER theory:
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;  TH_STUTTER.BM
;;;
;;; This file contains Stutter theory for BM.  It is supposed to be
;;; loaded directly when needed (i.e. not general enough to be
;;; stored in Lib/mlp).
;;;
;;; Our current double P-recursive def. of Stutter:
;;; Originally, it came from THETA-PRF-79 (done while babysitting
;;; for Caroline...) followed by MUCH experimentation and fiddling.

Definition: stut-r (x, y) = if empty (y) then x

elseif empty (p (y)) then b (x) elseif l (p (y)) then stut-r (x, p (y)) else b (stut-r (x, p (y))) endif

Definition: stut (x, y) = if empty (y) then e

elseif empty (p (y)) then a (e, h (x))

elseif l (p (y)) then a (stut (p (x), p (y)), l (stut (p (x), p (y))))

else a (stut (p (x), p (y)), h (stut-r (x, p (y)))) endif

; Stut-Induct inducts like stut, but without the case disjunction
; on LPx which is useless when we stutter on a line rather than an
; input. The resulting induction is not very different from a
; straight P induction, but it takes care of the empty Px case
; separately, and without bringing an elimination.

Definition: stut-induct (x) = if empty (x) then 0

elseif empty (p (x)) then 1 else stut-induct (p (x)) endif

;; Properties of Stut:

Theorem: stut-empty empty (stut (x, y)) = empty (y)

Theorem: stut-e (stut (x, y) = e) = empty (y)

Theorem: stut-p p (stut (x, y)) = stut (p (x), p (y))

;; Properties of Stut-R:
; Stut-R-E maybe shouldn't be enabled all the time, but when we're
; doing P inductions on Stut-R, this gives the base case.  The
; induction step is given by Stut-R-P.  Note that we don't have a
; full empty x hyp because Stut-R returns x and not sfix x in case
; y is empty...  Maybe we want to fix that at some point.

Theorem: stut-r-e stut-r (e, y) = e

Theorem: stut-r-p p (stut-r (x, y)) = stut-r (p (x), y)

Theorem: stut-r-len len (x) < (1 + (len (y) + len (stut-r (x, y))))

Theorem: stut-r-not-empty (len (y) < len (x)) → (¬ empty (stut-r (x, y)))

; Stut-Rem removes the trailing Ts of y, but ignores Ly (like R)
; and leaves one T: this weird def, so it works like Stut-R needs!

Definition: stut-rem (y) = if empty (y) then e

elseif empty (p (y)) then y

elseif l (p (y)) then stut-rem (p (y))

else y endif

Theorem: stut-rem-empty empty (stut-rem (x)) = empty (x)

Theorem: stut-rem-len len (stut-rem (x)) < (1 + len (x))

Theorem: stut-rem-len2 ((¬ empty (p (x))) ∧ l (p (x))) → (len (stut-rem (x)) < len (x))

; Stut-Num counts the number of F in y, ignoring Ly, and starts
; w/ 1, like Stut.

Definition: stut-num (y) = if empty (y) then 0

elseif empty (p (y)) then 1

elseif l (p (y)) then stut-num (p (y))

else 1 + stut-num (p (y)) endif

Theorem: stut-num-lessp stut-num (x) < (1 + len (x))

Theorem: stut-num-eq-0 (stut-num (x) = 0) = empty (x)

; Requires a small induction.

Theorem: stut-num-rem-len (¬ empty (x)) → (stut-num (p (stut-rem (x))) < len (x))

; From Stut-Num and Bn we get a CLOSED FORM for Stut-R !!!

Theorem: stut-r-closed stut-r (x, y) = bn (stut-num (y), x)

; Stut-inv is the key invariant property during Stuttering:

Theorem: stut-inv ((¬ empty (y)) ∧ (len (x) ≮ len (y))) → (l (stut (x, y)) = h (stut-r (x, p (stut-rem (y)))))

; but we only want to use it during the non-stuttering induction
; step, and not in general so:

Theorem: stut-inv0 ((¬ empty (y)) ∧ (len (x) ≮ len (y)) ∧ (¬ l (y))) → (l (stut (x, y)) = h (stut-r (x, p (stut-rem (y)))))

Event: Disable stut-inv.

