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(in-package "ACL2")
(include-book "land")
(include-book "lior")
(include-book "lxor")
(local (include-book "../arithmetic/top"))
(local (include-book "logand"))
(local (include-book "logior"))
(local (include-book "logxor"))
(local (include-book "merge"))
(local (include-book "bvecp"))
(local (include-book "bits"))
(defthmd lior-land-1
(equal (lior x (land y z n) n)
(land (lior x y n) (lior x z n) n))
:hints (("Goal" :use ((:instance logior-logand
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))
(z (bits z (1- n) 0))))
:in-theory (enable lior land))))
(defthmd lior-land-2
(equal (lior (land y z n) x n)
(land (lior x y n) (lior x z n) n))
:hints (("Goal" :use ((:instance logior-logand
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))
(z (bits z (1- n) 0))))
:in-theory (enable lior land))))
(defthmd land-lior-1
(equal (land x (lior y z n) n)
(lior (land x y n) (land x z n) n))
:hints (("Goal" :use ((:instance logand-logior
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))
(z (bits z (1- n) 0))))
:in-theory (enable lior land))))
(defthmd land-lior-2
(equal (land (lior y z n) x n)
(lior (land x y n) (land x z n) n))
:hints (("Goal" :use ((:instance logand-logior
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))
(z (bits z (1- n) 0))))
:in-theory (enable lior land))))
(defthmd lior-land-lxor
(equal (lior (land x y n) (lior (land x z n) (land y z n) n) n)
(lior (land x y n) (land (lxor x y n) z n) n))
:hints (("Goal" :use ((:instance log3
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))
(z (bits z (1- n) 0))))
:in-theory (enable lior land lxor))))
(defthmd lxor-rewrite
(equal (lxor x y n)
(lior (land x (lnot y n) n)
(land y (lnot x n) n)
n))
:hints (("Goal" :use ((:instance logxor-rewrite-2
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))))
:in-theory (enable lior land lxor))))
(defthmd lnot-lxor
(equal (lnot (lxor x y n) n)
(lxor (lnot x n) y n))
:hints (("Goal" :use ((:instance lnot-logxor
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))))
:in-theory (enable lxor))))
;move
(defthm bits-of-1-plus-double
(implies (and (integerp x)
(<= 0 i)
)
(equal (bits (+ 1 (* 2 x)) i 1)
(bits x (1- i) 0)))
:hints (("goal" :in-theory (enable bits expt-split mod-prod mod-sum-cases)
))
)
;move
;useful for inductions involvling lnot?
(defthm lnot-shift-plus-1
(implies (and (case-split (integerp x))
(case-split (< 0 n))
(case-split (integerp n))
)
(equal (lnot (+ 1 (* 2 x)) n)
(* 2 (lnot x (1- n)))))
:hints (("Goal" :use (:instance bits-plus-bits (x (+ 1 (* 2 X))) (n (1- N)) (m 0) (p 1))
:in-theory (enable lnot expt-split))))
;move?
;do we want this sort of theorem about lognot and logand too?
; Matt K.: The original proof fails using v2-8-alpha-12-30-03. I don't know
; why, but I do notice that the induction heuristics are getting in the way,
; because the problem goal (the one with the hint below) goes through in the
; proof-builder. So we use the proof-builder proof.
#|
(local
(defthm lnot-lior-aux
(implies (and (integerp x) ;gen?
(integerp y) ;gen?
)
(equal (lnot (lior x y n) n)
(land (lnot x n) (lnot y n) n)))
:hints (("subgoal *1/2" :use (lior-def
(:instance land-def (x (LNOT X N)) (y (LNOT Y N)))
(:instance mod012 (x x))
(:instance mod012 (x y))
(:instance lnot-fl (k 1))
(:instance lnot-fl (x y) (k 1))
)
)
("Goal" :in-theory (enable lnot-shift)
:do-not '(generalize)
:induct ( op-dist-induct x y n)))))
|#
(local
(DEFTHM LNOT-LIOR-AUX
(IMPLIES (AND (INTEGERP X) (INTEGERP Y))
(EQUAL (LNOT (LIOR X Y N) N)
(LAND (LNOT X N) (LNOT Y N) N)))
:INSTRUCTIONS
((:IN-THEORY (ENABLE LNOT-SHIFT))
(:INDUCT (OP-DIST-INDUCT X Y N))
:PROVE
(:PROVE :HINTS
(("Goal" :USE
(LIOR-DEF (:INSTANCE LAND-DEF (X (LNOT X N))
(Y (LNOT Y N)))
(:INSTANCE MOD012 (X X))
(:INSTANCE MOD012 (X Y))
(:INSTANCE LNOT-FL (K 1))
(:INSTANCE LNOT-FL (X Y) (K 1))))))
:PROVE)))
(defthm lnot-lior
(equal (lnot (lior x y n) n)
(land (lnot x n) (lnot y n) n))
:hints (("goal" :in-theory (disable lnot-lior-aux)
:use (:instance lnot-lior-aux (x (fl x)) (y (fl y)))))
)
; See comment above about v2-8-alpha-12-30-03. A similar situation applies
; just below.
#|
(local
(defthm lnot-land-aux
(implies (and (integerp x) ;gen?
(integerp y) ;gen?
)
(equal (lnot (land x y n) n)
(lior (lnot x n) (lnot y n) n)))
:hints (("subgoal *1/2" :use (land-def
(:instance lior-def (x (LNOT X N)) (y (LNOT Y N)))
(:instance mod012 (x x))
(:instance mod012 (x y))
(:instance lnot-fl (k 1))
(:instance lnot-fl (x y) (k 1))
)
)
("Goal" :in-theory (enable lnot-shift)
:do-not '(generalize)
:induct ( op-dist-induct x y n)))))
|#
(DEFTHM LNOT-LAND-AUX
(IMPLIES (AND (INTEGERP X) (INTEGERP Y))
(EQUAL (LNOT (LAND X Y N) N)
(LIOR (LNOT X N) (LNOT Y N) N)))
:INSTRUCTIONS
((:IN-THEORY (ENABLE LNOT-SHIFT))
(:INDUCT (OP-DIST-INDUCT X Y N))
:PROVE
(:PROVE :HINTS
(("Goal" :USE
(LAND-DEF (:INSTANCE LIOR-DEF (X (LNOT X N))
(Y (LNOT Y N)))
(:INSTANCE MOD012 (X X))
(:INSTANCE MOD012 (X Y))
(:INSTANCE LNOT-FL (K 1))
(:INSTANCE LNOT-FL (X Y) (K 1))))))
:PROVE))
(defthm lnot-land
(equal (lnot (land x y n) n)
(lior (lnot x n) (lnot y n) n))
:hints (("Goal" :in-theory (disable LNOT-LAND-aux)
:use (:instance lnot-land-aux (x (fl x)) (y (fl y)))))
)
|