; Now we relate Stut-R for Py and P Rem Py, to get the key to the
; induction step on main Stut inductions, in the non-stuttering
; case.
;
; It's a BAD rewrite (i.e. expanding, potentially self-applicable),
; and so are the preliminary lemmas needed to build to it.  This
; is not just an unfortunate construction.  It's inherent, because
; we're essentially giving an alternate definition via a Stut-Rem
; recursion.  And definitions are expanding, self-applicable,
; rewrites.  We get around the problem by lucking out: the
; hypotheses are sufficient to prevent successful self-applic.

Theorem: stut-r-indstep-num ((¬ empty (x)) ∧ l (x)) → (stut-num (x) = (1 + stut-num (p (stut-rem (x)))))

Event: Disable stut-r-indstep-num.

; OLD induction step prereq: not needed anymore.
;
;(prove-lemma Stut-R-indstep-Num-Rem (rewrite)
;(implies (and (not (empty y))
;       (not (empty (P y)))
;       (not (L (P y)))
;       )
;  (equal (Stut-Num (P (Stut-Rem (P y))))

; (sub1 (Stut-Num (P (Stut-Rem y))))
; ))
;((enable Stut-R-indstep-Num))
;)
;(disable Stut-R-indstep-Num-Rem)
; OLD induction step: not needed anymore.
;
;(prove-lemma Stut-indstep (rewrite)
;(implies (and (not (empty y))
;       (not (empty (P y)))
;       (not (L (P y)))
;       )
;  (equal (Stut-R x (P y))
; (B (Stut-R x (P (Stut-Rem (P y)))))
; ))
;((enable Stut-R-indstep-Num-Rem B-Bn-sub1)
; (disable Bn)
; )
;)
;(disable Stut-indstep) ; potentially self-looping...
;                       ; so we enable explicitely.

; NEW & GENERALIZED induction step hack , note: needs just ONE
; prereq! We're getting cleaner...

Theorem: stut-r-indstep ((¬ empty (y)) ∧ (¬ l (y))) → (stut-r (x, y) = b (stut-r (x, p (stut-rem (y)))))

Event: Disable stut-r-indstep.

; all the internal stuff shouldn't be needed outside:

Event: Disable stut-r.

Event: Disable stut-num.

Event: Disable stut-rem.

; Note: by leaving Stut, Stut-inv0, Stut-R-closed enabled, we get
;   the effect of an alternate recursive definition of Stut in the
;   most convenient form.  The remaining uncleanliness is that
;   Stut-R-indstep and Stut-R-closed match the same stuff, and need
;   to be used at different places in the main proof.  So far, we
;   survive by extreme cunning: they are in the right order, and
;   the hypothesis on Stut-R-indstep prevents wrong occurences.
;   This is neither clear nor robust...
;; eof: th_stutter.bm

;; BEGIN: ACTUAL Circuit CORRECTNESS modulo Stuttering.
; REVERSAL PROPERTY for sy-a:

Theorem: sy-a-reversal (¬ empty (bn (n, sy-a ('yout, p (x))))) → (h (b (bn (n, sy-a ('yout, x)))) = (1 + h (bn (n, sy-a ('yout, p (x))))))

Theorem: count-ac-l l (stut (sy-a ('yout, xst), xst)) = l (sy-c ('yout, xst))

; Now extending to strings.  For some unknow reason, compared to Funacc, we
; need BOTH splits here... Probably because of some weird non-triggering
; phenomenon in equality hyp usage.

Theorem: apl-split-cout (¬ empty (x)) → (sy-c ('yout, x) = a (p (sy-c ('yout, x)), l (sy-c ('yout, x))))

Theorem: apl-split-stuta (¬ empty (x)) → (stut (sy-a ('yout, x), x)

= a (p (stut (sy-a ('yout, x), x)), l (stut (sy-a ('yout, x), x))))

; and finally:

Theorem: count-ac-correct stut (sy-a ('yout, xst), xst) = sy-c ('yout, xst)

;; END: ACTUAL Circuit CORRECTNESS modulo Stuttering for A and C.

Event: Disable count-ac-l.

Event: Disable count-ac-correct.

; HCorr properties are the "Hand-Correctness" formulas... They are not
; necessary for the proof of Count-AB-L, but I'm trying to see if they help.

Theorem: hcorr-ab-t ((¬ empty (x)) ∧ (¬ empty (p (x))) ∧ l (sy-b ('yst, p (x)))) → (l (sy-b ('yout, x)) = l (sy-b ('yout, p (x))))

Event: Disable hcorr-ab-t.

; obviously..

Theorem: hcorr-ab-f ((¬ empty (x)) ∧ (¬ empty (p (x))) ∧ (¬ l (sy-b ('yst, p (x))))) → (l (sy-b ('yout, x)) = (1 + l (sy-b ('yout, p (x)))))

Event: Disable hcorr-ab-f.

; Count-AB-L succeeds with either HCorrs enabled, or the expansion hint
; therein.  The costs (time/clarity) seem equal.  In the future, if the
; effort involved in getting HCorrs is greater, the dichotomy may be useful.

Theorem: count-ab-l l (stut (sy-a ('yout, x), sy-b ('yst, x))) = l (sy-b ('yout, x))

Theorem: apl-split-bout (¬ empty (x)) → (sy-b ('yout, x) = a (p (sy-b ('yout, x)), l (sy-b ('yout, x))))

Theorem: apl-split-stuta2 (¬ empty (x)) → (stut (sy-a ('yout, x), sy-b ('yst, x))

= a (p (stut (sy-a ('yout, x), sy-b ('yst, x))), l (stut (sy-a ('yout, x), sy-b ('yst, x)))))

Theorem: count-ab-correct stut (sy-a ('yout, x), sy-b ('yst, x)) = sy-b ('yout, x)

;; END: ACTUAL Circuit CORRECTNESS modulo Stuttering for A and B.
; eof: countstut.bm
; ))

Index

a, 2, 4, 8, 13, 14 a2-bc-s-inc, 3 a2-bc-s-mux, 5 a2-bnc-s-inc, 3 a2-bnc-s-mux, 5 a2-e-s-inc, 2 a2-e-s-mux, 5 a2-e-sy-a, 4 a2-e-sy-b, 6 a2-e-sy-c, 7 a2-empty-s-inc, 2 a2-empty-s-mux, 5 a2-empty-sy-a, 4 a2-empty-sy-b, 6 a2-empty-sy-c, 7 a2-hc-s-inc, 3 a2-hc-s-mux, 5 a2-ic-s-inc, 3 a2-ic-s-mux, 5 a2-lc-s-inc, 3 a2-lc-s-mux, 5 a2-lp-s-inc, 3 a2-lp-s-mux, 5 a2-lp-sy-a, 4 a2-lp-sy-b, 6 a2-lp-sy-c, 7 a2-lpe-s-inc, 3 a2-lpe-s-mux, 5 a2-lpe-sy-a, 4 a2-lpe-sy-b, 6 a2-lpe-sy-c, 7 a2-pc-s-inc, 3 a2-pc-s-mux, 5 a2-pc-sy-a, 4 a2-pc-sy-b, 7 a2-pc-sy-c, 8 apl-split-bout, 14 apl-split-cout, 13 apl-split-stuta, 13 apl-split-stuta2, 14 b, 3, 5, 8, 12, 13 bn, 3, 5, 10, 13

count-ab-correct, 14 count-ab-l, 14 count-ac-correct, 13 count-ac-l, 13

e, 2–10 empty, 2–14 eqlen, 3–7

h, 3, 5, 8, 11, 13 hcorr-ab-f, 14 hcorr-ab-t, 14

i, 3, 5–7 inc, 2, 3

l, 2–5, 8, 10–14 len, 3–7, 9–11

mux, 4, 5

p, 2–14

s-if, 4

s-inc, 2, 3, 6, 7 s-mux, 4–7 s-not, 6 sfix, 3, 6, 7 smux-is-sif, 4 stut, 8, 9, 11, 13, 14 stut-e, 9 stut-empty, 9 stut-induct, 9 stut-inv, 11 stut-inv0, 11 stut-num, 10, 11 stut-num-eq-0, 10 stut-num-lessp, 10 stut-num-rem-len, 10 stut-p, 9 stut-r, 8–12 stut-r-closed, 10 stut-r-e, 9 stut-r-indstep, 12 stut-r-indstep-num, 11 stut-r-len, 9 stut-r-not-empty, 9 stut-r-p, 9 stut-rem, 10–12 stut-rem-empty, 10 stut-rem-len, 10 stut-rem-len2, 10 sy-a, 3, 4, 13, 14 sy-a-reversal, 13 sy-b, 6, 7, 14 sy-c, 7, 8, 13

topor-sy-a, 3 topor-sy-b, 6 topor-sy-c, 7